mm BSi;::^-fc5i?S;<5A<^;- V f^ a^atmll InittBrHttg SItbratg JIttfara, N»w ^nrb ALEXANDER GRAY MEMORIAL LIBRARY ELECTRICAL ENGINEERING THE GIFT OF F.£arap©to#f- Cornell University Library QA 37.C61 Handbook of mathematics for engineers an 3 1924 003 951 195 CQ Tl (Q CD (D O) CO CL (Ji Cornell University Library The original of tiiis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924003951195 CONTENTS vii PART lU GEOMETRY PAGE Definitions 251 Plane Geometry 253 Straight Lines 253 Polygons 258 Circle 265 Similar Polygons and the Measurement of Angles ' 271 Mensuration of Polygons 278 Regular Polygons and the Mensuration of the Circle 284 Solid Geometry 298 Planes 298 Julyhedral Angles, Polyhedrons, and Symmetry 305 The Cylinder, the Cone, and the Sphere 812 Similar Polyhedrons and the Measurement of Angles 319 Mensuration of Polyhedrons 821 Regular Polyhedrons and the Mensuration of Cylinders, Cones, and Spheres 825 Problems in Geometry 333 Angles, Triangles, Perpendiculars, and Parallels 833 Proportional Lines and Similar Polygons 347 The Division of Circles into Equal Parts, Regular Polygons 353 Areas of Polygons and Circles 362 PART IV TRIGOHOMETRY Plane Trigonometry 372 Determination of a Point 372 Determination of a Straight Line 375 Trigonometric Expressions and Their Use 877 Projection of Straight Lines 387 Formulas Expressing the Relations between the Trigonometric Func- tions 391 Calculation of Trigonometric Tables 402 Principles used in Solving Triangles 405 Solution of right Triangles 408 Solution of Plane Triangles 410 Table of Trigonometric Functions 422 Application of the Equation of the Third Degree and the Trigonome- tric Solution of the Irreducible Case 445 viii CONTENTS PAGBJ Spherical Trigonometry General rormulas ^^° Eight Spherical Triangles ^°^ Solution of the Oblique Spherical Triangles 45f Problems in Spherical Trigonometry 464 Angles Formed by the Faces of Regular Polyhedrons 466 PART V ANALYTIC GEOMETRY Analytic Geometry 471 Determination of a Line 471 Homogeneity 473 The Geometrical Construction of Algebraic Formulas ^ 476 General Construction of Curves Represented by Equations "482 The Straight Line 484 The Circle 487 The Ellipse 489 The Hyperbola 611 The Parabola 522 Curves of the Second Degree, or Conic Sections 536 Lemniscate, Cissoid, Strophoid, and LimaQon 539 The Spiral Archimedes 542 Involute, Evolute, Radius of Curvature 546 Cycloid 547 Epicycloid 550 Helix 553 Miscellaneous Curves 555 A Note on the Polar Coordinate System 561 PART VI ELEMENTS OF CALCULUS ALCULU Introduction Diffekential Calculus ggg 565 Differentials and Derivatives of Fundamental Functions 572 Theorems of Differentiation gng Tangents gg,^ Successive Derivatives "' an, „ . d04 Concavity and Convexity and Direction of Bending 606 Point of Inflection „„„ Taylor's Theorem „. Maxima and Minima Radii of Curvature ._. 630 CONTENTS ix PAGE Integral CALcnLus 634 Introduction 634 Eules for Integration 639 Integration by Series 664 Apphcahons or Integkal Calculus 656 Quadrature of Curves 656 Tlie Cubature of Solids 668 Eectification of Curves 669 Eectiflcation of Curves Expressed in Polar Coordinates 673 Area of Surfaces of Revolution 677 Cubature of Solids of Revolution 679 Center of Gravity 680 Radius of Gyration and Moment of Inertia 694 Moment of Inertia of Plane Surfaces 700 PAET I AEITHMETIC RULES AND DEFINITIONS* 1. The name quantity is given to everything which may be expressed in numbers by comparing it with a quantity of the same sort taken as unity. Lengths which are expressed in feet or meters; surfaces in square feet or square meters; volumes in cubic feet or cubic meters; weights and forces in pounds or kilo- grams; -prices in dollars and cents; time in days; angles in degrees, etc., are quantities. Number, space, and time are quantities of which everyone has an idea and need not be defined. 2. Mathematics is the science of quantities. 3. Arithmetic is the science of numbers. 4. Numeration is that part of arithmetic which deals with the formation, the reading, and the writing of numbers. It is di- vided into spoken numeration, or numeration which deals with the formation and reading of the numbers, and written numera- tion, or notation which has for a purpose the expression of num- bers by figures and letters. 5. The number one is the unit of numbers, to which the name simple unit or unit of the first order has been given; the number ten, which consists of ten simple units, is a number of the second order; one hundred is of . the third; one thousand of the fourth; ten thousand of the fifth, and so on. It may be noted that units of successive orders are each ten times that of the order immediately preceding. 6. The simple unit, the thousand, which is equal to one thou- sand simple units; the million, which is equal to one thousand thousands; the billion, which is equal to one thousand millions; * A number placed in parenthesis ( ) indicates cross reference to the article bearing tliat number. 1 2 ARITHMETIC the trillion, which is equal to one thousand billions; the quad- rillion; the quintillion, etc.; in a word, all the units, starting from simple units, which are one thousand tinjes greater than the one immediately preceding, are called principal units. 7. The first nine numbers are represented respectively by the nine figures 1, 2, 3, 4, 5, 6, 7, 8, 9; with the aid of these, together with the tenth figure, 0, which has no value in itself, all possible numbers may be written. To write a dictated number in figures, commencing at the left, write one after the other the figures which represent the num- ber of hundreds, of tens and of units of each principal unit dic- tated, replacing the units which are lacking by ciphers. For example, the number thirty million fifty thousand seven hundred eight is written 30,050,708. It is seen that in a whole number any figure placed, at the left of another expresses units ten times as great as that one. It is this convention which permits the writing of all possible num- bers with the aid of only ten figures. 8. All figures of a number have two values: one absolute, expressed by its form, the other relative, due to the position which it occupies; thus, in the number 508, the figure 5 has five for an absolute value, and five hundred for a relative value. The in a number has neither an absolute nor a relative value; it serves simply to place the other figures in the desired order, that is, to give them a determined relative value. It is for this reason that is not called a significative figure, a designation given to the other nine figures. 9. To pronounce a number written in figures, commencing at the right, separate them, in thought, or by commas, into peri- ods of three figures each, except the last period which may have one or two figures; then commencing at the left, pronounce suc- cessively the number of hundreds, tens and units of each period, giving the name of the principal units which they represent. Thus, the number 3,405,834,067 is pronounced three billion four hundred five million eight hundred thirty-four thousand sixty-seven. Instead of saying one ten, two tens . . ., nine tens, usage has made it: ten, twenty . . ., ninety. The same instead of say- ing ten one, ten two . . ., ten nine, we say eleven, twelve . . ., nineteen. 10. The base of a system of numeration is a constant number RULES AND DEFINITIONS 3 of any order, of which the unit of the immediately superior order (5) is composed. Thus, ten is the base of the system of numera- tion adopted; and for this reason it is called the decimal system. The number of figures employed in a system is equal to the base of the system. 11. Roman Notation. The Romans employed letters to rep- resent the numbers. They are still used, especially on monu- mental inscriptions. The letters employed are: I, V, X, L, C, D, M. They represent respectively: 1, 5, 10, 50, 100, 500, 1,000. The number I placed one, two, or three times at the right of the numbers I and V, increases these numbers by one, 'two, or three units; and if it is written at the left of V or X it decreases them by one tinit; thus the first ten whole numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, are respectively represented by: I, II, III, IV, V, VI, VII, VIII, IX, X. The number X written one, two, or three times at the right of the number X or L, increases these numbers by one, two, or three tens; and written at the left of L or C diminishes them by ten. Thus the numbers: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, are written: X, XX, XXX, XL, L, LX, LXX, LXXX, XC, C. To write the whole numbers comprised between two consecu- tive whole numbers of tens, it suffices to write the first nine num- bers at the right of each number of tens. Thus the numbers 13, 34, 56, 97 are written XIII, XXXIV, LVI, XCVII. The number C, placed after itself or the number D, or before D and M, permits the writing of the whole numbers of hundreds in the same manner as the whole numbers of tens were written. Thus the numbers: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, are written respectively: C, CC, CCC, CD, D, DC, DCC, DCCC, CM, M. 4 ARITHMETIC The first hundred numbers written after each number of hun- dreds give all the whole numbers comprised between one and ten hundreds. The number M written one, two, or three times at the right of itself gives the numbers 2000, 3000, 4000. To write the whole numbers comprised between two consecu- tive whole numbers of thousands, the first 999 numbers are written at the right of each number of thousands. The above conventions permit the writing of all the numbers under 5000. Thus the numbers 1856 and 4584 are written MDCCCLVI and MMMMDLXXXIV. 12. A number is concrete or abstract, according as it does or does not indicate the nature of the thing which it represents. Thus when we say seven o'clock, twelve dollars, 7 and 12 are concrete numbers; but when we say simply seven, twelve, they are abstract numbers. 13. An operation is a manner of transforming numbers. There are only four fundamental operations in arithmetic, because all the others are simply combinations of these four. They are: addition, subtraction, multiplication, and division. 14. A calculation is the sum and total of all the operations performed upon the numbers. 15. A theorem is a truth rendered evident by a course of rea- soning called a demonstration. 16. An axiom is a self-evident truth which is accepted with- out demonstration. 17. A problem is a question to be solved. 18. The theorem, the axiom, and the problem come under the common name of proposition. 19. An hypothesis is a preliminary proposition established to fit the demonstration of a theorem or problem. 20. A corollary is the consequence of one or several proposi- tions. 21. The proof of an operation is a second operation performed to verify the accuracy of the result obtained by the first; a proof establishes the probable but not the absolute correctness of a result. 22. Axioms of Arithmetic (16). 1st. Two quantities equal to a third quantity are equal to each other. RULES AND DEFINITIONS 5 2d. When the same operation is performed upon two equal quantities the results are equal. 3d. The value of a whole is not altered by changing the order of its parts. 23. Sign abbreviations : The sign = means i equal to. + plus. — minus. ± plus or minus, X or • times. -5- divided by. > greater than. < less than. 7 + 8 - 6 : =4x3-1 Thus means 7 plus 8 minus 6 equals 4 times 3 minus 6 divided by 2. The parenthesis ( ) expresses the result of the operations upon the quantities which it contains. Thus having 9-6 + 2x4 = 3 + 8= 11, we have 18 - (9 - 6 + 2 X 4) = 18 - 11 = 7, and 5 X (9 - 6 + 2 X 4) or 5 (9 - 6 + 2 X 4) = 5 X 11 = 55. 18 — 9 — 6 indicates that 9 is to be taken from 18 first, and then 6 from the remainder 9 ; which gives 18 — 9 — 6 = 3 ; which is. 18 - 9 - 6 = 18 - (9 + 6). BOOK I FUNDAMENTAL OPERATIONS ON WHOLE NUMBERS ADDITION 24. Addition is an operation by which several quantities are united in a single one, called the sum or total. 25. To add the whole numbers, 4805, 27, 446, 9: In general, to add given numbers, write the num- ^ bers one below the other in such a manner that the 446 figures which express units of the same order come 9 in the same vertical column, and underline the last 5287 number, 9, to separate it from the result. Then com- mencing at the right add successively the figures of each column; place the units of that order in the result and carry the tens to the next column. Thus the sum of the figures in the first column being 27 units, we place 7 units in the result and carry 2 tens to the next column. The operation is com- menced at the right because of the tens which have to be carried. In order to calculate rapidly, instead of saying, as ordinarily: 9 and 6 are 15, 15 and 7 are 22, 22 and 5 are 27, 7 in the result, and 2 to carry; 2 and 4 are 6 and 2 are 8, 8 in the result, etc., it is well to accustom oneself to saying : 9, 15, 22, 27 (write the 7 without pronouncing and pass to the Remainders 45,433 T^^"^" °^ *"°') ' f ' ^ ^^^^^ ^^' ^*^- ^^^^ KAaKR there are many figures to be added, it is 97864 ^^^^' especially if one is not accustomed to 39,518 it) to divide the operation into several 58,763 partial additions, and afterwards add the dfi'It^ partial results. It is also convenient, espe- 39 358 ^i^lly when one has long operations to 4 22635 make, to write the partial sums at one side in the order in which they are obtained. This permits one, in case of a distraction, to recommence the addition of the figures of a column, without SUBTRACTION 7 being obliged to repeat the whole operation. It permits also of the verification of the addition of any column without refer- ence to the others. The scheme shown here is very convenient. 26. To prove an addition, recommence, making the partial additions in the opposite direction. Thus add from top to bottom, or from bottom to top, according as the first operation was made from bottom to top, or top to bottom (97). SUBTRACTION 27. Subtraction is an operation by which the difference of two quantities is taken. These two quantities are the two terms of the difference. The larger one or the first term is called the minuend, the smaller or second term, the subtrahend, and the difference the remainder. 28. From these definitions it follows that: 1st. The first term is equal to the second term plus the re- mainder. 2d. When the first term is increased or decreased, the remainder is increased or decreased. 3d. When the second term is increased or decreased, the re- mainder is decreased or increased. 4th. The remainder is unchanged when both terms are in- creased or decreased by the same quantity. 5th. To subtract a sum from a quantity, subtract the first part of the sum from the quantity; the second part from this re- mainder, etc., until the last part has been subtracted. 6th. To subtract a quantity from a sum, subtract the quantity from one of the parts of the sum. 29. To subtract two whole numbers, 2935 and 372. 2935 372 2563 In general, to find the difference between two whole numbers, write the smaller number below the larger in such a manner that the figures which express units of the same order come in the same column; underline the smaller number 372 to separate it from the remainder. Then commencing at the right, take each figure of the second term from the corresponding figure in the first and place the remainder below. 8 ARITHMETIC When a figure such as 7 in the second term is larger than the corresponding figure 3 of the first term, the subtraction is made possible by adding 10 units of that order to the first term, this being compensated by adding one unit to the following figure of the second term (28, 4th). This adding of one imit to the fol- lowing figure of the second term is the reason for beginning at the right. In performing the operation one says, 2 from 5 leaves 3, 7 from 13 leaves 6, 4 from 9 leaves 5, from 2 leaves 2, writ- ing successively the partial remainders 3, 6, 5, 2 in the remainder. 30. Proof of subtraction. Adding the remainder 2563 to the second term 372, will give the first term 2935, if the work is cor- rect (28, 1st). Another proof is to subtract the remainder from the first term which should give the second terra. 31. When quantities are separated by the signs + or — (example: 3 + 4-5), 3 and 4 preceded by + are said to be positive and 5 preceded by — to be negative. When the first quantity is positive it is not necessary to write plus + before it, but if it is negative the sign — must precede it. If 7 is to be taken from 4, the smaller is taken from the larger and the negative sign placed before the result, thus: 4 - 7 = -3. 1st. 59,243 The result -3 indicates that the quantity could 87,564 not be subtracted. 32 932 8252 '^^ subtract the sum of several quantities 29,848 ^™™ ^^^ s^"^ of several other quantities, the - 3,624 sums are made separately and the difference of - 2,808 the results taken. 184,907 When all the quantities are written in a ~ ^^'^64 column, and one does not wish to rewrite them 145,543 in order to separate them, the sign - is placed before all those to be subtracted, so as to avoid confusion in making the two sums. (See the operation at the left.) The last number 2808 is underlined apd the two sums placed below, the sum to be subtracted coming last. Then the subtraction is made in the usual manner. In place of this method, the rule of subtraction may be ap- plied in a general way and the two partial sums be dispensed with. Commencing at the right the positive numbers are added and from each partial sum the negative numbers are succes- sively subtracted. Thus one says (operation 2) 3 and 4 7 and MULTIPLICATION 9 2, 9 and 8, 17; 17 less 2, 15, less 4, 11, less 8, 3, and 3 is written in the result. The same operation is repeated for 2d. 59,243 each column. It is seen that nothing is done — ^9 Q^2 except to follow the rule of subtraction (29) 8352 which is but a little extended in this case, since 29 848 several figures are subtracted in succession, and — 3,624 it is possible to have several units to add to or — 2,808 to subtract from the next column (96 and 403, 145,543 and the application of the preceding rule to the solution of any right triangles when logarithms are used. Part IV). The preceding rule naturally applies in the case where there is but one number to be taken from a sum of several others (see operation 3), and also where the 3d. 59 243 ^^^ of several numbers is to be taken from a 87,564 single number (see operation 4); in this last 8,252 case it is better to operate in the following man- 29,848 j^3j.. ~ ' Commencing at the right, the negative figures ' of each column are added and the partial sum taken from the corresponding positive figure, the ■_ oo'qqo latter being increased by 1, 2, 3, . . . times 10 — 3'624 ^s the case may be and adding 1, 2, 3 . . . units — 2,808 to the next column for compensation. Thus one 145,543 says: 8 and 4, 12 and 2, 14; from 17 leaves 3. 1 and 0, 1 and 2, 3 and 3, 6; from 10 leaves 4. 1 and 8, 9 and 6, 15 and 9, 24; from 29 leaves 5. 2 and 2, 4 and 3, 7 and 2, 9; from 14 leaves 5. 1 and 3, 4; from 8 leaves 4. from 1 leaves 1. MULTIPLICATION 32. Midtiplication is an operation by which a number called the multiplicand is repeated as many times as there are imits in another called the multiplier. The result is called the product. The multiplicand and the multiplier are the factors of the prod- uct. Multiplication is an abbreviated method of adding as many numbers equal to the multiplicand as there are units in the multiplier. From the definition of multiplication it follows: 1st. When one of the factors is 0, the product is 0, and when 10 ARITHMETIC one of the factors is unity 1, the product is equal to the other factor. 2d. In general the product is of the same sort as the multi- plicand, and the multiplier an abstract number (12). 33. From the definition of multiplication and from axiom 2 (22), it follows: 1st. The product of the sum of several quantities and a num- ber is equal to the sum of the products obtained by multiplying each part of the sum by the number: given 19 = 3 + 7 + 9, we have 19 X5 or 95 = (3 + 7-1- 9) 5 = 3X5 + 7X54-9X5. 2d. The product of a quantity with the sum of several numbers is equal to the sum of the products obtained by multiplying the quantity by each part of the sum : 5x 19 or 95 = 5x (3 + 7 + 9) = 5X3 + 5X7 + 5X9. 34. When the two terms 25 and 8 of a difference are multi- plied by the same number 4, the difference 17 is multiplied by that number 4: 25 X 4 - 8 X 4 = (25 - 8) X 4 = 17 + 4 = 68. 35. The following table, constructed by Pythagoras, contains all the products of two numbers of a single figure each: 1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 24 18 27 36 45 3 6 9 12 15 18 21 4 8 12 16 20 24 28 32 5 10 15 20 25 30 35 40 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 72 81 9 18 27 36 45 54 63 MULTIPLICATION 11 To find the product of two numbers of a single figure in the above table, 8x3, for instance, find the multiplicand 8 in the top horizontal row, and the multiplier 3 in the first vertical column; follow the vertical column which contains 8 down until it intersects the horizontal row, containing 3, and the block at this intersection will contain the product 24. 36. The result obtained by multiplying a series of numbers together in order of their positions ; the first by the second, the product by the third, the new product by the fourth, and so on, is called the product, and the numbers the factors. 37. A number is said to contain all the factors of another number when it is equal to the product of several factors, among which are the factors of the other number. Thus 2x5x3x7 = 210, contains all the factors of 5 X 7 = 35. 38. A quantity is a multiple of another when it is equal to the latter multiplied by a whole number. Thus 7x3 = 21 is a multiple of 7, also of 3. Conversely, when one quantity is a multiple of another, the latter is an under multiple of the first. 39. The sum, 7x4 + 7x3 + 7x5 = 7(4 + 3 + 5) = 7x12 = 84 of several multiples of the same quantity; 7 is a multiple of that quantity (33 and 38). 40. The difference, 7x9-7x4 = 7 (9 -4) =7x5 = 35 of two multiples of the same quantity ; 7 is a multiple of that quantity (34 and 38). 41. The product of any number of factors is not changed by any change in the order of the factors: 3x4x7x5 = 4x5x3x7 = 420 (36). 42. To multiply any number 9 by a product 3 X 4 X 7 = 84, instead of multiplying the number by the product 84, it is pos- sible to multiply it by the first factor 3, the product thus obtained by the second factor 4, and so on through until the last factor has been used as multiplier (36) : 9X 84 = 9X (3X4X7) =9x3x4x7 = 756. 43. When a factor of a product, 5 X 3 X 4 = 60, is mul- tiplied by a number 7, the product is multiplied by the same number : 5X(3X7)X4 = 5X3X4X7 = 60X7 = 420. 12 ARITHMETIC In multiplying several factors of a product by several numbers, the product is multiplied by the product of those numbers: (5 X 6) X (3 X 7) X 4 = (5 X 3 X 4) X (6x7) = 60 X 42 = 2520. 44. To multiply a whole number by a unit followed by one or more ciphers, it is only necessary to write as many ciphers after the number as there are at the right of the imit: 425 X 100 = 42,500. 45. To obtain the product of several numbers, all or part of which end with ciphers, it suffices to obtain the product of the numbers neglecting the ciphers and write at the right of the product as many ciphers as have been neglected in the operation. Thus, in multiplying 400 by 6000, one multiplies 4 by 6, and writes five ciphers to the right of the product 24 : 400 X 6000 = 2,400,000. 46. To multiply a number, 458, of several figures, by a number 6, of a single figure, 458 6 2748 Write the multiplier under the multiplicand, and imderline it to separate it from the result. Then commencing at the right, multiply successively each figure of the multiplicand by the multiplier; write the units of each partial product under the corresponding figure of the multiplicand, and add the tens to the next product (the carrying of the tens is what obliges one to commence at the right). Thus, one says: 6 times 8 are 48 (write 8, carry 4); 6 times 5 are 30, and 4 are 34 (write 4 and carry 3); and so on for all the figures of the multiplicand. 47. To multiply a number, 5736, of several figures, by another number, 743, of several figures, 5736 743 17208 22944 . 40152 4261848 MULTIPLICATION 13 Write as in the preceding case, the multiplier under the mul- tiplicand, so that units of the same order correspond, and underline the multiplier. Then multiply the multiplicand suc- cessively by each figure of the multiplier, starting at the right (46); write each partial product below in such a manner that the first figure at the right comes under the figure of the mul- tipher which has been used ; then add the partial products, which sum is the product desired. If the multiplier contains ciphers between significative figures, as ciphers give for a partial product, they are neglected, and the general rule is applied as before: 34256 3002 68512 102768 102836512 Remark. It may be noted that the number of partial products is always equal to the number of significative figures in the midtiplier. 48. To prove a multiplication, invert the order of the factors, that is, take the multiplier for the multiplicand- and reciprocally, and if the operation is correct, the same result will be obtained (41 and 99). Remark. It will be shown farther on, after the operation of division, that by dividing the product by one of the factors, the quotient will give the other factor if the work is correct. 49. The number of figures in the product is equal to the sum, of the number of figures in the multiplicand and multiplier, or equal to this sum, less one. Thus the multiplicand containing 5 figures and the multiplier 3, the product contains 8 or 7. 50. Short methods of multiplication (44 and 45). 1st. The operation is sensibly shortened by taking the factor which contains the least number of significative figures (8) for multiplier, and above all, when there are figures which appear several times in the multiplier. The number of partial products is less, and the partial products which are equal have to be cal- culated only once. 2d, When the multiplier is 11 or 12, operate as if it were com- 14 ARITHMETIC posed of but one figure (46). Thus in multiplying 97,648 by 11, one says; 97648 97648 11 117 1074128 683536 1074128 11424816 11 times 8, 88 (write 8 and carry 8); 8 and 11 times 4, 44, 52 (write 2 and carry 5); 5 and 66, 71 (write 1); 7 and 77, 84; 8 and 99, 107. With the multipUer 11, the product is equal to the sum of the multiplicand and itself, moved one place to the left. Thus in the preceding example, one says: 8 (write 8 in the result); 8 and 4 are 12 (write 2 and carry 1); 1 and 4, 5, and 6, 11; 1 and 6, 7, and 7, 14; 1 and 7, 8, and 9, 17; 1 and 9, 10. When two adjacent figures of the nmltiplier form the number 11 or 12, as in the second example shown above, multiply the multiplicand by 11 or 12 as by a single figure; which gives one partial product less. 3d. When the multiplier contains only 9s, except the last fig- ure at the right, which may be anything, to get the product, multiply the multiplicand by unity, followed by as many ciphers as there are figures in the multiplier, and from the result sub- tract the product of the multiplicand and the difference between 10 and the number at the right of the multiplier. Having, for example, 9998 = 10,000 - (10 - 8) = 10,000 - 2, to multiply with 65,873, we have 65,873 X 9998 = 65,873 X 10,000 - 65,873 X 2 ^ 658,730,000 - 131,746 = 658,598,254. In doing the operation, write simply 658,730,000 - 131,746 658,598,264 If instead of one figure at the right of the 9s there are 2, 3 . . ., figures, from the multiplicand, followed by as many ciphers as there are figures in the multiplier, subtract the product of the multiplicand and difference between 100, 1000 . . ., and the 2, 3 . . ., figures at the right of the multiplier. 4th. When a multiplier, such as 48,546, contains parts 54 = 6X9 and 48 = 6 X 8, which are multiples of one of its fig- DIVISION 15 ures 6, after having multiplied by 6, multiply the partial product by 9, which gives the product of the multiplicand and 54; the same partial product by 8 gives the product of the multiplicand and 48. 58453 48546 6 350718 54 = 6 X 9 3156462 48 = 6 X 8 2805744 2837659338 5th. Having 5 = -^) 25 = —r- and 125 = — -— > to multiply a number by 5, 25, or 125 multiply by 10, 100, or 1000 and divide the product by 2, 4, or 8. 1479 X 25 = 117^= 36,975, 4729 X 125 ^hl^ = 591,125. When adjacent figures of the multiplier form the numbers 25 or 125, the multiplicand may be multiplied by these numbers as above : 1479 257 7 10353 25 36975 380103 6th. Since the product of several factors is not changed by changing the order of the factors (41), and since several of the factors can be replaced by their product (42) many times by suitable grouping of the factors, an operation may be materially shortened, which would be very long if carried out in the way indicated. Example: 25X9X5X7X2X4 = 9X7 (25x4) X (5X2) = 63X100X10 = 63 X 1000 = 63,000. DIVISION 51. Division is an operation by which a quantity called the dividend is separated into as many equal parts as there are units in a whole number called the divisor; one of these parts is the quotient of the division. Division is a short method of performing a series of subtrac- tions. In subtracting successively the divisor from the dividend 16 ARITHMETIC and from the remainder until a remainder is obtained which is smaller than the divisor, the number of subtractions performed is the quotient. 52. From the definition of division it follows that the dividend is equal to the product of the quotient and the divisor (32). 53. A number is said to be divisible by another, when the quotient obtained by the division of first by the second is a whole number. The second number is said to be a divisor of the first. 54. All numbers are divisible by themselves and unity. The quotient is equal to one in the first case and to the dividend in the second. 55. A number is even or odd according as it is or is not divis- ible by 2. The numbers 2, 4, 6, 8, divisible by 2, are called even numbers, and is also considered even. The other numbers, 1, 3, 5, 7, 9, are odd. A number is odd or even according as its first figure at the right is odd or even (90). 56. When a number, 12, is a multiple of another, 4, the first is divisible by the second and conversely (52). 57. The product of several whole numbers is divisible by any one of its factors (38 and 56). 58. When a number contains all the factors of another num- ber the first is divisible by the second (37, 38, and 56). 59. Any divisor, 4, common to several numbers 36, 12, 16, divides their sum, 64 (39 and 56). 60. Any divisor, 7, common to two numbers, 42 and 14, divides their difference, 28 (40 and 56). 61. Any divisor, 5, of a number, 35, will divide any multiple, 35 X 3 = 105, of that number (39 and 56). 62. To divide a sum by a number, divide each part of the sum by the number (33), thus: 32 + 12 + 16 32 12 16 „ „ 4 =^ + -4+-4=8 + 3 + 4 = 15. 63. To divide a difference, 32 — 12, by a number, 4, divide each of the terms by the number 4 (34), thus: 32 - 12 32 12 ■ 4 " -^"T^^"^^^* DIVISION 17 64. To divide a whole number, 4,145,824, by another whole number, 845. 845 4145824 1690 3380 2535 7658 3380 7605 50?0 00^ 5915 -^ 6760 254 7605 845 4906 To divide one number by another, write the divisor at the right of the dividend, separate them by a vertical line, and under- line the divisor. Then, from the left of the dividend, point off just enough figures so that the number 4145 which results will contain the divisor; look in the table of the first nine multiples of the divisor to find how many times the divisor is contained in the part of the dividend which has been pointed off and this gives the first figure 4 at left of the quotient; write this figure under the divisor; subtract from the first partial dividend 4145 the product 3380 of the divisor and the figure obtained in the quotient, which gives 765 as a remainder, at the right of this partial remainder bring down, that is, write, the next figure 8 of the dividend; find how many times the divisor is contained in the number 7668 which results, thus determining the second figure 9 of the quotient; subtract from the second partial divi- dend 7658 the product 7605 of the divisor and the second figure of the quotient, giving a remainder of 53, at the right of which write the following figure 2 of the dividend. Since the divisor is not contained in the third partial dividend 532, the third figure of the quotient is 0. At the right of 532, write the fol- lowing figure 4 of the dividend; find how many times the divisor is contained in the fourth partial dividend 5324, and continue thus until all the figures of the dividend have been used. The last remainder obtained 254 is the remainder of the division. Generally one does not take the trouble to write the first nine multiples of the divisor. Then to find the number of times that the divisor is contained in the partial dividend 4145, consider simply the first figure 8 at the left of the divisor; neglect as many figures at the right of the partial dividend as have been sup- pressed in the divisor, and find how many times 8 is contained 18 ARITHMETIC in the number 41 which results; 8 being contained 5 times in 41, it is natural to suppose that 5 is the number of times the divisor 845 is contained in the partial dividend 4145; but in multiplying 5 by the figure 4 of the divisor there will be 2 to carry to the product of 8 by 5, which will give 42, showing that 5 is too large. Trying 4 as we have just done with 5, we find it to be the first figure at the left of the quotient. The product of this figure and the divisor need not be written but may be subtracted as fast as the figures are obtained. The preceding division would be performed in the following manner: 4145824 7658 5324 254 845 4906 and to perform the operation one says: How many times is 8 contained in 41? (trying 5, and saying 5 times 8 are 40, and 2, which results from 5 times 4, are 42, showing 5 to be too large) 4 times (write 4 in the quotient); 4 times 5, 20; 20 from 25, 5 remainder and 2 to carry; 4 times 4, 16, and 2, 18; 18 from 24, 6 and 2 to carry; 4 times 8, 32, and 2, 34; 34 from 41, 7. Bring down 8; how many times is 8 contained in 76? 9 times (write 9 in the quotient); 9 times 5, 45; 45 from 48, 3, and 4 to carry; 9 times 4, 36, and 4 are 40, from 45, 5; 9 times 8, 72, and 4, 76, from 76, (not necessary to write 0). Bring down 2; how many times is 8 contained in 5? No times (write in the quotient). Bring down 4; how many times is 8 contained in 53? 6 times, etc. When the divisor is very large, and the quotient is to have a large number of figures, or when there are many numbers to be divided by the same divisor, it is advantageous to construct a table of the nine first multiples of the divisor. Because in this way the successive figures of the qiiotient are obtained imme- ^diately, and the multiplication of the divisor by the figures is avoided. The work can be shortened still more by not writing the multiples of the divisor under the partial dividends when subtracting. ^ When the divisor has only one figure, 7 for instance, write simply the dividend, and remember that to divide a number by 7 is simply to take one-seventh of it (162), DIVISION 19 dividend 174,389 quotient 24,912 remainder 5, one says: a seventh of 17 is 2 (write 2 in the quotient under the dividend and carry 17 — 7 X 2 = 3); a seventh of 34, 4 (write 4 and carry 6); the seventh of 63, 9; of 8, 1; of 19, 2; the re- mainder of the division. is 5. Remaek 1. The dividend, 4,145,824, and the divisor, 845, being given, the number of figures which the quotient is to contain may be found by pointing off at the left of the dividend just enough figures, 4145, to contain the divisor, then the number of figures left in dividend increased by one will equal the number of figures in the quotient, thus, in the example above, 3 + 1 = 4 figures in the quotient. Remaek 2. A figure in the quotient is too large when its pro- duct with the divisor is larger than the corresponding partial dividend, that is, when it can not be subtracted from the partial dividend. If, however, the subtraction is possible and the remainder is larger than the divisor, then the figure in the quotient is too small. 65. To prove a division, multiply the divisor by the quotient and add the remainder, which is always smaller than the divisor, which will give the dividend if the work is correct (52); thus in the preceding example 4906 X 845 + 254 should equal 4,145,824 (100). 66. To divide a number by one followed by any number of ciphers, separate with a comma as many figures at the right of the divi- dend as there are ciphers in the divisor. The part at the left, expressing the simple Units, is the quotient, and the part at the right is the remainder. Thus: 5^^-847 35 100 ~ ^^^■'^^ 847 is the quotient, and 35 the remainder. In decimal numbers the quotient is 847.35, and the remainder (89 and 182). When ciphers at the right of a whole -number are suppressed, it is the same as dividing the number by one followed by as many ciphers as have been suppressed (44) : 8500 ■ioo" = ^^- 20 ARITHMETIC 10 100 , ,<,^ 1000 .^ , ,1 ,, , Having 5 = y, 25 = -^ and 125 = -^, it follows that when a number is to be divided by 5, 25, or 125 the operation may be shortened (164) by multiplying the number by 2, 4, or 8 and dividing the product by 10, 100, or 1000: 36,957 36,957X4 591,473 591,473 X 8 _ -25~ = 100 = ^^^^'^^' ^25" ~ 1000 -*'''*^''»*- The decimal numbers obtained are the exact quotients (91). 67. To divide a number, 504, by a product, 42, of several factors 2, 3, 7, divide the number by the first factor, 2, of the product, the quotient, 252, obtained by the second, 3; and so on until the last factor, 7, has been used as divisor, which will give the quo- tient, 12, desired (42) : 37,471 37,471 _ 374.71 _ . -WQ- = lool^ - "T- - ^^-^^ ^^^^^■ 68. When a factor, 8, of a product, 3X8X5 = 120, is divided by a number, 4, the product is divided by that number (43), thus: 3x|x5 = ^-A|^ = f = 30. 69. To divide a product by one of its factors, suppress this factor in the product. Thus (68) : ?2<|21^ = 3x|x5 = 3XlX5 = 3X5. 70. When a product contains all the factors of another pro- duct, the quotient of the first divided by the second may be ob- tained by suppressing in the first product all the factors of the second (67 and 69): 2X3X5X7 3X7 = 2X5. 71. When the dividend 54 is multiplied or divided b}^ a num- ber 3, without changing the divisor 6, the quotient 9 is mul- tiphed or divided by that number: 54X3 ^ 54 : 3 9 _ — g — = 9 X 3 = 27, and — x— = ;^ = 3. DIVISION 21 72. When the divisor 6 is multiplied or divided by a number 3, without changing the dividend 54, the quotient 9 is divided or multiplied by that number; ^* ^ 3, and -M^ = 9X3 = 27. 6X3 3 "' "- 6-1-3 73. When the dividend 54 and the divisor 6 are multiplied or divided by the same number 3, the quotient 9 remains imehanged : 54X3 „ , 54^3 _ ___ = 9, and -g^=9. 74. From (73) it follows that when the dividend and divisor have common factors, the operation may be shortened by elim- inating those factors: 7 X 324 X 23 _ 324 _ 324-5-4 _ 81 ^ „ 7 X 12 X 23 ~ 12 " 12-5-4 ~ 3 It follows also that when the dividend and divisor end with ciphers, the same number of ciphers may be suppressed at the right of each, without altering the quotient (66 and 73); 35,000 350 700 7 50. 75. All common divisors, 6, of the dividend, 48, and divisor, 18, divide the remainder, 12, of the division, and all common divisors of the remainder, 12, and the divisor, 18, divide the dividend, 48. 76. When the dividend 48 and the divisor 18 are multiplied or divided by the same number 6, the quotient remains un- changed; but the remainder is multiplied or divided by that number. 77. When the dividend 48 is increased or diminished by a certain number of times the divisor 9, the quotient 5 is increased or diminished a certain number of times unity; but the remainder is unaltered. Thus the sum 48 -1- 54 = 102 of two numbers is not divisible by a third number 9, when only one of the numbers 54 is divisible by 9. The sum 102 divided by 9 gives for a quotient the sum 5 + 6 = 11 of the quotients of 48 and 54 by 9, and for a remainder, the remainder 3 of 48 by 9. BOOK II PROPERTIES OF WHOLE DIVISORS 78. A number is a prime number when it is not divisible ex- cept by itself and one (53): 1, 2, 3, 5, 7, 11, 13, 17 . . . are prime numbers. 79. All numbers, 21, which are not prime numbers are the product of several prime factors larger than unity: 21 = 3 X 7. 80. Several numbers are said to be prime to each other when they have no other common divisor than unity (53): such are the numbers 4 and 9; also 6, 10, and 15. The numbers 6, 8, and 12 being all divisible by 2, are not prime to each other. 81. All prime numbers which do not divide a whole number are prime with that number : such are 7 and 15. 82. The greatest common divisor of several numbers is the largest number which will divide each of the numbers. Remark. The greatest common divisor of several numbers prime to each other is one. 83. The least common multiple of several numbers is the small- est number which is a multiple of each of the numbers (38). 84. The separation of a number into its factors, factoring, is to find several numbers, the product of which will equal the num- ber. Thus, having 24 = 2 X 3 X 4, the number 24 is separated into three factors 2, 3, and 4. 85. The product of several factors each equal to a given num- ber is a power of that number. Thus, having 27 = 3 X 3 X 3, and 81 = 3 X 3 X 3 X 3, 27 and 81 are powers of 3. 86. The degree of the power of a number is the number, of factors of that power. Thus 3 and 4 are the degrees of the powers 27 and 81 of the number 3. Remark. All powers of 10 are equal to one followed by as many ciphers as there are units in the degree of the power. Thus the third power of 10 is 1000; 10 X 10 X 10 = 1000 (44). 87. The second power, 7 X 7 = 49, of a number, 7, is the square of the number, 7; the third power, 4 X 4 X 4 = 64, of a number, 4, is the cube of the number, 4. 22 PROPERTIES OF WHOLE DIVISORS 23 88. The exponent of a number raised to a certain power is the degree of this power written to the right and a little above the number. Thus, to express, in an abbreviated manner, that the number 5 is raised to the fourth power, write 5^ in- stead of 5 X 5 X 5 X 5. Remark. The first power of a number is the number itself, which may be considered as having the exponent one, although properly speaking it is no power and has no exponent. 89. To obtain a quotient and a remainder by dividing a number by a power of 10, separate on the right of the number as many figures as there are units in the degree of the power; the part to the left and the part to the right considered as expressing simple units, are respectively the desired quotient and remainder. Thus having to divide 97,845 by 10^ = 1000, separate three figures, which will give 97.845; the quotient is then 97 and the remainder 845. Corollary. If a number be divisible by a power of 10, it must end in at least as many ciphers as there are units in the degree of the power (66). 90. To obtain the remainder in the division of a number by 2 or 5, it suffices to find the remainder in the division of the first figure at the right by 2 or 5. Thus the number 45,737 divided by 2 gives 1 for a remainder, and divided by 5 gives 2, because the first figure 7 divided by 2 or 5 gives respectively 1 or 2 for a remainder; the figure is considered as divisible by 2 and by 5(55). 91. In general, to obtain the remainder in the division of a num- ber by any power of 2 or 5, it suffices to find the remainder in the division of the number, obtained by pointing off as many figures on the right of the number as there are units in the degree of the power, by the power. Thus, to obtain the remainder in the division of 45,737 by 2' = 8, or by 5^ = 125, find the remainder in the division of 737 by 8 or by 125, which gives respectively 1 and 112 (50 and 66). In order that a number be divisible by any power of 2 or 5, the number, obtained by pointing off at the right of the number in question as many figures as there are units in the degree of the power, must be or divisible by the power. Thus, for ex- ample, a number is divisible by 125 if the three figures at the right form the numbers 000, 125, 250, 375, 500 .. . 24 ARITHMETIC 92. To obtain the remainder in the division of a number by 9, add the figures considering them as simple units; operate on this sum as upon the first number, and so on until a result is obtained which does not exceed 9. When this result is less than 9, it is the required remainder; and if it is 9, the remainder is 0. Thus to obtain the remainder in the division of 75,487 by 9, for instance, add 7 + 5 + 4 + 8 + 7 = 31; then add 3 + 1 = 4, and 4 is the required remainder. It is immaterial how the sum is made, commencing at the right or left. The operation is shortened by taking 9 from each succes- sive sum which is greater than or equal to 9. Thus, one says: 7 and 8, 15 (less 9), 6 and 4, 10 (less 9), 1 and 5, 6 and 7, 13 (less 9), 4. The operation may be shortened still more by neglecting the figures 9 and any group of which the sum is 9. Thus in the preceding example neglecting 4 and 5: 7 and 8, 15, 6 and 7, 13; 4. Finally, a step still more expeditive consists in neglecting the figures 9 and those of which the sum is 9 and continuing the addition imtil all the figures have been used, reducing the suc- cessive sums which are multiples of 9 to 0, and those which are not, to numbers in the tens. Thus according as a sum is 27, 29, or 20 it may be reduced to 0, 2, or 2. Given the following num- ber to fimd the remainder when dividing by 9: 8,562,647,683,568,697, one says: 7, 13, 21, 27; 5, 8, 16, 22, 29; 2, 6, 12, 14, 20; 2, 7, 15, 6. If for one reason or another the above short methods are not used and the successive sum becomes too large, it may be reduced by adding its figures and proceeding as before. If, for instance, one has 75, one says: 5 and 7, 12; 2 and 1, 3, and continues the addition with the number 3. 93. If a number is divisible by 9, the sum of the figures which express the simple units must be divisible by 9, that is, be a multiple of 9 (38 and 53). 94. To obtain the remainder in the division of a number by 3, firstly, find its remainder in its division by 9 (92); then the re- mainder in the division of this first remainder by 3. Thus the number 45,847 giving 4 for a remainder in its division by 9, and 4 divided by 3 giving 1 for a remainder, 1 is the required re- mainder in the division of the number in question by 3. PkOPEETiES OP WHOLE DIVISORS 25 95. If a number is divisible by 3, the sum of the figures which express the simple imits is divisible by 3, that is, must be a mul- tiple of 3 (38 and 53). 96. To obtain the remainder in the division of a number by 11, commencing at the right point off the figures in periods of two figures each; and add these numbers, considering them as ex- pressing simple units; operate on this sum as before and so on until a result is obtained which does not exceed 99 ; the remainder in the division of this last sum by 11 is the required remainder. Thus, it being given to find the remainder in the division of 7,345,798 by 11, separate the number into periods of two figures each, which gives 7, 34, 57, 98; adding, we get 98 + 57 -h 34 -I- 7 = 196, then 96 + 1 = 97; the remainder 9 in the division of 97 by 11 is the required re- mainder. It is evident that this sum of periods of two figures each may be obtained by adding them directly, in saying 98 and 57, 155 and 34, 189 and 7, 196, if one is accustomed to calculating, or one can add the right-hand figures considered as units, 8-1-7 + 4 -t- 7 = 26, and then the others taken as tens, 2-1-9-1-5 -1- 3 = 19, the 2 being carried from the first sum ; writing these according to their orders, that is, 19 before the 6, we get the same result 196; upon which the operation may be continued. If a number is divisible by 11, the sum of the periods of two figures each must be a multiple of 11, that is, divisible by 11. Another rule for finding the remainder in the division of a num- ber 7,395,748 by 11: commencing at the right with the first fig- ure, add every other figure, 8-1-7-1-9-1-7 = 31, then do the same thing, commencing with the second figure, 4 -|- 5 -|- 3 = 12; subtract the second result from the first, 31 — 12 = 19, and divide the difference by 11, which gives 8, the required remainder. Operating on this remainder 19 as on the original number, the required remainder is 9 — 1 = 8. If a number 7391 gives a sum 3-1-1=4, which is less than 7 -F 9 = 16, the subtraction is made possible by increasing the first by a number which is a multiple of 11. Thus [4 + 22] - 16 = 10, 10 being the re- mainder. Operating as in Ex. 2d (31), for the number 7,395,748, one would say without writing a single figure: 8, 15, 24, 31; less 26 ARITHMETIC 4, 27, less 5, 22, less 3, 19. Having obtained the difference 19, one says, 9 less 1, 8, and 8 is the required remainder. With this manner of operating, when applied to the number 7391, where 11 is added to make the subtraction possible, one says: 1, 4; (4 + 11 or 15) less 9, 6; 17 less 7, 10. 97. The proof of the addition of several whole numbers by the rule of 9. Find the remainders 8, 3, 1, 4, in the division of the numbers to be added by 9 ; add these remainders, and if the remainder 7 in the division of this sum 16 by 9 is equal to the remainder 7 in the division of the sum 2437 of the whole num- bers by 9, the result 2437 is correct (26). NUMBEKS Eemaih 827 453 325 832 8 3 1 4 2437 16 16 7 Remaek. This proof may be done more rapidly by adding the remainder of the first number directly to the figures of the sec- ond; the remainder obtained for the first two directly to the third and so on. Thus, using the abbreviations as in (92), one says (leaving out 7 and 2 in the first and 5 and 4 in the second) : 8, 11, 16, 18; 3, 5, 8, 16; 7, which ought to be equal to the re- mainder in the division of 2437 by 9. 98. The proof of the subtraction of two whole numbers by the rule of 9. Consider the larger number, 845, as being the sum of the smaller, 258, and the remainder, 587, then proceed as in addition (97). Thus the sum 6 -1- 2 = 8 of the remainders in the 845 8 division of the smaller number and the difference by 258 6 9 being equal to the remainder 8 in the division of the 587 2 larger number 845 by 9, the operation is correct (30). 8 The remark under (97) applies here as well, but the ordinary proof of subtraction being so simple, the proof by 9 is seldom used. 99. The proof of the multiplication of two whole numbers by 5063 85 4 813 59 5 48 3 20 2- 3 5 PROPERTIES OF WHOLE DIVISORS 27 the rule of 9. Find the remainders 6 and 2 in the division of the numbers 357 and 65 by 9 (92); multiply 357 6 these two remainders together, and the remainder 65 _2 3, in the division of the product 12 by 9, is equal 1785 12 3 to the remainder in the division of the product ^^^^ 23,205 by 9, if the calculations are correct (48). 23205 3 Remark. This proof is often used. Like all proofs by 9, it does not show errors equal to a multiple of 9. It is a probability but not a mathematical certainty. 100. The proof of the division of two whole numbers by the rule of 9. Consider the dividend as being the product of the divisor 85, and the quotient 59 plus the remainder 48, the proof is a combination of the proof for addition and that for multiplication (97 and 99). Thus, find the remainders 4 and 5 in the division of the divisor and quotient by 9; multiply them together, and the remainder 2, in the division of this product 20 by 9, increased by the remainder 3, in the division of the remainder 48 by 9, should equal the remainder 5 in the division of the dividend by 9 (65). Instead of finding the remainders 2 and 3 in the division of the product 20 and the remainder 48 by 9 and adding them 2 + 3 = 5, the same result may be obtained by finding the remainder in the division of the sum 48 + 20 by 9. One says (97): 2, 10, 14; 5. 101. The proof of the four operations is the same by the rule of 11 as by that of 9 (97 to 100), but is rarely used. However, if the correctness of the results is of very great importance, both methods of proof may be used. 102. To find the greatest common divisor of two whole numbers, 876 and 360 (82), divide the greater number by the smaller, writing the quotient obtained, 2, 2 2 3 4 and those following over the corre- 876 360 156 48 ~12 sponding divisors; then divide the 156 48 12 smaller number by the remainder obtained 156; and this first re- mainder by the next 48 ; and so on until a remainder of is obtained. The last divisor 12 is the greatest common divisor (125). 2 2 3 4 876 156 360 48 156 12 48 12 73 30 13 4 1 28 ARITHMETIC Generally the greatest common divisor of two numbers is found simply to determine the quotient of these numbers by their greatest common divisor (146). In performing the opera- tion of finding the greatest common divisor, these quotients are easily obtained, as are also those of the remainders or successive divi- sors 156, 48, and 12. Thus, on a horizontal line under 12 write 1; under the divisor 48, on the same horizontal line, write the last quo- tient obtained 4 ; under the divisor 156, the number 4X3 + 1 = 13, obtained by adding the preceding number 1 to the prod- uct of the number 4, just written, and the quotient 3 written above in the same column; under the divisor 360, the number 13 X 2 + 4 = 30, obtained by adding the preceding number 4 to the product of the last number obtained, 13-, and the quotient 2 in the same column, and under the number 876, the numbers 30 X 2 + 13 = 73, obtained in the same manner. The number 1, 4, 13, 30, and 73 are respectively the quotients in the divisions of the divisors 12, 48, 156, and the given numbers 360 and 876 by the greatest common divisor 12. Remark. The greatest common divisor of two numbers, 36 and 144, of which one divides the other, is the smaller, 36, of the numbers. 103. All divisors, 3, common to two numbers, 384 and 36, divide their greatest common divisor, 12, also the successive remainders, 24, 12, obtained in the process of finding the greatest common divisor. 104. To find the greatest common divisor of any number of numbers, find the greatest common divisor of two of the num- bers (102), then the greatest common divisor of that greatest common divisor and another of the numbers, and so on until all of the numbers have been used; the last greatest common divisor is the one desired (125). 105. The greatest common divisor of several numbers is mul- tiplied or divided by a number when those numbers are mul- tiplied or divided by the same number. It follows that the quotients of several numbers divided by their greatest common divisor are prime to each other. 106. Any number, 4, which divides a product, 7 X 16, of two PROPERTIES OF WHOLE DIVISORS 29 factors, and which is prime to one of the factors, 7, divides the other factor, 16. 107. Any prime number, 5, which divides a product, 12 X 13 X 25, divides at least one of the factors of the product; and all prime numbers which divide a power, 15^, of a number, 15, divide the number. 108. Any number, 4, prime to each factor of a product, 7 X 15 15 X 23, is prime to the product. Any number, 4, prime with another, 15, is prime to any power of that number. 109. When two numbers, 4 and 15, are prime to each other, all powers of one are prime to any power of the other. 110. Any number, 720, divisible by two numbers, 4 and 9, prime to each other (80), is divisible by their product, 36. Ill: Any number, 7200, divisible by several numbers, 4, 9, 25, prime to each other in pairs, is divisible by their product. 112. The least common multiple of several whole numbers, 4, 9, 25, prime to each other in pairs, is equal to their product, 4 X 9 X 25 = 900 (83). 113. Any common multiple, 192, of two numbers, 24 and 16, is a multiple of the product, 8X3X2, whose factors are the greatest common divisor, 8, of these numbers and the quotients, 3 and 2, of their division by this greatest common divisor; and, conversely, any multiple of this product is a common multiple of the two numbers, 24 and 16. 114. The least common multiple of two numbers, 24 and 16, is equal to the product, 8 X 3 X 2 = 48, whose factors are the greatest common divisor, 8, of these numbers and the quotients, 3 and 2, of their division by this greatest common divisor. In the same manner the least common multiple may be determined (112 and 126). 115. Any common multiple of two, 24 and 16, is a multiple of their least common multiple, 48. 116. The least common multiple, 48, of two numbers, 24 and 16, is equal to the product of either one of the numbers and the quotient of the division of the other number by their greatest common divisor, 8 (114). 117. The product of the greatest common divisor, 8, of two numbers, 24 and 16, and their least common multiple, 48, is equal to the product, 24 X 16, of the two numbers. 118. When two numbers, 24 and 16, are multiplied or divided 30 ARITHMETIC by the same number, their least common multiple, 48, is mul- tiplied or divided by that number. 119. To find the least common multiple of several whole num- bers, 6, 8, 9, 10, find the least common multiple, 24, of the first two, 6 and 8 (114), then the least common multiple, 72, of that least common multiple, 24, and the third number, 9, and so on; the last least common multiple, 360, is the one required (126). 120. When the least common multiple, 72, of several numbers 8, 12, 18, is divided by each one of the numbers, the quotients, 9, 6, 4, are prime to each other; and, conversely, when a num- ber, 72, is such that in dividing it by several others, 8, 12, 18, quotients, 9, 6, 4, are obtained which are prime to each other, this number is the least common multiple of all the others. 121. Any whole number, 43, is prime when, being between the squares, 25 and 49, of two consecutive prime numbers, 5 and 7, it is neither divisible by the smaller of these prime numbers, nor by any number which precedes it, except one. 122. In general, to determine a prime number, divide by 2, 3, 5, 7, etc., until a quotient is obtained which is equal to or less than the last prime number used as divisor (121). 123. The series of prime numbers is unlimited. In the follow- ing tables on the next pages are given: 1st. Prime numbers from 1 to 10,000. 2d. Numbers less than 10,000 which do not contain the prime factors 2, 3, 5, 7, and 11, and their prime factors. PROPERTIES OF WHOLE DIVISORS Table of Prime Numbers between 1 and 10,000 31 1 367 839 1367 1907 2467 3061 3643 4243 4889 5501 6121 6761 7433 8069 8713 9349 2 73 53 73 13 73 67 59 53 4903 03 31 63 51 81 19 71 3 79 57 81 31 77 79 71 59 09 07 33 79 57 87 31 77 5 83 59 99 33 2503 83 73 61 19 19 43 81 59 89 37 91 7 89 63 1409 49 21 89 77 71 31 21 51 91 77 93 41 97 11 97 77 23 51 31 3109 91 73 33 27 63 93 81 8101 47 9403 13 401 81 27 73 39 19 97 83 37 31 73 6803 87 11 53 13 17 09 83 29 79 43 21 3701 89 43 57 97 23 89 17 61 19 19 19 87 33 87 49 37 09 97 51 63 99 27 99 23 79 21 23 21 907 39 93 51 63 19 4327 57 69 6203 29 7507 47 83 31 29 31 11 47 97 57 67 27 37 67 73 11 33 17 61 8803 33 31 33 19 51 99 79 69 33 39 69 81 17 41 23 67 07 37 37 39 29 S3 2003 91 81 39 49 73 91 21 57 29 71 19 39 41 43 37 59 11 93 87 61 57 87 5623 29 63 37 79 21 61 43 49 41 71 17 2609 91 67 63 93 39 47 69 41 91 31 63 47 57 47 81 27 17 3203 69 73 99 41 57 71 47 8209 37 67 53 61 53 83 29 21 09 79 91 5003 47 63 83 49 19 39 73 59 63 67 87 39 33 17 93 97 09 51 69 99 59 21 49 79 61 67 71 89 53 47 21 97 4409 11 53 71 6907 61 31 61 91 67 79 77 93 63 57 29 3803 21 21 57 77 11 73 33 63 97 71 87 83 99 69 59 51 21 23 23 59 87 17 77 37 67 9511 73 91 91 1511 81 63 53 23 41 39 69 99 47 83 43 87 21 79 99 97 23 83 71 57 33 47 51 83 6301 49 89 63 93 33 83 503 1009 31 87 77 59 47 51 59 89 11 59 91 69 8923 39 89 09 13 43 89 83 71 51 57 77 93 17 61 7603 73 29 47 97 21 19 49 99 87 99 53 63 81 5701 23 67 07 87 33 51 101 23 21 53 2111 89 3301 63 81 87 11 29 71 21 91 41 87 03 41 31 59 13 93 07 77 83 99 17 37 77 39 93 51 9601 07 47 33 67 29 99 13 81 93 5101 37 43 83 43 97 63 13 09 57 39 71 31 2707 19 89 4507 07 41 53 91 49 8311 69 19 13 63 49 79 37 11 23 3907 13 13 43 59 97 69 17 71 23 27 69 51 83 41 13 29 11 17 19 49 61 7001 73 29 99 29 31 71 61 97 43 19 31 17 19 47 79 67 13 81 53 9001 31 37 77 63 1601 53 29 43 19 23 53 83 73 19 87 63 07 43 39 87 69 07 61 31 47 23 47 67 91 79 27 91 69 11 49 49 93 87 09 79 41 59 29 49 71 5801 89 39 99 77 13 61 51 99 91 13 2203 49 61 31 61 79 07 97 43 7703 87 29 77 57 601 93 19 07 53 71 43 67 89 13 6421 57 17 89 41 79 63' 07 97 21 13 67 73 47 83 97 21 27 69 23 8419 43 89 ^ 13 1103 27 21 77 89 67 91 5209 27 49 79 27 23 49 97 n 17 09 37 37 89 91 89 97 27 39 51 7103 41 29 59 9719 79 19 17 57 39 91 3407 4001 4603 31 43 69 09 53 31 67 21 81 31 23 63 43 97 13 03 21 33 49 73 21 57 43 91 33 91 41 29 67 51 2801 33 07 37 37 51 81 27 59 47 9103 39 93 43 51 69 67 03 49 13 39 61 57 91 29 89 61 09 43 97 47 53 93 69 19 57 19 43 73 61 6521 51 93 67 27 49 99 53 63 97 73 33 61 21 49 79 67 29 59 7817 8501 33 67 211 59 71 99 81 37 63 27 51 81 69 47 77 23 13 37 69 23 61 81 1709 87 43 67 49 57 97 79 51 87 29 21 51 81 27 73 87 21 93 51 69 51 63 5303 81 53 93 41 27 57 87 29 77 93 23 97 57 91 57 73 09 97 63 7207 53 37 61 91 33 83 1201 33 2309 61 99 73 79 23 5903 69 11 67 39 73 9803 39 91 13 41 11 79 3511 79 91 33 23 71 13 73 43 81 11 41 701 17 47 33 87 17 91 4703 47 27 77 19 77 63 87 17 51 09 23 53 39 97 27 93 21 51 39 81 29 79 73 99 29 57 19 29 59 41 2903 29 99 23 81 53 99 37 83 81 9203 33 63 27 31 77 47 09 33 4111 29 87 81 6607 43 7901 97 09 39 69 33 37 83 51 17 39 27 33 93 87 19 47 07 99 21 51 71 39 49 87 57 27 41 29 51 99 6007 37 53 19 8609 27 57 77 43 59 89 71 39 47 33 59 5407 11 53 83 27 23 39 59 81 51 77 1801 77 53 57 39 83 13 29 59 97 33 27 41 71 83 57 79 11 81 57 59 53 87 17 37 61 7307 37 29 57 83 93 61 83 23 83 63 71 57 89 19 43 73 09 49 41 77 87 307 69 89 31 89 69 81 59 93 31 47 79 21 51 47 81 9901 11 73 91 47 93 71 83 77 99 37 53 89 31 63 63 83 07 13 87 97 61 99 99 93 4201 4801 41 67 91 33 93 69 93 23 17 97 1301 67 2411 3001 3607 11 13 43 73 6701 49 8009 77 9311 29 31 809 03 71 17 11 13 17 17 49 79 03 51 11 81 19 31 37 11 07 73 23 19 17 19 31 71 89 09 69 17 89 23 41 47 21 19 77 37 23 23 29 61 77 91 19 93 39 93 37 49 49 23 21 79 41 37 31 31 71 79 6101 33 7411 53 99 41 67 S3 27 27 89 47 41 37 41 77 83 13 37 17 59 8707 43 73 59 29 61 1901 59 49 32 ARITHMETIC Table of Numbers between 1 and 10,000 which do not Contain the Prime Factors S, 3, S, 7, and 11 and Their Prime Factors. No. Factors. No. Factors. No. Factors. No. Factors. 169 13 X13 1333 31 X43 2171 13 X 167 2951 13 X 227 221 13 X17 39 13 X 103 73 41 X53 77 13 X 229 47 13 X19 43 17 X79 83 37 X59 83 19 X 157 89 17 X17 49 19 X71 97 13 X 13 X 13 87 29 X 103 99 13 X23 57 23 X59 2201 31 X71 93 41 X73 323 17 X 19 63 29 X47 09 47 X47 3007 31 X97 61 19 X19 69 37 X37 27 17 X 131 13 23 X 131 77 13 X29 87 19 X73 31 23 X 97 29 13 X 233 91 17 X23 91 13 X 107 49 13 X 173 43 17 X 179 403 13 X31 1403 23 X61 57 37 X61 53 43 X71 37 19 X23 11 17 X83 63 31 X73 71 37 X83 81 13 X37 17 13 X 109 79 43 X53 77 17 X 181 93 17 X29 57 31 X47 91 29 X79 97 19 X 163 527 17 X31 69 13 X113 2323 23 X 101 3103 29 X 107 29 23 X 23 1501 19 X79 27 13 X 179 07 13 X 239 33 13 X41 13 17 X89 29 17 X 137 27 53 X59 51 19 X29 17 37 X41 S3 13 X 181 31 31 X 101 59 13 X43 37 29 X53 63 17 X 139 33 13 X 241 89 • 19 X 31 41 23 X67 69 23 X 103 39 43 X73 611 13 X47 77 19 X83 2407 29 XS3 49 47 X67 29 17 X37 91 37 X43 13 19 X 127 51 23 X 137 67 23 X29 1633 23 X71 19 41 X59 61 29 X 109 89 13 X53 43 31 X53 49 31 X79 73 19 X 167 97 17 X41 49 17 X97 61 23 X 107 93 31 X 103 703 19 X37 51 13 X 127 79 37 X67 97 23 X 139 13 23 X31 79 23 X73 83 13 X 191 3211 13 X 13X 19 31 17 X43 81 41 X41 89 19 X 131 33 53 X61 67 13 X59 91 19 X89 91 47 XS3 39 41 X79 79 19 X41 1703 13 X 131 2501 41 X61 47 17 X 191 93 13 X61 11 29 X59 07 23 X 109 63 13 X 251 99 17 X47 17 17 X 101 09 13 X 193 77 29 X 113 817 19 X43 39 37 X47 33 17 X 149 81 17 X 193 41 29 X29 51 17 X 103 37 43 X59 87 19 X 173 51 23 X37 63 41 X43 61 13 X 197 93 37 X89 71 13 X67 69 29 X61 67 17 X 151 3317 31 X 107 93 19 X47 81 13 X 137 73 31 X83 37 47 X 71 99 29 X31 1807 13 X 139 81 29 X89 41 13 X 257 901 17 X53 17 23 X79 87 13 X 199 49 17 X 197 23 13 X71 19 17 X 107 99 23 X 113 79 31 X 109 43 23 X41 29 31 X59 2603 19 X 137 83 17 X 199 49 13 X73 43 19 X97 23 43 X61 97 43 X 79 61 31 X31 49 43 X43 27 37 X71 3401 19 X 179 89 23 X43 53 17 X 109 41 19 X 139 03 41 X 83 1003 17 X59 91 31 X61 69 17 X 157 19 13 X 263 07 19 X53 1909 23 X83 2701 37 X73 27 23 X 149 27 13 X79 19 19 X 101 43 13 X 211 31 47 X 73 37 17 X61 21 17 X 113 47 41 X67 39 19 X 181 73 29 X37 27 41 X47 59 31 X89 73 23 X 151 79 13 X83 37 13 X 149 71 17 X 163 81 59 X 59 81 23 X47 43 29 X67 73 47 X59 97 13 X 269 1121 19 X59 57 19 X 103 2809 53 X53 3503 31 X 113 39 17 X67 61 37 X53 13 29 X97 23 13 X 271 47 31 X37 63 13 X 151 31 19 X 149 51 53 X 67 57 13 X89 2021 43 X47 39 17 X 167 69 43 X83 59 19 X61 33 19 X 107 67 47 X61 87 17 X 211 89 29 X41 41 13 X 157 69 19 X 151 89 37 X 97 1207 17 X71 47 23 X89 73 13 X 13 X 17 99 59 X 61 19 23 X 63 59 29 X71 81 43 X67 3601 13 X 277 41 17 X 73 71 19 X 109 99 13 X 223 11 23 X 157 47 29 X 43 77 31 X67 2911 41 X71 29 19 X 191 61 13 X 97 2117 29 X73 21 23 X 127 49 41 X 89 71 31 X 41 19 13 X 163 23 37 X79 53 13 X 281 73 19 X 67 47 19 X 113 29 29 X 101 67 19 X 193 1313 13 X 101 59 17 X 127 41 17 X 173 79 13 X 283 PROPERTIES OF WHOLE DIVISORS 33 Table of Numbers between 1 and 10,000 which do not Contain the Prime Factors — Continued. No. Factors. No. Factors. No. Factors. No. Factors. 3683 29 X 127 4453 61 X73 5207 41 X 127 5947 19 X 313 3713 47 X79 69 41 X 109 13 13 X401 59 59 X 101 21 61 X61 71 17 X 263 19 17 X 307 63 67 X89 37 37 X 101 89 67 X67 21 23 X 227 69 47 X 127 43 19 X 197 4511 13 X 347 39 13 X 13 X 31 77 43 X 139 49 23 X 163 31 23 X 197 49 29 X 181 83 31 X 193 67 13 X 17 X 17 37 13 X 349 51 59 X89 89 53 X 113 63 53 X71 41 19 X 239 63 19 X 277 93 13 X 461 81 19 X 199 53 29 X 157 67 23 X 229 6001 17 X 353 91 17 X 223 59 47 X97 87 17 X 311 19 13 X 463 99 29 X 131 73 17 X 269 93 67 X79 23 19 X 317 3809 13 X 293 77 23 X 199 5311 47 X 113 31 37 X 163 11 37 X 103 79 19 X241 17 13 X 409 49 23 X 263 27 43 X89 89 13 X 353 21 17 X 313 59 73 X83 41 23 X 167 4601 43 X 107 29 73 X73 71 13 X 467 59 17 X 227 07 17 X 271 39 19 X281 77 59 X 103 69 53 X73 19 31 X 149 53 63 X 101 6103 17 X 359 87 13 X 13 X 23 33 41 X 113 59 23 X 233 07 31 X 197 93 17 X 229 61 59 X79 63 31 X 173 09 41 X 149 3901 47 X83 67 13 X 359 71 41 X 131 19 29 X 211 37 31 X 127 81 31 X 151 77 19 X 283 37 17 X 19 X 19 53 59 X67 87 43 X 109 89 17 X 317 57 47 X 131 59 37 X 107 93 13 X 19 X 19 5429 61 XS9 61 61 X 101 61 17 X 233 99 37 X 127 47 13 X 419 69 31 X 199 73 29 X 137 4709 17 X 277 59 53 X 103 79 37 X 167 77 41 X97 17 53 X89 61 43 X 127 87 23 X 269 79 23 X 173 27 29 X 163 73 13 X 421 91 41 X 151 91 13 X 307 47 47 X 101 91 17 X 17 X 19 6227 13 X 479 4009 19 X 211 57 67 X71 97 23 X 239 33 23 X 271 31 29 X 139 69 19 X 251 5513 37 X 149 39 17 X 367 33 37 X 109 71 13 X 367 39 29 X 191 41 79 X79 43 13 X 311 77 17 X 281 43 23 X241 53 13 X 13 X 37 61 31 X 131 4811 17 X 283 49 31 X 179 83 61 X 103 63 17 X 239 19 61 X79 61 67 X83 89 19 X 331 69 13 X 313 41 47 X 103 67 19 X 293 6313 59 X 107 87 61 X67 43 29 X 167 87 37 X 151 19 71 X89 97 17 X 241 47 37 X 131 97 29 X 193 31 13 X 487 4117 23 X 179 49 13 X 373 5603 13 X 431 41 17 X 373 21 13 X 317 53 23 X 211 09 71 X79 71 23 X 277 41 41 X 101 59 43 X 113 11 31 X 181 83 13 X 491 63 23 X 181 67 31 X 157 17 41 X 137 6401 37 X 173 71 43 X97 83 19 X 257 27 17 X 331 03 19 X 337 81 37 X 113 91 67 X73 29 13 X 433 07 43 X 149 83 47 X89 97 59 X83 33 43 X 131 09 13 X 17 X 29 87 53 X79 4901 13 X 13 X 29 71 53 X 107 31 59 X 109 89 59 X71 13 17 X 17 X 17 81 13 X 19 X 23 37 41 X 157 99 13 X 17 X 19 27 13 X 379 99 41 X 139 39 47 X 137 4223 41 X 103 79 13 X 383 5707 13 X 439 43 17 X 379 37 19 X 223 81 17 X 293 13 29 X 197 63 23 X 281 47 31 X 137 97 19 X 263 23 59 X97 67 29 X 223 67 17 X 251 5017 29 X 173 29 17 X 337 87 13 X 499 4303 13 X 331 29 47 X 107 59 13 X443 93 43 X 151 07 59 X73 41 71 X71 67 73 X79 97 73 X89 09 31 X 139 53 31 X 163 71 29 X 199 99 67 X97 13 19 X 227 57 13 X 389 73 23 X 251 6509 23 X 283 21 29 X 149 63 61 X83 77 53 X 109 11 17 X 383 31 61 X71 69 37 X 137 5809 37 X 157 27 61 X 107 43 43 X 101 83 13 X 17 X 23 33 19 X 307 33 47 X 139 51 19 X 229 5111 19 X 269 37 13 X 449 39 13 X 503 69 17 X 257 23 47 X 109 91 43 X 137 41 31 X 211 79 29 X 151 29 23 X 223 93 71 X83 57 79 X83 81 13 X 337 41 53 X97 99 17 X 347 83 29 X 227 87 41 X 107 43 37 X 139 5909 19 X 311 93 19 X 347 93 23 X 191 49 19 X 271 11 23 X 257 6613 17 X 389 99 53 X 83 61 13 X 397 17 61 X97 17 13 X 509 4427 19 X 233 77 31 X 167 21 31 X 191 23 77 X 179 29 43 X 103 83 71 X73 33 17 X 349 31 19 X 349 23 X 193 91 29 X 179 41 13 X 457 41 29 X 229 34 ARITHMETIC Table of Numbers between 1 and 10,000 which do not Contain the Prime Factors — Continued. No. Factors. No. Factors. No. Factors. No. Factors. 6647 17 X 17 X 23 7363 37 X 199 8033 29 X 277 8759 19 X 461 49 61 X 109 67 53 X 139 47 13 X 619 73 31 X 283 67 59 X 113 73 73 X 101 51 83 X97 77 67 X 131 83 41 X 163 79 47 X 157 77 41 X 197 91 59 X 149 97 37 X 181 87 83 X89 83 69 X 137 97 19 X 463 6707 19 X 353 91 19 X 389 8119 23 X 353 8801 13 X 677 31 53 X 127 97 13 X 569 31 47 X 173 09 23 X 383 39 23 X 293 7409 31 X 239 37 79 X 103 43 37 X 239 49 17 X 397 21 41 X 181 43 17 X 479 51 53 X 167 51 43 X 157 23 13 X 571 49 29 X 281 57 17 X 521 57 29 X 233 29 17 X 19 X 23 63 31 X 263 73 19 X 467 67 67 X 101 39 43 X 173 69 41 X 199 79 13 X 683 73 13 X 521 53 29 X 257 77 13 X 17 X 37 81 83 X 107 99 13 X 523 63 17 X 439 89 19 X 431 91 17 X 523 6817 17 X 401 71 31 X241 8201 59 X 139 8903 29 X 307 21 19 X 359 93 59 X 127 03 13 X 631 09 69 X 151 47 41 X 167 7501 13 X 577 07 29 X 283 17 37 X241 51 13 X 17 X 31 19 73 X 103 13 43 X 191 27 79 X 113 59 19 X 19 X 19 31 17 X 443 27 19 X 433 47 23 X 389 77 13 X 23 X 23 43 19 X 397 49 73 X 113 57 13 X 13 X 53 87 71 X97 71 67 X 113 51 37 X 223 59 17 X 17 X 31 89 83 X83 97 71 X 107 57 23 X 369 77 47 X 191 93 61 X 113 7613 23 X 331 79 17 X 487 83 13 X 691 6901 67 X 103 19 19 X 401 99 43 X 193 89 89 X 101 13 31 X 223 27 29 X 263 8303 19 X 19 X 23 93 17 X 23 X 23 29 13 X 13 X 41 31 18 X587 17 X 449 21 53 X 167 9017 71 X 127 31 29 X 239 33 33 13 X 641 19 29 X 311 43 53 X 131 57 13 X 19 X 31 39 31 X 269 47 83 X 109 S3 17 X 409 61 47 X 163 41 19 X 439 61 13 X 17 X 41 73 19 X 367 63 79 X97 47 17 X 491 71 47 X 193 89 29 X 241 97 43 X 179 57 61 X 137 73 43 X 211 7003 47 X 149 7709 13 X 593 59 13 X 643 77 29 X 313 09 43 X 163 29 59 X 131 81 17 X 17 X 29 83 31 X 293 31 79 X89 39 71 X 109 83 83 X 101 89 61 X 149 33 13 X 541 47 61 X 127 99 37 X 227 9101 19 X 479 37 31 X 227 61 23 X 337 8401 31 X 271 13 13 X 701 61 23 X 307 69 17 X 457 11 13 X 647 31 23 X 397 67 37 X 191 71 19 X 409 13 47 X 179 39 13 X 19 X 37 81 73 X97 81 31 X 251 17 19 X 443 43 41 X 223 87 19 X 373 83 43 X 181 41 23 X 367 67 89 X 103 93 41 X 173 87 13 X 599 63 79 X 107 69 53 X 173 97 47 X 151 7801 29 X 269 71 43 X 197 79 67 X 137 99 31 X 229 07 37 X 211 73 37 X 229 93 29 X 317 7111 13 X 547 11 73 X 107 79 61 X 139 97 17 X 541 23 17 X 419 13 13 X 601 83 17 X 499 9211 61 X 151 41 37 X 193 31 41 X 191 89 13 X 663 17 13 X 709 53 23 X 311 37 17 X 461 97 29 X 293 23 23 X 401 67 17 X 421 49 47 X 167 8507 47 X 181 53 19 X 487 63 13 X 19 X 29 59 29 X 271 09 67 X 127 59 47 X 197 69 67 X 107 71 17 X 463 31 19 X 449 63 59 X 157 71 71 X 101 91 13 X 607 49 83 X 103 69 13 X 23 X 31 81 43 X 167 97 53 X 149 51 17 X 503 71 73 X 127 99 23 X 313 7913 41 X 193 57 43 X 199 87 37 X 251 7201 19 X 379 21 89 X89 67 13 X 659 99 17 X 547 23 31 X 233 39 17 X 467 79 23 X 373 9301 71 X 131 41 13 X 557 43 13 X 13 X 47 87 31 X 277 07 41 X 227 61 53 X 137 57 73 X 109 93 13 X 661 13 67 X 139 67 13 X 13 X 43 81 19 X 419 8611 79 X 109 29 19 X 491 77 19 X 383 67 31 X257 21 37 X 233 47 13 X 719 79 29 X 251 69 13 X613 33 89 X97 53 47 X 199 89 37 X 197 79 79 X 101 39 53 X 163 67 17 X 19 X 29 91 23 X 317 81 23 X 347 51 41 X 211 79 83 X 113 7303 67 X 109 91 61 X 131 53 17 X 509 89 41 X 229 13 71 X 103 99 19 X 421 71 13 X 23 X 29 9407 23 X 409 19 13 X 563 8003 53 X 151 83 19 X 457 09 97 X97 27 17 X 431 21 13 X 617 8711 31 X 281 51 13 X 727 39 41 X 179 23 71 X 113 17 23 X 379 69 17 X 557 61 17 X 433 27 23 X 349 49 13 X 673 81 19 X 499 PROPERTIES OF WHOLE DIVISORS 35 Table of Numbers between 1 and 10,000 which do not Contain the Prime Factors — Continued. No. Factors. No. Factors. No. Factors. No. Factors. 9487 53 X 179 9599 29 X 331 9731 37 X 263 9893 13 X 761 9503 13 X 17 X 43 9607 13 X 739 61 43 X 227 99 19 X 521 09 37 X 257 17 . 59 X 163 63 13 X 751 9913 23 X 431 17 31 X 307 37 23 X 419 73 29 X 337 17 47 X 211 23 89 X 107 41 31 X 311 97 97 X 101 37 19 X 523 29 13 X 733 59 13 X 743 99 41 X239 43 61 X 163 53 41 X 233 71 19 X 509 9809 17 X 577 53 37 X 263 57 19 X 503 73 17 X 669 27 31 X 317 59 23 X 433 63 73 X 131 83 23 X 421 41 13 X 757 71 13 X 13 X 59 71 17 X 563 9701 89 X 109 47 43 X 229 79 17 X 587 77 61 X 157 03 31 X 313 53 59 X 167 83 67 X 149 89 43 X 223 07 17 X 571 69 71 X 139 91 97 X 103 93 53 X 181 27 71 X 137 81 41 X 241 97 13 X 769 540 2 270 2 135 3 45 3 15 3 5 5 1 124. The general rule for separating a number into its prime factors greater than one. Divide successively, as many times as possible, by each of the numbers 2, 3, 5, 7 . . . which may be used as divisors, until a prime number is obtained in the quotient; this last quotient and all the numbers which have been used as divisors are the prime factors of the number. For example, to separate the number 540 into its prime factors, the calculation is arranged as shown, which gives the fac- tors 2, 2, 3, 3, 3, 5; or 540 = 2 X 2 X 3 X 3 X 3 X 5 = 2^ X 3^ X 5. The table on page 32 permits of an easy separation into its factors of a number, 2,031,810 for instance, which contains only •prime factors 2, 3, 5, 7, and 11, and other prime factors of which the product is not greater than 10,000. It is seen immediately that the number contains the factors 2 and 5 (90), then the factor 3 (95), and the factor 11. The last quotient, 6157, may be found in the table, which indicates that it does not contain any of the factors 2, 3, 5, 7, and 11, and gives its prime factors 47 and 131, which could not have been obtained without proving that the number did not contain any prime number less than 47. The prime factors are: 2,031,810 = 2X3X5X11X47X 131. Remark 1. When a number, 8100, is the product of known numbers, 81 and 100, the process of separating it into its prime 2,031,810 2X5 203,181 3 67,727 11 6,157 47 X 131 56 ARlTtiMETlC factors may be shortened by finding the prime factors of 81 and of 100. 81 = 3S 100 = 2^ X 5\ 8100 = 2^ X 3^ X 5^ Remark 2. This last example shows that when a number, 8100 = 90^ is an exact power, the exponents of its prime factors are divisible by the degree of the power. 125. The greatest common divisor of several numbers, 240, 180, 72, is equal to the product of the prime factors common to these numbers, each of these factors being raised to the power corresponding to the smallest exponent which it bears as a factor of the numbers. Thus, having given: 240 = 2^ X 3 X 5, 180 = 2^ X 3^ X 5, 72 = 2' X d\ the greatest common divisor of these numbers is 2^ X 3 = 12. This gives another method for determining the greatest common divisor of several numbers (102 and 104). 126. The least common multiple of several numbers is equal to the product of their prime factors, each of the factors being raised to the power corresponding to the largest exponent which it bears as a factor of the numbers. Thus the least common multiple of the numbers in the above example, 240, 180, and 72, is 2* X 3^ X 5. This being another method of finding the least common multiple of several numbers (114 and 119). 127. To find all the divisors of a number, 360, separate the number into its prime factors (124), writing them in a vertical column; multiply the first factor 2 by the second 2, the first two factors and their product 4 by the third, omitting the multiplica- tions which would give the products already obtained; multiply in the same manner the first three factors and the products ob- tained by the fourth factor, and so on until the last factor has been used as multiplier; all the unequal prime factors of the number, and the products that have been obtained, are the re- quired divisors. The operation is carried on as follows; the PROPERTIES OF WHOLE DIVISORS 37 number 1 being always a divisor, is written at the top of the table: 360 180 90 45 15 5 1 2 2, 4 2, 8 3, 6, 12, 24 3, 9, 18, 36, 72 5, 10, 20, 40, 15, 30, 60, 120, 45, 90, 180, 360. 1 3 9 5 2 10 20 4 6 8 12 24 40 15 18 45 36 30 72 60 120 90 180 360 The prime factors of a number being known, given for example 360 = 2^ X 3^ X 6, it is simpler, in obtaining all its divisors, to write 1 and the successive powers 2, 4, 8, of 2 con- tained in the number in the first column; in the second the products of the numbers in the first with the powers 3 and 9 of 3 contained in 360, and in the third column the products of the numbers in the first two columns with the first power 5 of 5 contained in 360. The numbers forming this table, when com- pleted, are all the divisors of 360. The number of divisors of a number is equal to the prod- the sums obtained by increasing the exponent of each Thus, given 360 = 2' X 3^ X 5, the 128. uct of prime factor by 1 (124) number of divisors counting 1 and 360 is (3 + 1) (2 + 1) (1 + 1) = 24. 129. To find all the common divisors of several numbers, find the greatest common divisor of the numbers, then all the divi- sors of this greatest common divisor (125 and 127). BOOK III FRACTIONS AND DECIMALS FRACTIONS 130. A fraction or a fractional number is one or several parts of a unit which has been divided into equal parts. Thus, a unit having been divided into 9 equal parts, the number formed with 5 of these parts is a fraction. 131. The denominator of a fraction is the number which indi- cates into how many parts the unit has been divided. The numerator is the number which indicates how many of these equal parts are contained in the fraction. Thus, in the preceding example, 9 is the denominator and 5 the numerator. The numerator and denominator are the two terms of the fraction. Conceive that a fraction may contain all the parts of one or several units, and even all the parts of one or several units plus the parts of another unit these units; being the same and being all divided into the same number of equal parts. When a fraction does not contain all the parts of one, that is, when its numerator is less than its denominator, it is less than imity. If it contains all the parts of one, its terms are equal, and it is equal to unity. Finally, if the numerator is greater than the denominator, the fraction is larger than unity. According as a fraction is smaller or larger than unity, it is called a proper or an improper fraction (130). 132. To pronounce a fraction, pronounce the numerator, then the denominator, adding the termination th. Thus the fraction in (130) is pronounced five ninths. There are exceptions for the denominators 2, 3, and 4; thus we say one half, one third, one quarter, or fourth. 133. In writing a fraction, write the numerator above the denominator and separate them by a line. Thus five ninths is written - • 134. A fraction represents the quotient of the division of its 38 FRACTIONS 39 5 numerator by its denominator (51). Thus - is equal to 5 divided by 9. 7 Any whole number, 7, may be considered as a fraction, - . with the number 7 for a numerator and unity 1 for a denominator. 135. To reduce an improper fraction to a whole number and a proper fraction, or to a mixed number, divide the numerator by the denominator, and add to the quotient a fraction, having the remainder for a numerator and the denominator of the improper fraction for a denominator. Thus : 63 ^ ^ 37 ^ 2 -9='' ^^^ T = ^+5- 136. To reduce a whole number to an equivalent fraction having a given denominator 9; for the numerator of the fraction take the product 63 of its denominator 9 with the whole number 7. Thus: 7 X 9 63 9 9 ■ 137. In adding the terms of several equal fractions, the resulting fraction is equal to any one of those fractions: 3 3 3 3 12 4 10 14 4+10 + 14 28 7~7~7~7~28' 6 ~ 15 21 ~ 6 + 15 + 21 ~ 42 In subtracting the terms of two equal fractions which have not the same terms, . a resulting fraction is obtained which is equal to both of the given fractions: 28 10 28 - 10 18 42 15 42 - 15 27 138. When the terms of any two unequal fractions are added, generally the value of the resulting fraction lies between that of the two fractions added: 4 4+9 9 4 4+8+9 9 7^7+1^5' 7^7 + 5 + 5"^5' 139. When the same quantity is added to both terms of a fraction, the fraction is increased or diminished according as the fraction 4o Arithmetic is proper or improper (131). In each case unity is the limit which it approaches as the terms become larger, but which can never be attained because the terms can never become equal: 5 6 + 3 , 11 11 + 2 9<9T3' "^'^ T>4 + 2- On the contrary, if the same quantity is subtracted from both terms of a fraction, the fraction is diminished or increased ac- cording as the fraction is proper or improper. In each case the fraction departs farther and farther from unity: 8 8-3 ^ 13 13-2 12>12^:3' ^^"^ -6<-r^- When the fraction is equal to unity its value is not altered by adding to, or subtracting the same quantity from each term. 140. To multiply a fraction by a whole number, multiply the numerator, or, if it is possible without a remainder, divide the denominator by the number. Thus: 3 , 3 3 ^^^ 8^* = 8T4 = 2- 141. To divide a fraction by a whole number, multiply the denominator, or, if it is possible without a remainder, divide the numerator by the number. Thus: 3, 3 3 ,8,8-f-42 7 = ^ = r^ = 2-8' ^""^ r-^ = -T- = r 142. It does not alter the value of a fraction to multiply or divide both its terms by the same number (73) : 3 3x2 6 8 8--4 2 4 "4x2 8' and 12 12 -- 4 ~ 3 IRREDUCIBLE FRACTIONS 143. To simplify or reduce a fraction to a simpler form, is to diminish the value of its terms without changing value as a fraction. 144. A fraction is irreducible, or reduced to its simplest form, IRREDUCIBLE FRACTIONS 41 1 3 when it cannot be made simpler. Such are the fractions ^ , t; -r (146). 145. The terms of an irreducible fraction, _, are prime to each other (80). 3^ ^ 146. To reduce a fraction, jp; to a simpler form, divide the two terms by a common divisor (142): 30 _ 30 H- 3 _ 10 46 ""46^ 3 ~ 15' 30 To reduce a fraction, j^, to its simplest form, divide its terms by their greatest common divisor, 15 (102): 30 _ 30 ^ 15 _ 2 . 45 ~ 45 -^ 15 ~ 3 ' or cancel all the prime factors common to the two terms (125): 30_ 2 X 3 X 5 _2 46~3x3x5~3' Applying what was said in (102), not only the greatest common divisor, 12, of the terms of the fraction, — --> is obtained, but also the quotient, 30 and 73, of the two terms divided by 12, and it may be written 360_30 876~73' 1 68 In practice, to reduce a fraction, krS' to a simpler form, its 168 _ 84 _ 42 _ 14 _ 2 terms being even, divide by 2; for 252 ~ 126 ~ 63 ~ 21 ~ 3 the same reason divide the terms 84 of the resulting fraction, -— ^> by 2 ; it is now seen that the 42 terms of the resulting fraction, tttt, are divisible by 3 (95), and 14 those of the fraction ^r by 7. Thus a fraction may often be re- duced to its simpler form by dividing out its common factors. 42 ARITHMETIC 147. The least common multiple, 36, of the denominators of 5 4 7.^,,, several irreducible fractions, g, g> ^> is the least common de- nominator to which the fractions may be reduced (151). 148. The greatest common divisor of several irreducible fractions, -, -, — , is the fraction, zttt.) whose numerator 3 is the greatest common divisor of the numerators (104), and whose denominator is the least common multiple 140 of their denominators (119). 149. The least common multiple of several irreducible fractions, a' s' Tn> is the irreducible fraction -^, whose numerator is the least common multiple 140 of the numerators, and whose denom- inator is the greatest common divisor 3 of the denominators. REDUCTION OF FRACTIONS TO THE SAME DENOMINATOR 150. To reduce fractions to the same denominator is to find fractions equal to the given fractions, with denominators equal to each other (131). 151. To reduce two fractions to the same denominator, mul- tiply the terms of each fraction by the denominator of the other. And, in general, to reduce several fractions to the same denomi- nator, multiply each numerator by the product of the denomi- nators of the others, and as -common denominator use the prod- uct of all the denominators: 2 2x6 12 1 3x5x6 90 3' 3x6 18 2~ 2x3x5x6 180 5 6^ 5x3 6x3 15 18 2 3~" 2x2x6x6 180 120 180 4 4x2x3x6 144 5 180 180 5 5x2x3x6 150 6 180 "180 When it is seen that a number is divisible by all of the de- nominators of the given fractions, that is, is common multiple of the denominators (126), it is taken as common denominator, and the numerator of each fraction is multiplied by the quotient obtained in dividing this commo;i denominator by the denomi- ADDITION OF FRACTIONS 43 nator of the fraction. Thus, in the preceding examples, it is seen immediately that 6 and 30 may be taken as common de- nominators, and then we have: 2 2x2 3 6 4 ~6 1 1 x 15 15 2 30 30 5 5 6= 6 5 ~6 2 2 x 10 20 3 30 ~ 30 4 4x6 24 5 ~ 30 " 30 5 5x5 25 6 30 ~ 30 It is always possible to find the least common multiple of the denominators (126), and use it as common denominator as was done above. The number, 2 X 3^ by which the numerator of the fraction 7 K7^ must be multiplied, for ex- 7 7 7 X 2 X 32 126 20 ^^ ample, is obtained simply by -iH -|^ 11 X 3 X 5 165 canceling in the common de- 2^X5 11 23 X 3 23 2='x 3- 17 7 X 2 X • 32 2= X 32 X 6 11 X 3 X 5 23 X 32 X 6 23 X 2 X 6 23 X 32 X 6 17 X 23 20 24 23 X 3 23 X 32 X 6 360 23 23 23 X 2 X 6 230 nominator 23 X 3^ X 5, the factors of the denominator 36 2== X 3- 23 X 32 X 6 360 2^ X 5 of the fraction X X 5. 17 17 17 X 23 136 ^ ^, . , , ^ 45 ^ 3^x 5 ^ 23 X 3^ X 5 "" 360 example the general rule would have given 777,600 for the common denominator. When the denominators of the given fractions are prime to each other (80), their least common multiple is equal to their product, and then to reduce the fractions to the same denomina- tor, follow the general rule without any possible simplification (147). ADDITION OF FRACTIONS 152. To add fractions, reduce them, if necessary, to the same common denominator (151); and add the numerators which result; then the result of the operation is a fraction whose num- erator is the sum of the reduced numerators and whose denomi- nator is the common denominator. Example: 44 ARITHMETIC 5 2_20 12 3 ~30 7 7 _ 42 + 12 +5 ~ 30 17 ^_25 + 12 + 6 ~ 30 6 + 7 + 17 29 87 1^- = I2^^'"- 30'"- 153. To add a whole number and a fraction, reduce the whole number to an equivalent fraction, having for a denominator the denominator of the fraction (136), and proceed as in the preced- ing case. This amounts to adding to the numerator of the given fraction the product of the denominator and the whole number: 4 4 + 5 X 7 39 5 + V = = — 5 6 154. To add any number of fractions and whole numbers to- gether, add the fractions and whole numbers separately, and then operate with the sums as in the preceding case: 1 , I. , o , 2 ,^ , „, . /I . 2\ „ . 11 _ 107 12 "" l2"" 2 + 5 + 3 + |=(5 + 3)+(i + |) = 8 + SUBTRACTION OF FRACTIONS 155. To obtain the difference between two fractions, reduce them, if necessary, to the same denominator (151); subtract the numer- ators of the reduced fractions, and the required result will have this difference for a numerator and the common denominator for a denominator: 7 2 7-2 5 3 1 21 4 17 9 9 9 9' 4 7 28 28 "~ 28 156. To subtract a fraction from a whole number, or conversely, reduce the whole number to an equivalent fraction, having for a denominator that of the fraction (136), and proceed as in the preceding case. Thus: ^ _ 4 _ 56 4 _ 56 - 4 52 15 15 12 3 7 7 7 7-7' T~'^^ T~T=4' MULTIPLICATION OF FRACTIONS 45 157. To subtract a whole number plus a fraction from a whole 1 3 number plus a fraction, 4 + -^ from 7 + ^ for example, reduce each of the quantities to an equivalent fraction (153), then take the difference of the fractions obtained (155 and 156). However, it is simpler to subtract, first the fractions 5~3~15~i5~15' 4 then the whole numbers, 7 — 4 = 3; which gives 3 + j^ • When the fraction from which the subtracting is to be done is the lesser, it is increased by a unit, which means that the numerator is to be increased by a number equal to the denomi- nator, and to compensate this the whole number to be subtracted is reduced by one unit. As a special case, the fraction may be zero. Examples: ^4 ^-5 ^+3 ^+15 ^4 or or *4 *4 8 7+1 ^+-5 ^+5 4 Difference 3 + zn^ 15 Difference 2 +~^ 15 3 Difference 4 + - 5 To subtract several whole numbers plus fractions from several whole numbers plus fractions, reduce all the plus quantities to one whole number and fraction (154), the same with the nega- tive quantities, and proceed as in the preceding case. MULTIPLICATION OF FRACTIONS 158. To multiply a quantity by a fraction, multiply it by the numerator of the fraction and divide the product by the denom- inator. Remark. In multiplying a quantity by a fraction, the prod- uct is equal to, greater or less than the multiplicand, according as the fraction multiplier is equal to, greater, or less than unity. 159. To multiply a whole number by a fraction, is the same as a fraction by a whole number (140). Thus: 3 _ 9x3 _ 27 7 7_ _ 7 ^^4- 4 -4' "^^g^g-s 3' 46 ARITHMETIC 160. To multiply one fraction by another, multiply the numera- tors together for the numerator, and the denominators for the denominator: 3 7 3 X 7 21 4^5^4x5^ 20* 161. The jrroduct of any number of whole numbers and fractions is a fraction whose numerator is the product of the whole num- bers and the numerators of the given fractions, and whose de- nominator is equal to the product of their denominators: 3 2 5x3x2x2 60 5XjX^X^- ^^ 28 ■ In practice, before going through the calculations, write out the multiplication and cancel the common factors of the two terms (146). This shortens the operation, and gives a product reduced to its lowest terms providing all common factors are canceled. In the preceding example, canceling 2 X 2 in the 5X3 15 numerator and 4 in the denominator, we have — = — = -=- for a result. In the example 4 6 42 11 ^ X ^ X ^^ X 11 11 9^7^ 35^ 8 ~px;x^^x^~2l' 3 7 ^ cancel 4 in the numerator and replace 8 by 2 in the denominator (confusion is avoided by drawing a line through the canceled factors); cancel 5 in the numerator and replace 35 by 7 in the denominator; then a 7 in the denominator, replacing the 42 by 6 in the numerator; finally 6 in the numerator, by canceling 2 and replacing 9 by 3 in the denominator. The result is 11 11 3 X 7 "21' 162. To find a certain fraction of a fraction of any quantity, multiply the quantity by the product of the fractions (161). Thus: 2 . ^ „ 2 10 30f5are5x3 = ^. 1 „ 2 - „ „ 1 2 10 jof 3 0f6are5xJX3 = J2• 3„1.2.„ _ 2 13 30 ^of -of -0f5are5x3XjX^ = gj. DIVISION OF FRACTIONS 47 Remark. To multiply a fraction which has unity 1 for a nu- merator by a quantity is to divide the quantity by the denomi- nator of the fraction. Thus : 1 1*1 6 of 15 = -I (64). 163. Articles (33, 34, 41, 42, 43) are equally true with whole numbers and fractions. DIVISION OF FRACTIONS 164. To divide a quantity by a fraction, multiply by the divisor fraction inverted (159 and 160). 7^^_ 4_28 4_2_4 5_20 "4 ^3 3'7'5 72 14' Remark. The quotient is equal to, less, or greater than the dividend according as the divisor fraction is equal to, greater, or less than imity. 165. The articles (56, 59, 60, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73), and some which are immediate consequences of them, being founded upon principles applicable to fractions as well as whole numbers, apply to both sorts of numbers. 166. To divide whole numbers plv^ fractions by whole numbers flus fractions, reduce the dividend to one fraction (154 and 157), and the divisor to another, and divide, proceeding as in the preceding case (164). Thus: (-^(-1) 17^9 _ 17 4^68 5 ■4'59"45' DECIMAL NUMBERS 167. A decimal fraction is a fraction whose denominator is a 3 278 power of 10 (85 and 86). Such are the fractions zr^ and it^- 168. A decimal number is a number composed of a whole number, which may be zero, and one or several decimal fractions, whose numerators are less than the base, 10, and whose denomi- nators are powers of that base. Such are: f37 + ^ + 10 1000 8 \ / 3 5 7 \ loooj' ^""^ (lo + Too + 1000/ ' 48 ARITHMETIC 169. Numeration of decimals. To simplify the writing of a deci- mal number, the several figures composing the number are written on a horizontal line and separated into two parts by a period ; the part at the left expresses whole units; the first figure at the right of the period expresses tenths, or decimals of the first order; the second, hundredths, or decimals of the second order, and so on; thus in a decimal, as in a whole number, any figure placed at the left of another figure expresses units ten times as great as those at its right (7). According to this method, the number (37 + ^ + j^) is written 37.508, and ( ^g + Joq + Jqqq) is written 0.357. To pronounce a decimal number written in figures, pronounce successively the part at the left and right of the period, adding to each the units expressed by the first figure to the right of each part. Thus the number 37.508 is pronounced thirty-seven units five hundred eight thousandths, and 0.357 is pronounced no units, three hundred fifty-seven thousandths. When the decimal part contains more than 5 or 6 figures, in pronouncing, it is convenient to divide it into periods of 3 figures each, com- mencing at the decimal point; then, commencing at the left, pronounce successively each "period of figures, giving each the name of the units expressed by the figure at the right. Thus, the number 37.32504645769 is pronounced: 37 units, 325 thousandths, 46 millionths, 457 billionths, 69 hundred billionths, or 690 trillionths, adding a cipher in the last period. 170. Each figure placed at the right of the decimal point, or period, is a decimal, or decimal figure of the given number. Its form indicates its absolute value, and its position its relative value (8). 171. It does not alter the value of a decimal to suppress or add ciphers at the right: 32.45 = 32.4500, and 3.12500 = 3.125. 172. To reduce a decimal to the form of a decimal fraction (167), take the given number for numerator, omitting the decimal DECIMAL NUMBERS 49 point, and for denominator 1 followed by as many ciphers as there are decimals in the given number: 173. Conversely, to reduce a decimal fraction to the form of a decimal number, write the numerator and separate on the right as many decimal figures as there are ciphers in the denominator. In the case where there are less figures in the numerator than ciphers in the denominator, write ciphers at the left of the figures : 1000 = 2-348. and j^oo = 0.037. 174. The value of a given quantity is near the value of an- other quantity by less than a third quantity, when the difference of the first two is less than the third quantity. Thus 24.37 is less than a hundredth, .01, smaller than 24.376, because 24.376 - 24.37 = 0.006 is less than 0.01. 175. The nearest value of a decimal, at least of a decimal of a certain order, is the result which is obtained by suppressing in the given number all the decimals written at the right of the figure which expresses the units of the given order. Thus the value of the number 7.46537 to the thousandths place is 7.465. 176. In getting the nearest possible value of a decimal, retaining a certain number of decimal figures, there are three cases: First, if the first figure which follows the last which is to be retained is less than 5, suppress the 5 with the figures which follow; sec- ond, if it is larger than 5, or if it is 5 followed by other significa- tive figures, suppress it with those which follow and increase the last figure by 1 ; third, finally, if it is 5 and not followed by other figures, suppress it, and add either one or nothing to the last figure. In any case the error can not be greater than a half a unit of the order of the last figure. The value of 4.8365 to the first decimal place is 4.8; to the second decimal place, 4.84; to the third place, 4.836 or 4.837. 177. To multiply or divide a decimal by one, followed by several ciphers, move the decimal point to the right or left as many places as there are ciphers after the one: 3.127 X 100 = 312.7; 25.83 -^ 1000 = 0,02583. 50 ARITHMETIC Remark. The same rule applies where the dividend is a whole number, 453 ^ 100 = 4.53. THE FOUR FUNDAMENTAL OPERATIONS ON DECIMAL NUMBERS 178. To add decimals, proceed in the same manner as in the addition of whole numbers (25), placing the point in the result on the same vertical line with the points in the numbers. (This rule applies equally well where some of the numbers are whole numbers.) 37.425 8.72 436 0.54 68.034 550.719 179. To find the difference of two decimals, or of a whole number and a decimal, operate as with whole numbers (29), placing the decimal point in the result on the same vertical line with the points in the numbers. (When there are more decimals in one of the numbers than in the other, write or imagine to be written at the right of the number ciphers sufficient to make the number of decimal figures the same in each number.) 68.740 837 53.837 73.534 14.903 763.466 180. To multiply several decimal numbers or decimals and whole numbers together, disregard the decimal points and operate as with whole numbers (47), pointing off at the right of the result as many decimal figures as there are decimals in all the factors : 3.27 0.2 8.75 X 4 X 6.3 = 220.500 = 220.5. 4.005 0.3 1635 0.06 130800 13.09635 Remaeic. Since all decimals may be reduced to the form of decimal fractions (172), all rules and principles which apply to fractions apply also to decimals (163). Thus, for example, the value of a product of several decimals is not changed by chang- ing the order of its factors. 181. To divide a decimal by a whole number, write the figures FOUR FUNDAMENTAL OPERATIONS ON DECIMALS 51 the same as in the operation on whole numbers (64). Then divide the whole number part of the dividend by the divisor, which gives the whole part of the quotient; reduce the remainder to tenths, adding the tenths in the dividend by placing the tenths figure at the right of the remainder; divide this number by the divisor, which gives the first decimal (tenths) of the quotient; reduce this remainder to hundredths and proceed as before until a remainder zero is obtained or a figure expressing units of an indicated order. If the remainder is less than one-half the divisor, it is neglected; if it is greater, the last figure of the quo- tient is increased by 1; and if it is equal to half the divisor, the last figure may be increased by one or left as it is (176). This rule still holds where the dividend is a whole number and it is desired to have decimals in the quotient: 12 35.427 12 135 15 114 2.95225 62 30 27 60 30 60 11.25 If in the first example a quotient to the thousandths place had been desired, the operation would have been completed when 2.952 was obtained in the quotient. The last remainder 3 being smaller than half the divisor 12, 2.952 is the nearest true value to the thousandths place. 182. To divide a whole number or a decimal by a decimal, take the given divisor for a divisor, removing the decimal point; and the given dividend multiplied by 1 followed by as many ciphers as there are decimals in the divisor (177) for a dividend, and proceed as in the division of whole numbers (181). Thus to divide 3.3756 by 0.45, operate in the following manner: 45 337.56 22 5 060 15 7.501 Remark 1. Article (165) apphes to decimals. Remark 2. The proof of the operations with decimals is the same as with whole numbers (26, 30, 48, 65). In the proofs by the rule of 9 and 11 neglect the decimal point (97, 98, 99, 100, 101). 52 ARITHMETIC 183. Two numbers are reciprocals of each other when their product is equal to unity 1. Thus the reciprocal of the number 7is ^• THE REDUCTION OF FRACTIONS TO DECIMALS 184. A decimal number is periodic, when one or several deci- mal figures leappear in the same order indefinitely: such is the number 2.37474 . . . The number 74, formed by the figures 7 and 4, reappears in the same order indefinitely, and is the period of the decimal. 185. A decimal number is simple periodic or mixed periodic, according as it commences or not with the tenths figure. Thus the number 3.4545 ... is simple periodic, and 2.37474 ... is mixed periodic. 186. A constant quantity is the limit of a variable quantity, when the difference of the two quantities may become infinitely small without reaching zero. The unit 1 is the limit of the deci- mal 0.9999 . . . Because by taking an infinite number of 9's the difference between the resulting number and 1 will be infinitely small, but never can equal zero (38, 139). Remark. A variable quantity can have but one limit. 187. To reduce a fraction to decimals, is to put the fraction in the form of a decimal. 188. To reduce a fraction to decimals, divide its numerator by its denominator, operating as in the division of a decimal by a whole number (182): ¥ = 3.375. o 189. When the denominator of an irreducible fraction (144) contains only the factor 2 and 5, the reduction of the fraction to decimals will give an exact quotient, in which the number of decimal figures is equal to or greater than the exponents of the factors 2 and 5 in the denominator. 127 127 190. Any irreducible fraction of which the denominator con- tains one or several prime factors other than 2 and 5, cannot be REDUCTION OF FRACTIONS TO DECIMALS 53 reduced exactly to decimals, and the division of its numerator by its denominator gives a periodic quotient (184): 127 127 30 2x3x5 4.23333 127 .. . 191. Any fraction, -ofr ' is the limit (186) of the periodic quo- tient 4.2333, . . . , obtained in reducing the fraction to decimals (187). o 192. When the denominator of an irreducible fraction, -, does o not contain the factors, 2 nor 5, the reductions of the fraction to decimals gives a simple periodic quotient (185): I = 2.666 . . . o 193. When the denominator of an irreducible fraction con- tains one or several of the factors 2 and 5, together with other prime factors, the reduction of the fraction to decimals gives a mixed periodic quotient in which the number of non-periodic decimal figures is equal to or greater than the exponents of the factors 2 and 5 in the denominator. Thus the irreducible frac- tion — = gives two non-periodic decimals. 194. The number of figures contained in the period can not exceed the product of the prime factors of the denominator other than 2 and 5, less 1. Thus in the preceding example it cannot exceed 3 X 7 — 1 = 20. 195. The generant of any simple periodic decimal 0.2727 less 27 than unity and whose period is not 9, is that fraction — which has the period for a numerator and as many 9's as there are figures in the period for a denominator. Thus: 27 ^ = 0.2727 . . . (197, Remark). 196. Any simple periodic decimal 4.2727 . . . greater than unity and whose period is not 9, results from the reduction of a fraction to decimals. The same holds true for any mixed periodic deci- mal 4.342727 . . . whose period is not 9. To obtain the generant fraction of a simple periodic decimal 54 ARITHMETIC 4.2727 greater than unity, take the difference between the whole part followed by the period and the whole part for the numerator, and as many 9's as there are figures in the period for the denomi- nator. Thus: 427 - 4 _ 423 99 ~ 99 ■ To obtain the generant fraction of a mixed decimal 15.273434 . . .. for the numerator take the whole number followed by the non- periodic figures and the first period less the whole number fol- lowed by the non-periodic part, and for a denominator as many 9's as there are figures in the period followed by as many ciphers as there are figures in the non-periodic part of the decimal. Thus, 152,734 - 1527 ^ 151,207 9900 ~ 9900 ■ Remark. When the period is the figure 9, the decimal has no generant; the limit is obtained by suppressing the periods and increasing the last figure to the right by one. Thus: q 4Q _ 4 0.999 . . . = I = 1; 4.999 . . . = \ = 5; 4.34999 . . . ^ ^^^^00^' = 4.35. OPERATIONS OK COMBINED FRACTIONS AND DECIMALS, COMPLEX DECIMALS 197. To add complex decimals, reduce each decimal to the form of a fraction (172), and proceed as in the addition of frac- tions (152). Remark. When given decimals have a limited number of figures, and the fractions are exactly reducible to decimals (188), operate as in the addition of decimals. The same methods hold true for the subtraction, multiplica- tion and division of complex decimals. NUMERICAL APPROXIMATIONS. SHORT METHODS OF OPERATING 198. When a quantity is replaced by an approximate value, the difference between the exact value and the approximate value is called the absolute error, and the quotient obtained by dividing the absolute error by the exact value is called the rela- NUMERICAL APPROXIMATIONS 55 tive error. Thus, the distance between two points being 40 meters, if we suppose it to be 42 or 38 the absolute error is two meters, 42"- 40°= 2», 40°^- 38° = 2'», and the relative error 2 1 jq = 27)' The relative error is the error in each unit of the exact number. 199. When a whole number 314,159 is replaced by 314,100, or a decimal 3.14159 by 3.141, or 0.0314159 by 0.03141, that is, when figures at the right are replaced by ciphers if the number is whole or a decimal, the absolute error is respectively 59, 0.00059, ' 0.0000059, numbers formed by the suppressed figures, and the relative error is 59 0.00059 0.0000059 314,159 3.14159 0.0314159 From the foregoing examples it is seen that for numbers, which differ simply in position of the decimal point, the rela- tive error depends only upon the suppressed figures and not upon the position of the point; but the absolute error depends both upon the figures suppressed and the position of the point. The absolute error is respectively less than 100, 0.001, 0.000001, that is, than a unit of the order of the last figure retained, and the , ,. • , .1, 100 0.001 0.000001 relatzve error ^s less than ^^^-^ = 3^^^^ = ^;q^^^^, and evidently less than ^^^ = -^^ and less than j^ = 0.001, that is, than a decimal unit of an order, which is one less than the number of figures retained, not counting the ciphers at the left of the first significative figure. It follows that in order to obtain an approximate value of a whole or decimal number, which is less than the number, and has a relative error less than 0.1, 0.01, 0.001, 0.00001 . . . , retain at the left 2, 3, 4, 5 . . . figures com- mencing with the first significative figure. Thus the approxi- mate value of the numbers 314,159, 31415.9, 3.14159, 0.0314159 with a relative error less than 0.001 is respectively, 314,100, 31,410, 3.141, and 0.03141. Remark 1. When the first significative figure at the left of the number is greater than 1, the relative error as found by the preceding rule is less than half a decimal unit of an order, which is one less than the number of figures retained. In replacing 56 ARITHMETIC the number 0.0314159 by 0.03141, the relative error being less than ^^ , is evidently less than ^qqq or than a half a thousandth. Remark 2. When the first significative figure at the left of the part retained is 1, and the first figiu-e at the left of the part suppressed is less than 5 or is 5 not followed by significative figures, the relative error is less than one-half a decimal unit of an order, which is one less than the number of units retained. In replacing the number 1.14137 by 1.141, the absolute error, 0.00037, is less than one-half of a thousandth, and as the given number exceeds 1000 thousandths the relative error is less than a half a thousandth divided by 1000 thousandths or by 1, that is, than a half a thousandth. Remark 3. From the two preceding remarks, it follows that in the majority of cases, the relative error of a whole or decimal number, at the right of which one or several figures have been suppressed, is less than half of a decimal unit of an order, which is one less than the number of figures retained commencing with the first significative figure at the left. Remark 4. In retaining a certain number of figures, it is evi- dent that the relative error will be as much smaller as the abso- lute error is less; therefore, approximate values should be taken which give the smallest absolute error (177). 200. Addition. The absolute error of the sum of several num- bers, whose values are approximate, is equal to the sum of the absolute errors of the numbers. When the numbers have approximate values, some greater and some smaller than the number, add the plus and minus errors separately, and the difference of the two sums will be the absolute error of the sum, bearing the sign of the greater sura. The relative error of the sum of several numbers is equal to the absolute error divided by the sum. To find the sum of less than 11 numbers, with an absolute error of less than a unit of a certain order, add the numbers in- cluding the figures of the next lower order, neglecting all others at the right. Thus, to find the sum of the following numbers with an absolute error less than 0.1, 5.347 + 8.7537 + 0.0425 = 14.1432, take simply 6.34 + 8.75 + 0.04 = 14.13. Numerical approximations m The absolute error of each number is less than 0.01, and there being less than 11 numbers, the absolute error of the sum will be less than 0.01 X 10 = 0.1. If there are more than 10 numbers and less than 101, take one more still in making the addition. Given, the numbers 75.347, 8.7537, 0.6435, to find their sum with a relative error less than 0.01. 75.347 + 8.7537 + 0.6435 = 84.7432. First add: 70 + 8 + 0.6 = 78.6, the first figures at the left of each number; divide this sum by 100, formed by one followed by as many ciphers as indicated by the order desired (0.01), which gives 0.786; divide this sum by the number 3 of numbers to be added, and the first figure at the left of the quotient 0.262 expressing tenths it shows that it is sufficient to take each of the given numbers with one decimal only. If the first figure to the left had expressed hundreds, the given number would have to be taken with two decimals, and so on. Thus in the given example: 75.3 + 8.7 + 0.6 = 84.6. Since the relative error of the sum of the numbers is less than 0.01 when the absolute error is less than the hundredth part of the sum, as the sum of the given numbers is greater than 70 + 8 + 0.6 = 78.6, and, therefore, the hundredth part is greater than 78.6 X 0.01 = 0.786, in taking each of the given numbers with an absolute error less than -^ — = 0.262, and certainly less than 0.1 by taking a decimal figure, the absolute error is certainly less than 0.786 and evidently less than a hun- dredth of the sum. Therefore, the sum thus obtained satisfies the conditions given. 201. Subtraction. The greater number being the sum of the smaller and the difference, according as the absolute errors of the two numbers have or have not the same sign, the absolute error is equal to the difference or the sum of the absolute errors of the two numbers: 8.67 8.6 0.07 8.7 0.03 3.24 32 Om 3^ OM 5.43 5.4 0.03 5.5 0.07 58 ARITHMETIC It follows from what was said concerning addition (200), that to find the difference of two numbers with a relative error less than 0.01, for example, 75.3478 - 26.5363 = 48.8115, take the difference, 70 - 20 = 50, of the numbers formed by the first figures at the left of the num- bers; multiply this difference by the given error 0.01, which gives 0.5; take half 0.25 of the product, and since the first figure 2 at the left of this half expresses tens, one decimal is all that need be retained in the operation; which gives for a result, 75.3 - 26.5 = 48.8. 202. MuLtiflication. 1st. The absolute error of the -product of two factors, one of whose values has been approximated to a certain de- gree, is equal to the absolute error of the approximated factor multiplied by the other factor. The relative error of the product is equal to the relative error of the approximated factor (200). Calculate, correct to 0.01, the product, 3.1415926 ... X 271.8. The absolute error of the product being equal to the absolute error of the multiplicand multiplied by the multiplier 271.8, it suffices to take the multiplicand with an absolute error less than ^ifj-s and even better if less than '^„ = 0.00001 ; which gives 3.14159. This amounts to taking the approximated number with a number 2 + 3 = 5 of decimal figures equal to the. number of decimal figures 2 desired in the product plus the number 3 of whole number figures of the other factor. To find the same product with a relative error less than 0.01, take the approximated factor with a relative error less than O.Ol, that is, with 3 decimal figures (199), which gives 3.141 X 271.8. 2d. When the two factors of a product are replaced by approxi- mate values, one of which is less than the exact value, the absolute error of the produx:t is less than the sum of the products of each of the factors and the absolute error of the other factor, by the product of the absolute errors of the factors. NUMERICAL APPROXIMATIONS 69 The relative error of a product is less than the sum of the rela- tive errors of the two factors. Calculate, with an absolute error less than 0.01, the product, 314.15926 ... X 27.18281828 . . . The problem is satisfied when the absolute error of the product is less than 0.005 0.005 "28~'^'315"' Therefore, taking the first factor with four decimal figures and the second with five, we have an absolute error less than Instead of dividing the absolute error 0.01 into two parts, it may be divided in- any manner as long as the sum of the two parts is equal to 0.01. To find the preceding product with a relative error less than 0.01, it suffices if the relative error of each factor is less than 0.005, and still more if less than 0.001, which would be the case in taking four figures at the left of each of the factors, and we have 314.1 X 27.18. The relative error of the product of several approximated factors, whose approximate values are less than the exact values, is less than the sum of the relative errors of all the factors; the relative error of a power of an approximated number, whose approximate value is less than the exact, is less than the relative error of the number multipUed by the degree of the power. Calculate, with a relative error less than 0.01, the product, 314.15926 ... X 27.18281828 ... X 2.34246735 . . . It suffices if the sum of the relative errors of the factors is less than 0.01 ; consequently, taking each of the factors with a relative error less than -^ or less than 0.001, 314.1 X 27.18 X 2.342, one is sure of satisfying the conditions of the problem (199). For a product of approximate values, 314.15 X 27.18 X 2.34, 60 ARITHMETIC the relative errors of the factors being respectively less than 0.0001, 0.001, and 0.01, the sum of which is 0.0111, the relative error of the product is less than 0.1, and probably even less than 0.01. If the product 314.15926 ... X 27.18281828 ... X 2.34246735 . . . is desired with an absolute error less than 0.1, it suffices if the relative error is less than 0.1 divided by a number 320 X 30 X 3 = 28,800 greater than the product; this gives a relative error for each factor of less than g^^^^QQ = ^ ' and when each factor is taken with seven figures to the left, the relative error is less than ^^ = 0.000001, 314.1592 X 27.18281 X 2.342467. Remark. The relative error of the product of several approxi- mated factors, which approximations are greater than the exact values, is greater than the sum of the relative errors of all the factors; the relative error of a power of an approximated number, which approximation is greater than the exact value, is greater than the relative error of the number multiplied by the degree of the power. 203. Oughtred's short method of multiplication. To calculate a product of two whole numbers or decimals, 3.1415926 . . . X 32.18642 (see below), with an absolute error (198) less than a whole or decimal unit, 0.1 for example, write in an inverse order, the figures of the multiplier under the multiplicand in such a manner that the figure 2 of the simple units in the multipHer corresponds to the figure 1 in the multiplicand which expresses units (0.001) one himdred times smaller than those of the order desired, 0.1 ; then commencing at the right multiply successively the multiplicand by each figure of the multiplier, neglecting the figures of the multiplicand which are at the right of the figure which serves as multiplier (for the figure 3, for example, neglect 926 . . .); this leads to the fact that no figures in the multiplier at the left of the last figure 3 in the multiplicand are used as multi- pliers. Write the partial products under the multiplier, placing the first right-hand figures in the same vertical column; in adding, NUMERICAL APPROXIMATIONS 61 consider them to express units of an order one hundred times smaller than that desired, 0.1; in this example two figures, 07, are suppressed at the right of the result, and the last figure on the left is increased by one unit. Thus the product is 101.2. 3.1 4 159 26 . . . . . . 46 8 1 23 9 42 45 6 2 82 3 14 2 48 18 10 1.1 07 10 1.2 Remark. The preceding rule is given for a general case. The case where the sum 3 + 2 + 1+8 + 6 = 20 of the figures employed in the multiplier, plus the first figure, 4, which was neglected, gives 24, which is greater than 10 and less than 101. In the case where this sum is less than 10, and in that one where it is between lOCI and 1001, operate in the same manner as above, but writing the imits figure of the multiplier respectively under the figure of the multiplicand which expresses units ten or one thousand times smaller than those of the order desired in the result. 204. Division. When the dividend is replaced by an approxi- mate value, which is greater or less than the exact value, the abso- lute error of the quotient is equal to the absolute error of the dividend divided by the divisor, and its relative error is equal to that of the dividend. Thus replacing 3.14159 , 3.14 the absolute and the relative error of the quotient are respec- tively, 0.00159 0.00159 —38—^""^ 314159" When the divisor is replaced by an approximate value, which is larger or smaller than the exact value, the absolute error of the quotient is equal to the quotient multiphed by the absolute error of the divisor divided by the approximate value, and the relative 62 ARITHMETIC error is equal to the absolute error of the divisor divided by its approximate value. Thus replacing 38 , _38_ 3.14159 ^ 3.14' the absolute and relative error are respectively, 38 0.00159 , 0.00159 X - n-, A ^^^ 3.14159 -^ 3.14 3.14 From the form of the relative error, it follows that according as the approximation is less or greater than the exact value, the relative error of the quotient is greater or less than that of the divisor; and from the form of the absolute error, it follows that when the whole part of the divisor is greater than the quotient multiplied by a number a, if the divisor is replaced by its whole part, the absolute error of the quotient is less than - • Thus, in replacing -r^ by ^. as 6 > ^ X 5, the absolute error will be less than -=' 5 The dividend being equal to the product of the divisor and the quotient, the relative error of the quotient may he considered as being equal to the difference between the relative errors of the divi- dend and divisor (2d, 202), and consequently, at least, less than one of them. Therefore, to obtain a quotient with an error less than 0.1, 0.01, 0.001 . . ., the relative errors of the two numbers must be taken less than these same quantities, that is, respec- tively the 2, 3, 4 . . ., first figures at the left of the dividend and the divisor. Thus, to find the quotient of 3.1415926 . . . divided by 32.1864 . . ., with a relative error less than 0.001, divide 3.141 by 32.18. 205. Short method of division. To find the quotient of a whole or decimal number divided by a whole or decimal number, with an absolute error (198) less than a given whole or decimal xmit, 0.001 for instance (see example below), commence by determin- ing the number of figures 1 in the whole part of the quotient (64), and then, the total number n = l + 3 = 4of figures in the required quotient. If the whole part were 0, n would equal 3; if the figure in tenths place were 0, n would equal 2 ; and if the NUMERICAL APPROXIMATIONS 63 figure in hundredths place were 0, n would equal 1 (the highest order of units in the quotient is easily determined by inspection, and thxis the value of n). Then, removing the decimal points, take, at the left of the divisor, just enough figures so that the number 32 which results is at least equal to n = 4; at the right of 32 write the.n = 4 following figures of the divisor, and the resulting number 321,864 is the first partial divisor. To form the first partial dividend, separate at the left of the dividend just enough figures so that the decimal number 3,141,592.65 . . . which results is at least equal to the decimal number 321,864.18 . . ., formed by placing in the given divisor a point at the right of the first partial divisor, and the part 3,141,592 separated at the left of the dividend is the first partial dividend. The quotient 9 in the division of the first partial dividend by the first partial divisor, is the first left-hand figure in the required quotient. Take the remainder 244,816 obtained for a second partial dividend, and neg- lecting the first figure 4 at the right of the first partial divisor, the number 32,186 thus formed is the second partial divisor; divid- ing the second partial dividend by the second partial divisor, the second figure 7 of the required quotient is obtained. Taking the new remainder, 19,514, for the third partial dividend, and the number 3218, obtained by suppressing the first right-hand figure in the second partial divisor, for the third partial divisor, and continuing thus until the n = 4 figures of the quotient have been obtained, the required quotient is correct to the given place (0.001), when the decimal point is so placed that the first figure on the right expresses units of the given order. 3.141 592 65 . 0.321 864 18 . ■ . 3 141 592 244 816 19 514 206 321i8i6'4 9.760 It can happen that a partial dividend contains a corresponding partial divisor 10 times; then take 10 for a partial quotient, that is, write in the quotient and increase the figure immediately preceding by one unit; continuing the process ciphers are obtained for all the following figxires. The quotient obtained in this case is always larger than the exact value by less than a unit of the given order. An example of this case is: 26.389292 . . . divided by 3.1415926 correct to the third place (0.001), 64 ARITHMETIC 2 638 929 125 657 31412 2 31441519 83 10 8.400 206. The relative error of the power of an approximate number, which approximation is greater than the exact value, being greater than the product e X n of the relative error e of the number and the degree n of the power (202, Remark), it follows that the relative error of the root of an approximate number, which approximation is greater than the exact value, is less than the rela- tive error e' of the number divided by the index n of the root. Example. Extract -^65.36874 . . . If four figures at the left are taken, increasing the last by one unit, we have 65.37, which gives a relative error, e'<^-, ^6000' and for the root, ^<^<6J^ t^' oo' nA' ^'Hd varies so little in the different countries, that it may be considered as universal. One carat is equal to 205.5000 mg. or 4 grains. The approximate value of rough diamonds in dollars is obtained by multiplying the price of one carat by the square of their weight in carats. Thus a rough diamond of three carats is worth 40 X 3 X 3 = $360.00, one carat being worth $40.00. Formerly the value of cut diamonds was also calculated from the price of a one-carat stone, but, owing to an abnormal demand for small stones and a supply of very large ones, the large diamonds are most often cut up into smaller sizes. This process entails loss, so that a one-carat diamond more often costs more by weight than either a one and one-half or a two-carat diamond. 5th. For money: Eagle, dollar, dime, cent, mill, which in dollars is: $1 $0.1 $0.01 $0,001 The coins of the United States are: Gold: double eagle, eagle, half-eagle, quarter-eagle, three-dollar and one-dollar piece. Silver: dollar, half-dollar, quarter-dollar, and ten-cent piece. Nickel: five-cent piece. Bronze: one-cent piece. 220. Real or effective measwes are those which exist in the form of instruments or objects authorized by law. Effective measures, marked with the official stamp, are estab- lished with certain forms and dimensions which are best suited to facilitate their use. I. The effective measures of length, which are most commonly used, are: 1st. The chain, which is ordinarily one decameter (10 m.) and sometimes a double decameter (20 m.) long. i'HE METRIC SYSTEM 69 2d. The tape, which is rolled on an axle and protected by a housing made of leather or paper, is divided into meters which are subdivided into decimeters and centimeters, and the first decimeter is even divided into millimeters. Dressmakers and others use tapes 1 m., 1.5 m., 2 m. long. Civil engineers, etc., use tapes 5 m., 10 m., and 20 m. long. 3d. The double meter is a rule of wood or metal, sometimes jointed so as to be carried in the pocket, and generally divided into decimeters and centimeters. 4th. The meter, a straight rule, sometimes jointed in 2, 5, or 10 parts. It is divided into centimeters and ordinarily into milli- meters on the first decimeter. It is made of wood, whalebone, bone, ivory, and metal. 5th. The half-meter, a straight rule, of one piece or jointed in the middle. 6th. The double decimeter and the decimeter, made of boxwood, bone, or ivory. They are divided into millimeters and some- times into half -millimeters. 7th. The scale is made of steel and generally J or 1 decimeter long, and divided into millimeters and half-millimeters. II. There are no effective measures of surfaces; their measure is obtained by the use of geometry (Part III). III. The effective measures of volumes. In measuring the solids it is necessary to have recourse to geometry (Part III); but for the liquids and the grains there are effective measures. For the liquids there are 13 effective measures, of which five are called large measures and eight small measures. The 5 large measures are cylindrical vessels, the depth of which is equal to the interior diameter. According to their use they are made of copper, galvanized iron, and tin plate, Table of the Five Large Liquid Measures Name. Capacity IN Liters. Depth AND Diameter IX MM. Hectoliter 100 50 20 10 5 50.3.1 Half-hectoliter .... .... 399.3 Double-decaliter Decaliter .... . . . . 294.2 2.33.5 Half-decaliter 185.3 70 ARITHMETIC The 8 small measures for liquids other than milk and oil are made of an alloy containing 95 parts tin and 5 parts lead; the tin alone would be too breakable, and lead alone would be poi- sonous. They are hollow cylinders whose depth is twice their interior diameter. For milk and oil these 8 measures are made of tin plate, and their depth is equal to their interior diameter. Table of Eight Small Liquid Measures Name. Capacity IN Litems. Depth, KM. Diam- eters, MM. Milk and Oil, Depth AND Diam- eter, MM. Double-liter Liter Half-liter . Double-deciliter Deciliter Half-deciliter Double-centiliter .... Centiliter 2 1 0.5 0.2 0.1 0.05 0.02 0.01 216.8 172.1 136.6 100.6 79.9 63.4 46.7 37.1 108.4 86.0 68.3 50.3 39.9 31.7 23.4 18.5 136.6 108.4 86.0 63.4 50.3 39.9 29.4 23.4 For the grains, etc., there are 11 effective measures, which according to their use are constructed of wood, copper, or iron. They are ordinarily made of oak staves secured by metal fasten- ings. All are cylindrical in form and have an internal diameter equal to the depth. Table of Dry Measure Di- Di- Capa- ameters Capa- ameters Name. city, ANB Name. city, AND Liters. Depths, MM. Liters. Depths, MM. Hectoliter . . . 100 503.1 Liter 1 108.4 Half-hectoliter 50 399.3 Half-liter . . . 0.5 86.0 Double-decaliter . 20 294.2 Double-deciliter . 0.2 63.4 Decaliter . . . 10 233.5 Deciliter . . . 0.1 50.3 Half-decaliter . . 5 185.3 Half-deciliter . . 0.05 39.9 Double-liter . . 2 136.6 Prices of grains are usually based upon the hectoliter or metric quintal. In measuring grains, seeds, and small fruits, the meas- ure is level full or stricken. The mean weight of a hectoliter of wheat is 75 kg. ; of barley, 64 kg. ; of oats, 47 kg. Coal is measured in half-hectoliters, hectoliters, and tons. Fire-wood is measured in half-decasteres, double steres, and steres, which are respectively, 5, 2, and 1 cubic meters. THE METRIC SYSTEM 71 Each of these measures is constructed of wood, in the following manner. Upon a rectangular base two upright ends are fastened and braced. The distance between the uprights is respectively, 1, 2, or 3 meters for the stere, double stere, and half-decastere; the height varies with the length of the pieces of wood. 4th. Effective measures of weight. The 24 official weights which are used in commerce and industry are divided according to the following table into 5 large weights, 9 medium weights, and 10 small weights. Labge Weights. Medium Weights. Small Weights. kilog. kilog. gramme. 50 1 kilogr. 1 =1 1 gramme 1 =1 20 5 heotogr. 0.5 = ^ 5 decigr. 0.5 = , 10 2 hectogr. 0.2 = i 2 decigr. 0.2 = ^ 5 1 hectogr. 0.1 = T-V 1 decigr. 0.1 = „ 2 5 deoagr. 0.05 = ^V 5 centigr. 0.05 = jV 2 decagr. 0.02 = ^V 2 centigr. 0.02 = ^^ 1 decagr. 0.01 =^j 1 centigr. 0.01 = ^^ 5 grammes 0.005 = ^^ 5 milligr. 0.005= ^ 2 grammes 0.002 = ^i^ 2 milligr. 1 milligr. 0.002= ji^ 0.001 = „'t^ Ten of these weights, from 50 kg. to 5 decagrammes or a half- hectogramme, are made of cast iron. The 50 and 20 kg. weights have the form of a frustum of a rectangular pyramid with rounded edges, the 8 others have the form of a frustum of a hexagonal pyramid. All of these weights are supplied with a ring on top which lies below the surface when not in use, and thus does not interfere with the piling of the weights one upon the other. Fourteen weights, from 20 kg. to 1 gramme, are made of brass. They are cylindrical in form and have a button on top to take hold of. The height of the cylinder is equal to its diameter, and the height of the button is half of that. The diameter of the double gramme and gramme is often greater than the height. Weights are also made in the form of conical goblets which fit one over the other, and are inclosed in a box of the same form. The box itself represents a legal weight. The nine weights under the half-gramme are made of little, thin square or octagonal pieces of brass, aluminum, silver, or plati- num. One corner is slightly raised so as to facilitate handling with pincers. They are mostly employed in chemical analysis and experimental physics. 72 ARITHMETIC 221. Units of time. The different units of time are not of the decimal order, and do not belong to the metric system. The solar day is the time included between two consecutive crossings of a certain meridian by the sun. The solar year is the time required by the earth to make one complete revolution around the sun, and is equal to a number of solar days which lies between 365 and 366. The solar year is constant, but the solar days are not, for the two following rea- sons: f,rst, the non-imiform velocity of the earth in its orbit, by which the apparent diurnal movement of the sun is more rapid in winter than in summer; second, the obliquity of the ecliptic, which makes the apparent diurnal movement of the sun in right ascension, that is, in the plane of the terrestrial equator, slower at the equinoxes than at the solstices. The principal unit of time is the mean day, or the mean value of the 365 solar days. The mean day is divided into 24 equal parts called hours, the hour into 60 equal parts called minutes, the minute into 60 seconds, the second into fifths, tenths, or hundredths. In writing units which express time, write the abbreviations for the different units after each number. The minutes and seconds are sometimes denoted by ' or ". Thus 3 da. 8 hr. 35 min. 45 sec. or 3 da. 8 hrs. 35' 45" represents 3 days 8 hours 35 min- utes 45 seconds. The sidereal day is the interval of time between two consecu- tive transits of a certain meridian by a star. Its duration is constant, and equal to 23 hrs. 56' 4" mean time. Remark. The solar year contains approximately 365.24225 mean days. The civil year is the legal year; the solar year is increased or decreased enough so that it contains exactly 365 or 366 days. One hundred consecutive years form a century. The civil year is divided into twelve parts called months, the names of which are January, February, March, April, May, June, July, August, September, October, November, December. The number of days in each month is easily remembered by memorizing the following: "Thirty days has September, April, June, and November; All the rest have thirty-one, Except February, which has but twenty-eight in fine, Until leap year gives it twenty-nine." THE METRIC SYSTEM 73 The solar year is 0.24225 mean day longer than the civil year, and if the civil always had 365 days, at the end of 4 years it would be 0.969 day ahead of the solar year; it is to compen- sate for this that one day is added every fourth year, such a year being called leap year. From this correction it follows that every four years the civil year is placed 1 — .969 = 0.031 days behind the solar year, and at the end of a century is 0.031 X 25 = 0.775 day behind; for this reason the last year of each cen- tury is not leap year. From this it again follows that at the end of each century the civil day is 1 — 0.775 = 0.225 day ahead of the solar year, and every fourth century is 0.225 X 4 = 0.9 = 1 — 0.1 ahead; thus it is that we have a leap year every fourth century. After this third correction the civil year is 0.1 day behind the solar year every 400 years, which is 1 day at the end of 4000 years ; thus by suppressing a leap year every 4000 years, the civil year terminates at the same instant as the solar year if we accept 365.24225 as the exact value, which in reality is only an approximation. These four successive corrections may be represented by put- ting the ratio of the solar year to the mean day in the form o^r i 1 1 1 366 + -r - :rKH + 4 100 ' 400 4000 The Julian calendar was established by Julius Csesar forty-six years before Christ, and was in use in the Roman world until 1582, at which time the pope, Gregory, instituted the Gregorian calendar, which is in use to-day in nearly every country. To-day the Julian dates are 12 days behind the Gregorian dates; and when writing to countries which still employ the Julian calendar (Russia and Greece), it is customary to write 1 9 Y5 Jan., ^ Feb., which gives the dates according to both calen- dars. 222. The circumference of a circle is divided into 360 equal parts called degrees; the degree into 60 equal parts called min- utes; the minute into 60 equal parts called seconds. The quad- rant of a circumference is 90 degrees. Degrees, minutes, and seconds are units used to measure angles and arcs (see Geometry). In writing degrees, minutes, and seconds, the signs °, ', and ", 74 ARITHMETIC respectively, are placed above and a little to the right of the number; thus 3° 17' 28" is read 3 degrees 17 minutes 28 seconds. Often the circumference of a circle is divided into 400 equal parts called grades, and each grade into 100 equal parts, which parts are again divided by 100. The quadrant equals 100 grades. These measures conform with the law of decimals. Thus 74.3705 g. reads 74 grades 37 hundredths of a grade 5 hundredths of a hundredth of a grade. 223. A complex quantity is a quantity composed of several parts, compared with different units of its kind. Such are the quantities 7 da. 16 hr. 34 m. and 42° 21' 15". PROBLEMS RELATING TO MEASURES 224. In general, concrete decimals may be operated upon in the same manner as abstract decimals (178 to 182). 225. Application to the payment of workmen. A workman earns $4.75 per day; in a month of 26 working days he will earn $4.75 X 26 = $123.50. The following table gives the sum earned by a workman, working 10 hours a day for a certain number of days at a certain wage. To find what is due a man for a certain number of hours, 7, for example, at $4.75 per day, take as many days as there are hours and divide by ten, which in this case (referring to the table) gives $3.33. Therefore in 26 days and 7 hours the workman will earn $123.50 + 3.33 = $126.83. THE METRIC SYSTEM Wage Table 75 Days. so.so $0.60 $0.70 $0.75 $0.90 $1.00 $1.25 $1.50 $1.76 $2.00 1 0.50 0.60 0.70 0.75 0.90 1.00 1.25 1.50 1.75 2.00 2 1.00 1.20 1.40 1.50 1.80 2.00 2.50 3.00 3.60 4.00 3 1.50 1.80 2.10 2.26 2.70 3.00 3.75 4.50 5.25 6.00 4 2.00 2.40 2.80 3.00 3.60 4.00 5.00 6.00 7.00 8.00 6 2.50 3.00 3.50 3.75 4.50 6.00 6.26 7.50 8.75 10.00 6 3.00 3.60 4.20 4.50 5.40 6.00 7.50 9.00 10.50 12.00 7 3.50 4.20 4.90 5.25 6.30 7.00 8.75 10.50 12.25 14.00 8 4.00 4.80 5.60 6.00 7.20 8.00 10.00 12.00 14.00 16.00 9 4.50 5.40 6.30 6.75 8.10 9.00 11.25 13.50 15.75 18.00 10 5.00 6.00 7.00 7.50 9.00 10.00 12.50 15.00 17.50 20.00 11 5.50 6.60 7.70 8.26 9.90 11.00 13.75 16.50 19.25 22.00 12 6.00 7.20 8.40 9.00 10.80 12.00 15.00 18.00 21.00 24.00 13 6.50 7.80 9.10 9.75 11.70 13.00 16.25 19.50 22.75 26.00 14 7.00 8.40 9.80 10.50 12.60 14.00 17.50 21.00 24.60 28.00 15 7.50 9.00 10.50 11.25 13.50 16.00 18.75 22.50 26.26 30.00 16 8.00 9.60 11.20 12.00 14.40 16.00 20.00 24.00 28,00 32.00 17 8.50 10.20 11.90 12.75 15.30 17.00 21.25 25,50 29,75 34.00 18 9.00 10.80 12.60 13.50 16.20 18.00 22.50 27,00 31,50 36.00 19 9.60 11.40 13.30 14.26 17.10 19.00 23.75 28,50 33.25 38.00 20 10.00 12.00 14.00 15.00 18.00 20.00 26.00 30.00 35.00 40.00 21 10.50 12.60 14.70 16.75 18.90 21.00 26.25 31.50 36.75 42.00 22 11.00 13.20 15.40 16.50 19.80 22.00 27.50 33.00 38.50 44.00 23 11.50 13.80 16.10 17.26 20.70 23.00 28.75 34.50 40.25 46.00 24 12.00 14.40 16.80 18.00 21.60 24.00 30.00 36,00 42.00 48,00 26 12.50 15.00 17.50 18.75 22.50 25.00 31.26 37.50 43.75 60.00 26 13.00 15.60 1S.20 19.50 23.40 26.00 32.50 39.00 45.60 62.00 27 13.50 16.20 18.90 20.25 24.30 27.00 33.76 40.50 47.26 64.00 28 14.00 16.80 19.60 21.00 25.20 28.00 35.00 42.00 49.00 66.00 29 14.50 17.40 20.30 21.75 26.10 29.00 36.25 43.50 50.76 68.00 30 15.00 18.00 21.00 22.50 27.00 30.00 37.60 45.00 52.60 60.00 Days $2.25 $2.50 $2.75 $3.00 $3.25 $3.50 $3.75 $4.00 $4.25 $4.50 1 2.25 2.50 2.75 3.00 3.25 3.50 3.76 4.00 4.26 4.50 2 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.60 9.00 3 6.75 7.50 8.25 9.00 9.75 10.50 11.25 12.00 12.75 13.50 4 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 6 11.25 12.50 13.75 15.00 16.25 17.50 18.76 20.00 21.26 22.50 6 13.50 15.00 16.50 18.00 19.50 21.00 22.50 24.00 25,60 27.00 7 15.75 17.50 19.25 21.00 22,75 24.50 26.25 28.00 29.75 31.50 8 18.00 20.00 22.00 24.00 26.00 28.00 30.00 32.00 34.00 36.00 9 20.25 22.50 24.75 27.00 29.26 31.50 33.76 36.00 38,25 40.50 10 22.50 25.00 27.60 30.00 32.50 35.00 37.50 40.00 42.50 45.00 11 24.75 27.50 30.25 33.00 35.75 38.50 41.25 44.00 46.75 49.60 12 27.00 30.00 33.00 36.00 39.00 42.00 45.00 48.00 51.00 54.00 13 29.25 32.50 35.76 39.00 42.26 45.50 48.75 52.00 66.25 58.50 14 31.50 35.00 38.60 42.00 45.50 49.00 52.50 66.00 69.50 63,00 IS 33.75 37.50 41.25 45.00 48.75 62.50 56.25 60.00 63.75 67.50 16 36.00 40.00 44.00 48.00 52.00 56.00 60.00 64.00 68.00 72.00 17 38.25 42.50 46.75 61.00 55.25 59.50 63.75 68.00 72.25 76.50 18 40.50 45.00 49.50 54.00 58.50 63.00 67.60 72,00 76.50 81.00 19 42.75 47.50 52.25 57.00 61.75 66.50 71.25 76.00 80,75 85.50 20 45.00 50.00 56.00 60.00 65.00 70.00 75.00 80.00 85.00 90.00 21 47.25 52.50 57.75 63.00 68.25 73.50 78.75 84.00 89,25 94.50 22 49.50 55.00 60.50 66.00 71.50 77.00 82,50 88.00 93,50 99.00 23 51.75 57.50 63.25 69.00 74.76 80.50 86.25 92.00 97.75 103.60 24 54.00 60.00 66.00 72.00 78.00 84.00 90,00 96.00 102.00 108.00 25 56.25 62.50 68.75 75.00 81.25 87.60 93.75 100.00 106.26 112.50 26 58.50 65.00 71.50 78.00 84.50 91.00 97.50 104.00 110.50 117.00 27 60.75 67.50 74.25 81.00 87.76 94.50 101.25 108.00 114.75 121.50 28 63.00 70.00 77.00 84.00 91.00 98.00 105,00 112.00 119.00 126.00 29 65.25 72.50 79.75 87.00 94.25 101. .50 108.75 116.00 123.25 130.50 30 67.50 75.00 82.50 90.00 97.50 105.00 112.50 120.00 127.50 135.00 76 ARITHMETIC Wage Table — {Continued) Days. S4.7S S5.00 J5.25 $6.60 $5.75 $6.00 $6.25 $6.50 $6.76 $7.00 1 4.75 5.00 5.50 5.50 6.76 6.00 6.26 6.60 6.76 7.00 2 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.60 14.00 3 14.25 15.00 15.75 16.50 17.25 18.00 18.76 19.60 20.25 21.00 4 19.00 20.00 21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00 S 23.76 25.00 26.25 27.50 28.75 30.00 31.26 32.60 33.75 36.00 6 28.50 30.00 31.50 33.00 34.50 36.00 37.50 39.00 40.60 42.00 7 33.25 35.00 36.75 38.50 40.25 <2.00 47.75 45.50 47.25 49.00 8 38.00 40.00 42.00 44.00 46.00 48.00 50.00 52.00 54.00 66.00 9 42.75 45.00 47.25 49.50 51.76 54.00 56.25 58.50 60.75 63.00 10 47.50 50.00 52.50 55.00 57.50 60.00 62.50 65.00 67.50 70.00 11 52.25 55.00 57.75 60.50 63.26 66.00 68.76 71.50 74.25 77.00 12 57.00 60.00 63.00 66.00 69 00 72.00 76.00 78.00 81.00 84.00 13 61.75 65.00 68.25 71.50 74.75 78.00 81.25 84.50 87.75 91.00 14 66.50 70.00 73.50 77.00 80.50 84.00 87.60 91.00 94.50 98.00 15 71.25 75.00 78.75 82.50 86.25 90.00 93.75 97.50 101.25 105.00 16 76.00 80.00 84.00 88.00 92.00 96.00 100.00 104.00 108.00 112.00 17 80.75 85.00 89.25 93.50 97.76 102.00 106.25 110.50 114.75 119.00 18 85.50 90.00 94.50 99.00 103.60 108.00 112.50 117.00 121.50 126.00 19 90.25 95.00 99.75 104.50 109.25 114.00 118.75 123.50 128.25 133.00 20 95.00 100.00 105.00 110.00 115.00 120.00 125.00 130.00 136.00 140.00 21 99.75 105.00 110.25 115.50 120.75 126.00 131.25 136.50 141.76 147.00 22 104.50 110.00 115.50 121.00 126.50 132.00 137.50 143.00 148.60 154.00 23 109.25 115.00 120.75 126.50 132.25 138.00 143.75 149.60 155.26 161.00 24 114.00 120.00 126.00 132.00 138.00 144.00 150.00 166.00 162.00 168.00 25 118.75 125.00 131.25 137.60 143.76 160.00 156.25 162.60 168.76 175.00 26 123.50 130.00 136.50 143.00 149.50 156.00 162.50 169.00 175.50 182.00 27 128.25 135.00 141.75 148.50 155.25 162.00 168.75 175.60 182.25 189.00 28 133.00 140.00 147.00 154.00 161.00 168.00 175.00 182.00 189.00 196.00 29 137.75 145.00 152.25 169.50 166.75 174.00 181.26 188.50 195.75 203.00 30 142.50 150.00 157.50 166.00 172.50 180.00 187.60 195.00 202.50 210.00 Days. $7.25 It7.60 $7.75 $8.00 $8.25 $8.60 $8.76 $9.00 $9.50 $10.00 1 7.25 7.50 7.75 8.00 8.25 8.50 8.76 9.00 9.50 10.00 2 14.50 15.00 15.50 16.00 16.60 17.00 17.60 18.00 19.00 20.00 3 21.75 22.50 23.25 24.00 24.75 25.50 26.26 27.00 28.50 30.00 4 29.00 30.00 31.00 32.00 33.00 34.00 35.00 36.00 38.00 40.00 6 36.25 37.50 38.75 40.00 41.25 42.60 43.75 45.00 47.50 50.00 6 43.50 45.00 46.50 48.00 49.50 51.00 52.50 54.00 57.00 60.00 7 50.75 52.50 54.25 56.00 57.76 69.50 61.25 63.00 66.50 70.00 8 58.00 60.00 62.00 64.00 66.00 68.00 70.00 72.00 76.00 80.00 9 65.25 67.50 69.75 72.00 74.25 76.60 78.75 81.00 85.50 90.00 10 72.50 75.00 77.50 80.00 82.60 86.00 87.50 90.00 95.00 100.00 11 79.75 82.50 85.25 88.00 90.75 93.60 96.25 99.00 104.50 110.00 12 87.00 90.00 93.00 96.00 99.00 102.00 105.00 108.00 114.00 120.00 13 94.25 97.50 100.75 104.00 107.25 110.50 113.75 117.00 123.50 130.00 14 101.50 105.00 108.50 112.00 115.50 119.00 122.50 126.00 133.00 140.00 16 108.75 112.50 116.25 120.00 123.75 127.50 131.25 135.00 142.50 150.00 16 116.00 120.00 124.00 128.00 132.00 136.00 140.00 144.00 152.00 160.00 17 123.25 127.50 131.75 136.00 140.25 144.50 148.75 153.00 161.50 170.00 18 130.50 135.00 139.50 144.00 148.60 163.00 157.50 162.00 171.00 180.00 19 137.75 142.50 147.25 162.00 156.76 161.50 166.25 171.00 180.50 190.00 20 145.00 150.00 155.00 160.00 166.00 170.00 175.00 180.00 190.00 200.00 21 152.25 157.50 162.75 168.00 173.26 178.50 183.75 189.00 199.50 210.00 22 159.50 165.00 170.50 176.00 181.60 187.00 192.50 198.00 209.00 220.00 23 166.75 172.50 178.25 184.00 189.75 195.50 201.25 207.00 218.50 230.00 24 174.00 180.00 186.00 192.00 198.00 204.00 210.00 216.00 228.00 240.00 25 181.25 187.50 193.75 200.00 206.25 212.50 218.76 225.00 237.50 260.00 26 188.50 195.00 201.60 208.00 214.50 221.00 227.50 234.00 247.00 260.00 27 195.75 202.50 209.25 216.00 222.76 229.50 236.25 243.00 256.50 270.00 28 203.00 210.00 217.00 224.00 231.00 238.00 245.00 252.00 266.00 280.00 29 210.25 217.50 224.75 232.00 239.25 246.60 253.75 261.00 275.50 290.00 30 217.50 225.00 232.60 240.00 247.50 255.00 262.50 270.00 285.00 300.00 THE METRIC SYSTEM 77 226. To compare a quantity expressed by a concrete decimal with one of the units of its kind, remove the decimal point to the right of the figure which represents the imits. Thus, to express the quantity 365.867 m. in centimeters, advance the decimal point two places towards the right, giving 36586.7 cm., that is, a number one hundred times greater and which expresses units a hundred times smaller than the given number. 227. To reduce a complex quantity 5 years, 7 months, and 8 days to one of its units. Let it be required to reduce the given quantity to years. The year has 12 months, 5 yrs. + 7 mo. = 5 X 12 + 7 = 67 mo., and as a month has 30 days, 67 mo. + 8 da. = 67 X 30 + 8 = 2018 da. But 1 yr. = 12 X 30 = 360 da., therefore, or) -to 6 yrs. + 7 mo. + 8 da. = ^^yrs. = 6.60555 yrs. . . . (181). Since the month contains 30 days, 2018 5 yrs. + 7 mo. + 8 da. = —^^ mo. = 67.2666 mo. . . . 228. The inverse of the preceding problem. Reduce 2018 days to years, months, days, etc. Divide 2018 by 360: 2018 da. 218 12 360 5 yrs. 7 mo. 8 da. 436 218 2616 96 30 2880 00 The division of 2018 by 360 gives 5 for the quotient and 218 for the remainder, thus: 2018 ^ 218 , , 218 X 12, ^^yrs. = 5yrs.+3g5yrs. = 5yrs.+ ^^^ 2616 „ , „ 96 3g^mo. = 5yrs. + 7mo.3g^ mo. = 5 yrs. + ^^g^mo. = 5 yrs. + 7 mc^r^mo. = 5 yrs. + 7 mo. + ^^^^da. = 5yrs. + 7mo. + 8da. 229. The same problem, the number of years 5.60555 ... yrs. being expressed in decimals. ^^^ ^„„ 660,555 Putting the decimal in the form of a decimal fraction j^qq^qOO' 78 ARITHMETIC proceed as before. In this case the division by 1 followed by ciphers renders the operation more simple, as we have but one series of multiplications (177). Thus: 5.60555 yrs. = 5 yrs. + 0.60555 yrs. = 5 yrs. + 0.60555 X 12 mo. = 5 yrs. + 7.2666 mo. = 5 yrs. + 7 mo. + 0.2666 X 30 da. = 5 yrs. 7 mo. 8 da. 230. The four operations on the complex numbers are performed by following the same methods as with whole or decimal numbers, remembering that the different units are no longer equal to 10 of the units of next lower order, when reducing the partial results to units of the next higher order (addition and multiplication), or next lower order (division), and when a number has to be in- creased in order to make a subtraction possible. It may be noted also that the numbers of each order of units may have more than one figure. Additiok SUBTEACTIOir 7 hrs. 5 min. 64.8 sec. 9 hrs. 25 min. 14.8 sec. 2 10 40.4 3 31 30.4 5 18 47.6 6 hrs. 53 min. 44.4 sec. 14 hrs. 35 min. 22.8 sec. Multiplication 8 da. 3 hrs. 19 min. 16.3 sec. 7 37 da. 10 hrs. 14 min. 54.1 sec. Division (231) 7 hrs. 18 min. 13.5 sec. 4 3 60 1 hr. 49 min. 33.375 sec. 180 18 198 38 2 60 120 13.5 133.5 13 15 30 20 BRITISH SYSTEM OF WEIGHTS AND MEASURES 79 // the multiplier is a fraction, multiply the complex quantity by the numerator and divide the product by the denominator. If the divisor is a fraction, multiply the complex dividend by the fraction inverted. When a problem involves the multiplica- tion or division of one complex number by another, reduce one of them to a common unit (227), and proceed as when dividing a complex number by a fraction. An example in division. A movement takes 5 hrs. 10 m. 3 s. to turn 2° 18' 15"; how long will it take for it to turn 1°, its velocity being constant? From the question it is seen that 2° 18' 15" should be reduced to degrees, which gives ' ' = tha ii dividing by the common factors 3 and 5. The time is then ,„ . „ , 240 1240 hrs. 12 min. (5 hrs. 10 mm. 3 sec.) x ^fo 653 653 = 2 hrs. 14 min. 33.63 sec. BRITISH SYSTEM OF WEIGHTS AND MEASURES 231. Although the British system of measures is in general use in this country, the values of the individual units, in some cases, differ from those used in Great Britain. Therefore, in the tables that follow the values, assigned to the units apply to those used in the United States unless otherwise stated. MEASURES OF LENGTH 232. Linear measure has but one dimension, and is used for comparing lines and distances. Table of Common Linear Measure 12 inches (in. ") = 1 foot (ft. '). 3 feet = 1 yard (yd.) = 36 in. 5J yards = 1 rod (rd.) = 16J ft. = 198". 320 rods = 1 mile (mi.) = 1760 yds. = 5280 ft. = 63,360 in. 233. Table of Surveyor's Linear Measure. 7.92 inches (in.) = 1 link Q..). 25 links = 1 rod (rd.) = 198 in. 4 rods = 1 chain (ch.) = 100 1. = 792 in. 80 chains = 1 mUe (mi.) = 320 rds. 8000 1. = 63,360 in. 80 ARITHMETIC 234. Miscellaneous Units ■}^ inch = 1 line. \ incli = 1 barleycora or size (boot and shoe measure). 3 inches = 1 palm. 4 inches = 1 hand (for measuring the height of horses). 9 inches = 1 span. 18 inches = 1 cubit. 28 inches = 1 pace (military pace). 3 feet = 1 pace (ordinary). 6 feet = 1 fathom (for measuring depths at sea). 120 fathoms = 1 cable length. 1.15 statute mile = 1 nautical or geographical mUe. 1 nautical mile = 1 knot (for measuring speed of vessels). 3 knots = 1 league (for measuring distances at sea). 60 nautical miles = 1 degree = 69.16 statute miles. \ statute mile = 1 furlong. 360 degrees = 1 circumference of the earth. MEASURES OF SURFACE 235. Surface has two linear dimensions, length and breadth. Table of Common Square Measure 144 square inches (sq. in., D") = 1 square foot (sq. ft.). 9 square feet = 1 square yard (sq. yd.) = 1296 sq. in. SOJ square yards = 1 square rod (sq. rd.) = 272J sq. ft. = 39,204 sq. in. 160 square rods = 1 acre (A.) = 4840 sq. yds. = 43,660 sq.ft. 640 acres = 1 square mile (sq. mi.) = 102,400 sq. rds. = 3,097,600 sq. yds. = 27,878,400 sq. ft. 236. Tahle of Surveyor's Square Measure 625 square links (sq. 1.) = 1 square rod (sq. rd.). 160 square rods = 1 acre (A.) = 100,000 sq. 1. 640 acres = 1 section (sec.) = 102,400 sq. rds. = 64,000,000 sq. 1. 36 sections = 1 township (Tp.) = 23,040 A. = 368,640 sq. rds. = 2,304,000,000 sq. 1. MEASURES OF VOLUME 237. Volume has three linear dimensions, length, breadth, and thickness. Table of Common Cubic Measure 1728 cubic inches (eu. in.) = 1 cubic foot (eu. ft.). 27 cubic feet = 1 cubic yard (cu. yd.) = 46,656 cu. in. 238. Table of Wood Measure 16 cubic feet = 1 cord foot (cd. ft.). 8 cord feet = 1 cord (cd.) = 128 cu. ft. BRITISH SYSTEM OF WEIGHTS AND MEASURES 81 These measures are also used in measuring small, irregular stones. A cord is a pile 8 ft. long, 4 ft. wide, and 4 ft. high. Wood cut in lengths of 4 feet is called cord wood. 239 Stone Measure 24f cubic feet = 1 perch. A perch of stone in masonry is 16^ feet long, 1^ feet wide, and 1 foot high. MEASURES OF CAPACITY 240. Measures of capacity are divided into liquid and dry measures. Liquid measures are used for measuring liquids. There are two kinds of liquid measure, namely, common liquid measure, used for measuring water, milk, etc., and apothecaries' liquid measure, used for measuring liquid medicines. 241. Table of Common Liquid Measure 4 gills (gi.) = 1 pint (pt.). 2 pints = 1 quart (qt.). 4 quarts = 1 gallon (gal.) = 8 pts. = 231 cu. in. 31i gallons = 1 barrel (bbl.) = 126 qts. = 252 pts. 2 barrels = 1 hogshead (hhd.) = 63 gal. = 252 qts. = 504 pts. Remark. Casks holding from 28 gal. to 43 gal. are called barrels, and those holding from 54 gal. to 63 gal. are called hogs- heads, but whenever barrels or hogsheads are used as measures, a barrel means 31^ g^-l- ^nd a hogshead 63 gal. 242. Table of Apothecaries' Liquid Measure 60 minims (M.) = 1 fluid dram (/ 5). 8 fluid drams = 1 fluid ounce (/ §). 16 fluid ounces = 1 pint (0). 8 pints = 1 gallon (cong.) = 231 cu. in. 243. Dry measure is used for measuring grains, seeds, fruit, vegetables, etc. Table of Dry Measure 2 pints (pt.) = 1 quart (qt.). 8 quarts = 1 peck (pk.) =16 pts. 4 pecks = 1 bushel (bu.) = 32 qts. = 64 pts. Remark. In measuring grains, seeds, and small fruits, the measure must be even full; but in measuring apples, potatoes, and other large articles, it must be heaping full. 82 ARITHMETIC 244. Comparative Table U. 8. liquid measure, 1 gal. = 231 cu. in. U. S. liquid measure, 1 qt. = 57i cu. in. U. S. dry measure, J pk. = 268J cu. in. U. S. dry measure, 1 qt. = 67^ cu. In. U. 8. apothecaries' liquid measure, 1 gal. = 231 cu. in. Great Britain liquid measure, 1 qt. = 69.3185 cu. in. Great Britain liquid measure, 1 gal. = 277.274 cu. in. Great Britain dry measure, 1 qt. = 69.3185 cu. in. Great Britain dry measure, 1 bu. = 2218.192 cu. in. MEASURES OF WEIGHT 245. There are three systems of units used for measuring weights, namely, avoirdupois, apothecaries' , and troy. 246. Avoirdupois weight is used in weighing all ordinary articles. Table 16 ounces (oz.) = 1 pound (lb.). 100 pounds = 1 hundredweight (cwt.). 20 hundredweight = 1 ton (T.) = 2000 lbs. 247. Apothecaries' weight is used in weighing dry medicines and drugs. Table 20 grains (gr.) = 1 scruple (sc. or 9). 3 scruples = 1 dram (dr. or 5). 8 drams = 1 ounce (oz. or §). 12 ounces = 1 pound (lb. or lb). 248. Troy weight is used in weighing precious stones and metals, such as gold, silver, etc. Tabk 24 grains (gr.) = 1 pennyweight (pwt.). 20 penny-weights = 1 oimce (oz.). 12 ounces = 1 pound (lb.). 249. Comparative Table Avoirdupois Apothecaries' Troy 1 pound = 7000 gr. 5760 gr. 6760 gr. 1 ounce = 437.5 gr. 480 gr. 480 gr. CONVERSION TABLES 83 CONVERSION TABLES Metric-English and English-Metric Linear Measure 250. Common h Inear measure 1 inch = 25.40 mm. 1 meter = 39.37 in. Ifoot = 0.30 m. 1 meter = 3.28 ft. 1 yard = 0.91 m. 1 meter = 1.09 yds. 1 mUe = 1.61 km. 1km. = 0.62 mi. 251. Surveyors' linear measure llink = 20.12 cm. 1 meter = 4.97 1. Irod = 5.03 m. 1 meter = 0.19 rds. 1 chain = 20.12 m. 1 km. = 0.05 ch. 1 nautical mile = 1.85 km. Square 1 km. 1 Measure 0.54 n. mi. 252. Common square measure 1 sq. inch = 6.45 cm.^ 1 sq. cm. = 0.16 sq. in. 1 sq. foot = : 9.29 dm.2 1 sq. m. = 10.76 sq. ft. 1 sq. yard = 0.84 m.' 1 sq. m. = 1.20 sq. yd. 1 sq. mile = ■■ 2.59 km.2 1 sq. km. = 0.39 sq. mi. 253. Surveyors^ square measure 1 sq. link = ■■ 404.81 cm.2 1 m.} = 22.23 sq. 1. 1 sq. rod = ■- 25.30 m.' 1 km.2 = 247.11 acres 1 acre = 0.41 hectares 1 hectare = 2.47 acres Measures of Volume 254. 1 cubic inch = 16.39 cm.' 1 cm.' = 0.06 cu. m. 1 cubic foot = 0.03 m.3 1 dm.' = 0.04 cu. ft. 1 cubic yard = 0.77 m.' 1 m.' = 1.31 cu. yds. Measures of Capacity 255. Dry measure 1 pint = 0.55 1. 1 liter = 1 dm.' 1 _ 0.91 qts. 1 quart = 1.10 1 1 liter = 61.02 cu. in. 1 peck = 8.81 1. 1 decoliter = 1.13 pks. 1 bushel = ; 35.24 1. 256. Liquid measure. (Common) 1 pint = 0.47 1. 1 liter = 2.11 pts. 1 quart = 0.95 1. 1 liter = 1.06 qts. 1 gaUon (U. S.) = 3.79 1. 1 hectoliter = 26.42 gals (U.S.) 1 gaUon (Br.) = 4.55 1. 1 hectoliter = 22.00 gal. (Br.) 84 ARITHMETIC Liquid measure. {Apothecaries') 1 dram = 3.66 cm.' 1 cm.' 1 ounce = 29.37 cm.' 1 liter = 1 dm.' 1 pint = 0.47 1. 1 liter 1 gallon = 3.79 1. 1 deciliter 257. 1 ounce 1 poxmd 1 hundredweight 1 ton (short) 258. 1 grain 1 pennyweight 1 ounce 1 poimd 259. 1 grain 1 scruple 1 dram 1 ounce 1 pound Weights Avoirdupois weights ■ 28.35 g. 1 hectogramme ■■ 0.45 kg. 1 kilogramme ■■ 50.80 kg. 1 ton : 0.91 t. 1 ton Troy weights 1 gramme 1 gramme 1 hectogramme 1 kilogramme = 64.79 mg, = 1.55 g. = 31.10 g. = 0.37 kg. Apothecaries' weights ■ 64.79 mg. 1 milligramme' 1.30 g. 3.89 g. 3.10 g. 0.37 kg. 1 gramme 1 gramme 1 gramme 1 kilogramme 0.27 fl. J 34.48 fl. g 2.12 0. 2.64 gal. 3.53 oz. 2.21 lbs. 2204.62 lbs. 1.10 tons. 15.43 gr. 0.65 pwt. 3.22 07.. 2.68 lbs. 0.015 gr. 15.432 gr. 0.77 sc. 0.25 dr. 2.68 lbs. BOOK IV POWERS AND ROOTS DEFINITIONS 260. That which was said in articles 85 to 88, concerning powers of whole numbers, appTies to any number, fraction, decimal, or complex. Thus, --. ©■ (^ X ?)* M)' 3 are respectively the square of 3.25, the cube of —, the fourth 5 5 power of 4 X s' and the fifth power of 4 + =■ 261. Any number which has a given number for a power is the root of that number. 262. If, of two numbers, the first is a power, of a certain de- gree, of the second, the second is the root, of the same degree, of the first. Thus, 3 giving 3, 9, 27, 81 . . . for 1st, 2d, 3d, 4th powers, these respective numbers have 3 for 1st, 2d, 3d, 4th roots. 263. The roots of the second and third degree are designated as square root and cube root. 264. To indicate the root of a number, write the number under the sign "y/ , called a radical, at the upper left-hand corner of which the index of the root is written. Thus, ^9, y^n^, ^4Ti6, y/J|, express respectively the square, cube, fourth, and fifth roots of 35 the quantities 9, 27 X 3, 4 + 16, and ^ • Remark. The first root of a number being equal to the num- ber, the radical sign and index are discarded. For the square root it is customary to write simply _the radical sign without the index. Thus, instead of writing \^9, write simply V9. 85 86 ARITHMETIC SQUARES AND SQUARE ROOTS 265. To square a number, and, in general, to raise a number to any power, multiply the number by itself and the successive products until as many multiplications have been performed as are indicated by the index of the power. Thus, to square multiply I by itself (160): 266. Directions for using a table of squares and cubes, of the consecutive whole numbers from 1 to 1000, for squaring or cubing whole, decimal, or fractional numbers. Assume that the table gives directly the square and cube of numbers not greater than 1000, which covers all cases in general practice. In an abstract or concrete decimal, if, neglecting the decimal point, the whole number which results is not greater than 1000, by the use of the table find the square or cube of this whole number, and separate on the right two or three times as many decimal figures as there are decimals in the given number. 1. Example. Determine the area of a square the side of which is 7.96 m. Taking the centimeter as unity, we have the length of the side equal to 796 cm., and from the table the area is 633,616 sq. cm. = 63.3616 m^ 2. Find the volume of a cube whose side is 0.796 m. Taking the millimeter as unity, the side of the cube is 796 mm., and the table gives the volume as 504,358,336 mm^ = 0.504358336 m'. If the given number, on removing the decimal point, is larger than 1000, reduce it to units such that the whole part will be as large as possible without exceeding 1000, and the square or cube of this whole part as given by the table may be taken as an approximation, which in ordinary cases is quite sufficiently accurate. Thus, in the first example the side of the square being 7.963 m., take the centimeter for unity, which gives 796.3 cm. Neglecting the 3 millimeters, proceed as in the above example, which gives 633,616 cm^ = 63.3616 m^, or the square of 7.96 m., and may be taken as an approximation to the square of 7.963 m. If the side were 7.968 m., instead of taking 7.96 m. take 7.97 m., SQUARES AND SQUARE ROOTS 87 so as to have the nearest approximation. For a fraction find the square or cube of each of the terms (265). 267. Table of cubes and squares of whole • numbers between 1 and 10. Roots, 012345 6 7 8 9 10 Squares, 1 4 9 16 25 36 49 64 81 100 Cubes, 1 8 27 64 125 216 343 512 729 1000 268. The square of a whole number, of a single figure, has two figiures; that of one having two figures has three or four; that of three has five or six, etc. From this it follows that in order to obtain the number of figures in the square root of a given number, separate the number into periods of two figures each, commencing at the simple units. The number of periods gives the number of figures in the square root. 269. The square of a quantity composed of two parts is made up of the following: 1. The square of the first; 2. Twice the product of the first and the second; 3. The square of the second. Thus: (3 + 5)== = 3^ + 2 X 3 X 5 + 52 = 9 + 30 + 25 =64. As a special case, the square of a number composed of tens and units is made up as follows : 1. Of the square of the tens; 2. Of twice the product of the tens and miits ; 3. Of the square of the units. 54== = 50' + 2 X 50 X 4 + 42 = 2500 + 400 + 16 = 2916; 2732 = 270' + 2 X 270 X 3 + 3' = 72,900 + 1620 + 9 = 74,529. 270. The difference of the squares of two consecutive whole num- bers is equal to twice the smaller of the numbers, plus one. (26 + 1)' - 26' = 26 X 2 + 1 = 53. 271. To extract the square root of a whole number, 74,529 for example, commencing at the right separate the number into periods of two figures each (the number of periods is the number of figures in the root) (268), and draw a vertical line at the right, to separate it from the root. Take the square root, 2, of the greatest square, 4, contained in the first period at the left, 7; this root, 2, which can have but one figure (268), is the first figure at the left of the root. Subtract the square of the first figure found from the first period at the left; at the right of the remainder. ARITHMETIC 3, write the next following period, 45; separate the first figure, 5, at the right of the resulting number; divide the part at the left, 34, considered as expressing simple units, by twice the number obtained in the root, which gives 8 for a quotient; this quotient is either the next figure of the root, or it is too large. To prove it, write it at the right of double that part of the root already obtained; multiply the number 48 which results by 8, and the product 384, being greater than the number 345, shows that 8 is too large. Operating on the figure 7 as on the figure 8, the product 329, obtained by multiplying the number made up of double the part of the root already found and 7 by 7, being less than 345, 7 is the next figure in the root. Subtract the product 329 from 345; at the right of the remainder, 16, write the next period, 29; separate the figure 9 from the others, and divide the part at the left, 162, considered as expressing simple units, by double, 27 X 2 = 54, the part of the root already ob- tained, which gives as quotient the next figure in the root or one too large. This is proved as was the preceding figure, and so on until all the periods of the number have been operated upon. 7.46.29 273 4 48 8 ■ 47 7 643 34.5 3 32.9 384 329 1629 162.9 162 9 7.4 6.29 3 4.6 1 162.9 273 48 47 643 Generally the products of the figures and the root are not written, but they are subtracted as fast as they arie obtained; this was done in the second operation shown above. 272. Limit of the remainder of a square root. In the operation of extracting the square root, if the remainder which corresponds to the part of the root already obtained is not less than twice that part of the root plus one, that part of the root is too small by at least one unit; and when the remainder is less than twice that part of the root plus one, that part of the root cannot be increased by one. Thus the last remainder should always be less than twice the root, plus one. When it is less than the root, the root is correct to half a unit and is less than the exact value. In the opposite CUBES AND CUBE ROOTS 89 case, the root is increased by one and is then correct to a half unit, but is greater than the exact value (175, 206). 273. If, as in the last example, a remainder of zero is obtained, the given number is a perfect square. If, on the contrary, the last remainder is not zero, the given number is not a perfect square. The root obtained is the root of the number, but less than the exact root by less than one unit (272), that is, of the whole part (175). It is the exact root of the largest perfect square contained in the given number, and the remainder is the difference between this number and the largest perfect square. The exact root of the given number cannot be expressed exactly by any number, whole, fractional, or decimal; it is incommensurable (213), and consequently cannot ■ be expressed by a periodic decimal (195, 196, 206). It can be expressed only by approximation. 274. A ivhole number is not a perfect square: 1st. When it does not contain all the prime factors of a power of an even degree (124, 273). 2d. When, being an even number, it is not divisible by 2^ = 4. 3d. When the zeros which terminate it are not in even num- bers. 4th. When it is terminated by one of the four figures 2, 3, 7, 8. 5th. When, terminating with 5, it has not the figure 2 in tens' place. CUBES AND CUBE ROOTS 275. Since the cube of a number of a single figure does not contain more than three figures; of one of two figures contains four, five, or six, etc., it follows that in order to obtain the number of figures in the cube root of a whole number, the number is divided into periods of three figures each, the number of periods giving the number of figures in the root (268). In general, to obtain the number of figures in the mth root of a whole number, divide the number into periods of m figures, and the number of periods will be the number of figures in the root. 276. The cube of- a quantity composed of two parts is made up of the following: First, the cube of the first part; second, the triple product of the square of the first and the second; third, the triple product of the first and square of the second; fourth, the cube of the second. Thus: 90 ARITHMETIC (4 + 5)3 = 43 + 3x42x5 + 3X4X52 + 5^ = 64 + 240 + 300+125 = 729. As a special case, the cube of a number composed of tens and units is made up of four parts: First, the cube of the tens; second, the triple product of the square of the tens and the units; third, the triple product of the tens and the square of the units; fourth, the cube of the units. Thus: 145' = 140' + 3 X 1402 X 5 + 3 x 140 X 5^ + 52 = 2,744,000 + 204,000 + 10,500 + 125 = 3,048,625. 277. The difference of the cubes of two consecutive whole numbers is equal to the triple square of the smaller, plus the triple of the smaller, plus one : (26 + 1)' - 26' = 262 X 3 + 26 X 3 + 1. 278. To extract the cube root of a whole number, 3,048,625 for example, commencing at the right, separate the number into periods of three figures each (the number of periods indicates the number of figures in the root) (275). Take the cube root, 1, of the greatest cube 1, contained in the first period, 3, at the left; the root, which can have but one figure (275), is the first figure at the left of the root. 3.048.6 26 145 1 3x l'=3 3x 12 = 3 3x 142 = 588 20.48 17 44 3 04 6.25 3 04 6 25 3xl0''x6=1800 3x10 X 62 = 1080 6'= 216 3096 3x102x4 =1200 3x10 x42= 480 4'= 64 1744 3x1402x5=294000 3x140 X 52= 10500 5== 125 304625 Subtract the cube of the first figure, 1, from the first period at the left; at the right of the remainder, 2, write the next period, 048; separate two figures at the right of the resulting number; divide the part at the left, 20, considered as expressing simple units, by three times the square of that part of the root already obtained, which gives for a quotient a figure . 6, which is either the next figure of the root or too large. To determine which, finish the operation of constructing the cube, that is, since the cube of the tens has been subtracted, three times the square of the tens times the units 3.102.6 = 1800, three times the tens times the square of the units 3.10.62 = 1080, and the cube of CUBES AND CUBE ROOTS 91 the units 6' = 216; adding, the sum 3096 being greater than 2048 shows that 6 is too large. By the same process it is found that 5 is also too large. Try- ing 4 the sum of the three parts, 1744, being less than 2048, 4 is taken as the next figure in the root. Subtract 1744 from 2048; at the right of the remainder, 304, write the figures 625 of the next period; separate two figures, 25, on the right of the resulting number; divide the part at the left, 3046, considered as expressing simple units, by three times the square of that part of the root already obtained, 3 X 14^ = 588, which gives 5 for a quotient, this being either too large or the next figure in the root. The truth may be established in the same manner as above, considering 140 as one part and 5 as the other, and constructing the three parts: [3 X 140^ X 5 = 294,000] + [3 X 140 X 5' = 10,500] + [5' = 125] = 304,625; since this sum is not greater than 304,625, 5 is the next figure of the root. Continue thus until all the periods of the root have been used. 279. Limit of the remainder of a cube root. The largest re- mainder which can be obtained in the process of extracting the cube root of a number, cannot be as great as three times the square of that part of the root already obtained, plus three times that part of the root, plus one. If the remainder is equal to or greater than this sum, the last figure in the root is too small, and should be increased. 280. At any point in the operation of extracting the cube root of a number, the remainder, followed by all the periods which have not been operated upon, is equal to the number of which the root is desired less the cube of that part of the root already obtained. Analogous to the square root (273), if the cube root falls be- tween two consecutive whole numbers, it cannot be expressed by any number, whole, fractional, or decimal ; it is incommensur- able. This root can only be expressed by approximation. 281. An even number cannot be a perfect cube unless it is divisible by 2' = 8. A number termin ating with ciphers cannot be a perfect cube unless the number of ciphers be a multiple of 3 (274). 282. Proof by the rule of 9. A power of a number being the result of the multiplication of this number taken several times as factor, the proof by 9 of the raising of a number to a certain 92 ARITHMETIC power, is made in the same manner as the proof by 9 in mul- tipUcation (99). To prove by 9 the extraction of a root, the given number being equal to a certain power of the root, plus the remainder, proceed in the same manner as in the proof by 9 of a division (100). Thus, to prove by 9 the example in (278), find the remainder 1 of the root 145 by 9, take the cube 1 of this remainder, and the re- mainder 1 of this cube by 9, added to the remainder by 9 of the remainder obtained in the extraction of the root, gives the sum 1, of which the remainder, 1 by 9, should be equal to the remainder by 9 of the given number 3,048,625. The proofs by 11 of powers and roots are calculated in the same manner as the proofs by 9 (101). SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS OF FRACTIONS AND DECIMALS 283. The square of a fraction being the product of the fraction and itself, it is obtained by squaring each of the terms (160, 266): /4Y_ 4? _ 16 284. The cube of a fraction being the product obtained by using the fraction three times as a factor, it is obtained by cubing each of the terms (266) : 285. From the manner in which the squares and cubes of fractions are formed, it follows that in order to extract the square or cube root of a fraction, it suffices to extract the square or cube root of its terms (262). Thus : V AQ ^^ - f d '^/ ^^ - ■^^^ - * 49 V49 7 V 125 -^125 5 286. The extraction of the square or cube root of a fraction may be reduced to the extraction of the root of but one number. To do this, multiply the two terms of the given fraction by the ROOTS OF FRACTIONS AND DECIMALS 93 denominator, for the square root, or by the square of the de- nominator for the cube root. Thus, /I _ /4 X 7 _ V28 V28 V7~V7v7~ \/72 and n-^] 7 X 7 V72 7 '4 X 52 -^4 X 6 X 5 -v'lOO 5 V 5 X 52 ■ a/53 5 It is seen that in this method of operating, the denominator of the root is the same as that of the given fraction (275). This method holds for all fractions; but if the denominator of the given fraction is not a prime number, it may be better to reduce it to a perfect square or cube, by multiplying the two terms by any convenient factors: /266 ^W / 19 / 19 X 2 X 7 _ Vl9 X 2 X 7 504~V23x32x7^V2^x32x72"" 2x3x7 T9 ■ yi9 X 3 X 72 -^2793 _ -s/2793 2' x 32 X 7 V 2" X 33 X 7^! 2x3x7 42 19 Thus the square of ^^ expressed in 84ths and the cube root in 42ds are obtained (269). 287. The square of a decimal number being the number mul- tiplied by itself, and the cube the number taken three times as a factor, the squares and cubes of numbers are found according to the rules given for multiplication of decimals (180) : 3.546^ = 3.546 X 3.546 = 12.574116; 23.73 = 23.7 X 23.7 X 23.7 = 13,312.053. 288. Since in multiplying a decimal the point is dropped and as many places pointed off in the product as the sum of the decimals in the two numbers, it follows in squaring a number the number of decimals in the square must always be even, be- cause they are obtained by multiplying the number of places in the given number by two. In the same manner it may be shown that the cube of a decimal contains three times as many places as the given number. 289. From the rules concerning the formation of the squares and cubes of decimal numbers (287), the following conclusions may be derived ; 94 ARITHMETIC 1st. To extract the square root of a decimal number, drop the decimal point and proceed as though the number were whole, separating at the right of the root half as many places as there are in the given number: V54:76 = ^^^2 = — = r.4 (172 and 259). VlOO 10 ^ 2d. To extract the cube rodt of a decimal number, drop the decimal point and proceed as though the number were whole, separating at the right of the root one-third as many decimal places as there are in the given number: 73,048,625 145 -n/3.048625 = ;,l!izr_ =^^ = 1.45. -v/1,000,000 100 290. To obtain the square root of any number correct to a given decimal (175), the number must contain twice as many decimals as are desired in the root, and if it has not that number, ciphers must be added at the right; thus, if the square is desired correct to one unit, one tenth, one hundredth, or one thousandth, etc., the given number must contain zero, two, four, or six, etc., decimals. Then dropping the decimal point the root is extracted in the usual manner (271), pointing off at the right of the result the required number of decimals. Thus it is found that the square root of 247 correct to one unit is 15; that the square root of the same number to the hun- dredths place is V247.0000 = 15.71 ; that of 2.5 to the hundredths place is slil = V2:6000 = 1.58 ; 5 that of — to the thousandths place is ^ = VO.464546 . . . = 0.674. 291. Extracting the square root of 0.454545 correct to the thousandths place is the same as extracting the square root of 454545 correct to a unit (290) and pointing off three decimal figures at the right of the result; also the rule in (316) may be ap- plied; thus, calculate Vo.454500, which gives 0.674 for the root and 0.224 for the remainder, and the nearest root to the one- thousandth place is 0.675, which is slightly greater than the exact value. ROOTS OF FRACTIONS AND DECIMALS 95 292. To obtain the cube root of any number correct to a given decimal, operate in the same manner as when finding the square root, except that instead of taking twice as many decimals in the given number as are required in the root, three times as many are talcen. The cube root of 12.5 to the hundredths place is -^12.500000 = 2.32 ; that of 0.000012755427 to the thousandths place, -^0.000012755 = 0.023 ; 71 that of ^ to the hundredths. = 3^ = 3', \/3'2orA/3 = 3t = 3l, n ''/|oi-^3="i = 3-i 307. Remark. The rules given in the preceding chapters show that the extraction of the square or cube root of any num- ber, whole, decimal, or fractional, leads to the extraction of the square or cube root of a whole number, correct to imits' place (271, 278, 290, 292). 308. Use of a table of squares of consecutive whole numbers from 1 to 1000, in shortening the process of extracting the squa/re root of any number, whole, decimal, or fractional : 1st, Correct to the first whole unit; 2d, Correct to a decimal of a given order. 1st. To extract the square root of any number, correct to the first whole unit. The operation consisting of extracting the square root, correct to the first whole imit, of a whole number, the whole part of a decimal number or the whole part of a fraction reduced to deci- mals (290), it is not necessary to consider more than the whole numbers ; and there are two cases, one where the number is not greater than the greatest number in the table, 1,000,000, and' one where it is. First Case. Extract the square root, correct to one unit, of the whole number 786,545. Looking in the column of squares,* the square 784,996 is the nearest to the given square, which is less than the given square, that is, it is the largest whole square contained in the given number; the root, 886, is found in the first column, and is the root of the given number correct to one unit. This root is slightly less than the exact; 887 is also the correct root to one unit, but is slightly larger. The difference, 1549, between the given number and the largest square which it contains, is the * Eeferenoe may be had to almost any handbook for a table of powers and roots. POWERS AND ROOTS OF THE Nth DEGREE 99 remainder which would be obtained in extracting the square root of that number, correct to one imit: 786,545 - 784,996 = 1549. Any decimal number, 786,545.273 for example, having 786,545 for a whole part, would have 886 for its square root, to the first whole unit, with 1549.273 for a remainder. Second Case. Extract the square root, correct to one whole unit, of the whole number 7,875,127,437. Separate at the right of the number an even number, 4, of figures so that the part at the left will be the largest possible number less than the square of 1000; this part coming under the first case, 887 is given for its square root, to a whole unit, with 743 for a remainder. This number, 887, forms the first three left-hand figures of the required root (271), and to obtain the remaining figures, operate according to the rule of (271): 78 75 12.74.37 88 741 78 67 69 17 744 177 481 7 43 7.4 4 1 7 0976 70 976 177 481 33 9 83.7 17 7 48 1 16 2 35 6 Thus at the right of the remainder, 743, write the next period 74, separate the figure 4 on the right, and divide the part at the left, 7437, by twice 887, that part of the root already obtained, which gives 4 as the next figure of the root if not too large. The correctness of 4 is proved and the work continued as per (271). Thus it is found that 88,741 is the square root and 162,356 the remainder. It is seen that the table gives directly the first three figures of the root. Any decimal number, 7,875,127,437.45 for example, having for a whole part the preceding number, would have the same root; the remainder being 162,356.45. 2d. To extract the square root of any number to a given decimal place. From the rule in (290), it follows that these calculations are the same as those given in 1st, and that there are two cases to be considered. 100 ARITHMETIC First Case. Extract the square root, correct to one hiindredth, of the number 78.6545273. Retaining four decimal places, we have 78.6545; dropping the decimal point and extracting the root to one unit as in the first case of 1st, the table gives 886 for the root and 1549 for the remainder; therefore 8.86 is the required root and 0.1549273 is the remainder. Second Case. Extract the square root, correct to one thou- sandth, of the number 7875.1274. Add ciphers to complete the number to 6 decimal places; neglect the decimal point, which gives the number 7,875,127,400; find the square root of this whole number, correct to one whole unit, as in the second case of the 1st. This gives 88,741 for root and 162,319 for a remainder; pointing off the decimals, 88.741 is the required root, and 0.162319 the remainder. 309. Use of the table of cubes of the consecutive whole numbers from 1 to 1000, to shorten the process of extracting the cube root of any number, whole, decimal, or fractional: 1st, Correct to a whole unit; 2d, Correct to a given decimal. 1st. Extract the cube root of any number, correct to one whole unit. The operation consisting of extracting the cube root, correct to the first whole unit, of a whole number, the whole part of a decimal number or the whole part of a fraction reduced to deci- mals (292), it is not necessary to consider more than the whole numbers; and there are two cases, one where the number is not greater than the greatest cube in the table, and one where it is. First Case. Extract the cube root, correct to one whole unit, of the number 97,062,526. Looking in the column of cubes,* the cube 96,702,579 is the nearest value to the given cube that does not exceed it, that is, it is the largest whole cube contained in the number; the root, 459, is found in the first column, and is the root of the given number correct to one unit. This root is slightly less than the exact value; 460 is also correct to one whole unit, but is slightly larger. The difference, 97,062,526 - 96,702,579 = 359,947, between the given number and the largest square which it con- * Eeferenoe may be had to almost any handbook for a table of powers and roots. POWERS AND ROOTS OF THE Nth DEGREE 101 tains, is the remainder which woiild be obtained in extracting the cube root of that number, correct to one whole unit. Any decimal number, 97,062,526.38 for example, having 97,062,526 for a whole part, would have 459 for its cube root, to one whole unit, and 359,947.38 for the remainder. Second Case. Extract the cube root, correct to one whole unit, of the number 97,062,526,893,127. Separate at the right of the number of figures, 6, whose mul- tiple is 3, such that the part at the left will be the largest pos- sible number which is less than the cube of 1000; this part comes under the first case; and from the table we have 459 as the first three figures of the root, and 359,947 as the remainder (278). To obtain the following figures of the root, continue the operation according to the rule in (278), as was done in the second case of 1st for the square root: 97 062 526.8 93.1 27 96 702 579 45 956 63 204 300 X 5 68 850 X 5 25 X 5 6 334207 500 x 6 827 100 X 6 36 X 6 63 273 175 x 5 316 365 875 6 335 034 636 x 6 38 010207816 359 947 8.93 316365 8 75 43 582 0181.27 38 010 2 07 8 16 5 671 8 10 3 11 Thus it is found that the cube root of the given number is 45,956, and the remainder 5,571,810,311. It is seen, that as in the case of the square root (308), the table gives directly the first three figures of the root. As in the first case, any decimal number having the number given in the above example would have 45,956 as its cube root, correct to one imit; and the remainder would be the same as found above, followed by the decimal part of the given number. 2d. To extract the cube root of any number, correct to a given decimal place. From the rule in (292), it follows that these calculations are the same as those, given in 1st, and that there are two cases to be considered. First Case. Extract the cube root, correct to one hundredth, of the number 97.06252632. Retaining six decimal places, and dropping the decimal point, we have 97,062,526; operating upon this number as in first case. 102 ARITHMETIC 1st, the table gives the root 459 and the remainder 359,947; pointing off the decimals, we have 4.59 for the root and 0.359947 for the remainder. Second Case. Extract the cube root, correct to one thousandth, of the number 97,062.52689. Add ciphers to complete the number to 9 decimal places, and neglecting the decimal point we have 97,062,526,890,000, the cube root of which is found precisely as in second case of 1st. This gives 45,956 as root and 5,571,807,184 as remainder, and pointing off we have 45.956 for the root and 5.571807184 for the remainder. EXTRACTION OF SQUARE AND CUBE ROOTS BY MEANS OF SUCCESSIVE ADDITIONS 310. Some of the properties of squares of whole numbers. Write the three following series, one immediately beneath the other: first, the successive odd numbers, commencing at unity; second, the successive whole numbers (n); third, the squares (c) of these successive whole numbers: 15 17 19 21 23 25... 8 9 10 11 12 13... 64 81 100 121 144 169... third series, of any number n, which is directly above in the second series, is equal to the sum of the first n terms of the first series (3d). Thus, the square, c = 25 of n = 5, is equal to the sum of the first five terms in the first series; which is easily proved. 2d. The first series is an arithmetical progression commenc- ing at unity, of which the constant difference is 2, the nth term t. t = 1 + 2 {n - I) = 2n - 1. (359) Thus the whole square, 49, having 7 for its root, is the sum of the first seven terms of the first series, and the seventh term of this series is < = 2 X 7 - 1 = 13. 3d. The sum c of the first n terms in the first series, consid- ered as an arithmetical progression, being equal to one-half the product of the first term plus the nth term t and the number of terms n, we have c=(i^^ (361) 13 5 7 9 11 13 (n) 1 2 3 4 5 6 7 (c) 1 4 9 16 25 36 49 1st. The square c, in the EXTRACTION OF SQUARE AND CUBE ROOTS 103 In substituting for t the value in 2d, c becomes equal to v?, as was stated in 1st. The sum s of the first n terms in the second series is . = 2:+2^;forn=5, «=(iJ^^)-S = 15. The sum S of the first n terms of the third series, that is, the squares of the first n consecutive whole numbers, is equal to twice the root, 2n, of the largest square, plus one, divided by one-third the sum s of the roots : Substituting for s the value given above, we have S = ^n(n+r) (2n + 1). (Algebra, Book III.) Find the sum s of the first n = 13 consecutive whole num- bers. According as the sum, s = ~ = ^^ ^ = 91, has or has not been calculated, the first or second expression for the value of s should be used: ,S = (2 X 13 -M) X g- = 819 or ^ = ^ x 13 X 14 x 27 = 819. 4th. When a series of whole consecutive squares does not commence with unity, for example, the first square is n'^ = c', and the last n^ = c; the sum s' of the corresponding roots is equal to the difference c — c' between the largest and smallest square, plus the sum n + n' oi the two square roots and the whole ex- pression divided by 2. Thus we have c — c' + n + n' In fact, the second series considered as an arithmetical pro- gression the first term of which is n' and the last n, the number of terms is n — n' + 1, giving _ (n' + n) (n-n' + 1) ^ which is the same as the preceding equation when n' and n" are substituted for c and c'. 104 ARITHMETIC If the first square of the series c' = 9, the last c = 64, and n' = 3 and n = 8, then the sum of the series of roots is 64-9 + 8 + 3 s, = 2 = ^^• 5th. To obtain the sum of the squares of the consecutive whole numbers of which the smallest is n' and the largest n, calculate, as in 3d, the sum S of the squares of the first n con- secutive whole numbers, then the sum S' of the first n' — I consecutive whole numbers, a.nd subtract S' from S, which will give the desired sum. 311. Some of the properties of cubes of whole numbers (310). Write the four following series one immediately beneath the other: first, the successive numbers forming an arithmetical pro- gression, whose common difference is 6 and whose first term is 3; second, the successive whole numbers, n ; third, the cubes c of these successive whole numbers; fourth, the sums of the successive whole numbers : 3 9 15 21 27 33 39 45 51 57... (n) 1 2 3 4 5 6 7 8 9 10... (C) 1 8 27 64 125 216 343 512 729 1000... 13 6 10 15 21 28 36 45 55... 1st. The cube C, in the third series, of any whole number, n, in the second series, is equal to one-third of the sum of the first n terms of the first series, multiplied by the number n of terms (3d). Thus, the cube C = 125 of n = 5 is equal to one- third 25, of the sum s' = 75 of the first five terms in the first series, multiplied by 5; which can be easily proved. 2d. The first series being an arithmetical progression, of which the first term is 3 and the common difference 6, the nth term t is i = 3 + 6 (n - 1) = 6ra - 3. (359) Thus the whole cube, 343, having 7 for a cube root, is a third of the first seven terms in the first series, multiplied by 7; and the seventh term of this series is j = 6 X 7 - 3 = 39. 3d. The sum s', of first n terms of the first series, considered as an arithmetical progression, is equal to one-half the product EXTRACTION OF SQUARE AND CUBE ROOTS 105 of the sum of the first term 3 and the nth term t, and the number n of terms. Thus, , (3 + 0^ «^= 2 (361) Substituting for t the value found above, s' = Sn', whence n^ = —, o and therefore, in multiplying the two terms by n, 4th. Any cube, C, of a whole number, n, is equal to 6 times the sum of the first n — \ terms in the fourth series, plus the number n of terms. Thus, ^3 = 8^ = 6 (1 + 3 + 6 + 10 + 15 + 21 +28) +8 = 6X84 + 8 = 512. 5th. The sum, S, of the cubes of the n consecutive whole numbers, commencing at 1 or the first n terms of the third series, is equal to the square of half the sum of n^, and n. Thus, S = (^~^- (Algebra, Book III.) Putting n = 8 we have S = (^-^) = 36== = 1296. 6th. To obtain the sum of the cubes of the consecutive whole numbers, commencing with n' and ending with n, calculate, as in 5th, the sum s of the cubes of the first n consecutive numbers and the sum S' of the first n' — 1 consecutive numbers, and then subtract the two sums, which will give the required sum. 312. Extraction of the square root by successive additions. This method of operating rests upon the fact that the square of a whole number, n, increased by twice the number, n, and by 1, is equal to the square of the next larger whole number, n + 1 (270). The first three figures of the root may be taken from the table, as in (308), and the remaining figures calculated according to the method of successive squares, which will be sufficient to demon- strate the method so that the entire root could be obtained by its use. Given the number 787,512.74 to extract the square root, cor- 106 ARITHMETIC rect to one hundredth. The operation is the same as (282, 2d case, 2d); that is, find the square root of 7,875,127,400, correct to 1 unit, and point off two places in the result. The table gives 887 for the first three figures, the square, 786,769, of which is the greatest whole square contained in the number 787,512. Writing 786,769 below, and proceeding according to the rule given in (270), we have: The square of 8870 Twice the root 8870, plus 1 . The sum or the square of 8871 Twice the root 8871, plus 1 . The sum or the square of 8872 Twice the root 8872, plus 1 . The sum or the square of 8873 Twice the root 8873, plus 1 . The sum or the square of 8874 Twice the root 8874, plus 1 . The sum or the square of 8875 The last square being greater than the number formed by the first four periods at the left of the given number, 8874 is the great- est whole square contained in the number, and 4 is the fourth figure of the root. To calculate the 5th, operate in the same manner. The square of 87,740 .... Twice the root 87,740, plus 1 . . The sum or the square of 88,741 Twice the root 88,741, plus 1 . . The sum or the square of 88,742 7,867,690 17,741 78,694,641 17,743 78,712,384 17,745 78,730,129 17,747 78,747,876 17,749 78,765,625 7,874,787,600 177,481 7,874,965,081 177,483 87,75,142,564 The last square being greater than the number formed by the first five periods at the left of the given number, 88,741 is the greatest whole square contained in the number, and 1 is the fifth figure of the root; pointing off, we have 887.41 as the required root. The remainder is obtained by subtracting the largest square found, from the number formed by all the periods of the given number, with twice as many decimal places pointed off at the right as there are decimals in the root. The remainder in the above example is 16.2319. Noting that twice the roots plus one, which are successively added, increase by a common difference EXTRACTION OF SQUARE AND CUBE ROOTS 107 of 2, it is seen that the extraction of the root is reduced to a series of very simple additions ; and as for each figure of the root, the number of these additions averages 5 and is never greater than 9, it follows that in less than an hour the root of a number containing 60 figures could be extracted, which, according to the ordinary way, would take at least a half a day (271). 313. The cube of a whole number n being given, required to find that of (n + 1). (276.) (n + ly = n^ + Sn" + 3n + 1. Since 3n^ is equal to the sum s^ of the first n terms in the first series (311, 3d), for example, to obtain the cube of 21, know- ing that of 20, operate thus: Cube of 20 ... 8000 Sum of the terms s - Bn^ or ^^ = 3 X 20^ or ^t, 1200 n 20 3 times the root n = 20 . . 60 Unity .... 1 The cube of 21 9261 314. The cubes of two consecutive whole numbers, n and n+l, being given; to find that of the next consecutive number, n + 2. Let d be the difference between the cubes (n + ly and n^ (313). d = 3n^ + 3n + 1. Writing (313) (n + 2y = (n + 1)' + 3 (n + 1)^ + 3 (n + 1) + 1; expanding (n + 2y = (n + ly + 3n^ + 6n + 3 + 3n + 3 + 1; substituting {n + 2y = (n + ly + (3n^ + 3n + 1) + 6 (ra + 1) = (n + ly + d +6(n + 1). For example, having 20' = 8000 and 21« = 9261 given, to find the cube of 22, then of 23, etc., operate as follows: Cube of 21 (313) 9261 Difference, d = 2P - 20^ ... 1261 6 (n + 1), or 6 times the root, 21 126 The sum or the cube of 22 . . . 10,648 Difference, 22== - 2P 1387 6 times the root 22 ... . 132 The sum or cube of 23 .... 12,167 108 ARITHMETIC 315. Extraction of the cube root by successive additions. It follows from the two preceding articles that the cube root may be extracted by means of successive additions, as was the square root (312). Let it be given to find the cube root, correct to one thousandth, of the number 97,062.52689. The operation resolves itself (309, 2d, 2d case) into the extraction of the cube root, to one whole unit, of the number 97,062,526,890,000, and separating three decimal figures at the right of the result. The table gives 459 as the first three figures of the root, the cube 96,702,579 of which is the largest whole cube contained in the three periods at the left. The remaining figures are obtained as follows : Cube of 4590 96,702,579,000 Three times the square of the root 4590 .... 63,204,300 Three times the root 4590 13,770 Unity 1 The sum or cube of 4591 (313) 96,765,797,071 Difference between this cube and the preceding, 63,218,071 6 times the root 4591 27,546 Sum or cube of 4592 (314) 96,829,042,688 Difference between this cube and the preceding, 63,245,617 6 times the root 4592 27,552 Sum or cube of 4593 96,892,315,857 Difference between this cube and the preceding, 63,273,169 6 times the root 4593 27,558 Sum or cube of 4594 96,955,616,584 Difference between this and preceding cube . . 63,300,727 6 times the root 4594 27,564 Sum or cube of 4595 , 97,018,944,875 Difference between this and preceding cube . . 63,328,291 6 times the root 4595 , 27,570 Cube of 4596 97,082,300,736 The last cube being greater than the number formed by the first four periods of the given number, 4595 is the greatest whole cube contained in the nimiber, and 5 is the fourth figure in the root. To get the fifth figure, operate' as before; but it may be noted that in finding three times the square of 45,950, the cal- EXTRACTION OF SQUARE AND CUBE ROOTS 109 dilations may be greatly simplified by resolving the number into 45,900 and 50 (269); thus: The square of 45,900 is obtained by writing four ciphers at the right of the square of 459, which is taken from the table 2,106,810,000 45,900 X 50 X- 2 4,590,000 Square of 50 2,500 Sum or square of 45,950 2,111,402,500 Multiplying by 3, we have 3 times the square . . 6,334,207,500 This method of calculating the square, or three times the square of a number formed by writing figures at the right of a number of which the square is known, shortens long and tedious operations, especially in extracting the cube root where the triple square of that part of the root already found is so often used (278, 309). Continuing the example: The cube of 45,950 . . Triple square of the root 45,950 Three times the root 45,950 Unity The cube of 45,951 Difference, 45,951 - 45,950 6 times the root 45,951 The cube of 45,952. Difference .... 6 times the root . The cube of 45,953 . Difference . . . 6 times the root . Cube of 45,954 . . Difference . . 6 times the root . . Cube of 45,955 . Difference .... 6 times the root . Cube of 45,956 . . 97,018,944,875,000 6,334,207,500 137,850 1 97,025,279,220,351 6,334,345,351 275,706 97,031,613,841,408 6,334,621,057 275,712 97,037,948,738,177 6,334,896,769 275,718 97,044,283,910,664 6,335,172,487 275,724 97,050,619,358,875 6,335,448,211 275,730 97,056,955,082,816 Continuing thus, it is found that the cube of 45,957 is greater than the number 97,062,526,890,000, formed by the first five no ARITHMETIC periods; therefore 6 is the fifth figure of the root, and pointing off, we have 45.956, the required root. The remainder is found by subtracting from the number formed by all the periods the largest cube which is contained in that number, and separating at the right of the difference three times as many decimal figures as there are in the root. Thus the remainder in the given example is 5.571807184. No matter how many figures there are in the root, they may all be calculated in the same manner as 5 and 6 in the above example. It may be noted that the above operations are simply addi- tions; thus the difference of two consecutive cubes is equal to the sum of the two numbers written between these cubes, and 6 times the root is obtained simply by adding 6 to the latter of these numbers. SHORT METHODS OF CALCULATING THE SQUARE AND CUBE ROOT 316. To extract the mth root of a whole number. A, with an error less than one whole unit, it suffices to retain more than the mth part of the figures in A; which is more than half for the square root, and more than one-third for the cube root. Since the error tends to decrease the root, it follows that in order to extract the root of a number correct to one whole unit, 71+1 take figures at the left and complete the n figures by adding ciphers to this part; then extract the mth root, which will be correct to one whole unit and slightly larger than the exact value. Thus: 1st. The square root, 274, of the number 74,600, greater, by less than one whole unit, than the exact root, is in general the square root of any number containing 5 figures, the first 3 of which are 746. Likewise the square root, 88,742, of the number 7,875,120,000, greater, by less than one whole unit, than the exact root, is the square root of 7,875,127,400, correct to one whole unit (308). 2d. The cube root, 460, of the number 97,000,000, greater, by less than one whole unit, than the exact root, is the cube root of the number 97,062,526, correct to one unit. Likewise the cube root, 45,957, of the number 97,062,000,000,000, greater, by SHORT METHODS FOR SQUARE AND CUBE ROOTS 111 less than one whole unit, than the exact root, is the cube root of the number 97,062,526,893,127, correct to one unit (309). Remark 1. That which has been said, applies equally to the extraction of the square, cube, or mth root of a number, correct to any given decimal (290, 292, 308, 309). Thus: 1st. The square root, 2.74, of the number 7.4600, greater, by less than one hundredth, than the exact root, is the square root of the number 7.467342, correct to one himdredth. 2d. The cube root, 45.957, of a number, 97,062.000000000, greater, by less than one thousandth, than the exact root, is the cube root of the number 97,062.52689, correct to one thousandth. Remark 2. From the above it follows that when the number, the root of which is to be found, has to be calculated, as is the case with fractions (290, 292), only those figures which are de- sired at the left need be obtained. 317. When in extracting the square root of a number, correct to a unit, more than half of the figures of the root have been obtained, the rest may be obtained by dividing the given number, less the square of that part of the root already obtained, that is, the number formed by the last remainder followed by the periods which have not been operated upon, by twice that part of the root already obtained. Thus, in the example (308, 1st, 2d case), having obtained the first three figures of the root, the last two figures are found as shown here below: 743, the last remainder, followed by the periods which have not been operated upon, 7437, gives the number 7,437,437 as the dividend, and the quotient 41 is obtained by dividing this dividend by twice that part of the root already obtained, 88,700: 7 43 74 37 34 14 16 40 37 17 74 00 41 The square root thus obtained is equal to, greater or less than, the exact, according as the square of the quotient 41 is equal to, greater or less than, the remainder 164,037. Thus, in the above example, having 41^ = 1681 < 164,037, the root 88,741 is less than the exact root. As may be seen, the quotient 41 may be obtained by writing only half the figures of the imused periods after 743 and divid- 112 ARITHMETIC ing the resulting number, 74,374, by twice the root already obtained, considered as simple units, 1774. Writing at the right of the remainder 1640 the figures which were not employed, the remainder 164,037 is obtained. Applying simultaneously this rule and the one preceding: 7 43 00 00 33 40 15 66 00 17 74 00 41 which gives 88,742 for the square root, greater, by less than one unit, than the exact root. 318. When in extracting the cube root of a number, correct to a whole unit, more than half of the figures plus one have been obtained, the rest may be obtained by dividing the given num- ber, less the cube of that part of the root already obtained, that is, the number formed by writing the remaining unused periods after the remainder, by the triple square of the root already obtained. Thus, in the example (311, 1st, 2d case), having obtained the first four figures of the root, the remaining figures are found as shown below: Dividing the number 43,582,018,127 by the triple square 6,334,- 207,500 of that part of the root already obtained, 45,950, the last figure, 6, of the root is obtained. Thus: 43 582 018 127 6 334 207 500 5 576 773 127 6 The cube root thus obtained is equal to, greater or less than, the exact, according as the product of 3 times that part of the root already obtained, plus the quotient 6 and the square of the quotient, is respectively equal to, greater or less than, the re- mainder 5,576,773,127. Thus in the given example, (3 X 45,950 + 6) X 6^ = 4,962,816 < 5,576,773,127, the root is less than the exact root. Analogous with the square root (317), the quotient 6 may be obtained by writing at the right of the last remainder, 43,582,018, one-third of the figures not employed, and dividing the resulting number 435,820,181 by 63,342,075. Writing the rest of the figures in the given number at the right of the remainder, we have the required remainder, 5,576,773,127. SHORT METHODS FOR SQUARE AND CUBE ROOTS 113 Applying simultaneously this rule and the one preceding: 43 055 125 000 5 049 880 000 6 334 207 500 6 which gives 45,957 for the cube root, greater by less than one unit. If the root should have six figures, as for the number 97,062,256,893,127,463 for example, after having determined the first four figures, 4595, the two others, 6 and 8, are obtained by the following divisions : 4 358 201 812 557 677 312 50 940 712 63 342 075 68 4 355 512 500 554 988 000 48 251 400 63 342 075 68 The root is 459,569, greater than the exact root by less than one unit. 319. Remark. The rules in the two preceding articles apply also to the extraction of the square and cube roots of any number, correct to a given decimal, provided the number contains 2 or 3 times as many decimals as are required in the root (316, Remarks). BOOK V RATIOS, PROPORTIONS AND PROGRESSIONS DEFIHITIOHS 321. Ratio is the result of the comparison of two numbers of the same kind. This comparison is made by taking the differ- ence of the two quantities or dividing one by the other. The arithmetical ratio is the difference of two quantities. Thus the arithmetical ratio of 6 and 18 is written 18-6, and pronounced 18 to 6 or 18 less 6. In the case where the second number is larger than the first, the difference is preceded by the negative sign — , which indicates that the quantity could not be subtracted (31). Thus: 6 - 18 = - 12. A geometrical ratio is the quotient obtained by dividing the first quantity by the second. Thus, the geometrical ratio of 18 and 6 is the quotient 3 (207). Written 18 : 6 or ^ , and pronounced 18 is to 6, or 18 divided by 6, or the ratio of 18 to 6. Remark. When the word ratio is used alone, a geometrical ratio is always understood. 322. In the preceding arithmetical and geometrical ratios (321), 18 and 6 are the two terms of the ratio, the first term 18 is the antecedent, and the second 6 the consequent. 323. An arithmetical ratio being the difference of two quan- tities, the properties given in articles 28, 34, and 63 hold here. Thus, for example, an arithmetical ratio is not altered by increas- ing or decreasing both its terms by the same number. Likewise a geometrical ratio being a quotient, the properties given in articles 71, 72, 73, 74, and 77 also apply here. Thus, 114 DEFINITIONS 115 for example, a geometrical ratio is unaltered when both its terms are multiplied or divided by the same number. 324. Two equal arithmetical ratios form an arithmetical pro- portion. The ratio 8 to 4 being equal to 13 to 9, these numbers form a proportion, which is written 8 - 4 = 13 - 9, and pronounced 8 is to 4 as 13 to 9, or 8 less 4 equals 13 less 9. 325. Likewise, two equal geometrical ratios form a geometrical proportion. Thus, the geometrical ratio 8 to 4 being equal to 12 to 6, these four numbers form a geometrical proportion, which is written 8 : 4 : : 12 : 6 or 8 : 4 = 12 : 6 or f = ^ , 4 6 and is pronounced 8 is to 4 as 12 to 6, or 8 divided by 4 equals 12 divided by 6, or the ratio of 8 to 4 equals the ratio of 12 to 6. Remark 1. Two incommensurable ratios are equal when the antecedent of the first ratio contains a fraction, as small as desired, of its consequent, as many times as the antecedent of the second ratio contains the same fraction of its consequent (162, 213). Remark 2. The word proportion used alone means geometrical proportion. 326. Four quantities are said to be proportional or in propor- tion when the ratio of the first to the second is equal to the ratio of the third to the fourth. Thus, given the four proportional quantities 8, 4, 12, 6; 8 : 4 = 12 : 6. In this case the first two or the last two are in direct proportion to the two others. If four quantities of a proportion are so related that an increase in one of the four causes a corresponding decrease in another, the two quantities are said to be inversely proportional to each other. Thus, in the proportion, 8 : 4 = 12 : 6, the quantity 8 is inversely proportional to the quantity 6, while the quantity 8 is directly proportional to the quantity 12. 327. In any arithmetical or geometrical proportion, the an- tecedent of the first ratio, that of the second ratio, the conse- quent of the first ratio and that of the second, are called respec- tively the first antecedent, the second antecedent, the first conse- 116 ARITHMETIC qumt, and the second consequent. The first and fourth terms of the proportion are called the extremes, and the second and third terms the means. 328. The fourth term of a proportion is called the fourth proportional of the other three terms (326). It is a fourth arith- metical or a fourth geometrical, according as the proportion is arithmetical or geoinetrical. 329. In an arithmetical proportion, such as 5-7:7-9, where the means are equal, the term 7 is an arithmetical mean between the two others, 5 and 9, and the term 9 is the third arithmetical of the two, 5 and 7. Such a proportion is written 5 • 7 • 9. 330. Likewise, in a geometrical proportion, 4 : 12 = 12 : 36, where the means are equal, the mean, 12, is the mean propor- tional of the two others, 4 and 36, and 36 is the third propor- tional of 4 and 12. Such a proportion is written 4 : 12 : 36. 331. Remark. 1st, when the antecedents or the consequents of an arithmetical or geometrical proportion are equal to one another, the consequents or antecedents are equal; 2d, when two arithmetical or geometrical proportions have a common ratio, the ratios which are not common form a proportion, that is, are equal. ARITHMETICAL PROPORTIONS 332. In all arithmetical proportions the sum of the extreme is equal to that of the means. Thus, having 9 - 4 = 13 - 8, we have 9 + 8 = 4 + 13. 333. When the sum, 9 + 8, of two numbers is equal to the sum, 4 + 13, of two others, the four numbers form an arithmet- ical proportion in which the two numbers forming one of the sums are the extremes or the means, and the other two numbers forming the second sum are the means or extremes. GEOMETRICAL PROPORTIONS 117 334. When four numbers are not in arithmetical proportion, the sum of the means does not equal the sum of the extremes. 335. An arithmetical proportion is not altered by: 1st, increas- ing or diminishing an extreme and a mean by the same quantity; 2d, dividing or multiplying all the terms by the same number. Thus, the preceding proportion gives: (9 + 2) - (4 + 2) = 13 - 8, (9 + 2) - 4 = (13 + 2) - 8, etc., and (9 X 2) - (4 X 2) = (13 X 2) - (8 X 2), etc. 336. In any arithmetical proportion each extreme is equal to the sum of the means less the other extreme, and each mean is equal to the sum of the extremes diminished by the other mean. Thus, the proportion 8 — 4 = 13 — 9 gives 8 = 4 + 13 - 9 and 13 = 8 + 9 - 4. From this it follows that if three terms of an arithmetical pro- portion are known, the fourth is easily found. 337. The arithmetical mean of two numbers, 5 and 9, is half, 7, of the sum, 14, of these numbers: 5-7 = 7-9. 338. An arithmetical proportion may be transformed as much as desired so long as the equality between the sum of the means and that of the extremes is not destroyed (333). Thus, having 9 + 8 = 4 + 13, the 8 following proportions may be constructed: 9-4=13-8, 9-13 = 4-8, 8-4=13-9, 8-13 = 4-9, 4-9 = 8-13, 4-8 = 9-13, 13-9 = 8-4, 13-8 = 9-4. The remarks in (345) apply to arithmetical as well as to geometri- cal proportions. GEOMETRICAL PROPORTIONS 339. In all geometrical proportions the product of the extremes is equal to the product of the means. Thus, in the proportion 8 : 4 = 12 : 6, we have 8 X 6 = 4 X 12. 340. When the product, 8 X 6, of two numbers is equal to the product, 4 X 12, of two other numbers, the four numbers form a proportion, of which the two factors of one of the prod- ucts are the extremes or the means, and the two factors of the other product the means or extremes. 118 ARITHMETIC 341. When .four numbers are not in proportion, the product of the means is not equal to that of the extremes. 342. A geometrical proportion is not altered by multiplying or dividing one of the extremes and one of the means by the same number. Thus, the preceding proportion gives 8X2 12 8X2 12 X2 ^ 4Y2 = -6' -4- = -6-'^*^- 343. In any proportion, each extreme is equal to the product of the means divided by the other extreme, and each mean is equal to the product of the extremes divided by the other mean. From this it follows that the fourth term, x, of the proportion, ^ ^. ■ 2 X 24 „ 6 : 2 = 24 : a;, is x = — g — = 8. 344. The geometrical mean, x, of two numbers, 4 and 36, is the square root of the product of the two numbers (330). The proportion 4 : a; = a; : 36 gives a;^ = 4 X 36, or x = V4 X 36 = 12. 4 : 12 = 12 : 36. 345. A proportion may be transformed as much as desired so long as the equality between the product of the means and that of the extremes is not destroyed. Thus, having 8 X 3 = 2 X 12, the 8 following proportions may be constructed: 8 : 2 = 12 : 3, 8 : 12 = 2 : 3, 3 : 2 = 12 : 8, 3 : 12 = 2 : 8, 2:8= 3 : 12, 2 : 3 = 8:12, 12 : 8 = 3 : 2, 12 : 3 = 8:2. Remarks: 1. The first four of the above proportions show that when four numbers are in proportion they will be in pro- portion when their means or extremes are transposed (340). 2. The last four of these proportions show that a proportion is not destroyed when the means and extremes are interchanged. 3. The first proportion, 8 : 2 = 12 : 3, giving 8 : 12 = 2 : 3, it follows that in any proportion the first antecedent is to the second antecedent as the first consequent is to the second. 346. A proportion is not destroyed by multiplying or dividing the four terms or only an extreme and a mean by the same num- ber (323). Thus, having 1 : 4 = 3 : 18 gives | ; ¥=^ J:18 = ^:8 gives ^ : --f. GEOMETRICAL PROPORTIONS 119 8 : 2 = 12 : 3, we have also 8x3:2X3 = 12X3:3X3. 347. From this it follows that fractional terms may be reduced. Thus, reduce the terms to the same denominator and suppress the denominator: 1 1 o 2 . 3 1 12 4 o , ,^ . 2 = 6 = 2=3 S^^^^ 6 = 6 = -6=6 "' 3:1 = 12:4. When only one extreme or one mean is a fraction or one ex- treme and one mean, two terms are all that need be reduced to a common denominator (323, 340): 18 or 2 : 12 = 3 : 18; :8 or 9: 18 ='4 : 8. The terms of a proportion may be simplified by multiplying or dividing the four terms or only an extreme and a mean by the same number : 9 : 3 = 36 : 12 gives 3 : 1 = 12 : 4. 348. When two proportions have the same antecedents or the same consequents, their consequents or their antecedents are proportional (331, 327): 3 : 9 = 15 : 45 and 3 : 6 = 15 : 30 give 9 : 45 = 6 : 30. 349. In any proportion, 8 : 4 = 6 : 3, for example: 1st. The sum or difference of the first two terms is to the first or second term as the sum or difference of the last two terms is to the third or fourth. Thus, (8 + 4) : 4 = (6 + 3) : 3 and (8 + 4) : 8 = (6 + 3) : 6; (8 - 4) : 4 = (6 - 3) : 3 and (8 - 4) : 8 = (6 - 3) : 6. 2d. The sum of the first two terms is to the sum of the last two terms as the difference of the first two is to the difference of the last two : (8 + 4) : (6 + 3) = (8 - 4) : (6- 3); or by interchanging the means : (8 + 4) : (8 - 4) = (6 + 3) : (6 - 3). 3d. The sum or difference of the two antecedents is to the 120 ARITHMETIC second or first antecedent as the sum or difference of the con- sequents is to the second or first consequent: (8 + 6) : 6 = (4 + 3) : 3 and (8 + 6) : 8 = (4 + 3) : 4; (8 _ 6) : 6 = (4 - 3) : 3 and (8 - 6) : 8 = (4 - 3) : 4. 4th. The sunti of the antecedents is to that of the consequents as the difference of the antecedents is to that of the consequents: (8 + 6) : (4 + 3) = (8 - 6) : (4 - 3). 5th. The sum or difference of the antecedents is to the sum or difference of the consequents as any antecedent is to its con- sequent: (8 + 6) : (4 + 3) = 8 : 4 = 6 : 3, (8 - 6) : (4 - 3) = 8 : 4 = 6 : 3. 350. When the terms of several proportions are multiplied together in order, the four products form a proportion. . Thus, having 4:2 = 6:3, 7 : 5 = 14 : 10, 3 : 9 = 6 : 18, we have 4X7X3:2X5X9 = 6X14X6:3X10X18. 351. The quotients obtained by dividing, in order, the terms of one proportion by those of another, are in proportion : 4 2 _ ^ ^ 7 ■ 5" 14 'lO' 352. Similar powers and roots of the four terms of a propor- tion form a proportion. Thus, having 3 : 7 = 6 : 14, we have also 33 . 73 _ 63 . 143^ and V3 : V7 = V6 : ■\/i4. 353. In a series of equal ratios, the sum of any number of antecedents is to the sum of their consequents as any antecedent is to its consequent. Thus, having 3 : 6 = 4 : 8 = 7 : 14 = 5 : 10, 3_4_ 7 _ 5 ""■ 6 - 8 " 14 - 10' we have (3 -)- 4 + 7) : (6 + 8 -|- 14) = 3 : 6 = 5 : 10. (137) 354. In a proportion, and in general in a series of equal ratios, ARITHMETICAL PROGRESSIONS 121 the square root of the sum of the squares of a certain number of antecedents is to the square root of the sum of the squares of their consequents as any antecedent is to its consequent. Thus, the above series gives V32 + 42 + 72 + 52 : V62 + 82 + 142 + 102 = 3 : 6. That which is true for the square root of the sum of the squares is true for any root, mth, of the sum of the mth powers: ^33 + 43 + 73 . ^Qs + 83 + 143 = 3 : 6. 355. In any proportion, the product of the antecedents is to the product of the consequents as the square of one antecedent is to the square of its consequent: 3 : 7 = 6 : 14 gives 3 X 6 : 7 X 14 = 3^ : 72. 356. In a series of equal ratios, the product of a certain num- ber of antecedents is to the product of their consequents as any antecedent raised to a power of a degree equal to the number of antecedent factors is to its consequent raised to the same power: 3 : 6 = 4 : 8 = 7 : 14 = 5 : 10, 3X4X7:6X8X14 = 3=:63 = 5': 10^ ARITHMETICAL PROGRESSIONS 357. A series of numbers increasing or decreasing, such that the arithmetical ratio of each term to the term which immediately precedes it is constant (321), forms an arithmetical progression. These numbers are the terms of the progression, and the constant ratio of each term to the one immediately preceding is the com- mon difference. Thus the numbers 4, 7, 10, 13, 16 form an ascend- ing arithmetical progression of which the common difference is 7-4 = 3. It is written 4 • 7 • 10 • 13 ■ 16, (a) and pronounced, as 4 is to 7 is to 10 is to 13, etc. Remark. The same numbers written in the inverse order would form a descending arithmetical progression: 16 • 13 ■ 10 • 7 • 4. (6) 358. An arithmetical progression is not altered when all its terms are increased or decreased by the same quantity (28, 4th). A progression is not altered when all its terms are multiplied or 122 ARITHMETIC divided by the same number; but the common difference is mul- tiplied or divided by that number (34, 63). 359. According as an arithmetical progression is ascending or descending, each term is equal to the first plus or minus the common difference, taken as many times as there are terms before the one under consideration. Thus, in the progression (a) the 5th term is 4 + (3 X 4) = 16, and in the progression (b) the third term is 16 — (3 X 2) = 10 (310, 311). 360. The sum of two terms equally distant from the extremes is equal to the sum of the extremes in the arithmetical progression. Thus, 4 • 7 • 10 • 13 • 16 gives 4 + 16 = 7 + 13 = 10 + 10. 361. The sum, s, of the terms of an arithmetical progression is equal to the sum of the extremes, times the number of terms divided by 2. The progression above gives s = i±l^x5 = 50. (310,311) 362. To insert a certain number of arithmetical means between two given numbers, determine the common difference in the desired progression thus : take the difference between the two given num- bers and divide this difference by the number of means plus one. Having the common difference, add it to the first number, and then to the successive sums obtained, which sums are the means. Given the numbers 4 and 28, required to insert three means between them: rrx. A-ff .28-4 24 „ The common difference is -r; 5- = -r = 6 ; 3 + 1 4 ' and adding 6 to 4 and successively to the sums, we have 4 • 10 ■ 16 • 22 ■ 28. The same result is obtained by commencing with the larger num- ber and subtracting the common difference. 363. When the number of arithmetical means to be inserted is equal to a power of 2 less 1, these arithmetical means may be found directly by taking an arithmetical mean between the two given numbers (337); then an arithmetical mean between each of the given numbers and the term which has been found, and so on. GEOMETRICAL PROGRESSIONS 123 Let it be required to insert 2^ - 1 = 3 means between and 1. Taking the arithmetical mean 0.5 between and 1, we have the progression • ■ 5 • 1 ; then inserting an arithmetical mean between each of the successive terms of this progression, the required progression is obtained: 0. ■ 0.25 • 0.5 • 0.75 • 1. 364. In inserting the same number of means between the consecutive terms of an arithmetical progression, the whole forms a new arithmetical progression. Inserting three means between the consecutive terms of the arithmetical progression 2 ■ 14 • 26, we obtain the new progression, 2 • 5 • 8 • 11 ■ 14 • 17 ■ 20 • 23 • 26. 365. The sums of the corresponding terms of several arithmetical progressions form an arithmetical progression of which the com- mon difference is the sum of the common differences of the sev- eral progressions the terms of which have been added. In sub- tracting the terms of an arithmetical progression from the corre- sponding terms of another arithmetical progression, the remainders form an arithmetical progression of which the common difference is the difference of the common differences of the given progressions. GEOMETRICAL PROGRESSIONS 366. An ascending or descending series of numbers, such that the geometrical ratio of each one to the one which precedes it is constant, forms a geometrical progression. These numbers are the terms of the progression, and the constant ratio of each term to the one which precedes is called the multiplier (321). Thus the numbers 2, 6, 18, 54, 162 form an ascending geomet- rical progression, of which the multiplier is 3. It is written 2 : 6 : 18 : 54 : 162, and pronounced, as 2 is to 6 is to 18 is to 54, etc. Remark. The same numbers written in an inverse order give a descending geometrical progression, of which the multiplier is ^• 367. A geometrical progression is not altered when all its terms are multiplied or divided by the same number (323). 368. In an ascending or descending geometrical progression, any term is equal to the first multiplied by the multiplier raised to a power of a degree equal to the number of terms which precede the 124 ARITHMETIC term in question. Thus, in the preceding progression, the fifth term is equal to 2 X 3^ = 2 X 81 = 162. 369. The product of two terms equally distant from the extremes is equal to the product of the extremes. The example of (366) gives 2 X 162 = 6 X 54 = 18 X 18. 370. The product, p, of the terms of a geometrical progression is equal to the square root of the product of the extremes raised to a power of a degree equal to the number of terms in the pro- gression. Thus, the above example gives p = V( 2X162)= = 1,889,568. 371. The sum, s, of the terms of a geometrical progression is obtained by subtracting the first term from the product of the last term and the multiplier and dividing this difference by the multiplier less one. The progression of (366) gives (162x3) -2 ^ s 3 _ 1 // the progression were descending, the sum of the terms would be obtained by dividing the first term diminished by the product of the last term and the multiplier, by one less the multiplier. Thus, the progression 162 : 54 : 18 : 6 : 2 gives ^^^-^X3- '''--3 162x3-2 ,,, s = — — ^=-^= 2 = 242. 3 3 372. To insert a certain number of geometrical means between two given numbers, determine the multiplier of the progression which is desired thus: Divide the second of the numbers by the first, and extract the root, of an index equal to the number of means plus one, of the quotient. Now multiply the first number by the multiplier thus obtained, and the product will be the first mean, or the second term of the progression, which in turn mul- tiplied by the multiplier will give the third term, and so on. Let it be required to insert three geometrical means between the numbers 2 and 162. The multiplier is */l|? = ^81 = VVl = V9 = 3. (298) f GEOMETRICAL PROGRESSIONS 126 Multiplying the first term, then the successive products, by 3, the following progression is obtained: 2 : 6 : IS : 54 : 162. 373. When, as in the preceding example, the number of geo- metrical means to be inserted is equal to a power of 2 less 1, the means may be foimd by first finding a mean between the given numbers (319), then the mean between each of the given num- bers and the mean already found, and so on. Let it be required to insert 2^—1 means between 2 and 162. Taking the geo- metrical mean V2 X 162 = 18, between 2 and 162, the progres- sion 2 : 18 : 162 is obtained. Inserting a geometrical mean between each of the consecutive terms of this progression 2 and 18, 18 and 162, the required progression is obtained: 2 : 6 : 18 : 54 : 162. 374. In inserting the same number of geometrical means between the consecutive terms of a geometrical progression, the whole forms a new geometrical progression. Thus, in inserting three means between each of the consecutive terms of the pro- gression 1 : 81 : 6561, the following progression is obtained: 1 : 3 : 9 : 27 : 81 : 243 : 729 : 2187 : 6561. 375. The products of the corresponding terms of several geo- metrical progressions form a new progression, of which the mul- tiplier is equal to the product of the multipliers of the progres- sions. In dividing the terms of a geometrical progression iy the corre- sponding terms of another progression, the quotients form a geo- metrical -progression, of which the multiplier is equal to the mul- tiplier of the first progression divided by the multiplier of the second. In raising all the terms of a progression to the same power, a new geometrical progression is obtained, of which the multiplier is equal to the multiplier of the given progression raised to the given power. In extracting the same root of all the terms of a progression, an- other progression is obtained, of which the multiplier is equal to the same root of the multiplier of the given progression. 126 ARITHMETIC o 0) US Q V ^jBnn'Bf ^ .laqraaoao; % jaqo^oo S jeqraa^dgg o annp |udy qoj'Bin jCiBtuqa^g; C5 U3 j9qraao9Q; g^ lO lO j9qniaA0j>i § CO lO 'ji •« j9qo?oo § CO CO lO lO tH -* jaqms^dag CO iS fe^ U5 tH ^ CO CO }stiSnv' CD CO CO CO S CM I3 -* ^ CO G^ <© O tr~ CO CO CO (M (M CO CO rS O Y-t 1—i 2 S 00 T-H 1— t § CD 1— i CO eptembe I— 1 CO 1— t CO 1— I (M rH I— 1 CD rH CO 00 1—1 o i-O CM I— 1 1—1 OS 1— ( CD § 1-3 r-< O G<1 (M T-H rH s 1—1 OS CD CO [3 1-5 O I-H OS T-\ O ^ CO S o OS ,_| Fh o. .CO < ^ OS 00 LO G^ ^ >> ^ CO 5 d i^ p e -0 P) Cfl § fi o II 1:3 5 a 5 ^ e: & rt cd ^ s 1 1 .a a S 1 & a & g s^ CG g S a ^ ,Q H B 3 !? 5 S>3 ■rt ^ g •^ ■S !N -. hQ tH fH Si S Si 3 S S 3 S g -g « I S'^ „• ja s> ■" S Kl S9 S fc 5 S X " a 9< ■s S. f^ ^ I ^ & s . IN ^3 i 1 1 ill w « O ^ O rfl ^ CD r sS.s 2.115 So .a ■ ■2 I o £. i Q ■^ -0 K 6D I ^ B ' 13 eg 09 ■ .s a " a g + » BOOK VI DIVERSE RULES RULE OF THREE 376. A rule of three is a rule by which a problem may be solved, that is, an unknown value determined by means of several pro- portions (325). 377. The rule of three is simple when it consists in the deter- mination of the fourth term of a proportion, of which three terms are known (343). If, on the contrary, the three terms are not given directly, but have to be determined by applying the rule of three several times, the rule is called the compound rule of three. 378. Any problem, which may be solved by the rule of three, contains two known quantities of the same kind, and two other quantities of the same kind only one of which is known. A ratio can exist only between like quantities; and according as the ratio of the like quantities, one of which is unknown, is the direct or inverse of that of the other two (326), the rule of three is said to be direct or inverse. 379. Simple direct rule of three. If 5 workmen construct 25 meters of road, how many meters would 7 workmen construct in the same time ? It is evident that the number of meters is directly propor- tional to the number of workmen which do the work; therefore, designating the number of meters constructed by 7 men, by x, we have (326): 7 X 25 5 : 7 = 25 : a;, from which x = — z — = 35 meters. 5 This problem, or any problem involving the simple or composite rule of three, may be solved by the method of reduction to unity, using proportions. Thus, if 5 workmen do 25 meters of road, one man will do-^ = 5 meters in the same time, and 7 men will 5 do seven times as much, or 127 128 ARITHMETIC 7 X 25 „_ — = — = 35 meters. 5 380. The simple inverse rule of three (378). 1st Problem. If it takes 20 hours for A. 'men to do a certain piece of work, how long would it take 10 men to do the same work ? The number of hours being inversely proportional to the num- ber of men, and letting x be the number of hours it takes 10 men to do it, we have 20 X 4 4 : 10 = a; : 20, from which x = — -- — = 8 hours Method of reduction to unity. Since it takes 4 men 20 hours, it would take one man 4 X 20 hours, and 10 men 20 X 4 _ , — zr^ — = 8 hours. 3 2d Problem. How many yards of cloth j of a yard wide will it 7 take to line a piece 45 yards long and ^ of a yard wide ? The lengths being inversely proportional to the widths, we have: 3 7 from which 6 ^^ = ^' 7 45 X - 6 45x7x4^„ — 3 — ^ — 3x6 — =5x7x2 = 70 yards. 4 Method of reduction to unity. 45 yards of cloth ^ of a yard ide is equivalent t yard wide would be 7 ^ ■? wide IS equivalent to 45 X g yards, one yard wide, and j of a 45 x^ — X — = 70 yards long. 4 381. Examples of the compound rule of three (377). 1st Example. 2 men working 3 hours per day for 5 days, con- struct 90 yards of road; how many yards would 3 men working 7 hours per day for 2 days construct ? RULE OF THREE 129 Solution by proportions. Writing the knowns and the unknowns as follows : 2 men 3 hr. 5 da. 90 yds. 3 men 7 hr. 2 da. x yds. the problem may be solved by a series of simple rules of three or proportions; but it is more convenient to reduce the problem to a simple rule of three as follows: 2 men, working 3 hours a day, do as much as 2 X 3 men work- ing one hour, and 2X3 men working 1 hour a day for 5 days, do as much as 2 X 3 X 5 men working one hour. Likewise, 3 men working 7 hours per day for 2 days do as much work as 3 X 7 X 2 men working one hour. The problem is now: If 2 X 3 X 5 men do 90 yards of construction, how many yards will 3X7X2 men do in the same time? This may be solved by a simple direct proportion, thus (379): 2x3x5:3x7x2 = 90: a:, from which 90 X 3 X 7 X 2 2x3x6 18 X 7 = 126 yards. The terms should be written with all their factors so as to facili- tate cancellation. Method of reduction to unity. Since 2 men, working 3 hours a day for 5 days, have made 90 yards, 1 man, working 1 hour 90 a day for 1 day, would make - yards, and therefore, ^ X o X o 3 men working 7 hours a day for 2 days would make 90 X 3 X 7 X 2 2x3x5 = 126 yards. 2d Example. 2 men, working 3 hours a day for 5 days, make 90 yards of road ; how nmny days would 3 men, working 7 hours a day, have to work in order to do the same amount f Solution by proportions. 2 men 3 hr. 5 da. 90 yds. 3 men 7 hr. x da. 90 yds. Proceeding as in the 1st example, the above is reduced to the simple inverse proportion: 130 ARITHMETIC 2X3 men having taken 5 days to do a certain piece of work, how many days will it take 3X7 men to do the same work? We have (380): Sx2x3 , (2 X 3) • (3 X 7) = a; : 5, from which x = — g ^ ^ days. Method of reduction to unity. From the problem it follows that 1 man working 1 hour a day would take 5X2X3 days to do 90 yards of construction; therefore 3 men working 7 hours a day would take 6x2x3, 3d Example. If the men working 7 hours a day were obliged to make 126 yards of road instead of 90 yards, for instance, 2 men 3 hr. 5 da. 90 yds. 3 men 7 hr. x da. 126 yds. the operation would have been divided into two parts, first find- ing the number of days it would take them to do 90 yards as was done above; and then we have: A certain number of men working — = — days construct 90 yards of road; how many days will it take them to make 126 yards ? This is again a simple proportion (379); 5x2x3 90 : 126 3x7 5 X 2 X 3 X 126 126 14 . , ^= 3x7x90 =r^^T=^^^^'- Method of reduction to unity. 1 man working 1 hour a day 5X2x3 would take ^ days to do 1 yard of work; therefore 3 men working 7 hours a day would make 126 yards in 5 X 2 X 3 X 126 - , 3 X 7 X 90 = ^ ^^y'- 382. A general rule for solving a simple or a compound rule of three (379. 380, 381). The quantities which enter into the problem are like in pairs, and the ratio of the unknown to the known quantity of the same kind is equal to the product of the direct or inverse ratios of the others; thus, in the 3d problem (381) the ratios of the number of INTEREST RULES 131 workmen and the number of hours being inverse to that of the number of days, and that of the number of yards being direct, we have: X 2 3 126 . , . , „ 2 X 3 X 126 . ^ 5 = 3^7^-90-'^^°°^ ^^^"^^-^>^ 3x7x90 = ^ '^^^^ INTEREST RULES 383. Interest is the sum paid for the use of money. The sum which draws the interest is called the capital or principal. 384.. The interest on $100 for one year is the rate of interest. Thus, when $100 brings $5 per year, the rate of interest is 5 per cent, which is written 5%. Legal interest is interest according to a rate fixed by law. This differs in different states. If no rate is specified, legal rate is understood. 385. Interest is said to be simple when the principal remains the same throughout the duration of the loan. 386. Interest is compound when the interest is added to the principal at the end of each year or other fixed period and bears interest with it. Savings banks furnish an example of this kind of interest. 387. The solution of the various problems in interest depends upon the two following principles: 1st. The simple interest on a principal is proportional to the time for which the loan is made (326). 2d. Two principals loaned at the same rate, for the same time, are directly proportional to their interests (326). 388. Problems in simple interest. Let C be the capital loaned, T the duration of the loan in years, / the simple interest on the principal C for the time T, and i the rate of interest; then from 1st, it follows that i X T is equal to the simple interest on $100 for the time T, and from 2d we have the proportion I :iXT = C : 100; from which : . , , C xixT ^''- ^ = 100 ' . ^ 7 X 100 ,,, .. / X 100 , _, / X 100 . 2d. ^x^=— ^, or4th, * = -^3^ andr = -^3^; 3d. C=l^ I X T 132 ARITHMETIC Time must always be expressed in years (229). Thus, 5 months c 105 = r = -K ' and 125 days = ^ = 360' ^^*^ *^^ ^^^ °^ *^^ P™" portion, given above, or the 4 equations, all problems in simple interest may be solved (391, 395). Problem 1. What is the interest, I, on $45,000, loaned for 4 years at 5%? Substituting in formula 1, I = 45,000^ X 5 X 4 ^ ^g^Q^^^^ which shows that in order to find the interest on a principal loaned for a certain number of years, multiply the principal by the rate and by the number of years, and divide the product by 100. After 4 years, the amount is C + / = 45,000 + 9000 = 154,000.00. The value of / and of / + C may be found directly by the method of reducing to unity. Thus, in one year $100 would bear $5.00 interest, and $1.00 would bear $0.05; in 4 years, $1.00 would bear $0.05 X 4, and at the end of this time the amount would be (1 + 0.05 X 4) dollars; thus, 7= 45,000 X 0.05 X 4 = $9000.00 C + 7 = 45,000 (1 + 0.05 X 4) = $54,000.00. Problem 2. What is the interest, I, on $45,000, loaned at 5% for 4 years and 3 months ? 51 4 years and 3 months are 12X4 + 3 = 51 months or j^ years; substituting in formula 1 (388): 61 45,000 X 5 X q-s . (, „ „ „ „ „ T 12 45,000 X 5 X 51 «Q^„„KA ^ = 100 = 100 X 12 = $9562.50. Thus, to obtain the interest on a principal loaned for a certain number of months, multiply the principal by the rate and by the number of months, and divide the product by 1200. At the end of 4 years 3 months the amount is C + I = 45,000 + 9562.50 = $54,562.50. INTEREST RULES 133 Proceeding as in Problem 1, the method of reducing to unity gives : 51 12 I = 46,000 X 0.05 X T^ = $9562.50. C + 1 = 45,000^1 + 0.05 + j^) = $54,562.60. Problem 3. What is the interest, I, on $45,000, loaned at 5% for 48 days ? One day is equal to ^^ of a year, and therefore 48 days is 48 equal to ^^ years; and substituting in formula 1 (388): ot)U 48 4o,uw X o X ygQ ^g^Q^Q X 6 X 48 45,000 x 48 _ ^ - Too ~ SpOO ~ 7200 ~ *"^""- „, . 45,000 X 5 X 48 , ,, , . , , , The expression, WTM) ' ^'^°^^ *"^* ^^ order to cal- culate the interest on a loaned principal for a certain number of days, multiply the principal by the rate and by the number of days, and divide the product by 36,000. The expression, — ' „„„ — , shows that when the rate is 5% the interest may be obtained by multiplying the principal by the number of days and dividing the product by 7200. At the end of 48 days the amount is : 45,000 +7= 45,000 + 300 = $45,300.00. The method of reduction to unity (Problems 1 and 2) gives: 48 I = 45,000 X 0.05 X K777. = $300.00. C + 7 = 45,000^1 + 0.05 X^) = $46,300.00. ^. ^ 36,000 In commercial calculations of mterest, the quotient, — g— ; 134 ARITHMETIC obtained in dividing 36,000 by the rate, is called the constant divisor. If the rate were 6%, ^ 45,000 X 6 X 48 45,000 X 48 .„ „. „„ ^ = gpOO = —6000 = ^'^^"•""' which shows that the interest is obtained by substituting the constant divisor, 6000, for 7200. Table of Constant Divisors for the Rates in Most Common Use Eate. DiTISOE. Bate. DiVISOB. Bate. 5.50 Divisor. Eate. Divisor. Eate. DIVISOK. 1 36,000 3.25 11,077 6,545 7.75 4,645 10 3,600 1.25 28,800 3.50 10,286 5.75 6,261 8 4,500 10.25 3,512 1.50 24,000 3.75 9,600 6 6,000 8.25 4,364 10.50 3,429 1.75 20,571 4 9,000 6.25 5,760 8.50 4,235 10.75 3,349 2 18,000 4.25 8,470 6.50 5,538 8.75 4,114 11 3,273 2.25 16,000 4.50 8,000 6.75 5,333 9. 4,000 11.25 3,200 2.50 14,400 4.75 7,579 7 5,143 9,25 3,892 U.50 3,130 2.75 13,091 o 7,200 7.25 4,965 9.50 3,789 11.76 3,064 3 12,000 5.25 6,857 7.50 4,800 9.75 3,692 12 3,000 In obtaining the interest, instead of dividing the product of the principal and the number of days by the constant divisor, this product may be multiplied by the reciprocal of the constant divisor, which is called the constant multiplier. Thus, in the preceding example: 7 = 45,000 X 48 = 45,000 X 48 X 6000 45,000 X 48 X 0.00016666 6000 . = $360.00. This method has been and is still used to a certain extent, but the best method is that of aliquot parts, which involves the following steps: 1st. Take one hundredth of the principal, which is equal to the interest at 6% for 60 days. The interest on $2400.00 at 6% for 60 days is 2400 X 60 / = 6000 2400 Too" = $24.00. INTEREST RULES 136 2d. By the method of aHquot parts, find the interest for the given number of days, knowing it for 60 days. 3d. From this interest found for 6% subtract 1111 6' 4' 3' 2' according as the given rate is 5, 4.5, 4, 3. Thus, to obtain the interest on $2400 for 175 days at 4.5%: Interest at 6% for 60 days = $24.00 " 6% " 60 " = 24.00 " 6% " 30 " = 12.00 " 6% " 20 " = 8.00 " 6% " _5 " = 2.00 175 " $70.00 One fourth of 70 17.50 The required interest . . . $52.50 The quotient obtained in dividing 360 by the rate -^ = 60 is called the base, and expresses the number of days which the principal must be loaned in order that the interest equal one hundredth of the principal. For the following rates: 6, 5, 4.5, 4, 3, it is 60, 72, 80, 90, 120. Instead of commencing with the base, 60, as above, which has the advantage of having a large number of aliquot parts, the base which corresponds to the rate given in the problem may be used. Thus, find the interest on $2400 at 4.5% for 175 days. Interest for 80 $24.00 "80 24.00 " "10 3.00 "5 1.50 Required interest $52.50 JS6 ARITHMETIC Problem 4. If the interest on $45,000, flaced for 4 years 3 months, is 19562.50, what is the rate f Substituting in formula (2) (388): . _ 9562.50 X 100 _ 9562.50 X 100 X 12 ^ *~ 51 ~ 45,000 X 51 ^°' 45,000 X ^ Using the method of reduction to unity, the interest on $1.00 for 4 years 3 months being ^ dollars, that on $100.00 for .1. ' .■ ij u 9562.50 X 100 , , , the same time would be . _ „„„ > and tor 1 year 4o,UUU 9562.50 X 100 12 ^^ .„ , . , . .„ 45,000 ^ 5l = ^^-00' ^^'""^ '' ^^- Problem 5. For how long will the principal, $45,000, have to be loaned at 5% in order that the interest be $9562.50 ? Substituting in formula (2) (388): „ 9562.50 X 100. . ^_ An /ooos = 45 000 X 5 = ^-^^ ^^■' °^ ^ y'"^-' ^ ^°^- (^^^^• Problem 6. What principal loaned for 4 years 3 months at 5% will bring $9562.50 interest f Substituting in formula (3) (388): „ 9562.50 X 100 9562.50 X 100 X 12 ^..„„„„„ C = s^ = ^^^^ = $45,000.00. ^'^12 Problem 7. What principal must be placed at 5% to amount to $54,562.50 in 4 years 3 months ? In 4 years 3 months $1.00 would bring (formula 1): 51 Therefore the amount of $1.00 placed for 4 years 3 months is $1.2125, and the required principal is 5i^ = ,45,000.™. INTEREST RULES 137 3'89. Problems in compound interest (361, 365). Problem 1. What would be the amount of $45,000 loaned for 4 years at 5% compound interest f At the end of one year the amount of 11.00 would be $1.05, and that of $45,000, 45,000 X 1.05. This, taken as a new principal, at the end of the second year would give 45,000 X 1.05 X 1.05 = 45,000 X LO?. In like manner, at the end of the third year the amount would be 45,000 X 1.05 X 1.05 = 45,000 X 1.05 , and so on. From this it follows that the amount of a principal, at the end of a whole number of years at compound interest, is equal to the principal multiplied by the amount of $1.00 at the end of 1 year raised to a power the degree of which is equal to the number of years. Thus, at the end of 4 years the principal $45,000 would be 45,000 X 1.05 = 45,000 X 1.215506 = $54,697.77. If the rate-had been 4.5, for example, the number 1.05 would have been replaced by 1.045. The table given on the following pages contains, in column a, the successive powers of these numbers up to the 60th for the different rates of interest, that is, the successive amounts of $1.00 from 1 to 60 years at compound interest. To solve the foregoing problem, find the value of $1.00 at the end of 4 years at 5%, then multiply 45,000 by that number. Problem 2. What principal must be placed at compound inter- est of 5% for 4 years in order that the amount be $54,697.77? If $1.00 amounts to $1.05* or $1.215506 at the end of 4 years, then it would take as many dollars in the principal as 1.215506 is contained in the given amount, thus: In column b of the tables, the principals, for different amounts at different rates and covering a period of 60 years, are given. 138 ARITHMETIC Thus, in the above, the principal corresponding to 4 years and 5% is 0.822703. Therefore the required principal is 54,697.77 X 0.822703 = $45,000. Problem 3. What is the amount of $45,000 loaned at 5% compound interest for 4 years 3 months ? First find the amount at the end of 4 years as in Problem 1. Then find the simple interest at 5% for that amount, 54,697.77, taken as principal for 3 months (Problem 2, 388): 54,697.77 (l + 0.05 X ^W $55,381.49. Problem 4. What principal must be placed at 5% compound interest for 4 years 3 months to give $55,381.49 as the amount ? At the end of 4 years $1.00 becomes (1.05)*; and at the end of 4 3'ears 3 months $1.00 becomes L.05' i 1 + 0.05 X ^) = $1.2307. Therefore the principal is the quotient obtained in dividing the amount 55,381.49 by the value of $1.00 at the end of 4 years 3 months: 55,381.49 ^,.„„„ ^^230^ = $45,000. This problem may also be solved by using the table. Let x be the principal placed for 3 months which will give $1.00 as the amount : $1.00 = X ^1 + 0.05 X ^) = a; X 1.0125, 1 X — 1.0125 From the column h of the table, and corresponding to 5% and 4 years, the principal which will give $1.00 as amount is found, and then the principal for 4 years 3 months is 0.822703 X rrrrTrF' 1.0125 and the principal which will give $55,381.49 is: 55,381.49 X 0.822703 r:ol25 = 145,000. INTEREST RULES 139 Problem 5. How long must 145,000 be placed at 5% com- pound interest, in order to obtain an amount equal to $55,381.49 ? The problem consists in finding how long $1.00 would have to be placed in order to obtain the amount : 55,381.49 45,000 -551-2307. Calculating, as in Problem 1, the value of $1.00 at the end of the first, second, third, etc., years, it is found that the duration of the loan is between 4 and 5 years. This may also be taken directly from the tables, column a. At the end of 4 years $1.00 becomes $1.215506, and now it must be found how long it will take $1.215506 to bear 1.2307 - 1.215506 = $0.015194, which is done as in Problem 5 (363). The time is „ 0.015194 X 100 ___ o .u ^ = 1.215506 X 5 = ^-2^ ^"^^ °' 2 '"°^'^'- Therefore the total duration is 4 years 3 months. 390. Interest Tables. The following compound interest tables contain : 1st. Column a, the amount of $1.00 at the end of each year of the loan. Each value is equal to the value of $1.00 at the end of 1 year raised to a power with an exponent equal to the dura- tion of the loan. Thus, at the end of 4 years, at 5%, the value is $r05*= $1.215506 (Problem 1, 389). 2d. Column b, the principal which will produce an amount equal to $1.00 in 1, 2, 3, etc., years. For example, the principal which will produce an amount equal to $1.00 in 7 years, at 5%, is equal to =-, = 0.710681, that is, the value of $1.00 divided 1.05' by its value at the end of 1 year raised to the power the ex- ponent of which is equal to the number of years (Problem 2, 389). 3d. Column c, the amount at the end of each year where there is a yearly deposit of $1.00. It is to be noted that the amount at the end of 5 years, at 5%, is equal to the sum 5.801913 of the first 5 values in colunm a. 4th. Column d, the principal which will produce a yearly income of $1.00 per year payable during 1, 2, ... 60 years. 140 ARITHMETIC 3%. 3i%. 47 4S 49 50 51 52 53 64 55 56 57 58 59 60 1.03 1.060900 1.092727 1.125509 1.159274 1.194052 1.229874 1.266770 1.304773 1.343916 1.384234 1.425761 1.468534 1.512590 1.557967 1.604706 1.652848 1.702433 1.753506 1.806111 1.860295 1.916103 1.973587 2.032794 2.093778 2.156591 2.221289 2.287928 2.356566 2.427262 2.500080 2.575083 2.652335 2.731905 2.813862 2.898278 2.985227 3.074783 3.167027 3.262038 3.359899 3.460696 3.564517 3.671452 3.781596 3.895044 4.011895 4.132252 4.256219 4.383906 4.515423 4.650886 4.790412 4.934125 5.082149 5.234613 5.391651 5.553401 5.720003 0.970874 0.942596 0.915142 0.888487 0.862609 0.837484 0.813092 0.789409 0.766417 0.744094 0.722421 0.701380 0.680951 0.661118 0.641862 0.623167 0.605016 0.587395 0.570286 0.553676 0.537549 0.521893 0.506692 0.491934 0.477606 0.463695 0.450189 0.437077 0.424346 0.411987 0.399987 0.388337 0.377026 0.366045 0.355383 0.345032 0.334983 0.325226 0.315754 0.306557 0.297628 0.288959 0.280543 0.272372 0.264439 0.256737 0.249259 0.241999 0.234950 0.228107 0.221463 0.215013 0.208750 0.202670 0.196767 0.191036 0.185472 0.180070 0.174825 1.03 2.090900 3.183627 4.309136 5.468410 6.662462 7.892336 9.159106 10.463879 11.807796 13.192030 14.617790 16.086324 17.598914 19.156881 20.761588 22.414435 24.116868 25.870374 27.676486 29.536780 31.452884 33.426470 35.459264 37.553042 39.709634 41 .930923 44.218850 46.575416 49.002678 51.50276 54.07784 56.73018 59.46208 62.27594 65.17422 68.15945 71.23423 74.40126 77.66330 81.02320 84.48389 88.04841 91.71986 95.50146 99.39650 103.40840 107.54065 111.79687 116.18077 120.69620 125.34708 130.13749 135.07162 140.15377 145.38838 150.78003 156.33343 162.05344 5.891603 0.169733 167.94504 0.970874 1.913470 2.828611 3.717098 4.579707 5.417191 6.230283 7.019692 7.786109 8.530203 9.252624 9.954004 10.634955 11.296073 11.937935 12.561102 13.166119 13.753513 14.323799 14.877475 15.415024 15.936917 16.443608 16.935542 17.413148 17.876842 18.327032 18.764108 19.188455 19.600441 20.000429 20.388766 20.765792 21.131837 21.487220 21.832253 22.167235 22.492462 22.808215 23.114772 23.412400 23.701359 23.981902 24.254274 24.518713 24.775449 25.024708 25.266707 25.501657 25.729764 25.951227 26.166240 26.374990 26.577661 26.774428 26.965464 27.150936 27.331006 27.505831 27.675564 1.035 1.071225 1.108718 1.147523 1.187686 1.229255 1.272279 1.316809 1.362897 1.410599 1.459970 1.511069 1.563956 1.618695 1.675349 1.733986 1.794676 1.857489 1.922501 1.989789 2.059431 2.131512 2.206114 2.283328 2.363245 2.445959 2.531567 2.620172 2.711878 2.806794 2.905031 3.006708 3.111942 3.220860 3.333590 3.450266 3.571025 3.696011 3.825372 3.959260 4.097834 4.241258 4.389702 4.543342 4.702359 4.866941 5.037284 5.213589 5.396065 5.584927 5.780399 5.982713 6.192108 6.408832 6.633141 6.865301 7.105587 7.354282 7.631682 7.878091 0.966184 0.933511 0.901943 0.871442 0.841973 0.813501 0.785991 0.759412 0.733731 0.708919 0.684946 0.661783 0.639404 0.617782 0.596891 0.576706 0.557204 0.538361 0.520156 0.502566 0.485571 0.469151 0.453286 0.437957 0.423147 0.408838 0.395012 0.381654 0.368748 0.356278 0.344230 0.332590 0.321343 0.310476 0.299977 0.289833 0.280032 0.270562 0.261413 0.252573 0.244031 0.235779 0.227806 0.220102 0.212669 0.205468 0.198520 0.191807 0.185320 0.179053 0.172998 0.167148 0.161496 0.156035 0.160758 0.14.5660 0.140734 0.135975 0.131377 0.126934 1.035 2.106225 3.214943 4.362466 5.550152 6.779408 8.051687 9.368496 10.731393 12.141992 13.601962 15.113030 16.676986 18.295681 19.971030 21.706016 23.499691 26.357180 27.279682 29.269471 31.328902 33.460414 36.666528 37.949857 40.313102 42.769060 46.290627 47.910799 60.622677 63.429471 66.33450 59.34121 62.46315 65.67401 69.00760 72.45787 76.02890 79.72491 83.55028 87.50964 91.60737 95.84863 100.23833 . 104.78167 109.48403 114.35097 119.38826 124.60185 129.99791 136.58284 141.36324 147.34595 153.53806 169.94689 166.58003 173.44533 180.66092 187.90520 195.51688 203.39497 0.966184 1.899894 2.801637 3.673079 4.515052 5,328553 6.114544 6.873956 7.607687 8.316605 9.001651 9.663334 10.302739 10.920520 11.517411 12.094117 12.651321 13.189682 13.709837 14.212403 14.697974 16.167125 15.620411 16.058368 16.481615 16.890352 17.285365 17.667019 18.035767 18.392045 18.736276 19.068866 19.390208 19.700684 20.000661 20.290494 20.570526 20.841087 21,102500 21.365072 21.599104 21.834883 22.062689 22.282791 22.495450 22.700918 22.899438 23.091244 23.276566 23.455618 23.628616 23:795765 23.957260 24.113295 24.264053 24.409713 24.550448 24.686423 24.817800 24.944734 INTEREST RULES 141 ai B 4%. i 4i%. >< u b V d a - 5 > ^ C ; d ; 1 1.04 0.961539 1.04 0.961539 1 1.046 0.956938 1.045 0.956938 2 1.081600 0.924566 2.121600 1.886095 2 1.092026 0.915730 2.137026 1,872668 3 1.124864 0.888996 3.246464 2.775091 3 1.141166 0.876297 3.278191 2.748964 4 1.169859 0.854804 4.416323 3.629895 4 1.192519 0.838561 4.470710 3.687526 5 1.216663 0.821927 5.632976 4.451822 5 1.246182 0.802451 5.716892 4.389977 6 1.266319 0.790315 6.898294 5.242137 6 1.302260 0.767896 7.019152 6.157873 7 1.316932 0.759918 8.214226 6.002055 7 1.360862 0.734829 8.380014 6.892701 8 1.368569 0.730690 9.582795 6.732745 8 1 422101 0.703186 9.802114 6.695886 9 1.423312 0.702587 11.006107 7.435332 9 1.486095 0.672904 11.288209 7.268791 10 1.480244 0.675564 12.486351 8.110896 10 1.552969 0.643928 12.841 1Y9 7.912718 11 1.539454 0.649681 14.025806 8.760477 11 1.622853 0.616199 14.464032 8.528917 12 1.601032 0.624697 15.626838 9.385074 12 1.695881 0.589664 16.159913 9.118581 13 1.665074 0.600574 17.291911 9.985648 13 1.772196 0.664272 17.932109 9.682852 14 1.731676 0.577475 19.023688 10.563123 14 1.851945 0.539973 19.784054 10.222825 15 1.800944 0.555265 20.824631 11.118387 15 1.935282 0.616720 21.719337 10.739546 16 1.872981 0.533908 22.697512 11.662296 16 2.022370 0.494469 23.741707 11.234015 17 1.947900 0.613373 24.645413 12.166669 17 2.113377 0.473176 25.866084 11.707191 18- 2.026817 0.493628 26.671229 12.669297 18 2.208479 0.452800 28.063562 12.159992 19 2.106849 0.474642 28.778079 13.133939 19 2.307860 0.433302 30.371423 12.593294 20 2.191123 0.456387 30.969202 13.590326 20 2.411714 0.414643 32.783137 13.007937 21 2.278768 0.438834 33.247970 14.029160 21 2.620241 0.396787 35.303378 13.404724 22 2.369919 0.421955 35.617889 14.451116 22 2.633662 0.379701 37.937030 13.784425 23 2.464716 0.406726 38.082604 14.856842 23 2.762166 0.363350 40.689196 14.147776 24 2.563304 0.390122 40.645908 15.246963 24 2.876014 0.347704 43.665210 14.495478 26 2.665836 0.376117 43.311745 15.622080 25 3.005434 0.332731 46.670645 14.828209 26 2.772470 0.360689 46.084214 15.982769 26 3.140679 0.318403 49.711324 15.146611 27 2.883369 0.346817 48.967583 16.329586 27 3.282010 0.304691 62.993333 16.451303 28 2.998703 0.333478 51.966286 16.663063 28 3.429700 0.291571 56.423033 15.742874 29 3.118651 0.320651 56.084938 16.983716 29 3.584036 0.279015 60.007070 16,021889 30 3.243398 0.308319 58.328335 17.292033 30 3.745318 0.267000 63.762388 16.288889 31 3.373133 0.296460 61.70147 17.588494 31 3.913857 0.255602 67.66626 16,5443i91 32 3.508059 0.285058 65.20953 17.873552 32 4.089981 0.244500 71.75623 16,788891 33 3.648381 0.274094 68.85791 18.147646 33 4.274030 0.233971 76.03026 17,022862 34 3.794316 0.263552 72.65223 18.411198 34 4.466362 0.223896 80.49662 17,246758 35 3.946089 0.253416 76.59831 18.664613 35 4.667348 0.214264 86.16397 17,461012 36 4.103933 0.243669 80.70225 18.908282 36 4.877378 0.205028 90.04134 17.666041 37 4.268090 0.234297 84.97034 19.142579 37 6.096860 0.196199 96.13821 17.862240 38 4.438813 0.225286 89.40916 19.367864 38 5.326219 0.187750 100.46442 18.049990 39 4.616366 0.216621 94.02552 19.584486 39 5.565899 0.179666 106.03032 18.229656 40 4.801021 0.208289 98.82654 19.792774 40 5.816365 0.171929 111.84669 18.401684 41 4.993061 0.200278 103.81960 19.993052 41 6.078101 0.164525 117.92479 18.666110 42 5.192784 0.192575 109.01238 20.185627 42 6.351615 0.157440 124.27640 18.723550 43 5.400496 0.185168 114.41288 20.370795 43 6.637438 0.150661 130.91384 18.874210 44 6.616515 0.178046 120.02939 20.548841 44 6.936123 0.144173 137.84997 19.018383 45 5.841176 0.171198 125.87067 20.720040 45 7.248248 0.137964 146.09821 19.156347 46 6.074823 0.164614 131.94539 20.884654 46 7.574420 0.132023 162.67263 19.288371 47 6.317816 0.158283 138.26321 21.042936 47 7.915268 0.126338 160.58790 19.414709 48 6.570528 0.152196 144.83373 21.195131 48 8.271456 0.120898 168.85936 19.535607 49 6.833349 0.146341 151.66708 21.341472 49 8.643671 0.115692 177.60303 19.651298 50 7.106683 0.140713 158.77377 21.482185 50 9.032636 0.110710 186.53667 19.762008 51 7.390951 0.136301 166.16472 21.617486 51 9.439105 0.105942 195.97477 19.867950 52 7.686689 0,130097 173.85131 21.747582 52 9.863865 0.101380 205.83863 19.969330 S3 7.994062 0.126093 181.84536 21.872675 53 10.307739 0.097015 216.14637 20.066345 54 8.313814 0.120282 190.15917 21.992957 54 10.771587 0.092837 226.91796 20.159182 55 8.646367 0.115656 198.80554 22.108612 55 11.256308 0.088839 238.17427 20.248021 56 8.992222 0.111207 207.79776 22.219819 56 11.762842 0.085014 249.9.3711 20.333034 67 9.361910 0.106930 217.14967 22.326749 57 12.292170 0.081353 262.22928 20.414387 58 9.726987 0.102817 226.87566 22.429567 68 12.845318 0.077849 276.07460 20.492236 59 10.115026 0.098863 236.99069 22.528430 59 13.423357 0.074497 288,49796 20.566733 60 10.519627 0.095060 247.61031 22.623490 60 14.027408 0.071289 302.52536 20.638022 , 142 ARITHMETIC 5%. 1.05 1.102500 1.157625 1.215506 1.276282 1.340096 1.407100 1.477455 1.551328 1.628895 1.710339 1.795856 1.885649 1.979932 2.0/8928 2.182875 2.292018 2.406619 2.526950 2.653298 2.785963 2.925261 3.071524 3.225100 3.386355 3.555673 3.733456 3.920129 4.116136 4.321942 4.538039 4.764941 5.003189 5.253348 5.516015 5.791816 6.081407 6.385477 6.704751 7.039989 7.391988 7.761588 8.149567 8.557150 8.985008 9.434258 9.905971 10.401270 10.921333 11.467400 12.040770 12.642808 13.274949 13.938696 14.635631 15.367412 16.135783 16.942572 17.789701 18.679186 0.952381 0.907030 0.863838 0.822703 0.783526 0.746215 0.710881 0.676839 0.644609 0.613913 0.584679 0.556837 0.530321 0.505068 0.481017 0.458112 0.436297 0.415521 0.395734 0.376890 0.358942 0.341850 0.325571 0.310068 0.295303 0.281241 0.267848 0.255094 0.242946 0.231377 0.220360 0.209866 0.199873 0.190355 0.181290 0.172657 0.164436 0.156605 0.149148 0.142046 0.135282 0.128840 0.122704 0.116861 0.111297 0.105997 0.100949 0.096142 0.091564 0.087204 0.083051 0.079096 0.075330 0.071743 0.068326 0.065073 0.061974 0.059023 0.056212 0.053536 1.05 2.152500 3.310125 4.525631 5.801913 7.142008 8.549109 10.026564 11.577893 13.206787 14.917127 16.712983 18.598632 20.578564 22.657492 24.840366 27.132385 29.539004 32.065954 34.719252 37.505214 40.430475 43.501999 46.727099 50.113454 53.669126 57.402583 61.322712 65.438848 69.760790 74.29883 79.06377 84.06696 89.32031 94.83632 100.62814 106.70955 113.09502 119.79977 126.83976 134.23175 141.99334 150.14301 158.70016 167.68516 177.11942 187.02539 197.42666 208.34800 219.81540 231.85617 244.49897 257.77392 271.71262 286.34825 301.71566 317.85144 334.79402 352.58372 371.26290 0.952381 1.859410 2.723248 3.545951 4.329477 5.075692 5.786373 6.463213 7.107822 7.721735 8.306414 8.863252 9.393573 9.898641 10.379658 10.837770 11.274066 11.689587 12.085321 12.462210 12.821153 13.163003 13.488574 13.798642 14.093945 14.375185 14.643034 14.898127 15.141074 15.372451 15.592811 15.802677 16.002549 16.192904 16.374194 16.546852 16.711287 16.867893 17.017041 17.159086 17.294368 17.423208 17.545912 17.662773 17.774070 17.880067 17.981016 18.077158 18.168722 18.255926 18.338977 18.418073 18.493403 18.565146 18.633472 18.698545 18.760519 18.819542 18.875754 18.929290 6%. 1.06 1.123600 1.191016 1.262477 1.338226 1.418519 1.503630 1.593848 1.689479 1.790848 1.898299 2.012196 2.132928 2.260904 2.396558 2.540352 2.692773 2.854339 3.025600 3.207135 3.399564 3.603537 3.819750 4.048935 4.291871 4.549383 4.822346 5.111687 5.418388 5.743491 6.088101 6.453387 6.840590 7.251025 7.686087 8.147252 8.636087 9.154252 9.703507 10.285718 10.902861 11.557033 12.250455 44 12.985482 45 13.764611 14.590487 15.465917 16.,393872 17.377504 18.420154 19.525364 20.696885 21.938698 23.255020 24.650322 26.129341 27.697101 29.358927 31.120463 32.987691 0.943396 0.889996 0.839619 0.792094 0.747258 0.704961 0.665067 0.627412 0.591899 0.558395 0.S26788 0.496969 0.468839 0.442301 0.417265 0.393646 0.371364 0.350344 0.330513 0.311805 0.294155 0.277505 0.261797 0.246979 0.232999 0.219810 0.207368 0.195630 0.184557 0.174110 0.164255 0.154957 0.146186 0.137912 0.130105 0.122741 0.115793 0.109239 0.103056 0.097222 0.091719 0.086527 0.081630 0.077009 0.072650 0.068538 0.064658 0.060998 0.057546 0.054288 0.051215 0.048316 0.045582 0.043002 0.040567 0.038271 0.036105 0.034061 0.032133 0.030314 1.06 2.x83600 3.3V 4616 4.637093 5.975319 7.393838 8.897468 10.491316 12.180795 13.971643 15.869941 17.882138 20.015066 22.275970 24.672528 27.212880 29.905653 32.759992 35.785591 38.992727 42.392290 45.995828 49.815577 53.864512 68.156383 62.705766 67.528112 72.639798 78.058186 83.801677 89.88978 96.34317 103.18376 110.43478 118.12087 126.26812 134.90421 144.05846 153.76197 164.04768 174.95055 186.50758 198.75803 211.74351 225.50813 240.09861 256.56463 271.95840 289.33591 307.75606 327.28142 347.97831 369.91701 393.17203 417.82235 443.96169 471.64879 501.00772 532.12818 566.11587 0.943396 1.833393 2.673012 3.465106 4.212364 4.917324 5.582381 6.209794 6.801692 7.360087 7.886875 8.383844 8.852683 9.294984 9.712249 10.106895 10.477260 10.827604 11.158117 11.469921 11.764077 12.041582 12.303379 12.650358 12.783366 13.003166 13.210634 13.406164 13.590721 13.764831 13.929086 14.084043 14.230230 14.368141 14.498246 14.620987 14.736780 14.846019 14.949075 15.046297 16.138016 16.224543 16.306173 16.383182 16.455832 15.624370 15.589028 15.650027 15.707572 15.761861 15.813076 15.861393 15.906974 15.949976 15.990543 16.028814 16.064919 16.098980 16.131113 16.161428 BOOK VII LOGARITHMS 391. Definition. When two progressions, ••• ll- T7-- I- 1=1=3:9:27:81... 8--6--4--2-0-2-4- 6- S.-- one, geometrical and containing the term 1 ; and the other arithmeti- cal and containing the term 0, are written one beneath the other so that the terms and 1 come in the same column (332 and 341), then each term of the arithmetical progression is the logarithm of the corresponding term of the geometrical progression. Thus the logarithm of 27, which is written log 27, is equal to 6 or log 27 = 6. 392. The multiplier of the geometrical progression is the base of the system of logarithms. 393. Instead of considering logarithms as the terms of a progression, they may be considered as degrees of a power of a constant number. This constant number is the base of the system, and any power of this base has the degree of the power for its logarithm. Thus, 3== = 9, 3' = 27, 3" = 1, 3-^ = ^ = ^ (305), have respectively 2, 3, 0, and — 2 for logarithms in the system whose base is 3. 394. Common logarithms. The base of this system is 10. The system was first published by Henry Briggs, and is sometimes called the Briggs system. In this system the two progressions of (391) are replaced by I — : ^— : J- : — : 1 : 10 : 100 : 1000 : 10,000 : 100,000 . • . 10,000 1000 100 10 ' ' ... -4. -3-2.-1-0. 1- 2- 3- 4. 5..- Considering the logarithms as exponents as in (393), we have .••10-^ 10-^ 10-' 10-1 10° 10' 10' 10= 10* 10'. .. which means, according to the definition (391), 143 144 ARITHMETIC log 1 = ; log 10 = 1 ; log 100 = 2 ; log 1000 = 3, etc. log^ = -i; logiJo = -2; l°gOT = -3^«t<'- 395. Hoiv the two fundamental progressions can give the loga- rithms of all the numbers. This series of powers infinitely prolonged in both directions, or the two progressions continued in the same manner, give only the numbers which have whole, positive, or negative numbers for logarithms ; but as many geometrical means may be inserted between the terms of the geometrical progression as desired, and in this manner, by inserting an equal number of arithmetical means between the terms of the arithmetical progression, the terms of the new arithmetical progression are the logarithms of the corresponding terms of the geometrical progression. Thus the logarithms of any number may be foxmd (263 and 273). Likewise, numbers, which differ from one another by an in- finitely small amount, may be taken as exponents in the preced- ing series, and the successive powers will differ from one another also by an infinitely small amount. Thus it is seen that any given number may be a term of the geometrical progression or one of the powers in the series given above, and that its logarithm is the corresponding term of the arithmetical progression, or the exponent of the power. Like- wise any given number may be a term of the arithmetical series or an exponent of a power, and is the logarithm of the corre- sponding term of the geometrical progression or of the power. Thus any positive number has a logarithm, and any number, positive or negative, is the logarithm of a positive number. It is evident that a table cannot be constructed which contains all the numbers, neither as numbers nor as logarithms, but there are tables which contain enough so that the differences between the successive numbers are so small that the values obtained may be considered exact. 396. The properties of a system of logarithms. The properties given below for the common system hold true for any system when the base of the given system is substituted for the base 10. Considering the two progressions or the powers of the base (394), we have: 1st. The logarithm of the base 10 is unity. 2d, The logarithm of unity is zero. LOGARITHMS 145 3d. The logarithm of a number greater than unity is positive. 4th. The logarithm of a number less than imity is negative. 5th. A negative number has no logarithm. 6th. The logarithm of the product of several factors, 10-^ = Jqq, 10' = 10, and 10* = 10,000, is equal to the sum, -2+1 + 4 = 3, of the logarithms of the factors : log (10-^X101X100 = log 10-^+i + ^ = log 10'= -2 + 1+4 = 3 (296). The logarithm 3 corresponds to 10' = 1000, that is, 1000 is the product of the factors — , 10 and 10,000. Thus, multiplication is accomplished by aid of addition. 7th. The logarithm of a power, (10^', of a number, 10^ = 100, is equal to the logarithm 2 of the number multiplied by the degree 3 of the power: log (10^)' = log 102 >< 3 = 2 X 3 = 6. (297) The logarithm 6 corresponds to 10° = 1,000,000, that is, 100' = 1,000,000. Therefore a number mxiy be raised to any power by a simple multiplication. 8th. The logarithm of the quotient obtained by dividing one number, 10= = 100,000, by another, 10^ = 100, is the logarithm 5 of the dividend less the logarithm 2 of the divisor: 10' log JqJ = log 10^-2 =5-2 = 3. (305) 3 being the logarithm of 1000, 1000 = ' , and it is seen that a division may be performed by means of a siMr action. 9th. The logarithm of a root of a number, 10°, is equal to the logarithm 6 of the number divided by the index 2 of the root: log VlO° = log 10'^ = log 10' = 5 = 3. (306) The logarithm 3 corresponds to 1000, that is. V1,000,000 = 1000. Therefore roots may be extracted by means of a simple division. 10th. According as a number lies between 1 and 10, 10 and 100, 100 and 1000, etc., its logarithm lies respectively between 146 ARITHMETIC and 1, 1 and 2, 2 and 3, etc.; from which it follows that since the logarithms are expressed in decimals, the whole part of the log- arithm of a whole number or a decimal number greater than unity, contains as many units less one as there are figures in the whole part of the given number. Thus the whole part is 3 for the num- ber 4725, and 2 for the number 827.34, Likewise, for a number lying between 1 and 0.1, 0.1 and 0.01, 0.01 and 0.001, etc., whose logarithm hes between and — 1, — 1 and — 2, — 2 and — 3, etc., the whole part of a negative logarithm of a decimal number less than unity, contains as many units as there are ciphers between the decimal point and the first significative figure in the given number. Thus the whole part is for the number 0.236 and — 2 for the number 0.00326. 397. The whole part of a positive or negative logarithm is called the characteristic, and the decimal part is called the man- tissa. 398. The logarithm of a number multiplied or divided by a power of 10. From (396) it follows that knowing the logarithm of a number, in order to find the logarithm of a product or quo- tient of the given number and unity followed by several ciphers, it suffices to increase or decrease the given logarithm by as many units as there are ciphers at the right of the 1. Thus, having log 68 = 1.8325089, we have log 6800 = 3.8325089, and having log 5657 = 3.7525862, we have log 5.657 = 0.7525862. In fact (396, 6t'hand8th): log (68 X 100) = log 68 + log 100 = log 68 + 2, log g^ = log 5657 - log 1000 = log 5657 - 3. Thus it is seen that when the logarithm is increased or dimin- ished by one or several units, the result is the logarithm of the product or the quotient of the given number and a power of 10 of a degree equal to the number of units by which the given logarithm has been increased or diminished. It is also seen that the logarithms, of the products or quotients of a certain number and the different powers of 10, differ only in the characteristic, which is increased or decreased by as many LOGARITHMS 147 units as there are units in the exponents of the powers of 10; the mantissa remains the same. 399. From what was said in (398) it follows that in order to determine the logarithm of a decimal number, neglect the decimal point and take the logarithm of the number, and subtract as many units from characteristic as there are decimal figures in 1827 the given number. Thus, having 18.27 = -^ (396, 8th), we have: log 18.27 = log 1827 - 2 = 3.2617385 - 2 = 1.2617385. 826 Likewise, having 0.826 = tt^' we have log 0.826 = log 826 - 3 = 2.91698005 - 3. 400. Logarithm of which the characteristic alone is negative. The logarithm of 826 being less than 3, it is seen, as was shown in (396), that the logarithm of 0.826, and in general of any number less than one, is negative. To express the value of the logarithm of 0.826, subtract 2.91698005 from 3 and place the negative sign — before the result. Thus: log 0.826 = - (3 - 2.91698005) = - 0.08301995. It is convenient not to have the mantissa negative (405). In order to obtain this, subtract only the characteristics 2 and 3, and take 1 for the characteristic and write the negative sign above it to indicate that it alone is negative. Thus : log 0.826 = 1.91698005. Likewise, log 0.0826 = 2.91698005, and log 0.00826 = 3.91698005. Thus the number of negative units in the characteristic is equal to the order of the first significative figure after the decimal point. 401. The complement of a positive number is that number which, if added to the given number, would give a whole number equal to unity followed by as many ciphers as there are figures in the whole part of the given number. Thus we have : c* 375.8762 = 1000 - 375.8762 = 624.1238. The complement of a positive number is easily obtained: subtract each of the significative figures except the last from 9, it is written thus: 148 ARITHMETIC and the last from 10, and place as many ciphers at the right of the number obtained as there are at the right of the given num- ^^^'- c* 587,300 = 412,700. As the whole part of a logarithm generally does not contain more than one figure, the complement of a positive logarithm is the result obtained in subtracting the logarithm from 10. Thus, c* log 826 = 10 - 2.91698005 = 7.08301995. Since it is so easy to obtain the complement, in operations where there is a logarithm to be subtracted, add it to its com- plement and subtract 10 from the result. Thus: 127 X 39 Having — — , instead of writing 8^D 1 97 V ^Q log g^ = log 127 + log 39 - log 826 = 2.10380372 + 1.59106461 - 2.91698005 = 0.77788828 log 127 = 2.10380372 log 39 = 1.59106461 c* log 826 = 7.08301995 0.77788828 The required result is the number 5.9964, corresponding to the logarithm 0.77788828 (see Rule 31). 402. Logarithmic tables. There are many logarithmic tables. The smaller ones give the logarithms of all the whole numbers up to 10,000; the larger ones up to 108,000. Often the char- acteristics are omitted, as they are easily supplied (397, 10th). The logarithms of the numbers between 1 and 10, 10 and 100, etc., being incommensurable, it is impossible to put their exact values in the tables. In Callet's tables the values are given to 8 decimal places for the whole numbers less than 1200 and those between 100,000 and 108,000, and to 7 decimal places for the numbers between 1200 and 100,000 (176). The tables by Jerome Lalande give the logarithms of all the whole numbers up to 10,000, correct to 5 decimal places. M. Marie has carried this table to 8 decimals for the numbers up to 990 and from there to 10,000 to 7 places. The tables have the numbers in the first column, the logarithms in the second, and the difference of the consecutive logarithms in the third. Supposing that we have a large table of logarithms at our LOGARITHMS 149 disposal^ that of Lalande for example, we will solve the follow- ing problems: 403. Problem 1. Find the logarithm of a given number : 1st. Of a whole number, 847, which may be found in the table, that is less than 10,000. Looking in the first column, the num- ber 847 is foimd; then in the same horizontal line in the second column will be found the logarithm 292,788,341. 2d. Of a whole number, 487,346, which is not found in the table. Separate on the right of the number just enough decimal figures so that the part on the left will be the largest possible number less than 10,000, the upper limit of the table. Thus, having 487,346 = 4873.46 X 100, we have (398 and 399): log 487,346 = log 4873.46 + log 100 = log 4873.46 + 2, which reduces to finding the logarithm of 4873.46. The number 4873.46 lies between 4873 and 4874, and therefore its logarithm hes between the tabular values 3.6877964 and 3.6878855. To obtain the quantity x which must be added to the log 4873 in order to get that of 4873.46, take the difference 0.0000891 be- tween the logarithms of 4873 and 4874, as found in the third column; this difference represents a difference of unity in the numbers; therefore for the difference 4873.46 — 4873 = 0.46, as- suming that the differences of the logarithms are proportional to the differences of the numbers, for such small values, we have X = 0.0000891 X 0.46 = 0.0000410. Therefore log 4873.46 = 3.677964 + 0.0000410 = 3.6878374, and log 487346 = 5.6878374. In this manner the logarithm of any number may be obtained. Callet's table gives, besides the differences, the nearest approx- imate values of the products of this difference and the first 9 multiples of 0.1, retaining 7 decimals, which greatly ^ shortens the calculation of x. Thus, to obtain the product of 891 ten millionths and 0.46, since 891 X 0.46 = 891 X 0.4 + 891 X 0.06 (33), taking 356 ten millionths in the column under 891 and at the right of 4 as the product of 891 and 0.4, and then 535 ten millionths opposite 6 as the product, of 891 and 0.6 or 54 ten millionths as the product of 891 and 0.06, x = 0.0000356 + 0.0000054 = 0.0000410. The calculations for the preceding example are written as follows: 1 89 2 178 3 267 4 356 5 445 6 535 7 624 8 713 9 802 150 ARITHMETIC Number 487,346 log 4873 = 3.6877964 for 0.4 356 for 0.06 54 log 4873.46 = 3.6878374 log 487 346 = 5.6878374 Assuming proportionality between the increments of the num- bers and the logarithms does not permit of the use of more than two decimals, and even these two are not exact. 7 3d. Of a fraction -■ According to (396, 8th), we have: log ^ = log 7 - log 4 = 0.84509804 - 0.60205999 = 0.24303805. If the fraction was less than unity, the logarithm of its de- nominator would be larger than that of its numerator, therefore the sign would be negative. Thus, according to (400), 24 log=| =log 24-log 47 = 1.38021124-1.67209786= -0.29188662, or - 1 + 1 - 0.29188662 = T.70811338. 4th. Of a decimal. A decimal number may be considered as a fraction whose numerator is the given number, omitting the decimal point, and whose denominator is unity followed by as many ciphers as there are decimal figures in the given number. The rule given in (399) is deduced from Problem 1, 3d. Thus we have, log 4.873 = log 4873 - 3 = 3.6877964 - 3 = 0.6877964. Likewise, log 0.0487346 = log 487,346 - 7 = 5.6878374 - 7 = 2.6878374. 404. Problem 2. To find the number corresponding to a given logarithm. 1st. When the given logarithm can be found in the table, the corresponding number is found in the column at the left. Thus the number which has 1.91907809 for a logarithm is 83. 2d. • When a logarithm differs only in the characteristic from a logarithm given in the table, multiply or divide the corresponding number by 1 followed by as many ciphers as the number of units in the given logarithm exceeds or is exceeded by that in the logarithm found in the table. Thus, to find the number whose logarithm is 4.91907809, we find 8300 in the table whose loga- rithm is 3.91907809, and multiplying by 10 we have 83,000 whose LOGARITHMS 151 logarithm is 4.91907809. The same result would have been obtained if the log of 830 or 83 had been found, which are re- spectively 2.91907809 and 1.91907809. 3d. When the given logarithm cannot he found in the tables, and its characteristic is the largest in the table, as, for example, 3.2733127, find between what logarithms the given logarithm lies, in this case, between 3.2732328 and 3.2734643, and the number corresponding to the given logarithm lies between 1876 and 1877. Evidently the whole part of this number is 1876; to obtain the decimal part x, take the difference 0.0002315, given in the third column, between the logarithms of 1876 and 1877; then find the difference between 3.2733127 - 3.2732328 = 0.0000799, the given logarithm and the next lower found in the table. The difference of the numbers being 1 for 0.0002315, for a difference of 0.0000799 it will be, ^ 0.0000799 _ 799^ ^ "^ 0.0002315 ~ 2315 ~ The number whose logarithm is 3.2733127 is therefore 1876.345. The products of the difference 2315 and the first 9 multiples of 0.1, given in Callet's table (403, 2d), may be used to shorten 2315 the above operation. Thus, in taking 694, the largest difference which is not greater than 799, the figure 3 at the left is the tenths figure of the re- quired number. Taking the difference 799 — 694 = 105, the product 926 X 0.1 = 92.6 being the largest difference contained in 105, the figure 4 is the hundredths figure in the required number. Now taking the difference 105 - 92.6 = 12, the product 1157 X 0.01 = 11.57 is the largest dif- ference contained in 12, and gives 5 as the thousandths figure. Therefore, x = 0.345. The calculations may be tabulated thus : log ... . 3.2733127 for ... . 3.2732328 1876 1 231 2 463 3 694 4 926 5 1157 6 1389 7 •1620 8 1852 9 2083 1st remainder 799 for ... . 694 0.3 2d remainder . 105 for ... . 93 0.04 3d remainder . 12 for .... 12 0.005 Number . . 1876.345 152 ARITHMETIC Assuming proportionality between the increments of the log^ arithms and the numbers, only two decimals can be taken as exact and the third as an approximation. If the table gives 5 decimals, then not more than one should be counted on in the above calculation. 4th. When the given logarithm cannot be found in the table, and its characteristic is not the largest in the table, reduce the char- acteristic to 3, the largest in the table, by adding or subtracting the proper number of units, and proceed as in the preceding 3d example. The characteristic is reduced to 3 so as to have the largest number of figures possible. The decimal point in the number found is moved to the right or left as many places as there were units subtracted from or added to the given logarithm. Thus, to find the number whose logarithm is 1.2733127, reduce the characteristic to 3 by adding 2, and proceeding as in 3d we have the corresponding number 1876.345; dividing this by 100, we have 18.76345, or the number corresponding to the given logarithm. 5th. When the given logarithm is entirely negative, add enough units to make it entirely positive, and to give it the largest char- acteristic 3 in the table. Find the number corresponding to the resulting logarithm, and move the decimal point to the left as many places as there were units added to the characteristic of the given logarithm. Thus, to find the number whose logarithm is — 2.3121626, add 6 units to this logarithm, which gives 3.6878374. The number corresponding to the latter is 4873.46, therefore the number corresponding to the given logarithm is 0.00487346. 6th. When only the characteristic of the given logarithm is negative, add enough imits to the characteristic to make it posi- tive and equal to the largest characteristic 3 in the table; find the number corresponding to the resulting logarithm, and move the decimal point as many places to the left as there were units added to the given characteristic, and the number thus obtained will correspond to the given logarithm. Thus, to find the number corresponding to the logarithm 2.6878374, add 5 units to the characteristic — 2, which gives 3.6878374, and the corresponding number is 4873.46; moving the decimal point 5 places to the left, we have the number 0.0487346, corresponding to the given logarithm. LOGARITHMS 153 405. The use of logarithms. 1st. To multiply 5736 by 743 (396, 6th). log (5736 X 743) = log 573.6 + log 743 = 3.7586091 + 2.8709888 = 6.6295979. The number 4,261,848 which corresponds to this logarithm is the required product. 2d. To divide 4,261,848 by 743 (396, 8th) : log (^^^f^) = log 4,201,848 - log 743 = 6.6295979 - 2.8709888 = 3.7586091. The number 5736 which corresponds to this logarithm is the required quotient. 3d. Raise a number 17 to the third power (396, 7th). log (17«) = 3 (log 17) = 3 X 1.23044892 = 3.69134676. The number 4913 which corresponds to this logarithm is the cube of 17. Calculate the cube of ^" „. - 0.529 log f^^)' = (log 0.042 - log 0.529) X 3 = (2.6232493 -1.7234557) X 3 = 2.8997936 X 3 = 4.6993808; then 0.00050047. 0.042Y_ ,0.529/ In this example the logarithm 2.8997936 is multiplied by 3. Multiply the decimal part separately and add the 2 units to the product 3X2 = 6, which gives 2 + 6 = 4 for the characteristic of the required logarithm (31). Instead of operating as above, reduce the logarithm to an en- tirely negative logarithm and multiply by 3, thus (400) : 2.8997936 X 3 =- 1.1002064 X 3 =- 3.3006192 = 4.6993808, which is not as convenient as the first method. 4th. Extract the fifth root of 243 (396, 9th). log V243 = "^ = ^-'''t'''' = 0.47712125. 154 ARITHMETIC The number 3 which corresponds to this logarithm is the required root. Calculate the cube root of ." _- • (J.ozy ^ In ciAo inor n r\A9 - log Y o.042 _ log 0.042 - log 0.529 _ 2.6232493 - 1.7234557 V 0.529 ~ 3 ~ 3 ^•^T''= 1.6332645; then H = "■-«• In this example the logarithm 2.8997936 is divided by 3. Reduce the characteristic to a multiple of 3 by adding 1, which gives 3, and this is compensated for by adding 1 to the decimal part. This is all done without writing anything, and continuing one-third of 3 is 1, of 18 is 6, of 9 is 3, etc. As in the multipli- cation (3d), the logarithm may be reduced to an entirely nega- tive logarithm. 406. From 3d and 4th in the preceding article, it is seen that any power or root of any number may be found with the aid of logarithms. Let it be required to raise 125 to the ^ power. o , lo^(l25^)= I (log 125) = ^-^^^f^ = 0.69897000, . ^: The number 5, corresponding to this logarithm, is the ^ power of 125. Thus it is seen that raising a number to the ^ power is the same as taking the cube root of it (306). In .general, to raise a number to a fractional power, extract the root whose index is the reciprocal of the degree of the power; and, conversely, to extract a fractional root, raise the number to the power -the degree of which is the reciprocal of the index of the root. Thus, log ^64 = log ^64*) = I X 1.80617997 = 2.70926996. 2 The number 512, corresponding to this logarithm, is the „ 3 "^ root or the ^ power of 64. LOGARITHMS 155 This example shows that in order to raise a given number to 3 a fractional power, the ^ power for instance, raise the number to the power 3 equal to the numerator, and extract the root indicated by the denominator of the power obtained. It is also seen that 2 in order to extract a fractional root, the ^ for instance, extract o the root of the number indicated by the numerator, and raise this root to the power 3 indicated by the denominator; which is 3 the same as raising the given number to - power, that is, cubing the number and then extracting the square root of the cube. 407. NaTperian or hyperbolic logarithms. This system was in- vented by the Scottish baron John Napier and published by him in 1614. The base of the system is the number 2.718281828459 . . . The common logarithms are better adapted to ordinary numer- ical calculations, but the hyperbolic or natural logarithms are used in higher mathematics (see Part V). 408. The logarithms log A and loge A, of the same number A, in two systems which have respectively h and h' for their base, are inversely proportional to the logarithms of these bases taken in any system. Thus, taking, for example, the logarithms 6 and b' in the system log A, log A __ log b' log, A ~ log 6 ' whence log^ = log,A 5-^ and log, A = log A^,, or, noting that log & = 1 (396, 1st), log A = log, A X log b', and log, A = log A X j^/ • The above makes it possible to change the logarithm of any number A in a system to a logarithm of this same number in another system. For example, the hyperboHc log log, A= 6.6106960 of the number A = 743 being given; find the common logarithm of the same number A. The base 6'= 2.7182818 of the natural system has for common logarithm log b'= 0.4342945; therefore, log 743 = 6.6106960 X 0.4342945 = 2.8709888. 166 ARITHMETIC Thus the 'product of the natural logarithm of a number and 0.4342945 is the common logarithm of the number. We have also, 1 . ««in«o«n ^°g^ 2.8709888 log„4 or 6.6106960 = log V 0.4342945 == 2.8709888 X 2.302585. The natural logarithm of a number is equal to the quotient ob- tained by dividing the common logarithm of the number by 0.4342945, or the product of the common logarithm and 2.302585, or 2.3026. log, 10 = 2.302585. 1 decimals KiLlX J U.XOIJ U IllUlUl^lCO \JL \jLiX} i-yjp. logo' 1 log?)' log 6' r 0.4342944819 T 2.3025850930 2 0.8685889638 2 4.6051701860 3 1.3028834457 3 6.9077552790 4 1.7371779276 4 9.2103403720 5 2.1714724095 5 11.5129254650 6 2.6057668914 6 13.8155105580 7 3.0400613733 7 16.1180956510 8 3.4748558552 8 18.4206807440 9 3.9086503371 9 20.7232658369 to 10 409. A general formula for the calculation of compound interest. The calculation of compound interest was given in (389). The general formula is developed as follows : Let r be the interest on $1.00 for one year. After one year, $1 is worth 1 -{-r = v^, %2 are worth (1 + r) 2 . . . etc. If Vi is taken as a new principal placed at simple interest for the second year, at the end of the second year the principal v^ will be, ^2 = (1 + r) (1 + r) = (1 + r)\ Likewise, if v^ is taken as a new principal for the next year, at the end of the third year ^8 = (1 + ry (1 + r) = (1 + ry and so on. Thus the principal of $1.00 placed for n years will become v^={l + r)" at the end of the nth year. Therefore, a principal C placed at LOGARITHMS 157 compoiind interest at the rate r for n years would at the end of the nth year amount to F = C (1 + rf, from which, taking the logarithms (1), log V = log C + n log (1 + r). By the aid of the formula (1) the diverse problems of com- poimd interest may be solved. Example 1. What principal must be placed at 4.5% com- pound interest in order that the amount be $290,818.00 after 40 years? Solution. The formula (1) gives: C= ^ or C = (1 + rf 290,818 (1 + 0.045)* whence log C = log 290,818 + C" 40 log (1.045). The logarithmic calculations : 40 log (1.045) = 0.7646516 C 40 log (1.045) = 9.2353484 log 290,818 = 5.4989700 C 40 log (1.045) = 9.2353484 - 10.0000000 log C = 4.6989700 C = $50,000. Example 2. How many years must $50,000.00 be placed at 4.5% compound interest in order that the amount equal $290,- 818.00? Solution. Substituting in formula (1) : log 7 = log C + n • log (1 + r), log 7 - log C n = n = log (1 + r) 5.463 6216 - 4.6989700 0.0191163 = 40 years. Example 3. How many years will it take for a certain principal to double itself when placed at 5% compound interest? 158 ARITHMETIC Solution. According to the statement of the problem, V = 2 C; then substituting in the formula (1): 2C = C(l +rT; dividing by C, 2 = (1 + r)»; taking the logarithms of the two numbers, log 2 = wlog(l + r), for r = 0.05, log 2 0.3010300 log 1.05 0.0211893 n = 14 years, 207 ; or reducing to days, w = 14 years, 75 days. The preceding calculation presupposes that the compounding holds for fractions of a year, which is not the case. Therefore the number of years is all that should be used; and to calculate the number of days, find the value of $1.00 after 14 years, thus: (1.05)" = $1.9799; then find how many days this amount must be placed at 5% simple interest to become equal to $2.00 or to give the interest 2 - 1.9799 = $0.0201. $1.00 brings in 360 days $0.05 0.05 and in 1 day in n days 360 0.05 • n 360 Therefore, $1.9799 after n days will amount to '■'' >lif Q^^ = 0.0201 360 n = 73 days. It is seen that the two results differ but little, and therefore it is generally sufficiently accurate to use the general rule for compound interest even for fractions of a year. 410. General formula for annuity. The general formula is de- veloped below: The capital C is loaned at compound interest and must be fully repaid at the end of n years, paying a constant sum each year, called an annuity. LOGARITHMS 169 Let r be the interest on $1.00 for 1 year. According to article (407), the final value of C is F = C (1 + r)». The sum of the final values of the different payments A is equal to the final value V. The first payment can be placed at compound interest for n - 1 years; therefore, this payment represents a final value of: Vi = a{l + r)"-'; likewise the second payment represents a final value V2 = a{l + r-)"-2; the third, the next to the last, and finally the last. Vs = a(l + r)"- v^i = a(l + r); v„ = a. Summing these different final values, the final value F of C is obtained : a + ail + r) + ail + ry + ...ail + r)"'^ = C (1 + r)\ The first member: a[l + il + r) + il + ry + . . . il + r)--']. Writing it in this manner, we see that the annuity is multiplied by the sum of the terms of a geometrical progression whose first term is 1, whose multiplier is (1 + r), and whose last term is (1 + r)"~S and according to article (371) the sum is (1 + r)'-^ (1 + r) - 1 (1 +r)''-l (1 + r) - 1 ~ r a\^:^±Jp^]=Cil+ry r- qi + ry ^= (l+r)«-l- ^^) This is the value of the annuity. This formula cannot be calculated by logarithms. In using logarithms, commence with the term (1 + ry, writing (1 + r)" = 7 log F = nlog(l + r), 160 ARITHMETIC then y — 1 is the denominator in (1), giving r • cV which may be calculated by logarithms. If the annuity a, the rate r, and the number of years n, are given, the capital C is found by substituting in the formula (1), _ ajl+ry-a ^ r(l+r)» ^-^^ The determination of the number of years n, when the capital C, the rate r, and the annxiity are given, from the formula (2), Cr(l + r)" = a(l + r)" - a; transposing, a = a = (1 + r)»(a - Cr); and taking the logarithms, log a = « ■ log (1 + r) + log (a — Cr) _ log g — log (g — Cr) ~ log (1 + r-) In order that the problem be possible, it is necessary that the difference (g — Cr) be positive, because a negative number has no logarithm. Thus the annuity a should always be greater than Cr the simple interest on the capital. It is possible to find a fractional number of years, 15f years for example, then take either 15 or 16 years and calculate the corresponding annuity, which is a practical solution of the problem. Determine the rate when the capital C, the annuity a, and the number of years n are given. Solution. Write the formula (1) : transposing, _ r . C (1 + r)° '^ ~ (1 + r)» - 1 ' g (1 + r)" - g = r . C (1 + r)» Cr (1 + r)" = g (1 + r)" - a a a r = C C (1 + r)" ' ^^^ r can only be calculated by a method of successive approxima- tions. SINKING FUNDS 411. A sinking fund is a sum set aside annually at compound SINKING FUNDS 161 interest to liquidate a debt, or replace an equipment which has a limited life. Let The debt = C. The rate of interest = r. The sum set aside = S. The number of years = n. Then we have the following relations : Sum at the end of the first year = S. Sum at the end of the second year = S + s {1 +r). Sum at the end of the third year = s + s (1 + r) + s (1 + ry. Sum at the end of the nth year = s + s(l + r) + . . . s(l + r)"~'^ Summing this series (371), we have ^_ s[(l+r)"-l] r Example 1. If a government owes $500,000, what sum must be set aside annually as a sinking fund to liquidate the debt at the end of 10 years, money being worth 5%? _ Cr _ 500,000 ■ 0.05 _ 25,000 _ __. _. . •^ " (1+ r)"-! ~ (1.05)i» - 1 " 0.628 ~ ^^^y'^""- Example 2. If $10,000 is set aside each year as a sinking fund with which to renew a $110,000 equipment, how long will it take to accumulate the required sum, money being worth 5%? Putting C = $110,000, S = $10,000, r = 0.05, and n = the number of years, we have, ^ s[(l+rr-l] ^ r Cr = s (1 + r)» - s, — ; — = (1 + r)", s Cr + s log = log (Cr + s) —log s = n log (1 + r), s _ log ( Cr + s) — log s ~ log (1 + r) log (110,000 • 0.05 + 10,000) - log 10,000 ~~ log 1.05 log 15,500 - log 10,000 log 1.05 4.1903 - 4.0000 _ 0.1903 0.0212 "" 0.0212 = 9 years. 162 ARITHMETIC STOCKS AND BONDS 412. A corporation is an association of individuals transacting business as a single person under rights and limitations granted by statide or charter. 413. The capital stock of a corporation is the amount of money- invested, and is represented by a certain number of equal shares; each share generally represents $100. 414. A stock certificate is a written evidence of the holder's title to a described share or interest in stock. 415. The gross earnings are the total receipts from the business, and deducting the expenses from these the net earnings are ob- tained. 416. A dividend is an apportionment of a certain part of the earnings, and is generally declared at a certain per cent. 417. An assessment is a sum levied upon the stock to meet expenses. 418. The face value of the stock is called the par value; and when the company is prosperous and declares large dividends, its stock is quoted above par; and on the other hand, when the company must levy an assessment, it is not considered prosper- ous, and its stock falls helow par. 419. Market value is the selling price of the stock. 420. Preferred stock is stock that does not share in the general dividends, but is entitled to its share of the profits before the regular stock. 421. Watered stock is the inflation of the capital stock by the issue of stock for which no payment is made. 422. Bonds are written agreements under seal to pay a speci- fied amount on or before a specified date. 423. Coupon bonds are bonds which have coupons or certifi- cates of interest attached. 424. Government bonds are bonds issued by the government. They usually take their name from the rate and date they bear; thus, 4^'s of '91 means 4^% bonds payable in 1891. 425. Persons who buy and sell stocks and bonds are called stock brokers. They receive a commission called brokerage, which is reckoned on the par value of the stock. STOCKS AND BONDS 426. In operations with stocks, let 163 The par value . . . = C. Per cent premium Per cent discount Per cent assessment ■ = r. Per cent dividend Premium Discount = 1. Assessment " Dividend Market value . . = A. Number of shares = n. Then the relations between these various quantities are expressed by the following formulas: nCr = / and nC ± I = A. With the aid of these formulas any problem in stocks can be performed, providing the brokerage is deducted, always bearing in mind that brokerage is computed upon the par value of the stock. 427. Examples : 1. A business man meets an assessment of $83.25 levied at 1\% on his stock. How many shares has he ? Putting shares = n, assessment = I, per cent assessment = r, we obtain, nCr = I or n = 83.25 Cr 100 • 0.0225 = 37 shares. 2. If a 7% dividend is declared upon 50 shares Chicago City R. R. stock, what is the amount of the dividend ? Putting n = 50, r = 7%, dividend = /, we have, I = nCr = 50.100.0.07 = $350. 3. A broker bought stock for a party at 124f and immediately sold the same for 143i, remitting $1341 as net proceeds. How many shares did he buy the brokerage being „% ? Putting ^ = 124f , ^2 = 143i, n = number of shares, then the 164 ARITHMETIC total brokerage is, 2 ■ n ■ C ■ 0.00^ = brokerage, and the net proceeds, $1341. $1331 = w^j - nAi - 2 • n • C • O.OOJ = n[i2-(^ + 2C0.00i)] = n[l43i-(l24f + ?)J- 1341 n = 18.625 = 72 shares. 428. In operations with bonds, let Market price = C. Years yet to run = n. Rate of interest = r. Face of bond = C". Current rate of interest = r'. Rate of interest on investment = x. Then (409) C(l + x)" is the value of the purchase money at the end of n years (409); and if the interest received on the bond is put immediately at compound interest at r'%, the amount of money received is (371), C'r (1 + r')"-' + C'r (1 + r'f-^ + C'r + C C'r[(l+rr-l] Therefore, = C' + c(i+.)-=c- + ^^-^(^+;^"-^^ , C'r [jl +rT- l]\n l + -=[^+ cr' 1 C'r' + C'r (1 + r-)" - C'r 'y ~c? /■ Example. At what price must 7% bonds, running 12 years with interest payable semi-annually, be bought in order that the purchaser may receive 5% on his investment semi-annually, which is the current rate of interest 9 Putting C = 100, and since the interest is paid semi-annually r' = 0.025, r = 0.035, n = 24, and x = 0.025. BANK DISCOUNT 165 Substituting these values in the above formulas, r' ^ ^ C'r' + C'r{l +r'f-C'r r' (1 + xf we obtain, ^_ 2.5 + 3.5 (1.025)^^ -3.5 0.025(1.025)" which, when solved by logarithms, gives C = 118. BANK DISCOUNT 429. A bank is an institution for the deposit, discount, or circulation of money. 430. A note is a written evidence of debt coupled with a prom- ise to pay. 431. The maker is the one who promises to pay, and the payee is the one to whom the promise is made. 432. A draft is an order on one person to pay another. The party who writes the draft is the drawer, the one to whom it is given is the payee, and the one on whom it is drawn is the drawee. 433. Writing on the back of commercial paper constitutes an indorsement. If the draft is acknowledged by the drawee, it is said to be accepted. 434. Bank discount is simple interest computed upon the sum due at a futui'e date and paid in advance. 435. The sum named in the note is the face, and the face less the discount is the proceeds. 436. The time from the date of discount to the date of ma- turity is called the term of discount. In non-interest-bearing notes, the face is the sum to be dis- counted. In interest-bearing notes, the face plus the interest due at maturity is the sum to be discounted. 437. The operations with notes and relations between the dif- ferent factors are expressed by the following formulas: 166 ARITHMETIC Let Face of note = C. Face plus interest due at maturity = C. Rate of interest = r'. Rate of discount = r. Discount = /. Proceeds = A. Term = T. Interest = /'. Then C = C + F CrT or C'rT = I. C - IotC - I =- A. 438. Example: $890.00 New York, N. Y., Mat 18, 1896. Five months after date, we promise to pay Harper Bros, and Co. Eight Hundred Ninety dollars. Value received with interest at 6% per annum. Bkown, Smith & Co. Find the proceeds, discounted on the date of issue at 7%. Referring to table (p. 126), we find the term of discount, which is 153 days, and adding the three days' grace we have 156 days. Now substituting in the formulas, 1 "ifi C = (C + r) = C + Cr'T= 890 + 890 • 0.06 • ^^i = $912.82. ^ ' 365 1 "ifi ^=C" = C"rr = 912.82 -912.82 -0.07^^^ = 912.82-27.36= $885.52. 365 INSURANCE 439. Insurance is an indemnity in case of loss. The insurance business is carried on by corporations regulated by state laws. They may be mutual companies, stock companies, or both. The contract between the insured and the insurer is called the 'policy. The sum paid for the insurance is called the 'premium. 440. Life insurance is an agreement of a company to pay a certain sum of money in event of the death of a person or at the expiration of a term of years. 441. A straight life 'policy continues during the life of the insured. INSURANCE 167 442. An endowment policy is payable to the insured at the ex- piration of a term of years, or to his estate if he dies sooner. 443. The expectation of life is the probability of life as deduced from the mortality tables compiled from statistics. 444. The rate of life insurance is expressed as a given sum on each $1000, and is determined by the expectation of life which the insured has at the time of taking out the policy. Thus, referring to the table we see that a man of a certain age has an expectation of life of n years; then, letting the premium be c, the rate of interest be r, and the face of the policy be A, we have, c [(1 + r)" - 1] _ c [(1 + r)" - 1] A^ (1 + r) - 1 Ar (371) and c=^^^^^„_^- Of course, in practice, charges have to be added to cover ex- penses, etc., but the above formula forms a basis of comparison, and illustrates the principle upon which life insurance is grounded. Expectation Table Constructed from the American Experience Table of Mortality. Age. Expecta- Age. Expecta- Age, Expecta- tion, Tears. tion, Tears. tion, Tears. 10 48,7 37 30,4 64 11.7 11 48.1 38 29,6 65 11,1 12 47.4 39 28,9 66 10.5 13 46.8 40 28,2 67 10,0 14 46,2 41 27.5 68 9,5 15 45.5 42 26,7 69 9.0 16 44,9 43 26.0 70 8,5 17 44.2 44 25.3 71 8.0 18 43,5 45 24.5 72 7.6 19 42,9 46 23,8 73 7,1 20 42,2 47 23,1 74 6,7 21 41.. 5 48 22,4 75 6,3 22 40.9 49 21.6 76 5,9 23 40.2 50 20,9 77 5,5 24 39.5 51 20,2 78 5,1 25 38,8 52 19,5 79 4,8 26 38,1 53 18,8 80 4.4 27 37.4 54 18,1 81 4.1 28 36,7 55 17 4 82 3.7 29 36,0 56 16,7 83 3.4 30 35,3 57 16.1 84 3.1 31 34.6 58 15.4 85 2,8 32 33.9 59 14,7 86 2,5 33 33.2 60 14.1 87 2.2 34 32.5 61 13,5 88 1.9 35 31,8 62 12,9 89 1.7 36 31,1 63 12.3 90 1,4 PAET II ALGEBEA DEFINITIONS AND PRINCIPLES 445. Algebra is a generalized arithmetic. In algebraic oper- ations the result of a certain problem is not desired, but a general solution which may be applied to all analogous propositions (2, 3, 18). In algebra, known and unknown quantities are expressed by means of letters, and the relations which exist between them by signs. Having written a number of such quantities and ex- pressed the relations between them, they are transformed to simpler forms and each unknown expressed in terms of the known quantities. Such a general expression is called a formula (503), and the value of the unknown quantities is obtained by substi- tuting the values of the known quantities in the formula and performing the arithmetical operations as indicated. 446. Characters and signs used in algebra are: 1st. The letters of the alphabet, which are used to represent quantities. Ordinarily the first letters of the alphabet are used to represent known quantities, and the last letters imknown quantities. The notations a', a", a'", etc., are pronounced a prime, a double prime, a third, etc.; and a^, a^, a^, etc., are pronounced a sub one, a sub two, a sub three, etc.; both are used to express analogous quantities of different values in the same proposition. 2d. The signs given in Art. 24, Part I, are the same in algebra as in arithmetic; thus, a + b — c= dX e 9 reads, a plus b minus c equals d times e minus b divided by g. Generally the product of several letters a, b, 36ora<3& should the time x be reckoned in the future or the past. 45 — 33 For a = 45 and 6 = 11, we have x = ^ = 6 yrs.; that is, in six years the father will be three times as old as his son. 55 _ 69 For a = 55 and h = 23, we have x = ^ = — 7 yrs.; 200 ALGEBRA that is, seven years ago the father was three times as old as his son. 2d. Impossible Roots. One-half plus .one-third of a certain number plus 5 equals ^ of the same number plus 7; what is the number? From inspection it is seen that this problem is impossible, since ij + - = - we cannot have: ^ o O Solving this equation, we have: 12 3 x + 2a; + 30 = 5a; + 42 or (3-1-2-5) a;=42-30, and x= -^ -, that is, 12 0xa;=12 or x = -jr- = or < form an inequality. These two expressions are the members of the inequality. It is understood that in a general way a quantity, A, is greater than a quantity, B, when the difference, A — B, is positive; and that A < B when the difference is negative. From this it follows that all positive quantities are greater than zero, and that all negative quantities are less than zero, being as much less as their absolute value is greater. Thus we have, i > 0, > - 6, 3 > - 4, - 3 > - 7, because 1-0=^, 0-(-6)=6, 3 -(-4) = 7, -3- (-7) = 4. 528. With a few modifications, the principles given in (509) for equations apply as well to inequalities. 1st. An inequality is not reversed when the same quantity is added to or subtracted from both its members. Thus: 5 > 3, we have 5-7>3-7 or -2>-4. 202 ALGEBRA It follows also that a term may be transposed from one mem- ber to the other by changing its sign. But if the signs of all the terms are changed, the inequahty is reversed. Thus : 5 > 3, and - 5 < - 3. 2d. An inequality is not reversed when both members p.re multiplied or divided by the same positive number, but is re- versed when the number is negative. Thus: 12>4gives: 12x2>4x2, y>|' 12x-2<4x-2; having 12 — 4 = 8, we have, 12x2-4x2 = 8x2, Y~l^i' 12x-2-(4x -2)=8X -2. 3d. The sum of the members of several inequalities in the same sense gives an inequality also in that sense. 4th. According as the two members of an inequality are positive or negative, their squares form an inequality in the same sense as the first or reversed: a > b gives a? > ¥ and — a> — b gives a^ < ¥. 529. By aid of these principles an inequality may be solved, following the same steps as in solving an equation (511). The X which should satisfy the condition 3a; „ ^ ,2 -2--7>a: + 3, we have successively: 9x-42>6x + 4, 9a;-6x>42 + 4, 3a;>46, a;>^. o 46 Any quantity greater than -^ fulfills the conditions of the given inequality. BOOK III POWERS AND ROOTS OF ALGEBRAIC QUANTITIES SQUARE ROOTS 530. The powers and roots in Algebra have the same significa- tion as in Arithmetic (85, 236, 430, 444). 531. The square of a product is equal to the product of the squares of the factors: (3 aWcf = 9 a'h^(?. (299, 465) 532. A fraction is squared by squaring its terms: [-w)--W (300) 533. The square of a binomial is equal to the square of the first term plus twice the product of the first term and the second, plus the square of the second. The double product is positive or negative according as the terms have like or unlike signs (479, 480). (See Art. 485 for square of any polynomial.) 534. Since in forming the square of the square root of a quan- tity the quantity is obtained, it follows from (465) that in order to extract the square root of a monomial, extract the square root of its coefficient and divide its exponents by 2: V36 aWc" = 6 a'b<^. From this rule it follows that a monomial is not a perfect square, and tliat its square root cannot be extracted when its coefficient is not a perfect square (248), and its exponents even numbers. When the square root of an imperfect square is to be extracted, simply indicate the operation by putting the quantity under a radical. Thus, having to extract the square root of 35 a*b, write simply V35a^6. Such quantities are called irrational monomials (447), or surds. 535. The square root of the product of two or any number of fac- tors is equal to the product of the square roots of these factors (301, 531). V36 aWc^ = ^/36 X ^/a^ X Vft* X Vc^ 203 204 ALGEBRA 536. From this it follows that in order to simplify an irrational monomial (534), separate it into factors and extract the rbot of the perfect squares, leaving the surds under the radical. Thus, V36 aWc^ = 6 a VbV, and V8 ab*c^ 2 W V2a?. In the above expressions 6 a and 2 6^ are the coefficients of the surd, and the second member is called a mixed surd. 537. The square of a positive or negative quantity being always positive (465), it follows that a positive monomial has two equal square roots opposite in sign. Thus, V4aW = ± 2 a^h. 538. The square of any quantity being positive (465), it fol- lows that the extraction of the square root of a negative quantity is impossible. Thus, V^^ = 4V^, V- 4 aV = 2a^b V'^T, V- Sa¥=b VSa V^ are algebraic symbols which represent impossible operations. They are called imaginary expressions. Problems in the second degree often conduct to these results. The general form of an imaginary quantity is a V— 1, in which a is real. Any imaginary root in an equation of the second degree may he put in the form a ± 6 V— 1, in which a and h are real quantities (572). 539. The square root of a fraction is obtained by extracting the square root of each of its terms: . fa Va .„„„ ,„„^ Vi: = ;r.- (302,532) V6 540. Two radicals are similar when they differ only in their coefficients (536). Such are: 3 V^^ (c + d) -^ab^, 2 (c + 2 d) V^. 541. The combination of similar radicals by addition or sub- traction. Perform the operations upon the coefficients and use the result as coefficient of the radical. Thus, 3 ■s/a6"' + (c + rf) ^/a6^ = (3 + c + d) ^/ab\ 3 VoS' -(c + d) \l^ = (3 - c - d) ^fc^. SQUARE ROOTS 205 If the radicals were not similar, the operations would simply be indicated. Tlius, adding Va and 3 VB, we have: and subtracting we have: Va - 3 Vb. 542. To multiply a radical of the second degree by another, mul- tiply the quantities under the radicals together, and for coefficient of the product take the product of the coefficients of the given radicals. Thus, Va X V6 = ^/ab, 3 VSo^ X - 5 Va6 = - 15 V5~^, 2 V3 a + 6^ X 5 c V3 a + 6^ = 10 c V(3 a + b^Y =10c(3a+¥). It is evident, that if the radicals are similar, as in this last case, the product is obtained by neglecting the ■>/" sign and mul- tiplying the quantity under it by the product of the coefficients of the given radicals. 543. To divide a radical of the second degree by another. Divide the quantities under the radical separately, taking the quotient of the coefficients for the coefficient of the result. Thus, Va ./a 5a V5 5a. lb 12acV66c ^ .rr- — = = V - ' F = — V -' F=- = 3 a VS c. VB ^ b 2b\/c 2b^ c Ac V2& 544. To remove factors which are perfect squares from under the radical, write their square root outside of the radical as factors of the coefficient (536). Thus, V3 aWc = ab^ V3c, 8 d ^a%^c* = 8 bcH Vo^. To place a factor of the coefficient under the radical, square it and write it under the sign ^y~ as a factor of the radical. Thus, 3 Va = VOo^, a Vb = V^, 4a V6T^ = 4 ^a\b-\-c) = ^Jma^{b + c). 545. A calculation involving irrational expressions may often 206 ALGEBRA be simplified by eliminating the radicals from the denominators. Examples : 7 7 V5 m misfa- V6) Nm 12n/22 - 20 ^Ima+mb 2V5 10 ' V^ + 3VIT V6 a-& ' 3 VlT(4 V2 - 2 V3) 16 X 2 - 4 X 3 6V22 - 3 V33 a — b ■6V33 4\^ + 2 V3 (V^ + V6) - V3 or yfma + Vm6-f • ^/mc 10 The two terms of the fractions were multiplied respectively by ■n/5, Va - V6> Va + V&, 4 V2 - 2 VS, so as to rationalize the denominators (484). In the following example, the two terms of the given fraction are first multiplied by (Va + Vfc) + Vc; then the terms of the fraction thus obtained by {a + b — c) - 2 Va6: Vm Va + V6 — Vc or Vm '^fmi + VmB + ^4mc _ "slma + Vm6 + Vmc (Va + '^bf - c a + b- c + 2Va6 (Vma + ^/mb + \fmc)(a + b-c — 2Va6) (a + 6-)+2Va6~ (a + 6 - c)^ - 4a6 546. From what was said in Art. 485 concerning the square Of any polynomial, it follows that in order to extract the square root of a polynomial, the expression must be arranged according to the powers of some letter (see example below); extract the square root of the first term at the left, 4 a°, which gives the first term, 2 a^, of the root; neglect the first term of the polynomial and divide the first term, 28 a', of the remainder by twice the first term of the root, 4 a', which gives the second term, 7 a^, of the root; subtract from the first remainder the double product, 28 a^, of the first term of the root and the second, and the square, 49 a*, of the second; divide the first term, 12 a', of the second remainder, by twice the first term of the root, which gives the third term of the root; subtract from the second remainder the double products, 12 a' and 42 a^, of the first and second term of the root by the POWERS AND ROOTS OF ALGEBRAIC QUANTITIES 207 third, and the square, 9, of the third term; divide the first term of the third remainder by twice the first term of the root, which gives the fourth term of the root, and so on. Given, for example, the polynomial 49 a* + 12 a' + 9 + 4 a^ + 42 a^ + 28 a^ to ex- tract the square root, which is done as follows: Square 4ta^ + 28a^ + 4:9a* + 12a^ + 4:2a' + 9 -4a= 1st remainder 28 a^ + 49 a^ 12 oH 42 a^ + 9 -28a=-49a* 2d remainder 12a= + 42aH9 -\2a?-42a?-9 2a?+ 7a2 + 3 root. 4a3+ 7a' la? 28a= + 49a^ 3d remainder The root may have either the sign + or 4a3+14a2 + 3 12a= + 42a2+9 (537). POWERS AND ROOTS OF ALGEBRAIC QUANTITIES OF ANY DEGREE 547. To raise a monomial to the mth power, raise its coefficient to the with power and multiply the exponent of each letter by m (465). If m is an even number, the mth power has always the sign + ; but if m is odd, the mth power has the sign of the given monomial (463): 3)"'a^"'6""c"'. (3 aWc)'' = 3"'a2'"6'"'c'", ( - 3 aVc)"' = ( ■ Remark. These examples show that the mth power of a prod- uct is equal to the product of the mth powers of the factors (531). 548. The mth power of a fraction is obtained by raising each of the terms to the mth power (532): /day _ 3V 549. In general, designating the absolute value of V^ by a' (450), we have: when m is even, 7a = ± a', (547) when m is even, 7-a=a'V— 1 imaginary ; (538) when w is odd. ^a = a', when m is odd, 7 — a = — a'- Thus: V4 = ±2 , VT6=±2, ^-16 = 2V-1, ^27 = = 3, — = a-p, \/— = ««. o^ V a™ 555. Positive and negative fractional exponents are operated upon in the same manner as whole exponents, and as the exponent 2, for example. The following examples show the manner of oper- ating in the different cases : 1st. ^^3 X -s/a^ = a^ X a^ = a*"" ^ = o^°^ , /I X ^/^ = a-^ X at = a-^^^ = a^, n/| ah *c-i X 0^1)^0^= ah^c ^ ; 2d. J:„-i^^f-(-i) = ^?-f = „H o^6* xa'h^ = a^b-K 3d. To raise a monomial having any exponent to any power, multiply the exponent of each letter by the exponent of the power. Thus, (ay = a", (a'bj = a"6^, \a^) = a^^ =ar, \a-^) =a , {2a-ih^y=64:a-'>b% (m\r r, a^j s= a' r mr 4th. To extract any root of a monomial, divide the exponent of each letter by the index of the root. Thus, ^f^ = a, -Va^" = ab% VJ = a% \/7-i = a-i, '^JT'^ah-h THE USE OF LOGARITHMS IN ALGEBRAIC CALCULATIONS 556. What was said in Arithmetic in regard to logarithms may be repeated here (396). The following examples sum up the uses which may be made of logarithms in shortening the arithmetical calculations which may arise in algebraic opera- tions: 210 ALGEBRA 1st. Log (abc) = log a + log 6 + log c; 2d. Log f-T- j = log a + log 6 - log c — log d ; 3d. Log (a'^b^c") = mloga + nlogb + p log c; 4th. Log ( -;^ 1 = log a + m log 6 — n log c ; 5tli. Log (a^ - ¥) = log [(a + 6) (a - 6)] = log {a + b)+ log (a - 6) ; 6th. Log VCa^" - V) = ilog (a + 6) + ^log (a - b); (484) 7th. Log (a' -v^a'') = log a'+log \'a^ = 31oga + jloga= -ploga; ,, m 8th. Log V(a3 - bT = "^log [(a - 6) {a? + ab + 6==)] = —log (a - 6) - —log (a^^ + 06 + 6^); 9th. Log ^^f~^y^ = Jlog (a + 6) + ilog (a - 6) - 2 log(a+6) 1 3 = glog {a-b)~ -log (a + b). ARRANGEMENTS, PERMUTATIONS, COMBINATIONS 557. Having m distinct objects, m letters for example: 1st. An arrangement of these m letters, in groups containing n letters, is made by taking n of them in as many different ways as possible and placing them in a horizontal line. Any two arrangements differ by their letters or only by the order which they occupy. The three letters, a, b, c, taken in groups of 2, give six arrange- ments : ab, ac, ba, be, ca, cb. 2d. The different groups which may be formed with these m letters, placing one by the other on the same line, are called permutations. Each permutation contains all the letters, and therefore any two permutations can differ only in the order of the letters. The three letters, a, b, c, give six permutations: abc, acb, cab, bac, bca, cba. 3d. All the possible different groups of n letters, which can ARRANGEMENTS, PERMUTATIONS, COMBINATIONS 211 be made with these m letters, in such a manner that each group differs from the others by at least one letter, are called combi- nations. No attention is paid to the order of the letters, so that if the letters represent different quantities, the combinations rep- resent all the different products which may be obtained by taking n of the m quantities in all possible manners as factors. The letters, a, b, c, taken in twos, give three combinations, ab, ac, be. 558. The following series of m letters are arrangements in groups of 1, of m letters: a,b,c,d, , fc, ■ and their number, A^ = m. The arrangements of m letters in groups of 2 are obtained by writing at the right of the letter a of the preceding series suc- cessively each of the m — 1 other letters; then at the right of the letter b each of the m — 1 other letters, and so on. The ar- rangements thus obtained are given in the table below: ab, ac, ad, . . ., ak, ba, be, bd, . . ., bk, ca, cb, cd, . . ., ck. ka, kb, kc, . . ., kh, and their number, A^ = m{m — 1). The arrangements of m letters in groups of 3 are obtained in the same manner, by writing at the right of each arrangement in the preceding table successively each of the m — 2 other letters which do not appear in that particular arrangement; which gives; abc, abd, abe, . . ., abk, acb, acd, ace, . . ., aek, bac, bad, bae, . . ., bak, The number of these arrangements is A^ = m {m — 1) (m — 2). Therefore the number of arrangements of m letters n in a group is: A" = m (m - 1) im - 2) . . . (m - n + 1). 212 ALGEBRA Example. How many different numbers may be formed with 4 significative figures ? m = 9 and w = 4: A^ = 9X8X7X6 = 3024. 559. The permutations of m letters are simply the arrange- ments of these m letters in groups containing all the letters. The number of permutations is: p^= A^ = m (m - 1) (m - 2) . . . 3 • 2 . 1 = 1 ■ 2 . 3 • 4 • . . m. With 1 letter we have P™ = 1. With 2 letters we have Pm = 1-2. ab ha To form the permutations of 3 letters, introduce the letter c at the right, in the middle, and at the left of the preceding permu- tations of 2 letters, which gives : ahc, acb, cab, hoc, bca, cba, and P„ = 1 . 2 . 3. Thus it is seen that in general the perinutations of any number of letters is formed as here below: Pm= 1 . 2 . 3 . 4 . . . m. Example. In how many ways may 5 soldiers be lined up? From the preceding formula: Ps = 1.2. 3. 4. 5 = 120. 560. Suppose that all the combinations of m letters n in a group have been made, if permutations are made of the letters in each combination, the arrangements of m letters n in a group wiU be formed, and the number of arrangements will be equal to the number of combinations of m letters n in a group multiplied by the number of permutations of n letters. Thus we have: A" =(r X P,andC" =^. Replacing A^ and P^ by their values (558, 559), we have; ^ _ m (m — 1) (m — 2) • • • (m — w + 1) »»" 1.2.3---n For n = m this formula gives c™ = 1. NEWTON'S BINOMIAL THEOREM 213 For m = 7 and n = 3, we have : ' 1 • 2 . 3 ~ '^^• It is seen that the successive numbers from 1 to n are found in the denominator, and that the numerator contains the same number of successive numbers, starting at n and descending. 561. The number of combinations of m objects in groups of n is equal to the number of combinations of m objects, m — n, in a group: n m — n C = C , which is easily proved by aid of the formula in the preceding article. 562. The number of combinations of m objects in groups of n is equal to the number of combinations of m — 1 objects n in a group plus the number of combinations of m— In— 1 in a group : n n , n — 1 c = c , + c , m m— 1 wi — 1* NEWTON'S BINOMIAL THEOREM 563. From the rule for obtaining the product of any number of polynomials (468, 469), it follows that this product is the sum of the products obtained by taking in all possible ways one term in each of the polynomial factors. Find the product {x + a)ix + b){x + c) ...{x + h) {x + k), of m binomials which have the same first term x, arranged accord- ing to the descending powers of x. Taking the first term x in each of the binomial factors, we have the first term x'" of the product. Taking successively the second term a in the first binomial with the first term x in all the others, the second term b of the second binomial with the first term x in all the others, and so on, the partial products ax"-\ bx-\ . . . kx''''^ are obtained, and their sum (a + b + C+ ■■■ +k):r-'or S^x'"-' is the second term of the product. Taking successively the second terms in any two binomial 214 ALGEBRA factors and the first x in the m — 2 others, the partial products aftx"*"*, acx"'~^, bcx'"~^, are obtained, and their sum {ab + ac + ---) x""' or S^^'^ is the third term of the product. The fourth term is: «m — 3 {abc + abd + ■ ■ ■) a;""-', or SsPiT Any term of the m — n degree is obtained by taking succes- sively the second terms in any n factors and the first x in the others and summing these partial products. The result may be written in the following manner: SnX""". The next to the last term is : S„,-iX. Finally the last term is simply the product abc . . . k, OT S„, of all the second terms of the binomial factors. Therefore the desired product is : x™ + S^x'"-' + S'x""-' +■■■+ Snx""-" + ■••+ S^^^x + S^. It may be noted that Oi, 02> Os> ■•■) "iiJ ■■■) ^m—l> ^m are nothing but the sums of the combinations obtained by taking the m second terms of the binomial factors respectively, 1, 2, 3, ... n, ... m — 1, m in a group (557). 564. To raise a binomial (x + a) to the mth power. This is done by supposing each of the second terms, a, b, c, . . .k oi the binomial factors, to be equal to a; then we have (560): Si = a + a + a+--- = Cja, = ma, S,= a' + a' + a^+... = C'^a^ = ^'^V^a^ S, = a« + a- + a» +... = 0^= "^^^ ~ '^^.^^ " '^ a«, o„ = a = V a = a , NEWTON'S BINOMIAL THEOREM 2l6 therefore, (x +aT = a;"" + max""-' + ^^f ~ ^^ a'x"'-^ + "^"r-2-r~'^ "^""'"^+ • • • + "^""""^ + «"• This formula is known as Newton's binomial theorem, and has the follovping properties: 1st. {x + a)™ is composed of m + 1 terms, of which the first is aj™ and the last a". 2d. The exponent of x decreases by 1 in passing from one term to the next, and therefore becomes for the last term; the exponent of a increases by 1 from one term to the next, starting at the first term as 0, and becoming m for the last term. Thus it is seen that in any term the sum of the exponents of x and a is equal to m. 3d. The coefficient of any term is obtained by multiplying the coefficient of preceding term by the exponent of x in that term, and dividing the product 1 plus the exponent of a in the same term. 4th. The coefficients of two terms equally distant from the extremes are equal, and therefore the coefficients of two terms equally distant from the middle term if m is even, and from the middle if m is odd, are equal. Thus, having calculated at least half of the terms, we may write the coefficients of the remaining terms without further calculation. Applying these rules to the two following examples, we have: ix + ay = a^ + 8 ax' + 28 a^x" + 56aV + 70a*x' + 56a'x^ + 28aV + 8a''x + a^; (x + ay = x'' + 7ax'' +21 aV + 35 aV + 35 aV + 21 a'x'' + 7a'x + a''. The term which we represented by SnX"'-", is : m (m - 1) (m - 2) . . . (m - n + 1) ^„^„ „ 1 • 2 • 3 . . .n This term is called a gmeral term, and having it any term may be calculated without having the others, by substituting the values of m and n in the above formula. Thus the (n + l)th = fourth term of the value of (x + a)"-' is: 8lll^a3^-8 = 56aV. 216 ALGEBRA If in the binomial formula we replace a by —a, we have: ,_i^ m(m-l) ^,^^_. (x - a)™ = a;™ - max'"-^+ Y- ^ '' «'^"'~'' ±a" which differs from the first in that the signs of the terms are alternately positive and negative. 565. In the following table, known as Pascal's triangle, the figures in the horizontal lines are the coefficients of Newton's binomial for different values of m. The vertical column 1 contains the number of combinations in groups of 1 of 1, 2, 3, . . . objects (560); column 2 contains the number of combinations in groups of 2 of 2, 3, 4, . . . objects; and in general the column n contains the number of combinations in groups ol n oi n, n + 1 , n + 2, . . . objects. 1st 2d 3d 4th 5th 6th 7th 8th 9th 10th m = 1 1 . , , , m = 2 2 1 . . m = 3 3 3 1 , m = 4 4 6 4 i m = 5 5 10 10 5 i m = 6 6 15 20 15 6 i m = 7 7 21 35 35 21 7 i m = 8 8 28 56 70 56 28 8 t m = 9 9 36 84 126 126 84 36 9 1 m = 10 10 45 120 210 252 210 120 45 10 L A number in the column n and the horizontal row m, expresses the number C^ of combinations of m objects in groups of n (500). Thus, 8 objects combined in groups of 5 give: C" =56. Any number in the arithmetical triangle is equal to the one above it plus the one at the left of that one. Thus, the number 56 in the 8th horizontal row is equal to 35 + 21. This follows from the relation, (562) ,From this relation the formation of the arithmetical triangle is easy. The mth number of any column is equal to the sum of the NEWTON'S BINOMIAL THEOREM 217 m first numbers of the preceding column. Thus, considering the 4th number 35 in the 4th colunm, we have: 35 = 15 + 20, 15 = 5 + 10, 5 = 1+4, and 35 = 20 + 10 + 4 + 1. In general, the mth number in the nth vertical coliunn is found in the (m + n — l)th row; that is, _ (m + n— l)(m + w — 2)- • ■m m (m+1)- • -(m+n — 1) m+n- 1-2-3 n 1-2-3- 566. The number of balls contained in a pile which has a tri- angular base. A triangle of m balls on a side being formed of m rows which contain respectively 1, 2, 3, . . . m balls, corresponds to the whole consecu- tive numbers contained in the first colunm of the arithmetical triangle. These numbers are called figurate numbers of the first order, and the triangle contains m (m + 1) Kg. 2 1 + 2 + 3+. m 1-2 balls, (565) a number which is the mth in the second column of the arith- metical triangle (565). For m = 6, there are 21 balls in the triangle. Thus the numbers 1, 3, 6 ... in the second column of the arith- metical triangle are the triangular or figurate numbers of the second order ; they represent the number of balls contained in the successive layers of a triangular pile; and the sum of the first m layers, that is, m(m + 1) (m + 2) 1-2-3 (565) Fig. 3 is the number of balls contained in the pyramid, and is represented by the mth number in the third column of the arithmetical triangle. For m = 6 there are 56 balls in the pyramid. Thus the numbers contained in the third column are the pyramidal numbers. 567. A pyramid with a square base having m balls- on a side may be considered as being formed of two tri- 218 ALGEBRA angular pyramids, the edges of which contain m and m — 1 balls; the total number of balls which it contains is (566): m (m + 1) (m + 2) (m — 1) m (m + 1) _ m (m + 1) (2 m + 1) 1-2-3 "^ 1-2-3 ~ 6 ' and this number is the sum, 12 + 2^ + 3^ + • • • + TO^ of the squares of the first m whole successive numbers, since these squares express the number of balls contained in the successive layers of the quadrangular pyramid. For m = 48, we have: P + 22 + 3^ + ■ ■ ■ + 48^ = 4S X 49 X 97 ^ ^^ ^^^^ D 568. Considering a pile with a rectangular base, one of the sides of the base containing m and the other n < m balls, as being made up of a pile with a square base n balls on a side, and a prism having a height equal to "^^^^ m — n balls and a triangular base of n balls on a side, ®' the number of balls which it contains is (566, 567): n(n+l){2n + l) n{n+l)(m-n) n(n + 1) (3m -w + 1) 6 ^ 2 ~ 6 For m = 25 and n = 10, the pile contains 10 X 11 X 66 i„.^, ,, t; = 1210 balls. 6 569. The sum S^ of the mth powers of n numbers, a,b, c . . . j, k in an arithmetical progression whose common difference is r (357). Having b = a + r, c = b + r,---,k = j + r, we have (564) : 1 1.^ 1 (k+rr^^ = k--^+ !!^rk-+^^^^±^r^k-^ + . . .+ !^rt+, NEWTON'S BINOMIAL THEOREM 219 Adding these equalities, and cancelling the terms 6""+', c^+i, • • • fc^+i^ which are common to both members of the resulting equa- tion, and making a"' + &"'+■•■ /b™ = S^, a"^'^ + fc™-! H +fc"'-i = (S„,_i ■•■, a + b + ---+k = S, and n = So, we have : from which By means of this formula, commencing with So = n, Si, S^, Ss, ... may be successively calculated. Per a = 1, r = 1, and m = 1, from which k = Sg = h, and 6 6 37 ■37\2 12 57 6 19 37 X" = 12 The roots are correct, since -k- + 3 = -r- j 6 6 Example 3. 4. a^ - 2 x^ + 2 ax = 18 ab Transposing and solAdng: 6 ~^- and -^ X 3 = -^ ■ 6 6 18 6^ x' — ax = 2 a' 9ab + Qb" and ^ =2+V4 + 2a2-9a6 + 9 6=' = 2a-36 I - y J + 2 a^ - 9 ab + 9¥ = - a + 3b. In obtaining these values of x, it ma^ be noted that the quantity 9 3 under the radical j a' — 9 ab + 9b^ is the square of - a — 3 6 (479). The roots are correct, since 2a — 3b — a + 3b = a, and (2 a - 3 &) (- a + 3 6) = -20" + 9ab - 9V (468). Example 4. ax' + bx = 0. This equation, in which the known quantity is 0, dividing by a gives:" ___b ~ 2. x' + -X = 0, from which a ^v/.^ = o _6^ 2a b^ 4.a' 576. The roots of the complete quadratic ax^ + bx = c may be obtained without reducing the equation to the form x^ + px = q, b c that is, without making - = p and - — q (572). b c Substituting p = - and q = - m the following: -iW? + ». 224 ALGEBRA we have; x = -^±\/^, + -= "-^^ ' ^■^" . (1) 2a y 4: a' a ^ " ^ ' - 6 ± V62 + 4ac 2a This formula is more generally used than the former because the calculations are simpler. In this case we have: b -, c x' + x" = ) and x'x" = a a When the coefficient 6 of a; is even, we can write 6 = 2 6' in the formula, which gives : - 2b' ± V46'' + 4ac - b' ± Vb'^ + ac X = -■ X = 2a a In this form the arithmetical calculations are still simpler. From the equation It -^'^T^+H-'n^^ 3 a;2 - 28 X = - 49, we have: ' _ 14 ± Vl4'-13 X49 _ 14 ± 7 . ^~ 3 ~ 3 ' 7 that is, x' = 7 and a; = ^ • When a = 1, the formula (1) becomes: _ - 6 ± Vft' + 4c X-- 2 • which is the same as the general formula in article (572). 577. To resolve a trinomial of the second degree x^ + px+ q = into two factors of the first degree. 1st. Since this trinomial comes from the equation 3^ + px = - q,we have (572, 573): a;' + x" = — p or — (x' + a/') = p and a;'a;" = q. Substituting these values for p and q in the trinomial, we have: 3^ - {x' + x") x + x'x" = 0, or (a; — x') {x — x") = 0, and in general, a;' + pa; + g = (a; — x') {x — a/'). (1) For example, the equation a;' + 4 a; — 12 = 0, giving a;' = — 6 and a/' = 2, we have : a;2 + 4 a; - 12 = (a; + 6) (a; - 2). EQUATIONS INVOLVING ONE UNKNOWN 225 The trinomial aa^ + 6a; + c = in the same manner gives; ax^ + bx + c = a{x - x') (x - a;"). Thus the roots of the equation 3a;^ — 7a; + 2 = 0, being x' = 2 and x" = -, we have: 3 a;^ - 7 X + 2 = 3 (a; -2) (x - |V 2d. Having the trinomial, x^ + px + q = P. (2) Adding and subtracting ^ in the first member, we haveg x^ + px + ^ + q-'^ = P, hf)"-(?-^)=^. or hij-m^^h Designating the two roots of the trinomial (2) by x' and x", when the trinomial is made equal to zero, the difference of these two squares may be written: (-+I + v/?^')('+|-\/?^)-^. , and (x — x') (x — x") = P. In the same manner, having: ax^ + hx + c = P. Designating the roots of this trinomial by xf and a;" and making it equal to zero, we may write: P = a (a; - a;') (a; - a;") = ax' -^ hx + c, (3) in which expression x' and x" have certain fixed values and x has absolutely any value. In giving x a positive or negative value, calculate the corresponding value P of the trinomial ax^ + hx + c ^ P. Example. Given the trinomial, P = 3 a!=' - 6 a; - 45, 226 ALGEBRA to be resolved into factors of the first degree. Find the roots of the equation: 3x2- 6 a; -45 =0, or a^- 2 a;- 15 = 0, a; = + 1 ± Vl + 15, x'= 5, x"= - 3. Therefore, the given trinomial may be written in the form: P = 3 (a; - 5) (a: + 3). In this form we can study the values of P corresponding to different values of x. Some of these values are given below. For a; = P = = - 45 1 - 48 2 -45 3 - 36 4 - 21 5 - 1 - 36 - 2 - 21 -3 -4 + 27 EQUATIONS OF THE SECOND DEGREE INVOLVING SEVERAL UNKNOWNS 578. The solution of a system of two simultaneous equations, involving two unknowns, one or both of which are of the second degree. 1st. If one of them is of the first degree, express one of the un- knowns in terms of the other and substitute in the other equation, which will give a second degree equation involving only one unknown; this may be solved and the value obtained substi- tuted in the first equation, which in turn will give the value of the other unknown. Thus, having ax + by = 2 s and xy = t, from the first equation (511): 2s — aa; y = — r— • EQUATIONS INVOLVING SEVERAL UNKNOWNS 227 Substituting this value of y in the second, /2 s — ax\ a „ , 2 s ehminating the denominators and changing the signs, ax' - 2 sx + bt = 0, and therefore (576), s±y/s^ - abt X = a Substituting this value of x in the first of the given equations, and solving: SztVs^- abt y= — 6 — The syste m of equ ations has two direct solutions, because evi- dently s> Vs^ — abt; but in order that they be real, s' must be greater than, or equal to, abt. These two solutions when separated are: s + Vs' - abt s - ^/s^ - abt a b , s — ^/s^ — abt s + Vs^ - abt and x = > y = j • a For a = 6 = 1, the given equations become x + y = 2 s, xy = t, and the values of x and y are reduced to : X = s rt Vs^ — t and j/ = s T Vs^ — t, which shows that the two values of y are equal to those of x taken in an inverse order, that is, if s + Vs^ — t is the value of a;, s — Vs^ — t is the corresponding value of y, and conversely. Special Method. Noting that the solution of the system X + y = 2 s and xy = t amounts to finding two numbers x and y, the sum and product of which are known, it is seen that they are the roots of the equa- tion (573, 574): z'' -2sz + t = Qi, which gives directly (542) : z' = & + Vs^ - t and z" = s - \ls' - t. 228 ALGEBRA The solutions of the equation are therefore, putting succes- sively X = z' and y = 2", X = s + Vs^ — t, y = 2s — x = s — Vs^ — t and X = s — Vs^ — t, y = 2s — x,= s + Vs^ — t, values found by the general method. This special method may be applied to the systemj X — y = 2, xy = 15. Putting 2/ = - 2/1, X +2/1 = 2, xy^ = -15 X and j/i, being the roots of the equation z^ -2z = 15, which gives, 2' = 5 and 2" = - 3 ; X = 5, 2/1 = 2 - 5 = - 3; X = - 3, 2/1 = 2 + 3 = 5. Therefore the solutions of the given system are: X = 5, 2/ = 3; a; = — 3, y = — 5. This special method may also be applied to the system? X + y = 8, 3? + f = 34:. If the first equation is squared, x^ -\- 2 xy -[■ y'^ = 64, and the second one subtracted from it, we have: 2xy = 30 or xy = 15; and we have again, a; + 2/ = 8 and xy = 15, X and y being the roots of the equation 2^ - 8 2 = - 15, which gives 2' = 5 and 2" = 3, and the solutions of the system are : X = 5, 2/ = 8 — 5 = 3; x = 3, 2/ = 8-3 = 5. 2d. When one of the equations is of the first degree with reference to one of its letters only, solve for the value of this imknown and EQUATIONS INVOLVING SEVERAL UNKNOWNS 229 substituting in the other equation an equation of the third degree is obtained. Thus, having ax' + by = 2 s and xy = t, from the first equation; 2 s aa? Substituting in the second equation, 2s a , _ b ^ b ' eliminating the denominators and changing the signs, aa^ — 2 sx + bt = 0. 3d. A system of two simultaneous equations of the second degree involving two unknowns. a;2 + j/2 = 25, xy = 12. 12 The second equation gives y = — , and substituting this in the first, we have: 144 ar' + ^ = 25, or a;* - 25 x' + 144 = 0. x^ Thus we have an equation of the fom-th degree; but this equation, being a quadratic, is easily solved (579). Thus the system may be solved by multiplying the second equation by 2 and adding it to the first: x' + 2xy + y'' or (x + yY = 49, from which x + i/ = i 7. (1) Subtracting the second multiplied by 2 from the first, we have: 3? — 2xy + y^ or {x — yY = 1, from which a; — 1 = =t 1. (2) The equations (1) and (2) giving the sum and difference of the quantities x and y, the quantities themselves may be easily foimd. These equations added and subtracted give: rt7±l , ±7T1 X = — 2 — ■ ^^'^ y = — 2 — ° The roots of the given system are: a; = 4, 2/ = 3; a; = 3, 2/ = 4; a: = - 4, 2/ = - 3; x = - 3, 2/ = - 4. These roots satisfy the system. 230 ALGEBRA The elimination of one (ff the unknowns in two complete quadratic equations involving two unknowns gives an equation of the fourth degree. Considering the following: ay^ + bxy +cx' + dy + fx + g =0, a'y^ + b'xy + c'x^ + d'y + fx + g' = 0, arranged according to x, cx^ + (by + f)x + ay^ + dy + g = 0, c'u? + {b'y + f')x + a'f + d'y + gf' = 0. If the coefficients of x^ were the same in the two equations, by subtracting them an equation of the first degree of x, which could be substituted in one of the given equations would be obtained; from this equation the value of x in the terms of y may be found, and substituting this value in one of the given equations, an equation is obtained which contains only one unknown y (520, 3d). Or if each term of the first equation is multiplied by c', and those of the second by c, we have: cc'a;' + {Tay + /) c'x + {af + dy + g) c' = 0, cc'x^ + {b'y + /') ex + {a'f + d'y + g') c = 0. Subtracting one from the other, we obtain : {{be' -cb')y+fc' -cf''\x + {ac' -ca')y''-]-{dc' -cd')y + gc' -eg' = 0, which gives : _ {ca' — ac') y^ + {ed' — dc') y + eg' — gc' ^~ (&c' - eb') y + fc'-cf This value of x substituted in one of the given equations will give the fimal equation for y. Without making this substitution, which would be somewhat complicated, it is easily seen that the equation in y would be of the fourth degree. TRINOMIAL EQUATIONS 579. The trinomial equations are of a degree greater than the second, and their solution may be brought to that of an equation of the second degree involving one unknown. The general form is; oa;''" + bx^ = c. TRINOMIAL EQUATIONS 231 rhey are called trinomial equations because they involve three Idnds of terms : the terms in x "", the terms in x", and the known terms. Putting aj"* = y, the equation is of the second degree: ay^ — hy ^ c. Having calculated the values of y from this equation, those of x are given by the formula: X = Vj/. If m ts an even number, all positive real values of y give two equal real values of opposite sign for x; while the negative values of y give imaginary values of x (514). If m is odd, all real values of y give but one value of x, which is real and of the same sign as y. Given the trinomial equation, a;* - 25 a;2 + 144 = 0. Putting x^ = y, we have, 2/^ - 25 2/ + 144 = 0, and 25 rt V252 - 4 X 144 y = (572, 576) But x^ = y and a; = ± \fy^ . ./25±V25^-4X 144 x=±V 2 which shows that the equation has 4 roots, equal in pairs and opposite in sign. Effecting the calculations, we find first: 2/ = 16 and y = ^■ Then a; = ± 4 and a; = =t 3. Which values satisfy the given equation. EQUATIONS OF ANY DEGREE 580. A graphical method of obtaining an approximate solution of an equation of any degree. Given, the equation, a;= + 5 a;< + a;' - 16 x^ - 20 a; - 16 = with all its terms in the first member. 232 ALGEBRA Draw two axes Ox and Oy perpendicular to one another. The different values given to x in the equation are laid off to a con- venient scale on the axis Ox or Ox' according as they are positive or negative. Per- pendiculars are raised at the points thus obtained on xx', and on these the values y of the first mem- ber for different values of x are laid off to a convenient scale, which need not be the same as the first. Hav- ing obtained a sufficient number of points, a smooth curve is drawn through them, and the distances from to the points where this curve crosses xx' are the roots of the equation. For X = 0, the value y of the first member of the equation is — 16, which gives OA = — 16. For X = Ov = 0.5, 2/ = w = 0.5^5X0.5* + 0.5'- 16X0.5^-20X0.5 -16 =-29.53. For X = Ou = - 2.5, 2/ = ui!, = 2.5= + 5X2.5^- 2.5'- 16X2.5H 20X2.5 -16 = 16. According as x is positive or negative, the different terms which enter in the value of y will have the signs of the first or the second of these last two inequalities, which makes it necessary to find the signs but once for each sign of x. Constructing a table of these values, we have : -4, 0. Having obtained y = ior the values 2,-2 and — 4 of a;, these are the real roots of the equation. If the curve is plotted, it will cut the axis xx' at the points for which x = 2, x = — 2, and x = — i. An examination of the equation shows that for a;= 0.5 1 1.5 2 -0.5 -1 -1.5 -2 -2.5 - 3 -3.5 2/=- -16 -29.53 -45 -45.72 -9.84 -9 -7.65 16 ~85 40 EQUATIONS OF ANY DEGREE 233 values of x greater than 2, the values of y are all greater than and positive; and furthermore, since the curve does not cut the axis xx' between x = 2 and a; = 0, 2 is the only positive real root of the equation. In the same manner it is shown that — 2 and — 4 are the only real negative roots. When, as in the preceding example, the roots are whole, they may be obtained rapidly enough without tracing the curve. Having obtained a value of y which approaches 0, upon aug- menting or diminishing x, y = is quickly found, and the cor- responding value of X is the required root. In engineering practice the positive root is the one which is most often used. In this case the negative values of x are not used, and no curve is plotted on the negative end of the xx' axis. Furthermore, the nature of the problem generally permits of a fair guess as to the value of x, and the curve need be drawn only near this point. The graphical method is most useful when the roots are not whole or when they contain a great number of figures. Given, to solve the equation. x' - 3 x^ + 7 a; - 40 = 0. For x=0, we have y = AO =-40; x=l = Ov, y = va = 1- 3+ 7- -40= -35; x = 2 = 0v', y = v'a' = 8-12 + 14- -40= -30; x = s = Ov", y = v"a" =27-27 + 21- -40=-19; x = 4: = 0v"', w = i;"'a"' = 64 -48 + 28 - -40 = 4. y having become positive indicates that the equation has a positive root between 3 and 4. Further, the equation shows that for values of x greater than 4, y would always be positive and greater than 0. Thus there is only one pos- itive real root, and this is shown by the curve. The point c where the curve intersects Ox gives, with the exactitude furnished by a plotted curve, v"c = 0.9 of v"v"', or of 1 in practice, and we have 3.9 for the root of the equation. If it is desired to prove the correctness of this root or to determine it more accurately, the following method is employed: / For X = 39, the equation gives y = 0.99. / y This indicates that 3.9 is too great. Fig. 6 234 ALGEBRA For X = 3.8, we have, y = — 1.85. Therefore, x has between 3.8 and 3.9. The value of x augmenting from 3.8 to 3.9 = 0.1, for an aug- mentation of 1.85 + 0.99 = 2.84 of y, supposing that the incre- ments remain proportional, which amounts to supposing the curve to be a straight line between those points, for the augmentation 1 85 1.85 of y, x would augment 0.1 ^^ = 0.065. Therefore, the required root is 3.865; and substituting in the equation, we have y = 0.0234, which is more than accurate enough for ordinary practice. X = 3.866 gives y = + 0.005. If the negative roots are desired, they may be obtained in the same manner. 581. Solution of an equation of any degree by successive approxi- mations. Given, the equation, a:= + 200 X = 5000, or a;= + 200 a; - 5000 = 0. Suppose a; = in all the terms which contain x except one; ordinarily the term which contains x with the largest exponent is excepted, because the value of x is more rapidly approached when the coefficients of the other terms of an elevated degree are not very great. Making a; = in the second term of the given equation, 3^ = 5000, or 5 log a; = log 5000, and x = 5.4928 Substitute this value for x in the terms which were first made equal to zero. 3^ + 200 X 5.4928 = 5000; and x = 5.2269. Substituting this new value in the equation a^ + 200 X 5.2269 = 5000; and x = 5.2411. This value when substituted gives a fourth x = 5.2403, which gives a fifth x = 5.2403 The value x = 5.240 may be taken as the root; and substitu- ting, we have : y =- 1.45. X = 5.241. y = 2.51. MAXIMA AND MINIMA 235 Instead of starting with a; = 0, it is possible to start with any value which the nature of the problem may indicate as being near the true value. MAXIMA AND MINIMA 582. When an expression takes different successive values, it is said to have reached a maximum or minimum when its value is less or greater than the values which immediately precede or follow it. A maximum or a minim,um is said to be absolute when the ex- pression has no value which is larger than this maximum and none which is smaller than the minimum. In other cases it is a relative maximum or minimum. At this point, only problems which may be solved by means of second degree equations will be treated, leaving the general treatment of maxima and minima for a later chapter. 583. The maximum of the product xy = z of two variable factors X, y, whose sum x + y = ais constant, occurs when these two factors are equal, that is, when x = y =-^- 1st. Having (481) {x + yY — {x — yY = 4 x;/ (x + yY being a positive constant quantity, the product 4 xy, and therefore, xy will increase in proportion as x — y decreases in absolute value, and will be a maximum when , . a x — y = 0, that is, x = y = -^■ 2d. Having x + y = a, and xy = z, it follows (574) that x and y are the roots of the equation u' — au + z = 0, which gives (572): a , .la^ a .jo? ^ ^ = 2 + V4-^' 2/ = 2-V4-^- If X and y are to have real values, z = xy should not be greater 2 Q- than — ) which is the maximum value. But when z = xy = -j, 4 * the two roots x and y of the equation are equal, and we have as in 1st, a 236 ALGEBRA 3d. On a straight line AB, take successively the lengths AC and CB, representing to some chosen scale the numbers x and y, the sum of which x + y = a = AB is constant; on AB as diameter describe a semicircle, and at C erect a perpendicular to C 2^ B AB. Representing z by CD, we have, no matter '^' what the position of C may be, that is, what the values of x and y may be, z^ = xy. The maximum of xy corresponds, therefore, to that of z^ or z; but 2 is a maximum when s is at the center of the semicircle, and we have: a z = x=y = 2' 584. From the preceding article (583), it follows: 1st. That of all rectangles of the same perimeter, the square has the maximum area. 2d. That of all the right triangles the sum of whose legs is con- stant, the isosceles has the maximum, area. 3d. That of all triangles of the same base a, and the same peri- meter 2 p, the isosceles has the maximum area. The expression for the area s of a triangle being (see Trigo- nometry) s= Vp {p - a)ip - b) {p - c), the factors p and p — a being constants, s will be a maximum when the product {p — b) (p — c) is a maximum; and since the sum 2p — 6 — c = a is a constant, this will be when p — b — p — c ov b = c. 585. The product of any number of n positive factors, the sum a of which is constant, is a maximum, when all the factors are equal. Because if only two factors are unequal, replacing each by their arithmetical mean (337), the product of the factors is increased, but the sum remains unchanged. From this it follows: 1st. That the arithmetical mean of n positive numbers which are not equal is greater than their geometrical mean. Thus, having , ^ /a + b + c -\ \" , a + b + c-i „ /-r abc- ■ ■< 1 — ■ ■ , we have ' ■ > ^abc-- ■ \ n J n MAXIMA AND MINIMA 237 2d. That of all triangles having the same perimeter 2 p, the equilateral triangle has the maximum area. Thus, having (584) s = Vp {p - a) (p - b) {p ~ c), since p is positive, each of the three factors should be positive; because if one or all of them were negative, s would have an imaginary value; and if two were negative, p — b and p — c, for example, we would have 2 p < b + c, which is impossible, p being constant, s will be a maximum when the product of the three other factors is a maximum, that is, when p — a = p — b = p — c, or a = b = c. 586. The product abc . . . of any number n of positive factors, the sum a™ + 6™ + c™ -| — -of the mth powers of which is constant, is a maximum when the factors are equal. Let it be given to find the rectangle of maximum area which may be inscribed in a given circle. S being the area of the inscribed rectangle, x and y the dimen- sions, and d the diameter of the given circle or the diagonal of the rectangle, we have: xy = s, or x^y^ = s^, and x^ + y^ = d^. The sum d^ of the factors x^ + y' being constant, in order that s^, and therefore the area of the rectangle, be a maximum, x' must equal y^ and x = y. Thus the square is the largest rectangle which may be inscribed in a circle. 587. The sum x + y = a, of two positive numbers x and y, being given, find the maximum of the product x™2/", wherein m and n are whole positive numbers. We have: m'^n" being a constant, the product af'y" will be a maximum when — X — is a maximum. But this last product is composed »Y" 'it If tl of m factors — and n factors - > the sum m \-n- = x + yoi m n m n which is constant; therefore, it is a maximum when all these factors are equal, that is, when m n 238 ALGEBRA Thus the product o^'if is a maximum when x and y are propor- tional to their exponents m and n. This applies, no matter how many factors there may be. From the two equations X + y = a and m we deduce (520); X ma and y = na m + n " m + n Example 1. Inscribe an isosceles triangle ABC of a maximum area in a circle of a given radius r. Let 2 a; be the base of the triangle, y its height, s its area, and CD the diameter perpendicular to the base AB. Then we have xy = s and x'' = y (2r — y). The second equation expresses that a; is a mean proportional be- tween the two segments of the diameter. (See Geometry.) s wiU be a maximum when xy or x'^y'^ =y^{2r — y) is a maximum. But in this last product, which is obtained by multiplying the value of x^ by y'', the sum y + {2r — y) = 2r is constant. Therefore, 3 being the exponent of the first factor y^, and 1 that of (2 r— y), we have for a maximum: 2/3,3 — -^ — =-, and y = -^r. 2r — y 1 ^ 2 This value of y indicates that the maximum triangle is an equi- lateral. Example 2. Construct a box having a maximum capacity, with a square ABCD of cardboard. To construct such a box, draw parallel lines at equal distances from the sides; remove the four squares at the corners and fold the four rectangles, such as EFLK, so as to form the sides of the box. The base of the box is the square EFGH. Designating the constant AB by 2 I and the variable AK by x, the capacity c of the box i c = (2 Z - 2 xyx = 4 (Z - xfx, ': H G E F — 1 — 1 — I.— A K r-B Kg. 9 MAXIMA AND MINIMA 239 and the sum (Z — xy+ x being constant, the maximum of c corresponds to I- X 2 , I 21 -^=-, andx = -=-. Thus, to obtain the largest box divide AB and AD into six equal parts and draw parallels through the first points of division. Example 3. In an analogous manner find the largest cylinder which can be inscribed in a sphere. Let r be the radius of the sphere, x the radius of the base of the cylinder, and 2 y the height, then r r /2 y = — = or 2y = r — = ? and a; = r \/ _ - ^ V3 ^ V3 V3 Example 4. Circumscribe a given cylinder by a cone of mini- mum volume. Let h be the height of the cylinder, r the radius of its base, y the height of the cone, and x the radius of its base, then we find that for a minimum volume, y = Sh and x = -r. 588. Resolve a given number into two factors x and y, the sum z of which should he a minimum. Having X + y = z, and xy = a, X and y are, for any value of z, the roots of the equat.on u^ — zu = - a, which gives (572, 573): s . f^ z . [z^ ^ = 2+Vi-^' y = 2-Mi.-"'- z' If X and y shotJd have real values, — should at least be equal to a, or 3 = 2 Va; at this lower limit, the two roots are equal, and we have: ^ 1- X = y =-^ = 'yJa. Thus the minimum of the sum x + y of two variable positive factors, the product xy = a of which is a constant, occurs when each of these factors is equal to the square root of the given product (583). From this it follows: 1st. That of all rectangles, which have the same area, the square has the shortest perimeter. 240 ALGEBRA 2d. That of all the right triangles, which have the same area, the sum of the legs of the isosceles is the least. 589. The minimum of the sum of any number n of variable positive factors, of which the product a is constant, occurs when all the factors are equal, that is, when each of them is equal to Va. Because if only two of the factors were unequal, replacing each by their geometrical mean, their sum would be diminished, as would also the total sum, without changing the product of the factors (585). 590. The sum x^ -\- -if = z of the squares of two variable quan- tities X and y, the sum x + y = a of which is constant, when the two quantities are equal, and therefore, each equM to -^ . Squaring both members of the equation, X + y = a, we have x'^ + y^ = a^ — 2 xy, and it is seen that x'^ + y^ will be a minimum when xy is a maxi- mum, that is, when (583) x = y = -. From this it follows that: 1st. Of all right triangles, of which the sum of the legs is con- stant, the isosceles has the shortest hypotenuse. 2d. Of all the rectangles having the same perimeter, the square has the shortest diagonal. 3d. Of all the squares inscribed in a given square, the one whose corners bisect the sides of the given square is the smallest. 591. The preceding comes under the general head of finding the maximum and minimum of a trinomial ax^ + bx + c. Designating the variable value of the trinomial by y, we have: ax^ + bx + c = y or ax' + bx + c — y = 0, from which (576): _ -b ± V4ay- (4ac - j^ ^~ 2^ ■ Thus, in order to obtain a real value of x, the following condi- tion must be fulfilled: 4:ay^4:ac — ¥; (1) and there are two cases, according as the coefficient of x^ is posi- tive or negative. PROPERTIES OF TRINOMIALS OF SECOND DEGREE 241 Case 1. For a > 0, the relation (1) gives: _ 4 ac — 6^ It is seen that in this case for real values of x the smallest value 4 ac — 6^ o/ y is — , and since for this minimum value the radical becomes 0, we have x = — ■^—■ 2a ^ Thus the trinomials 3 x^ - 7 X + 2 and x^ + X + 1, in which the coefficient of x^ is positive, have respectively for their absolute minimum values, 4X3X2-7X7 25 , . , , ^ -77 ^^^-3 = - - , which corresponds to x = - ^^^ = - ; 4X1X1-1X13 , . , , ^ 1 TT-—. = -7 , which corresponds to x = — - • 4X1 4 ^ 2 Case 2. For a < 0, the relation (1) gives: 4 Q^Q A2 y ^ — 2 (since 4 a is negative). 4 ac — h"^ It is seen that the greatest value of y is — -r , and this maxi- mum corresponds to x = — -^r— • 2 a Thus the trinomial — 9x^ + 6a;— 1, in which the coefficient of x' is negative, has for an absolute maximum, value, 4X-9X-1-6X6 36-36 which corresponds to x = — 4 X - 9 - 36 6 1 = 0, 2X-9 3 PROPERTIES OF TRINOMIALS OF THE SECOND DEGREE The properties of the trinomials of the second degree written, in the form y = ax^ + bx + c may be summed up as follows : First property. (Unequal roots.) If in making a trinomial of the second degree equal to zero, two real unequal roots are obtained, any quantity lying between these two roots, substi- 242 ALGEBRA tuted for x in the trinomial, will give signs which are the opposite of that of the coefficient a of the first term of the second degree; and any quantity lying outside of the roots, that is greater or less than the roots, substituted for x in the trinomial, gives to this trinomial the same sign as that of the coefficient a of its first term. To demonstrate this, assume that a is positive, and let x' and a;" be the roots of the trinomial; then from the transformation in article (543) we may write: y = a {x — x') (x — x"). Replacing a; by a number a, which lies between the roots, that is, a;' > a > x" and a — x' < 0, u. - x" < 0, we have the product a (a — x') (a — x") = y, with the opposite sign to that of the coefficient a of its first term. From the above relations : LL - a;' > . a - x' <0, a. - x" > °^ a - x" < 0, we have the product a (« - a;') (a - x") = y, with, the same sign as that of a, since the two factors (a — x') and (a — x") are of the same sign, and the value of y approaches infinity as the value of a increases. Second property. (Equal roots.) If the roots of the trino- mial are equal, any number a substituted for x in the trinomial will give the same sign as that of the coefficient a of the first term. The trinomial may be written in the form y = a(x - x'y, and will always have the same sign as a for any value positive or negative given to x, and will approach infinity for increasing values of a = x. Third property. (Imaginary roots.) In case the roots are imaginary, any value substituted for x in the trinomial will give the same sign as that of the coefficient a of the first term. PROPERTIES OF TRINOMIALS OF SECOND DEGREE 243 Solving the equation, ax' + bx + c = 0, (1) we obtain, a;^ - b Ji Vb^ - 4ac .. 2a ' since the roots are imaginary, we have: 4 ac > b\ and c _6^ o 4a^ The quantity - being greater than a positive quantity, we may write: ^ = -^ +jfc^ (2) The relation (1) may be written: a(x' +-a; + -) = 0. \ a aj n Substituting the value of - (2) In this form it is seen that by replacing x by any value, a result y of the same sign as a would be obtained; therefore, in the case of imaginary roots, the trinomial ax? + bx + c = y always retains the same sign as the coefficient a of its first term; when X is replaced by any value, positive or negative, and the value of the trinomial approaches infinity, a = a; is increased. For X = — 7:;— ^the trinomial has a minimum value. 2 a " — Example 1. It is desired to study the following fraction; find its maximum and its minimum when x is varied. 244 ALGEBRA Write; a^ - 2 a; + 21 _ 6 a; - 14 ^' then a;' - 2 a; + 21 = 6 ail/ - 14 2/ or a;2 - 2 a; (1 + 3 2/) + 14 2/ + 21 = 0, X = 1 + 3 2/ ± V9 2/2 - 82/ - 20. If a; is to be real, the trinomial 9 y' — 8y — 20 must be positive; the roots of this trinomial are: y 9 2/' - 8 2/ - 20 = 0, 4 ± V16 + 9 X 20 2/' = 2; 2/"+-^ Thus two unequal roots are obtained, and the first property of trinomials of the second degree is applicable, and gives, for all values of y between y' and y", a negative trinomial and imaginary x; and for all values not between y' and y", a positive trinomial and a real y; therefore, y may be varied from 2 to + infinity and from — ^ to — infinity, and 2 is the minimum and — ^ the maximum value of the given fraction. It remains to determine the corresponding values of x. The maximum and minimum of y were deduced from the relation 9 2/' - 8 2/ - 20 = 0, which does away with the radical and gives for x: X = 1 +3y. Substituting successively 2/' = 2 and y" = - — for y, we obtain: x' = 7 for y' — 2 (minimum) 7 10 and x" = — 5 for y" = — pr- (maximum) Example 2. Study the variation of the expression, 2/ = a;±V2ar' — a;; PROPERTIES OP TRINOMIALS OF SECOND DEGREE 245 that is, determine the maximum and minimum of y when the quantity x varies in all possible manners. Find the roots of the polynomial 2 a;2 - x = 0, which may be written, a; (2 X - 1) = 0, x' = and x" = k • Thus two unequal roots are obtained, and the first property must be applied in order to study the variation of the quantity 2 7? — x; any quantity between and J substituted for x would make the quantity 2 x'^ — x negative, and thus give an imaginary value to y, while- any quantity not lying between those values would make the quantity 2 x^ — x positive; from this it follows that the quantity x can vary from - to + oo and from to — oo for all real values of y, and that x" = ^ is a minimum, and x = a maximum; therefore, the corresponding values of y may be calculated, which give: y' = x' = 0, corresponding to the maximum of x, y" = x" = ^, corresponding to the minimum of x. As to the maxima or minima of y, it is seen that the relation 2/ = + a; ± ^12 3? - X = + a; ± V a; (2 a; - 1) gives greater absolute values of y for greater absolute values of x, therefore, y varies from to + oo and from ^ to — °o • Example 3. Study the variation, 2/ = a^ + 6a; + 9. The roots of the trinomial are: a; = - 3 ± V,9 - 9 = - 3. These roots being equal, the above trinomial may be written, y = {x + Z){x + S) = {x + 2.)\ In this form it is seen that any value positive or negative would give a positive value to y; but for a; = — 3 the quantity 246 ALGEBRA y equals 0; therefore, y varies from to + oo , and x varies from + 00 to — 00 . Example 4. Study the variation, y = 3? — 4lX + 15. Putting the trinomial equal to and solving for x, ar^ - 4 a; + 15 = 0, a; = 2 ± V4 - 15 = 2 ± V^ni. The values of x being imaginary, the third property of tri- nomials must be applied in order to study the variation of the trinomial, that is, that any value substituted for x will give the trinomial the same sign as that of the coefScient of x^. The above trinomial may be written: a;^ - 4 a; + 15 = (a; - 2)2 - 4 + 15, y = {x-2y + 11. In this form it is seen that y is positive for all values of x, posi- tive or negative, and that the value of y increases with that of X] but for a; = 2, the quantity j/ is a minimum and is equal to; y- 11. From this minimum, y varies to + oo . Example 5. Study the variation, 2/ = 3 a; - 1 ± Vaj^ - 4 a; + 15, a;2 - 4 a; + 15 = 0, a; = 2 ± V^ni. Referring to Example 4, we may write, 2/ = 3 a; - 1 ± V(a; - 2)^ + 11. In this form the radical is positive for any value, positive or negative, given to x; and x may vary from — oo through to + 00. As to y, its maximum and minimum are obtained by making the radical as small as possible, that is, taking a; = 2, which gives for y. y = 3 X 2 - 1 ± Vn, 2/' = 5 + VTT (minimum), ^' = 5 — Vll (maximum). EQUATION OF THE THIRD DEGREE 247 These values are the Umits; therefore, y varies from y' to + oo, and from y" to — oo , but there is no value of x which can make y = o. EQUATION OF THE THIRD DEGREE 592. Transformations which permit the solution of an equa- tion of the third degree. The most general form of an equation of the third degree is: ax^ + bx^ + ex + d ^ 0. (1) All the terms may be divided by a, which will give a^ + Bx'' + Cx + D = (2) The term x' may be eliminated by proceeding as in the follow- ing special case. Given: a;3 _ 4 a;2 + 5 a; - 2 = 0. (3) Let x = y + h; h being indeterminate, and y a new unknown. Then substituting this value of x in equation (3), f + 3y% + Syh^ + h^ - 4:y^ - 8yh - 4:h^ + 5y + 5h - 2 = 0, or f + y'i3h-4:) +yi3¥-8h + 5) + ¥ - 'k¥ + 5h - 2 = 0. This relation is true for all values of h; therefore, we can put 3/1-4 = -I- Then substituting this value for h in all the terms of the last equation, we have an equation of the form: y" + py + q = 0, (4) wherein p is the numerical coefficient of the term y, and q the sum of all the known terms. It is in this form (4) that an equa- tion of the third degree is most often solved, or, which is the same thing, in the form: a^ + px + q = 0. The solution of third degree equations. 3? -{- px + q = Q. (a) 248 ALGEBRA Let X be replaced by the sum of two unknowns. X = y + z. (6) Substituting in (a), y^ + Sy^'z + Syz" + ^ + p{y + z) + q = 0, or y^ + ^ + {y + z) (3 yz + p) + q= 0. (c) The unknowns y and z should satisfy only the relation (6), therefore the following condition may be imposed: 3yz + p = 0. (d) ■f + z^ + q = 0. (e) Then reducing (c), From equation (d), and from equation (e), y^ + z^ = — From these it follows that the quantities y^ and z^ are the roots of the following equation, substituting x = y + z, -'/-i+v/f^+v^-i-v/?^;- <^) When the square root is positive, the calculation may be effected without difficulty and the roots of the equation deter- mined. The other roots are imaginary, and are calculated from the following formulas : Let A and B be the values of the two cubic radicals, then the three roots of the equation of the third degree are: Xi = A + B, Xi = Aa + Bo?, Xi = Aa? + Ba, wherein a represents one of the two imaginary cubic roots of EQUATION OF THE THIRD DEGREE 249 unity, or one of the roots of the equation x^ = 1, which gives be- sides a = 1, the two roots: Note. — See examples at end of Trigonometry (1072) Remark. When the quantity ^ + ;^ is negative, the square roots are imaginary, and consequently so are the cube roots, and it appears that the roots should be imaginary. But here is a peculiarity of the third degree equation, because the three roots are real. It is called the irreducible case of the third de- gree equation, and trigonometric transformations must be used to express the roots. (See end of Trigonometry.) In many cases numerical equations of the third degree may be solved without recourse to the general formula (A), by a process similar to that in (580). Thus, having given: write then make 3 a;=' - 4 a^ -I- 5 a; - 18 = 0, y = 3x^ - ix^ + 5x - 18, a; = 0, 1, 2, 3, - 1, - 2, - 3, etc. and calculate the corresponding values of y, and plot the graph of the equation (546). The points where the graph cuts the x- axis will determine the roots of the equation with a sufficient degree of accuracy. 593. The solution of an equation in annuities by the graphic method. Calculate the rate of an annuity, a = $11,986, corresponding to a loan of c = $200,000 for 50 years. Referring to article (410), it is seen that the solution of this problem is expressed by the formula (3). Therefore, the relation r=^ ^ (1) c c (1 - r)" ^ ' wherein r = rate (unknown), c = $200,000, a = $11,986, n = 50 years, is to be solved. 250 ALGEBRA It is noted that the second term of the second member of the equation is smaller than the first — j if the second term is neg- lected, the value of r will be too large. 200,000 ^■^^y^'^- Substituting this value or 0.06 in equation (1), 11,986 r = 0.05993 - ■ 200,000 (1.06)5° then with logarithms, r = 0.05669. To find if this value is too large or too small, write (1) in the form _ a a . y~c c(l+r)» *■• ^"'^ Substituting 0.05669 for r, y = - r = - 0.05669. Now it is seen that this value is too large; try r = 0.056, the equation (2) gives: y = - 0.0000007. This very small value indicates that the value of r is very nearly correct. If r is taken as 0.055, we find y = + 0.0008089, which shows that the value of r lies between 0.056 and 0.055. Below are the various values obtained in the trials: Values of r Values of y 0.05993 - 0.05669 0.056 - 0.0000007 0.0555 + O.00041 Thus the method of trial and error consists in giving values to r which give opposite signs to y, and in the given example, it is foimd that the value of r lies between 0.056 and 0.055. Try- ing r = 0.0558, we still get a positive value for y, which shows that r lies between 0.056 and 0.0558, and so on. The same is found to be true for r = 0.0559; thus the value r = 0.056 is correct to less than one thousandth. PART III GEOMETRY DEFINITIONS 594. The volume of a body is that portion of space occupied by the body. The Umit of a body or its volume is the surface of the body or the volume. The limit of a portion of the surface is a line. The extremities of a portion of a line are called points. Remabk. a volume has three dimensions: Length, breadth, and thickness; a surface has two, length and breadth; a line has only one, length; a point has none. 595. Volumes, surfaces, and lines come imder the common head of geometrical figures. Geometrical figures are represented to the eye by material objects; but geometry has nothing to do with the material, it is simply the shape and size which are studied. 596. Two figures coincide when they have the same shape and size and are superposed one upon the other. Two equal figures have the same shape and size, and coincide throughout their extent when superposed one upon the other. Two equivalent figures have the same size. Remahk. Two equal figures are always equivalent, but two equivalent figures are not necessarily equal. 597. A straight or right fine may be thought of as a thread tightly stretched between two points. A straight line is the shortest dis- ^ ^ tance between two points A and B. ' ; Only one straight line can be drawn between two points A and B; two straight lines which have two points in common coincide throughout their length, and two points are sufficient to determine a straight fine. 251 262 GEOMETRY 598. The direction of any straight line AB is the line itself prolonged indefinitely from its extremities A and B. 599. Directions of a line. Every straight line may be con- sidered as having two directions: thus in Fig. 10 we have the directions AB and BA, which are distinguished by the order of the letters. 600. A broken line ACDB is composed of a series of different successive straight lines. 601. A curved line AmnB is a line no part of which is straight. It is the limit which a broken line approaches when the number of its elements is indefinitely increased (136). 602. A plane is an indefinite surface, such that a straight line joining any two points in that surface will lie wholly in the surface. 603. A jilane may he constructed to contain: First, any three points not in a straight line; Second, any two intersecting straight lines; Third, any line and a point which lies outside of the line; but only one such plane can be constructed, because all planes containing three points, two intersecting lines or a point and a line, coincide and are one. 604. The intersection of a plane and a line is a point. The intersection of two planes is a straight line, which contains all the points common to both. 605. A figure is a plane figure when it has all its points in the same plane. 606. The contour or perimeter of a surface is the line which bounds the surface on all sides. 607. A broken surface is a surface composed of several plane surfaces not situated in the same plane (600). 608. A curved surface is a surface no part of which is plane. It is the limit approached by a broken surface when the number of its elements is indefinitely increased (601). 609. A figure which contains all the points that fulfill a certain set of conditions is called a geometrical locus (585). 610. Geometry is the science which treats of position, form, and magnitude. Plane geometry treats of plane figures. Solid geometry treats of solids and space. PLANE GEOMETEY BOOK I STRAIGHT LINES 611. Two straight lines AB and AC drawn from the same point A and in different directions form a geometrical figiire called an angle; the lines AB and AC, which may be prolonged indefinitely, are the sides of the angle; and the common point A is the vertex of the angle. The magnitude of an angle is independent of that of the sides. A very clear idea of an angle and its magnitude may be obtained by supposing the lines '^'®' ^^ to coincide first, and then that they be spread apart like a compass; the angle, at first 0, increases in value as the legs of the compass are separated. A single angle is designated by the letter at its vertex; thus, one would say the angle A. But when there are several angles which have the same vertex, each is designated by the three letters BAC or CAB, with the letter which represents the vertex in the middle. The angle A is the angle between the two straight lines AB and AC (Fig. 12); and, in general, the angle between the two straight lines AB and CD (Fig. 13), which may or may not be situated in the same plane, is the angle BCD' formed by one of the lines AB and a line CD' paral- ■^s- ^^ lei to CD and intersecting AB in any point C. It is seen that an- angle between two straight lines is deter- mined by the direction of the lines; thus, for the direction AB and CD the angle would be AC'D'. 612. Two angles BAC and CAD are adjacent when they have the same vertex A, and one side common, and are exterior to one another (Fig. 14). 613. Two angles are vertical angles when they have the same 253 254 GEOMETRY vertex and the sides of one are prolongations of the other. Such are angles AOC and BOD, AOD and BOC (Fig. 15) sides of the Fig. 14 'A >C' B rig. 15 A B Fig. 18 Vertical angles are equal. 614. A straight line is perpendicular to another when by the intersection of one with the other equal adjacent angles are formed. Thus (Fig. 16), supposing AOC = BOC, CD is per- pendicular to AB; and therefore, AB is also perpendicular to CD. When one line is perpendicular to another, the latter is also perpendicular to the former. Lines which intersect and are not perpendicular are oblique lines. Such are AB and CD in (Fig. 15). 615. A vertical line is one if prolonged would pass through the center of the earth. All straight lines perpendicular to a vertical are horizontal (766). 616. The angles formed by the intersection of two lines per- pendicular to one another are called right angles. Such are AOC and BOC in (Fig. 16). All right angles are equal. All angles BOD (Fig. 15), less than a right angle, are acute angles; and all angles AOD (Fig. 15), greater than a right angle, are obtuse angles. 617. Two angles are complementary or complements when their sum is equal to a right angle; such are the angles BAC and CAD (Fig. 14), supposing their sum BAD to be a right angle. Two angles are supplementary or supplements when their sum is equal to two right angles or a straight angle. Such are the two angles AOD, BOD (Fig. 15). B' B A! Fig. 17 618. The sum of all the consecutive adjacent angles AOB, BOB', B'OB", B"OA', about a point A on one side of a straight line A' A, is equal to a straight angle or two right angles. The perpendicular PO erected at the point STRAIGHT LINES 255 on AA' determines two right angles AOP and POA' which are equal to the sum of AOB, BOB', B'OB", B"OA'. If two angles AOB, BOA', are supplementary (617), the exterior sides OA, OA', form a straight line. The sum of all the consecutive adjacent angles AOB, BOB', B'OB" . . ., formed about a point by any number of straight lines radiating from the point, is equal to four right angles. 619. From any point a perpendicular may be drawn to a given line, but only one can be drawn from that given point. To erect a perpendicular OC upon a straight line AB (Fig. 16), is to draw a perpendicular through B' the line at a point taken on the line. To drop a perpendicular CO upon a straight line jj AB (Fig. 16), is to draw a perpendicular to the line passing through a given point C outside of the line. 620. From a point A outside of a given straight line BC, drop a perpendicular AD and several obliques AE, AF, and AG; then: First, the perpendicular is shorter than any oblique; second, the two obliques AE, AF, which cut off equal dis- tances at the foot of the perpendicular, are equal; third, of the two obliques AE, AG, the one AE, which cuts off the shorter distance from the base of the perpendicular, is the shorter line. '^' The converse holds for all these statements. The perpendicular AD, being the shortest distance from the point A to the straight line, is the distance from the point to the line. 621. A perpendicular CD erected at the middle of a line AB is the geometrical locus of all points equidistant from the extremities of the line (609). That is, that any point C, taken ^e on CD, gives AC = BC, and any point E not on the q Hne CD, we have AE > BE or AE < BE, according as E is on the right or left of CD. 622. The bisector of an angle is.^ a straight line which divides the rig. 20 angle into two equal parts. The bisector AD of an angle BAC is the geomet- rical locus of all the points within the angle and equidistant from the sides (609). That is, if from any point E taken on AD the perpendiculars 256 GEOMETRY EG and EF are drawn to the sides, these perpendiculars are equal; if a point H is taken outside of AD, the perpendicular HF will be greater than HI. The bisectors of two vertical angles form a straight line (613). The bisectors of two supplementary adjacent angles are per- pendicular to one another and form a right angle (612, 614, 617). 623. Two straight lines AB and CD (Fig. 22) are parallel when being in the same plane they may be indefinitely prolonged without meeting (598). Through a point A (Fig. 22) exterior to a given line CD, one and only one parallel to this ^' line can be drawn. 624. When any two straight Hues AB, CD, situated in the same plane, are cut by a third straight line EF, called a trans- versal, we have the folloAving angles formed: 1st. Interior angles, each of the four angles formed between the two given lines. Such are AGH, BGH, CHG, DHG. 2d. Exterior angles, each of the four angles formed outside of the two given lines. Such are AGE, BGE, CHF, DHF. 3d. The alternate-interior angles are the two angles formed on opposite sides of the transversal, interior and not adjacent. Such are AGH and DHG, BGH and CHG. 4th. The interior-exterior angles are two angles, one exterior and one interior, both on the same side of the transversal and not adjacent. Such are AGH and CHF, BGH and DHF, CHG and AGE, DGH and BGE. 5th. The alternate-exterior angles are the two angles formed on opposite sides of the transversal, exterior and not adjacent. Such are AGE and DHF, BGE and CHF. 625. When the two lines AB and CD are parallel (Fig. 22): 1st. The sum of the two interior angles on the same side of the transversal is equal to two right angles ; and conversely, if the sum of two interior angles situated on the same side of a trans- versal is equal to two right angles the lines are parallel. 2d. The sum of the two exterior angles on the same side of the transversal is equal to two right angles, and conversely. 3d. Any two angles of the same name, alternate-interior or alternate-exterior, are equal, and conversely. STRAIGHT LINES 257 626. Two straight lines AB and A'B', perpendiciilar to a third straight Une CD, are parallel to one another (614 and 623). 627. Any straight line CD perpendicular to one of two parallels is perpendicular to the other. .0 . B A' The part intercepted by the two parallels on the perpendicular CD is a constant, that is, ' g the parallels are everywhere equidistant from ^„ 23 one another. 628. The two straight lines AB and A'B' being parallel to one another (Fig. 24), any straight hne EF, which is parallel to one, is also parallel to the other. 629. Two angles whose sides are perpendicular are either equal or supplementary (617). OA being perpendicular to OC, and OB to OD, we have AOB = COD or EOF, and AOB is the supplement of DOE or COF. Remark. The same holds where the angles have not the same vertex. 630. Two angles whose sides are parallel each to each, are either equal or supplementary. AB being parallel to DE, and BC to EF, we have ABC = DEF or D'EF', and ABC is sup- plementary to DEF' or D'EF. The two angles are equal when their sides extend in the same direction or in opposite Fig. 26 \'^ directions from their vertices, and supplemen- tary when two of the parallel sides extend in one direction and two in the other. BOOK II POLYGONS 631. A polygon is a plane figure bounded on all sides by a broken line (600, 605), Such is the figure ABODE. Each of the straight lines AB, BC, . . ., which form the perimeter of the polygon, is a side of the polygon. Each of the angles EAB, ABC . . ., formed by two adjacent sides of the polygon, is an angle of the polygon. Any line AC joining two vertices not adjacent is a diagonal of the polygon. 632. A polygon of three sides is called a triangle; one of four sides, a qiiadrilateral; one of five, a pentagon; one of six, a hexa- gon; one of seven, a heptagon; one of eight, an . octagon; one of nine, an enneagon; one of ten, a decagon; one of eleven, endecagon; one of twelve, a dodecagon ; one of fifteen, a pentadecagon ; one of twenty, an icosagon. 633. A triangle ABC is a right triangle when one of its angles is a right angle (616). The hypotenuse of a right triangle is the side AC opposite the right angle ABC. 634. A triangle is an obtuse triangle when one of its angles is obtuse (616). A triangle is an acute triangle when all of its angles are acute. 635. A triangle ABC is an isosceles triangle when two of its sides AB and AC are equal. Remark. In an isosceles triangle, the angles B and C opposite the equal sides are equal; and con- versely, if in a triangle two angles B and C are equal, the sides opposite these angles are equal and the triangle is isosceles. In an isosceles tri- angle the altitude Am bisects the angle A and the Fig. 29 base 5C (639). 258 POLYGONS 259 636. A triangle is equilateral when its three sides are equal. Remaek. In an equilateral triangle the angles are all equal; and, conversely, if all the angles are equal, the triangle is equi- lateral. A triangle is a scalene triangle when none of its sides nor angles are equal. 637. In any triangle ABC, any side AC is smaller than the sum AB + BC of the other two sides and greater than their difference AB - BC. B^ Kg. 30 Kg. 31 638. In a triangle ABC (Fig. 30), of two unequal sides AB Emd AC, the smaller side is opposite the smaller angle; and, con- versely, the side AB being smaller than the side AC, the angle C is smaller than the angle B. 639. The base of a triangle may be any side. In the isosceles triangle (Fig. 29), the side BC which is not equal to the others is taken as the base. The vertex of a triangle is the vertex of the ^ E p angle opposite the base. / \ / The altitude of a triangle is the perpendicular f/- f — ^ distance from the base to the vertex. Fig. 32 Thus, having BC as base (Fig. 31), the vertex is A, the altitude is AD. 640. A parallelogram is a quadrilateral whose opposite sides are parallel. Such is ABCD. In a parallelogram the opposite sides and angles are equal. In order that a quadrilateral be a parallelogram, two opposite sides must be equal and parallel. It is also a parallelogram when the. opposite sides, are equal each to each, or when the opposite angles are equal each to each. n E g 641. Any side may be taken as the base of a \ parallelogram. -i"' The altitude of a parallelogram is the dis- 'B SHg.'aa tance from the base to the opposite side. 260 GEOMETRY Thus, having taken AB for the base (Fig. 32), the altitude is the perpendicular EF intercepted by the base and the side DC (627). 642. A trapezoid is a quadrilateral which has two sides and only two sides parallel. Such is ABCD (Fig. 33). The bases of a trapezoid are the two parallel sides AB and DC. The altitude of a trapezoid is the distance EF between the two bases (627). A trapezoid is rectangular when one of the non-parallel sides is perpendicular to the base, C Pig. 34 Fig. 36 A trapezoid is isosceles or symmetrical when its non-parallel sides or legs are equal. 643. A rectangle is a parallelogram ABCD whose angles are right angles (Fig. 34). 644. The base of a rectangle may be any side.' The altitude of a rectangle is the length of either side adjacent to the base. 645. A rhombus is a parallelogram ABCD whose sides are all equal (Fig. 35). Fig. 37 Pig. 38 Any side may be taken as the base of the rhombus. (641) The altitude of the rhombus is the distance from the base to the opposite side (627). 646. A square is a rectangle ABCD with equal sides (Fig. 36). The base is any one of the sides, and the altitude the adjacent side. 647. A polygon is equiangular when all its angles are equal. Such are the equilateral triangle and the rectangle (636, 643). POLYGONS 261 A polygon is equilateral when all its sides are equal (600, 609). Remark. A polygon can be equiangular and equilateral at tlie same time. Such are the equilateral triangle and the square. 648. A broken line or a curved line (600, 601) is said to be convex when it lies entirely on one side of any one of its straight line elements, finite in (Fig. 37) and infinitely small in (Fig. 38). A straight line can not cut a convex line in more than two points. A polygon is convex when bounded by a convex line. 649. A certain convex line AEFGB is greater than any other convex line ACDB which is included by the first when the two have their extremities at the same points A and B. Since DB < DIB, CI < CHGI, and AH < AEFH, we have ACDB < ACIB < AHGB < AEFGB. The exterior line may be formed by two sides of a triangle, and the interior line by two lines joining a point with- in the triangle to the extremities of the base. When the exterior convex line A'EFGB meets the line AB prolonged in A' so that the perpendicular AL < A'L, we still have ACDB < A'EFGB. A closed convex line is greater than any convex line totally included by it. Remark. All which has been said applies to convex lines which are wholly or only partly composed of curves as well as to broken lines. J A B D Fig. 41 650. Angles formed by one side of a polygon and the prolonga- tion of an adjacent side are called exterior angles of the polygon. Such is the angle DCH, formed by the side CD and the prolonga- tion CH of the adjacent side CB. EDI, AEK, etc., are exterior angles (653). 651. The two angles of a triangle not adjacent to the exterior angle are called opposite interior angles. Such are A and C with reference to the exterior angle CBD (653). 262 GEOMETRY 652. The sum of the interior angles of a polygon is equal to two right angles taken as many times less two as the figure has sides. Thus, s being the sum of the angles, and n the number of sides of a polygon, we have: & = 2 (n — 2) = (2 n — 4) rt /4 (right angles). For the triangle n = 3, s = 2 (3 - 2) = 2 rM. For the quadrilateral n = 4, s = 2 (4 — 2) = 4 r< ^. For the pentagon n = 5, s = 2 {5 — 2) = 6 rt A. For the hexagon n = 6, s = 2 (6 - 2) = 8 rt ^. and so on for any number of sides. Remark. The sum of the angles of a triangle being equal to two right angles, it follows that if one of the three angles is right or obtuse, the two others are acute. The two acute angles of a right triangle are coniDlementary (617, 633). 653. The exterior angle CBD (Fig. 41) of a triangle is equal to the sum of the two opposite interior angles A and C, and con- sequently greater than either of them. When the successive sides of a polygon are prolonged as in (Fig. 40), the sum CBG + DCH+ EDI + ... of the exterior angles is always equal to four right angles. 654. Any two triangles ABC, A'B'C, are equal: 1st. When two sides and the included angle of one are equal to two sides and the included angle of the other: ZA=ZA',AB = A'B', AC = A'C. 2d. When one side and the ad- jacent angles of one are equal to one side and the adjacent angles of the other: A5 = A'B', Z A = Z A', Z B = Z B'. 3d. When they have three sides equal each to each (663). 655. Two right triangles ABC, A'B'C, are equal: 1st. When the hypotenuse and an acute angle of one are equal to the hypotenuse and an acute angle of the other: BC = B'C, ZB = ZB'. 2d. When the hypotenuse and one leg of one is equal to the hypotenuse and one leg of the other: B'C = BC, A'B' = AB. 656. Two parallelograms are equal when two adjacent, sides POLYGONS 263 and the included angle of one are equal to two adjacent sides and the included angle of the other (640). Two rectangles are equal when two adjacent sides of one are equal to two adjacent sides of the other (643). Two rhombuses are equal when one side and one angle of one are equal to one side and one angle of the other (645). Two squares are equal when one side of one is equal to one side of the other (646). 657. Two polygons of n sides are equal when they have n — 2 angles or sides equal each to each, and situated in the same order, and respectively n — 1 sides or angles equal each to each, and situated in the same order. The number of conditions necessary for the equality of two polygons of n sides is, therefore, (w — 2) + (w — 1) = 2 n — 3, and these conditions suffice when they are properly chosen. 658. When two triangles have two sides of one equal respec- tively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first is greater than the third side of the second. Conversely, when two sides of a triangle are equal respectively to two sides of another, but the third side of the first is greater than the third side of the second, then the angle opposite the third side of the first is greater than the angle opposite the third side of the second. 659. In an isosceles triangle (Fig. 29), the line Am drawn from the vertex to the middle of the base is perpendicular to the base and bisects the angle at the vertex. Hg. 44 Fig. 45 660. The diagonals of a parallelogram bisect each other; con- versely, if the diagonals of a quadrilateral bisect each other, the figure is a parallelogram (Fig. 44). Besides these properties of a parallelogram: 264 GEOMETRY 1st. The diagonals of a rectangle are equal (Fig. 45); from this it follows that in a right triangle BCD, the middle point of the hypotenuse is equidistant from the three vertices, B, C, D. 2d. The diagonals of a rhombus are perpendicular to one another (Fig. 46). 3d. The diagonals of a square are equal and perpendicular to each other. The converse statements of the above are true. 661. The diagonal of a parallelogram divides the figure into two equal triangles (Fig. 44). The diagonals of a rhombus and a square divide the figure into four equal right triangles (Fig. 46). The point of intersection of the two diagonals of any paral- lelogram is the center of the figure (Figs. 44-46), that is, the point lies in the middle of any transversal which contains it and terminates in the perimeter of the parallelogram. Drawing two such transversals and connecting their extremities by straight lines, we have a parallelogram. All transversals which pass through the point divide the parallelogram into two equal polygons. 662. In any trapezoid: First, the straight line MN, which joins the middles of the opposite non-parallel sides, or legs, is parallel to the bases and equal to half their sum, MN = ^ ; second, the straight line EF, which joins the middles of the diag- onals, coincides with MN and is equal to half the difference of , , „^ AB-DC the bases, EF = ^ „ In any trapezoid the middles of the bases, the '''^^^ ^^ „ point of intersection of the diagonals, and the ^E_l2\^ vertex of the angle formed by producing the A- E legs, lie in the same straight line. '^' 663. A triangle may be constructed: 1st. When two sides and the included angle are given. 2d. When one side and two angles are given. 3d. When the three sides are given. 4th. When two sides and an angle opposite one of the sides are given (654). (See problems in Geometry.) 664. A parallelogram may be constructed when two adjacent sides and the included angle are given; a rectangle, when two adjacent sides are given; a rhombus, when one side and one angle are given; a square, when one side is given (656). BOOK m THE CIRCLE 665. The circle is a plane surface bounded by a curved line called the circumference, all points of which are equally distant from a point within, called the center. Any straight line drawn from the center to the circum- ference is called a radius. Thus the circumference is the geometrical locus of all points situated at a distance equal to the radius from the center (609). Two circles of the same radius are equal, and their circumferences are equal. 666. An arc of a circle is a portion BmC of the circumference. The chord of an arc is a straight line BC joining the extremi- ties of the arc. Any chord BD which passes through the center, is called a diameter, and divides the circle and its circumference- into two equal parts. The diameter is equal to two radii; and since the radii of the same circle are all equal, so are the diameters. Any chord BC, which does not pass through the center, is less than the diameter. The diameter divides the circle and circumference into two equal parts; and any chord, other than a diameter, divides them into two unequal parts. 667. Any angle AOD, whose vertex is at the center, is called an angle at the center. An arc is intercepted by an angle at the center when the radii which form the sides of the angle are drawn to the extremities of the arc. 668. That part of a circle BmC, bounded by an arc and its chord, is called a segment of the circle. The chord is the base of the segment. That part AOD of a circle bounded by an arc and two radii is called a sector of a circle. The arc is the base of the sector; the center of the circle is the vertex. 265 266 GEOMETRY 669. The longest chord wliich can be drawn through a point m, which lies within the circle, is the diameter DD' which passes through the point; the shortest chord is the chord AB perpendicular to the diameter DD'. 670. The shortest and the longest line which can be drawn from a point to the circumference of a circle have the same direction as a line drawn from the given point to the center of the circle, whether the point be within or without the circle. Thus (Fig. 49), the longest line from the point m to the circumference is mD', and the shortest is mD. Kg. 49 Fig. 60 Fig. 51 Fig. 52 671. Any diameter DD' (Fig. 49), perpendicular to a chord AB, divides the chord and each of its subtended arcs into two equal parts, mA = mB, DA = DB, D'A = D'B. A perpendicular erected at the middle of a chord passes through the center of the circle (621). 672. In the same circle or two equal circles: 1st. Two equal arcs ADB, A' D'B' (Fig. 50), not greater than a semicircumference, are subtended by equal chords AB, A'B', and conversely. 2d. Of two arcs the greater is subtended by the greater chord, and conversely. 3d. Two equal chords AB, A'B', are equally distant from the center, OD = OD' (Fig. 51), and conversely. 4th. Of the two chords AB, AB' (Fig. 52), the longer is nearer the center, OD < OD', and conversely. 5th. Equal arcs ADB, A'D'B', are subtended by equal angles at the center (Fig. 50), and conversely. 6th. A greater arc is subtended by a greater angle at the center, and conversely. 7th. The two equal chords AB A'B' (Fig. 50), are the bases of equal segments, and conversely. THE CIRCLE 267 8th. Two equal arcs ADB, A'D'B' (Fig. 50), are the bases of equal sectors, and conversely. 673. A straight line BC is inscribed in a circle (Fig. 48) when it has its extremities in the circumference of that circle. The angle CBD formed by two chords which meet at the cir- cumference is called an inscribed angle (Fig. 48). An angle is inscribed in a segment when its vertex Hes in the circumference and its sides pass through the ends of the base of the segment. All angles inscribed in the same segment are equal (684). A polygon is inscribed in a circle when its sides are inscribed in the circle (Fig. 62). The polygon is circumscribed by the circle. 674. A straight line can not cut a circumference in more than two points, and all lines which cut the circumference in two points are called secants. 675. A straight line AB is tangent to a circle when they have but one point m in common. A tangent may be thought of as being the limit of a secant where the two points of intersection ap- proach each other and finally coincide. The perpendicular AB erected at the extremity of a radius Om is tangent to the circle. The perpendicular Om erected at the point of contact of the tangent AB is normal to the circumference at the point m. All normals to the circumference pass through the center, and all radii are normal to the circumference. The shortest and longest distance from a point to the circumference of a circle are the normals to the circumference which pass through the point C670). fig. 53 Fig. 54, Fig. B5 Two circles and 0' are tangent to each other when they have one point m in common. They are externally or internally tan- gent according as one lies wholly without or within the other. Two circles tangent to the same line at the same point are tangent to each other. The point common to the tangent and FI I^ F A/ ^ 1 ■-> 1 \B ' ^ \ 1 1 ^ 1 ^ / G %. H 268 ' GEOMETRY the circumference (Tig. 53), or to the two tangent circumferences (Figs. 54 and 55), is called the point of contact or point of tangency. 676. Two parallels intercept equal arcs upon the circumference; this is true when they are two tangents EF, GH, or chords AB, CD, or a chord and a tangent AB, EF. Conversely, two chords, two tangents, or a chord and a tangent which intercept equal arcs, are parallel. 677. A polygon is circumscr-^ed about a circle when each of its sides is tangent to the circle at a point between the extremities (Fig. 63). The circle is inscribed in the polygon. ' Fig. 56 ir- J o ^ 678. A straight line is normal or oblique to a circumference or to an arc which it meets in a point, according as it is perpendicular or oblique to the tangent drawn to the cir- cumference or arc at that point (675). 679. Two circles are concentric when they have the same center. When two non-concentric circles are in the same plane, a line passing through their centers is called the line of centers. 680. A point can always be found which is equi- ^' distant from three others not in a straight line, and in the same plane with them; but only one can be found, and this is the center of a circle, whose circumference passes through the three points. (See Problems.) A circle, and only one, can be drawn through three points which are not in the same straight line (688). Two circles can not intersect in more than two points. The center is the only point from which more than two equal lines can be drawn to the circumference. 681. When two circles are tangent externally (Fig. 54), the distance between centers is equal to the sum of the radii. If the two circles are tangent internally (Fig. 55), the distance between centers is equal to the difference of the radii. The line of centers passes through the point of contact. 682. When two circles have no point in common (Figs. 58 and 59), the distance between centers is either greater than the sum of the radii or less than the difference, according as one circle lies wholly without or within the other. THE CIRCLE 269 683. When two circles intersect (Fig. 60), the line of centers is the perpendicular bisector of the chord mp which joins the points common to both, and the distance between centers is less than the sum of the radii and greater than their difference; we have 00' < Om + O'm and 00' > Om - O'm (637). Fig. 58 Fig. 59 Fig. 60 Conversely, when the distance between centers is less than the sum of the radii and greater than their difference, the circles in- tersect each other. When two circles have a common point m outside of the line of centers, they cut each other in a second point p, situated on the other side of the line of centers on a perpendicular to the line of centers and the same distance from it as the other point. 684. Any inscribed angle BCD is equal to half the angle at the center BOD, which intercepts the same arc jBD (673). All angles inscribed in a semicircle are right angles. A circle drawn upon a given line as diameter is the locus of the vertices of all the right trian- gles which have the given line for hypotenuse (609). Any angle ACB formed by a tangent AC and a chord CB is equal to half the angle at the center COB, which is subtended by the same arc CB; and therefore, it is equal to any angle inscribed in the seg- ment CDB which has the chord CB for a base. Fig. 61 ® B Fig. 62 Fig. 63 685. The opposite angles of any quadrilateral inscribed in a circle are supplementary, A + C = B + D = 2 right angles, and conversely. 270 GEOMETRY Fig. 65 686. The sum AB + DC of the opposite sides of a quadrilat- eral circumscribed about a circle (677) is equal to the sum AD + BC of the other two sides, and conversely. 687. The three bisectors of the an- gles of a triangle intersect in the same point (Fig. 64), which is the center of a circle inscribed in the triangle. The three bisectors of the exterior angles of a triangle (Fig. 65) meet in pairs on each of the bisectors of the interior angles produced, and these points of intersection 0', 0", 0'", are centers of circles each tangent to one of the sides of the triangle and the other two sides prolonged. These circles are called escribed circles. 688. The perpendicular bisectors of the sides of a triangle intersect in a point (Fig. 66), which is the center of a circum- scribed circle (680). 689. The three medians, that is, the three lines joining the vertices and the middles of the opposite sides, meet in the same point, which is the center of gravity of the triangle. 690. The radical axis of two circles (Fig. 67) is a geometrical locus XX', such that if tangents MT and MT' to the circles be drawn from any point M on the line they will be equal, XX' being perpendicular to the line of centers 00'. Draw- ing a common exterior tan- gent KK' to the two circles and bisecting it, we can construct the locus by drawing a perpendicular to the line of centers through the middle point of the common tangent. If the two circles are internally or externally tangent, the radical axis is the common tangent drawn through the point of contact; and if the two circles intersect each other, the radical axis is the common chord indefinitely produced in both directions. Fig. 67 BOOK IV SIMILAR POLYGONS AND THE MEASUREMENT OF ANGLES 691. Two lengths are proportional to two other lengths when their ratio is equal to that of the others (326). Lengths being measured in certain fixed units, these units may- be substituted in the ratios and the arithmetical operations performed. 692. To divide a line in extreme and mean ratio, is to divide it into two parts such that the larger part is the mean propor- tional between the whole line and the other part (330, 344, and Problems). 693. The parallels AA', BB', CC . . ., intercept proportional segments on the transversals PQ, RS. Thus: AB BC CD 'S Fig. 68 a/ ' if- b/ a'/b' .c/ B'Ac D/ cAiy' E/ d/Ie' ■7 EM' A'B' B'C CD' AE A'E' of any segments These ratios are also equal to that such as AE and A'E'. If the segments or intercepts on one trans- versal are equal, AB = BC = CD . . ., those on another transversal ,are also equal, A'B' = B'C = CD' = . . . 694. All lines OA, OB, OC . . ., meeting in a common point 0, intercept proportional seg- ments on two parallels AE, A'E'. Thus: AB BC^ CD B'C E C P Kg. 69 A'B' and these ratios are also equal to CD' AD A'D' the ratio of any two cor- responding segments AD and A'D'. 271 272 GEOMETRY Fig. 70 695. Two polygons ABODE, A'B'C'D'E', are similar -whew the angles of one are equal to the angles of the other and in the same order (A = A' , B = B', C = C . . .), and homologous sides are proportional. (AB _ BC_^ DC^ \ \A'B' B'C D'C ■■■■)• In two similar polygons: First, when the angles A, B . . . of one polygon are respectively equal to the angles A',B' ... of another, they are said to be homolo- gous angles; Second, the adjacent sides AB and A'B', BC and B'C of homol- ogous angles are homologous sides; Third, the vertices of homologous angles are homologous vertices; Fourth, diagonals AC and A'C . . . which join homologous Vertices are homolo- gous diagonals; Fifth, triangles ABC and A'B'C, ACD and A'C'D', which have homologous vertices, are homologous triangles. The ratio of the homologous sides of two similar polygons is the ratio of the symmetry of the two figures. 696. The straight lines Aa, Bb, . . ., which join the vertices of two similar polygons, meet in a point when prolonged; this point is called the center of symmetry. We have : OA^OB AB Oa Ob"" Ab' ratio of symmetry. If the figures have equal an- gles and proportional sides, but placed in an inverse order, they ^ still have a center of symmetry 0; and we have; OA^OB AB Oa Ob"'Ab' Fig 71. Two points p and p' in two similar figures (Fig. 71), such that a line joining them passes through the center of symmetry when prolonged, are said to be homologous points. The same is true in (Fig. 72). MEASUREMENT OF ANGLES 273 Two circles have two centers of symmetry, one between them 0', and one external to them 0, which are located at the inter- sections of their common tangents. 697. All transversals which cut the three sides of a triangle B Fig. 72 Kg. 73 ABC, determine six segments such that the product of any three which are not consecutive, equals the product of the other three. Thus, the consecutive segments being BD, DA, AE, EC, CF, FB, we have : BD X AE X CF = DA X EC X FB. The six segments are said to be in involution. The transversal may cut the sides of the triangle prolonged. Conversely, if three points taken on the sides of a triangle determine six segments in involution, these three points are in a straight line. 698. If three unequal but similar figures have their homologous dimen- sions parallel (Fig. 75), the three centers of symmetry 0, 0', 0", are in a straight line, and this line is called the axis of symmetry. If one of the figures has its dimensions situated in the inverse order of the others (Fig. 76), the centers of symmetry still fall in one straight line. Three circles have in general, six centers of symmetry, situated in threes, on four axes of symmetry (Fig. 77). 699. In any triangle ABC (Fig. 78) a straight line DE drawn 274 GEOMETRY parallel to the base, First, divides the sides proportionally, — - = -— ; = -— , and conversely; Second, forms, together with the adjacent sides of the triangle, a triangle ADE which is similar to the first ^5C (693, 695). 700. Two triangles ABC and A'B' -^ C are similar: 1st. When the angles are equal each to each: A = A', B = B', C' = C When two angles are equal, the third must be, and, therefore, two triangles are similar when two angles are equal each to each, 2d. When their sides are proportional: AB BC CA A'B' B'C C'A' 3d. When they have equal angles between adjacent pro- portional sides: /A - / A' A?_ =A^ . ' A'B' A'C 4th. When they have sides parallel (Fig. 79) or perpendicu- lar (Fig. 80) each to each. 5th. When they are right triangles and have the hypotenuse and one leg proportional each to each. Remark 1. In two similar triangles the homologous sides are opposite equal angles. Remark 2. In two triangles which have their sides parallel Fig. 78 Fig. 79 Fig. 80 or perpendicular each to each (4th), the homologous sides are parallel or perpendicular each to each. 701. Two parallelograms are similar when they have equal angles between adjacent proportional sides (695). MEASUREMENT OF ANGLES 275 702. Two polygons are similar (Fig. 70) when they can be divided into the same number of similar triangles situated in the same order, and conversely. Two polygons similar to a third are similar to each other. 703. In two similar polygons the perimeters and homologous diameters are proportional to the homologous sides; thus we have (Fig. 70): AB + BC + CD + DE + EA AC AB A'B' + B'C + CD' + D'E' + E'A' A'C A'B' (695) 704. The bisector of the vertex angle of a triangle divides the base BC into two segments proportional -..^ G to the adjacent sides, -^ = -j-^> and ^---■■^l/^Vs" , . ^n conversely. I " ^ The bisector of the exterior angle GAB cuts the opposite side produced so as to form segments which are proportional to the adjacent sides, -^ = -j^ = ytt' and conversely. From the proportion BP BI , , CT'^'CV ^"^ we have, CI' XBI = BI' y CI, which shows that the product of the whole line CI' and the mid- dle segment BI is equal to the product of the two extreme segments BI' and CI. The proportion (a) is said to be a harmonical proportion; the points /', B, I, C, form a harmonical system; the points I, I', are called conjugate -har- monics; the line BC is harmonically divided by the Kg. 82 two points I, I'. Since, for the same line BC, the position of the points I and I' AB depends upon the ratio -j-^ > it is seen that the line BC may be harmonically divided in an infinite number of ways; but the problem is determinate when AB and AC or their ratio is given. When AB = AC the bisector AI bisects the base BC, and AI, is parallel to the base and cuts it in infinity. 276 GEOMETRY 705. If in a right triangle ABC a perpendicular AD is drawn from the vertex A of the right angle to the hypotenuse BC: First, the triangles ABD, ADC, are similar to each other and similar to the original triangle ABC; Second, each leg of the right triangle is a mean proportional between the hypotenuse and its adjacent segment (330). Thus we have: BC -.AB = AB : BD and BC : AC = AC : CD; Third, the perpendicular is a mean proportional between the segments of the hypotenuse : BD -.AD == AD : CD. 706. When a perpendicular is drawn from any point A in a circumference of a circle to the diameter BC, and chords AB and AC are drawn between this point and the ex- B, — ^D tremities of the diameter (648, and Fig. 82): First, each chord is a mean proportional between the diameter and the adjacent segment; Second, the perpendicular is a mean proportional between the segments of the diameter. 707. The parts of two chords BC and DE, which intersect, are inversely proportional (326); thus: AB -.AD = AE : AC; AB X.AC = AD X AE. From the last equation it is seen that the product of the two parts of all the chords which can be drawn through ^ the same point A are equal. This product is equal to the square of half the chord which is perpen- dicular to the diameter drawn through the given point. 708. If from a fixed point A without a circle, two secants AB and AC, which terminate in the circumference of the circle, are drawn, they are proportional to their external segments; thus: AB :AC = AD : AE; and AB X AE = AC X AD. Fig. 83 If from a fixed point A without a circle, a tangent AF and a secant AB, which terminate in the circumference, are drawn, the MEASUREMENT OF ANGLES 277 tangent is a mean proportional between the secant and its ex- terior segment: AB lAF = AF : AE and AB x AE = AF^^ Thus for a certain point A without the circle, the product of the secant and its external segment is constant and equal to the square of the tangent drawn from that point. This result is analogous to the one obtained when the point was within the circle (707). 709. In the same or equal circles, two angles at the center are to each other as their intercepted arcs (667). All angles at the center are measured by their intercepted arcs. That is, that the angle contains the unit angle as many times as the arc contains the unit arc. Generally the arc of one degree is taken as the unit arc (222); therefore, the unit angle intercepts an arc of one degree, which is the 360th part of four right angles. The angle of one degree is divided, as is the arc, into 60 equal parts called minutes, and these in turn are subdivided into 60 equal parts called seconds. It should be noted that when an arc of a certain number of degrees is specified, no length is designated, but simply the num- ber of times this arc contains one 360th part of the circumfer- ence which has the same radius as the arc. Thus, arcs of the same number of degrees may be imequal. On the contrary angles of the same number of degrees are always equal. 710. An angle inscribed in a circle is measured by one-half its intercepted arc. The same is true of an angle formed by a tangent and a chord (684, 709). 711. The angle formed by two chords (Fig. 83) intersecting within the circumference is measured by one-half the sum EC + BD . ^, . ^ , , ■ of the intercepted arcs. 712. An angle formed by two tangents, two secants, or a tan- gent and a secant, intersecting without the circumference, is measured by one-half the difference of the intercepted arcs. Thus (Fig. 84), the angle BAC is measured by ^ , and angle FAC is measured by ^ BOOK V THE MENSURATION OF POLYGONS 713. The length of a line is the measure of the line, that is, the ratio of the whole line to one of xinit length (216, 321). The area of a surface is the measure of the surface, that is, the ratio of that surface to the unit surface. 714. The product of two lines is the product of their lengths. ^( i 715. The 'projection of a point A on a line CD f, i ! n is the foot £ of a perpendicular drawn from that Fig. 85 point to the line. The projection of a line AB on another CD is that part of the latter EF which lies between the projections of the extremities of the first AB on the second CD. 716. The area of a rectangle is equal to the product of its base and its altitude (644) : S = B XH. This expression for the area indicates that the surface contains as manj' units of surface, which have the unit of length for a side used in expressing ~B~ B and H, as the product B X H contains units. , ^'s- ^ Having B = 3.5' and H = 2.15', we have: —pP the squares of the sides is equal to the sum of / ^E^^jr \ the squares of the diagonals, plus four times the /^ ^\\ square of the line which joins the middle points ^ „ ^ of the diagonals EF: AF + BC' + Ci^ + DA" = ic' + fiF + 4 EF\ 738. In any trapezoid, the sum of the squares of the legs is equal to the sum of the squares of the diagonals, less twice the product of the bases. Referring to Fig. 47: ad" + BC' = AC^ + BW -2ABX DC. 739. In all parallelograms, the sum of the squares of the sides is equal to the sum of the squares of the diagonals, and con- versely. BOOK VI REGULAR POLYGONS AND THE MENSURATION OF THE CIRCLE 740. A regular polygon is a polygon which is equilateral and equiangular (647). The center and the radius of a regular polygon are the center and the radius OA of a circle circumscribed about the polygon (673). The apothem of a regular polygon is the radius OP of a circle inscribed in the polygon (677). The angle between the radii drawn to the extremities of any side is called the angle at the center of the polygon. The part OABC, included between two consecutive radii OA and OC, is called the sector of the polygon. 741. A circumference being divided into three or more equal parts: First, the chords which join the con- secutive points of division form a regular in- \„ scribed polygon; Second, the tangents drawn at the points of division form a regular cir- cumscribed polygon. Conversely: First, the vertices of a regular inscribed polygon divide the circumference into equal parts; Second, the points of contact of the sides of a regu- lar circumscribed polygon divide the circumference into equal parts (673, 677). The circle inscribed in and the circle circumscribed about the same regular polygon are concentric (679). When a regular polygon is circumscribed about a circle, each side is divided into two equal parts by the point of contact. 742. One circle, and only one, may be circumscribed about any regular polygon (741). One circle, and only one, may be inscribed in any regular polygon. 743. The area of a regular polygon is equal to one-half the product of its perimeter and its apothem OP (724, 740). 284 MENSURATION OF THE CIRCLE 285 744. Two regular polygons having the same number of sides are similar. Their perimeters are to each other as any two homologous linear dimensions; and their surfaces are to each other as the squares of these same dimensions (695, 703, 726, 740). 745. The side of a square circumscribed about a circle is equal to the diameter of the circle. The side c of a square inscribed in a circle of radius R is equal to V^ 2? (695). c:R = \/2:l and c = R V2. The side of a regular hexagon inscribed in a circle is equal to the radius of the circle. The side c of an equilateral triangle inscribed in a circle of radius R is equal to Vs R. c : i? = V3 : 1 and c = RVS. The side C of an equilateral triangle circumscribed about a circle is equal to double the side of an equilateral triangle inscribed in the same circle. C = 2c = VSR. The side C of a regular hexagon circumscribed about a circle is equal to one-third the side of a circumscribed equilateral triangle about the same circle. The side of a regular decagon inscribed in a circle is equal to the greater segment of a radius divided in extreme and mean ratio (632, 692). The side of a regular inscribed pentadecagon is. equal to the chord which subtends an arc, which is equal to the difference of the arcs subtended by the sides of a regular inscribed hexagon and decagon. The difference between the arcs subtended by the sides of a regular inscribed pentagon and hexagon, is subtended by the side of a regular inscribed polygon of thirty sides. (See Problems.) 286 GEOMETRY Sides and Apothem of Regular Polygons Inscribed in a Circle of Radius B Sides. Apothems. Equilateral triangle .... Square E\/2 IrV2 Pentagon Hexagon I R VlO - 2\/5 R IrV-s Octagon B V2- Vl lij(N/5_l) 1 E ■\/2 + \/2 Decagon i R VlO + 2 \/5 4 Dodecagon ijV2-\/3 or Ir{V6-V2) 2i |flV2+V3 or Ir(V2 + V6) 1 Pentadecagon side =jiJLVlO + 2V l-VS (\/5 - 1)] Radii and Apothems of Regular Polygons of the Side c Eadii. Apothems. Equilateral triangle . . . Square 1 3' 1 2' 1 10 c 1 2' 1 2' c ' 1 or -c V3 V2 1 6' 1 2" 1 10 1 2' 1 2' 1 2' 1 2' V3 Pentagon Hexagon c VSO + 10 Vs c V25 + 10 V5 V3 Octagon V4 + 2\/2 (I + V5) (I + V2) Decagon V5 + 2V6 Dodecagon v'2+\/3 (V2 + V6) (2 + V3) MENSURATION OF THE CIRCLE 287 Areas of Regular Polygons Inscribed in a Circle OF Radios B. Op side c. Equilateral triangle .... Square 4 2ff 4 Pentagon Hexagon Octagon Decagon Dodecagon 5BW10 + 2V5 Ir'V-s 2iJ2 \/2 ^ij2 Vio -2V5 4 3iJ2 ic'V26 + lO\/5 4 |c^V3- 2 c2 (1 + V2) |c2V5 + 2V5 3 c' (2 + VS) V2 = 1.4142135623... \/3 = 1.7320508075... v/g- 2. 2360679774... log 2 = 0.3010300 log 3 = 0.4771213 log 5 =0.6989700 Table 1. The values of the radius, the apothems, and the area of a regular polygon, whose side is taken as unity. Table 2. The values of the side of a regular polygon, accord- ing as the radius, the apothem, or the area of the polygon are taken as unity. First. The Side C=l. Second. Value of the Side Cfob, OF Sides OF THE POIiTaON. Badins Apothem Surface Radius =1 Apotliem=l Surface = 1 3 0.577350 0.288675 0.433013 1.732050 3.464101 1.619671 4 0.707107 0.500000 1.000000 1.414214 2.000000 1.000000 5 0.850651 0.688191 1.720477 1.175570 1.453085 0.762387 6 1.000000 0.866025 2.598076 1.000000 1.154701 0.620403 7 1.152382 1.038261 3.638912 0.867767 0.963149 0.524581 8 1.306563 1.207107 3.828428 0.765367 0.828427 0.455090 9 1.461902 1.373739 6.181823 0.684040 0.727940 0.402200 10 1.618034 1.538842 7.694207 0.618034 0.649839 0.390611 11 1.774732 1.702844 9.365640 0.563465 0.587253 0.326762 12 1.931852 1.866025 11.196150 0.617638 0.535898 0.298858 15 2.404867 2.352315 17.642360 0.415823 0.425113 0.238079 18 2.879385 2.835641 25.520770 0.34'7296 0.352654 0.197949 20 3.196227 3.156876 31.568760 0.312869 0.316769 0.177980 For the same number of sides, the sides, the apothems, and the radii vary in the same ratio, and the areas vary as the squares of these lengths (744). 288 GEOMETRY Example. Construct a prismatic reservoir which is to con- tain 36.75 cubic feet, to be 3 feet deep, and its base is to be a regular octagon. The area of the base — ^ = 12.25 square feet. Then from the table (2d) c^ : 0.4550r = 12.25 : 1 ; and c = 0.45509 V12.25 = 0.45509 X 3.5 = 1.592815 feet. From the table (1st) R : 1.306563 = 1.592815 : 1; and R = 1.306563 X 1.592815 = 2.081 feet. Therefore, describe a circle of 2.081 feet radius and lay off the chord 1.592815 feet, eight times, which wiU give the regular octagon that is to serve as base to the reservoir. 746. Having a regular inscribed polygon, to inscribe a regular polygon of twice the number of sides, join the vertices of the first to the middles of the arcs subtended by the sides of the first. Having a regular inscribed polygon of an even number of sides, to inscribe a regular polygon of half that number of sides, draw lines connecting every other vertex of the given polygon. Having a regular circumscribed polygon, to circumscribe a regular polygon of twice the number of sides, draw tangents to the circle at the middle points of the arcs intercepted by the sides of the given polygon. Having a regular circumscribed polygon of an even number of sides greater than four, to circumscribe a regular polygon of half the number of sides, erase every other side of the given polygon and prolong the remaining sides until they meet. 747. Let p and P be the perimeter of two regular similar polygons, one inscribed in and the other circumscribed about the same circle, designating by p' and P' the perimeters of regular inscribed and circumscribed polygons of double the number of sides, we have: P'=|^,andp'=VP^ = v/|?^- P + p ^ ^ VP + p 748. The circumference is greater than the perimeter of any inscribed polygon and less than that of any circumscribed poly- MENSURATION OF THE CIRCLE 289 gon. It is the limit which they approach as their sides become smaller and smaller, that is as the number of sides becomes greater (601, 649). 749. Two circles are always similar. Their circumferences C and c are to each other as their radii R and r, or as their diame- ter D and d, and their areas are to each other as the squares of their linear dimensions: - = - = - and - = — - — c r d ' s r^ ~ d' (744) 750. In two different circles arcs, sectors, and segments are said to be similar when they correspond to the same angles at the center (667). Similar arcs are to each other as their radii, their diameters, and the chords which subtend them. Similar sectors and segments are to each other as the squares of their radii, diameters, arcs, and chords (749). 751. The ratio of a circumference C to its diameter D is a con- stant uncommensurable number, which is commonly represented by T. TT = ^ = 3.141 592 653 589 793 238 462 643 .. . In practice generally not more than four places are expressed thus: TT = 3.1416. Tables of the nearest values to the seventh decimal place of the First 9 multiples of v, -k^, tt", Vtt, \lir, -, -^> -^) V - and \/ -> which are often met with in formulas. It jr2 tt' V^ r- 1 3.1415927 1 9.8696044 1 31.0062767 1 1.7724539 1 1.4645919 2 6.2831853 2 19.7392088 2 62.0125534 2 3.5449077 2 2.9291838 3 9.4247780 3 29.6088132 3 93.0188300 3 5.3173616 3 4.3937756 4 12.5663706 4 39.4784176 4 124.0251067 4 7.0898154 4 5.8583675 5 15 7079633 5 49.3480220 5 155.0313834 5 8.8622693 5 7.3229594 6 18.8495559 6 59.2176264 6 186.0.376601 6 10.6347231 6 8.7875513 7 21.9911486 7 69.0872308 7 217.0439368 7 12.4071770 7 10.2521432 8 25.1327412 8 78.9568352 8 248.0502134 8 14.1796308 8 11.7167351 9 28.2743339 9 88.8264396 9 279.0564901 9 15.9520847 9 13.1813269 290 GEOMETRY 1 IT 1 1 v'; n 1 0.3183099 1 0.1013210 1 0.0322515 1 0.5641896 1 0.6827841 2 0.6366198 2 0.2026420 2 0.0645030 2 1.1283792 2 1.3655681 3 0.9549297 8 0.3039631 3 0.0967545 3 1.6925688 3 2.0483522 4 1.2732396 4 0.4052841 4 0.1290060 4 2.2567583 4 2.7311363 6 1.5915494 6 0.5066051 5 0.1612575 5 2.8209479 6 3.4139203 6 1.9098593 6 0.6079261 6 0.1935090 6 3.3851375 6 4.0967044 7 2.2281692 7 0.7092471 7 0.2257605 7 3.9493271 7 4.7794885 8 2.5464791 8 0.8105682 8 0.2580120 8 4.5135167 8 5.4622725 9 2.8647890 9 0.9118892 9 0.2902635 9 5.0777063 9 6.1450566 Log ir = 0.4971499, log tt^ =0.9942997, log 7r'= 1.4914496, log V^ =0.2485749, Log -^^=0.1657166, logi =1.5028501, log i= 1.0057003, log -3=2.5085504, 1°^^/^= 1-7514251, logy^ = 1.8342834. 752. The expression of the length C of the circumference as a fiinction of its diameter D or its radius B. Having (751) then and C C = ttD or C = 2 irie, D = - and R = -r-- According as D = 1 or iE = 1, we have: C = TT or C = 2w. 753. The area S of a circle is equal to the product of its cir- cumference C and half its radius R, which is equivalent to area of a triangle whose base is equal to the circumference, and whose altitude is equal to the radius (718, 743). S D ,rZ)= TTU -r = 4 R -^ or ;S = 2irE^ = 7ri22; = 2V^and7? = yf. then D According as D = 1 or i2 = 1, we have: S = -:, or S = IT. (a) MENSURATION OF THE CIRCLE 291 Substituting for R in (a) its value in terms of the circumference C (752), we have: C = 4 nS. 754. Problems. 1st. What is the length of the circumference of a circle whose radius is 13 inches? From (716) C = 2 ttR = 2 ■ 3.1416 • 13 = 81.68 inches. 2d. What is the area of a circle whose radius is 13 inches? Having calculated the circumference, it is only necessary to multiply it by one-half the radius. Otherwise, according to (753) we have: S = nE^ = 3.1416 • 13 ■ 13 = 530.9 square inches. 3d. What is the radius of a circle whose area is equal to 530.9 square inches? From (751, 753) R = \/- = \/-X y/S = 0.5642 V530:9 = 13.0 inches. 755. The solution of the preceding problems using a table, which contains, to two decimal figures, the lengths of the circumferences and the areas of circles of whole diameters from 1 to 1000. 1st. The radius R or the diameter D of a circle being given, to calculate the length of the circumference and the area of the surface. Converting the given diameter into units of an order such that the whole part is the greatest possible number less than 1000; if the decimal part of this number is zero, the length of the cir- cumference may be read directly from the table in units of the order given and correct to within one hundredth of these units, and the area may be read directly in units of surface correct to within one hundredth of the chosen units. Example 1. For D = 2.5 inches, multiply by 10, which gives 25, then the table gives : For D = 25, C = 78.5, and — (252)^ "~ ^ = 49,875.92 . 100 = 4,987,592 square feet. Example 3. For d = 0.0252 inches, multiply by 10,000, then from the table C = 791.68 inches and \C angles of the dihedral angles of the other (792). Thus, Z. A'S'B' is the supplement of the plane angle A'OB'oi the dihedral angle SC; and Z ASB is the supplement of the plane angle of the dihedral angle S'C. 817. A solid bounded on all sides by polygons is a polyhedron (631). These polygons are the faces of the poly- hedron, the intersections of the faces are the edges, and the in- tersections of the edges are the vertices of the polj-^hedrons. A straight line joining any two vertices not in the same face is a diagonal of a polyhedron. 818. A polyhedron is called respectively a tetrahedron, a penta- hedron, a h-exahedron, . . . according as it has 4, 5, 6 . . . faces (632). 819. A prism is a polyhedron of which two opposite faces, called bases, are parallel, and the other faces, called lateral faces, intersect in parallel lines, called lateral edges. The altitude of 308 GEOMETRY the prism is the distance between the bases (779). In any prism (Fig. 123) the lateral edges AG, BH, CI, . . . are equal (779), and the lateral faces ABHG, BCIII, . . . are parallelograms (640). A prism is a right or an oblique prism, according as its lateral edges are perpendicular or oblique to the planes of the bases D c (762). A prism is triangular, quadrangular, pentagonal, . . . according as its bases are triangles, quadrila- 3 terals, pentagons . . . (632). / /- / A regular prism is a right prism whose bases are * ^ regular polygons (740). ig. 121 g2Q rpj^g sections of a prism made by parallel planes are equal polygons; thus the bases of a prism are equal, and any section made by a plane parallel to the bases is equal to the bases. A section of a prism made by a plane perpendicular to the lateral edges is a right section. 821. A truncated prism is that part of a prism included be- tween one base and a section made by a plane not parallel to the base. This base and the section are called the bases of the trun- cated prism (894). 822. A prism whose bases are parallelograms EFGH, DABC, (Fig. 124), is a parallelopiped (640). Thus, a parallelopiped is a hexahedron made up of six parallelograms, which are equal in pairs. Any face may be the base of the parallelopiped, and the dis- tance between the base and the opposite face is the g altitude. A A rectangular parallelopiped is one whose faces /|i\\ are all parallelograms. The three edges ED, EH, / 1 JjA EF, which meet in any one vertex E, are perpendi- ^\"| (^ A^, cular to each other. \S^^^ The three dimensions of a rectangular parallelopiped -^^ ^^ are the two dimensions of its base and its altitude, that is, the three adjacent edges which meet in any vertex. 823. The cube is a rectangular parallelopiped whose faces are squares. All its edges are equal. 824. A pyramid is a polyhedron (Fig. 125) of which one face ABCD, called the base, is a polygon, and the other faces SAB, SBC, . . . called lateral faces, are triangles having a common POLYHEDRAL ANGLES— POLYHEDRONS— SYMMETRY 309 vertex S, called the vertex of the pyramid. The intersections of the lateral faces are called lateral edges. Such are: SA SB The altitude is the perpendicular drawn from the vertex to the base. A pyramid is triangular, quadrangular, pentagonal, . . . according as its base is a triangle, quadrilateral, pentagon, . . . (632). A pyramid is regular when its base is a regular polygon and its lateral edges are equal. The lateral faces are equal isosceles triangles, the altitude of which is called the slant height of the pyramid. 825. A plane P parallel to the plane of the base ABODE of a pyramid (Fig. 126): 1st. Divides the edges SA, SB, . . . and the altitude Sh pro- portionally. Thus, SA^^SB^ ^Sh SA' SB'"" Sh' ' 2d. The section A'B'C'D'E' is similar to the base, and the ratio of the two polygons is equal to the ratio of the squares of the lateral edges and altitude. Thus, ABODE _^,g (,99^ ,23) A'B'O'D'E' SA' Sh'' 826. If two pyramids of the same altitude are cut by planes parallel to their bases, and at equal distances from their vertices, the sections will have the same ratio as their bases. If the bases are equal or equivalent, the sections are also. 827. The frustum of a pyramid is the por- tion of a pyramid included between the base and a section made by a plane parallel to the base. The base of the pyramid and the sec- tion are the bases of the frustum (Fig. 126) (895). 828. A polyhedron is convex when it is situa- ted totally on one side of the plane of any one of its faces (648). 829. Two polyhedrons are of the same kind when their surfaces are composed of the same number of triangles, quadrilaterals, pentagons, . . . placed in the same order. Thus, two pyramids 310 GEOMETRY or prisms are of the same kind when their bases have the same number of sides. 830. Two tetrahedrons are equal (818): First, when three adja- cent edges and the included polyhedral angle of one are equal to three adjacent edges and the included polyhedral angle of the other and placed in the same order; Second, when two faces and the included dihedral angle of one are equal to two faces and the included dihedral angle of the other and placed in the same order; Third, when one face and the three adjacent dihedral angles of one are equal to one face and the three adjacent dihedral angles of the other and arranged in the same order; Fourth, when the edges of one are equal to the edges of the other and are ar- ranged in the same order (809). 831. Two prisms are equal if three faces, including a trihedral angle of one, are respectively equal to three faces, including a trihedral angle of the other, and are similarly placed. Two right prisms of the same base and altitude are equal. All cubes which have an equal side are equal. 832. In any polyhedron the number of vertices plus the num- ber of faces is equal to the number of edges plus 2. Thus, V + F = E + 2, wherein V is the number of vertices, F the number of faces, and E the number of edges. 833. The number of conditions necessary for the equality of two polyhedrons of the same kind (829) is equal to the number E of edges. 834. The sum of all the face angles of a polyhedron is equal to as. many times four right angles as there are vertices in the polyhedron less two. Thus, s = 4 (7 - 2); for 7 = 8, s = 4 (8 - 2) = 24 rt 4 (652) wherein s is the sum of the face angles expressed in right angles, and V the number of vertices. 835. In any parallelopiped (822): First, the diagonals bisect each other; Second, the sum of the squares of the diagonals is equal to the sum of the squares of the sides. Thus, A, B, C, D, being the diagonals, and a, h, c, the three adjacent sides, we have: ^2 + 52 + (7 + 2)2 = 4 a2 + 4 52 + 4 c2. (739) POLYHEDRAL ANGLES — PLYHEDRONS — SYMMETRY 311 In any rectangular parallelopiped, the four diagonals are equal, and we have: 4 D^ = 4: a" + 4:V + 4: c" or D^ = a^ + V + c', that is, the square of one diagonal is equal to the sum of the squares of three sides. If the parallelopiped is a cube, the three sides are equal, and we have: D^ = 3 c^ and — = Vs. (731) Thus the ratio of the diagonal D to one side c of the cube is equal to the square root of three VS. 836. Two points are symmetrical with respect to a third point if this third point bisects the straight line which joins them. Two points are symmetrical with respect to a line or plane when this line or plane bisects at right angles the line which joins the two points. 837. Two straight lines are symmetrical with respect to a point, a line, or a plane when their extremities are symmetrical with respect to the point, line, or plane. The point, line, and plane are respectively called center of symmetry, axis of symmetry, and plane of symmetry. 838. Two polygons or two polyhedrons are symmetrical with respect to a point, a line, or a plane when each vertex of one has a symmetrical vertex in the other with respect to the point, the line, or the plane. 839. Two straight lines, two polygons symmetrical with re- spect to a straight line, are equal each to each. Two straight lines, two polygons symmetrical with respect to a point or a plane, are equal. Two polyhedral angles, or two polyhedrons symmetrical with respect to a point or a plane, have homologous dihedral angles equal and arranged in inverse order. In general they cannot be made to coincide. BOOK III THE CYLINDER— THE CONE— THE SPHERE 840. A right circular cylinder, or cylinder of revolution, is a solid generated by a rectangle ABCD, which makes one entire revolu- tion about one of its sides as an axis. The side AB which serves as axis is called the axis of the cylinder. The bases of ^313?^ the cylinder are the circles described by the sides AB and DC perpendicular to the axis. The altitude of the cylinder is the distance AB between the two The lateral surface of the cylinder is the sur- face generated by the side CD parallel to the axis. CD is called the generatrix. Any position of the gen- eratrix is an element of the surface. 841. A right circular cone, or cone of revolution, is a solid gen- erated by the revolution of a right triangle ABS about one of its legs as an axis. The side SB which serves as an axis is called the axis or the altitude of the cone. The base of the cone is the circle generated by the side AB perpendicular to the axis. The slant height of the cone is the hypotenuse of the generating tri- angle. The vertex of the cone is the point where the lateral surface meets the axis. The lateral sur- face is generated by the hypotenuse SA. SA is the generatrix. Any position of the generatrix is an element of the surface. 842. The section of a right circular cylinder made by a plane: First, parallel to the bases is a circle equal to the bases; Second, parallel to the axis is a rectangle whose opposite sides are two elements of the cylinder. 843. The section of a right circular cone made by a plane: First, parallel to the base is a circle; Second, passing through the vertex perpendicular to the base is an isosceles triangle whose sides are two elements of the cone. 312 THE CYLINDER — THE CONE — THE SPHERE 313 844. The frustum of a cone is that part of a cone included between the base and a section parallel to the base. The base of the cone and the section are the bases of the frustum. The slant height of the frustum of a cone of revolution is that part AB of the generatrix included between the two bases (Fig. 138), and the altitude is the distance CD between the bases (827). 845. A cylindrical surface is a curved surface generated by a moving straight line AB, called a generatrix, which moves parallel to itself and con- stantly touches a fixed curve CDS called the b7 directrix. When the directrix is a closed plane curve, all sections made by planes cutting the surface which are parallel to the plane of the directrix are equal to the directrix, and a cylinder is a solid CDEC'D'E' included by the parallel planes, which are limited by the curves equal to the directrix and that portion of the cylindrical surface included between these parallel planes. The bases of the cylinder are the two parallel planes CDE and C'D'E', and the distance HH' between the bases is the altitude. A cylinder is right or oblique, according as the generatrix is or is not perpendicular to the plane of the bases. In a right circular cylinder the directrix is a circle (840). The right section of a cylinder is a section made by a plane per- pendicular to the generatrix (820). 846. A prism and a cylinder are inscribed in or circumscribed about one another, according as their bases are in- scribed in or circumscribed about one another (673, 677). Just as a circle, or in general any plane sur- face limited by a curve, may be regarded as the limit approached by any inscribed or circumscribed polygon when the number of sides is indefinitely increased (601), the cylinder may be considered as being the limit approached by any inscribed or circumscribed prisms which have these polygons for bases. Thus, the right cyhnder may be considered as a right prism, and an oblique cylin- der as an oblique prism. Therefore all properties of. the surfaces or volumes of prisms apply as well to cylinders, provided that these properties are independent of the number of sides, and 314 GEOMETRY that the bases and altitude of the cyhnder are substituted for the bases and altitude of the prism. 847. The development of the lateral surface of a prism is a plane surface. If the prism is a right prism, the development is a rectangle, whose altitude is the altitude of the prism and whose base is the perimeter of the base of the prism. Likewise, the development of the lateral surface of a cylinder is a plane surface, and when the cylinder is a right cylinder, it is a rectangle whose altitude is the altitude of the cylinder and whose base is the perimeter of the base of the cylinder. 848. A conical surface is the surface generated by a moving straight line SA, called the generatrix, passing through a fixed point S, called the vertex, and constantly touching a fixed curve BCD, called the directrix. When the directrix BCD is the boundary of a '^ plane surface, the solid SBCD, included between this surface and the vertex, is called a cone. The plane surface is the base of the cone, and the altitude is the distance SH from the vertex to the plane of the base. When the directrix is a circle, and the vertex lies on a perpen- dicular erected at its center, the cone is a right circular cone (841). When these conditions are not fulfilled the cone is oblique. 849. A pyramid and a cone are inscribed in or circumscribed about one another, according as, having the same vertex, their bases are inscribed in or circumscribed about one another. The cone may be considered as the limit of inscribed or cir- cumscribed pyramids when the number of sides is indefinitely increased (846). Thus the right circular cone (841) may be considered as a regular pyramid (824) whose slant height is the side of the cone, and whose base is a circle; and, in general, any cone may be considered as being, a pyramid. Therefore all properties of surfaces or volumes of pyramids apply as well to cones, provided that they be independent of the number of sides of the base of the pyramid. 850. The development of the lateral surface of a pyramid is a plane surface, as is also that of the lateral surface of a cone. When the cone is one of revolution, the development of the lateral surface is the sector of a circle whose radius is the side THE CYLINDER— THE CONE— THE SPHERE 315 of the cone, and whose base is an arc equal to the circumference of the base of the cone (760). 851. A plane is tangent to a cylinder or to a cone of revolution when it touches only one element of the surface of the solid, that is, when it contains a tangent EF to the base and the element (840, 841) which passes through the point of contact E (Figs. 127 and 128). The above statement applies to any cone or cylinder whose base is a convex polygon. Any plane tangent to a cylinder or to a cone of revolution is perpendicular to a plane passing through the axis of the cone and the element (841) of the sur- face at the point of contact. 852. A sphere is a solid bounded by a surface every point of which is equally distant from a point called the center (665). A sphere may be considered as being generated by a semi- circle KCH, revolving on its axis KH. All straight lines OA, drawm from the center to the surface, are called radii. A straight line AB, which has its extremities in the surface of the sphere, is a chord. A chord CD which passes through the center is a diameter. All diameters are equal to two radii and consequently equal to each other. All sections CED, made by planes passing through the center, are called great circles. A quarter CE = ED of a great circle is called a quadrant (222). All great circles divide the sphere into two equal parts (666). A section AFB, made by a plane which does not pass through the center, is a small circle. 853. In the same sphere or in equal spheres two circles equally distant from the center are equal, and of two circles tmequally distant from the center, the smaller one is the farther. The converse statements of the above are also true (672). 854. The distance between two points on the surface of a sphere is the arc of the great circle joining these two points. 855. The extremities H and K of the diameter perpendicular to the plane of a circle AFB are the poles of this circle. Each of the poles H and K of a circle AFB is equally distant from all points in the circumference of the circle, that is, all the 316 GEOMETRY arcs of the great circles passing through the pole and the cir- cumference are equal. Conversely, if all points on a line drawn on the surface of a sphere are equidistant from one fixed point in the circumference, the line is the circumference of the circle which has this point for its pole. 856. The angle formed by the arcs AB, AC, of two great A V. circles which meet in a point A, is called a spherv- -D cal angle. The point of meeting is the vertex, and the arcs the sides. 857. A lune is a portion ABFCA of the sur- face of a sphere, bounded by two semi-circumfer- ences of great circles. The angle of the lune is the angle DAE between the semi-circumferences which form its boundaries. A spherical wedge is a portion AOFBC of a sphere bounded by a lune and two great semicircles. The dihedral angle formed by the planes of the semicircles is the angle of the wedge. The plane angle of this dihedral angle is the angle DAE (792). A spherical lune or wedge is right, acute, or obtuse, according as its angles are right, acute, or obtuse (788). Two great circles the planes of which are perpendicular to each other divide the sphere into four equal right wedges, and the surface into four equal right lunes. 858. A part ABC of the surface of a sphere boimded by three or more arcs of great circles is called a spherical polygon. The arcs are the sides of the polygon. 859. A spherical triangle is right, isosceles, or equilateral, under the same conditions as a plane triangle (633, 635, 636). A spherical triangle is bi-rectangular or tri-rectangular according as it has two or three right angles. 860. A spherical triangle is the polar triangle of another when the vertices of the second are the poles of the first (855). 861. A spherical pyramid is a solid OABC, bounded by a spherical polygon ABC, and the circular sectors OAB, OAC, OBC, whose bases are the different sides of the polygon and whose vertex is the center of the sphere (Fig. 133). The poly- gon ABC is the base of the pyramid, and the center of the sphere is the vertex. A spherical pyramid is bi-rectangular or tri-rectangular accord- THE CYLINDER — THE CONE— THE SPHERE 317 ing as its base is a bi-rectangular or tri-rectangular triangle (859). Three great circles, such that the plane of each is perpendicu- lar to the planes of the two others, divide the sphere into eight tri-rectangular pyramids equal each to each, and the surface into eight equal tri-rectangular spherical triangles. 862. In any spherical triangle any side is less than the sum of the other two and greater than their difference (601). Articles (635, 636, 638, 658) apply as well to spherical triangles as to plane triangles. 863. The sum of the sides ot any spherical polygon is less than the circumference of a great circle. 864. The angle of two arcs of great circles (856) is equal to the plane angle of the dihedral angle formed by the planes of the two arcs. The angles of a spherical polygon are the plane angles of the dihedral angles formed by the planes of the sides (792). 865. The sum of the angles of a spherical triangle are less than six and greater than two right angles (813). 866. Two spherical triangles on the same or equal spheres are equal: First, when two sides and the included angle of one are equal to two sides and the included angle of the other and simi- larly placed; Second, when one side and the adjacent angles of one are equal to one side and the adjacent angles of the other and similarly placed; Third, when they have three sides equal each to each and similarly placed; Fourth, when they have three angles equal each to each and similarly placed (654, 809). 867. A spherical triangle may be constructed: First, when two sides and the included angle are given; Second, when one side and the adjacent angles are given; Third, when three sides are given; Fourth, when three angles are given (663). 868. A zone is that portion of the surface of a sphere included between two parallel planes CED, AFB (Fig. 132). The bases of the zone are the two circumferences CED and AFB, which include the zone. When one of the two planes is tangent to the sphere, the zone has only one base. The distance between the bases is the altitude of the zone. 869. A line is inscribed in a sphere when it terminates in the surface of the sphere. Such is AB (Fig. 132). A polyhedron is inscribed in a sphere when all its sides are 318 GEOMETRY inscribed in the sphere. A sphere is circumscribed about a poly- gon when the polygon is inscribed in the sphere (673). 870. A sphere, and only one, may be passed through four points in space not in the same plane (680). The six planes drawn perpendicular to the middles of the edges of a tetrahedron meet in a single point equally distant from the four vertices of the tetrahedron. This point is the center of a sphere, which may be circumscribed about the tetrahedron (688). 871. A straight line AE and a sphere are tangent when they have only one point A in common (Fig. 133). A plane DAE is tangent to a sphere when they have but one point A in common. Any plane DAE perpendicular to a radius OA at its extremity is tangent to the sphere (673). Any straight line AD perpen- dicular to the radius OA is tangent to the sphere, and lies in the plane which is tangent to the sphere at that point A. The perpendicular OA erected to the tangent plane DAE at the point of contact is normal to the sphere 0. Any line normal to the surface passes through the center of the sphere, and all radii are normal to the surface of the sphere. The shortest and longest distances from a given fixed point to the surface of a sphere is the normal to the surface of the sphere passing through the point (675). 872. A polyhedron is circumscribed about a sphere when each of its faces is tangent to the surface of the sphere. A sphere is inscribed in a polyhedron when the polyhedron is circumscribed about the sphere (677). 873. The six planes which bisect the dihedral angles of a tetra- hedron meet in a single point equally distant from the four faces of the tetrahedron. This point is the center of a sphere which may be inscribed in the tetrahedron (687). 874. Two spheres are tangent when they have but one point in common (675). Two spheres which have their common point on the line of centers are either tangent externally or internally, according as the point is situated between the centers or on the prolongation of the line of centers. Articles (681 to 683) apply to the surfaces of spheres as well as to circles, except that the surfaces cut each other in circles. BOOK IV SIMILAR POLYHEDRONS AND THE MEASURE- MENT OF ANGLES 875. Two polyhedrons are similar when their dihedral angles are equal each to each and are similarly placed, and the homolo- gotis faces are similar (695). 876. A plane P (Fig. 126), drawn parallel to the plane of the base of a pyramid, cuts off a pyramid SA'B'C'D'E', which is similar to the original pyramid SABCDE (825). 877. Two tetrahedrons are similar: First, when they have an equal polyhedral angle included between proportional edges and similarly placed; Second, when they have an equal dihedral angle included between two faces similar each to each and similarly placed; Third, when they have a similar face and three adjacent dihedral angles equal each to each and situated in the same order; Fourth, when their edges are proportional each to each and similarly placed (700). 878. Two prisms or two pyramids are similar when they have an equal dihedral angle at the base included between two faces similar each to each and similarly placed. Two prisms or two regular pyramids (819, 824) are similar when their bases are similar polygons and their altitudes to each other as the sides of their bases, or as the radii of the circles inscribed in or circumscribed about the bases. Two right prisms are similar when their bases are similar and their altitudes are to each other as the homologous sides of the Two rectangular parallelopipeds are similar when their dimen- sions are proportional. All cubes are similar. 879. Two polyhedrons composed of the same number of tetra- hedrons similar each to each and similarly placed, are similar; and the converse is also true (702). Two polyhedrons similar to a third are similar to each other. 319 320 GEOMETRY 880. All dihedral angles are measured by their plane angles (792), that is, they contain as many right dihedral angles as their plane angles contain right plane angles. 881. A spherical lune is measured by twice its angle (857), that is, it contains as many tri-rectangular spherical triangles (861) as twice its angle contains right plane angles (918). A spherical wedge is measured by twice its plane angle, that is, it contains the tri-rectangular spherical pyramid as many times as its plane angle contains right angles (857, 861, 928). 882. Taking the tri-rectangular spherical triangle and the right triangle as imits (881): 1st. A spherical triangle is measured by the excess of the sum of its angles over two right angles. 2d. Any spherical polygon is measured by the excess of the sum of its angles over as many times two right angles as there are sides less two (858, 864, 919). 883. Taking the spherical tri-rectangular pyramid and the right angle as units (881): 1st. A spherical triangular pyramid is measured by the excess of the sum of its angles over two right angles. Any spherical pyramid is measured by the excess of the sum of the angles of its base over as many times two right angles as there are sides to the base less two (861, 864, 929). 884. Any trihedral angle is measured by the excess of the sum of its plane angles over two right angles. Taking the tri-rectangu- lar trihedral angle and the plane right angle as units (792, 805). BOOK V MENSURATION OF POLYHEDRONS (781) 885. The volume of a body is the ratio of that body to another taken as unity (216). Thus, supposing a cube whose side is equal to one foot is taken as iinity, when a body, of any form whatever, contains the tenth part of the foot cube twelve times, the volume of the body is equal to 1.2 cubic feet. 886. The product of a surface and a line is the product of the area of the surface by the length of the line (713). The area is expressed in units of surface one side of which is the unit of length. 887. The volume of a rectangular parallelopiped is equal to the product of its base and its altitude, or the product of its three di- mensions (822). The volume of a cube is equal to the cube of its edge (823). 888. Two parallelepipeds are to each other as the products of their three dimensions, or as the products of their bases and altitudes. If they have an equal dimension, they are to each other as the products of their other two dimensions, and if they have two dimensions equal they are to each other as their third dimension (717). Two cubes are to each other as the cubes of their edges (823). 889. The volume of a prism is equal to the product of its base and its altitude (819). When the prism is a right prism, the altitude is equal to one of the lateral edges. The volume of a prism is also equal to the product of its right section and one of its lateral edges (820). Any parallelopiped, being simply a special case of the prism, is measured the same as a prism (887). Any two prisms are to each other as the products of their bases and their altitudes, and according as two prisms have equiva- lent bases or equal altitudes they are to each other as their alti- tudes or their bases. They are equivalent if they have the same altitudes and equivalent bases. 890. The lateral surface of a right prism is equal to the perim- 321 322 GEOMETRY eter of the base times the altitude, and the lateral surface of any prism is equal to the perimeter of its right section times one of the lateral edges (820). 891. The volume of any pyramid is equal to one-third the prod- uct B X H of the base and the altitude. It is equal to one- third the volume of a prism of equivalent base and equal alti- tude (824, 889). Any two pyramids are to each other as the products of their bases and their altitudes, and according as the two pyramids have the equivalent bases or the same altitude they are to each other as their altitudes or their bases. They are equivalent if they have the same altitudes and equivalent bases. 892. The lateral surface of a regular pyramid is equal to half the product of the perimeter of the base and the altitude of one of the lateral faces (824). 893. Two tetrahedrons, triangular prisms, or parallelepipeds, which have an equal polyhedral angle, are to each other as the products of the sides which include the equal angle (725). 894. The volume of a truncated triangular prism ABCDEF F Fig. 134 (821) is equal to the sum of the volumes of the three pyramids whose common base is the lower base of the prism and whose vertices are the vertices A, B, C, of the upper base of the prism. F = |B(a + 6 + c), wherein V is the volume of a truncated prism; B is the lower base; and a, b, c, the altitudes of the various vertices A, B, C, with respect to the base B. 895. The volume of the frustum of a pyramid ABCDEFGH (827) is equal to the sum of the volumes of three pyramids having an altitude equal to the altitude of the frustum and their MENSURATION OF POLYHEDRONS 323 bases respectively, the lower base EFGH, the upper base ABCD, and a mean proportional between these two bases of the frustum (344). Thus, V = ^HXB + ^HXb + ^H\fBb = ^H(B + b + V56), wherein V is the volume of the frustum, B the lower base, and b the upper base. 896. The volumes of two similar polyhedrons are to each other as the cubes of their homologous linear dimensions, and their sur- faces are to each other as the squares of these dimensions. 897. The volume of a pile of stones or the capacity of dump-cart. Suppose a pile of crushed stone to be piled so that its upper and lower bases are rectangles, then its volume is (Fig. 136) : .. h, 6 [b(2a + a') + b' (2 a' + a)], wherein h is the height of the pile, a and b the dimensions of the lower base, and a' and b' those of the upper. The same formula may be xised for the calculation of the capacity of a dump-cart. If b' should equal zero, as is sometimes the case, we have: V = ^b(2a + a'). When the bases are similar, the solid is the frustum of a pyra- mid, and its volume may be calculated from the formula in ar- ticle (895). 898. Excavations. To calculate the total volume of an excavation, divide it into parts bounded laterally by vertical planes, on the bottom by any quadrilateral ABCD (Fig. 137), and on the top by the surface of the soil, which has no geometrical form but which may be supposed to be generated by a straight line which moves on the two opposite lines EF and GH, or EH and FG, the points E, F, G, H, all being on the surface of the soil. Since the area of a trapezium is expressed in triangles, and designating respectively the areas of the triangles Fig. 137 by ABC, b, ABD, V, CDA, CDB, b'". 324 GEOMETRY the volume of the solid is equal to : b{h+h'+h")+b'(h+h'+h"') + h"(h,+h" + h"') + h"'{h'+h"+h"') 6 When ABCD is a trapezoid, AB being parallel to CD, we have b = h' and &" = V" , and the preceding formula becomes: _ b (2 fe + 2 /t^ + fe" + /t"0 + h" (h + h' +2h" + 2 h'") 6 If ABCD is a parallelogram, we have 1 = 6' = 6" = 6'", and the formula becomes: -- 6 (/i + /i' + /i" + h'" ^ h + h' + h" + h'" V = 2 =^ 4 ^' B = 2 & being the total surface of the base ABCD. When the upper base EFGH is plane, we have further h + h" = h' + h"', and therefore: V_B^ + ^" -B ^' + ^"' When the base ABCD is reduced to a triangle ABC, the soHd becomes a truncated triangular prism, and we have (894), B being the surface of the triangle ABC, y ^^ h + h' + h" _ It is possible that the upper base may become reduced to a single edge EF, the altitudes h" and h'" becoming zero. In this case, according as the base is a trapezium, a trapezoid, or a paral- lelogram, we have respectively, making h" and h'" = in the preceding formulas: „_ h(b + b' + b") +h'ib + V + b'") ^ 6 ' T7_ b{2h + 2h') + b" (h + h') _(h + h') (2 6 + b") 6 6 ' ^. b(h + h') ^h + h' ^= 2 =^-^- Finally, if the upper base become reduced to a single point E, we have a pyramid, and the volume is : v=4 BOOK VI REGULAR POLYHEDRONS AND THE MENSURA- TION OF CYLINDERS, CONES, AND SPHERES 899. A regular polyhedral angle is one which has all its dihedral angles equal and all its face angles equal (803). 900. A regular polyhedron is one whose dihedral angles are all ec^ual and whose faces are regular polygons, equal each to each (740, 817). Thus all cubes are regular polyhedrons (823). The center and the radius of a regular polyhedron are the center and the radius of the sphere circumscribed about the polyhedron. The apothem of a regular polyhedron is the radius of the sphere inscribed in the polyhedron (743, 869, 872). 901. In any regular polyhedron a single sphere may be in- scribed, and about any regular polyhedron a single sphere may be circumscribed (900). 902. Two polyhedrons of the same kind are always similar (829, 875). 903. The volume of a regular polyhedron is equal to its sur- face times one- third its apothem (900). Table of Five Regular Polyhedrons Giving the number and kind oj their faces, their surfaces, and their volumes; their edges being taken as unity (745). POLYHEDEOSS. FACES. SURFACE. VOLUME. Tetrahedron Cube or hexahedron .... Octahedron Dodecahedron loosahedron 4 triangles. 6 squares. 8 triangles. 12 pentagons. 20 triangles. 1.732051 6.000000 3,464102 20.645779 8,660254 0.117851 1.000000 0,471404 7.663119 2,181695 From this table an octahedron whose edge is 2.5 feet has respectively 3.464102 X (2.5)2 ^ 21.6506 sq. ft. 0.471404 X (2.5)' = 7.365687 cu. ft. for its surface and volume. 325 326 GEOMETRY 904. Two cylinders or cones of revolution (804, 805) are simi- lar when the altitude h and radius r of the base of the first are proportional to the altitude h' and the radius r' of the base of the second, that is, when h:h' ^r:r'. 905. Two spheres are always similar. 906. The lateral surface of a cylinder of revolution (840, 845) is equal to the perimeter of the base times the altitude. Thus, for a circular cylinder: S = 2 ttRH, wherein S is the surface, 2 irR the perimeter of the base, R the radius of the base, and H the altitude of the cylinder. The lateral surface of any cylinder is equal to the perimeter of its right section times its generatrix (845, 890). 907. The volume of any cylinder is equal to its base times its altitude. Thus, for a circular cylinder, V = ■ttR'H, wherein V is the volume, R the radius of the base, nR^ the area of the base, and H the altitude of the cylinder. 908. The lateral surface of a cone of revolution is equal to half the product of the circumference of its base by its slant height (718, 841). Thus, for a circular cone, S = ttRC, wherein S is the surface, R the radius of the base, tR half the circumference of the base, and C the slant height. 909. The volume of any cone is equal to one-third the product of its base and its altitude. Thus, for a circular cone, 7= i ^R'H, wherein F is the volume, R the radius of the base, irR^ the area of the base, and H the altitude of the cone. Thus the volume of a cone is one-third that of a cylinder of an equivalent base and the same altitude (907). 910. Two cylinders or two cones are to each other as the prod- ucts of their bases and their altitudes. If they have the same MENSURATION OF CYLINDERS, ETC. 327 altitudes they are to each other as their bases; if they have equiv- alent bases they are to each other as their altitudes, and if they have equal altitudes and equivalent bases they are equivalent (889, 891, 907, 909). 911. Two similar cylinders or cones of revolution (904) are to each other as the cubes of any of their homologous linear dimen- sions. Thus, y W C^ R^ D^ V " H'^ " C'^ ~ R'^~ D'^' wherein V is the volume, H the altitude, C the slant height, R the radius of the base, and D the diameter of the base. The lateral surfaces and the total surfaces of similar cones or cylinders are to each other as the squares of these same dimensions. 912. The lateral surface of the frustum of a right cone (836) is equal to the slant height, times half the sum of the circumferences of its bases. Thus, ,2irR + 2Trr Fig. 138 s = c- = C^{R+r), wherein (S is the surface, C the slant height, R the radius of the lower base, and r the radius of the upper base. 913. The volume of the frustum of a cone is equal to the sum of the volumes of three cones which have a common altitude equal to the altitude of the frustum, and their bases equal respec- tively to the lower base, the upper base, and the mean propor- tional between the two (895). V= i ^Rm + lm-m+ I V«-2 X ^R'H = i TTff {R' + r^ + Rr), o o o wherein V is the volume, R the radius of the lower base, r the radius of the upper base, and H the altitude of the frustum. 914. The surface generated by the base BC of an isosceles triangle ABC, revolving about an axis MN, which passes through the vertex exter- nal to the triangle and in the same plane, is equal to the pro- jection PQ = p oi the base upon the axis MN, times the cir- II A l> QS Kg. 139 328 GEOMETRY cumference 2 nr, of the circle whose radius is equal to the alti- tude, AD = r^ of the triangle. Thus, *S = p X 2 Trri. The surface generated by a sector of a regular polygon under the same conditions is found in the same manner, p being the projection of the entire base upon the axis. 915. The surface of a zone is equal to the altitude H of the zone times the circumference 2 irii of a great circle (852, 868). Thus, S = 2 tRH. 916. On the same or equal spheres, two zones are to each other as their altitudes, and on spheres of different radii two zones of the same altitude are to each other as the radii or diameters of the spheres (915). 917. The surface oj a sphere of radius R = —> when considered as a zone, is equal to (915), S = 2irRX2R = 47rR^ = 7rD\ Thus the surface of a sphere is equal to the area of four great circles, or of a circle whose radius is equul to the diameter of the sphere (753). The surfaces S and s of two spheres are to each other as the squares of their radii R and r or their diameters D and d. Thus, S = 4:^R^ and s = 4 tt", and S IS = R' ir'' = D^ : d\ 918. The surface of a spherical lune is equal to the arc a corre- sponding to its angle a times the diameter 2R oi the sphere (881). Thus: and S = 2iBa=4^i22 " . * 919. The surface of any spherical triangle is equal to the radius of the sphere times the excess of the sum of the arcs a, b, c, corre- sponding to the angles over the semi-circumference (882). Thus the surface of a triangle is S = Ria + b + c- ■n-R). MENSURATION OF CYLINDERS, ETC. 329 The area of any spherical polygon is equal to the radius of the sphere times the excess of the sum of the arcs corresponding to its angles over as many times a semi-circumference as there are sides less two. 920. The volume of a solid generated by the revolution of any triangle ABC about a straight line MN, drawn through its vertex in the same plane and external to the D . triangle, is equal to the surface generated by the base BC times a third of the altitude ,1^^ ' AD = hoi the triangle. ¥~A^ S The surface generated by the base of a ^' triangle is the lateral surface of the frustum of a cone (Fig. 140) (912); it is that of a cone when AC or AB coincide with MN (908), and that of a cylinder when 5C is parallel to MN (906). In any case this surface may be measured, and if it be repre- sented by *S, the volume generated by the triangle ABC is : v = lsh. 921. The volume of a solid generated by an isosceles triangle ABC (Fig. 139) revolving about a straight line drawn through its vertex, in its plane and external to it, is equal to the projection p of the base BC on the axis multiplied by two-thirds of the area of a circle whose radius is the altitude AD = r^ of the triangle. Thus, 2 r P X ^TJ-i^. The volume of a solid generated by the revolution of a sector of a regular polygon about a straight line MN drawn through the ver- tex, in the same plane and external to it, is equal to the pro- jection p of the base on the axis times two-thirds the area of a circle inscribed to the base. The sector may be a semi-poly- gon revolving on its diameter. In any case, r^ being the radius of the inscribed circle, the generated volume is 2 V = p X ^■TT r^. 922. The volume generated by the revolution of a regular poly- gon about one of its sides as an axis: expressed in terms of its radius B, and in terms of its side c (745): 330 GEOMETRY Triangle Square . . Peutagon . Hexagon . Octagon Decagon . Dodecagon 3 27riJ3V2 f 7rijW5 + ?\/5 irija 2 TT B' vr+Tvi ^TRWb \ T E' (Ve + V2) 1 , 4 ^7rcS(5 + 2\/6) 2 7rc3(3 + 2\/2) 3 TT c' (7 + 4 VS) 923. A spherical sector is a solid generated by the revolution of a circular sector OAB \ about a diameter OQ, external to the sector j and in the same plane with it. The base of 1 the spherical sector is the zone described by the base AB of the circular sector (868). The volume of a spherical sector is equal to ^ig- 1« the altitude H = MQ of the zone, which serves as base, times two-thirds the area of a great circle of radius R. Thus, F = rRm. 924. Considering the sphere as a spherical sector whose alti- tude is equal to the diameter of the sphere 2 R = D, from the preceding article, we have the volume of the sphere equal to its diameter, times two-thirds the area of a great circle. rR' 6 925. The volume of any spherical sector is equal to one-third of the area of the zone, which serves as base, times the radius (891, 915). Thus, V = lx 2rrRH XR = ^TvRm. o o 926. The volume of a sphere is also equal to one- third of the product of its surface and its radius. Thus, 7 = 5 X 47ri?2 XR = 5 7rM o o MENSURATION OF CYLINDERS, ETC. 331 Writing R in terms of the surface *S (917), we have: S' = 36 wV\ 927. Two spheres are to each other as the cubes of their radii or diameters. V and v being the volumes of the two spheres, we have (924): 4 4 V = K'^R^ and v = -irr^ ; o o then V :v ■= R^ -.r^ = D^ -.d?. (917) 928. The volume of a spherical wedge is equal to the arc a corresponding to its angle a times two-thirds the square of its radius R. Thus, F = I aR^. (881, 918) 929. The volume of any spherical pyramid is equal to the product B X ^R of the base, times one-third the radius (883, 919). 930. The volume of a solid generated by the revolution of a circular segment CDm, about a diameter AB, ex- C vn ternal to the segment, is equal to the projection /t ^"^^^O) PQ = p oi its base CD = b upon the axis, multi- I , . il plied by one-sixth of the area of a circle whose j,; j^g ^ radius is equal to the base b. Thus, 931. The volume of any spherical segment is equal to half the sum of its bases, times its altitude, plus the volume of a sphere whose diameter is equal to the altitude of the segment. Thus, H being the altitude, and r and r' the radii of the bases, we have (753, 868, 924): V = 2 -^ + 6 ^ ~2~ ^ *" ^ 6 When the segment has only one base, half the sum of the bases is replaced by half the base; thus, 2 6 Considering the sphere as being a segment the altitude H of which is equal to the diameter 2 i? = D of the sphere, the first 332 GEOMETRY term in the second member of the above equation becomes zero, and we have: 932. A right cylinder is equilateral when its height is equal to the diameter of its base (840). A right cone is equilateral when its slant height is equal to the diameter of its base (841). A right cylinder is inscribed in a sphere when its bases are little circles of the sphere (852). An equilateral cylinder ADBC is circum- scribed about a sphere (Fig. 143) when its axis is a diameter of the sphere. / ^"^ \ A cone is inscribed in a sphere when its ver- /^\^ y \ tex and the circumference of its base lie on K C G the surface of the sphere. An equilateral Fig. 143 (.Q^g EFG is circumscribed about a sphere (Fig. 143) when its axis is the altitude of an equilateral triangle cir- cumscribed about a great circle of the sphere. 933. The total surfaces of a sphere, of a circumscribed cylinder of a circumscribed equilateral cone, are to each other as the numbers 4, 6, 9; and their volumes are to each other as these same numbers (906, 907, 908, 909, 917, 924). Remark 1. The lateral surface of the cylinder is equivalent to the total surface of the sphere. Remark 2. The total surface of the cylinder is a mean pro- portional between that of the sphere and the cone (344). Remark 3. The volume of the cylinder is a mean propor- tional between that of the cylinder and the cone. The total surfaces of the sphere, of the inscribed cylinder and equilateral cone, are to each other as the numbers 16, 12, 9; and their volumes are to each other as the numbers 32, 12\/2, 9. Thus the total surface of the cylinder is the mean proportional between that of the sphere and the cone; and its volume is also a mean proportional between those two solids. PROBLEMS IN GEOMETRY DRAWING OF THE FIGURES 934. Figures which are drawn simply to aid in following the demonstration of a problem, may be done free hand; but when measurements are to be obtained by a certain construction, the figures must be drawn accurately and to scale. In order to do this, instruments are necessary. 935. All the instruments which are necessary to construct all the figures of elementary geometry are the rule and the compass. The first is used for drawing straight lines, the second for describing circles, and both of them in combination for con- structing angles. Besides these two instruments we have several others, which, though not necessary, are almost indispensable; these are: the T-square, the triangles, the protractor, the reducing compass. The T-square is used for drawing parallel horizontal lines. The triangles, which are generally, one 60° and one 45°, right triangle, are used to draw parallels and perpendiculars. The protractor is used for laying off and measuring angles. The reducing compass is used for constructing similar figures according to a given proportion, having one figure given. Remark. When a point is to be determined by the intersec- tion of two lines, these lines should intersect as nearly at right angles as possible. ANGLES — TRIANGLES — PERPENDICULARS — PARALLELS 936. To construct an angle equal to a given angle E (Fig. 144), from the point £' as a center, with any radius EG, describe the arc GH; from the point on the line AB, with the same radius, describe the indefinite arc CL; take CD = GH and draw the side OD; then the angle DOC is equal to the angle E. Remark. The angle may be constructed by aid of the pro- tractor or with the triangles, by drawing lines parallel to the sides and intersecting in the point (630). 333 334 GEOMETRY To construct an angle equal to the sum of two given angles A, B, construct angle GOE = angle A, then angle HOG = angle B, and then angle HOE is equal to angle A plus angle B. In the same manner the sum of any number of angles may be constructed, and, in general, the angles may be added or sub- tracted. To construct the supplement of a given angle GOE, prolong one side EO, then the angle GOF is the supplement (617). To construct the complement of a given angle GOE, erect a per- D, «E> A 0- C B E Fig. 144 G r^?\ F E Fig. 145 pendicular OH to one side OE at the vertex 0, and the angle GOH is the complement. Two angles, A and B, of a triangle being given to find the third, construct the angle HOE equal to the sum of A and B, then HOP is the required angle (652). 937. To draw a straight line AB through a given point A, so as to make a given angle ABC with another line BC. Through D Kg. 146 any point D, taken on the straight line BC, draw the line DE, making the angle CDE equal to the given angle (Fig. 146); then draw the line AB through A parallel to ED (625). 938. Two sides a and, b and the included angle C of a triangle being given to construct the triangle (663). Construct an angle equal to the given angle C; lay off a distance on one leg equal to a, and on the other equal to b; then join the two by the line AB which completes the triangle ABC (654). C\A/B' PROBLEMS IN GEOMETRY 335 In the same manner a parallelogram may be constructed when two sides and the included angle are given. 939. One side a, and the two adjacent angles B and C, being given to construct the triangle (663). Draw BC equal to a; then at the extremities construct the angles ABC g C = B and ACB = C; the point A where the .^ IV A prolonged sides of these angles meet deter- f ""^ mines the triangle ABC (654). If the angle opposite the side had been given, the third angle would have been deter- mined according to article (936), and the prob- b' ' «""" *Q lem would be the same as the one preceding. ^s- ws 940. The three sides a, h, c, of a triangle being given to con- struct the triangle (663). Draw the line BC equal to the side a; then from the extremities with b and c respectively as radii, arcs of circles are described, and their point of intersection A determines the triangle; drawing AB and AC, we have the re- quired triangle ABC (Fig. 149) (654). 941. Two sides a and b, and an opposite angle A, of a triangle being given to construct the triangle. Construct the given angle A (Figs. 150 to 152) ; on one of the legs of this angle lay off AC = b; with C as center describe an arc of radius equal to a which cuts the line AB in B and B'; joining these two points to the vertex C we have one or two triangles which satisfy the conditions (663). A/ Fig. 119 Bt.t 1st. When the angle A is right or obtuse, angle B is acute (652), and a> b (638); the arc BB' cuts AB in two points, but the triangle ABC is the only one which satisfies the conditions, because the angle CAB' is less than a right angle. 2d. If the angle A is acute and a>b (Fig. 151) ZA> ZB, there is still but one solution, and that is the triangle ABC. In case a = b there is still but one solution, because the point B' falls upon the vertex of the angle A, 336 GEOMETRY 3d. When A is acute and a < 6, we have Z.AT ^"^ CDS> and OA as a radius; then the circumference of this circle should pass through vertices fi- B, C, D. 983. Inscribe a regular octagon in a given circle. 'B Having inscribed a square A BCD in the circle (981), divide each of the arcs subtended by these sides into two equal parts (946), and, joining these points of division to the adjacent vertices of the square, the octagon AEBFCGDH is obtained. If an octagon had been inscribed in the circle, joining every other vertex, a square would have been obtained. Remark. From the above it may be deduced that, in general, having a regular polygon inscribed in a circle, to inscribe a polygon of double the number of sides, bisect the arcs subtended by the sides and connect these points to the extremities of the chords. Having a polygon inscribed in a circle, to inscribe another of half the number of sides, connect every other vertex. A regular octagon may be inscribed in a circle without in- scribing a square. Operating as in (981) with , * , the 45° triangle, the circumference may be divided into 8 equal parts, which is indicated by the 4 diameters HF, EG, AC, BD, although it is not necessary to draw them; the sides may also be drawn with the triangle; noting J^t^ that the side AE is parallel to the chord HB, and commencing at this chord, the four sides AE, GC, HD, BF, are drawn; then, starting at the chord AF, the four other sides EB, DG, AH, FC, are drawn. 984. Draw a regular octagon when one side A is given. Describe a circle with any radius OA; in this circle inscribe a regular octagon (983), or simply one side AB of this octagon; / (^M>1 PROBLEMS IN GEOMETRY 357 through the point I, taken on the prolongation of one of the radii OA, OB, draw IL parallel to AB and equal to the given side A of the octagon; then draw La parallel to OB and inter- secting OA in a. From as a center, and with Oa as a radius, describe a circle, and the octagon ahcd . . . inscribed in this circle is the one required. This construction applies to all regular polygons which may be geometrically inscribed in a circle, but it may be greatly sim- plified for some polygons. Thus for an octagon, after having drawn the straight lines OA and OB, making an angle of 45°, take OA = OB; through a point / draw IL parallel to AB, and con- /'" /[ tinue as in the preceding example. / / \q Erecting a perpendicular CO at the middle of 1^ //''r"~N the side AB of the octagon which is to be con- \y'' ^ structed, take CD = CB and DO = DA; the point AC B is the center of a circle which may be circum- '^' scribed about the octagon in question, which is then easily constructed (983). Angle ODA = DCA + DAC = 90° + 45° = 135° (653); .,_„ 180° - 135° 45°, ZAOC = 2 = "2" therefore Z.AOB = 45° = ^ ■ 985. Inscribe a regular hexagon in a given circle. Laying the radius of the given circle off successively as chord, P B _ D these six chords will form the ;~/V ^y^" '\ six sides of a regular inscribed s2S^^ rl'--^'-X h^^^g"^ ABCDEF (978) (Fig. ' W '^ /A / 207). '^\l / ^--jL i/' A hexagon may be inscribed — ^B A ^ with a 60° triangle in the same ^s- 207 Fig- 208 manner that an octagon was in- scribed with a 45° triangle (983). Draw the diameter FC with one triangle, then with another triangle slide this one parallel to itself until it is below the figure; then, resting the short side of the 60° triangle against the first, the diameters FB and AD are drawn, and, joining the extremities of these diameters, we 358 GEOMETRY have the required hexagon; but, noting that each diameter is parallel to two sides of the hexagon, the sides may be drawn in directly with the triangles without drawing the diameters. It is thus that hexagonal bolt-heads and nuts are constructed. &86. Construct a regular hexagon whose side is given (Fig. 208). Describe a circle with the given side for a radius, and inscribe a regular hexagon, which fulfills the conditions of the problem (985). To construct a regular hexagon on a given straight line AB as a side, from the extremities A, B, as centers, with a radius equal to AB, describe the arijs BF and AC, which intersect in the center of the circle circumscribed about the hexagon; with the points C and F as centers, and the same radius AB, describe two other arcs, thus obtaining the points D and E; then DECBAF are the vertices of the required hexagon. 987. Inscribe an equilateral triangle in a given circle. Inscribe first a hexagon and join every other vertex; thus the triangle ACE (Fig. 207) is obtained. 988. Construct an equilateral triangle when one side is given. Operate as in (940) and make each side equal to the given side. The 60° triangle may also be used for constructing an equilateral triangle, the 60° angle being equal to the angle of the required triangle. Having inscribed a hexagon or an equilateral triangle, poly- gons of 12, 24, 48, . . . sides may be successively inscribed as indicated in (951). 989. If perpendiculars are dropped from the vertices of an equilateral triangle upon any diameter DE of the circum- scribed circle (Fig. 209), the sum AF + BG of the two perpen- diculars on one side of the diameter is equal to the perpendicular CH on the other side. Drawing the radius CO at C, it is perpendicular to the middle point of AB (621, 671). The rhombus ALBO gives 01= ^ OC = -75- , and drawing IK perpendicular to DE, since the triangles nil lOK and COH are similar, we have IK = -^ • But in the trape- zoid AFGB we have: AW-i- Tin IK = J" (662); therefore AF + BG = CH. PROBLEMS IN GEOMETRY 359 990. Construct a dodecagon whose side AB is given (632). Erect a perpendicular CO at the middle point of the given side AB; from ^4 as a center, with AB as a radius, describe an arc DB, and take DO = DB = AB; the point is the center of the circle circumscribed about the required dodecagon. Angle ODA = DCA + DAC = 90° + 60° = 150° (653) (Fig. 210); ">6 ZAOC 180° - 150° 30° 2 therefore ZAOB = 30° = 360° 12 991. Inscribe in a given circle : ¥ivst, a regular decagon; Second, a regular pentagon; Third, a regular pentadecagon; Fourth, a regular polygon of 30 sides (632). 1st. AB being the side of the decagon, the angle at the center 36°. Drawing the bisector AG of the angle A, we ^=10 - have (704): OA:OG = AB : GB. Since OA = OB and AB = AG = OG, we have: OB :0G = 0G: GB, which shows that the side AB is equal to the longer segment OG of the radius OB, divided in extreme and mean ratio (971). r- \ '/ i '^... A C B Fig. 210 To determine the side of a decagon, draw two radii OA, OB, perpendicular to one another; on OB as a diameter describe a circle; draw AO', and AD = AC is the side of the required deca- gon, because it is equal to the longer segment of the radius OA divided in extreme and mean ratio. Laying off the chord AD around the circumference, the re- qmred decagon is obtained. 360 GEOMETRY 2d. Joining every other vertex of the regular inscribed deca- gon, a regular inscribed pentagon is obtained (Fig. 213). If it is desired to obtain the side of the pentagon directly, the arc DC may be prolonged to E (Fig. 212), then DE is the required side. 3d. The difference between the arcs subtended by the sides of a regular inscribed hexagon and decagon being equal to - — — l5 of the circumference, the chord which subtends this dif- 4th. Having ^ - - ^ Fig. 212 Fig. 213 0^. \ ^rlL ■\E ference is the side of a regular pentadecagon. Having the side and laying it off aroimd the cir- cumference of the circle, the re- quired pentadecagon is obtained. k' '^ '' seen that the side of a regular in- scribed polygon of 30 sides is the chord which subtends the arc equal to the difference of the arcs subtended by the sides of the regular inscribed pentagon and hexagon. To construct a regular decagon on a given side AB, erect a per- pendicular CO at the middle point of AB, and at B erect another perpendicular BD = BC; take DE = DB, and from the point A as cen- ter, and a radius equal to AE, describe an arc intersecting the perpendicular bisector of AB in 0, the center of the circle which may be circum- scribed about the decagon (Fig. 214). Drawing EF perpendicular to AE, the two right triangles ABD and AEF are similar; BD being the half of AB, EF is half oi AE; furthermore, since FE = FB, being tangents drawn from the point F to the same arc, from F as a center and a radius FE, describe an arc through B; thus it is seen that AB is equal to the longer segment of a radius AE divided in extreme and mean ratio. 992. Inscribe a polygon of any number of sides in a circle. Di- vide the circumference into as many parts as the polygon has sides, and join the points of division (967, 975), which will give the required polygon. 9\ C B Fig. 214 F PROBLEMS IN GEOMETRY 361 3'o construct a regular polygon of any number of sides, the same method as was used in Fig. 205, the construction of the regular octagon, may always be pursued. 993. Circumscribe a regular polygon about a given circle. In- scribe the required polygon in the given circle; draw tangents to the middle points of the arcs subtended by the sides of the inscribed poly- gon; these tangents are parallel to the sides of the polygon, and form the polygon A'B'C . . . which was required. In general, the circum- scribed polygon is constructed in the same manner as the inscribed, it being necessary only to divide the circumference into the required number of parts and draw the tangents. 994. Inscribe a regular octagon in a given square ABCD. Draw the diagonals of the square, and from the vertices A, D^— „ rr-TiC ^! C, D, as centers, and radii equal to OA, describe arcs which determine the 8 vertices of the octagon on the sides of the square. 995. Cover a plane surface with regular polygons. The sum of the consecutive adjacent angles which may be formed about a point in a plane being equal to 4 right angles or 360 (618), any regular polygon whose angle is contained a whole number of times in 4 right angles may be used to cover a plane surface (652). Therefore the fol- lowing may be used: Fig. 116 i i>^t-i Fig. 217 Fig. 218 Fig. 219 Fig. 220 1st. The equilateral triangle, whose angle = -^ = ^ of a right angle (Fig. 217); 4 2d. The square, whose angle = -of a right angle (Fig. 218); 2X4 4 3d. The regular hexagon, whose angle = — ^ — = ^ of a right angle (Fig. 219). 362 GEOMETRY 2X6 3 The angle of a regular octagon, being equal to — - — = ^ of a right angle, is not contained a whole number of times in 4 right angles, and consequently an octagon can not be used; but com- bining an octagon and a square in such a manner that two angles of the octagons and one of the square have the same vertex. we have - X 2 + 1 face (Fig. 220). 4 right angles, which will cover the sur- AREAS OF POLYGONS AND CIRCLES 996. Find the area of any polygon. The polygon is divided into triangles by drawing all the diagonals through one vertex, or by joining a point taken within the polygon to all the vertices; find the area of each triangle (718), and the sum of these results will give the area of the polygon. Ordinarily the polygon is divided into right triangles and right trapezoids by drawing a diagonal and dropping perpendiculars from the vertices upon this diagonal. 997. To change any polygon ABODE to an equivalent polygon having one less side. Whether the polygon be convex (Fig. \ "^ f f y \ VSD J 1 1 E F ^ A J3 Pig. 222 Kg. 223 221), or have a re-entrant angle (Fig. 222), join C and E, draw DF parallel to CE, and join C and F, then the triangles CED and CEF are equivalent (720), and consequently the polygons ABODE and ABOF are also equivalent. Remark. In this manner any polygon may be transformed into an equivalent triangle. 998. Oonstruct a square equivalent to the difference of two given squares. Draw two straight lines AB and AC perpendicular to each other; on one take AB = a < h, where a and h are the sides of PROBLEMS IN GEOMETRY 363 D^ E the given squares, and from B as a center, and 6 as a radius, describe an arc, cutting AC in C, thus determining the side x of the required square. From the right-angled triangle ABC (730): AC" = BC" - IF = b^ - a\ The same result would have been obtained by describing a semicircle on the side BC = 6 as diameter, drawing in the chord BA = a from B, and connecting A and C (684). Having the side AC, the square is con- structed as in article (982). 999. Construct a sqware equivalent to the sum or difference of any number of squares, a, b, c, d, being the sides of the given squares. Let k be the side of the equivalent square, and /i;2 = a^ + &2 + 0=* - (f . Draw two perpendiculars AB, AC, equal to a, b, and join c and B; at C draw CD = c perpendicular to CB, join D and B; on BD as a diameter, describe a semi-circumference, and lay off DE = d &s chord; then, joining E and B, the required side k is determined. The successive right triangles give (730, 998): BC^ = a" + b\ W = BC^ + (? = a^ + b^ + c"". Pig. 224 BE" BD'' - d^ + b^ + c" - d^. Having the side k, the square is constructed as in article 982. 1000. Find the side x of a square which bears a given ratio m : n to a given square a?. Take AB = m and BC = n; on AC as a diameter, describe a semicircle ; at the point B erect n c'^v a perpendicular BD to the line Kg. 226 AC; draw DA and DC, on DC prolonged beyond C if it is necessary; take DE = a, and, draw- ing EF parallel to CA, we have DF = x. From (352, 699, 732): x:a = DA:DC or 7? -.a" = 01 -.DC^ = AB :BC = m:n. If the ratio m : n had been that of two numbers, 3:5 for example, take AS = 3 times and BC = 5 times some length taken as unity. 364 GEOMETRY Construct a square which is a fractional part of a given square. 3 Let r be the fraction, that is, the squares are to each other as 5 3 is to 5. Instead of operating as above, describe a semicircle 3 on AB as diameter, take AE = ^AB (967); at E erect a perpen- dicular EF to AB, and draw the chord AF, which is the side of the required square (Fig. 226). Having (732) AF' = AB X AE, 7-02 -ro2^ AE , AF' AE 3 AF = AB X ■ — and -=-, = = - • AB AB^ AB 5 10.01. Two similar polygons p and p' being given, construct a third polygon P, which is similar to them and equivalent, \st, to their sum; 2d, to their differ- ence. 1st. Construct a right triangle ABC (Fig. 224) with its legs equal to two homologous sides, a and h, of the polygons p and p', and then the hypotenuse will be equal to x, the homologous side to a and h of the similar polygon F; on this side the polygon P is constructed similar to p and p' (972) and is equivalent to their sum. From (726), p:p' = a" : ¥, and (p + p') : {a^ + b^) = p -.a? (349) P : x^ = p : a^, these two proportions having three equal terms, x^ = a^ + ¥, and we have P = p + p'. 2d. Taking the longer side, b, as the hypotenuse of the right triangle (Fig. 223), and constructing P on the leg AC = x, for the same reasons as in the first case we would have P = p' — p. 1002. Construct a polygon p, similar to a given polygon P, and make the areas bear a given ratio, m : n, to each other. a being one of the sides of the polygon P, find the side x of the equare, such that x'^ : a^ = m : n (1000), and on a; as a homo- logous side to a, construct a polygon p, similar to P (972). In order that the perimeters of the polygons have the ratio m : n, we must have x : a = -m : n (703, 969). In order that a circle of a radius x, bear a ratio m : n to a circle PROBLEMS IN GEOMETRY 365 of given radius a, we must have x' : a^ = m : n, and for the cir- cumferences to bear the same ratio, we must have x : a = m : n. 1003. Construct a square equivalent to a given parallelogram or triangle, x being the side of the square, and h and h the base and altitude of the given figure, according as the figure is a par- allelogram or a triangle, we have (718, 721): 3? = hXh or oi? = hX% which shows that x is the mean proportional between the base and altitude in the first case and between the base and half the altitude in the second case (970). Remark. From this article and (997), a method for construct- ing a square equivalent to any given polygon may be deduced. Then article (999) gives the means of constructing of a square ■equivalent to any number of polygons combined in addition or subtraction. 1004. Construct a rectangle on a given straight line c, equiva- lent to a given rectangle whose dimensions are a and b. The fourth proportional x, of the three lines c, a, b, is the second dimension of the required rectangle (969). From c : a — b : x, we have c X x = b X a. (339) 1005. Construct a rectangle equivalent to a given square, and the sum of whose dimensions is equal to a given line AB. On AB as a diameter, describe a semicircle; ^ draw the perpendicular CD equal to the side c -r-5- of the given square, then drawing DE parallel (^ ^ j and EF perpendicular to AB, the two segments A - C FB AF and BF are the dimensions of the required ^' rectangle. From (706) : EF^ or (? ^ AF X BF. AB The problem is only possible when c < -^ , and it is seen that of all the rectangles of the same perimeter the square is the maxi- mum (584). 1006. Construct a rectangle equivalent to a given square, the difference of whose dimensions is equal to a given line AB. On AB as a diameter, describe a circle; at one extremity A, erect a perpendicular AC, equal to the side c of the given square, 366 GEOMETRY and drawing CO, the dimensions of the required rectangle are CD and Ci;. From (708): CD -.0 = c:CE, CDXCE = c\ In any quadrilateral A BCD: The middle points of the four sides are the vertices of a parallelogram MNPQ (640, 699); 2d. The area of the parallelogram MNPQ is equal to one-half that of the quadrilateral ABCD. This follows from the fact that the four triangles, OMN, ONP, OPQ, OQM, are respectively equivalent to the four triangles, BMN, CNP, DPQ, AQM, having the same base and equal altitudes. 1008. The lunes of Hippocrates. Describing semicircles on the three sides, a, b, c, of a right triangle as diameters, the area of the two shaded lunes is equal be to that, -^ ) of the triangle. Noting that the area of the lunes is equal to the sum of the areas of the two semicircles described on the diameters b and c Pig. 231 and the triangle ABC less the area of the semicircle described on the diameter a, we have from (718, 730, 753) : -S^ be 2 va'' be 2 ^(&^ -a') = ~- There are other portions of a circle which may be measured exactly, but they are not contained a whole number of times in the entire circle; if such had been the case, the determination of the quadrature of a circle could have been easily solved (1017). 1009. The area S of the ring included between the two concentric circles of radii OA and OB, is equivalent to the area irAB of a circle whose diameter is equal to the chord AC of the external circle tangent to the interior circle, PROBLEMS IN GEOMETRY 367 From (730, 753): S = ttOI' - irOW = TT (OT - OW) = TriS". From this it follows that in order to divide a circle of radius OA into two equivalent parts by a concentric circle, draw the chord AC, making an angle of 45° with the radius OA, and the perpen- dicular OB to AC will be the radius of the required circle. To divide a circle of radius OA by a concentric circle in such a manner that they bear a certain ratio to each other, for example, so that the area of the internal circle be to that of the ring as 3 : 2, divide OA so that OD -.DA = 3 : 2 (967); at the point D erect a perpendicular on OA and prolong it to the semi-circumference described on OA as a diameter, then OB is the radius of the in- ternal circle. From (732, 749) : B:2 = 0D:AD = 0B-. AB' = ttOB" tAB. In dividing OA into a certain number of equal parts and mak- ing the same construction for each point of division that has just been made for the point D, the circle of radius OA will be divided into the same number of equivalent parts by the concentric circles. 1010. Dividing the diameter AB = D oi a circle into any number of parts, d, d', d" , equal or unequal, the sum s of the circumferences of the circles which have the diameters d, d', d", is constant and equal to the circumference of the circle whose diameter is D. From (752): S = -n-d + -n-d' + ird" = -ir {d + d' + d") = irD. This is also true for semicircles. 1011. Dividing the diameter AB = D into a certain number of equal parts, 3 for example, upon which as diameters semicircles are de- scribed, as shown in Fig. 233, then the circle of diameter D is divided into the same number 3 of equal parts, the perimeter of each being equivalent to the circumference of the circle whose diameter is D (1010), and the area equal to :^ that of the circle of diameter D. Thus, we have, 368 GEOMETRY 1st. Noting that S is equal to a semicircle of diameter AC = — , plus a semicircle of diameter AB = D, less a semicircle o 2 of diameter C5 = - D, we have from (753): 1 1 /DV 11 11/2 7)\2 ^=2Xin3)+2X4-^^-2X4n-#) 7 J_ 4- 1 _ A^ - i — Al8 2 18/ ~ 3 4 ■ ttDVI 4 2d. (S' being equal to twice the remainder obtained in sub- tracting a semicircle of diameter AC = -^ from a semicircle of o 2 diameter ylD = - D, we have: O «'-Exj'(W-5xl'(?)' From (Fig. 233) it is seen that: ^/4_ n_ 1 ^ 4 U gj" 3 4 ' REGULAR POLYHEDRONS AND SPHERES 1012. The figures shown below are the developments of five regular polyhedrons; they show clearly enough how these devel- opments are drawn when a side of the polyhedron is given. Fig. 234 Kg. 235 Fig. 236 For (Figs. 234, 236, and 238) the 60° triangle is used. As to the dodecahedron, after having constructed the pentagon P on the length given as one side, the sides of this polygon are PROBLEMS IN GEOMETRY 369 prolonged and a circle drawn through the points of intersection, and by drawing parallels to the sides of the pentagon P one- half of the development is determined. For the second half Fig. 238 Kg. 239 prolong ab and take cd = ab; on cd as a chord describe a circle of the same radius as the first, and in this circle by drawing parallels to the sides in the first half of the development, the construction is completed. 1013. A sphere being given, find its radius. Take two points, A and B, on the surface of the sphere; from these points as centers, or rather as poles, with any convenient radius, describe two arcs which intersect in two points, D, D' ; with another radius determine a third point, D". D, D', and L" being equally distant from the points A and B, they lie in the circumference of a great circle whose plane is perpendicular to the middle of AB, and it follows that if a triangle whose sides are equal to the dis- tances between the three points, D, D', D" (940), is constructed, that its circumscribed circle will be equal to the great circle of the sphere, and its radius will be equal to that of the sphere (952). 1014. Two points, A, B, on the surface of a sphere being given, describe a great circle through the points. From the points A and B as poles, with a radius equal to the chord of a quadrant (852, 916), describe two arcs which intersect in the point C, and from this point as a pole with the same radius describe a circle, which is the required great circle. It is seen that the same construction may be used to find the poles of the circumference or an arc of a great circle. Kg. 240 370 GEOMETRY 1015. Describe a small circle passing through three points, A,B,C, on the surface of the sphere. Operating as in (1013), two points, D, D', equidistant from A and B are determined, and through the points D, D', a great circle is described, whose plane is perpendicular to the line AB at its middle point, since it contains the points D, D', and whose center, 0, is eq^aidistant from the points A and B (768). In the same manner a great circle is determined whose plane is per- pendicular to BC at its middle point; this circle intersects the first in the line PP', and the extremities P and P' are the poles from which as centers the required small circle may be described. 1016. Through a point A, taken on the surface of a sphere, draw a great circle perpendicular to the circumference or arc of another great circle BD. From the point A, taken on BD or outside of BD, as pole, and the chord of a quadrant as radius (1013), describe an arc of a great circle cutting the given circle in P; from the point P as pole, with the same radius, gf describe a great circle, which will pass through the point A. When the point A is the pole of BD, any great circle which passes through A satisfies the conditions, but in all other cases there is but one solution. 1017. There are three problems which appear to belong to elementary geometry, and which may be solved with a rule and a compass (935). They are: 1st. The trisection of an angle, that is, the division of an angle or an arc into three equal parts (976). By the following construction an angle C is obtained equal to a third of a given angle AOB; but the problem is not solved geometrically, since the method of trial and error is used to determine the line CDB. From the vertex as center, with a radius equal to OA, a semicircle is de- scribed; on the edge of a rule or a piece of paper CD is laid off equal to the radius OA, then the rule is so manipulated that the points C and D fall respectively upon the Fig. 242 Fig. 213 PROBLEMS IN GEOMETRY 371 line AC aad the semi-circumference DBA, while the line CD ex- tended will pass through B; when this is the case, draw CDB, 1 and the angle C will be equal to ^ of the angle AOB. o Since the exterior angle AOB = B + C (653), and B = BDO (635), and BDO = C + COD = 2 C, we have Z AOB = 3 Z C. 2d. r/ie quadrature of a circle, which consists in constructing a square which has the same area as a given circle (1008). The following method gives the solution correct to one decimal unit of the fifth order. Draw a diameter AB, and a tangent BC; take OG = -^ oi the radius 6 OA ; from the point G as a center and a radius equal to twice the diameter AB, describe an arc which cuts the tangent in C; join A and C, and the chord BD is the side of the required square BDEF. 3d. Duplication of a cube, which consists in finding the side of a cube which is double that of a given cube. The solution is obtained by calculation. PAET IV TRIGONOMETEY PLANE TRIGONOMETRY 1018. The special object of trigonometry is to furnish methods for the calculation of the unknown parts of a triangle (angles and sides) when enough is given to determine them (938 to 942). Any polygon being composed of triangles, it follows that the more general purpose of trigonometry is to calculate the unknown parts of any polygon which is sufficiently determined. DETERMINATION OF A POINT 1019. The means of fixing the position of a point on a line. Since from a certain point in a given line the same distance may be measured in two directions (599), it follows that it is .not sufficient for the determination of a point to know its distance from a certain point in a given line, but the direction in which the distance is taken must also be known. To simplify the expressions and facilitate the calculations, it is agreed to consider the distances measured in one direction as positive, and in the opposite direction as negative, and these are designated in the calculation by the usual signs + and - (449). Generally the lines drawn from left to right and from down to up are considered positive, and those from right to left and up to down as negative. The fixed point of a line from which all . distances on the line are measured is called the origin. When the line on which the distances are measured is a straight line, it is called an axis. The distance of any point on the axis to the origin is called the abscissa; it is generally designated by + x or — a;, according as it is measured in one direction or the opposite' 1020. Two directions, xx' and yy', perpendicular to each other being given, the position of any point in the plane of these two 372 DETERMINATION OP A POINT 373 — I q SE' iF^ U^'" U / Fig. 245 directions is determined when the projections of the point on the straight lines xx' and yy' are known (715). p and q being the projections of a point on the Hnes xx' and yij', erecting the perpendiculars pM and qM, each of these per- pendiculars contains the point, therefore it must be at their intersection, M. The point M being determined when its projections, p and q, upon two rectangular lines are known, and these projections being determined by their distance from the origin taken on the axes xx' and yy' (1019), therefore a point in a plane is determined when the abscissas of its projections upon two rectangular axes, drawn in the same plane, are known. The common origin of the two axes xx' and yy' is taken at their intersection 0. The axes are called coordinate axes. The axis xx' is called the x-axis. The axis yy' is called the y-axis. The distances measured on the a;-axis are called abscissas, and those on the y-axis are called ordinates. The abscissa Op of the projection p is also the abscissa of the point M. Since Op = Mq, it is seen that the abscissa of a point is the distance of the point from the y-axis. The abscissa, which is designated by x, is positive or negative, according as it is measured on Ox or Ox'; that is, according as the point is at the right or the left of the y-axis. In a like manner, since Oq = Mp, the ordinate of a point is the distance of the point from the a;-axis. The ordinate is desig- nated by y, and is positive or negative, according as the pomt is located above or below the a;-axis. The abscissa and ordinate of a point are the coordinates of the point. Thus a point is determined by the algebraic values of its coordi- nates X and y (450). For M, X = + Op and y = +0q; M', x= - Op' and y=+Oq: M", x = - Op' and 2/ = - Oq'; M'", x= +0p and y= - Oq'. 374 TRIGONOMETRY When X = 0, the point lies on the y-axis; when y = 0, it lies on the a;-axis; and when both x and y are equal to 0, it lies on both, that is, at the origin. Remaric 1. That which has been said of the rectangular axes xx' and yy', holds likewise when the axes make any angle with each other; but then the lines Mp, Mq . . ., which remain parallel to the axes yy', xx' , are oblique to the axes aia;' and yy'. Remark 2. In the case where the axes are rectangular, join- ing and M, the right triangle OMf gives (730): OM' = x + f. The distance of any other point, M' , M" . . ., from the origin gives the same relation with the coordinates of the point con- sidered. 1021. Means of fixing the position of a point in space. In the same manner as a point in a plane is determined by its projections on two straight rectangular axes drawn in the plane (1020), the position of any point in space is determined when its projec- tions on three planes, each perpendicular to the other two, are known (763). a, b, and c being the projections of a point M on the three planes xOy, xOz, and yOz, determined by the rectangular axes xx', yy', and zz', -which are the intersections of the planes, if at each of these points a perpendicular to the corresponding plane is erected, they will all three meet in the point M. Thus a point is clearly determined by its projections on the three planes. Each of the projections, a, b, c, being deterniined when its respective projections, p and q, p and *S, q and S, on two axes are known, it follows that these three projections, and consequently the point M, are determined when the points p, q, and s are known, which is nothing other than the projections of the point M upon the three axes, xx', yy', and zz' (715, 790). The three points, p, q, s, on the axes, being determined by their abscissas with reference to the origin (1019), a point M is therefore determined when the abscissas of its projections on three rectangular axes are known. The three rectangular axes, xx', yy', and zz', are likewise called coordinate axes; xx' being the x-axis, yy' the y-axis, and zz' the z-axis. DETERMINATION OF A STRAIGHT LINE 375 The three planes determined by these axes are called the coordinate planes. The abscissas Op, Oq, Os, of the projections of the point M on the axes, are called the coordinates of the point M; Op is the abscissa x, Op is the ^/-ordinate, and Oz the 2-ordinate. Thus a point is determined by its coordinates (1020). Since Op = Mc, Oq = Mb, and Os = Ma, the coordinates X, y, and 2 of a point are equal to the distances of this from the coordinate planes. These coordinates are positive or negative, according as the projections of the point upon the axes lie upon 'the parts Ox, Oy, and Oz, or upon Ox', Oy' , and Oz'. Thus x will be positive or negative, according as the point M lies at the right or left of the plane yOz; y will be positive or negative, according as the point lies in front of or behind the plane xOz; and finally, z will be positive or negative, according as the point M is above or below the plane xOy. When a; = 0, the point is in the plane yOz; if j/ = or 2 = 0, the point is respectively in the plane xOz or xOy. When two of the coordinates are equal to zero, the point lies on one of the axes; thus, for x = y = 0, the point is on the 2-axis. If a; = y = 2 = 0, the point is on all three axes, and must be at the origin. Remark 1. That which has been said of planes or axes which are perpendicular to each other applies as well when they are inclined to each other, except that the perpendiculars Ma, Mb, Mc, to the planes of projection remain parallel to the axes. The projections, a, b, c, on the coordinate planes, or those, p, q, s, on the axes, instead of being orthogonal projections, are then oblique projections. Remark 2. The distance OM from the point M to the origin being the diagonal of a parallelopiped whose edges are the coordinates of the point, in case the axes are rectangular, the parallelopiped is rectangular, and we have, Oiir = x^ + 2/^ + z\ (835) This relation exists no matter where the point is located about the origin. DETERMINATION OF A STRAIGHT LINE 1022. The position of a straight line is fixed by that of its ex~ tremities, and therefore by the coordinates of its extremities (1021), 376 TRIGONOMETRY A straight line may also be defined by the conditions which deter- mine: First, one extremity; Second, its length; Third, its direction. 1st. The position of one extremity of a straight line is deter- mined by the algebraic values of the coordinates of this extrem- ity. 2d. The length of a straight line is determined, without re- gard to the sign, by the ratio of it and the linear unit (713). 3d. It remains to fix the direction and sign of the line. No matter what the position of the line with reference to the axes is, its direction and sign with reference to these axes will be known, when its direction and sign with reference to a system of axes parallel to the first and passing through the known extremity of the given line are known. 1023. This last part of the question is therefore reduced to the determination of what is necessary to fix the direction and sign of a straight line with reference to a system of coordinate axes whose origin is at one extremity of the given line (598, 599). At first, consider the most simple case, namely, where the straight line is in the same plane as the axes, that of xy for example (1021). Let Ou be the straight line, then its sign is indicated by the order of its extremities and u; the direction of this line will be determined when the angle uOx, which the line makes with the part Ox of a;-axis, is known, and it is indicated upon which side of the a;-axis this angle is to be taken because it is easily seen that two equal angles may be drawn with Ox as one side. In order to dispense with the necessity of designating whether an angle is to be measured from one side or the other of Ox, a conventional system analogous to that in (1019) for fixing the position of a point has been adopted. Thus it has been agreed to consider as positive all the angles described by the straight line Ox in turning about the point in the direction indicated by the arrow AB, and as negative all the angles described in turning in the opposite direction AC. The positive angle is zero when Ou coincides with Ox; then it takes all the values between 0° and 90° in turning from Ox to Oy; when it coincides with Oy, it makes a positive angle of 90° TRIGONOMETRIC EXPRESSIONS 377 with Ox. In turning from Oy to Ox' it takes all the values from 90° to 180°, from Ox' to Oy' all the values from 180° to 270°, from Oy' to Ox all the values from 270° to 360°, and from Ox on, all the values from 360° up. If Ou had revolved in the negative direction, it would have described all the negative angles just as it has the positive. It should be noted that the angles + a, + (360° + a), + (720 + a), etc.; - (360 - a), - (720 - a), etc., all designate the same straight line, both in direction and sign. Remark. As the line Ou describes angles about the point 0, the points in the line describe arcs corresponding to these angles (667)-, and according as these angles are positive or negative, the arcs are also positive and negative. Thus an angle is determined when its corresponding arc is Icnown, and vice versa; it is, of course, assumed that the arc is preceded by its sign + or — , according to the conventions adopted. TRIGONOMETRIC EXPRESSIONS — THEIR USE FOR THE EX- PRESSION OF THE VALUE OF ANY ANGLE OR ARC, POSITIVE OR NEGATIVE 1024. In the case where the straight line Ou has one of its extremities at the origin 0, the line is determined when the alge- braic values of the coordinates y = Mp, and x = Mq, of its other extremity are known (1020). The ratios between the quantities x, y, and OM are constant, no matter what the position of M on Ou may be, that is, no matter what the value OM = r may be. The quan- tity OM = r is always positive since it is the distance of the point M from the origin 0, and is measured in the positive direction along the generatrix Ou of the angle uOx. From this it follows that the direction of the line is determined when the algebraic values of two of the constant ratios between x, y, and r are known; because, assuming any value of r, these ratios give the corresponding values of x and y (516). Six different ratios or trigonometric expressions or functions may be formed with the quantities x, y, and r; ^. ■ %; f] [T ? \ ^ \ "{ X X / A '^ mf /' i" T Fig. 248 378 TRIGONOMETRY - . ratio of the ordinate Mp to the radius of the arc AB passing r through the point M, is the sine of the angle uOx = a, and of the are AM, which is also designated by a. It has the same sign as the ordinate y (1020); - , ratio of the abscissa Op to the radius, is the cosine of the angle r and arc u. Its sign is the same as that of x; — , ratio of the ordinate to the abscissa, is the tangent of the angle X and arc a. It is positive or negative according as y and x have the same or opposite signs; - , the reciprocal of the sine, of the same sign, is called the co- secant of the angle and arc a; -, the reciprocal of the cosine, of the same sign, is called the X secant of the angle and arc a; — , the reciprocal of the tangent, is called the cotangent of the angle and arc u.. It is positive or negative according as x and y have like or unlike signs; consequently it has the same sign as the tangent. The above functions are written: V ^ i V sin a = - , cosa = - , tana = -, r r x V T X CSC a = — ) sec a = — » cot a = — ■ y X y 1025. Other forms of these functions. Trigonometric lines. y Mp 1st. We have sin ^ = - = — i- , ratio of the radius r to half r r ' Mp of the chord which subtends the arc corresponding to double the angle u. X Ot) 2d. Cos a = - = — i- . As is shown in Fie;. 248, the cosine and r r a I the sine of a are respectively equal to the sine and cosine of the com- plement of the angle a. 3d. Drawing the tangent AT (Fig. 248), the two similar triangles OAT, OpM, give (700, 1024): AT y ^ = - = tan a. p X TRIGONOMETRIC EXPRESSIONS 379 Thus the tangent of an angle a is also represented by the ratio of the positive or negative tangent AT, drawn from the origin A of the arc described with the radius r, and prolonged to meet the other side of the angle a, to the radius r. This is why the expression - is called tangent. 4th. The same similar triangles OAT and OpM give: OT r — = - = sec a. r X The secant is therefore represented by the ratio of that portion of the secant OT, measured on the second side of the angle and included between the center and the tangent, and the radius r. This gives the function its name secant. 5th. Drawing the tangent BS from the point B until it meets Ou, the two similar triangles OBS and OqM give: BS X = - = cot a, r y which shows that the cotangent of an angle is represented by the ratio of the tangent BS to the radius. This formula and the Fig. 248 show that a cotangent of an angle is nothing other than the tangent of its complement. This is where it gets its name cotangent. 6th. From the two similar triangles OBS and OqM: OS r ■ — - = - = CSC a. r y Thus the cosecant of an angle is represented by the ratio of that portion OS of the secant to the radius. From this formula and the figure, it is seen that the cosecant of an angle is nothing other than the secant of its complement, and hence its name cosecant. We have therefore: Mp sm a = — - , r Op cos a = —^ , r AT tan a = r OS CSC a = < r OT sec u. = — > r BS cot a = r atting r = 1, sin u. = Mp, cos a = Op, tana = AT CSC a = OS, sec a = OT, cot a = BS. 380 TRIGONOMETRY These last values of the trigonometric functions are represented by lines, and are called trigonometric lines. 1026. There are still two trigonometric functions which we will simply define, since they are not frequently used. = — — is the versed sine of the angle and arc a. For r = 1, r r the versed sine, vers sin a, is equal to Ap. = — i is the coversed sine of the angle and arc a. For r r r = 1, the coversed sine, covers sin u,, is equal to Bq. 1027. Signs of trigonometric functions. Since the only variables which enter in the trigonometric functions of (1024) are the co- ordinates X and y, it is very easy to determine the signs of these variables no matter what the value of a may be (487, 1020). For the values of a between 0° and 90°, a; -and y are positive, x varies from r to 0, and y from to r; therefore (1024) : and varies from to + 1 ; and varies from + 1 to ; and varies from to + oo ; . and varies from + oo to 1 ; , and varies from 1 to + oo ; cot a = -\ — , and varies from + oo to 0. y For the values of u, between + 90° and + 180°, y is positive and varies from r to 0, while x is negative and varies from to — r; therefore: sin a = + - , and varies from + 1 to ; r ri' '>. cos a = = , and varies from to — 1 ; r r ' '^ V V tan u, = — - = — - , and varies from — oo to ; — XX r esc a = 4 — , and varies from 1 to + oo ; y sin a -+f' cos a =+-:> tana =+l- esc a -->• sec u, X TRIGONOMETRIC EXPRESSIONS 381 r V sec u, = = , and varies from oo to — 1 : — XX ' — X X cot u. = = , and varies from to — oo. y y For the values of a between + 180° and + 270°, y is negative and varies from to — r, and x is also negative and varies from — r to ; therefore : sm 1 = — ~ = — ^ , and varies from to — 1 : r r ' — X X cos a = = , and varies from — 1 to ; r r - y y tan a = — - = + ^ , and varies from to + oo : — XX V T CSC a = = , and varies from — oo to — 1 ; - y y T T sec a = = . and varies from — 1 to — a> : — XX' cot u = = H — , and varies from + oo to 0. -y y For the values of u, between + 270° and + 360°, y is negative and varies from — r to 0, while x is positive and varies from to + r; therefore: sin a = — - = — ^ , and varies from — 1 to ; r r X cos a = H — , and varies from to + 1 ; r tan a = — - = — S- and varies from — oo to ; + x X r r esc a = = , and varies from — 1 to — oo ; - y y r sec u, = - , and varies from + oo to + 1 : X _I_ rr X cot u, = = . and varies from to — oo. -y y For values of u greater than 360°, these values and signs are repeated and so on; thus, the trigonometric functions of the angles (360° + 30), (360° X 2 + 30), etc., are the same as those of an angle of 30°. 382 TRIGONOMETRY By inspection of Fig. 248 it is seen that for any negative angle — a (1023), the trigonometric functions have the same values and the same signs as for the positive angle 360 — a. From this it follows that if a table of the values for the negative angles were constructed, we would have the same as in the one given above, but in an inverse order. Thus, for the angles from 0° to — 90°, we would have the same values as for the positive angles from 360° to 270°. The figure (249) below, indicates the signs of the trigonometric functions for the different values of the angle or the arc o. 1/ V m sine and esc cos and sec Fig. 249 tan and cot 1028. It should be noted that the absolute values of the co- ordinates y and x, and therefore, those of the trigonometric functions of any angle uOx (1024), are equal to those of the acute angle which the line Ou makes with Ox or its prolongation Ox' (Fig. 248), this acute angle being always considered as positive. From this it follows that in forming the table (1071) of the values of the trigonometric functions of all the positive angles included between 0° and 90°, it will contain also the absolute values of all the angles greater than 90°; having the absolute value, the sign may be prefixed which belongs to the given angle according to the table (1027) or the figure 249. If it is desired to have the sine of the angle uOx = + 215°, for example. Noting that Ou makes an angle of 215 — 180 = 35° with Ox'] look in the table (1071) for the sine 0.57358 of the angle of 35°, and prefixing the minus sign before this absolute value which corresponds to the angle 215°, we have: sin 215° = - 0.57358. Any angle being given, the algebraic values of its trigono- metric functions may be determined. 1029. A single trigonometric function does not determine the angle a, since for a given value + /S of the sine there are two TRIGONOMETRIC EXPRESSIONS 383 angles a and 180° — u., and for sin ^ = — >S there are two angles 180° + a and 360° - a. Since an acute angle a corresponds to a positive cosine, while its supplement 180° — a corresponds to a negative cosine, an angle is determined when the value and sign of its sine and the sign of its cosine are given. In the same manner there are two values of the angle for one value and sign of the cosine, and in order to determine an angle, the value and sign of its cosine and the sign of its sine must be known. + t being the value of the tangent of the angle a, we have t = - and t = — ^ , equations which may be satisfied by the two lines Ou and Ou", directly opposed to one another and mak- ing the angles a and 180° + a with the line Ox. Thus an angle is not determined by its tangent; but it becomes determined when besides its tangent the sign of one of its coordinates x or y, or, which is the same thing, its sine or cosine, is known. If the given tangent were — t, we would have — t = — - and — X — t = — - > which values are satisfied by the lines Ou' and Ou'", + X directly opposed to each other and making the angles 90° + a, and 270° + a with Ox. Thus the angle is not determined, but will be when, besides the tangent, the sign of the sine or cosine is known. In general, for each algebraic value of the principal trigono- metric functions, sine, cosine, and tangent, there corresponds, for each of the two other functions, two equal values opposite in sign; this is shown in Fig. 249. It follows then that having the value of any one of the trigonometric functions, the angle is determined if the sign of one of the other two is known. 1030. Designation of an angle by the words batter and grade. In masonry the batter of a wall is said to be so and so many feet per a certain number of feet in height, meaning that the face of the wall is inclined to the vertical by an angle whose tangent is equal to the ratio of the given numbers. For instance, if the batter of a wall is 1 : 10, the tangent of the angle is 0.1. The grade of a road is the height which the road rises from the horizontal in a given distance; it is generally expressed in per cent. Thus, a grade of 3%= r^ =tan a =0.03 is expressed by the tangent 384 TRIGONOMETRY of the angle which the surface of the road makes with the hori- zontal. If the distance is taken on the surface of the road, this ratio is then the sine of the slope angle a, but in any case the slope is generally so small that there is little difference between the tangent and the sine. 1031. We have seen how, having a table containing the values of the trigonometric functions of the angles from 0° to 90°, the functions of any angles may be found (1028). Noting that the sine, the cosine, the tangent, the cotangent, the secant and the co- secant of an acute angle are respectively equal to the cosine, the sine, the cotangent, the tangent, the cosecant and the secant of its complement, it is seen that the functions of the angles from 0° to 45° are all that are necessary to determine those of all the angles. For example, if it is desired to have the sine of 70°, look for the cosine of 90° - 70° = 20° in the table (1043). The absolute value of the cosine of an angle of 125° is 0.57358, the cosine of 180° - 125° = 55° (1071) and the sine of 90°- 55° = 35° ; its algebraic value is - 0.57358 (1027). General Rule. When the value of a trigonometric function of an angle between 90° and 180° is to be determined, find the value corresponding to the supplement of the angle and prefix the sign corresponding to the given angle (1027), which gives the required value. In practice, it is rarely required to find the func- tions of angles greater than two right angles, but, even if it should be, it offers no difficulties that have not been explained above. 1032. Trigonometric tables. In practice, use is scarcely ever made of functions other than the sine, cosine, tangent, and co- tangent, and therefore the tables contain only these values. The tables are so arranged that each absolute value may be read as a function of an angle and its complement. For in- stance, the sine of one angle is the cosine of its complement. Referring to the table (1071), the numbers in the second column are sines of the angle whose number of degrees is read at the top and minutes at the left in the first column, and at the same time these same values are the cosines of the angles (comple- ments of the above) whose decrees are written at the bottom and minutes in the last column at the right. Reading from the top, the functions of all the angles expressed in minutes up to 45° are given, then reading from the bottom the functions of the angles from 45° to 90° are found. TRIGONOMETRIC EXPRESSIONS 385 1033. Determination of the position of a straight line in space. We have just seen how, by means of the trigonometric func- tions, the position of a Hne in the plane of the coordinates is fixed. Let us now examine the case where the straight line lies outside of these planes. Assume that one extremity of the line Ou lies at the origin of the coordinate system. The position of the line will be determined when the coordinates x, y, and z of a point M situated in the line at any distance + OM ^^^' ^^ = r from the origin (1021). This position will, therefore, be determined when the ratios -> —> and - are known. The signs r r r of the ratios are determined by the signs of x, y, and z, because r is always positive. Let the angles which the line Ou makes with the axes Ox, Oy, and Oz be respectively <*, ^8, and y. Mp being the perpen- dicular to Ox (770), Op is the abscissa x of the point M, and, in the plane Oux, we have: Ov X 7^ = - = cos a. OM r Likewise in the planes uOy and uOz we have: (1024) ft ^ - = cos B and - = cos y, r r which shows that, knowing the cosines of the angles which . the line makes with the coordinate axes, the algebraic ratios -) -} and - are known, and therefore the line is determined. r r r 1034. We have: therefore that is, a;2 + 2/2 + 22 = OM' = r^; X^ 1^ 2^ p- •+ ^ + ^ = 1' cos^ a -h cos^ /3 + cos^ y = 1, (1021) (a) which shows that the sum of the squares of the cosines of the angles which a straight line makes with the rectangular axes of a system of coordinates is eqvnl to one. 386 TRIGONOMETRY Remark 1. This relation shows that the cosines of the angles which a line makes with the three axes of a rectangular coordi- nate system cannot be arbitrarily chosen; but that the algebraic values of the cosines of two of the angles and the sign of the third cosine being given, the third cosine and the position of the line may be determined by means of the equation (a). Remark 2. The cosine of an angle which a straight line makes with an axis determines the surface of a cone of revolution of which the straight line is the generatrix. The cosines which the straight line makes with two axes of the coordinate system determine two lines, namely, the intersections of two conical sur- faces of revolution, one line making an acute and the other an obtuse angle with the third axis; now if the sign of the cosine of the angle which the line makes with the third axis is known, it is determined which of the intersections is the required line, and thus the position of the line is fixed. Remark 3. If the line is situated in the plane of two of the axes, the formula (o) becomes, cos^ a + cos^ /8 = 1. (1020) 1035. The circumference of a circle whose radius r = 1, being expressed by 2 tt (752), the quantity •"• corresponds to 180°, and it is evident that it may be used as a unit in measuring arcs and angles. An arc a being expressed as a function of tt, the value x of this same arc in degrees is 180 , . X = a (a) 7r Conversely, if a is expressed in degrees, its value x in function of IT is ^ = "180- ■ (^) Thus, according as TT TT TT TT TT 2 IT O'TT 6' 5' V 3' 2' "3"' "' ~2 , 2 T, the same arc expressed in degrees is respectively: 30° 36° 45° 60° 90° 120° 180° 270° 360°. M" X.' 4= PROJECTION OF STRAIGHT LINES 387 PROJECTION OF STRAIGHT LINES 1036. A straight line having two directions (599), the length of a finite line will take the + or — sign, according as the length was taken in the positive or negative direction. When a straight line is considered independently, either of its directions may be taken as positive, the opposite being nega- tive. But when the line is referred to a given axis or system of axes, its sign is determined by its position with reference to these axes. The direction of the projection of a straight line upon an axis is indicated by the order of the letters of two of its points, and the sign of each direction is the same as that for the same direc- tion of the axis (1019). To make this clear, the absolute length of the line M'M" or M"M' being 30 feet, the algebraic value of M'M" is + 30 feet, and that of M"M' is — 30. feet. In the same way the absolute value of the projection p'p" or q p"p' of M'M" on the axis Ox being 22 / j /^' ^^ feet, the algebraic value of p'p" is + 22, / '^'' and that of p"p' is — 22 feet. jrig. 251 1037. The algebraic expression of the projection of a straight line upon an axis. Having Op" = x" , abscissa of the point M", and Op' = x', abscissa of the point M', it follows that p'p" = + (x" - x'), and p"p' = - (x" - x'.). Analogous expressions are obtained for the projections on each of the other axes Oy and Oz. These expressions apply equally in the cases where x' and x" have like or unlike signs. Thus, the values of x' and x" both being negative, which is the case when M' and M" lie at the left of the yz plane, we have: p'p" = + [- x" - (x')] =-(- x" + x'), p"p' =-{- x" - (-a;')] = -{-x" -F x'). (426) If x' were negative and x" positive, the preceding formulas would give: p'p" = + [+ x" - {-x')\ = + {x" + x'), and p"p' = - [+ x" - {-x')\ = - (x" + x'). 388 TRIGONOMETRY 1038. Relation between a straight line and its projections (1040). If through the point M' (Fig. 281) axes parallel to the first sys- tem are drawn, the projections of M'M" on these axes would be respectively equal to the projections on the first; furthermore, these projections would be the coordinates of the point M". If the axes are rectangular, taking the length of M'M" equal to u, the formula of (1021) may be applied thus: u^ = {x" - x'Y + {y" - y'Y + {z" - i'f. In case one of the projections is zero, which is the case when the line is situated in one of the coordinate planes or parallel to it, the preceding formula becomes, V? = (x" - x'Y + {y" - y'y, when the line is parallel to the xy plane. This formula is the same an given in (1020). If the line were in the two planes yx and xz, for example, or parallel to them, it would coincide with the axis x or be parallel to it. Then its true length would be projected upon the x-axis, while the projections on the other two axes would be zero, and the preceding formula would become, v? = (x" — x'Y or u = {x" — x'), which is the same as in (1037). 1039. The algebraic sum of the projections of the several por- tions of a broken line ACDE on any axis, that is, the projection of the broken line on the axis, is equal to the projec- tion of the line AE, which joins the extremities of the broken line, upon the same axis (1040). x' being the abscissa of the point A, x" that of the points B and D, x'" that of C, and x^ that of E, we have successively (1037): Projection of AB = x" — x' , BC = x'"- x", CD = x" - x'", DE=x'^-x". Adding all the projections, and reducing, we have (458): Projection of ACDE = x^- x' , Fig, 252 PROJECTION OF STRAIGHT LINES 389 which is nothing other than the projection of the straight line AE joining the extremities of the hne ACDE. Remark. Considering a curved line as a broken line whose segments are infinitely small (601), it follows that the above statement applies also to curves, or, in gen- eral, any line. 1040. Projection of a straight line, and, in general, any line, upon an axis, expressed in terms of its trigonometric functions (1037). 1st. Let a straight line M'M" be situated in the plane xy, with its extremity M' at the origin of the axes. From (1024), by re- presenting the length of M'M" by u, the projections M'p and M!q of the line on the axes by Px and Py, and noting that these projections are the coordinates of the point M": y' y M' - I ^^■' W f ^ ' Jf' Fig. 253 U = cos a, and -^ = sin a ; u Px = u cos a, and Py = u sin a. 2d. These expressions apply also in the case where the line M'M" being in the plane x'y', does not have its extremity at the origin. The angles a and /8 which the line M'M" makes with the axes being the same as those which it makes with the parallel axes M'x and M'y, and, moreover, since the projections x" — x' and y"— y' are respectively equal to P^ and Py, we may write: x" — x'= u cos a, and y" — y'= u sin a. 3d. It remains to consider the case where the line M'M" is not in the plane of the axes (Fig. 253). The angle a which M'M" makes with Ox is equal to the angle M"M'x' which it makes with the axis Mx' parallel to. Ox (611); moreover, the projection M'N of M"M' on M'x' is equal to the projection 'p''p" = P^ of this same line of Ox, and we may write : in o. -' M I" TWx Kg. 254 U COS a. Thus, no matter what the position of a line with reference to an axis may be, the algebraic value of the projection of the line 390 TRIGONOMETRY upon the axis is equal to the absolute length of the line multiplied by the cosine of the positive angle included between the positive side of the axis and the line-(1019, 1023). Remark. We have said (3d) that the projections M'N and p'-p" were equal to each other. Proof. — The perpendiculars M"N, M'p' and M"p" drawn to the axes being in the' planes which pass through M'M" per- pendicular to the parallel axes, since these planes cut the axes in M', N, p' and p", and parallels comprehended between par- allels are equal, we have M'N = p'p". 4th. For a broken hne ACDE (Fig. 252), making AC = u', CD = u" . . ., and designating the positive angles which AC, CD . . . make with Ox (3d) by a', a" . . ., we have: Projection of AB = AB cos a', i( if BC = BC cos a'. Adding, we have: Projection of u' = u' cos a', Cl (( u" = u" cos a", it it U"'= u"'cOSa"\ Adding all three, we have: a;IT_ ^' _ y^' (jQg „/ ^ y// gQg jj// ^ y^"/ pQg g^t/r^ x^^— x' being the projection of the line AE (2d) joining the ex- tremities of the broken line. Representing the sum of the products by S w cos a, the distance between the two extremities by U, and the angle which the line joining the extremities makes with the axis by a, the preceding equation becomes: U cos a = 2 M cos o. Remark 1. Considering a curve as an infinite number of straight lines, this last equation applies also to curves. Remark 2. a' being the angle which AC makes with a par- allel to Ox drawn through A, and not through C (Fig. 252), its value Ues between 270° and 360°; a" being the angle which CD makes with a parallel to Ox drawn through C, its value lies be- tween 90° and 180°; a"' lies between 0° and 90°. The angle o', formed by AC and Ox, being between 270° and 360°, its cosine is algebraically equal to that of the acute angle 360° — o', which is the smaller of the two angles which the line RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS 391 AC makes with Ox. likewise the cosine of an angle a^, which Ues between 180° and 270°, is algebraically equal to that of the obtuse angle 360 — Oj, which is the smaller of the two angles which the line forms with thq axis Ox. To determine the projec- tion of a straight line or a series of straight lines on an axis, the calculations may be facilitated by taking the cosine of the smaller angle which the line makes with the axis Ox. FORMULAS EXPRESSING THE RELATIONS BETWEEN THE TRIGONOMETRIC FUNCTIONS 1041. Relations between the trigonometric functions of the same angle or arc a. From (1024): 1st. sin a = - J from which y = r sin a ; r X and cosa= -j from which x = r cos a. r Substituting these values of x and y in the equation y2 + a.2_^2_ (1020) we obtain r^ sin^ a + r^ cos^ a = r', or sin^ a + cos^ a == 1 ; from which sin a = ± \/l — cos^ a and cos a = ± Vl — sin^ a. „ , , V r- sin a sin a . 2d. tan a = - = = , X r cos a cos a from which sma cos a sma tana ± VI - sin^ a ± Vl -htan^a J ^ =1= Vl — COS^ a and tana = , COS a 3d cot a ^ '■''°^" 1 or COM a — 1 , /, , , 2 ±V1 + tan-'a y r sm a Thus, cot a = , or tan a = — — -; tan a cot a 392 TRIGONOMETRY from which 4- Vl - sin^ a or sm a = or cos a ■■ cot a rt Vl - Sin^ a sin a id cot a cos a =1= Vl - COS^ a 4th. sec r r X r cos a 1 =t Vl + COt^ a ' cot a ± Vl + cot' a ' 1 1 ) or cos a sec a rt Vsec^ a — 1 ov-^ ^ — I . or sin a = , =t V 1 — sin' a sec a sec a = = zt Vl + tan' a, or tan a = ± Vsec' a — 1 , cos a 1 =!r Vl + cot' a ^ 1 sec a = = ) or cot a = cos a cot a ± Vsec' a - 1 rxi. r r 1 .1 5th. CSC o = - = — -. — = —. — > or sm a y r sm a sin a esc a 1 zt VCSC' a - 1 CSC a = -7- — =) or cos a = , ± Vl — cos' a CSC a 1 =t Vl + tan' a ^ 1 CSC a = -; — = > or tan a = -p — ^, sin a tan a ± Vcsc' a — 1 CSC a = -: = ± Vl + cot' a, or COt a = =t: VcSc' a — 1, sma 1 sec a CSC a CSC a = — — = -; — , = ) or sec a sin a ± Vsec' a - 1 ± VCSC^ a - 1 1042. Relations between the trigonometric functions of two eqvM angles or arcs of unlike signs, a and — a. For the same value of r, the lines making the angles a and — a with Ox will give (1024): 1st. For y, two values, y and — y, equal and of unlike signs; consequently the sines - and — - will be equal and of unlike signs; and sin ( — a) = — sin a. Thus, two equal angles of unlike signs have equal sines also of unlike signs. RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS 393 2d. For X, two values, equal and of the same sign ; consequently X X the cosines will both be - or ; and cos ( — a) = COS a. Thus, two equal angles of like signs have the same cosines. 3d. Since the values of x are equal and of the same sign, while those of y are equal and of unlike signs, it follows that the tan- gents - and — - are always equal and of unlike signs; and X X tan ( — a) = — tan a. Thus, two equal angles of unlike signs have equal tangents also of unlike signs. From the above we may deduce : 4th. CSC ( — a) = — CSC a; sec (— a) = sec a; cot ( — a) = — cot a. 1043. Relations between the trigonometric functions of two com- plementary angles or arcs, that is, whose sum a + a' = 90°. Let a = uOx and a' = uOy. ^ ' y and x being the coordinates of the point M, q ■ and r being the radius OM, we have for angle a (1041): Oq OT y = r sin a, and Op or x = r cos a. On the contrary, for the positive angle a', the same values of x and y give: y = r cos a' and x = r sin a'. Putting these two values of x and y equal to each other, and cancelling r, we have: sin a = cos a' and cos a = sin a'. Dividing, S/^ that is (1041, 2d), tan a = Also (1041, 3d), sin a cos a cos a' Sin a' or tan a tan a'= 1. tan a tan a = cot a'. 394 TRIGONOMETRY Thus, the angles a and a' being complementary, the sines, cosines, and tangents of one are respectively equal to the cosines, sines, and cotangents of the other. This is easily verified with the aid of Fig. 255 (1031). 1044. Relations between the trigonometric functions of two angles or arcs, whose difference a — a' = 90°. Since two angles are complementary when their algebraic sum is equal to a right angle, by considering a' as negative we have the same case as the one preceding (1043). I^et M'Ox = a', the smaller of the two angles, and MOx = u., the larger. The angles being measured in the positive direction from Ox, the angle MOM' = a n K / > V p^ \ •v p' 3> / V ^ ^ / Ma Fig. 256 From the relations which exist between a and a', the value of the remainder MOM' must be a right angle; therefore, the right triangles MOp, M'Op' , are equal and Mp or y = Op' or x', and Op or x = M'p' or y'. Noting that y and x' have like signs and x and y' have unlike signs, no matter what the values of a and a' may be, that is, no matter what the position of the angle MOM' about the point 0, as shown in the Fig. 256, MOM'= MfiM^'= M.pM^'= M^OM^', may be, it follows that: y = x' and x = — y'. Replacing, as in the preceding article, y, x, y' and x' by their values as given in article (1041), sin a = cos a' and cos a = — sin a'. Thus, for two angles whose difference is equal to a right angle, the sine of the greater is equal to the cosine of the smaller, and has the same sign, and its cosine is equal to the sine of the latter but has a different sign. Dividing the two equations, sin a cos a' from which and tan a = — cos a 1 tan a' ' tan a = sin a' ' tan a tan a' = — 1, - cot a'. \ RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS 395 Example. What is the sine, cosine, and tangent of an angle of 165°? The relation a - a' = 90° becomes 165° - a' = 90°, and a' = 165° - 90° = 75°. From the table (1071), cos 75° = 0.25882, sin 75° = 0.96593, and the cot 75° = 0.26795, we have then, y- sin 165° = 0.25882, cos 165° = - 0.96593, and i tan 165° = - 0.26795. 1045. Relations between the trigonometric . ■ ^ functions of two angles or arcs a and b and A those of their sum {a + b). \^ Let mOA = b, MOm = a, and then MO A OXQ ^/""A' = (a + 6). ^'e-^-! Investigating the relations which exist between the coordinates MQ = Y,OQ = X, and those OP = x' and MP = y' of the same point M with reference to the two systems of rectangular Coordi- nates X, y, and x'y', Y being the projection of 0PM on Oy, and X being that of OPM on Ox, we find (1040) : Y = x' cos POy + y' cos PbO, X = x' cos b + y' cos Mcx. The angle Mcx = y'Ox, the difference between which and a right angle is equal to x'Ox or b; then cos Mcx = — sin b. (1044) This relation exists no matter what the position of M may be, that is, regardless of the values of a and b. The angle POy is the complement of the angle b; then cos POy = sin&. (1043) This relation exists no matter what value b may have; because, this angle being obtuse, the difference between it and a right angle is the angle POy, and we have again: cos POy = sin b. (1044) The angle PbO = b (629); and cos PbO = cos b. Substituting these values of the cosines of Mcx, POy, and PbO in the equations of X and Y: F = a;' sin & + y' cos b, X = x' cosb — y' sin b. 396 TRIGONOMETRY Since (1041): F = r sin (a + 6), X = r cos (a + b), y' = r sin a, x' = r cos a, the preceding equations become, r sin (a + b) = r cos a sin b + r sin a cos 6, r cos (a + &) = »" cos a cos b — r sin a sin b. Cancelling r, sin (a + b) = sin a cos 6 + cos a sin 6, (!) cos (a + b) = cos a cos & — sin a sin b. (2) . , ,, sin (a + 6) sin a cos 6 + cos a sin 6 ,,„„> tan {a + b)= -. — -^ = r ^ ^—r • (1041) cos (a + 6) cos a cos 6 — sin a sin 6 ^ ' Dividing both terms by cos a cos b, and substituting the tangent for the sine divided by the cosine, , , , , tan a + tan b tan (a + &) = ^ ^ r • (3) 1 — tan a tan o ^ ' 1046. The trigonometric functions of the difference {a — b) of two angles a and b expressed in terms of the functions of the two angles. Retaining the same value of b, given in the formulas (1) and (2) of the preceding article, and making {a + b) = a', which gives a = {a' — b), we have: sin a' = sin (a' — b) cos 6 + cos (a' — b) sin b, (1) cos a' = cos (a' — b) cos b — sin (a' — 6) sin b. (2) Putting a' = a and reducing the equation (2) (511): , , . cos a , . . ,. sin b ,., cos (a — 6) = ; 4- sin (a — b) r • (3) cos- b cos 6 ^ Substituting this value in equation (1), . , 7 , / , , sin^6"\ . cos a sin & ,,, sin (a — b) cos 6 -\ =- = sin a ; (4) V cos 6/ cos 6 ^ ' From (509, 4th): , , sin^6 cos^fe + sin^b 1 /,n.,x cos 6 + = = . (1041) cos cos cos Substituting this value in equation (4), sin (a — 6) _ sin a cos b — cos a sin b . cos b cos 6 ' that is, sin (a — 6) = sin a cos b — cos a sin b. RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS 397 Substituting this value of sin (a — b) in equation (3), cos (a — 6) •= cos a cos 6 + sin a sin b. Dividing one by the other, sin (a — b) sin a cos b — cos a sin b tan (a — o) = -. ^r = j— — -. — - — -. — r • cos (a — o) cos a cos o + sm a sm o Dividing both terms by cos a cos b, , , . tan a — tan 1) tan (a — 6) = — — — - . 1 + tan a tan o 1047. Relations between the trigonometric functions of an angle a and those of one of twice its value .2 a. Making b = a in the values given for sin (a + b), cos (a + b), and tan (a + b) (1045) : 1st. sin {a + b) = sin 2 a = sin a cos a + cos a sin a, that is, sin 2 a = 2 sin a cos a; (a) 2d. cos (a + 6) = cos 2 a = cos' a — sin^ a. (1) From (1041), cos' a = 1 — sin' a. Substituting this value in equation (1), cos 2 a = 1 — 2 sin' a. (b) If, instead of eliminating cos' a from equation (1), sin' a is elim- inated : cos 2 a = 2 cos' a — 1; (6') , ,s , ^ 2 tan a . . 3d. tan {a + b) = tan 2 a = — 5— • (c) 1048. Relations between the trigonometric functions of an angle a and those of another of half its value ■^■ Substituting a for 2 a and - a for a in the formulas of the pre- ceding article: 1st. Formula (a) gives: _ . 1 1 sin a = 2 sm ^ a cos ^ a. From formula (&), cos o = 1 — 2 sin' - a, 398 TRIGONOMETRY and (571), ... cos a siri; 3in-o = ± y- 2d. Formula (b') gives: cos a = 2 cos^ ^ a — 1, 3d. Formula (c) becomes: , i . I ^ + cos a and cos -a 2 tan - a tan a = 1 — tan^ ;r a Transposing, Solving, tan^ ~ a + tan ^a=\. 2 tan a 2 tan J a = - -^ ± v/-^ + 1 = r^" (" 1 ± Vl +tan^a). 2 tan a V tan^ a tan a Also from (1041): 1 Vl .1 2 r i /I — cos a tan-a = -— = ± \/i-i 2 1 V 1 + cos a cos- a 1049. To obtain the trigonometric functions of 3 a in terms of those of a, put 6 = 2 a in the formulas (1), (2), and (3) of (1045), which gives : sin 3 a = sin a cos 2 a + cos a sin 2 a, cos 3 ffl = cos a cos 2a — sin a sin 2 a, tan a + tan 2 a tan 3 a = 1 — tan a tan 2 a Substituting the values of sin 2 a, cos 2 a, and tan 2 a given in formulas (a), (&), and (c) (1047), and simplifying, we have: sin 3 a = 3 sin a — 4 sin' a, (1) cos 3 a = 4 cos' a — 3 cos a, (2) RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS 399 „ 3 tan a — tan^ a ,„, tan 3 o = — j ^- — -. (3) 1 — 3 tair a 1050. By making b = 3 a, then b = Aa, etc., in the formulas of (1045), the relations which exist between the trigonometric functions of any multiple of a and those of a may be obtained. 1051. Changing a to-;: a, the formulas (1), (2), and (3) of (1049) give: sin a = 3 sin - a — 4 sin^ - a, o o COS a = 4: COS'' - a — 3 cos ^ a, O O 3 tan is^ ~ ^^^^ o ^ tan a = z 1 — 3 tan^ - a O These formulas express the relations which exist between the sine, cosine, and tangent of an angle, which is equal to three times an- other, and the sine, cosine, and tangent of the latter. 1052. Other relations between the trigonometric expressions, which are frequently used in practice. 1st. By addition and subtraction of the values of the sine and cosine of (a + b) and (a — b) (1045, 1046), we obtain: sin (a + b) + sin (a — 6) = 2 sin a cos b, sin {a + b) — sin (a — b) = 2 cos a sin b, cos (a — 6) + cos {a + b) = 2 cos a cos b, cos {a — b) — cos {a + b) = 2 sin a sin b. These formulas may be used to transform the product of two trig- onometric expressions to a sum or difference. 2d. Putting {a + b) = p and (a — b) = q in the pi;eceding formulas, from which (520) a = -^ (p + q) and b = - (p — q), we have : sin p + sin g = 2 sin - (p + q) cos g (P — ?)' sin p — sin 5 = 2 cos 2 (p + q) sin o (P ~ ?)' 400 TRIGONOMETRY cos p + cos 2 = 2 cos - (p + q) cos ^ (P ~ 3)' cos g' — cos p = 2 sin - (p + q) sin- (p — g). These formulas are frequently used in logarithmic calculations, to change a sum or difference to a product. 3d. From these last formulas, by division; noting that '-^ = tanA = -4T= (1041) cos A cot A ^ ' , • sin - (p + 5) cos - (p — g) tan ^ (P + ?) sm p + sm g _ 2 2 _ 2 -^ ^' sin p — sin 5 1,1 . 1 / cos 2 (p + ?)sin-(p - g) tan-(p-g)- . sin - (p + g) sin p + sin o '2 ^' ^ 1 , , , , = ^ = tan H (P + S)? cosp + cosg 1 , , . 2 ■^ ^' cos - (p + q) , ■ cos ^ (p — q) , sin p + sm o 2 ^ ^ ^ 1 , . ^ = -■, = cot t; (p — q), cos o — cos p . I , . 2 ^ sin 2 (P - 3) Sin p — Sin a 2 • ^ , 1 . ^— ; ^ = \ = tan H (P — ih cosp + coso 1 , , 2^^ ^" cos - (p - g) cos - (p + g) sm p — sm g 2 ^ , 1 , = 5 -■ = cot 7c (p + g), cos g — cos p . 1 , . 2 ^ * sin 2 (P + g) , ■ cos- (p + g) cos -(p — g) , . cos p + cos g 2^^ ^' 2^^ ^' A , , -. A , \ ■ = — ^ = = cot7r(p+g)cot-(p-g). cosg— cosp .1 , , . 1, > 2^'^ ^ 2^^^ ^ sm^Cp +g)sin -(p-g) From the first formula it is seen that the sum of the sines of two angles is to their difference as the tangent of half the sum of these angles is to .half their difference, RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS 401 4th. Some other convenient transformations of products, sums, and differences are given below: , sin a , sin b sin a cos b ± sin b cos a sin (a ± b) tan a ± tan b = ± r = ; = — ^^ r> cos a cos cos a cos b cos a cos b 11 , T 2cos-(« + 6)cos-(a — 6 , 1,1 cosa+coso 2 2 seca+seco= 1-- r = ;— = ; ) cos a cos cos a cos o cos a cos o 1 1 J, 2sin-(a — 6)sin-(a + 6) , 1-1 coso — cosa 2 2 seca— seco= r= 7~ = ; j cos a coso cos a cos cos a cos o sin a + cos 6 = sin a + sin (90° - &) = 2 sin ^45° + ^^] sin ^45° + ^^\ Bina + cosa=2sin 45° sin (45° + a) = '\/2sin(45° + a), sina-cos6=sina-sin(90°-6)=-2sin^45°-^]sin/'45°-^V sin a — cos a = — 2 sin (45° — a) sin 45° = — V2 sin (45° — a) , sin^ a — sin^ b = sin (a + b) sin (a — &), cos^ a + cos^ b— I = cos (a + b) cos (a — b), 1 + sina = 1 + cos (90° - a) = 2 cos^ ^45° - |\ 1 - sin a = 1 - cos (90° - a) = 2 sin^ (45° - |\ \f\ 2sin^ - cos a I ' 2 ^ a tan- > + cos a I „ , a 2 2 cos^- '(«--i) 2sm2 2cos2f45° , , _, V2sin(45°±a) 1 ± tan a = ^ • cos a, V-—— . = / -/ f = 'tan 45°-- ; Vl+sma J o„_2/,^o a\ \ 2 402 TRIGONOMETRY For a -\-h + c = IT = 180°, we have: tan a + tan h + tan c = tan a, tan b, tan c, ,•!.,• ^ <* ^ c sin a + sin + sin c = 4cos ^r cos - cos -> Ji ^ Ji COt^ + COt^ + cot- = cot ;t cot ^ cot-) . ,a , . -b , . ,c , ^ . a . b . c , Sin^2 +^^"2 2 + 2^^°2 ^'"^ 2 ^^"^ 2 ^ CALCULATION OF THE TRIGONOMETRIC TABLES 1053. The trigonometric tables were described in article (1032). It will now be shown how they are calculated. 1st. When an angle less than 90° is de- creased, the ratio of the arc, which measures the angle, to the sine diminishes and approaches one as a limit (186). Supposing OM or r = 1, we have (1025) Mp = sin u. Op = cos a, and AT = tan ■*. Letting a equal the length of the arc AM, a > sin a and a < tan a. Since the sin u, or Mp is half the chord subtended by an arc twice as great as a, we have (649) : u. > sm a. (1) Furthermore, the surface of the sector 0AM being less than that of the triangle OAT, we have: ^OAX a<^OA X tano, and a < tan a or (1041) a < sm a cos a (718, 760) (2) From the inequalities (1) and (2), we have respectively: > 1 and TT- — < sm a sin u. cos a CALCULATION OF TRIGONOMETRIC TABLES 403 which shows that the ratio of the length of the arc to the sine is in- cluded between 1 and the quantity always greater than 1. Since, as a decreases, decreases and a approaches 1 as a cos a limit, it follows that —. , which is smaller than , may also be ' ' sm a cos a considered as having 1 for a limit. 2d. From the inequalities a < tan a and a > sin u or a > tan a cos a, (1041) we deduce: < 1 and > cos a, tan a tan a which shows that the ratio , always greater than cos a, lies between 1 and cos u,, and consequently has 1 for its limit. 3d. It will now be shown that the difference between the length a of the arc and the sine is less than one-fourth of the cube of the arc a. From the inequality (1st) 1 ''A" COS -a .11 1 we have: sin^a > -a cos-a. Multiplying this inequality by the equation sin a = 2 sin - a cos - a, (1058) and cancelling the common factor - a, sm a > a cos^ -r a, or sin a > a (1 — sin^ - a) , ■ • ,1 or sm a > a — a sin'' - «., 404 TRIGONOMETRY and a — sin a < a sin^ - a. Multiplying this inequality by (siniai> cos (a + V) = cos a cos 6 — sin a sin & ) the sines and cosines of all the angles from 0° to 45° may be found. The tangent and cotangent of each of these angles may be obtained from the formulas , sin a J , cos a /in.,\ tan a = and cot a = -. (1041) cos a sm a PRINCIPLES USED IN SOLVING TRIANGLES 405 5th. The trigonometric functions of the angles from 0° to 45° give those from 45° to 90°, as was shown in (1031) and (1043). Finally, having the trigonometric functions for the angles up to 90°, from what was said in (1028), they can be determined for any angle larger. It is evident that this method of calculating the trigonometric functions is long and fatiguing; it has been simplified by proceed- ing in another manner, but since it is not our purpose to calcu- late tables, this simpler method will not be given. In practice, the engineer scarcely ever deals with angles smaller than 1', therefore no angles smaller than 1' are given in the tables (1071). In case it is desired to work with smaller angles, the method of interpolation as used in the logarithmic tables may be resorted to. PRINCIPLES USED IN SOLVING TRIANGLES 1054. Remark. For the sake of simplicity in that which follows, the angles of the triangles will be represented by the letters A, B, and C written at the vertices and the sides respec- tively opposed to these angles by the letters a, b, and c, written near the middle of these sides. In the case of a right triangle the right angle is designated by A and the hypotenuse by a. 1055. Theorem 1. In any right triangle, each leg is equal to the hypotenuse multiplied by the cosine of the adjacent angle. Since b and c may be considered as projections of a upon the legs, we have C & = and c =a —. — 7 > sm A sm A or log b = log a + log sin B — log sin A, log c = log a + log sin C — log sin A . The area of the triangle can be calculated from the formula: a' sin B si n C or log B, we should also have a > b. There is always a solution, but there is only one, which may be seen from the Fig. 262. 2d. The given angle A being acute and a > b, then A > B, and it follows that B is acute, and there is but one solution. The C A 'B Fig. 263 Fig. 264 Fig. 263 shows that the angle A would be obtuse in the second triangle AB'C which has a and b for its sides. In the case where A is acute and a = b, B' coincides with A, and the only solution is an isosceles triangle. 3d. The given angle A acute and a < b. In this case B > A may be acute or obtuse, therefore there are two solutions, as indicated in (Fig. 264). In the triangle ABC, which satisfies the given conditions, the angle B is acute; in the triangle AB'C, which also satisfies the given conditions, B' = 180° — 5 is obtuse. There are two solutions when a < 6 is greater than CD = 6 sin ^1, that is, when ^ I. • A b sin A ^ ^ a> b sm A or < 1. SOLUTION OF PLANE TRIANGLES 413 When a = CD = b sin A, the arc BB' is tangent to AB at the point D, the two triangles ABC and AB'C coincide with the right triangle ADC, and there is but one solution. Finally, if a < CD or a < 6 sin A, the arc BB' would have no point common with AB, and there would be no solution. If, instead of commencing by determining the angles B and C, it had been desired to first determine the side c : a? = Z)2 + c2 _ 2 6c cos A, (1057) from which c^ — 2h cos A X c = a^ — V, and therefore, c = 6 cos A ±\/a^ — 6^ 4- t^ cos^ A, (572) or c = h cos A ±Va? — V sin^ A. (1041) 1063. Case 3. Having two sides a and b and the included angle C given, to find c. A, and B. 1st. We have, c = Va^ + b^ - 2 ab cos C ; (1057) c being known, sin A = , (1056) then 5 = 180 - (A + C). 2d. Commencing by determining A : sin A : sin B = a : b. In this proportion there are two unknowns, sin A and sin B; one is eliminated by writing (349) : (sin A + sin B) : (sin A — sin B) = {a + b) : (a — b), or substituting an equal ratio for the first member' (1052, 3d) : tan 1{A+B): tan^ (A - B) = {a +b) -.(a - h). i (A + S) = ^ (180 - C) = m° being known, this proportion contains only one unknown, namely, tan - {A — B), whose value is: tan \{A-B)= ^ tan^ (A + B). Putting - {A — B) =n°, then having half the sum m° and half 414 TRIGONOMETRY the difference n° of the angles A and B, from (520), A = m° + n° and B = m ~ n°. Having found A and B (1056), asinC c = sin A This solution is to be preferred where logarithms are to be used (1061). The area is given by the formula: 5="-^. (1065) 1064. Case 4. The three sides a, h, and c being given, to deter- mine the three angles A, B, and C. Writing a^ = b'^ + c^ - 2bc cos A, (1057) ^2 -(- C' _ ^2 we have, cos A = ^--r • (a) 2 be ^ ' Similar formulas will give B and C, or having determined A and B, C = 180 - (A + B), which in any case should be used as a check. If logarithms are to be used, a more convenient formula than (a) can be used (1061), which is developed as follows: 2 sin^i A = 1 - cos A. (1048, 1st) Substituting the value of cos A given in (a), „ . . 1 . , b^ + c^-a^ a^-¥-c' + 2bc 2 sm^ ~A = 1 -r-, = -— ■ 2 2 6c 2 6c g^ - (6 - c)^ (g + 6 - c) (g - 6 + c) ~ 26^ 26^ ' (728,729) from which siniA = v/(^ + 6 - c) (ra - 6 + c) 4 6c This formula may be simplified by making the following sub- stitutions : o + 6 + c = 2p, SOLUTION OF PLANE TRIANGLES 415 then a + b — c = 2(p — c), and a — b+c = 2{p — b), which gives - A being necessarily an acute angle, its value is determined by its sine, as is likewise that of the angle A. In the same manner, sinl5=v/ _ (p - g) (p - c) 2 '■ ■ J • 1 ^ 4 /(P — a) (P — fc) and sm - C = V ^^^- ^-r^- • 2 V ao As proof we may write, C= 180°- (4 + B). In the same manner the values of cos 7; A and tan - A mav be found, since: cos ^ A = y 1 - sin^ i A . (1041) Substituting the value given in (6) for sin - A, 1 A J\ (p - 6) (p - c) cos-A = V/l-^^^ 1^^^ -^ or reducing to the same denominator, and simplifying, 1 , 4 /p (P — a) From (1041, 2d): 1^ JiV - &) (P - c) sm tan^l- "'"^^^ = ^ ^^ _ 4 /(P - &) (p"^^ # 2 V be Analogous formulas may be obtained for the angles B and C, by proceeding in the same manner. The area may be calculated from the formula which is devel- oped in (3d) of the next article. 416 TRIGONOMETRY 1065. The area of a triangle may be expressed in terms of two sides and the included angle, or one side and two angles or three sides. - 1st. Letting S represent the area of the triangle, ^ = ^- (718) \a. ^ \ Substituting b sin A (1078) for CD, A D B be sin 4 . Pig. 266 (S = 2 ' W which shows that the area of a triangle is equal to half the product of any two sides and the sine of the included angle. c sin S 2d. Writing h = —. — ^ in the preceding expression (1056), we have : „ _ c^ sin .4 sin B _ c^ sin A sin B 2sinC ~ 2 sin {A + B)' 3d. Having sin A = 2 sin - A cos - A, (1048) substituting from (1087) for sin^ A and cos ^ A, sin A = 2 y ^ iP -a) ip - b) (p-c) bV then substituting this value of sin A in equation (a), S == Vp (p — a) (p — b) (p — c). For a = 200 ft., b = 180 ft., and c = 170 ft., S = V27.5 (275 - 200) (275 - 180) (275 - 170) = 14,343 sq.ft. EXAMPLES 1066. In trigonometry all problems are reduced to the deter- mination of triangles, or rather the sides and angles of these tri- angles. 1067. Find the height CD of a building, the base of which is accessible. On the groimd, which is level, measure the base DE, making it EXAMPLES All about equal to the height of the building so as to avoid two small angles; let DE = c == 10 yards, place the instrument at E and measure the angle B, which is 41°, and let the height of the in- strument be BE = AD = 1.2 yards. This done, the problem is reduced to determining the side 6 of a right triangle ABC, when the side c and the angle B are known. Or, from (1055, corollary) : 6 = c tan 5 = 10 X tan 41°= 10 X 0.86929 = 8.693 yds.; CD = 8.693 + 1.2 = 9.893 yds. In case the ground is not level, the point A can be determined and AD measured, then we have the same as in the first case. C ODD DDD □no C \ V c. -^^ X \ •\, ■\J Jb Fig. 266 Fig. 267 Solution 2. At the extremity E of the base, a stake of known height BE is driven. Then at D' in line with D and E a second stake is held so that C is in line with B and C, and measuring A'C and A'B, the two similar triangles ABC and A'BC give: 1^' = ^' ^^^ AC^ABXJ^; or, making CD = h, AB = d, and BE = a, AT" h = dX^ + a. (1) If one has an instrument for measuring angles, the angle ZBC is measured, and we have: AC = d cot ZBC, and h = d cot zBC + a. (la) 1068. To find the distance AC from the point A to an inaccessible but visible point C. Lay off a base AB = 100 j'-ards, for example; then measure the 418 TRIGONOMETRY angles A = 65° and B = 42°. The problem is now reduced to determining the side b of an oblique triangle when one side c and the two adjacent angles A and B are known (1061). First, C = 180° - {A + B) = 180 - (65° -f- 42°) = 73°, then sin C : sin 5 = c : b, and b = c sin B 100 X 0.669 sin C 0.956 70 yds. 1069. To determine the height of a building or mountain, the base of which is inaccessible. In this case the angle B = 43° is all that can be measured directly in the triangle ABC, and this is not sufficient for the Gal- s' 5fr — ~~^ .--' 7' Fig. 268 Fig. 269 culation of AC. Therefore the solution is commenced by deter- mining the side BC, which is done as in the preceding case (1068). C is an inaccessible point whose distance from B is found from the relations in the triangle BB'C, BB' and the adjacent angles being known. Having BC or a = 500 yards, for example (1055): & = a sin 5 = 500 X 0.682 = 341 yards, and therefore CD = 341 + 1.2 = 342.2 yards. SOLT7TION 2. The distance AB = d from the accessible point B to the vertical passing through the inaccessible point C is de- termined by measuring the base BB' , from B the angle between CB and BB' which the theodolite gives reduced to the horizontal, i.e., ABB' and from B' the angle AB'B. In the triangle ABB', the angle BAB' = 180 - {ABB' + AB'B) and (1056), AB = d EXAMPLES 419 BB' sin AB'B sin BAB' Substituting this value of d in formula (1067, la), the required height is obtained. 1070. Find the distance between two inaccessible points C and C. Determine the distances AC and AC between the point A and each of the inaccessible points C and C", according to the method in article (1068); then measuring the angle CAC, in the triangle CAC, we have two sides AC and AC and the included angle; therefore the " Fig. 270 side CC may be found from (1063). 1st. Determination of AC. Lay off the base AB = 100 yards, for example ; then the angle BAC = 66°, and ABC = 37°; and ACB = 180° - (66° + 37°) = 77°. Then we have: sin ACB : sin ABC = AB : AC, , ,„ AB sin ABC 100X0.6018 n. ^n ^ and AC = -. — ,^p = — ftk^tti = 61.76 yds. (a) sm ACB 0.9744 ■' ^ ' 2d. Determination of AC. Measure the angles BAC = 37° and ABC = 87°; then ACB = 180° - (37° + 87°) = 56°. In triangle ABC, sin ACB : sin ABC = AB : AC, , .„, AB sin ABC 100X0:9986 .,„„ ,^ , and AC = -. — .^,p ■ = ^r—^^ = 120.46 yds. sm ACB 0.829 ■ 3d. Determination of the angle CAC. When the four points A, B, C, and C are in the same plane, we have CAC = BAC - BAC = 66° - 37° = 29°. If these four points are not in the same plane, the angle is measured dire tly. 4th. Determination of CC. In the triangle ACC (1063), CC = y/^C' + AC"" -2X ACX ACX cos CAC, CC' = V^6r76' + i20:46'-2X 61. 76X120.46X0.87462 =72.88 yds. or 420 TRIGONOMETRY CC might also have been determined by the method in (1063, 2d). // logarithms are used in the solution, the following method is used. Let the angles of the triangle ACC be designated by the letters A, C, and C"; and the sides opposite these angles by the letters a, c, and c'. In the triangle ACC, C + C 180°-^. ,,, -~2~= 2 ' (^^ C + C A then from (1063, 2d), noting that tan ^ — = cot-^ (1043), that tan 45° = 1, and making — = tan ^, c r' 1 -- C -C c~ c' -A c A *^^ ^^ = ^T7' "^^ 2 = ^ ""^ 2 c _ tan 45 — tan 4> , -4 . ~ 1 + tan 45° tan <^^^^' from (1046), tan ^ ~^' = tan (45° - <^) cot 4 • (2) Having measured the angle A, the equations (1) and (2) give C and C (520). From (1056), c' sin A .,, With logarithms c' is calculated from the formula (a), and a from the formula (6). Distance AP from an accessible point A to an inaccessible straight line CC (Fig. 270). Since the angles ACP and ACC are supple- mentary, sin ACP = sin ACC or sin C; and consequently (1055), AP =c'sinC; c' and C being calculated as was demonstrated above. 1071. Table of the natural values of the trigonometric functions of the angles from 0° to 90°, for each minute. Starting at the tops of the pages, each angle in the last vertical column is the supplement respectively of the angle in the same horizontal row in the first vertical column; thus, 16° 51' + 163° 9' = 180°. EXAMPLES 421 Likewise, commencing at the bottoms of the pages, each angle in the first vertical column is the supplement of the corresponding angle in the last column; thus, 73° 52' + 106° 8' = 180°. Since supplementary angles have the same functions, it follows that the table contains the functions of the angles from 0° to 180° for each degree; the sign of the fimctions may be obtained from (1027) or Fig. 249. For angles between 180° and 360°, subtract 180°, thus obtaining an angle which is given in the table. Thus, tan 352° 46' = tan (352° 46' - 180°) = tan 172° 46'. From the table the tangent is 0.12692, and according to (1027) the tan 352° 46' is preceded by the sign — , so we have, tan 352° 46' = - 0.12692. The table also contains the lengths of the arcs which correspond to the angles from 0° to 90° when the radius r = 1. Thus the arc corresponding to the angle 23° 17' is 0.40637, and the arc corresponding to the complement 66° 43' of 23° 17' is 1.16442. According as an angle is greater than an angle in the table by 90°, 180° or 270°, its length is obtained by adding respectively: iir = 1.5707963 S 1.57080, 7r= 3.1415926 S 3.14160, I TT = 4.7123889 S 4.71240. Thus the length of the arc corresponding to the angle 66° 43' + 90° = 156° 43' is 1.16442 + 1.57080 = 2.73522. If the radius were 6 feet, the length of the arc in feet would be 6 X 2.73522 = 16.41132 feet. Remark. With the aid of the table, an arc which is a given fraction of the radius or diameter may be found. Thus, if it is 2 4 desired to find an arc equal to -^^ or -rr of the radius, in the sixth column under arc, find the number which is nearest to 0.4. The number 0.39997, which corresponds to 22° 55', is the nearest value. The next arc 0.40056 corresponds to 22° 56'; then by interpolation the angle which corresponds to the arc 0.4 is found to be 22° 55' 6". 422 TRIGONOMETRY 0° = 0' Sup. 179° = lOTW 1° Sup. 178° = 10680' 1 2 3 4 S 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 67 58 59 60 0.0 00000 00291 00582 00873 01164 01454 01745 02036 02327 02618 02909 03200 03491 03782 04072 04363 04654 04945 05236 05527 05818 06109 06400 06690 06981 07272 07563 07854 08145 08436 08727 0.0 09017 09308 09S99 09890 10181 10472 10763 11053 11344 11635 11926 12217 12508 12799 13090 13380 13671 13962 14253 14544 14835 15126 16416 16707 15998 16289 16680 16871 17162 17452 1.0 00000 00000 00000 00000 0.9 99999 99999 99997 99997 99996 99995 99994 99993 99992 99990 99989 99988 99986 99985 99984 99981 99979 99978 99976 99974 99971 99969 99967 99964 99962 0.9 99959 99957 99954 99951 99948 99945 99942 99939 99936 99932 99929 99926 99922 99918 99914 99910 99906 99902 99898 99894 99890 99886 99881 99877 99872 99867 99862 99858 99853 99848 0.0 00000 00291 00582 00873 01164 01454 01746 02036 0232' 02618 02909 03200 03491 03782 04072 04363 04664 04945 05236 05527 05818 06109 06400 06690 06981 07272 07663 07864 08145 08436 08727 0.0 09018 09309 09600 09891 10181 10472 10763 11054 11345 11636 11927 12218 12509 12800 13091 13382 13673 13964 14256 14546 14836 15127 15418 15709 16000 16291 16582 16873 17164 17455 Infinity 3437.7467 1718.8732 1146.9153 869.4363 687.5489 572.9672 491.1060 429.7176 381.9710 343.7737 312.5214 286.4777 264.4408 245.6620 229.1817 214.8576 202.2187 190.9842 180.9322 171.8854 163 7002 156.2691 149.4650 143.2371 137.5074 132.2186 127.3213 122.7740 118.5402 114.5886 110.8920 107.4265 104.1709 101.1069 98.2179 95.4896 92.9086 90.4633 88.1436 85.9398 83.8436 81.8470 79.9434 78.1263 76.3900 74.7292 73.1390 71.6151 70.1633 68.7501 67.4019 66.1055 64.8580 63.6567 62.4992 61.3829 60.3068 59.2669 68.2612 57.2900 0.0 00000 00291 00682 00873 01164 01454 01745 02036 02327 02618 02909 03200 03491 03782 04072 04363 04654 04945 06236 06527 05818 06109 06400 06690 06981 07272 07563 07854 08146 08436 08727 0.0 09018 09308 09599 09890 10181 10472 10763 11064 11346 11636 11926 12217 12508 12799 13090 13381 13672 13963 14254 14544 14835 16126 15417 16708 15999 16290 16581 16872 17162 17453 COM. OF ARC. COM. OF ARC. 1.5 70796 70505 70215 69924 69633 69342 69061 68760 68469 68178 67887 67597 67306 67015 66724 66433 66142 65861 65660 65269 64979 64688 64397 64106 63816 63524 63233 62942 62661 62361 62070 1.6 61779 61488 61197 60906 60615 60324 60033 59743 59452 59161 68870 68579 58288 67997 67706 57415 57125 56834 66643 66252 55961 55670 66379 65088 64797 54507 54216 53926 63634 63343 0.0 17462 17743 18034 18325 18616 18907 19197 19488 19779 20070 20361 20662 20942 21233 21524 21816 22106 22396 22687 22978 23269 23560 23851 24141 24432 24723 25014 25305 25595 25886 26177 0.0 26468 26768 27049 27340 27631 27922 28212 28603 28794 29086 29375 29666 29957 30248 30538 30829 31120 31411 31701 31992 32283 32674 32864 33156 33446 33737 34027 34318 34609 34899 0.9 99848 99843 99837 99832 99827 99821 99816 99810 99804 99799 99793 99787 99781 99774 99768 99762 99766 99749 99742 99736 99729 99722 99716 99708 99701 99694 99687 99680 99672 99666 99657 0.9 99650 99642 99634 99626 99618 99610 99602 99594 99585 99577 99668 99560 99551 99642 99634 99526 99516 99507 99497 99488 99479 99469 99460 99460 99441 99431 99421 99411 99401 99391 0.0 17455 17746 18037 18328 18619 18910 19201 19492 19783 20074 20365 20666 20947 21238 21529 21820 22111 22402 22693 22984 23275 23566 23857 24148 24439 24730 25022 25313 25604 25896 26186 0.0 26477 26768 27059 27350 27641 27932 28224 28516 28806 29097 29388 29679 29970 30262 30653 30844 31135 31426 31717 32009 32300 32591 32882 33173 33465 33756 34047 34338 34629 34921 57.28996 56.35069 55.44162 54.66133 63.70869 62.88211 52.08067 51.30316 50.64851 49.81573 49.10388 48.41208 47.73960 47.08534 46.44886 45.82935 45.22614 44.63860 44.06611 43.50812 42.96408 42.43346 41.91579 41.41059 40.91741 40.43584 39.96546 39.50589 39.05677 38.61774 38.18846 37.76861 37.35789 36.95600 36.56266 36.17760 35.80056 36.43128 36.06955 34.71511 34.36777 34.02730 33.69351 33.36619 33.04617 32.73026 32.42129 32.11810 31.82052 31.52839 31.24158 30.95993 30.68331 30.41158 30.14462 29.88230 29.62450 29.37111 29.12200 28.87709 28.63625 0.0 17453 17744 18035 18326 18617 18908 19199 19490 19780 20071 20362 20653 20944 21235 21526 21817 22108 22398 22689 22980 23271 23562 23853 24144 24435 24725 25016 25307 26598 26889 26180 0.0 26471 26762 27053 27343 27634 27926 28216 28507 28798 29089 29380 29671 29961 30252 30543 30834 31125 31416 31707 31998 32289 32579 32870 33161 33462 33743 34034 34325 34616 34907 1.5 53343 53052 52761 52470 52179 61889 51598 51307 51016 50725 50434 50143 49852 49561 49271 48980 48689 48398 48107 47816 47525 47234 46943 46653 46362 46071 45780 45489 45198 44907 44616 1.6 44326 44036 43744 43453 43162 42871 42580 42289 41998 41708 41417 41126 40835 40544 40253 39962 39671 39380 39090 38799 38508 38217 37926 37635 37344 37053 36762 36472 36181 35890 COM. OP AKC. ABC. I Sup. 90° = 5400' 9° = 5340' Sup. 91° = 5460' 88° = 5280/ EXAMPLES 423 zW Sup. 177° = 10620' 3° == 180' Sup. 176° = 10560' 1 2 3 i 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0.0 34899 35190 35481 35772 36062 36353 36644 36934 37225 37516 37806 0.9 99391 99381 99370 99360 99350 99339 99328 99318 99307 99296 99285 38097 99274 38388 99263 38678 99252 38969 99240 39258 99229 39550 39841 40132 40422 40713 41004 41294 41585 41876 42166 42457 42748 43038 43329 43619 0.0 43910 44201 44491 44782 45072 45363 45654 45944 46235 46525 46816 47106 47397 47688 47978 48269 48559 48850 49140 49431 49721 50012 50302 50593 50883 51174 51464 51755 52045 52336 99218 99206 99194 99183 99171 99159 99147 99135 99123 99111 99098 99086 99073 99061 99048 0.9 99036 99023 99010 98997 98984 98971 98957 98944 98931 98917 98904 98890 98876 98862 98848 98834 98820 98806 98792 98778 98763 98749 98734 98719 98705 98690 98675 98660 98645 98629 0.0 34921 35212 35503 35795 36086 36377 36668 36960 37251 37542 37834 38125 38416 38707 38999 39290 39581 39873 40164 40456 40747 41038 41330 41621 41912 42204 42495 42787 43078 43370 43661 0.0 43952 44244 44535 44827 45118 45410 45701 45993 46284 46576 46867 47159 47450 47742 48033 48325 48617 48908 49200 49491 49783 50075 50366 50658 50950 51241 51533 51824 52116 52408 28.63625 28.39940 28.16642 27.93723 27.71174 27.48985 27.27149 27.05656 26.84498 26.63669 26.43160 26.22964 26.03074 25.83482 25.64183 25.45170 25.26436 25.07976 24.89783 24.71851 24.54176 24.36751 24.19571 24.02632 23.85928 23.69454 23.53205 23.37178 23.21367 23.05768 22.90376 22.75189 22.60201 22.45410 22.30810 22.16398 22.02171 21.88125 21.74257 21.60563 21.47040 21.33685 21.20495 21.07466 20.94597 20.81883 20.69322 20.56911 20.44649 20.32531 20.20555 20.08720 19.97022 19.85459 19.74029 19.62730 19.51558 19.40513 19,29592 19.18793 19.08114 0.0 34907 35197 35488 35779 36070 36361 36652 36942 37234 37525 37815 38106 38397 38688 34979 39270 39561 39852 40143 40433 40724 41015 41306 41597 41888 42179 42470 42761 43051 43342 43633 0.0 43924 44215 44506 44797 45088 45379 45669 45960 46251 46542 46833 47124 47415 47706 47997 48287 48578 48869 49160 49451 '49742 50033 50324 50615 50905 51196 51487 51778 52069 52360 COM. OF ARC. COM. OF ARC. 1.5 35890 35599 35308 35017 34726 34435 34144 33854 33563 33272 32981 32690 32399 32108 31817 31526 31236 30945 30654 30363 30072 29781 29490 29199 28908 28618 28327 28036 27745 27454 27163 1.5 26872 26581 26290 26000 25709 25418 25127 24836 24545 24254 23963 23672 23382 23091 22800 22509 22218 21927 21636 21345 21054 20764 20473 20182 19891 19600 19309 19018 18727 18436 0.0 52336 52026 52917 53207 53498 53788 54079 54369 54660 54950 55241 55531 55822 56112 56402 56693 56983 57274 57564 57854 58146 58435 58726 59016 59306 59597 59887 60178 60468 60768 61049 0.0 61339 61629 61920 62210 62500 62791 63081 63371 63661 63952 64242 64532 64823 65113 65403 65693 65984 66274 66564 66854 67146 67435 67725 68015 68306 68596 69176 69466 69757 0.9 98629 98614 98599 98584 98568 98552 98537 98521 98505 98489 98473 98457 98441 98425 98408 98392 98375 98342 98325 98308 98291 98274 98267 98240 98223 98205 98188 98170 98163 98136 0.9 98U7 98081 98063 98045 98027 98008 97990 97972 97953 97934 97916 97897 97878 97859 97840 97821 97801 97782 97763 97743 97724 97704 97684 97664 97644 97624 97604 97584 97564 0.0 52408 52699 52991 53283 53575 53866 54158 54460 54742 55033 55325 55617 55909 56201 56492 56784 67076 67368 67660 57952 58243 58535 68827 69119 59411 59703 59995 60287 60579 60871 61163 0.0 61455 61747 62039 62331 62623 62915 63207 63499 63791 64083 64375 64667 64959 65251 65543 65836 66128 66420 66712 67004 67297 67589 67881 68173 68465 68758 69050 69342 69636 69927 19.08114 18.97552 18.87107 18.76775 18.66556 18.56447 18.46447 18.36554 18.26766 18.17081 18.07498 17.98015 17.88631 17.79344 17.70153 17.61056 17.52052 17.43138 17.34315 17.26581 17.16934 17.08372 16.99896 16.91502 16.83191 16.74961 16.66811 16.58740 16.60745 16.42828 16.34986 16.27217 16.19522 16.11900 16.04348 15.96867 15.89454 15.82110 16.74834 16.67623 16.60478 15.53398 15.46381 15.39428 15.32636 15.25705 15.18935 16.12224 16.05572 14.98978 14.92442 14.85961 14.79537 14.73168 14.66853 14.60592 14.54383 14.48227 14.42123 14.36070 14.30067 0.0 52360 52651 52942 53233 53523 53814 54105 54396 54687 54978 66269 55560 65851 56141 56432 56723 57014 57305 57696 57887 58178 68469 68759 69050 69341 69632 59923 60214 60505 60796 61087 0.0 61377 61668 61959 62250 62541 62832 63123 63414 63706 63995 64286 64577 64868 65159 65450 65741 66032 66323 66613 66904 67195 67486 67777 68068 68359 68650 68941 69231 69622 69813 COM. OF ARC. COM. OF ARC. 1.5 18436 18146 17855 17564 17173 16982 16691 16400 16109 15818 15528 15237 14946 14655 14364 14073 13782 13491 13200 12910 12619 12328 12037 11746 11455 11164 10873 10582 10292 10001 09710 1.6 09419 09128 08837 08546 08255 07964 07674 07383 07092 06801 06510 06219 05928 05637 05346 05056 04765 04474 04183 03892 03601 03310 03019 02728 02438 02147 01856 01565 01274 00983 Sup. 92° = 5520' 87° — 5'2SXr Sup. 93° =: 5580' 86° = 5160' 424 TRIGONOMETRY 4° = 240' Sup. 175= = 10500' B° = 300' Sup. 174° = 10440' 4 S 6 7 8 9 10 11 12 13 14 15 16 17 IS 19 20 21 22 23 24 25 261 27 28 29 So 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 51 62 53 54 55 66 67 58 69 60 0.0 89766 70047 70337 70627 70917 71207 71497 71788 72078 72368 72668 72948 73238 73528 73818 74 108 74399 74689 74979 75269 76559 7S849 76139 76429 76719 77009 77299 77589 77879 78169 78459 0.0 78749 79039 79329 79619 79909 80199 80489 80779 81069 81359 81649 81938 82228 82518 96589 0.9 97564 97544 97523 97603 97482 97461 97441 97420 97399 97378 97367 97336 97314 97293 97272 97250 97229 97207 97185 97163 97141 97119 97097 97075 67053 97030 97008 96985 96963 96940 96917 0.9 96894 96871 96848 96825 96802 96779 96755 96732 96708 96685 96661 96637 96613 82808 83098 83388 83678 83968 84258 84547 84837 85127 85417 85707 86997 86286 86576 86866 87156 96566 96641 96517 96493 96468 96444 96419 96395 96370 96345 96320 96295 96270 96242 96220 96195 0.0 69927 70219 70511 70804 71096 71388 71681 71973 72266 72558 72850 73143 73435 73728 74020 74313 74606 74898 75190 75483 76775 76068 76360 76653 76946 77238 77531 77824 78116 78409 78702 0.0 78994 79287 79580 79873 80166 80468 80751 81044 81336 81629 81922 82215 82508 82801 83094 83386 83679 83972 84265 84558 84851 85144 85437 85730 86023 86316 86609 86902 87196 87489 14.30067 14.24113 14.18209 14.12354 14.06546 14.00786 13.95072 13.89405 13.83783 13.78206 13.72674 13.67186 13.61741 13.56339 13.50980 13.45662 13.40387 13.35152 13.29957 13.24803 13.19688 13.14613 13.09576 13.04577 12.99616 12.94692 12.89806 12.84956 12.80142 12.75363 12.70621 12.65913 12.61239 12.56600 12.51994 12.47422 12.42883 12.38377 80576 12.33903 80867 0.0 69813 70104 70395 70686 70977 71268 71559 71849 72140 72431 72722 73013 73304 73595 73886 74176 74467 74768 76049 76340 76631 75922 76213 76504 76794 77085 77376 77667 77968 78249 78540 0.0 78831 79122 79412 79703 79994 80286 12.29461 12.25051 12.20672 12.16324 12.12006 12.07719 12.03462 11.99235 11,95037 11.90868 11.86728 11.82617 11.78533 11.74478 11.70460 11.66449 11.62476 11.58529 11.54609 11.50715 11.46847 11.43005 81158 81449 81740 82030 82321 82612 82903 83194 83485 83776 84067 84368 84648 84939 85230 86621 85812 COM. OF ARC. 1.5 00983 00692 00401 00110 1.4 99820 99529 99238 98947 98656 98365 98074 97783 97493 97202 96911 96620 96329 96038 95747 95456 95166 94875 94584 94293 94002 93711 93420 93129 92838 92547 92257 1.4 91966 91675 91384 91093 90802 90511 90220 89929 89639 89348 89057 88766 88475 88184 87893 87602 87311 87021 86730 86439 86148 85857 85566 85275 84984 86103 86394 86685 86976 87266 COM. OF ATiC. 84693 84403 84112 83821 83530 0.0 87166 87446 87735 88025 88316 88605 88894 89184 89474 89763 90053 90343 90633 90922 91212 91602 91791 92081 92371 92660 92950 93239 93529 93819 94108 94398 94687 94977 95267 95566 96846 0.0 96135 96425 96714 97004 97293 97583 97872 98162 98451 98741 99030 99320 99609 99899 0.1 00188 00477 00767 01056 01346 01635 0.9 0.0 96195 87489 96169 87782 9614488075 96118 88368 9609388661 96067 1 88954 9604l'89248 96015 89541 95989 89834 95963 90127 95937 90421 95911 95884 95858 96831 96805 90714 91007 91300 91694 91887 96778 92181 96751 92474 9572592767 95698 93061 95671 93354 95644 93647 96616 93941 95589 94234 9556294528 95534 94821 95507 95115 9547995408 9545295702 95424 95995 95396 0.9 96368 95340 95312 95284 95256 97757 95227198051 95199 98345 95170,98638 95142 98932 9611399226 95084199619 95055 99813 0.1 96027 00107 01924 02216 02503 02792 03082 03371 03660 03950 04259 04528 96289 0.0 96583 96876 97170 97463 94998 94968 94939 94910 94881 94861 94822 94792 94762 94733 94703 94673 94643 94613 94582 94552 94622 00401 00695 00988 01282 01576 01870 02164 02458 02752 03046 03340 03634 03928 04222 04516 04810 06104 11.43005 11.39189 11.35397 11.31630 11.27889 11.24171 11.20478 11.16809 11.13164 11.09542 11.05943 11.02367 10.98815 10.95286 10.91778 10.88292 10.84829 10.81387 10.77967 10.74569 10.71191 10.67835 10.64499 10.61184 10.57890 10.54615 10.61361 10.48126 10.44911 10.41716 10.38640 10.35383 10.32245 10.29126 10.26025 10.22943 10.19879 10.16833 10.13805 10.10795 10.07803 10.04828 10.01871 9.98931 9.96007 9.93101 9.90211 9.87388 9.84482 9.81641 .9.78817 9.76009 9.73217 9.70441 9.67680 9.64935 9.62205 9.59490 9.56791 9.64106 9.61436 0.0 87266 87657 87848 88139 88430 88721 89012 89303 89594 89884 90176 90466 90757 91048 91339 91630 91921 92212 92502 92793 93084 93375 93666 93957 94248 94639 94830 95120 95411 96702 96993 0.0 96284 96575 96866 97157 97448 97738 98029 98320 98611 98902 99193 99484 99775 0.1 00066 00356 00647 00938 01229 01620 01811 ARC. 02102 02393 02684 02974 03265 03566 03847 04138 04429 04720 1.4 83530 83239 82948 82657 82366 82075 81785 81494 81203 80912 80621 80330 80039 79748 79457 79167 78876 78585 78294 78003 77712 77421 77130 76839 76649 76258 75967 75676 75385 75094 74803 1.4 74512 74221 73931 73640 73349 73058 72767 72476 72185 71894 71603 71313 71022 70731 70440 70149 69858 69367 69276 68985 68695 68404 68113 67822 67531 COM. OF ARC. 67240 66949 66658 66367 66077 Sup. 94'= = 5040' 85° = 5100' Sup. 95° = 6700' 84° = 6040' EXAMPLES 425 6° = sec Sup. 176° = lOSSC 7° = 420' Sup. 172° = 10320' COM. COM. / BIN. 0.1 COS. 0.9 FAN. 0.1 COT. ARC. OF ARC. / SIN. 0.1 COS. 0.9 PAN. 0.1 COT. iRC. OF ARC. 1.4 "" 0.1 1.4 0.1 0453 9452 B510 9.51436 0472 6608 60 2187 9255 2278 8.144351 2217 4862 60 1 0482 J449 054O 9.48781 0501 6579 59 1 2216 9251 2308 8.12481 2246 4833 59 2 0511 9446 0569 9.46141 0530 6549 58 2 2245 9248 2338 8.10536 2275 4804 58 3 0540 9443 0599 9.43515 0559 6520 57 3 2274 9244 2367 8.08600 2305 4775- 67 4 0569 9440 0628 9.40904 0588 6491 56 4 2302 9240 2397 8.06674 2334 4746 56 fi 0597 9437 0658 9.38307 0617 6462 55 5 2331 9237 2426 8.04766 2363 4717 55 6 0626 9434 0687 9.35724 0646 6433 54 6 2360 9233 2456 8.02848 2392 4688 54 7 0655 9431 0716 9.33155 0676 6404 53 7 2389 9230 2485 8.00948 2421 4659 53 g 0684 9428 0746 9.30.599 0705 6375 52 8 2418 9226 2515 7.99058 2450 4630 52 g 0713 9424 0775 9.28058 0734 6346 51 9 2447 9222 2544 7.97176 2479 4600 51 10 0742 9421 0805 9.25530 0763 6317 50 10 2476 9219 2574 7.95302 2508 4571 60 11 0771 9418 0834 9.23016 0792 6288 49 11 2504 9215 2603 7.93438 2537 4542 49 12 0800 9415 0863 9.20516 0821 6259 48 12 2533 9211 2633 7.91582 2566 4513 48 13 0829 9412 0893 9.18028 0850 6229 47 13 2562 9208 2662 7.89734 2595 4484 47 14 0858 9409 0922 9.15554 0879 6200 46 14 2591 9204 2692 7.87895 2624 4455 46 16 0887 9406 0952 9.13093 0908 6171 45 15 2620 9200 2722 7.86064 2654 4426 45 16 0916 9402 0981 9.10646 0937 6142 44 16 2649 9197 2751 7.84242 2683 4397 44 17 0945 9399 1011 9.08211 0966 6113 43 17 2678 9193 2781 7.82428 2712 4368 43 18 0973 9396 1040 9.05789 0996 6084 42 18 2706 9189 2810 7.80622 2741 4339 42 19 1002 9393 1070 9.03379 1025 6055 41 19 2735 9186 2840 7.78825 2770 4310 41 20 1031 9390 1099 9.00983 1054 6026 40 20 2764 9182 2869 7.77035 2799 4281 40 21 1060 9386 1128 8.98598 1083 5997 39 21 2793 9178 2899 7.75254 2828 4251 39 22 1089 9383 1158 8.96227 1112 5968 38 22 2822 9175 2929 7.73480 2857 4222 38 23 1118 9380 1187 8.93867 1141 5939 37 23 2851 9171 2958 7.71715 2886 4193 37 24 1147 9377 1217 8.91520 1170 5909 36 24 2880 9167 2988 7.69957 2915 4164 36 25 1176 9374 1246 8.89185 1199 5880 35 25 2908 9163 3017 7.68208 2944 4135 35 26 1205 9370 1276 8.86862 1228 5851 34 26 2937 9160 3047 7.66466 2974 4106 34 27 1234 9367 1305 8.84551 1257 5822 33 27 2966 9156 3076 7.64732 3003 4077 33 28 1263 9364 1335 8.82252 1286 5793 32 28 2995 9152 3106 7.63005 3032 4048 32 29 1291 9360 1364 8.79964 1316 5764 31 29 3024 9148 3136 7.61287 3061 4019 31 30 1320 9357 1394 8.77689 1345 5735 30 30 3053 9144 3165 7.59575 3090 3990 30 0.1 0.9 0.1 0.1 1.4 0.1 0.9 0.1 0.1 1.4 31 1349 9354 1423 8.75425 1374 5706 29 31 3081 9141 3195 7.57872 3119 3961 29 32 1378 9351 1453 8.73172 1403 5677 28 32 3110 9137 3224 7.56176 3148 3931 28 33 1407 9347 1482 8.70931 1432 5648 27 33 3139 9133 3254 7.54487 3177 3902 27 34 1436 9344 1511 8.68701 1461 5619 26 34 3168 9129 3284 7.52806 3206 3873 26 35 1465 9341 1541 8.66482 1490 5589 25 35 3197 9125 3313 7.51132 3235 3844 25 36 1494 9337 1570 8.64275 1519 5560 24 36 3226 9122 3343 7.49465 3264 3815 24 37 1523 9334 1600 8.62078 1548 5531 23 37 3254 9118 3372 7.47806 3294 3786 23 38 1552 9331 1629 8.59893 1577 5502 22 38 3283 9114 3402 7.46154 3323 3757 22 39 1580 9327 1659 8.57718 1606 5473 21 39 3312 9110 3432 7.44509 3352 3728 21 40 1609 9324 1688 8.55555 1635 5444 20 40 3341 9106 3461 7.42871 3381 3699 20 41 1638 9320 1718 8.53402 1665 5415 19 41 3370 9102 3491 7.41240 3410 3670 19 42 1667 9317 1747 8.51259 1694 5386 18 42 3399 9098 3521 7.39616 3439 3641 18 43 1696 9314 1777 8.49128 1723 5357 17 43 3427 9094 3550 7.37999 3468 3611 17 44 1725 9310 1806 8.47007 1752 5328 16 44 3456 9091 3580 7.36389 3497 3582 16 45 1754 9307 1836 8.44896 1781 5299 15 45 3485 9087 3609 7.34786 3526 3553 15 46 1783 9303 1865 8.42795 1810 5270 14 46 3514 9083 3639 7.33190 3555 3524 14 47 1812 9300 1895 8.40706 1839 5240 13 47 3543 9079 3669 7.31600 3584 3495 13 48 1840 9297 1924 8.38625 1868 5211 12 48 3572 9075 3698 7.30018 3614 3466 12 49 1869 9293 1954 8.36555 1897 5182 11 49 3600 9071 3728 7.28442 3643 3437 11 50 1898 9290 1983 8.34496 1926 5153 10 50 3629 9067 3758 7.26873 3672 3408 10 51 1927 9286 2013 8.32446 1955 5124 9 51 3658 9063 3787 7.25310 3701 3379 9 52 1956 9283 2042 8.30406 1985 5095 8 52 3687 9059 3817 7.23754 3730 3350 8 53 1985 9279 2072 8.28376 2014 5066 7 53 3716 9055 3847 7.22204 3759 3321 7 54 2014 9276 2101 8.26355 2043 5037 6 54 3744 9051 3876 7.20661 3788 3291 6 55 2043 9272 2131 8.24345 2072 5008 5 55 3773 9047 3906 7.19125 3817 3262 5 56 2071 9269 2160 8.22345 2101 4979 4 56 3802 9043 3935 7.17.594 3846 3233 4 57 2100 9265 2190 8.20352 2130 4950 3 57 3831 9039 3965 7.16071 3875 3204 3 58 2129 9262 2219 8.18370 2159 4920 2 58 386019035 3995 7.14553 3904 3175 2 59 2158 9258 2249 8.16398 2188 4891 1 59 3889 9031 4024 7.13042 3933 3146 1 60 2187 9255 2278 8.14435 2217 4862 60 3917 9027 SIN 4054 COT 7.11537 3963 3117 cos Sllf. COT TAN. COM OF ARC. / COS TAN. COM OF ARC. ' AHC ' ARC Sup- 96° = 5760' 83° = 4980' Sup. 9T° = 5320' 426 TRIGONOMETRY v = 480' Sup. 171° = 10260' 9° = W!f Sup. 170° = 10200* COM. COM. t BIN. COS. rAN. COT. ABC. OF ARC. t BIN. COB. TAN. COT. ARC. OF ARC. 0.1 0.9 0.1 0.1 1.4 0.1 0.9 0.1 0.1 1.4 3917 9027 4054 7.11637 3963 3117 60 5643 8769 5838 6.31375 5708 1372 60 1 3946 9023 4084 7.10038 3992 3088 59 1 5672 8764 5868 6.30189 5737 1343 59 2 3975 9019 4113 7.08546 4021 3059 58 2 5701 8760 5898 6.29007 5766 1313 58 3 4004 9015 4143 7.07059 4050 3030 57 3 6730 8755 5928 6.27829 5795 1284 57 4 4033 9011 4173 7.05579 4079 3001 56 4 6758 8751 6958 6.26655 5824 1255 56 5 4061 9006 4202 7.04105 4108 2972 55 5 5787 8746 6988 6.25486 5853 122& 55 6 1090 9002 4232 7.02637 4137 2942 54 6 5816 8741 6017 6.24321 6882 1197 54 7 4119 8998 4262 7.01174 4166 2913 53 7 5C45 8737 6047 6.23160 5912 1168 53 8 4148 8994 4291 8.99718 4195 2884 52 8 5873 8732 6077 6.22003 5941 1139 52 9 4177 8990 4321 6.98268 4224 2865 51 9 6902 8728 6107 6.20851 5970 1110 61 10 4205 8986 4351 6.96823 4253 2826 50 10 5931 8723 6137 6.19703 6999 1081 50 11 4234 8982 4381 6.95385 4283 2797 49 11 5959 8718 6167 6.18669 6028 1062 49 12 4263 8978 4410 6.93952 4312 2768 48 12 6988 8714 6196 6.17419 6057 1023 48 13 4292 8973 4440 6.92525 4341 2739 47 13 6017 8709 6226 6.16283 6086 0993 47 14 4320 8969 4470 6.91104 4370 2710 46 14 6046 8704 6256 6.15151 6115 0964 46 15 4349 8966 4499 6.89688 4399 2681 45 15 6074 8700 6286 6.14023 6144 0935 46 16 4378 8961 4529 6.88278 4428 2652 44 16 6103 8695 6316 6.12899 6173 0906 44 17 4407 8957 4559 6.86874 4457 2622 43 17 6132 8690 6346 6.11779 6202 0877 43 18 4436 8953 4588 6.85476 4486 2693 42 18 6160 8686 6376 6.10664 6232 0848 42 19 4464 8948 4618 6.84082 4515 2564 41 19 6189 8681 6406 6.09552 6261 0819 41 20 4493 8944 4648 6.82694 4644 2535 40 20 6218 8676 6436 6.08444 6290 0790 40 21 4522 8940 4678 6.81312 4573 2606 39 21 6246 8671 6466 6.07340 6319 0761 39 22 4551 8936 4707 6.79936 4603 2477 38 22 6275 8667 6495 6.06240 6348 0732 38 23 4580 8931 4737 6.78564 4632 2448 37 23 6304 8662 6525 6.05143 6377 0703 37 24 4608 8927 4767 6.77199 4661 2419 36 24 6333 8657 6555 6.04061 6406 0673 36 25 4637 8923 4796 6.76838 4690 2390 35 25 6361 8652 6585 6.02962 6436 0644 35 26 4666 8919 4826 6.74483 4719 2361 34 26 6390 8648 6615 6.01878 6464 0615 34 27 4696 8914 4856 6.73133 4748 2332 33 27 6419 8643 6645 6.00797 6493 0586 33 28 4723 8910 4886 6.71789 4777 2302 32 28 6447 8638 6674 6.99720 6622 0557 32 29 4752 8906 4915 6.70450 4806 2273 31 29 6476 8633 6704 5.98646 6651 0528 31 30 4781 8902 4945 6.69116 4835 2244 30 30 6606 8629 6734 5.97576 6681 0499 30 0.1 0.9 0.1 0.1 1.4 0.1 0.9 0.1 0.1 1.4 31 4810 8897 4975 6.67787 4864 2215 29 31 6633 8624 6764 5.96510 6610 0470 29 32 4838 8893 5006 6.66463 4893 2186 28 32 6562 8619 6794 5.95448 6639 0441 28 33 4867 8889 5034 6.65144 4923 2157 27 33 6691 8614 6824 5.94390 6668 0412 27 34 4896 8884 5064 6.63831 4952 2128 26 34 6620 8609 6864 6.93335 6697 0383 26 35 4925 8880 6094 6.62523 4981 2099 25 36 6648 8604 6884 6.92283 6726 0364 25 36 4954 8876 5124 6.61219 6010 2070 24 36 6677 8600 6914 5.91236 6765 0324 24 37 4982 8871 5153 6.59921 5039 2041 23 37 6706 8595 6944 5.90191 6784 0295 23 38 5011 8867 5183 6.58627 5068 2012 22 38 6734 8590 6974 5.89161 6813 0266 22 39 5040 8863 5213 6.57339 5097 1982 21 39 6763 8685 7004 5.88114 6842 0237 21 40 6069 8858 5243 6.66055 5126 1953 20 40 6792 8580 7033 5.87080 6871 0208 20 41 5097 8854 5272 6.54777 5155 1924 19 41 6820 8575 7063 5.86051 6901 0179 19 42 5126 8849 6302 6.53503 5184 1895 18 42 6849 8670 7093 5.86024 6930 0160 18 43 5155 8845 5332 6.62234 5213 1866 17 43 6878 8665 7123 6.84001 6959 0121 17 44 5184 8841 5362 6.50970 6242 1837 16 44 6906 8561 7153 6.82982 6988 0092 16 46 6212 8836 5391 6.49710 5272 1808 16 45 6935 8556 7183 6'81966 7017 0063 15 46 6241 8832 5421 6.48456 5301 1779 14 46 6964 8551 7213 6.80963 7046 0034 14 47 5270 8827 6461 6.47206 5330 1750 13 47 6992 8546 7243 5.79944 7075 0004 1.3 9975 13 48 5299 8823 5481 6.45961 5359 1721 12 48 7021 8541 7273 5.78938 7104 12 49 5327 8818 5511 6.44720 5388 1692 11 49 7050 8536 7303 5.77936 7133 9946 11 50 5366 8814 5540 6.43484 5417 1663 10 60 7078 8531 7333 5.76937 7162 9917 10 51 6385 8809 5670 6.42253 5446 1633 9 51 7107 8526 7363 5.76941 7191 9888 9 52 6414 8805 6600 6.41026 5475 1604 8 52 7136 8521 7393 5.74949 7221 9869 8 53 5442 8800 6630 6.39804 5604 1575 7 53 7164 8516 7423 6.73960 7250 9830 7 54 5471 8796 6660 6.38587 6633 1546 6 54 7193 8511 7453 6.72974 7279 9801 6 65 5500 8791 6689 6.37374 5562 1517 5 55 7222 8506 7483 5.71992 7308 9772 5 66 5529 8787 5719 6.36166 5582 1488 4 56 7250 8501 7513 5.71013 7337 9743 4 57 6657 8782 5749 6.34961 5621 1459 3 57 7279 8496 7543 5.70037 7366 9714 3 58 6686 8778 5779 6.33761 5660 1430 2 58 7308 8491 7573 5.69064 7395 9684 2 69 6616 8773 5809 6.32666 5679 1401 1 69 7336 8486 7603 5.68094 7424 9666 1 60 6643 8769 5838 6.31376 5708 1372 60 7365 8481 7633 5.67128 7453 9626 COM. COM. cos SIN. COT. TAN. OF ARC. / COB. SIN. COT. TAN. OF ARC. ABC. Unc. Sup. 98° = SSSC 81° = 4860' Sup. 99° = 6940' 80° = 4800' EXAMPLES 427 10° = 600' Sup. 169° = 10140' 11° = = 660' Sup. 168° - 10080' COM. COM. t SIN. COB. TAN. COT. AKC. OF ARC. ' SIN. COS. TAN. COT. ABO. OF ARC. 0.1 0.9 0.1 0.1 1.3 0.1 0.9 0.1 0.1 1.3 7365 8481 7633 5.67128 7453 9626 60 9081 8163 9438 5.14455 9199 7881 60 1 7393 8476 7663 5.66165 7482 9597 59 1 9109 8157 9468 5.13658 9228 7852 59 2 7422 8471 7693 5.65205 7511 9568 58 2 9138 8152 9498 5.12862 9257 7823 58 3 7451 8466 7723 5.64248 7541 9539 57 3 9167 8146 9529 5.12069 9286 7794 57 4 7479 8461 7753 5.63295 7570 9510 56 4 9195 8140 9559 5.11279 9315 7765 56 5 7508 845S 7783 5.62344 7599 9481 55 5 9224 8135 9589 5.10490 9344 7736 55 6 7537 8450 7813 5.61397 7628 9452 54 6 9252 8129 9619 5.09704 9373 7706 54 7 7565 8445 7843 5.60452 7657 9423 53 7 9281 8124 9649 5.08921 9402 7677 53 S 7594 8440 7873 5.59511 7686 9394 52 8 9309 8118 9680 5.08139 9431 7648 52 9 7623 8435 7903 5.58573 7715 9364 51 9 9338 8112 9710 5.07360 9460 7619 51 10 7651 8430 7933 5.57638 7744 9335 50 10 9366 8107 9740 5.06584 9489 7590 50 11 7680 8425 7963 5.56706 7773 9306 49 11 9395 8101 9770 5.05809 9519 7561 49 12 7708 8420 7993 5.55777 7802 9277 48 12 9423 8096 9801 5.05037 9548 7532 48 13 7737 8414 8023 5.54851 7831 9248 47 13 9452 8090 9831 5.04267 9577 7503 47 14 7766 8409 8053 5.53927 7860 9219 46 14 9480 8084 9861 5.03499 9606 7474 46 15 7794 8404 8083 5.53007 7890 9190 45 15 9509 8079 9891 5.02734 9635 7445 45 16 7823 8399 8113 5.52090 7919 9161 44 16 9538 8073 9921 5.01971 9664 7416 44 17 7852 8394 8143 5.51176 7948 9132 43 17 9566 8067 9952 5.01210 9693 7386 43 18 7880 8389 8173 5.50264 7977 9103 42 18 9595 8061 9982 0.2 0012 5.00451 9722 7357 42 19 7909 8383 8203 5.49356 8006 9074 41 19 9623 8056 4.99695 9751 7328 41 20 7937 8378 8233 5.48451 8035 9045 40 20 9652 8050 0042 4.98940 9780 7299 40 21 7966 8373 8263 5.47548 8064 9015 39 21 9680 8044 0073 4.98188 9809 7270 39 22 7995 8368 8293 5.46648 8093 8986 38 22 9709 8039 0103 4.97438 9839 7241 38 23 8023 8362 8323 5.45761 8122 8957 37 23 9737 8033 0133 4.96690 9868 7212 37 24 8052 8357 8353 5.44857 8151 8928 36 24 9766 8027 0164 4.95945 9897 7183 36 25 8081 8352 8383 5.43966 8180 8899 35 25 9794 8021 0194 4.95201 9926 7154 35 26 8109 8347 8414 5.43077 8210 8870 34 26 9823 8016 0224 4.94460 9955 7125 34 27 8138 8341 8444 5.42192 8239 8841 33 27 9851 8010 0254 4.93721 9984 0.2 0013 7096 33 28 8166 8336 8474 5.41309 8268 8812 32 28 9880 8004 0285 4.92984 7066 32 29 8195 8331 8504 5.40429 8297 8783 31 29 9908 7998 0315 4.92249 0042 7037 31 30 8224 8325 8534 5.39552 8326 8754 30 30 9937 7992 0345 4.91516 0071 7008 30 31 8252 8320 8564 5.38677 8355 8725 29 31 9965 7987 0376 4.90785 0100 6979 29 32 8281 8315 8594 5.37805 8384 8695 28 32 9994 7981 0406 4.90056 0129 6950 28 0.1 0.9 0.1 0.1 1.3 0.2 0.9 0.2 0.2 1.3 33 8309 8310 8624 5.36936 8413 8666 27 33 0022 7975 0436 4.89330 0159 6921 27 34 8338 8304 8654 5.36070 8442 8637 26 34 0053 7969 0466 4.88605 0188 6892 26 3S 8367 8299 8684 5.35206 8471 8608 25 35 0079 7963 0497 4.87882 0217 6863 25 36 8395 8294 8714 5.34345 8500 8579 24 36 0108 7958 0527 4.87162 0246 6834 24 37 8424 8288 8745 5.33487 8530 8550 23 37 0)36 7952 0557 4.86444 0275 6805 23 38 8452 8283 8775 5.32631 8559 8521 22 38 0165 7946 0588 4.85727 0304 6776 22 39 8481 8277 8805 5.31778 8588 8492 21 39 0193 7940 0618 4.85013 0333 6746 21 40 8509 8272 8835 5.30928 8617 8463 20 40 0222 7934 0648 4.84300 0362 6717 20 41 8538 8267 8865 5.30080 8646 8434 19 41 0250 7928 0679 4.83590 0391 6688 ,19 42 8567 8261 8895 5.29235 8675 8405 18 42 0279 7922 0709 4.82882 0420 6659 18 43 8595 8256 8925 5.28393 8704 8375 17 43 0307 7916 0739 4.82175 0449 6630 17 44 8624 8250 8955 5.27553 8733 8346 16 44 0336 7910 0770 4.81471 0478 6601 16 45 8652 8245 8986 5.26715 8762 8317 15 45 0364 7905 0800 4.80769 0508 6572 15 46 8681 8240 9016 5.25880 8791 8288 14 46 0393 7899 0830 4.80068 0537 6543 14 47 8710 8234 9046 5.25048 8820 8259 13 47 0421 7893 0861 4.79370 0666 6514 13 48 8738 8229 9076 5.24218 8850 8230 12 48 0450 7887 0891 4.78673 0696 6485 12 49 8767 8223 9106 5.23391 8879 8201 11 49 0478 7881 0921 4.77978 0624 6456 11 50 8795 8218 9136 5.22566 8908 8172 10 50 0507 7875 0952 4.77286 0653 6427 10 51 8824 8212 9166 5.21744 8937 8143 9 51 0535 7869 0982 4.76595 0682 6397 9 52 8852 8207 9197 5.20925 8966 8114 8 52 0563 7863 1013 4.75906 0711 6368 8 53 8881 8201 9227 5.20107 8995 8085 7 S3 0592 7857 1043 4.75219 0740 6339 7 54 8910 8196 9257 5.19293 9024 8055 6 54 0620 7851 1073 4.74534 0769 6310 6 55 8938 8190 9287 5.18480 9053 8026 5 55 0649 7845 1104 4.73851 0798 6281 5 56 8967 8185 9317 5.17671 9082 7997 4 56 0677 7839 1134 4.73170 0828 6252 4 57 8995 8179 9347 5.16863 9111 7968 3 57 0706 7833 1164 4.72490 0857 6223 3 58 9024 8174 9378 5.16058 9140 7939 2 58 0734 7827 1195 4.71813 0886 6194 2 59 9052 8168 9408 5.15256 9169 7910 1 59 0763 7821 1225 4.71137 0915 6165 1 60 9081 8163 9438 5.14455 9199 7881 60 0791 7815 1256 4.70463 0944 6136 COM. COM. COS. SIN. COT. TAN. OF ARC. ARC. COS. SIN. COT. TAN. OF ARC. ARC. Sup. 100° — 6000' 79° — 4740' Sup. 101° = 6060' 78° = 4680 428 TRIGONOMETRY 12": = 720 ' Sup, 167° = 10020' 13° = 780' Sup. 166° = ggeo* COM. COM. SIN. COS. TAN. COT. ARC. OF ARC. SIN. COS. TAN. COT. ARC. OP ARC. 0.2 0.9 0.2 0.2 1.3 0.2 0.9 0.2 0.2 1.3 0791 7815 1256 4.70463 0944 6136 60 2495 7437 3087 4.33148 2689 4390 60 1 0820 7809 1286 4.69791 0973 6107 69 1 2523 7430 3117 4.32573 2718 4361 59 2 0848 7803'1316| 4.69121 1003 6077 68 2 2562 7424 3148 4.32001 2747 4332 58 3 0877 779711347! 4,68462 1031 6048 57 3 2680 7417 3179 4.31430 2776 4303 57 4 0905 7790 1377 4.67786 1060 6019 56 4 2608 7411 3209 4.30860 2806 4274 56 5 0933 7784 1408 4,67421 1089 5990 55 5 2637 7404 3240 4.30291 2836 4245 55 6 0962 7778 1438 4,66458 1118 5961 54 6 2665 7398 3271 4.29724 2864 4216 54 7 0990 7772 1469 4,65797 1148 5932 63 7 2693 7391 3301 4.29159 2893 4187 63 8 1019 7766 1499 4,66138 1177 5903 52 8 2722 7384 3332 4.28696 2922 4158 52 9 1047 7760 1529 4,64480 1206 5874 61 9 2750 7378 3363 4.28032 2951 4129 51 10 1076 7754 1560 4.63825 1235 5845 50 10 2778 7371 3393 4.27471 2980 4099 50 11 1104 7748 1590 4.63171 1264 5816 49 11 2807 7365 3424 4.26911 3009 4070 49 12 1132 7742 1621 4,62518 1293 6787 48 12 2835 7358 3455 4.26352 3038 4041 48 13 1161 7735 1661 4,61868 1322 5758 47 13 2863 7351 3485 4.25795 3067 4012 47 14 1189 7729 1682 4,61219 1351 5728 46 14 2892 7346 3516 4.25239 3096 3983 46 15 1218 7723 1712 4.60672 1380 5699 45 15 2920 7338 3547 4.24685 3126 3954 45 16 1246 7717 1743 4,59927 1409 5670 44 16 2948 7331 3578 4.24132 3155 3925. 44 17 1275 7711 1773 4.59283 1438 6641 43 17 2977 7325 3608 4.23580 3184 3896 43 18 1303 7705 1804 4.58641 1468 6612 42 18 3005 7318 3639 4,23030 3213 3867 42 19 1331 7698 1834 4.68001 1497 6683 41 19 3033 7311 3670 4,22481 3242 3838 41 20 1360 7692 1864 4.67363 1526 6554 40 20 3062 7304 3700 4,21933 3271 3809 40 21 1388 7686 1895 4,56726 1555 5525 39 21 3090 7298 3731 4,21387 3300 3779 39 22 1417 7680 1925 4,56091 1584 5496 38 22 3118 7291 3762 4.20842 3329 3750 38 23 1445 7673 1956 4,55458 1613 5467 37 23 3146 7284 3793 4,20298 3358 3721 37 24 1474 7667 1986 4,54826 1642 5437 36 24 3176 7278 3823 4.19756 3387 3692 36 25 1502 7661 2017 4.54196 1671 6408 36 25 3203 7271 3854 4.19215 3416 3663 35 26 1530 7655 2047 4.63568 1700 5379 34 26 3231 7264 3885 4.18676 3446 3634 34 27 1659 7648 2078 4.52941 1729 5350 33 27 3260 7257 3916 4,18137 3475 3605 33 28 1587 7642 2108 4.52316 1758 5321 32 28 3288 7251 3946 4.17600 3504 3576 32 29 1616 7636 2139 4.61693 1787 5292 31 29 3316 7244 3977 4.17064 3533 3647 31 30 1644 7630 2169 4.61071 1817 5263 30 30 3345 7237 4008 4.16530 3662 3618 30 0.2 0.9 0.2 0.2 1.3 0.2 0.9 0.2 0.2 1.3 31 1672 7623 2200 4.50451 1846 5234 29 31 3373 7230 4039 4.15997 3591 3489 29 32 1701 7617 2230 4,49832 1875 5205 28 32 3401 7223 4069 4.15465 3620 3459 28 33 1729 7611 2261 4.49215 1904 5176 27 33 3429 7217 4100 4.14934 3649 3430 27 34 1758 7604 2292 4.48600 1933 5147 26 34 3468 7210 4131 4.14405 3678 3401 26 35 1786 7598 2322 4.47986 9962 5118 25 35 3486 7203 4162 4.13877 3707 3372 25 36 1814 7592 2363 4.47374 1991 5088 24 36 3514 7196 4193 4.13350 3736 3343 24 37 1843 7585 2383 4.46764 2020 5059 23 37 3542 7189 4223 4.12825 3766 3314 23 38 1871 7579 2414 4.46155 2049 5030 22 38 3571 7182 4264 4.12301 3796 3285 22 39 1899 7573 2444 4.45648 2078 5001 21 39 3599 7176 4285 4.11778 3824 3256 21 40 1928 7566 2475 4.44942 2107 4972 20 40 3627 7169 4316 4.11266 3853 3227 20 41 1956 7560 2605 4.44338 2137 4943 19 41 3666 7162 4347 4.10736 3882 3198 19 42 1985 7663 2536 4.43735 2166 4914 18 42 3684 7155 4377 4.10216 3911 3169 18 43 2013 7647 2567 4.43134 2196 4885 17 43 3712 7148 4408 4,09699 3940 3139 17 44 2041 7641 2597 4.42634 2224 4856 16 44 3740 7141 4439 4.09182 3969 3110 16 45 2070 7634 2628 4.41936 2253 4827 15 45 3769 7134 4470 4.08666 3998 3081 15 46 2098 7528 2658 4.41340 2282 4798 14 46 3797 7127 4501 4.08152 4027 3052 14 47 2126 7521 2689 4,40745 2311 4768 13 47 3825 7120 4532 4.07639 4066 3023 13 48 2165 7515 2719 4,40152 2340 4739 12 48 3853 7113 4562 4.07127 4085 2994 12 49 2183 7608 2760 4,39560 2369 4710 11 49 3882 7106 4593 4.06616 4115 2965 11, 50 2212 7602 2781 4:38969 2398 4681 10 50 3910 7100 4624 4.06107 4144 2936 10 51 2240 7490 2811 4.38381 2427 4652 9 51 3938 7093 4655 4.05599 4173 2907 9 52 2268 7489 2842 4.37793 2457 4623 8 52 3966 7086 4686 4.05093 4202 2878 8 53 2297 7483 2872 4.37207 2486 4594 7 53 3995 7079 4717 4.04586 4231 2849 7 5( 2325 7476 2903 4.36623 2516 4665 6 54 4023 7072 4747 4.04081 4260 2820 6 55 2353 7470 2934 4.36040 2544 4636 5 55 4061 7066 4778 4.03578 4289 2790 5 56 2382 7463 2964 4;35469 2573 4507 4 66 4079 7058 4809 4.03076 4318 2761 4 57 2410 7457 2996 4.34879 2602 4478 3 67 4108 7051 8440 4,02574 4347 2732 3 58 2438 7450 3026 4.34300 2631 4448 2 68 4136 7044 4871 4.02074 4376 2703 2 69 2467 7444 3066 4.33723 2660 4419 1 59 4164 7037 4902 4.01576 4405 2674 1 60 2495 7437 3087 4.33148 2689 4390 60 4192 7030 4933 4.01078 4435 2645 cos SIN COT. TAN. COM. OF ARC. ARC. ' COS. SIN. COT. TAN. COM. OF ABC. ABC. / Sup. 102° == 0120' 77° = 4020' Sup. 103° = 6180' 76° — 4560' EXAMPLES 429 14° = 840' Sup. 165° = 9900' 15° = 900' 3up. 164° = 9840' COM. COM. / BIN. COS. TAN. COT. AHC. OP AKC. SIN. COS. TAN. COT. ARC. OF ARC. 0.2 0.9 0.2 0.2 1.3 0.2 0.9 0.2 0.2 1.3 4192 7030 4933 4.01078 4435 2645 60 5882 6593 6795 3.73205 6180 0900 60 1 4220 7023 4964 4.00582 4464 2616 59 1 5910 6585 6826 3.72771 6209 0871 59 2 4249 7015 4996 4.00086 4493 2587 58 2 5938 6578 6857 3.72338 6238 0841 58 3 4277 7008 5026 3.99592 4522 2558 57 3 5966 6570 6888 3.71907 6267 0812 57 4 4305 7001 5056 3.99099 4551 2529 56 4 5994 6562 6920 3.71476 6296 0783 56 5 4333 6994 5087 3.98607 4580 2500 55 5 6022 6555 6951 3.71046 6325 0754 55 6 4361 6987 5118 3.98117 4609 2470 54 6 6050 6547 6982 3.70616 6354 0725 54 7 4390 6980 5149 3.97627 4638 2441 53 7 6079 6540 7013 3.70188 6384 0696 53 8 4418 6973 5180 3.97139 4667 2412 52 8 6107 6532 7044 3.69761 6413 0667 52 9 4446 6966 5211 3.96651 4696 2383 51 9 6135 6524 7076 3.69335 6442 0638 51 10 4474 6959 5242 3.96165 4725 2354 50 10 6163 6517 7107 3.68909 6471 0609 50 11 4503 6952 5273 3.95680 4755 2325 49 11 6191 6509 7138 3.68485 6500 0580 49 12 4531 6945 5304 3.95196 4784 •2296 48 12 6219 6502 7169 3.68061 6529 0551 48 13 4559 6937 5335 3.94713 4813 2267 47 13 6247 6494 7201 3.67638 6558 0521 47 14 4587 6930 5366 3.94232 4842 2238 46 14 6275 6486 7232 3.67217 6587 0492 46 15 4615 6923 5397 3.93751 4871 2209 45 15 6303 6479 7263 3.66796 6616 0463 45 16 4644 6916 5428 3.93271 4900 2180 44 16 6331 6471 7294 3.66376 6646 0434 44 17 4672 6909 5459 3.92793 4929 2150 43 17 6359 6463 7326 3.65957 6674 0405 43 18 4700 6902 5490 3.92316 4958 2121 42 18 6387 6456 7357 3.65538 6703 0376 42 19 4728 6894 5521 3.91839 4987 2092 41 19 6415 6448 7388 3.65121 6733 0347 41 20 4756 6887 5552 3.91364 5016 2063 40 20 6443 6440 7419 3.64705 6762 0318 40 21 4784 6880 5583 3.90890 5045 2034 39 21 6471 6433 7451 3.64289 6791 0289 39 22 4813 6873 5614 3.90417 5075 2005 38 22 6500 6425 7482 3.63874 6820 0260 38 23 4841 6866 5645 3.89945 5104 1976 37 23 6528 6417 7513 3.63461 6849 0231 37 24 4869 6859 5676 3.89474 5133 1947 36 24 6556 6410 7545 3.63048 6878 0202 36 25 4897 6851 5707 3.89004 5162 1918 35 25 6584 6402 7576 3.62636 6907 0172 35 26 4925 6844 5738 3.88536 5191 1889 34 26 6612 6394 7607 3.62224 6936 0143 34 27 4953 6837 5769 3.88068 5220 1860 33 27 6640 6386 7638 3.61814 6965 0114 33 28 4982 6829 5800 3.87601 5249 1830 32 28 6668 6379 7670 3.61405 6994 0085 32 29 5010 6822 5831 3.87136 5278 1801 31 29 6696 6371 7701 3.60996 7023 0056 31 30 5038 6815 5862 3.86671 5307 1772 30 30 6724 6363 7732 3.60588 7053 0027 30 0.2 0.9 0.2 0.2 1.3 0.2 0.9 0.2 0.2 1.2 31 5066 6807 5893 3.86208 5336 1743 29 31 6752 6355 7764 3.60181 7082 9998 29 32 5094 6800 5924 3.85745 5365 1714 28 32 6780 6347 7795 3.59775 7111 9969 28 33 5122 6793 5955 3.85284 5394 1685 27 33 6808 6340 7826 3.59370 7140 9940 27 34 5151 6786 5986 3.84824 5424 1656 26 34 6836 6332 7858 3.58966 7169 9911 26 35 5179 6778 6017 3.84364 5453 1627 25 35 6864 6324 7889 3.58562 7198 9882 25 36 5207 6771 6048 3.83906 5482 1598 24 36 6892 6316 7920 3.58160 7227 9852 24 37 5235 6764 6079 3.83449 5511 1569 23 37 6920 6308 7952 3.57758 7256 9823 23 38 5263 6756 6110 3.82992 5540 1540 22 38 6948 6301 7983 3.57357 7285 9794 22 39 5291 6749 6141 3.82537 5569 1511 21 39 6976 6293 8015 3.56957 7314 9765 21 40 5320 6742 6172 3.82083 5598 1481 20 40 7004 6285 8046 3.56557 7343 9736 20 41 5348 6734 6203 3.81630 5627 1452 19 41 7032 6277 8077 3.56159 7373 9707 19 42 5376 6727 6235 3.81177 5656 1423 18 42 7060 6269 8109 3.55761 7402 9678 18 43 5404 6719 6266 3.80726 5685 1394 17 43 7088 6261 8140 3.55364 7431 9649 17 44 5432 6712 6297 3.80276 5714 1365 16 44 7116 6253 8172 3.54968 7460 9620 16 45 5460 6705 6328 3.79827 5744 1336 15 45 7144 6246 8203 3.54573 7489 9591 15 46 5488 6697 6359 3.79378 5773 1307 14 46 7172 6238 8234 3.54179 7518 9562 14 47 5516 6690 6390 3.78931 5802 1278 13 47 7200 6230 8266 3.53785 7547 9532 13 48 5545 6682 6421 3.78485 5831 1249 12 48 7228 6222 8297 3.53393 7576 9503 12 49 5573 6675 6452 3.78040 5860 1220 11 49 7256 6214 8329 3.53001 7605 9474 11 50 5601 6667 6483 3.77595 5889 1191 10 50 7284 6206 8360 3,52609 7634 9445 10 51 5629 6660 6515 3.77152 5918 1161 9 51 7312 6198 8391 3.52219 7663 9416 9 52 5657 6653 6546 3.76709 5947 1132 8 52 7340 6190 8423 3.51829 7693 9387 8 53 5685 6645 6577 3.76268 5976 1103 7 53 7368 6182 8454 3.51441 7722 9358 7 54 5713 6638 6608 3.75828 6005 1074 6 54 7396 6174 8486 3.51053 7751 9329 6 55 5741 6630 6639 3.75388 6034 1045 5 55 7424 6166 8517 3.50666 7780 9300 5 56 5769 6623 6670 3.74950 6064 1016 4 56 7452 6158 8549 3.50279 7809 9271 4 57 5798 6615 6701 3.74512 6093 0987 3 57 7480 6150 8580 3.49894 7838 9242 3 58 5826 6608 6733 3.74075 6122 0958 2 58 7508 6142 8612 3.49509 7867 9212 2 59 5854 6600 6764 3.73640 6151 0929 1 59 7536 6134 8643 3.49125 7896 9183 1 60 5882 6593 6795 3.73205 6180 0900 60 7564 6126 8675 3.48741 7925 9154 COM. COM. COS. SIN. COT. TAN. OF ARC. ARC. f COS. SIN. COT. TAN. OF ARC. ARC. Sup. 104° = 6240' 75° = 4500' Sup. 105° = 6300' 74° = 4440' 430 TRIGONOMETRY 16° = 9G0' Sup. 163° = 9780' 17° = 1020' Sup. 162° = 9720' COM. COM. BIN cos TAN COT. ARC OF ABC. SIN cos TAN. COT. ARC. OF ARC. 0.2 0.9 0.2 0.2 1.2 0.2 0.9 0.3 0.2 1.2 7564 6126 8675 3.48741 7925 9154 60 9237 5630 0537 3.27086 9671 7409 60 59 1 7592 6118 8706 3.48359 7964 9125 69 1 9265 5622 0606 3.26745 9700 7380 2 7620 6110 8738 3.47977 7983 9096 68 2 9293 5613 0637 3.26406 9729 7351 58 57 3 7648 6102 8769 3.47595 8012 9067 67 3 9321 6605 0669 3.26067 9768 7322 4 7676 6094 8800 3.47216 8042 9038 56 4 9348 5696 0700 3.26729 9787 7293 66 65 5 7704 6086 8832 3.46837 8071 9009 66 6 9376 6588 0732 3.26392 9816 7264 6 7731 6078 8864 3.46458 8100 8980 54 6 9404 5579 0764 3.25055 9846 7234 54 7 7759 6070 8896 3.46080 8129 8961 53 7 9432 6671 0796 3.24719 9874 7206 53 8 7787 6062 8927 3.46703 8158 8922 62 8 9460 6662 0828 3.24383 9903 7176 52 51 50 49 9 7815 6054 8958 3.46327 8187 8893 61 9 9487 6664 0860 3.24048 9932 7147 10 7843 6046 8990 3.44951 8216 8863 50 10 9615 6646 0891 3.23714 9661 7118 11 7871 6037 9021 3.44576 8245 8834 49 11 9643 5536 0923 3.23381 9991 0.3 0020 7089 12 7899 6029 9063 3.44202 8274 8805 48 12 9571 5528 0956 3.23048 7060 48 47 13 7927 6021 9084 3.43829 8303 8776 47 13 9699 5519 0987 3.22716 0049 7021 14 7955 6013 9116 3.43456 8332 . 8747 46 14 9626 6611 1019 3.22384 0078 6992 46 15 7983 6006 9147 3.43084 8362 8718 45 15 9654 6602 1061 3,22053 0107 6963 45 16 8011 6997 9179 3.42713 8391 8689 44 16 9682 5493 1083 3.21722 0136 6934 44 17 8039 5989 9210 3.42343 8420 8660 43 17 9710 5485 1115 3.21392 0166 6914 43 42 18 8067 5981 9242 3.41973 8449 8631 42 18 9737 5476 1147 3.21063 0194 6885 19 8096 5972 9274 3.41604 8478 8602 41 19 9766 5467 1178 3.20734 0223 6866 41 20 8123 5964 9305 3.41236 8607 8573 40 20 9793 6469 1210 3.20406 0262 6827 40 21 8160 5966 9337 3.40869 8536 8543 39 21 9821 5450 1242 3.20079 0281 6798 39 22 8178 6948 9368 3.40502 8566 8614 38 22 9849 6441 1274 3.19752 0311 6769 38 23 8206 5940 9400 3.40136 8694 8485 37 23 9876 5433 1306 3.19426 0340 6740 37 24 8234 5931 9432 3.39771 8623 8456 36 24 9904 5424 1338 3.19100 0369 6711 36 25 8262 6923 9463 3.39406 8662 8427 35 25 9932 6415 1370 3.18775 0398 6682 35 26 8290 5916 9495 3.39042 8682 8398 34 26 9960 5407 1402 3.18451 0427 6653 34 27 8318 6907 9526 3.38679 8711 8369 33 27 9987 5398 1434 3.18127 0456 6624 33 28 8346 5898 9668 3.38317 8740 8340 32 28 0015 5389 1466 3.17804 0485 6594 32 29 8374 5890 9590 3.37955 8769 8311 31 29 0043 5380 1498 3.17481 0514 6565 31 30 8402 6882 9621 3.37594 8798 8282 30 30 0071 5372 1530 3.17169 0643 6636 30 0.2 0.9 0.2 0.2 1.2 0.3 0.9 0.3 0.3 1.2 31 8429 6874 9653 3.37234 8827 8253 29 31 0098 5363 1562 3.16838 0572 6607 29 32 8457 5866 9685 3.36876 8866 8223 28 32 0126 6364 1594 3.16517 0601 6478 28 33 8486 6867 9716 3.36516 8886 8194 27 33 0154 6345 1626 3.16197 0630 6449 27 34 8513 6849 9748 3.36158 8914 8165 26 34 0182 633» 1658 3.16877 0660 6420 26 35 8541 5841 9780 3.35800 8943 8136 26 35 0209 5328 1690 3.15568 0689 6391 25 36 8569 6832 9811 3.35443 8972 8107 24 36 0237 5319 1722 3.15240 0718 6362 24 37 8597 5824 9843 3.35087 9002 8078 23 37 0265 5310 1754 3.14922 0747 6333 23 38 8625 5816 9875 3.34732 9031 8049 22 38 0292 6301 1786 3.14606 D776 6304 22 39 8662 6807 9906 3.34377 9060 8020 21 39 0320 6293 1818 3.14288 D806 6275 21 40 8680 6799 9938 3.34023 9089 7991 20 40 0348 6284 1860 3.13972 D834 6245 20 41 8708 5791 9970 0.3 3.33670 9118 7962 19 41 0376 6276 1882 3.13656 0863 6216 19 42 8736 5782 oo'oi 3.33317 9147 7933 18 42 0403 5266 1914 3.13341 0892 6187 18 43 8764 5774 0033 3.32965 9176 7903 17 43 0431 5267 1946 3.13027 0921 6158 17 44 8792 5766 0065 3.32614 9206 7874 16 44 0459 5248 1978 3.12713 D950 6129 16 45 8820 5767 0097 3.32264 9234 7845 16 46 0486 5240 2010 3.12400 D980 6100 15 46 8847 6749 0128 3.31914 9263 7816 14 46 0614 5231 2042 3.12087 1009 6071 14 47 8875 5740 0160 3.31566 9292 7787 13 47 0542 5222 2074 3.11776 1038 6042 13 48 8903 5732 0192 3.31216 9321 7758 12 48 0570 5213 2106 3.11464 1067 6013 12 49 8931 6724 0224 3.30868 9351 7729 11 49 0597 6204 2139 3.11163 1096 5984 u SO 8959 5716 0256 3.30521 9380 7700 10 50 0626 6195 2171 3.10842 1126 5965 10 51 8987 5707 0287 3.30174 9409 7671 9 51 0653 5186 2203 3.10632 1164 5925 9 62 9015 6698 0319 3.29829 9438 7642 8 52 0680 6177 2236 3.10223 183 5896 g 53 9042 6690 0351 3.29483 9467 7613 7 53 0708 5168 2267 3.09914 1212 5867 7 64 9070 5681 0382 3.29139 9496 7584 6 64 0736 5169 2299 3.09606 1241 5838 6 55 9098 5673 0414 3.28795 9526 7554 5 66 0763 5160 2331 3.09298 1270 5809 5 56 9126 6664 0446 3.28452 9654 7525 4 66 0791 5142 2363 3.08991 1300 5780 4 57 3154 5656 0478 3.28109 9583 7496 3 57 0819 5133 2396 3.08685 329 5751 3 58 9182 5647 0609 3.27767 9612 7467 2 58 0846 5124 2428 3.08379 1358 6722 2 59 9209 5639 0541 3.27426 9641 7438 1 59 0874 5115 2460 3.08073 1387 5693 1 60 9237 5630 0573 3.27085 9671 7409 60 0902 5106 2492 3.07768 1416 6664 COS. SIN. COT. TAN. COM. OF ARC. / COS. SIN. COT. TAN. COM. OF ARC. / ' ARC. ARC. Sup 106° = 63 30' 73° = 4 380' Sup. 107° = 642 0- 72° = 4 320' EXAMPLES 431 18° = 1080' Sup. 161° — 9660' 19° ^nw Sup. 160° = 9600' COM. COM. ' SIN COS TAN COT. ABC. OP ARC. r BIN. 0.3 COS. 0.9 TAN. 0.3 COT. ARC. OF ARC. 0.3 0.9 0.3 0.3 1.2 0.3 1.2 0902 5106 2492 3.07768 1416 5664 60 2557 4552 4433 2.90421 3161 3918 60 1 0929 5097 2524 3.07464 1445 5635 59 1 2584 4542 4465 2.90147 3190 3889 59 2 0957 5088 2556 3.07160 1474 5605 58 2 2612 4533 4498 2.89873 3219 3860 58 3 0985 5079 2588 3.06857 1503 5576 57 3 2639 4523 4530 2.89600 3248 3831 57 4 1012 5070 2621 3.06554 1532 5547 56 4 2667 4514 4563 2.89327 3278 3802 56 5 1040 5061 2653 3.06252 1561 5518 55 5 2694 4504 4596 2.89055 3307 3773 55 6 1068 5052 268S 3.05950 1590 5489 54 6 2722 4495 4628 2.88783 3336 3744 64 7 1095 5043 2717 3.05649 1619 5460 53 7 2749 4485 4661 2.88611 3365 3715 63 8 1123 5033 2749 3.05349 1649 5431 52 8 2777 4476 4693 2.88240 3394 3686 52 9 1151 5024 2782 3.05049 1678 5402 51 9 2804 4466 4726 2.87970 3423 3657 61 10 1178 5015 2814 3.04749 1707 5373 50 10 2832 4457 4758 2.87700 3452 3627 50 11 1206 5006 2846 3.04450 1736 5344 49 11 2859 4447 4791 2.87430 3481 3598 49 12 1233 4997 2878 3.04152 1765 5315 48 12 2887 4438 4824 2.87161 3610 3669 48 13 1261 4988 2911 3.03854 1794 5286 47 13 2914 4428 4856 2.86892 3539 3540 47 14 1289 4979 2943 3.03556 1823 5256 46 14 2942 4418 4889 2.86624 3568 3511 46 15 1316 4970 2975 3.03260 1852 5227 45 15 2969 4409 4922 2.86366 3598 3482 45 16 1344 4961 3007 3.02963 1881 5198 44 16 2997 4399 4954 2.86089 3627 3453 44 17 1372 4952 3040 3.02667 1910 5169 43 17 3024 4390 4987 2.86822 3656 3424 43 18 1399 4943 3072 3.02372 1939 5140 42 18 3051 4380 6019 2.85666 3685 3395 42 19 1427 4933 3104 3.02077 1969 5111 41 19 3079 4370 5052 2.85289 3714 3366 41 20 1454 4924 3136 3.01783 1998 5082 40 20 3106 4361 5085 2.85023 3743 3337 40 21 1482 4915 3169 3.01489 2027 5053 39 21 3134 4351 5117 2.84758 3772 3307 39 22 1510 4906 3201 3.01196 2056 5024 38 22 3161 4342 5150 2.84494 3801 3278 38 23 1537 4897 3233 3.00903 2085 4995 37 23 3189 4332 5183 2.84229 3830 3249 37 24 1565 4888 3266 3.00611 2114 4966 36 24 3216 4322 5216 2.83965 3859 3220 36 25 1592 4878 3298 3.00319 2143 4936 35 25 3244 4313 5248 2.83702 3888 3191 35 26 1620 4869 3330 3.00028 2172 4907 34 26 3271 4303 5281 2.83439 3918 3162 34 27 1648 4860 3363 2.99738 2201 4878 33 27 3298 4293 6314 2.83176 3947 3133 33 28 1675 4851 3395 2.99447 2230 4849 32 28 3326 4284 5346 2.82914 3976 3104 32 29 1703 4842 3427 2.99158 2259 4820 31 29 3353 4274 5379 2.82653 4005 3075 31 30 1730 4832 3460 2.98869 2289 4791 30 30 3381 4264 5412 2.82391 4034 3046 30 0.3 0.9 0.3 0.3 1.2 0.3 0.9 0.3 0.3 1.2 31 1758 4823 3491 2,98580 2318 4762 29 31 3408 4254 6445 2.82130 4063 3017 29 32 1786 4814 3524 2.98292 2347 4733 28 32 3436 4245 5477 2.81870 4092 2987 28 33 1813 4805 3557 2,98004 2376 4704 27 33 3463 4235 5510 2.81610 4121 2958 27 34 1841 4795 3589 2.97717 2405 4675 26 34 3490 4225 5543 2.81360 4150 2929 26 35 1868 4786 3621 2.97430 2434 4646 25 35 3518 4215 5576 2.81091 4179 2900 25 36 1896 4777 3654 2.97144 2463 4616 24 36 3545 4206 6608 2.80833 4208 2871 24 37 1923 4768 3686 2.96858 2492 4587 23 37 3573 4196 5641 2.80574 4237 2842 23 38 1951 4758 3718 2.96573 2521 4558 22 38 3600 4186 5674 2.80316 4267 2813 22 39 1979 4749 3751 2.96288 2550 4529 21 39 3627 4176 5707 2.80059 4296 2784 21 40 2006 4740 3783 2.96004 2579 4500 20 40 3655 4167 5740 2.79802 4325 2755 20 41 2034 4730 3816 2.95720 2609 4471 19 41 3682 4157 5772 2.79545 4354 2726 19 42 2061 4721 3848 2.95437 26.38 4442 18 42 3710 4147 5805 2.79289 4383 2697 18 43 2089 4712 3881 2.95155 2667 4413 17 43 3737 4137 5838 2.79033 4412 2668 17 44 2116 4702 3913 2.94872 2696 4384 16 44 3764 4127 5871 2.78778 4441 2638 16 45 2144 4693 3945 2.94590 2725 4355 15 45 3792 4118 5904 2.78523 4470 2609 15 46 2171 4684 3978 2.24309 2754 4326 14 46 3819 4108 5937 2.78269 4499 2680 14 47 2199 4674 4010 2.94028 2783 4296 13 47 3846 4098 5969 2.78014 4528 2551 13 48 2227 4665 4043 2.93748 2812 4267 12 48 3874 4088 6002 2.77761 4557 2522 12 49 2254 4656 4075 2.93468 2841 4238 11 49 3901 4078 6035 2.77607 4587 2493 11 50 2282 4646 4108 2.93189 2870 4209 10 SO 3929 4068 6068 2.77254 4616 2464 10 51 2309 4637 4140 2.92910 2899 4180 9 51 3956 4058 6101 2.77002 4645 2435 9 52 2337 4627 4173 2.92632 2928 4151 8 52 3983 4049 6134 2.76760 4674 2406 S 53 2364 4618 4205 2.92354 2958 4122 7 53 4011 4039 6167 2.76498 4703 2377 7 54 2392 4609 4238 2.92076 2987 4093 6 54 4038 4029 6199 2.76247 4732 2348 6 55 2419 4599 4270 2.91799 3016 4064 5 55 4065 4019 6232 2.75996 4761 2318 5 56 2447 4590 4303 2.91523 3045 4035 4 56 4093 4009 6265 2.75746 4790 2289 4 57 2474 4580 4335 2.91246 3074 4006 3 57 4120 3999 6298 2.76496 4819 2260 3 58 2502 4571 4368 2.90971 3103 3977 2 58 4147 3989 6331 2.76246 4848 2231 2 59 2529 4561 4400 2.90696 3132 3947 1 59 4175 3979 6364 2.74997 4877 2202 1 60 2557 COS. 4552 SIN. 4433 2.90421 3161 3918 60 4202 COS. 3969 SIN. 6397 2.74748 4907 2173 COT. TAN. COM. OF ARC. / COT. TAN. COM. OF ARC. / 1 ABC. 1 ARC. 71° = 4260' Sup. 109-'' = 6540' 70° = 4200' 432 TRIGONOMETRY 20° = 1200' Sup. 159° = 3540' 21° = 1260 Sup. 168° = S480, COM. COM. / SIN. COS. TAN. COT. ARC. OF ARC. / SIN. COS. TAN. COT. ARC. OF ARC. 0.3 0.9 0.3 0.3 1.2 0.3 0.9 0.3 0.3 1.2 4202 3969 6397 2.74748 4907 2173 60 6837 3368 8386 2.60509 6652 0428 60 1 4229 3959 6430 2.74499 4936 2144 59 1 5864 3348 8420 2.60283 6681 0399 59 2 4257 3949 6463 2.74251 4966 2116 58 2 5891 3337 8463 2.60057 6710 0369 58 3 4284 3939 6496 2.74004 4994 2086 57 3 5918 3327 8487 2.59831 6739 0340 57 4 4311 3929 6529 2.73766 5023 2057 56 4 6945 3316 8520 2.69606 6768 0311 56 5 4339 3919 6562 2.73609 5052 2028 65 5 6973 3306 8553 2.69381 6797 0282 55 6 4366 3909 6595 2.73263 6081 1998 54 6 6000 3295 8587 2.69156 6826 0253 54 7 4393 3899 6628 2.73017 5110 1969 63 7 6027 3285 8620 2.58932 6856 0224 63 8 4421 3889 6661 2.72771 6139 1940 62 8 6054 3274 8654 2.58708 6886 0195 52 9 4448 3879 6694 2.72526 5168 1911 61 9 6081 3264 8687 2.58484 6914 0166 51 10 4475 3869 6727 2.72281 5197 1882 60 10 6108 3253 8721 2.58261 6943 0137 50 11 4503 3859 6760 2.72036 5227 1863 49 11 6135 3243 8754 2.58038 6972 0108 49 13 4530 3849 6793 2.71792 5256 1824 48 12 6162 3232 8787 2.57816 7001 0079 48 13 4557 3839 6826 2.71548 5286 1796 47 13 6190 3222 8821 2.67593 7030 0050 47 14 4584 3829 6859 2.71305 6314 1766 46 14 6217 3211 8864 2.57371 7059 0020 1.1 9991 46 15 4612 3819 6892 2.71062 5343 1737 45 16 6244 3201 8888 2.57150 7088 45 16 4639 3809 6925 2.70819 5372 1708 44 16 6271 3190 8921 2.56928 7117 9962 44 17 4666 3799 6958 2.70577 5401 1678 43 17 P298 3180 8966 2.56707 7146 9933 43 18 4694 3789 6991 2.70335 5430 1649 42 18 6325 3169 8988 2.56487 7175 9904 42 19 4721 3779 7024 2.70094 6469 1620 41 19 6352 3159 9022 2.56266 7206 9875 41 20 4748 3769 7057 2.69853 6488 1691 40 20 6379 3148 9055 2.56046 7234 9846 40 21 4775 3759 7090 2.69612 6517 1562 39 21 6406 Sl37 9089 2.55827 7263 9817 39 22 4803 3748 7123 2.69371 6546 1533 38 22 6433 3127 9122 2.55608 7292 9788 38 23 4830 3738 7157 2.60131 5576 1504 37 23 6461 3116 9156 2.65389 7321 9759 37 24 4857 3728 7190 2.68892 5605 1475 36 24 6488 3106 9190 2.66170 7350 9730 36 25 4884 3718 7223 2.68653 5634 1446 35 25 6616 3095 9223 2.54952 7379 9700 35 26 4912 3708 7256 2.68414 5663 1417 34 26 6542 3084 9257 2.64734 7408 9671 34 27 4939 3698 7289 2.68176 5692 1388 33 27 6569 3074 9290 2.54516 7437 9642 33 28 4966 3688 7322 2.67937 5721 1359 32 28 6596 3063 9324 2.54299 7466 9613 32 29 4993 3677 7355 2.67700 5750 1329 31 29 6623 3052 9357 2.54082 7496 9584 31 30 5021 3667 7388 2.67462 5779 1300 30 30 6660 3042 9391 2.53865 7625 9555 30 0.3 0.9 0.3 0.3 1.2 0.3 0.9 0.3 0.3 1.1 31 5048 3657 7422 2.67225 5808 1271 29 31 6677 3031 9426 2.53648 7554 9526 29 32 5075 3647 7455 2.66989 5837 1242 28 32 6704 3020 9468 2.63432 7683 9497 28 33 5102 3637 7488 2.66752 5866 1213 27 33 6731 3010 9492 2.63217 7612 9468 27 34 5130 3626 7521 2.66516 5896 1184 26 34 6758 2999 9626 2.63001 7641 9439 26 35 5157 3616 7554 2.66281 5925 1155 25 35 6785 2988 9659 2.52786 7670 9410 25 36 5184 3606 7588 2.66046 5954 1126 24 36 6812 2978 9593 2.62571 7699 9380 24 37 5211 3596 7621 2.65811 5983 1097 23 37 6839 2967 9626 2.52367 7728 9351 23 38 5239 3585 7654 2.65576 6012 1068 22 38 6867 2966 9660 2.52142 7767 9322 22, 39 5266 3575 7687 2.65342 6041 1039 21 39 6894 2946 9694 2.51929 7786 9293 21 40 5293 3565 7720 2.65109 6070 1009 20 40 6921 2935 9727 2.51716 7816 9264 20 41 5320 3555 7754 2.64875 6099 0980 19 41 6948 2924 9761 2.51502 7845 9235 19 42 5347 3544 7787 2.64642 6128 0951 18 42 6975 2913 9795 2.61289 7874 9206 18 43 5375 3534 7820 2.64410 6157 0922 17 43 7002 2903 9829 2.61076 7903 9177 17 44 5402 3524 7853 2.64177 6186 0893 16 44 7029 2892 9862 2.60864 7932 9148 16 45 5429 3514 7887 2.63945 6216 0864 15 45 7056 2881 9896 2.60662 7961 9119 15 46 5456 3503 7920 2.63714 6246 0836 14 46 7083 2870 9930 2.50440 7990 9090 14 47 5483 3493 7953 2.63483 6274 0806 13 47 7110 2859 9963 2.50229 8019 9060 13 48 5511 3483 7986 2.63252 6303 0777 12 48 7137 2849 9997 0.4 0031 2.50018 8048 9031 12 49 5538 3472 8020 2.63021 6332 0748 11 49 7164 2838 2.49807 8077 9002 11 60 5565 3462 8053 2.62791 6361 0719 10 50 7191 2827 0066 2.49597 8106 8973 10 51 5592 3452 8086 2.62561 6390 0689 9 51 7218 2816 0098 2.49386 8135 8944 9 52 5619 3441 8120 2.62332 6419 0660 8 62 7246 2805 0132 2.49177 8164 8915 8 63 5647 3431 8153 2.62103 6448 0631 7 63 7272 2794 0166 2.48967 8194 8886 7 64 5674 3420 8l86 2.61874 6477 0602 6 54 7299 2784 0200 2.48758 8223 8857 6 65 5701 3410 8220 2.61646 6506 0573 5 56 7326 2773 0234 2.48549 8252 8828 5 56 5728 3400 8253 2.61418 6536 0544 4 66 7363 2762 0267 2.48340 8281 8799 4 57 5755 3389 8286 2.61190 6566 0615 3 57 7380 2751 0301 2.48132 8310 8770 3 68 5782 3379 8320 2.60963 6594 0486 2 58 7407 2740 0336 2.47924 8339 8741 2 59 5810 6368 8353 2.60736 6623 0457 1 59 7434 2729 0369 2.47716 8368 8711 1 60 5837 3358 8386 2.60509 6662 0428 60 7461 2718 0403 2.47509 8397 8682 COS. SIN. COT. TAN. COM. OF ARC. / COS. SIN. COT. TAN. COM. OF ARC. / ' Iahc. ARC. Sup. 110° = caoc 69° I 4140' Sup. 111° = ( 68° = 4080' EXAMPLES 433 22° = 1320' Sup. 157° = 9420' 23° = 1380' Sup. 156° = 9360' Sup. 112° = 6720' 67° = 4020' Sup. 113° = 6780' 66° = 3960' 434 TRIGONOMETRY 24° = 1440' Slip. 156° = 9300' 26° = IBOO' Sup. 154° = 9240* COM. COM. ' SIN. COS. TAN. COT. ARC. OF ARC. / SIN. COS. TAN. COT. ARC. OF ARC. 0.4 0.9 0.4 0.4 1.1 0.4 0.9 0.4 0.4 1.1 0674 1355 4523 2.24604 1888 5192 60 2262 0631 6631 2.14461 3633 3446 60 1 0700 1343 4558 2.24428 1917 5163 69 1 2288 0618 6666 2.14288 3662 3417 59 2 0727 1331 4593 2.24252 1946 5134 58 2 2316 0606 6702 2.14126 3691 3388 58 3 0753 1319 4627 2.24077 1975 5104 57 3 2341 0594 6737 2.13963 3720 3359 67 4 0780 1307 4662 2.23902 2004 5075 66 4 2367 0582 6773 2.13801 3760 3330 56 6 0806 1295 4697 2.23727 2033 5046 66 6 2394 0569 6808 2.13639 3779 3301 55 6 0833 1283 4732 2.23553 2062 5017 54 6 2420 0567 6843 2.13477 3808 3272 54 7 0860 1272 4767 2.23378 2091 4988 53 7 2446 0545 6879 2.13316 3837 3243 63 8 0886 1260 4802 2.23204 2121 4959 52 8 2473 0632 6914 2.13154 3866 3214 62 9 0913 1248 4837 2.23030 2150 4930 51 9 2499 0620 6950 2.12993 3895 3186 61 10 0939 1236 4872 2.22857 2179 4901 60 10 2625 0607 6985 2.12832 3924 3155 50 11 0966 1224 4907 2.22683 2208 4872 49 11 2552 0495 7021 2.12671 3953 3126 49 12 0992 1212 4942 2.22510 2237 4843 48 12 2678 0483 7056 2.12511 3982 3097 48 13 1019 1200 4977 2.22337 2266 4814 47 13 2604 0470 7092 2.12350 4011 3068 47 14 1045 1188 5012 2.22164 2295 4784 46 14 2631 04S8 7128 2.12190 4040 3039 46 15 1072 1176 5047 2.21992 2324 4755 45 15 2657 0446 7163 2.12030 4070 3010 45 16 1098 1164 5082 2.21819 2353 4726 44 16 2683 0433 7199 2.11871 4099 2981 44 17 1125 1152 5117 2.21647 2382 4697 43 17 2709 0421 7234 2.11711 4128 2952 43 18 1161 1140 5152 2.21475 2411 4668 42 IS 2736 0408 7270 2.11552 4157 2923 42 19 1178 1128 5187 2.21304 2441 4639 41 19 2762 0396 7306 2.11392 4186 2894 41 20 1204 1116 5222 2.21132 2470 4610 40 20 2788 0383 7341 2.11233 4215 2866 40 21 1231 1104 5257 2.20961 2499 4581 39 21 2816 0371 7377 2.11075 4244 2836 39 22 1257 1092 5292 2.20790 2528 4552 38 22 2844 0358 7412 2.10916 4273 2806 38 23 1284 1080 5327 2.20619 2557 4523 37 23 2867 0346 7448 2.10768 4302 2777 37 24 1310 1068 5362 2.20449 2586 4494 36 24 2894 0334 7483 2.10600 4331 2748 36 25 1337 1056 5397 2.20278 2615 4464 36 25 2920 0321 7519 2.10441 4360 2719 35 26 1363 1044 5432 2.20108 2644 4435 34 26 2946 0309 7556 2.10284 4389 2690 34 27 1390 1032 5467 2.19938 2673 4406 33 27 2972 0296 7590 2.10126 4419 2661 33 28 1416 1020 5502 2.19769 2702 4377 32 28 2999 0284 7626 2.09969 4448 2632 32 29 1443 1008 5538 2.19599 2731 3448 31 29 3025 0271 7662 2.09811 4477 2603 31 30 1469 0996 5573 2.19430 2761 4319 30 30 3051 0259 7698 2.09664 4506 2674 30 0.4 0.9 0.4 0.4 1.1 0.4 0.9 0.4 0.4 1.1 31 1496 0984 5608 2.19261 2790 4290 29 31 3077 0246 7733 2.09498 4535 2646 29 32 1522 0972 5643 2.19092 2819 4261 28 32 3104 0233 7769 2.09341 4664 2616 28 33 1549 0960 5678 2.18923 2848 4232 27 33 3130 0221 7805 2.09184 4593 2486 27 34 1575 0948 5713 2.18755 2877 4203 26 34 3166 0208 7840 2.09028 4622 2457 26 35 1602 0936 5748 2.18687 2906 4174 25 35 3182 0196 7876 2.08872 4661 2428 25 36 1628 0924 5784 2.18419 2935 4144 24 36 3209 0183 7912 2.08716 4680 2399 24 37 1655 0911 5819 2.18251 2964 4116 23 37 3235 0171 7948 2.08660 4709 2370 23 38 1681 0899 5854 2.18084 2993 4086 22 38 3261 0158 7984 2.08406 4739 2341 22 39 1707 0887 5889 2.17916 3022 4057 21 39 3287 0146 8019 2.08250 4768 2312 21 40 1734 0875 5924 2.17749 3051 4028 20 40 3313 0133 8056 2.08094 4797 2283 20 41 1760 0863 5960 2.17582 3080 3999 19 41 3340 0120 8091 2.07939 4826 2254 19 42 1787 0851 5995 2.17416 3110 3970 18 42 3366 0108 8127 2.07785 4855 2225 18 43 1813 0839 6030 2.17249 3139 3941 17 43 3392 0095 8163 2.07630 4884 2196 17 44 1840 0826 6065 2.17083 3168 3912 16 44 3418 0082 8198 2.07476 4913 2166 16 45 1866 0814 6101 2.16917 3197 3883 15 46 3445 0070 8234 2.07321 4942 2137 16 46 1892 0802 6136 2.16751 3226 3854 14 46 3471 0057 8270 2.07167 4971 2108 14 47 1919 0790 6171 2.16585 3255 3825 13 47 3497 0045 8306 2.07014 5000 2079 13 48 1945 0778 6206 2.16420 3284 3795 12 48 3623 0032 8342 2.06860 5029 2050 12 49 1972 0766 6242 2.16255 3313 3766 11 49 3649 0019 8378 2.06706 6059 2021 11 60 1998 0753 6277 2.16090 3342 3737 10 50 3575 0007 0.8 9994 8414 2.06653 6088 1992 10 51 2024 0741 6312 2.15925 3371 3708 9 51 3602 8450 2.06400 6117 1963 9 52 2051 0729 6348 2.15760 3400 3679 8 52 3628 9981 8486 2.06247 6146 1934 8 53 2077 0717 6383 2.15596 3430 3660 7 53 3654 9968 8521 2.06094 6175 1905 7 54 2104 0704 6418 2.15432 3459 3621 6 54 3680 9956 8557 2.05942 6204 1876 6 55 2130 0692 6454 2.15268 3488 3592 5 55 3706 9943 8593 2.05790 6233 1846 5 66 2156 0680 6489 2.15104 3517 3563 4 66 3732 9930 8629 2.05637 5262 1817 4 57 2183 0668 6525 2.14940 3546 3634 3 57 3759 9918 8665 2.05486 6291 1788 3 58 2209 0655 6560 2.14777 3575 3505 2 58 3785 9905 8701 2.05333 6320 1759 2 59 2235 0643 6595 2.14614 3604 3476 1 59 3811 9892 8737 2.05182 5349 1730 1 60 2262 0631 6631 2.14451 3633 3446 60 3837 9879 8773 2.05030 5379 1701 COS. BIN. COT. TAN. COM. OF ARC. ARC. ' COS. SIN. COT. TAN. COM. OF ARC. ARC. 1 Sup. 114° = 684C 66° =: 3900' Sup. 116° = 6900' 64° — 3840' 26° = ISM" EXAMPLES Sup. 163° = 9180' 27° = leW ./si 435 Sup. 182° = 9120' Sup.U6° = 6960' 63° = 3780' Sup. 117° = 7020' 62° = 3720' 436 TRIGONOMETRY 28° = 1680' Sup. 161° = 9060' 29° = 1740' Sup. 160° = 9000' COM. COM. ' SIN. COS. TAN COT. ARC. OF ARC. SIN. COS TAN. COT. ARC OF ARC. 0.4 0.8 0.5 f 0.4 1.0 0.4 0.8 0.6 0.6 1.0 6947 8295 3171 1.88073 8869 8210 60 8481 7462 5431 1.80406 0614 6465 60 1 6973 8281 3208 1.87941 8898 8181, 59 1 8506 7448 5469 1.80281 0644 6436 69 2 6999 8267 3246 1.87809 8927 8152 58 2 8532 7434 5507 1.80168 0673 6407 68 3 7024 8254 3283 1.87677 8956 8123 57 3 8557 7420 5545 1.80034 0702 ■6378; 57 4 7050 8240 3320 1.87546 8986 8094 66 4 8583 7405 5583 1.79911 0731 6349 66 6 7076 8226 3358 1.87415 9015 8065 65 6 8608 7391 5621 1.79788 0760 6320 66 6 ,7M1 8213 3395 1.87283 9044 8036 64 6 8634 7377 5659 1.79666 0789 6291 64 7 7127 8199 3432 1.87152 9073 8007 53 7 8659 7363 5697 1.79542 0818 6261 63 8 7152 8185 3470 1.87021 9102 7978 52 8 8684 7349 5736 1.79419 0847 6232 52 9 7178 8172 3507 1.86891 9131 7949 51 9 8710 7335 5774 1.79296 0876 6203 51 10 7204 8158 3545 1.86760 9160 7919 50 10 8735 7321 5812 1.79174 0905 6174 50 11 7229 8144 3582 1.86630 9189 7890 49 11 8761 7306 5850 1.79051 0934 6145 49 12 7255 8130 3620 1.86499 9218 7861 48 12 8786 7292 5888 1.78929 0964 6116 48 13 7281 8117 3657 1.86369 9247 7832 47 13 8811 7278 5926 1.78807 0993 6087 47 14 7306 8103 3694 1.86239 9276 7803 46 14 8837 7264 5964 1.78685 1022 6058 46 IS 7332 8089 3732 1.86109 9306 7774 45 15 8862 7250 6003 1.78663 1051 6029 45 16 7358 8075 3769 1.85979 9335 7745 44 16 8887 7235 6041 1.78441 1080 6000 44 17 7383 8062 3807 1.85850 9364 7716 43 17 8913 7221 6079 1.78319 1109 5971 43 18 7409 8048 3844 1.85720 9393 7687 42 18 8938 7207 6117 1.78198 1138 5941 42 19 7434 8034 3882 1.85591 9422 7658 41 19 8964 7193 6156 1.78077 1167 5912 41 20 7460 8020 3920 1.85462 9451 7629 40 20 8989 7178 6194 1.77965 1196 5883 40 21 7486 8006 3957 1.85333 9480 7599 39 21 9014 7164 6232 1.77834 1225 5854 39 22 7511 7993 3995 1.85204 9509 7570 38 22 9040 7150 6270 1.77713 1264 5825 38 23 7537 7979 4032 1.85075 9538 7541 37 23 9065 7136 6309 1.77592 1284 5796 37 24 7562 7965 4070 1.84946 9567 7512 36 24 9090 7121 6347 1.77471 1313 6767 36 25 7588 7951 4107 1.84818 9596 7483 35 25 9116 7107 6386 1.77351 1342 6738 35 26 7614 7937 4145 1.84689 9625 7454 34 26 9141 7093 6424 1.77230 1371 6709 34 27 7639 7923 4183 1.84561 9655 7426 33 27 9166 7079 6462 1.77110 1400 5680 33 28 7665 7909 4220 1.84433 9684 7396 32 28 9192 7064 6600 1.76990 1429 5651 32 29 7690 7896 4258 1.84305 9713 7367 31 29 9217 7060 6639 1.76869 1458 5621 31 30 7716 7882 4296 1.84177 9742 7338 30 30 9242 7036 6677 1.76749 1487 5592 30 0.4 0.8 0.6 0.4 1.0 0.4 0.8 0.6 0.6 1.0 31 7741 7868 4333 1.84049 9771 7309 29 31 9268 7021 6616 1.76629 1616 5563 29 32 7767 7854 4371 1.83922 9800 7280 28 32 9293 7007 6654 1.76610 1645 6534 28 33 7793 7840 4409 1.83794 9829 7250 27 33 931S 6993 6693 1.76390 1574 6506 27 34 7818 7826 4446^1.83667 9858 7221 26 34 9344 6978 6731 1.76271 1604 6476 26 35 7844 7812 4484:1.83540 9887 7192 25 36 9369 6964 6770 1.76151 1633 6447 25 36 7869 7798 4522 1.83413 9916 7163 24 36 9394 6949 6808 1.76032 1662 5418 24 37 7895 7784 4560 1.83286 9945 7134 23 37 9419 6935 6846 1.76913 1691 5389 23 38 7920 7770 4597 1.83159 9975 0.5 0004 7105 22 38 9446 6921 6886 1.76794 1720 6360 22 39 7946 77S6 4635 1.83033 7076 21 39 9470 6906 6923 1.75676 1749 6331 21 40 7971 7743 4673 1.82906 0033 7047 20 40 9495 6892 6962 1.75566 1778 5301 20 41 7997 7729 4711 1.82780 0062 7018 19 41 9521 6878 7000 1.76437 1807 5272 19 42 8022 7715 4748 1.82654 0091 6989 18 42 9546 6863 7039 1.76319 1836 5243 18 43 8048 7701 478611.82528 0120 6960 17 43 9571 6849 7078 1.75200 1865 5214, 17 44 8073 7687 4824 1.82402 0149 6930 16 44 9596 6834 7116 1.76082 1894 6186 16 45 8099 7673 4862 1 1.82276 0178 6901 15 45 9622 6820 7165 1.74964 1923 6166 15 46 8124 7659 4900 1.82150 0207 6872 14 46 9647 6805 7193 1.74846 1953 5127 14 47 8150 7645 4938 1.82025 0236 6843 13 47 9672 6791 7232 1.74728 1982 5098 13 48 8175 7631 4975il.81899 0265 6814 12 48 9697 6777 7271 1.74610 2011 5069 12 49 8201 7617 5013 1.81774 0295 6785 11 49 9723 6762 7309 1.74492 2040 5040 11 SO 8226 7603 5051 1.81649 0324 6756 10 60 9748 6748 7348 1.74375 2069 5011 10 51 8252 7589 5089 1.81524 0353 6727 9 51 9773 6733 7386 1.74257 2098 4982 9 52 8277 7575 5127 1.81399 0382 6698 8 52 9798 6719 7425 1.74140 2127 4952 8 53 8303 7560 5165 1.81274 0411 6669 7 53 9824 6704 7464 1.74022 2156 4923 7 54 18328 7546 5203 1.81150 0440 6640 6 54 9849 6690 7503 1.73905 2185 4894 6 5S 8354 7532 5241 1.81025 0469 6610 6 55 9874 6675 7541 1.73788 2214 4865 5 56 8379 7518 5279 1.80901 0498 6.581 4 56 9899 6661 7680 1.73671 2243 4836 4 57 8405 7504 5317 1.80777 0527 6552 3 57 9924 6646 7619 1.73555 2273 4807 3 58 8430 7490 5355 1.80653 0556 6523 2 58 9950 5632 7667 1.73438 2302 4778 2 59 8456 7476 5393 1.80529 0585 6494 1 59 9975 0.5 DOOO 6617 7696 1.73321 2331 4749 1 60 8481 7462 5431 1.80405 0614 6465 60 6603 7736 1.73205 2360 4720 COS. SIN. COT. TAN. COM. OF ARC. ARC. / COS. SIN. COT. TAN. COM. OF ARC. ARC. ' Sup. 118° = 7080' 61° = 3660' Sup. 119° = 7140' 60°=: EXAMPLES 437 30° = 1800' Sup. 149° = 8940' 31° = = 1860 ' Sup. 148° = 8880' COM.- COM. f SIN. COS. TAN. COT. ABC. OF ARC. SIN. COS. TAN. COT. ARC. OF ARC. 0.5 0.8 0.5 0.5 1.0 0.5 0.8 0.6 0.5 1.0 0^ 0000 0025 ^603 7735 1.73205 2360 4720 60 1504 5717 0086 1.66428 4105 2974 60 l"" 6588 7774 1.73089 2389 4691 59 1 1529 6702 0126 1.66318 4134 2945 59 ■ 2 0050 6573 7813 1.72973 2418 4662 58 2 1554 5687 0165 1.66209 4163 2916 58 3 0076 6559 7851 1.72857 2447 4632 57 3 1579 567^ 0205 1.66099 4192 2887 57 4 0101 6544 7890 1.72741 2476 4603 56 4 1604 5657 0245 1.65990 4222 2858 56 5 0126 6530 7929 1.72625 2505 4574 55 5 1628 5642 0284 1.65881 4251 2829 55 6 0151 6515 7968 1.72509 2534 4545 54 6 1653 5627 0324 1.65772 4280 2800 54 7 0176 6501 8007 1.72393 2563 4516 53 7 1678 5612 0364 1.65663 4309 2771 53 8 0201 6486 8046 1.72278 2593 4487 52 8 1703 5597 0403 1.65554 4338 2742 52 9 0227 6471 8085 1.72163 2622 4458 51 9 1728 5582 0443 1.65445 4367 2713 51 ).0 0252 6457 8124 1.72047 2651 4429 50 10 1753 5567 0483 1.65337 4396 2683 50 11 0277 6442 8162 1.71932 2680 4400 49 11 1778 5551 0522 1.65228 4425 2654 49 12 0302 6427 8201 1.71817 2709 4371 48 12 1803 5536 0562 1.65120 4454 2625 48 13 0327 6413 8240 1.71702 2738 4342 47 13 1828 5521 0602 1.65011 4483 2596 47 14 0352 6398 8279 1.71688 2767 4312 46 14 1852 5.506 0642 1.64903 4512 2567 46 15 0377 6384 8318 1.71473 2796 4283 45 15 1877 5491 0681 1.64795 4541 2538 45 16 0403 6369 8357 1.71358 2825 4254 44 16 1902 5476 0721 1.64687 4571 2509 44 17 0428 6354 8396 1.71244 2854 4225 43 17 1927 5461 0761 1.64579 4600 2480 43 18 0453 6340 8435 1.71129 2883 4196 42 18 1952 5446 0801 1.64471 4629 2451 42 19 0478 6325 8474 1.71015 2913 4167 41 19 1977 5431 0841 1.64363 4658 2422 41 20 0503 6310 8513 1.70901 2942 4138 40 20 2002 5416 0881 1.64256 4687 2393 40 21 0528 6295 8552 1.70787 2971 4109 39 21 2026 5400 0921 1.64148 4716 2364 39 22 0553 6281 8591 1.70673 3000 4080 38 22 2051 5385 0960 1.64041 4745 2334 38 23 0578 6266 8631 1.70560 3029 4051 37 23 2076 5370 1000 1.63934 4774 2305 37 24 0603 6251 8670 1.70446 3058 4022 36 24 2101 5355 1040 1.63826 4803 2276 36 25 0628 6237 8709 1.70332 3087 3992 35 25 2126 5340 1080 1.63719 4832 2247 35 26 0654 6222 8748 1.70219 3116 3963 34 26 2151 5325 1120 1.63612 4861 2218 34 27 0679 6207 8787 1.70106 3145 3934 33 27 2175 5310 1160 1.63505 4891 2189 33 28 0704 6192 8826 1.69992 3174 3905 32 28 2200 5294 1200 1.63398 4920 2160 32 29 0729 6178 8865 1.69879 3203 3876 31 29 2225 5279 1240 1.63292 4949 2131 31 30 0754 6163 8904 1.69766 3232 3847 30 30 2250 5264 1280 1.63185 4978 2102 30 0.5 0.8 0.5 0.5 1.0 0.5 0.8 0.6 O.B 1.0 31 0779 6148 8944 1.69653 3262 3818 29 31 2275 5229 1320 1.63079 5007 2073 29 32 0804 6133 8983 1.69541 3291 3789 28 32 2299 5234 1360 1.62972 5036 2044 28 33 0829 6119 9022 1.69428 3320 3760 27 33 2324 5218 1400 1 62866 5065 2014 27 34 0854 6104 9061 1.69315 3349 3731 26 34 2349 5203 1440 1.62760 5094 1985 26 35 0879 6089 9101 1.69203 3378 3702 25 35 2374 5188 1480 1.62654 5123 1956 25 36 0904 6074 9140 1.69091 3407 3673 24 36 2399 5173 1520 1.62548 5152 1927 24 37 0929 6059 9179 1.68979 3436 3643 23 37 2423 5157 1561 1.62442 5181 1898 23 38 0954 6045 9218 1.68866 3465 3614 22 38 2448 5142 1601 1.62336 5211 1869 22 ,39 0979 6030 9258 1.68754 3494 3585 21 39 2473 5127 1641 1.62230 5240 1840 21 40 1004 6015 9297 1.68643 3523 3556 20 40 2498 5112 1681 1.62125 5269 1811 20 41 1029 6000 9336 1.68531 3552 3527 19 41 2522 5096 1721 1.62019 5298 1782 19 42 1054 5985 9376 1.68419 3582 3498 18 42 2547 5081 1761 1.61914 5327 1753 18 43 1079 5970 9415 1.68308 3611 3469 17 43 2572 5066 1801 1.61808 5356 1724 17 44 1104 5956 9454 1.68196 3640 3440 16 44 2597 5051 1842 1.61703 5385 1694 16 45 1129 5941 9494 1.68085 3669 3411 15 45 2621 5035 1882 1.61598 5414 1665 15 46 1154 5926 9533 1.67974 3698 3382 14 46 2646 5020 1922 1.61493 5443 1636 14 47 1179 5911 9573 1.67863 3727 3353 13 47 2671 5005 1962 1.61388 5472 1607 13 48 1204 5896 9612 1.67752 3756 3323 12 48 2696 4989 2003 1.61283 5501 1578 12 49 1229 5881 9651 1.67641 3785 3294 11 49 2720 4974 2043 1.61179 5531 1549 11 50 1254 5866 9691 1.67530 3814 3265 10 50 2745 4959 2083 1.61074 5560 1520 10 51 1279 5851 9730 1.67419 3843 3236 9 51 2770 4943 2124 1.60970 5589 1491 9 52 1304 5836 9770 1.67309 3872 3207 8 52 2794 4928 2164 1.60865 5618 1462 8 53 1329 5821 9809 1.67198 3902 3178 7 53 2819 4913 2204 1.60761 5647 1433 7 54 1354 5806 9849 1.67088 3931 3149 6 54 2844 4897 2245 1.60657 5676 1404 6 55 1379 5792 9888 1.66978 3960 3120 5 55 2869 4882 2285 1.60553 5705 1374 5 56 1404 5777 9928 1.66867 3989 3091 4 56 2893 4866 2325 1.60449 5734 1345 4 57 1429 5762 9967 0.6 0007 1.66757 4018 3062 3 57 2918 4851 2366 1.60345 5763 1316 3 58 1454 5747 1.66647 4047 3033 2 58 2943 4836 2406 1.60241 5792 1287 2 59 1479 5732 0046 1.66538 4076 3003 1 59 2967 4820 2446 1.60137 .5821 1258 1 60 1504 5717 0086 1.66428 4105 2974 60 2992 4805 2487 1.60033 5850 1229 COM. COM. COS. SIN. COT. TAN. OF ABC. ABC. 1 COS. SIN. COT. TAN. OF ABC. ARC. f Sup. 120° = 7200' 59° = 3540' Sup. 121° = 726U' 58°: 438 TRIGONOMETRY 82° = 1920" Sup. 147° = 8820' 33° = = 1980* Sup. 146° = 8760" COM. COM. / SIN. COS. TAN. COT. ARC. OF AHC. SIN. 0.6 COS. TAN. COT. ARC. OF AHC. 0.6 0.8 0.6 O.S 1.0 0.8 0.6 0.6 0.9 2992 4805 2487 1.60033 5850 1229 60 4464 3867 4941 1.63986 7696 9484 60 1 3017 4789 2527 1.59930 5880 1200 59 1 4488 3851 4982 1.63888 7625 9455 59 2 3041 4774 2568 1 .59826 5909 1171 68 2 4513 3836 5023 1.53791 7654 9426 58 3 3066 4759 2608 1.59723 6938 1142 67 3 4637 3819 5065 1.53693 7683 9396 57 4 3091 4743 2649 1.59620 5967 1113 66 4 4661 3804 5106 1.53595 7712 9367 56 S 3115 4728 2689 1.59517 6996 1084 55 5 4586 3788 5148 1.53497 7741 9338 55 6 3140 4712 2730 1.59414 6025 1066 54 6 4610 3772 5189 1.53400 7770 9309 54 7 3164 4697 2770 1.59311 6054 1025 53 7 4635 3766 6231 1.53302 7799 9280 53 8 3189 4681 2811 1.59208 6083 0996 52 8 4659 3740 6272 1.53205 7829 9251 62 9 3214 4666 2852 1.59105 6112 0967 51 9 4683 3724 5314 1.63107 7858 9222 61 10 3238 4650 2892 1.59002 6141 0938 50 10 4708 3708 5365 1.53010 7887 9193 60 11 3263 4635 2933 1.68900 6170 0909 49 11 4732 3692 6397 1.52913 7916 9164 49 12 3288 4619 2973 1.68797 6200 0880 48 12 4756 3676 6438 1.6281« 7945 9135 48 13 3312 4604 3014 1.58696 6229 0851 47 13 4781 3660 5480 1.52719 7974 9106 47 14 3337 4588 3055 1.58693 6258 0822 46 14 4805 3646 5621 1.52622 8003 9076 46 15 3361 4573 3095 1.68490 6287 0793 45 15 4829 3629 5663 1.52525 8032 9047 45 16 3386 4557 3136 1.58388 6316 0764 44 16 4854 3613 5604 1.52429 8061 9018 44 17 3411 4542 3177 1.58286 6345 0736 43 17 4878 3597 6646 1.52332 8090 8989 43 18 3435 4526 3217 1.68184 6374 0706 42 18 4902 3581 6688 1.52236 8119 8960 42 19 3460 4511 3258 1.58083 6403 0676 41 19 4927 3565 5729 1.52139 8] 49 8931 41 20 3484 4495 3299 1.57981 6432 0647 40 20 4961 3549 6771 1.520*3 8178 8902 40 21 3509 4480 3340 1.57879 6461 0618 39 21 4975 3533 6813 1.51946 8207 8873 39 22 3534 4464 3380 1.57778 6490 0589 38 22 4999 3517 5854 1.51850 8236 8844 38 23 35.58 4448 3421 1.57676 6520 0560 37 23 5024 3601 5896 1.51754 82t)5 8815 37 .24 3583 4433 3462 1.67675 6549 0531 36 24 5048 3485 5938 1.61658 8294 8786 36 25 3607 4417 3503 1.S7474 6578 0502 35 25 5072 3469 5980 1.51562 8323 8756 35 26 3632 4402 3544 1.67372 6607 0473 34 26 5097 3453 6021 1.51466 8362 8727 34 27 3656 4386 3584 1.57271 6636 0444 33 27 5121 3437 6063 1.61370 8381 8698 33 28 3681 4370 3625 1.57170 6665 0415 32 28 6145 3421 6105 1.61276 8410 8669 32 29 3705 4355 3666 1.57069 6694 0385 31 29 6169 3405 6147 1.61179 8439 8640 31 30 3730 4339 3707 1^6969 67£3 0356 30 30 6194 3389 6189 1.51084 8468 8611 30 0.5 O.S 0.6 0.5 1.0 0.6 0.8 0.6 0.5 0.9 31 3754 4324 3748 1.66868 6752 0327 29 31 6218 3373 6230 1.50988 8498 8582 29 32 3779 4308 3789 1.56767 6781 0298 28 32 6242 3356 6272 1.50893 8627 8553 28 33 3804 4292 3830 1.56667 6810 0269 27 33 6266 3340 6314 1.5079' 8656 8524 27 34 3828 4277 3871 1.66666 6840 0240 26 34 5291 3324 6356 1.50''0'2 8585 8495 26 35 3854 4261 3912 1.56466 6869 0211 25 36 5316 3308 6398 1.50607 8614 8466 25 36 3877 4245 3953 1.66366 6898 0182 24 36 5339 3292 6440 1.50512 8643 8437 24 37 3902 4230 3994 1.66265 6927 0153 23 37 5363 3276 6482 1.50417 8672 8407 23 38 3926 4214 4035 1.56165 6956 0124 22 38 5388 3260 6524 1.50322 8701 8378 22 39 3951 4198 4076 1.56066 6986 0095 21 39 5412 3244 6666 1.50228 8730 8349 21 40 3975 4182 4117 1.55966 7014 0065 20 40 5436 3228 6608 1.50133 8759 8320 20 41 4000 4167 4158 1.55866 7043 0036 19 41 6460 3212 6650 1.50038 8788 8291 19 42 4024 4151 4199 1.65766 7072 0007 0.9 9978 18 42 6484 3195 6692 1.49944 8818 8262 18 43 4049 4135 4240 1.55666 7101 17 43 5509 3179 6734 1.49849 8847 8233 17 44 4073 4120 4281 1.55567 7130 9949 16 44 5533 3163 6776 1.49755 8876 8204 16 45 4097 4104 4322 1.66467 7169 9920 15 45 5557 3147 6818 1.49661 8906 8175 15 46 4122 4088 4363 1.55368 7189 9891 14 46 5581 3131 6860 1.49666 8934 8146 14 47 4146 4072 4404 1.55269 7218 9862 13 47 5606 3115 6902 1.49472 8963 8117 13 48 4171 4057 4446 1.55170 7247 9833 12 48 5630 3098 6944 1.49378 8992 8087 12 49 4195 4041 4487 1.55071 7276 9804 11 49 5654 3082 6986 1.49284 9021 8058 11 50 4220 4025 4528 1.64972 7305 9775 10 50 5678 3066 7028 1.49190 9050 8029 10 51 4244 4009 4569 1.54873 7334 9746 9 51 5702 3060 7071 1.49097 9079 8000 9 52 4269 3994 4610 1.54774 7363 9716 8 52 5726 3034 7113 1.49003 9108 7971 8 53 4293 3978 4652 1.54676 7392 9687 7 53 5750 3017 7155 1.48909 9138 7942 7 54 4317 3962 4693 1.54676 7421 9658 6 54 5775 3001 7197 1.48816 9167 7913 6 55 4342 3946 4734 1.54478 7460 9629 5 55 5799 2985 7239 1.48722 9196 7884 6 56 4366 3930 4775 1.54379 7479 9600 4 56 5823 2969 7282 1.48629 9225 7855 4 67 4391 3915 4817 1.54281 7509 9571 3 57 5847 2953 7324 1.48536 9254 7826 3 58 4415 3899 4858 1.54183 7538 9542 2 58 5871 2936 7366 1.48442 9283 7797 2 59 4439 3883 4899 1.54085 7567 9513 1 59 5895 2920 7409 1.48349 9312 7767 1 60 4464 3867 4941 1.53986 7596 9484 60 5919 COS. 2904 7461 1.48266 9341 7738 COS. SIN. COT. TAN. COM. OF ARC. , SIN. COT. TAN. COM. or AHC. ' AHC. ARC. 1 Sup. 122° = 7320' 57° 3420' Sup. 123° = 7380' 66' : 3360' EXAMPLES 439 84° = 2040' Sup. 146° = 8700' 36° = 2100' Sup. 144° = 8640' COM. COM. / SIN. COS. TAN. COT. ARC. OF ARC. r SIN. COS. TAN. COT. ABC. OF ABC. 0.6 0.8 0.6 0.6 0.9 0.5 0.8 0.7 0.6 0.9 5919 2904 7451 1.48256 9341 7738 60 7358 1915 0021 1.42815 1086 5993 60 1 6943 2887 7493 1.48163 9370 7709 59 1 7381 1899 0064 1.42726 1116 5964 59 2 6968 2871 7636 1.48070 9399 7680 68 2 7405 1882 0107 1.42638 1145 5935 58 3 6992 2866 7578 1.47977 9428 7661 57 3 7429 1865 0151 1.42550 1174 5906 57 4 6016 2839 7620 1.47885 9458 7622 66 4 7453 1848 0194 1.42462 1203 5877 66 5 6040 2822 7663 1.47792 9487 7593 55 6 7477 1832 0238 1.42374 1232 5848 56 6 6064 2806 7706 1.47699 9616 7564 54 6 7501 1815 0281 1.42286 1261 5819 54 7 6088 2790 7748 1.47607 9546 7635 53 7 7524 1798 0325 1.42198 1290 6789 63 8 6112 2773 7790 1.47614 9574 7606 52 8 7548 1781 0368 1.42110 1319 6760 62 g 6136 2757 7832 1.47422 9603 7477 51 9 7572 1765 0412 1.42022 1348 5731 61 10 6160 2741 7876 1.47330 9632 7448 50 10 7596 1748 0456 1.41934 1377 5702 50 11 6184 2724 7917 1.47238 9661 7418 49 11 7619 1731 0499 1.41847 1406 5673 49 12 6208 2708 7960 1.47146 9690 7389 48 12 7643 1714 0642 1.41759 1436 5644 48 13 6232 2692 8002 1.47064 9719 7360 47 13 7667 1698 0686 1.41672 1465 6615 47 14 6256 2675 8046 1.46962 9748 7331 46 14 7691 1681 0629 1.41584 1494 5586 46 IS 6280 2659 8088 1.46870 9777 7302 46 15 7715 1664 0673 1.41497 1623 5557 45 16 6305 2643 8130 1.46778 9807 7273 44 16 7738 1647 0717 1.41409 1552 5528 44 17 6329 2626 8173 1.46686 9836 7244 43 17 7762 1631 0760 1.41322 1581 5499 43 18 6363 2610 8215 1.46595 9866 7215 42 18 7786 1614 0804 1.41235 1610 5469 42 19 6377 2593 8258 1.46503 9894 7186 41 19 7809 1697 0848 1.41148 1639 5440 41 20 6401 2577 8301 1.46411 9923 7167 40 20 7833 1580 0891 1.41061 1668 5411 40 21 6425 2561 8343 1.46320 9952 7128 39 21 7867 1663 0936 1.40974 1697 5382 39 22 6449 2544 8386 1.46229 9981 0.6 0010 7098 38 22 7881 1646 0979 1.40887 1726 5363 38 23 6473 2628 8429 1.46137 7069 37 23 7904 1530 1023 1.40800 1756 6324 37 24 6497 2611 8471 1.46046 0039 7040 36 24 7928 1613 1066 1.40714 1786 6295 36 25 6621 2495 8514 1.45966 0068 7011 36 26 7952 1496 1110 1.40627 1814 6266 35 26 6545 2478 8657 1.45864 0097 6982 34 26 7976 1479 1154 1.40640 1843 6237 34 27 6569 2462 8699 1.46773 0127 6963 33 27 7999 1462 1198 1.40464 1872 5208 33 28 6593 2446 8642 1.45682 0166 6924 32 28 8023 1445 1242 1.40367 1901 5179 32 29 6617 2429 8686 1.46592 0186 6896 31 29 8047 1428 1285 1.40281 1930 5149 31 30 6641 2413 8728 1.45501 0214 6866 30 30 8070 1412 1329 1.40196 1969 5120 30 0.5 0.8 0.6 0.6 0.9 0.6 0.8 0.7 0.6 0.9 31 6666 2396 8771 1.45410 0243 6837 29 31 8094 1396 1373 1.40109 1988 5091 29 32 6689 2380 8814 1.45320 0272 6808 28 32 8118 1378 1417 1.40022 2017 6062 28 33 6713 2363 8857 1.45229 0301 6778 27 33 8141 1361 1461 1.39936 2046 6033 27 34 6736 2347 8900 1.45139 0330 6749 26 34 8166 1344 1505 1.39860 2075 6004 26 35 6760 2330 8942 1.45048 0359 6720 25 35 8189 1327 1549 1.39764 2105 4975 25 36 6784 2314 8986 1.44968 0388 6691 24 36 8212 1310 1593 1.39679 2134 4946 24 37 6808 2297 9028 1.44868 0417 6662 23 37 8236 1293 1637 1.39593 2163 4917 23 38 6832 2281 9071 1.44778 0447 6633 22 38 8260 1276 1681 1.39507 2192 4888 22 39 6856 2264 9114 1.44688 0476 6604 21 39 8283 1269 1725 1.39421 2221 4859 21 40 6889 2248 9157 1.44598 0505 6675 20 40 8307 1242 1769 1.39336 2250 4830 20 41 6904 2231 9200 1.44508 0534 6546 19 41 8330 1225 1813 1.39260 2279 4800 19 42 6928 2214 9243 1.44418 0563 6517 18 42 8364 1208 1867 1.39165 2308 4771 18 43 6952 2198 9286 1.44329 0692 6488 17 43 8378 1191 1901 1.39079 2337 4742 17 44 6976 2181 9329 1.44239 0621 6458 16 44 8401 1174 1946 1.38994 2366 4713 16 45 7000 2165 9372 1.44149 0650 6429 15 46 8426 1167 1990 1.38909 2395 4684 16 46 7024 2148 9416 1.44060 0679 6400 14 46 8449 1140 2034 1.38824 2425 4656 14 47 7047 2132 9459 1.4397C 0708 6371 13 47 8472 1123 2078 1.38738 2454 4626 13 48 7071 2U5 9502 1.43881 0737 6342 12 48 8496 1106 2122 1.38663 2483 4697 12 49 7095 2098 9645 1.43792 0766 6313 11 49 8519 1089 2166 1.38668 2612 4668 11 50 7119 2082 9688 1.43703 0796 6284 10 50 8543 1072 2211 1.38484 2541 4639 10 51 7143 2065 9631 1.43614 0825 6255 9 51 8567 1056 2255 1.38399 2570 4510 9 62 7167 2048 9675 1.43525 0864 6226 8 52 8590 1038 2299 1.38314 2599 4480 8 53 7191 2032 9718 1.43436 0883 6197 7 53 8614 1021 2344 1.38229 2628 4451 7 54 7215 2016 9761 1.43347 0912 6168 6 54 8637 1004 2388 1.38146 2657 4422 6 55 7238 1999 9804 1.43268 0941 6138 5 65 8661 0987 2432 1.38060 2686 4393 5 56 7262 1982 9847 1.43169 0970 6109 4 56 8684 0970 2477 1.37976 2715 4364 4 57 7286 1966 9891 1.43080 0999 6080 3 67 8708 0953 2521 1.37891 2745 4335 3 68 7310 1949 9934 1.42992 1028 6051 2 58 8731 0936 2565 1.37807 2774 4306 2 69 7334 1932 9977 0.7 0021 1.42903 1067 6022 1 59 8755 0919 2610 1.37722 2803 4277 1 60 7358 1916 1.42815 1086 5993 60 8779 0902 2664 1.37638 2832 4248 COM. COM. COS. SIN. COT. TAN. OF ABC. ARC. COS. SIN. COT. TAN. OF ABC. ARC. Sup. 124° = 7440' 65° = 3300' Sup. 126° = 7500' 54° 3240' 440 TRIGONOMETRY 36° = 2160' Sup. 143° = 8580' 37* = 2220' Sup. 142° = 8520' COM. COM. / SIN cos TAN. COT. ARC. OF 1 SIN cos TAN COT. AHO OF ARC. ARC- 0.6 0.8 0.7 0.6 0.9 0.6 0.7 0.7 0.6 0.9 8779 0902 2654 1.37638 2832 4248 60 0181 9864 5355 1.32704 4577 2502 60 1 8802 0885 2699 1.37554 2861 4219 69 1 0205 9846 5401 1.32624 4606 2473 59 2 8826 0867 2743 1.37470 2890 4190 58 2 0228 9829 5447 1.32544 463S 2444 58 3 8849 0850 2788 1.37386 2919 4160 57 3 0251 9811 5492 1.32464 4664 2415 57 4 8873 0833 2832 1.37302 2948 4131 56 4 0274 9793 5538 1.32384 4693 2386 56 5 8896 0816 2877 1.37218 2977 4102 55 5 0298 9776 5584 1.32304 4723 2357 55 6 8920 0799 2921 1.37134 3006 4073 54 6 0321 9758 5629 1.32224 4752 2328 54 7 8943 0782 2966 1.37050 3035 4044 53 7 0344 9741 5675 1.32144 4781 2299 53 8 8967 0765 3010 1.36967 3065 4015 52 8 0367 9723;5721 1.32064 4810 2270 62 g 8990 0748 3055 1.36883 3094 3986 51 9 0390 9706 5767 1.31984 4839 2241 51 10 9014 0730 3100 1.36800 3123 3957 50 10 0414 9688 5812 1.31904 4868 2212 50 11 9037 0713 3144 1.36716 3152 3928 49 11 0437 9671 5858 1.31825 4897 2182 49 12 9061 0696 3189 1.36633 3181 3899 48 12 0460 9653 5904 1.31746 4926 2153 48 13 9084 0679 3234 1.36549 3210 3870 47 13 0483 9635 5950 1.31666 4955 2124 47 14 9108 0662 3278 1.36466 3239 3840 46 14 0506 9618 5996 1.31586 4984 2095 46 15 9131 0644 3323 1.36383 3268 3811 45 IS 0529 9600 6042 1.31507 5013 2066 45 16 9154 0627 3368 1.36300 3297 3782 44 16 0553 9583 6088 1.31427 5043 2037 44 17 9178 0610 3413 1.36217 3326 3753 43 17 0576 9565 6134 1.31348 5072 2008 43 18 9201 0593 3457 1.36133 3355 3724 42 18 0599 9547 6180 1.31269 5101 1979 42 19 9225 0576 3502 1.36051 3384 3695 41 19 0622 9530 6226 1.31190 5130 1950 41 20 9248 0558 3547 1.35968 3414 3666 40 20 0645 9512 6272 1.31110 5159 1921 40 21 9272 0541 3592 1.35885 3443 3637 39 21 0668 9494 6318 1.31031 5188 1892 39 22 3295 0524 3637 1.35802 3472 3608 38 22 0691 9477 6364 1.30952 5217 1862 38 23 3318 0507 3681 1.35719 3501 3579 37 23 0714 9459 6410 1.30873 5246 1833 37 24 9342 0489 3726 1.35637 3530 3650 36 24 0738 9441 6456 1.30795 5275 1804 36 25 3365 0472 3771 1.35554 3559 3521 35 25 0761 9424 6502 1.30716 5304 .1775 35 26 9389 0455 3816 1.35472 3588 3491 34 26 0784 9406 6548 1.30637 5333 1746 34 27 3412 0438 3861 1.35389 3617 3462 33 27 0807 9388 6594 1.30568 5363 1717 33 28 3435 0420 3906 1.35307 3646 3433 32 28 0830 9371 6640 1.30480 5392 1688 32 29 3459 0403 3951 1.35224 3675 3404 31 29 0853 9353 6686 1.30401 5421 1659 31 30 9482 0386 3996 1.35142 3704 3375 30 30 0876 9335 6733 1.30323 5460 1630 30 0.5 0.8 0.7 0.6 0.9 0.6 0.7 0.7 0.6 0.9 31 3506 0368 4041 1.35060 3734 3346 29 31 0899 9318 6779 1.30244 6479 1601 29 32 3529 0351 4086 1.34978 3763 3317 28 32 0922 9300 6825 1.30166 5508 1572 28 33 9552 0334 4131 1.34896 3792 3288 27 33 0945 9282 6871 1.30087 5537 1542 27 34 9576 0316 4176 1.34814 3821 3259 26 34 0968 9264 6918 1.30009 5566 1513 26 35 9599 0299 4221 1.34732 3850 3230 25 35 0991 9247 6964 1.29931 5595 1484 25 36 9622 0282 4267 1.34650 3879 3201 24 36 1015 9229 7010 1.29853 5624 1455 24 37 9646 0264 4312 1.34568 3908 3171 23 37 1038 9211 7057 1.29775 5653 1426 23 38 9669 0247 4357 1.34487 3937 3142 22 38 1061 9193 7103 1.29696 5683 1397 22 39 9693 0230 4402 1.34405 3966 3113 21 39 1084 9176 7149 1.29618 5712 1368 21 40 9716 0212 4447 1.34323 3995 3084 20 40 1107 9158 7196 1.29541 5741 1339 20 41 9739 0195 4492 1.34242 4024 3055 19 41 1130 9140 7242 1.29463 5770 1310 19 42 9763 0178 4538 1.34160 4054 3026 18 42 1153 9122 7289 1.29385 5799 1281 18 43 9786 0160 4583 1.34079 4083 2997 17 43 1176 9105 7335 1.29307 5828 1252 17 44 9809 0143 4628 1.33998 4112 2968 16 44 1199 9087 7382 1.29229 5857 1222 16 45 9832 0125 4674 1.33916 4141 2939 15 45 1222 9069 7428 1.29152 5886 1193 15 46 9856 0108 4719 1.33835 4170 2910 14 46 1245 9051 7475 1.29074 5915 1164 14 47 9879 0091 4764 1.33754 4199 2881 13 47 1268 9033 7521 1.28997 5944 1135 13 48 9902 0073 4810 1.33673 4228 2851 12 48 1291 9015 7568 1.28919 5973 1106 12 49 9926 0056 4855 1.33592 4257 2822 11 49 1314 8998 7615 1.28842 B002 1077 11 50 9949 0038 4900 1.33511 4286 2793 10 50 1337 8980 7661 1.28764 3032 1048 10 51 9972 0021 4946 1.33430 4315 2764 9 51 1360 8962 7708 1.28687 3061 1019 9 52 9995 0.6 9019 0003 0.7 9986 4991 1.33349 4344 2735 8 52 1383 8944 7754 1.28610 3090 0990 8 53 5037 1.33268 4374 2706 7 53 1406 8926 7801 1.28533 3119 0961 7 54 9042 9968 5082 1.33187 4403 2677 6 54 1429 8908 7848 1.28456 3148 0932 6 55 9065 9951 5128 1.33107 4432 2648 5 55 1451 8891 7895 1.28379 3177 0903 5 56 9089 9934 5173 1.33026 4461 2619 4 56 1474 8873 7941 1.28302 3206 0873 4 57 D112 9916 5219 1.32050 4490 2590 3 57 1497 8855 7988 1.28225 3235 0844 3 58 0135 9899 5264 1.32865 4519 2561 2 58 1520 8837 8035 1.28148 3264 0815 2 59 9158 9881 5310 1.32785 4548 2531 1 59 1543 8819 8082 1.28071 3293 07S6 1 60 D181 9864 SIN. 5355 1.32704 4577 2502 60 1566 8801 8129 1.27994 3322 0757 COS. COT. TAN. COM. or ARC. f COS. RJN. COT. TAN. 30M. OF ARC. 1 ARC. 1 IHC. Sup. 126° = 7560' 53° = 3180' Sup. 127° = 7620' B2° ■■ 3120 EXAMPLES 441 38° = 2280* Sup. 141° = 8460' 39° = 2340' Sup. 140° = 8400' COM. COM. / SIN. COS. 0.7 TAN. 0.7 COT. ARC. OF ARC. SIN. COS. TAN. COT. ARC. OF ARC. 0.6 0.6 0.9 0.6 0.7 0.8 0.6 0.8 1566 8801 8129 1.27994 6322 0757 60 2932 7715 0978 1.23490 8068 9012 60 1 1589 8783 8175 1.27917 6352 0728 59 1 2956 7696 1027 1.23416 8097 8983 59 2 1612 8765 8222 1.27841 6381 0699 58 2 2977 7678 1075 1.23343 8126 8954 68 3 1635 8747 8269 1.27764 6410 0670 57 3 3000 7660 1123 1.23270 8155 8924 57 4 1658 8729 8316 1.27688 6439 0641 56 4 3022 7641 1171 1.23196 8184 8895 56 5 1681 8711 8363 1.27611 6468 0612 55 5 3045 7623 1220 1.23123 8213 8865 55 6 1704 8693 8410 1.27535 6497 0583 54 6 3068 7605 1268 1.23050 8242 8837 54 7 1726 3676 8457 1.27458 6526 0553 53 7 3090 7585 1316 1.22977 8271 8808 53 8 1749 8658 8504 1.27382 6555 0524 52 8 3113 7558 1364 1.22904 8301 8779 52 9 1772 8640 8551 1.27303 6584 0495 51 9 3135 7550 1413 1.22831 8330 8750 61 10 1795 8622 8599 1.27230 6613 0465 50 10 3158 7531 1461 1.22758 8369 8721 50 11 1818 8604 8645 1.27153 6642 0437 49 11 3180 7513 1510 1.22685 8388 8592 49 12 1841 8586 8592 1.27077 6672 0408 48 12 3203 7494 1558 1.22612 8417 8563 48 13 1864 8568 8739 1.27001 6701 0379 47 13 3225 7476 1606 1.22539 8446 8634 47 14 1887 8550 8786 1.26925 6730 0350 46 14 3248 7458 1665 1.22467 8475 8604 46 15 1909 8532 8834 1.26849 6759 0321 15 15 3271 7439 1703 1.22394 8504 8575 45 16 1932 8514 8881 1.26774 6788 0292 44 16 3293 7421 1752 1.22321 8533 8646 44 17 1955 8496 892S 1.25698 6817 0263 43 17 3316 7402 1800 1.22249 8562 8517 43 18 1978 8478 8975 1.25522 6846 0233 42 18 3338 7384 1849 1.22176 8591 8488 42 19 2001'8460 9022 1.26546 6875 0204 41 19 3361 7366 1898 1.22104 8620 8469 41 20 2024 8442 9070 1.26471 6904 0175 40 20 3383 7347 1946 1.22031 8650 8430 40 21 2046'8424 9117 1.26395 6933 0146 39 21 3406 7329 1995 1.21959 8679 8401 39 22 2069 8405 9164 1.25319 6952 0117 38 22 3428 7310 2044 1.21886 8708 8372 38 23 2092 8387 9212 1.25244 5992 0088 37 23 3451 7202 2092 1.21814 8737 8343 37 24 2115 8369 9259 1.28169 7021 0059 36 24 3473 7273 2141 1.21742 8766 8314 36 25 21388351 9308 1.25093 7050 0030 35 25 349G 7255 2190 1.21670 8795 8285 35 26 2160 8333 9354 1.26018 7079 0001 0.8 9972 34 26 3518 7236 2238 1.21598 8824 8265 34 27 2183 8315 9401 1.25943 7108 33 27 3540 7218 2287 1.21526 8853 8226 33 28 2206 8297 9449 1.25867 7137 9943 32 26 3563 7199 2336 1.21454 8882 8197 32 29 2229 8279 9498 1.25792 7166 9913 31 29 3585 7181 2385 1.21382 8911 8168 31 30 2251 8251 9544 1.25717 7195 9884 30 30 3608 7162 2434 1.21310 8940 8139 30 0.6 0.7 0.7 0.6 0.8 0.6 0.7 0.8 0.6 0.8 31 2274 8243 9591 1.25642 7224 9855 29 31 3630 7144 2483 1.21238 8970 8110 29 32 2297 i 8225 9639 1.25567 7253 9828 28 32 3653 7125 2531 1.21166 8999 8081 28 33 2320 8206 9686 1.25492 7282 9797 27 33 3675 7107 2580 1.21094 9028 8052 27 34 2342 8188 9734 1.25417 7311 9758 26 34 3698 7088 2629 1.21023 9057 8023 26 35 2385 8170 9781 1.25343 7341 9739 25 35 3720 7070 2678 1.20951 9086 7994 25 36 2388'8152 9829 1.25268 7370 9710 24 36 3742 7051 2727 1.20879 9115 7965 24 37 2411 8134 9877 1.25193 7399 9681 23 37 3765 7033 2776 1.20808 9144 7935 23 38 2433 8116 9924 1.25118 7428 9652 22 38 3787 7014 2825 1.20735 9173 7906 22 39 2456 8098 9972 0.8 !K)20 1.25044 7457 9623 21 39 3810 5996 2874 1.20665 9202 7877 21 40 2479 8079 1.24969 7486 9594 20 40 3832 6977 2923 1.20593 9231 7848 20 41 2502 8051 0067 1.24895 7515 9564 19 41 3854 6959 2972 1.20522 9250 7819 19 42 2524 8043 0115 1.24820 7544 9535 18 42 3877 5940 3022 1.20451 9290 7790 18 43 2547|8025 0163 1.24746 7573 9505 17 43 3899 5921 3071 1.20379 9319 7761 17 44 2570 8007 0211 1.24672 7602 9477 16 44 3922 6903 3120 1.20308 9348 7732 16 45 2592 7988 0258 1.24597 7631 9448 15 45 3944 6884 3169 1.20237 9377 7703 15 46 26157970 0306 1.24523 7661 9419 14 45 3956 6866 3218 1.20156 9406 7574 14 47 2638 7952 0354 1.24449 7690 9390 13 47 3989,6847 3268 1.20095 9435 7645 13 48 2660 7934 0402 1.24375 7719 9361 12 48 40116828 3317 1.20024 9464 7615 12 49 2683 7916 0450 1.24301 7748 9332 11 49 40336810 3365 1.19953 9493 7686 11 50 2706 7897 0498 1.24227 7777 9303 10 50 4056 6791 3415 1.19882 9522 7657 10 51 2728 7879 0546 1.24153 7806 9274' 9 51 4078'6772 3465 1.19811 9551 7528 9 52 2751 7851 0594 1.24079 7835 9244 8 52 4100 6754 3514 1.19740 9580 7499 8 53 2774 7843 0842 1.24005 7854 9215 7 53 4123 6735 3564 1.19669 9609 7470 7 54 2796 7824 0890 1.23931 7893 9186 6 52 4145 6717 3613 1.19599 9639 7441 6 55 2819 7806 0738 1.23858 7922 9157 5 55 4167 6698 3662 1.19528 9668 7412 5 56 2842 7788 0786 1.23784 7951 9128 4 56 4190 6679 3712 1.19457 9697 7383 4 57 2864 7769 0834 1.23710 7981 9099 3 57 4212 6561 3761 1.19387 9726 7354 3 58 2887 7751 0882 1.23637 8010 9070 2 58 4234 6642 3811 1.19315 9755 7325 2 59 2909 7733 0930 1.23563 8039 9041 1 59 4256 6623 3860 1.19246 9784 7295 1 60 2932 con. 7715 SIN. 0978 COT. 1.23490 8068 9012 60 4279 6604 SIN. 3910 1.19175 9813 7266 TAN. COM. OF ARC. 1 COS. COT. TAN. COM. OP ARC. / 1 1 ARC. 1 1 'Arc Sup. 128° = 7680' 61° = 3060' Sup. 129 = 7740' 60° = 3000' 442 TRIGONOMETRY 40° . 2400* Sup. 139° = 8340' 41° = 2460" Sup. 138° = 1 2 3 4 5 6 7 8 g 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0.6 4279 4301 4323 4346 4368 4390 4412 4435 4457 4479 4501 4524 4546 4568 4590 4612 4635 4657 4679 4701 4723 4746 4768 4790 4812 4834 0.7 6604 6686 6567 6548 6530 6511 6492 6473 6455 6436 6417 6398 6380 6361 6342 6323 6304 6286 6267 6248 6229 6210 6192 6173 6164 6135 0.8 3910 3960 4009 4059 4108 4158 4208 4258 4307 4357 4407 4457 4607 4566 4606 4666 4706 4756 4806 4856 4906 4956 5006 6067 5107 5157 4856 4878 4901 4923 4946 o.e 4967 4989 6011 6033 5055 6077 6099 5122 5144 5166 5188 5210 6232 5254 6276 5298 5320 5342 5364 5386 5408 5430 5462 6474 5496 5518 6640 5562 6116 6097 6078 6069 6041 0.7 6022 6003 6984 5965 6946 5927 5908 5889 5870 6851 5832 5813 5794 5776 6766 6738 6719 5699 6680 6661 5207 5267 6307 6368 5408 0.8 5458 6509 5559 5609 6660 5710 5761 5811 6862 5912 6963 6014 6064 6115 6166 6216 6267 6318 6368 6419 5642 6623 5604 5685 6666 5547 5528 5609 1.19175 1.19105 1.19035 1.18964 1.18894 1.18824 1.18764 1.18684 1.18614 1.18544 1.18474 1.18404 1.18334 1.18264 1.18194 1.18125 1.18056 1.17986 1.17916 1.17846 1.17777 1.17708 1.17638 1.17569 1.17600 1.17430 1.17361 1.17292 1.17223 1.17164 1.17085 1.17016 1.16947 1.16878 1.16809 1.16741 1.16672 1.16603 1.16535 1.16466 1.16398 0.6 9813 9842 9871 9900 9929 9959 9988 0.7 0017 0046 0075 0104 0133 0162 0191 0220 0249 0279 0308 0337 0366 0395 0424 0463 0482 0511 0540 0569 0599 0628 0657 0686 0.7 0715 0744 0773 0802 0831 0860 0889 0918 0948 0977 COM. OF ARC. 5584 6490 5606 5471 6470 6521 6672 6623 6674 6726 6776 6827 6878 6929 1.16329 1.16261 1.16192 1.16124 1.16066 1.15987 1.16919 1.15861 1.15783 1.15716 1.15647 1.15579 1.15511 1.15443 1.15376 1.15308 1.15240 1.15172 1.15104 1.15037 1006 1035 1064 1093 1122 1151 1180 1209 1238 1268 1297 1326 1355 1384 1413 1442 1471 1500 1529 1568 0.8 7266 7237 7208 7179 7150 7121 7092 7063 7034 7005- 6976 6946 6917 6888 6859 6830 6801 6772 6743 6714 6685 6656 6626 6697 6668 6639 6610 6481 6462 6423 6394 0.8 6365 6336 6306 6277 6248 6219 6190 6161 6132 6103 6074 6045 6016 5987 6957 5928 5899 6870 5841 6812 5783 6764 5725 5696 6667 6637 6608 5579 6550 5521 60 69 58 57 56 55 54 53 62 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 COM. OF ARC. arc! 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 51 52 53 54 55 56 67 58 69 0.6 6606 6628 5650 6672 6694 6716 5738 6769 6781 6803 5825 6847 5869 5891 6913 5935 5956 6978 6000 6022 6044 60 6088 6109 6131 6153 6175 6197 6218 6240 6262 0.6 6284 6306 6327 6349 6371 6393 6414 6436 6458 6480 6501 6523 6545 6566 6588 6610 6632 6653 6675 6697 6718 6740 6762 6783 6805 6827 6848 6870 6891 6913 0.7 6471 6462 6433 6414 5395 5375 6356 6337 5318 5299 5280 5261 5241 5222 5203 5184 5165 6146 6126 5107 6088 5069 5050 5030 .5011 4992 0.8 6929 6980 7031 7082 7133 7184 7236 7287 7338 7389 7441 7492 7543 7596 7646 7698 7749 7801 7862 7904 7955 8007 8059 8110 8162 8214 4973 4953 4934 4915 4896 0.7 4876 4857 4838 4818 4799 4780 4760 4741 4722 4703 4683 4664 4644 4625 4606 4586 4567 4648 4528 4609 4490 4470 4461 4431 4412 4392 4373 4353 4334 8265 8317 8369 8421 8473 0.8 8524 8576 8628 8680 8732 8784 8836 8888 8940 8992 9046 9097 9149 9201 9253 9306 9358 9410 9463 9516 1.15037 1.14970 1.14902 1.14834 1.14767 1.14699 1.14632 1.14566 1.14498 1.14430 1.14363 1.14296 1.14229 1.14162 1.14095 1.14028 1.13961 1.13894 1.13828 1.13761 1.13694 1.13627 1.13561 1.13494 1.13428 1.13361 1.13295 1.13228 1.13162 1.13096 1.13029 1.12963 1.12897 1.12831 1.12765 1.12699 9567 9620 9672 9726 9777 9830 9935 1.12633 1.12667 1.12601 1.12435 1.12369 1.12303 1.12238 1.12172 1.12106 1.12041 1.11975 1.11909 1.11844 1.11778 1.11714 1.11648 1.11682 1.11517 1.11452 1.11387 1.11321 1.11256 1.11191 1.11126 0.7 1658 1688 1617 1646 1676 1704 1733 1762 1791 1820 1849 1878 1908 1937 1966 1995 2024 2053 2082 2111 2140 2169 2198 2227 2267 2286 2318 2344 2373 2402 2431 0.7 2460 2489 2518 2547 2677 2606 2636 2664 2693 2722 2751 2780 2809 2838 2867 COM. OP ARC. 0.9 4314 0040 1.11061 COM. TAN. OF Iakc. 2897 2926 2955 2984 3013 3042 3071 3100 3129 3158 3187 3217 3246 3275 3304 0.8 5521 5492 5463 5434 6405 5376 5347 5317 5288 6259 6230 5201 5172 6143 6114 5086 5056 5027 4997 4968 4939 4910 4881 4862 4823 4794 4765 4736 4707 4678 4648 O.g 4619 4690 4561 4632 4503 4474 4445 4416 4387 4353 4328 4299 4270 4241 4212 4183 4154 4125 4096 4067 4038 4008 3979 3950 3921 3892 3863 3834 3778 60 59 58 57 66 55 54 S3 62 51 50 49 48 47 46 45 44 43 42 41 40 37 36 35 34 33 32 31 30 29 28 27 28 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 6 4 3 2 1 Sup. 130° = 7800' 49° = 2940' Sup. 131° = 7860' ilC^iSWf EXAMPLES 443 42° = 2520' 6 >up. 137° = 8220' 43° = 2580' Sup. 136° = 8160' COM. COM. / 81N. COS. TAN. COT. ARC. OI- ARC. f SIN. COS. 0.7 TAN. COT. ARC. OF ARC. 0.6 0.7 0.9 0.7 0.8 0.6 0.9 0.7 0.8 6913 4314 0040 1.11061 3304 3776 60 8200 3135 3252 1.07237 6049 2030 60 1 6935 4295 0093 1.10996 3333 3747 59 1 8221 3116 3306 1.07174 6078 2001 59 2 6956 4276 0146 1.10931 3362 3718 68 2 8242 3096 3360 1.07112 5107 1972 58 3 6978 4256 0199 1.10867 3391 3688 57 3 8264 3076 3415 1.07049 5136 1943 57 4 6999 4237 0251 1.10802 3420 3659 56 4 8285 3056 3469 1.06987 6165 1914 56 5 7021 4217 0304 1.10737 3449 3630 55 5 8306 3036 3524 1.06925 6195 1885 55 6 7043 4198 0357 1.10672 3478 3601 54 6 8327 3016 3578 1.06862 6224 1856 64 7 7064 4178 0410 1.10607 3507 3572 63 7 8349 2996 3633 1.06800 5253 1827 63 8 7086 4159 0463 1.10.543 3536 3543 62 8 8370 2976 3688 1.06738 5282 1798 62 9 7107 4139 0516 1.10478 3566 3514 51 9 8391 2957 3742 1.06676 5311 1769 61 10 7129 4120 0569 1.10414 3595 3485 50 10 8412 2937 3797 1.06613 5340 1740 50 11 7151 4100 0621 1.10349 3624 3456 49 11 8433 2917 3862 1.06551 5369 1710 49 12 7172 4080 0674 1.10285 3653 3427 48 12 8455 2997 3906 1.06489 5398 1681 48 13 7194 4061 0727 1.10220 3682 3398 47 13 8476 2877 3961 1.06427 5427 1662 47 14 7215 4041 0781 1.10156 3711 3369 46 14 8497 2857 4016 1.06366 5466 1623 46 15 7237 4022 0834 1.10091 3740 3339 45 15 8618 2837 4071 1.06303 5485 1594 45 16 7258 4002 0887 1.10027 3769 3310 44 16 8539 2817 4125 1.06241 5515 1565 44 17 7280 3983 0940 1.09963 3798 3281 43 17 8661 2797 4180 1.06179 5644 1536 43 18 7301 3963 0993 1.09899 3827 3252 42 18 8682 2777 4235 1.06117 6673 1507 42 19 7323 3944 1046 1.09834 3866 3223 41 19 8603 2757 4290 1.06066 6602 1478 41 20 7344 3924 1099 1.09770 3886 3194 40 20 8624 2737 4345 1.06994 5631 1449 40 21 7366 3904 1153 1.09706 3916 3165 39 21 8645 2717 4400 1.05932 5660 1420 39 22 7387 3885 1206 1.09642 3944 3136 38 22 8666 2697 4455 1.05870 5689 1390 38 23 7409 3865 1259 1.09578 3973 3107 37 23 8688 2677 4510 1.05809 5718 1361 37 24 7430 3846 1313 1.09514 4002 3078 36 24 8709 2657 4565 1.05747 5747 1332 36 25 7452 3826 1366 1.09450 4031 3049 35 25 8730 2637 4620 1.05686 5776 1303 35 26 7473 3806 1419 1.09386 4060 3019 34 26 8751 2617 4676 1.05624 5806 1274 34 27 7495 3787 1473 1.09322 4089 2990 33 27 8772 2597 4731 1.05562 6836 1245 33 28 7516 3767 1526 1.09258 4118 2961 32 28 8793 2577 4786 1.05501 6864 1216 32 29 7538 3747 1580 1.09195 4147 2932 31 29 8814 2557 4841 1.05439 6893 1187 31 30 7559 3728 1633 1.09131 4176 2903 30 30 8836 2637 4896 1.05378 6922 1168 30 0.6 0.7 0.9 0.7 O.S 0.6 0.7 0.9 0.7 0.8 31 7580 3708 1687 1.09067 4206 2874 29 31 8857 2517 4962 1.05317 5951 1129 29 32 7602 3688 1740 1.09003 4235 2845 28 32 8878 2497 5007 1.05255 5980 1100 28 33 7623 3669 1794 1.08940 4264 2816 27 33 8899 2477 5062 1.05194 6009 1070 27 34 7645 3649 1847 1.08876 4293 2787 26 34 8920 2457 5118 1.06133 6038 1041 26 35 7666 3629 1901 1.08813 4322 2768 25 35 8941 2437 5173 1.06072 6067 1012 25 36 7688 3610 1955 1.08749 4351 2729 24 36 8962 2417 5229 1.05010 6096 0983 24 37 7709 3590 2008 1.08686 4380 2699 23 37 8983 2397 5284 1.04949 3126 0954 23 38 7730 3570 2062 1.08622 4409 2670 22 38 9004 2377 5340 1.04888 3154 0925 22 39 7752 3551 2116 1.08559 4438 2641 21 39 9025 2367 5395 1.04827 6184 0896 21 40 7773 3531 2170 1.08496 4467 2612 20 40 9046 2337 5451 1.04766 6213 0867 20 41 7795 3511 2223 1.08432 4496 2583 19 41 9067 2317 5606 1.04705 6242 0838 19 42 7816 3491 2277 1.08369 4526 2564 18 42 9088 2297 6562 1.04644 6271 0809 18 43 7837 3472 2331 1.08306 4555 2525 17 43 9109 2277 5618 1.04583 6300 0780 17 44 7859 3452 2385 1.08243 4584 2496 16 44 9130 2257 6673 1.04522 6329 0751 16 45 7880 3432 2439 1.08179 4613 2467 15 45 9161 2236 6729 1.04461 6358 0721 15 46 7901 3412 2493 1.08116 4642 2438 14 46 9172 2216 5785 1.04401 6387 0692 14 47 7923 3393 2647 1.08053 4671 2409 13 47 9193 2196 5841 1.04340 6416 0663 13 48 7944 3373 2601 1.07990 4700 2379 12 48 9214 2176 5897 1.04279 6445 0634 12 49 7965 3353 2655 1.07927 4729 2350 11 49 9235 2156 6952 1.04218 6474 0605 11 50 7987 3333 2709 1.07864 4758 2321 10 50 9256 2136 6008 1.04158 6604 0576 10 51 8008 3314 2763 1.07801 4787 2292 9 51 9277 2116 6064 1.04097 6533 0547 9 52 8029 3294 2817 1.07738 4816 2263 8 52 9298 2095 6120 1.04036 6562 0518 8 53 8051 3274 2872 1.07676 4845 2234 7 53 9319 2075 6176 1.03976 6591 0489 7 54 8072 3254 2926 1.07613 4875 2206 6 54 9340 2055 6232 1.03916 6620 0460 6 55 8093 3234 2980 1.07550 4904 2176 5 55 9361 2035 6288 1.03856 6649 0431 5 56 8115 3215 3034 1.07487 4933 2147 4 56 9382 2015 6344 1.03794 6678 0401 4 57 8136 3195 3088 1.07425 4962 2118 3 57 9403 1995 6400 1.03734 6707 0372 3 58 8157 3175 3143 1.07362 4991 2089 2 58 9424 1974 64.57 1.03674 6736 0343 2 59 8179 3155 3197 1.07299 6020 2060 1 69 9446 1954 6513 1.03613 6766 0314 1 60 8200 COS. 3135 3252 COT. 1.07237 5049 2030 60 9466 1934 6569 1.03553 6794 0285 SIN. TAN. COM. OF ARC. / COS. SIN. COT. TAN. COM. OF ARC. / ARC. ARC. Sup. 132° = 7920* 47° = 2820' Sup. 133° = 7980' 46° = 2760' 444 44° = 2640' TRIGONOMETRY Sup. 136° = 8100' ' SIN. COS. TAN. COT. ARC. COM. OF ARC. 0.6 0.7 0.9 07 0.8 9466 1934 6669 1.03563 6794 0285 60 1 9487 1914 6625 1.03493 6824 0256 59 2 9508 1894 6681 1.03431 6853 0227 58 3 9529 1873 6738 1.03372 6882 0198 57 4 9549 1853 6794 1.03312 6911 0169 56 5 9570 1833 6850 1.03252 6940 0140 55 6 9591 1813 6907 1.03192 6969 0111 54 7 9612 1792 6963 1.03132 6998 0081 63 8 9633 1772 7020 1.03072 7027 0052 52 9 9654 1752 7076 1.03012 7056 0023 0.7 9994 51 10 9675 1732 7133 1.02952 7085 50 11 9696 1711 7189 1.02892 7114 9965 49 12 9717 1691 7246 1.02832 7144 9936 48 13 9737 1671 7302 1.02772 7173 9907 47 14 9758 1660 7359 1.02713 7202 9878 46 15 9779 1630 7416 1.02653 7231 9849 45 16 9800 1610 7472 1.02593 7260 9820 44 17 9821 1590 7529 1.02633 7289 9791 43 18 9842 1569 7586 1.02474 7318 9761 42 19 9862 1549 7643 1.02414 7347 9732 41 20 9883 1529 7700 1.02355 7376 9703 40 21 9904 1508 7756 1.02295 7405 9674 39 22 9925 1488 7813 1.02236 7434 9645 38 23 9946 1468 7870 1.02176 7463 9616 37 24 9966 1447 7927 1.02117 7493 9587 36 25 9987 0.7 0008 1427 7984 1.02057 7522 9558 35 26 1407 8041 1.01998 7551 9529 34 27 0029 1386 8098 1.01939 7580 9500 33 28 0049 1366 8156 1.01879 7609 9471 32 29 0070 1345 8213 1.01820 7638 9442 31 30 0091 1325 8270 1.01761 7667 9412 30 0.7 0.7 0.9 0.7 0.7 31 0112 1305 8327 1.01702 76961 9383 29 32 0132 1284 8384 1.01642 7725 9354 28 33 0153 1264 8441 1.01583 7754 9325 27 34 0174 1243 8499 1.01524 7783 9296 26 35 0195 1223 8556 1.01465 7813 9267 25 36 0215 1203 8613 1.01406 ' 7842 9238 24 37 0236 1182 8671 1.01347 7871 9209 23 38 0257 1162 8728 1.01288 7900 9180 22 39 0277 1141 8786 1.01229 7929 9151 21 40 0298 1121 8843 1.01170 7968 9122 20 41 0319 1100 8901 1.01112 7987 9092 19 42 0339 1080 8958 1.01063 8016 9063 18 43 0360 1059 9016 1.00994 8045 9034 17 44 0381 1039 9073 1.00936 8074 9005 16 45 0401 1019 9131 1.00876 8103 8976 15 46 0422 0998 9189 1.00818 8133 8947 14 47 0443 0978 9247 1.00759 8162 8918 13 48 0463 0957 9304 1.00701 8191 8889 12 49 0484 0937 9362 1.00642 8220 8860 11 50 0505 0916 9420 1.00583 8249 8831 10 51 0525 0896 9478 1.00626 8278 8802 9 52 0546 0875 9536 1.00467 8307 8772 8 53 0567 0856 9594 1.00408 8336 8743 7 54 0587 0834 9652 1.00350 8365 8714 6 55 0608 0813 9710 1.00291 8394 8685 5 56 0628 0793 9768 1.00233 8423 8656 4 57 0649 0772 9826 1.00175 8452 8627 3 58 0670 0752 9884 1.00116 8482 8598 2 59 0690 0731 9942 1.0 0000 1.00058 8511 8569 1 60 0711 0711 1.00000 8540 8540 COS. SIN. COT. TAN. COM. OF AHC. ARC. ' Sup. 134° = 8040' 45° = 2700' SOLUTION OF THE IRREDUCIBLE CASE 445 APPLICATION OF THE EQUATION OF THE THIRD DEGREE AND THE TRIGONOMETRIC SOLUTION OF THE IRREDUCIBLE CASE 1072. Cohtinuing from the point where we left off in (592) from the general equation x^ + px + q = 0, Case 1. One real and two imaginary roots. If the quantity 4 ^ 27 ^ ' the equation has only one real root of a sign opposite to that of its last term q, and two imaginary roots. Designating the values of the cubic radicals by A and B, the three roots of the equation are: xi= A + B (real), (2) X2= Aa + Ba? X« = Aa? + Ba. I (imaginary). (3) a is one of the two imaginary cube roots of one, that is, " 2 Example 1. Calculate the radius and altitude of a cylinder inscribed in a sphere, such that the area of its lateral surface is equal to the area of the two zones of one base, which are de- termined by the cylinder. Solution. Let R be the radius of the sphere, x the radius of the cylinder, and 2 y its altitude. Then the lateral surface of the cylinder equals 4 irxy and the surface of each zone 2 irR {R — y), and the equation of the problem is 4:irxy = 4:TrR{R — y), or xy = R{R - y). (1) The following relation exists between the three quantities R, X, and y\ R' = x' + f, (1022) and X = Vi2^ - y\ (2) 446 TRIGONOMETRY Dividing (1) by (2), Then , R^iR-yy _ R^R-y) men, y - ^, _^, - ^^^ . Transposing, y' + Rf + Rhj - m = 0. Taking R = \, this equation becomes: f + f + y-l = Q. (3) The term y'' may be eliminated by substituting,* 2/ = w-g- (4) After the substitution the equation (3) becomes: ». + ^^-i=0. ,5) Finally, to eliminate the denominators, write w = ^ in equa- tion (5), which then becomes: (6) ^ + Qz - 34 ■■ = 0. The equations (4) and (6) give: z 1 ^ = 3-3 2 - 1 3 It remains now to solve equation (6), which, according to the equation of the third degree, gives : 2 = Vl7 + V297 + •^17 - ^^97. Here the radical of the second degree is real, the equation has one real root and two imaginary ones; it is the first case, as explained above. * Let the general equation of tlie third degree be : 3^ + Ax''-\-Bx-\-C=ft. (1) Write x-=y-\-h\ then equation (1) becomes : j/» + j/2 (3 ft + ^) + 3/ (3 A2 + 2 ^A + 5) + AS + ^ft2 4- £ft + C= 0. The quantity ft being indeterminate, we may write, J 3ft+^ = 0, from which ft = — -■ Substituting this value of h in all the terms of the preceding equation, we get : yis^pyj^q — Q, SOLUTION OF THE IRREDUCIBLE CASE 447 Solving, z = 2.631, and y = ^-^ = 0.5436. The altitude of the cylinder is then 2y = 1.0872. The equation (2) will give the radius of the cylinder. X = V722 _ f = Vi _ 0.5436' = 0.8451. The other two roots of the equation (6) are imaginary; they are given by the equations (2) and (3) (see Case 1, page 445). Remark. If the radius of the sphere were R, the preceding solution would give the radius of the cylinder as: X = 0.8451 R, and the altitude as: 2y = 1.0872 R. Case 2. Three real roots of which two are equal. If the quantity 4 + 27 ~ "' the equation has two equal roots of the same sign as the in- dependent term q, and one root of sign opposite to that of q. The roots are, 2p (equal roots), Xi = — (single root). Remark. The absolute value of the last root is double that of the two equal ones. Example. The equation oi?-Zx + 2 = 0, gives the following values: - 3g -3X2 2p - 2 X 3 3g_ 6 p " -3 = +1, ,,= 3,^ 6 _2_ 448 TRIGONOMETRY Case 3. The irreducible case. Three real roots. If the quantity 4 + 27 ^ "' the equation has three real roots; but the value of x is com- posed of the sum of two imaginary quantities, which are calcu- lated by trigonometric formulas, as will be shown below. The trigonometric solution of the irreducible case of the equation of the third degree. The equation of the 3d degree being reduced to the form o(? + fx + q == 0, (1) the general value of x is (592), -vZ-l+v'FI-v'-l-v/?^- If the sum ~r + ^ <0, the value of x appears under the form of the sum of two imaginary quantities. Writing - I = p cos <^ and ^ + ^ = - p2 sin=' <^, we have p = 1/ — i^ and cos = -—^ ■ Then the values of the three roots are: Xi = 2Vp cos - ) Xj = - 2 -s/p cos (60° - I) , x,= +2V'^(l20°-|y Remark. If the last two roots are equal, we have: ^ = 0°. NoTE. If the cos <^ = -jr-^ is negative, the angle ' is found 2 p which is a supplement of <^ and has the same cosine with the sign + . This angle ^' should replace <^ in the values of the three roots. SOLUTION OF THE IRREDUCIBLE CASE 449 Example 1. Solve the equation : 0)5 + 5 a; + 1 = 0. Comparing with the general form, 3^ + px + q = 0, we have: _ + _ = ____< o. Thus, the example reduces to the irreducible case, and the for- mulas given above are to be applied. OALCni/ATION OF p CALCtTLATIOlSr OF ij> log 125 = 2.0969100 log 1 = 0.0000000 c' log 27 = 8.5686362 c' log 2 = 9.6989700 - 10 c' log p = 9.6672269 1 l^n aa^^Aao\ " 20.0000000 log P =h(0. 6655462) ■ = 2^ logcos<^= 1.3661969 or logp =0.3327731 ^=76° 33' 53" The value of cos 4> being negative, must be replaced by its supplement ', that is, ^' = 103° 26' 7" ; then ^=34° 28' 42.3", 60° - ^ = 25° 31' 17.7", 120° -^ = 85° 31' 17.7". Calculation of the three roots. Calculation of xi log 2 = 0.3010300 log \/p = 0.1109243 log cos^= 1.9161061 log Xi = 0.3280604 from which ajj = + 2.128 450 TRIGONOMETRY Calculation of x^ log 2 = 0.3010300 log Vp = 0.1109245 log cos ^60°- j\ = 1.9554101 log ( - Xj) = 0.3673646 from which x^ = —2.330 Calculation op x, log 2 - 0.3010300 log a/p= 0.1109245 log cos ^20° - |-) = 2.8925602 log xs = 1.3045147 Xs = 0.2016 Note. The calculations being so laborious, it is quite necessary to prove that the roots are correct by substituting their values in the given equation x^ - 5 X + 1 = 0, or in x^ — 5 X + 1 = y, and making sure that two consecutive values which differ by ) for example, give two values preceded by unlike signs for the sum y of the terms of the equation. Proof of Xi = 2.128: for xi = 2.128 y = - 0.0036 for Xj = 2.129 y = + 0.00499 Proof of Xj =- 2.330: for Xj = - 2.331 2/ = - 0.0010 for Xj = - 2.330 y = + 0.0007 Proof of xs = 0.2016: for xa = 0.201 y = + 0.0031 for Xs = 2.202 y = - 0.0018 We are assured that in taking X, = 2.128 Xj = - 2.330 x^ = 0.201 these values are correct to 0.001. ; SOLUTION OF THE IRREDUCIBLE CASE 451 Example 2. Divide a hemisphere into two equivalent parts by a plane parallel to the base. Solution. Let R be the radius of the sphere, then the volume 2 of the hemisphere is ^ ttE', and that of the spherical segment with one base, which should be equal to one-half the volume of the hemisphere, is (931): v = l.R^ If the altitude of the spherical segment is designated by x (931, Remark) : v^l-TX^ (SR -x) =l-^R\ ay' - 3 Rx" + E= = 0. Taking R = \, x' - 3 a;2 + 1 = 0. (1) To eliminate the term x^ take (see note (*) page 446) Equation (1) becomes: x = y+^^y+\. (2) 2/^ - 3 2/ - 1 = 0. (3) Comparing with the equation, it is seen that ^ + ^ < 0. 4 27 Thus we have the irreducible case of the third-degree equa- tion. The equation (3) has three real roots. Writing p = y — = 1, andcos ^ = y^ = —^ ' then <^ = 60° and ^ = 20°. o The three roots are: 2/i = 2V/3 cos-^j 2/2 = -2^p cos^60°-|j, 452 TRIGONOMETRY 2/8= 2Vp cos(l20°-|V Substituting the numerical values, 1/1 = + 1.8793, 2/2 = - 1.55208, ys = - 0.34729; then substituting in equation (2) : ^1 = 1+2/1 = 2.8793, ^2=1+2/2=- 0.55208, 3-3 = 1 + 2/3=+ 0.6527. The first value x.^ being greater than the radius R = 1, cannot be used as a solution. The second x^ being negative must also be rejected. The third value x^ being less than R = 1 and positive, is the solution which applies to the case in hand. ' 1 Remark. If the radius of the sphere were R, the altitude of the required segment will be Xi = 0.6527 R. SPHERICAL TRIGONOMETRY Properties of spherical triangles. 1073. A spherical triangle is determined by three axes T)f greai; circles drawn on the sphere. If the vertices are connected to the center of the sphere, a trihedral angle corresponding to the spherical triangle, the faces of which are measured by the sides of the spherical triangle, is formed. Each side of the spherical triangles, which are treated in trigo- nometry, is less than a semi-circumference. 1074. The measurement of the angles of a spherical triangle. The angles A, B, C, of a spherical triangle are measured by tangents drawn to the sides a, b, c, of the triangle. These angles measure the dihedral angles of the trihedral angle corresponding to the spherical triangle. A spherical triangle may be rectangular, bi-rectangular, or tri-rectangular. 1075. Lengths of the sides of a spherical triangle. R being the radius of the sphere, and n the number of degrees in the side of the triangle, we have: TrRn """ ISC?" 1076. General geometrical properties of spherical triangles. In a spherical triangle each side is smaller than the sum of the other two sides and greater than their difference. The sum of the three sides is less than the circumference, 360°, of a great circle. The sum of the three angles, A, B, C, lies between two and six right angles. 1077. Supplejnentary or polar spherical triangles. Two tri- angles are supplementary when the sides of the first are sup- plements of the angles of the second, and conversely. GENERAL FORMULAS 1078. Formula containing the three sides and an angle. Theorem. The cosine of any side a is equal to the product of the cosines of the other two sides, increased by the product 4fi3 454 TRIGONOMETRY of the sines of these two sides multiplied by the cosine of their included angle. Thus, cos a = cos h cos c + sin & sin c cos A. 1079. Formula containing the three angles and one side. This is the inverse of the preceding formula. Thus we have: cos A = — cos B cos C + sin 5 sin C cos a. 1080. Theorem. The sines of the sides of a spherical triangle are to each other as the sines of the opposite angles. sin A _ sin B _ sin C sin a sin b sin c 1081. Formulas containing two sides, the angle included by them and an angle opposite one of them. We have, cot a sin & = cos b cos C + sin C cot A, cot <2 sin c = cos c cos B + sin B cot A, cot b sin a = cos a cos C + sin C cot B, cot & sin c = cos c cos .A + sin 4 cot B, cot c sin a = cos a cos B + sin B cot C, cot c sin 6 = cos b cos 4 + sin A cot C. RIGHT SPHERICAL TRIANGLES In all cases that follow, A is the right angle, a, the hypotenuse, and B and C are the oblique angles of the spherical triangle. 1082. Theorem. The cosine of the hypotenuse is equal to the product of the cosines of the two sides. We have, cos a = cos b cos c. 1083. Theorem. The sine of each side is equal to the sine of the hypotenuse multiplied by the sine of the opposite angle. We have, sin b = sin a sin B, sin c = sin a sin C. 1084. Theorem. The tangent of each side is equal to the tangent of the hypotenuse multiplied by the cosine of the adjacent angle. We have, tan b = tan a cos C, tan c = tan a cos B. SOLUTION OF RIGHT SPHERICAL TRIANGLES 455 1085. Theorem. The tangent of each side is equal to the sine of the other side multiplied by the tangent of the angle opposite to the first side. We have, tan 6 = sin c tan B, tan c = sin 6 tan C. 1086. Theorem. The cosine of each oblique angle is equal to the cosine of the opposite side times the sine of the other oblique angle. We have, cos B = cos b sin C, cos C = cos c sin B. SOLUTION OF RIGHT SPHERICAL TRIANGLES 1087. These triangles have but one right angle. There are six cases to be considered. Case 1. Solve a right spherical triangle when the hypotenuse a and the side b are given. Given. Unknown. A = 90°; a, b c, B, C. Substituting in the formulas, cos a = cos b cos c, sin b = sin a sin B, tan b = tan a cos C, , ^ . cos a we obtain, cos c = j > cos . r, sin b sin B = -: > sin a _, tan b cos C = • • tan a Remark. The angle B and the side b are of the same species, that is, both are acute or obtuse. In order that the problem be possible, the hypotenuse must be included between the given side and its supplement. Another solution. The following formulas may also be used: tan - c = + Y tan ^{a + b) tan ^ (a — 6) > 456 TRIGONOMETRY tan (45° + i b) = ± tan^ (a+ 6) " — i ' tan jr (a — 6) £1 1 t / sm (g - b) tan jr C = + V ^ — 7 — TTR • 2 V sm (a + 6) 1088. Case 2. Solve a right spherical triangle having the hypotenuse a and one angle B given. Given. Unknowk. a,A = 90°, B. b, c, C. From the formulas sin 6 = sin a sin B, (1) tan c = tan a cos B. (2) The angle C may be deduced from cos a = cot B cot C (3) Transposing, ^ ~ cos a cot C = cot 5 Remark. The side b and the angle B are of the same species, that is, both acute or obtuse. The problem is always possible and has only one solution. It may be commenced by determining c and C from (2) and (3), and then 6 is determined from the equation tan 6 = sin c tan B. 1089. Case 3. Solve a right spherical triangle when two sides and the right angle are given. Given. Unknown. b,c,A = 90°. B, C, a. The following formulas give: cos a = cos & cos c, (1) , o tan b ,„, tan B = -. , (2) sm c tanC = ^. (3) sm 6 ^ SOLUTION OF RIGHT SPHERICAL TRIANGLES 457 Remark. The problem has only one solution and is always possible. The angles S and C may be determined by the formulas (2) and (3), and are calculated from one of the following: tan c = tan a cos B, tan b = tan a cos C. 1090. Case 4. Solve a right spherical triangle when a side b and the angle B opposite are given. Given. UintNOWN. b,B,A = 90°. C, a, c. The following formulas give: sin b . tan b . „ cosJB sin a = —. p; ) sin C = ;; ;;; Sm C = 7 • sin 5 tan 5 cos o The following may also be used: tani(S + 6) tan(45°+L) = ± / ? , (1) ^ ^ ' sj tani(5-6) tan(45°.,c) = ±v/ii?I- (2) tan (45° + ic) = ± ^cot^iB + b)cot^iB-b). (3) Remark. B and 6 are of the same kind: both are acute or obtuse. If 6 > 90°, then B > 90°, and in this case the radical (1) must be taken with a plus sign, +, and the two others (2) and (3) with minus signs, — . If 6 < 90°, then B < 90°, and in this case the radical (1) must be taken with a minus sign, — , and the two others (2) and (3) with plus signs, + . 1091. Case 5. Solve a right spherical triangle when one side b and the adjacent angle C is given. GiTEH. • UNKNOWir. b,C,A = 90°. a, c, B. 458 TRIGONOMETRY The following formulas give: cos B = cos h sin C, (1) tan 6 tan c = sin 6 tan C. (3) a and c may be determined first, and then B calculated from the following: cos a = cot B cot C, tan h = sin c tan 5. The problem is always possible and has but one solution. 1092. Case 6. Solve a right spherical triangle when the two oblique angles are given. Given. TJNKNO'wif. A = 90°, B, C. a, b, c. From the following formulas: cos a = cot B cot C, cos B cos b = cos c = sin C cosC sinB Another solution. The following formulas may also be used: , 1 , J -cos(BTC) tan ;r a = + y j~ — -^ , 2 Y cos {B — C) tan ^ 6 = + y tan (^^ + 45°) tan (^^^- 45°), tan i c = + y/tan (^^ + 45°) tan (^^ - 45°) • T) \ ft Remabk. In order that the problem be possible, — ^ — must B ~ C lie between 45° and 135°, and — ~ between - 45° and + 45°. There is but one solution. SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 1093. There are six cases. First and second case. Solve a spherical triangle when the three sides or three angles are given. SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 459 Case 1. Let the sides a, b, and c be given. From the following formulas: ^^ i^^^sg^^Wc^^ (1) 2 V sm p sm (p — a) 2 V SIR p sm [p — b) ^ ' tanAc = v/ '"-^~'^t^^";'^ - (3) 2 V sm p sm (p — c) ^ ' In these formulas we have, a + b + c V = 2 ' and the radical should be taken with the sign + . Remark. Each side should be less than the sum of the two others, and the whole sum less than 360°. Case 2. The three angles A, B, and C are given, and it fol- lows that the sides a', V, c', of the supplementary triangle are a' = 180° - A, b' = 180° - B, c' = 180° - C. The formulas (1), (2), and (3) with the sides a', b', and c', de- termine the angles A', B', and C" of the supplementary triangle; then the sides of the triangle in question are a = 180° - A', b = 180° - B', c = 180° - C". The triangle is then solved. But the following formulas may be used, which give the three sides directly: sin- A sin (4— ^AJ sm (5-lA)sin(c-iA) sin- AsinIB — ^^j sm (A-lA)sin(c-iA) 460 TRIGONOMETRY sin 2 ^^'"(^-^^) (A-iA)sin(5-iA) sin These radicals are taken with the sign +. In the preceding formulas A is the spherical excess ; that is, the difference be- tween the sum of the angles and 180°. Thus, A + B + C ~ 180° = A. A lies between and 360°. Remark. The sum of the three angles should he between two and six right angles. 1094. Third and fourth case. Solve a spherical triangle when two sides and the included angle or one side and the adjacent angles are given. The solution of these two problems is given by the formulas of Napier. Case 3. Two sides and the included angle given. GiTEN. TJNKNO-Vra. a, b, c. c, A, B. The following formulas, known as Napier's analogies, will be used. cos ^(a-b) tan i (A + B) = = cot^ C (1) COS -(a +6) sm ^(a-h) tan i (A - 5) = = cot^ C (2) sin 2 (a + &) J oos^(A-B) J tan^ (a + 6) = — ^j tan-c. (3) cos^(A + B) J sm^(A-B) J tan^ (a — b) = tan^c W sin ^{A + B) SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 461 The formulas (1) and (2) give A + B and A — B, from which A and B can be deduced. The values of A + B and A ~ B sub- stituted in (3) or (4) give c. Or c may be determined directly from cos c = cos a cos b + sin a sin b cos C, (5) which is easily solved by logarithms when written in the form: cos c = cos a (cos 6 + sin 6 tan a cos C). Let tan = tan a cos C, then cos c = cos a (cos 6 + sin 6 tan <^). Substituting -7 for tan <^, we have cos a cos (b — ) cos c = ^^ ^ • cos

, then cos C = — cos A (cos Sj— sin B cot <^). Substituting -: — rfor cot <^, we have: _ _ — cos A • sin {(p — B) _ cos 4 • sin (B — <^) cos O — : ; — : ', • sm sin 1095. i^i/)= ] .9963378 c' log cos ^= 10.10146724 - 10 log cos c = 1.97457370 c= 19°24'53.4" 466 TRIGONOMETRY To obtain the distance in miles, reduce the side c to seconds- thus, c = 69893.4", and 90° = 324000". Taking a quadrant as 6250 miles we have;: 90^ 6250 c ~ X ' 324000 6250 or 69893.4 X X = 1348 miles. AITGLES FORMED BY THE FACES OF REGULAR POLYHEDRONS Problem 3. There are only five regular polyhedrons (903): the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Tetrahedron. The polyhedral angle of a tetrahedron is a tri- hedral angle, the three equal faces of which are measured by the angle of an equilateral triangle. Therefore a spherical triangle, 2 the three sides of which are each equal to ^ of a right angle 60°, is to be solved. This is the first case in the solution of spherical triangles (1093). Let C be the required dihedral angle, then using the formula : ,^lc=\/^^^^^^^^, (1) 2 V sm a sm b we have a = b = c= 60°, a + b + c 60 X 3 „„„ p = 2 = ^^=90, p _ a = p - 6 = 90° - 60° = 30°. Formula (1) gives . 1„ , /(sin 30°)' sin 30° , „_» ^^^2^=V(ihr607^3^^^30^=*"^^^' and C = 70°31'43.6". Cube. The dihedral angle of a cube is 90°. Octahedron. This problem may be solved by spherical trigo- nometry, by dividing one of the polyhedral angles formed by four FACES OF REGULAR POLYHEDRONS 467 equilateral triangles into two trihedrons. A much simpler method is as follows: a being the edge of the octahedron, and C one of the dihedral angles, considering one of the two pyramids with a square base, which compose the octagon, we have, 1 h^ tan-C= ~ = V2, which gives C = 108° 28' 1.6". Dodecahedron. The polyhedral angle of this polyhedron is a trihedral angle, the three faces of which are measured by the angles 108° of a regular pentagon. Thus the dihedral angle of a dodecahedron is obtained by solving a spherical triangle whose three equal sides are each measured by 108°. The first case of spherical triangles (1093) gives: sin - r'=3i/ sin(p-a)sin(p-6) 2 V sin a sin b sin a sin b We have a=^b = c= 108°, a + b + c 108X3 P= 2 =-^—=162, p-a = p-b= 162° - 108° = 54°, .1 / (sin 54°)^ _ sin 54° . ™ 2 ^ V (sin 108°)^ ~ sin 108°' sin 108° = sin (180° - 108°) = sin 72^. Therefore, sin - C = ■ !" -„o ' 2 sm 72 and (7= 116° 33' 54". Icosahedron. It is readily seen that one of the dihedral angles of an icosahedron belong to a trihedral angle of which the three faces are known : two faces are formed by two equilateral triangles, and the third face is formed by a diagonal plane, which deter- mines an isosceles triangle whose angle at the vertex is equal to the interior angle of a regular pentagon. The three faces of the trihedron are known. a = 6 = I rt. ^ and c = 108°. 468 TRIGONOMETRY The formula in article (21) may be used. sm ^ G / sin (p — g) sin (p — b) V sin p sin 6 (^) P a + b + c 60° + 60° + 108° = 114° 2 2 p _ a = p - & = 114 - 60 = 54°. From formula (A) C = 138° 11' 22.8"- 1097. Formulas for transforming algebraic and trigonometric expressions into such a form that they may be solved by logarithms. Example 1. Let X = A ± B he given. 1st, considering we may write Putting we have and = A + B, -r = tan'' a, A 1 log tan a = - (log B - log A), x = Ail + tan^a) = a(i + ^^) • \ COS^a/ , ' _ . /cos'' o + sin^ a\ _ A ~ \ COS^ a / COS^ a log X = log A — 2 log COS a. 2d. If we consider x = A — B, and if B is less than A the ratio of B to A is less than unity, and we may write successively: -r = Sin'' a, A ' x== A (I — sin^ a) = ^ cos^ a, log x = log A + 2 log cos a. If B is greater than A we may write B , -r = tan a, A ' FACES OF REGULAR POLYHEDRONS 469 and therefore x = A {I - t&na) = a(i - ?ELl\ \ cos a/ . (cos a — sin a) A r ■ .„„ x=A ^- ^ = [sin (90 - o) - sin a]. cos a cos a "■ ^ ' -• Taking the formiilas (1052) sin p — sin g = 2 cos - (p + q) sin 5 (p — q), putting p = QO — a and q = a, then ^(P + 3) = 45°, i(p-3) = 45°-a, X = 2 cos 45° sin (45° - o). cos a ^ ' This formula is logarithmic. Example 2. Having given : X = tan a ± tan b, (1) we may write sin a , sin 6 sin a cos 6 ± cos a sin b cos a cos b cos a ■ cos b ' sin (o ± &) I or x = ^^ ^. I (2) cos a ■ cos |] ^ '^ Example 3. a; = cot 5 ± cot 4, (1) 1 , 1 tan A =t tan B or X = =, ± - tan B tan A tan A tan B 1 tan A tan B ^- .o. /... R (tan^^tan5). (2) ■NT X - , , D smA , sm5 Now tan A =fc tan B = 3- ± ^, cos A cos ii , . , ^ _ sin A cos B ± sin S cos A sin (A ± B) or tan A ± tan B = ^ = — — , = ■ \ '- . cos A cos B cos A • cos B Therefore from (2) we may write _ 1 sin (A ± 5) _ cos A cqsB sin (A ± B) tan A tan B cos A cos B ~ sin A sin B • cos A • cos 5 ' sin (A ± S) sm A sm 5 This formula is logarithmic 470 TRlGONOMHTkY Example 4. a; = "^2 + sin a, (i) or ' = ^(' + %°)' (2) T. XX- sin a , , , sin'6 Putting — j=r- = tan^ (b = ~ , f 3^ V2 cos^ / cos^ Formula (1) is therefore replaced by a logarithmic formula. The auxiliary angle <^ is calculated from the following formula deduced from (3) : log tan 4> = 7^ (log sin a — log "^2). Example 5. X = esc a + sec &, (1) 1 , 1 cos & + sin a or x= 1 r = —. r- , sm a cos o sm a cos o sin (90 — 6) + sin a or x= ^^-^ — r (2) sin a cos o ^ ' From (1052) we have sin p + sin 2 = 2 sin ^ (p + q) cos k (P — ?)• Putting 90 - 6 = p, a = q, we have -r{V + ^ = 45° ^ , 1 , . s . _o a + b 2(P-e) = 45 2~' and equation (2) becomes 2 sin (45°-^) cos (45°-^) sin a cos b which may be calculated by logarithms. PART V ANALYTIC GEOMETRY 1098. The purpose of analytic geometry is the study of geo- metrical figures by means of algebraic analysis. This branch of mathematics was invented by Descartes, who found that the properties of geometrical figures could be studied by algebraic methods; he also found graphic solutions for alge- braic calculations. The latter are the more useful to the engineer. Analytic geometry, like elementary geometry, is divided into two parts (610): plane geometry and solid geometry. DETERMINATION OF A LINE 1099. We have seen that the position of a point in a plane or in space is fixed when its coordinates are known (1020, 1021). In order that a line be determined, it suffices to know the coor- dinates of its points. When the same algebraic relation exists between the coordi- nates of each of the points of the line, as many points may be determined as one wishes, and therefore, by plotting the points which are thus obtained, the line may be drawn. Thus, if the relation between the coordinates of a plane curve are known, by assuming any value for one coordinate the corre- sponding value of the other is found from the given relation which determines a point on the curve (504). Suppose that the relation y = 3 x + 2 exists between the coordinates, then ii x = i, y = 3 x 4 + 2 = 14. Giving X a new value, another corresponding value of y is found, and so on. When the curve is not a plane curve, since only one coordinate may be chosen arbitrarily, the two others can only be determined when there are two equations (516). 471 472 ANALYTIC GEOMETRY 1100. Polar Coordinates. A point M is also determined in a plane MOx, when the angle MOx = a, which the line OM makes with the axis Ox, and the distance OM = p, called radius vector, from the pole are given. The two quantities a and p are- called polar coordinates. When the same algebraic relation exists between the polar coordinates of each of the points of a line, as in the preceding case, any number of points may be determined, and consequently, the line drawn. 1101. Focal coordinates. The position of a point M is also fixed in a plane, when the distances MF = p and MF' = p' from the point M to the two fixed points F and F' are known. The points F and F' are called foci, and the distances p and p' are called radium vectors or focal coordinates. These same coordi- ./* ^■7" F\ , M Pig. 271 rig. 272 Kg. 273 nates, p and p', determine a point M' in the same plane and symmetrical to M with respect to the axis FF', and an equation between p and p', considering them as variables, determines a line made up of two parts symmetrical to each other with re- spect to the axis FF' (504). A point M is also determined in a plane by the distances MF — p and MP = p', also called radius vectors, to the fixed point F and the fixed line Oy, which are respectively called the focus and the directrix. As in the preceding case, the two absolute lengths of the radius vectors determine two points, M and M', symmetrical to each other with respect to the axis Ox, drawn through the focus F perpendicular to the directrix Oy. Thus an equation between the two radius vectors, p and p', considered as variables, determines a line symmetrical with respect to the axis Ox. 1102. Curves are determined by the relations between their coordinates with respect to two axes (1099), or by those be- tween their polar coordinates (1100) or by those between their focal coordinates (1101). HOMOGENEITY 473 The study of the curves most often used in practice will make all this clear. The equation which expresses the relations be- tween the coordinates of a curve is called the equation of the curve. HOMOGENEITY 1103. A polynomial is said to be homogeneous when all its terms are of the same degree. The degree m of each term is the degree of homogeneity of the polynomial (455, 457). In general, we say that a function (504) is homogeneous and of the degree m, when in multiplying each of the letters which ap- pear in the expression by a constant k raised to the power of that particular letter, the function is multiplied by k'" (478). Such are: a? + 2ab, — - ^c, ah ^rj- a + y/ab a c a + c ' a^ — b^' of which the degree is respectively 2, 1, 0, and — 2. A monomial is always an homogeneous function of a degree equal to that of the monomial. If, in a function, letters appear which represent numerical coefficients, these letters are neglected in forming the degree of the homogeneity of the function. Thus, n being a numerical coefficient, the following function is homogeneous and of the first degree: a^ + (y + nx^) "^ab — {ny + xY The transcendental functions, sin, cos, ...., log, of homogeneous functions of the degree 0, such as e", in which u, is also an ho- mogeneous fimction of the degree 0, are considered as numerical coefficients. Such are: 1 a^ — e' ab ,6-1- Va2 - h^ ss- siii 2 I 1.2 ' log ■ —r ) e a^ -\-W a -{-b In multiplying each letter of a fimction of the degree o by k, the value of the function is not changed, and therefore k may be omitted; which, however, could not be done if the degree of the function were not 0. Thus the following function is homogeneous and of the degree ^ : a V6 -f- b Vc sin - a a + b 474 ANALYTIC GEOMETRY 1104. From the above and the operations on polynomials it follows : 1st. That the sum or difference of two homogeneous func- tions of the same degree is an homogeneous function of the same degree as the first (460, 461). 2d. That the product of several homogeneous functions of any degree is an homogeneous function of a degree equal to the sum of the degrees of the given functions (477). 3d. That the quotient obtained in dividing one homogeneous function by another is an homogeneous function of a degree equal to the degree of the first less that of the second (494). 4th. That a power of an homogeneous function is an homo- geneous function of a degree equal to the degree of the given function multiplied by the degree of the power (2d). 5th. That the root of an homogeneous function is an homo- geneous function of a degree equal to the degree of the given function divided by the index of the root (4th). 1105. An equation is said to be homogeneous when its two members are homogeneous and of the same degree, or when one of its members is zero and the other is homogeneous (1103). From this definition it follows : 1st. That an homogeneous equation remains homogeneous when all the letters which it contains are multiplied by the same factor k, with an exponent equal to that of each letter (1103). 2d. That an homogeneous equation between two concrete quan- tities of the same kind (12) — other quantities being considered as coefficients (1103) — is independent of the unit used to express these quantities. In changing the unit, all the concrete quan- tities are multiplied by the same factor whole or fractional. Conversely, if a whole algebraic equation — the only case which need be considered (447) '■ — between concrete quantities of the same kind exists, no matter what units are used, the equation is ho- mogeneous, or comes from the addition of several homogeneous equations of different degrees (1108). 1106. Any algebraic equation may be transformed to one in which one of the members is zero, and the other a whole rational quantity (447). If the equation is homogeneous and of the degree m, each of its terms contain m literal factors, not including the literal co- eflicients (1103). HOMOGENEITY 475 Thus in general an equation may be written in the form of the function / (a, b,x,y,....)= 0. 1107. In geometry, lengths are the only concrete quantities which have to be considered, because areas and volumes depend upon the linear dimensions. To express algebraically a relation between several lengths, they must first be reduced to the same units, which are generally arbitrarily chosen (1109). 1108. All equations in geometry are homogeneous when the unit is indeterminate. This is of the greatest importance in analytic geometry: it serves as a means of proof during the course of the calculations; it aids one in memorizing the formulas; it establishes analogies between the expressions, and may suggest methods of calculation which are more simple and elegant. Remark 1. When several homogeneous equations are com- bined by addition or subtraction, they should be of the same degree; because if they are not, the resulting equation, although exact, will not be homogeneous; and such a combination, in a well-conducted analysis, should be avoided. Remark 2. The theorem of homogeneity is applicable to all the equations of geometry; but in remembering that areas are the products of two lengths, and volumes the products of three lengths, therefore, according as a letter A or F represents an area or a volume, it must be considered as being of the second or third degree. Thus, h, h', b, h', expressing lengths, A and A' areas, and V and V volumes, the two following formulas are homogeneovis : A-A' = hh-h')Q)- h'), V-V' = \{h-h'){A-{-A' + VJZ^). In general, according as the unknown of a problem is an area or a volume, the expression which is obtained is homogeneous and of the second or third degree. Thus we have, A = ah ov V = ahc. 1109. In all which has been said, the unit has been taken as arbitrary. This hypothesis should hold for the solution of all geometrical problems; because, otherwise, if, for example, a cer- tain length was taken as unit, although homogeneous equations could be obtained they would not appear to be so. 476 ANALYTIC GEOMETRY Thus, taking an arbitrary unit, the area of a circle is: A = irr'. If, on the contrary, we take the radius equal to one, we have A' = TT X P = TT, equation in which the first member is of the second degree, and the second apparently of the degree 0, because ir is an abstract number. In order to give the equation its usual homogeneous aspect, the radius is expressed in arbitrary units; r is substituted for 1, and we have r^ in the second member. Thus, A'r^ or A = nr^. Taking the radius as unity, the volume of a sphere is: 4 4 F' = 3.X13 = -.. Substituting an arbitrary unit for the radius, which gives r instead of 1, the preceding equation becomes: 4 Y'j-3 Qp V = ~ tit'. o Half the major axis of an ellipse being taken as unity, the area of the ellipse is: A' = TT X 1 X &'. (1162) Substituting an arbitrary unit, a, for 1, and comparing all the lengths to this same arbitrary unit, we have,, A' a? = TT X a X ah', A == irab. THE GEOMETRICAL CONSTRUCTION OF ALGEBRAIC FORMULAS 1110. From the law of homogeneity it follows that any homo- geneous algebraic expression of the first degree, in which the different letters represent lengths, is an expression of a length X (1108, Remark 2), and this length may always be determined geometrically, that is, with the aid of a rule and compass: First, when the expression is rational (447) ; Second, when, being irra- tional, it contains only radicals whose index is 2 or a power of 2. 1111. Construction of rational expressions. To construct, x = a + h — c + d — e. CONSTRUCTION OF ALGEBRAIC FORMULAS 477 commencing at the point on an indefinite straight line, take OA = a, AB = 6, BC = d, CE = - c, and EF = - c. The distance - OF is value of x (Fig. 274). If we have ab construct the fourth proportional to the three lines, a, b, and m (969). „ abed For X = mnp Construct the fourth proportional x' = OX' = — to the three m hnes, m = OM, a = OA, and b = OB; then the fourth propor- x^ c cihc tional x" = OX" = — = — ■ to the three lines, n = ON, x'= n mn OX', and c = OC; finally, construct the fourth proportional, _ x"d abed X = Ox= = ) p mnp to the three lines, p = OP, x" = OX", and d = OD. The construction of the fourth proportionals in the preceding •E T-^ ^ r A B G X' A C Fig. 274 Fig. 275 example. After having drawn the indefinite lines OP and 0T>, lay off alternately on one and then the other, AO = a, OB = b, OC = c,OD = d, OM = m,ON = n, and OP = p; draw BM, CN, and DP, and AX', X'X", and X"x parallel respectively to the first; then Ox is the required length x. The expressions a' aa a^ aaa x = — = — I X = — ;= > etc., mm m' mm being the same as the above, except that the several factors are equal, x is found in the same way by constructing the fourth proportionals. 478 ANALYTIC GEOMETRY X being expressed by a fraction whose terms are polynomials, the construction is reduced to that given above by operating as follows : a^b + 4a^bc Let 5 ab^ - ¥c k being an arbitrary length, we may put the value of x in the form X = (a»b 4a^bc\ 5a^_Pc k' ¥ The exponent of k being one less than the degree of the terms which it divides, each of the resulting monomial fractions may ba constructed from the preceding rule, and A, B, M, N being the lengths found, we have, k{A+B) ^ M-N Determining A + B = a, and M — N = m, we have, ka x = — ; m and X, being the fourth proportional of the lengths k, a, and m, is constructed as shown above. Remark. In the preceding problems, as in those of the next article, if the given quantities instead of being lines were num- bers, taking a length as unity the given numbers could be rep- resented by lengths which, being submitted to the constructions indicated by the formula, would give a length, which, expressed in the chosen units, would be the required result. Thus, for example, 3X7 Taking the lengths a, b, m, equal respectively to 3, 7, and 5 times some chosen unit, and constructing the 4th proportional, ab X = — ) m the length x expressed on the given units would be, 3X7 X = — ? — CONSTRUCTION OF ALGEBRAIC FORMULAS 479 1112. Construction of irrational expressions. Since the degree of homogeneity should be 1 (1110), if the radical is of the second degree, the quantity placed luider the radical should be homogene- ous and of the second degree; thus, when this quantity is fractional the degree of the numerator is two units greater than that of the denominator, x = Vab is a mean proportional between the lines a and b ( 970). For X = V5X 7, taking a length as unity (1111, Remark), a and 6 being the lengths equal respectively to 5 and 7 times this unit, the mean proportional x = \/ab expressed in terms of the chosen unit is VS X 7. For x = VS, noting that Vs = V5 X 1, we have the same case as the preceding. X = Va^ + ¥ is the hypotenuse of a right triangle, the sides of which are a and b (703). X = Vd' — ¥ is one of the sides of a right triangle, having a for its hypotenuse and b for its second side (702); this is also a mean proportional VajS between the two lines, a + b and (3 = a — b. (729) X = a\/2, or a^ = 2 a^, is the hypotenuse of a right isosceles triangle, one leg of which is a (Fig. 276). X = Vab + &. After having constructed a mean proportional p = Vab, we have, X = v'jP' + &. a d^ X = — r-, from which a;^ = — , is the chord AB which subtends \/2 2 a quadrant whose diameter is a (706). 2 a 4:0? 3? 4: X = —;=.■ Squaring, we have x^ = — -j and — = -J which \/3 3 a^ 3 shows that the problem reduces to finding the side x a square -A-~. / / 1 \ jo a Fig. 277 Kg. 278 which is to another square a^ as 4 : 3. On a line MN, lay off lengths proportional to the numbers 4 and 3; on MN as a di- 480 ANALYTIC GEOMETRY ameter describe a semicircle; on AN take AC = a, and drawing CB parallel to MN, we have AB = x. From (1000), AB ■.a = AM : AN or AW : a" = IW : AN^ = 4:3. (732) a V2 x^ 2 X = — ^! and — = -) would also be solved by the preceding construction. If the quantity under the radical is a fraction, as . /a= + a¥ - 5b^if choosing an arbitrary length k, as in article (1111), we have. X = a' b^c The quantity written within the parentheses is reduced to a line a, and the denominator to a line m; such that m y m which shows that the construction of the 4th proportional u = — (1111), and the mean proportional x = y/ku (970), will give the required construction. If the index of the root were 2^ = 4, the quantity imder the radical would be homogeneous and of the 4th degree. Let ^ - ^/U^ZH. """V a' + bc To construct x, write a^ be k ^J This formula may be reduced as was the one in the preceding case. X = \/^ = sJkJk- = Vfc Vfc^ =• s/ki y m ^ y m ' CONSTRUCTION OF ALGEBRAIC FORMULAS 481 which shows that the 4th proportional of w = — , the mean m proportional v = \Jku, and the mean proportional x =Vkv must be con- structed. Finally, x may be expressed by a quantity, one part of which is rational and the other part irra- tional ; such as M'- w C AB }> ct ,^\\ ! . ^\l/ x = b±'^d' + V- c" X P' P'O P Kg. 279 First, the irrational part is constructed, AD = VoD^ - '•■i = "^n C = ''a' + b^ — &, which is only possible when a? + W > &. Subtract b from AD, and the point B, in the middle of AM, gives: 6 + \fcfTV^^ This first value of x, considered as positive, is laid off from the origin on OF and is equal to OP. If & > Va^ + 6^ - c^, the point M would be at M', and we AM' - = AB'. This value being negative, is laid would have x = off from in the direction OX equal to OP'. If the radical is preceded by the sign — , & is added to its value AD, and half of the line which results is the second value of x, which being negative is laid off in the direction OX from 0. Let it be required to construct a fa- 2±V4 4+«- (a) From the construction of the division of a in the extreme and mean ratio (971), we have the right triangle ABO (Fig. 280): AO then AI = AO 2^ V 4 + a'. 482 ANALYTIC GEOMETRY This is the first value of x; it is positive, and is laid off in the positive direction AY from the origin A. y The second value of X being -^-AO, The equation (a) becomes: it is negative, and is laid off from A in the negative direction AX. - a±^5a^ - a ± ay/5 a , ,- x= 2 = 2 = 2(±^^5-l). The two values of x represented by this expression are evi- dently the same as those represented by the expression (a), and are obtained by dividing a in the extreme and mean ratio. THE GENERAL CONSTRUCTION OF CURVES REPRESENTED BY EQUATIONS. 1113. An equation between two variables, x and y, being given, if these variables are considered as coordinates, each pair of real values of x and y which satisfies the equation determines a point; varying a; in a continuous manner between certain limits, the equation is ordinarily satisfied by real and continuous values of y, and then a continuous series of points, that is, a line, is obtained. Thus, in general, an equation between two coordinates represents a line (1099). 1114. To determine points of a curve, the values of x are ordinarily taken in arithmetical progression (357), and the corre- sponding values of y calculated from the equation. Above all, when the function is a whole algebraic function (447, 504), it is wise to take this precaution, because, in order to shorten the computations, the differences between the successive values of y may be used in getting new values. For example, let it be required to construct the equation y = an? + &, a form which is met with in equations relative to the determination of the curve taken by the cables in suspension bridges. Suppose we have a = 0.1 and b = 1, then j/ = 0.1 ar* -t- 1, CURVES REPRESENTED BY EQUATIONS 483 the following table shows that in giving successively to x the values 1, 2, 3, . . ., the values obtained for y are such that in taking their first differences, 0.1, 0.3, 0.5, . . ., the second differ- ences between the first differences are equal. Thus, taking suc- cessively a; = 0, X = 1, and x = 2, we have respectively y = 1, y = 1.1, and y = 1.4; the first differences are 0.1 and 0.3, and the constant second difference is 0.2. This second difference added to the last first difference gives the next following first difference, and each first difference added to the immediately preceding value of y gives the next following value of y; thus it is seen that by simple successive additions, the values of the first differences and then the values of the ordinates are ob- tained. abscissas x... 1 2 3 4 5 6 7 8... ordinates?/ ..1 1.1 1.4 1.9 2.6 3.5 4.6 5.9 7.4... 1st differences . . 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5... 2d differences . . 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2... The negative values of x would give the same values for y. According as the function is of the 2d, 3d, 4th, . . ., degree, the constant differences are respectively the second, third, fourth, etc., differences, which are obtained by calculating from the equation, 3, 4, 5, . . ., ordinates, and taking their successive differences. Having the constant difference, the process is re- versed as was done in the above example until the value of the next ordinate is obtained, and so on. 1115. Instead of calculating all the ordinates in constructing the curve y = ax? + h, the three first equidistant ordinates, AO, BP, CQ, may be calculated. Drawing the parallels AF and BG to the axis Ox, the two first differences, BH and CG, are determined, and prolonging AB, we have the second difference, CC = CG - CG = CG - BH, and is constant. To construct the P Q R Fig. 281 fourth, ordinate DR, prolong BC to D', and take D'D = CC. In the same manner the next ordinate ES, and all ordinates following, may be constructed, and then joining the points A, B, C, D, etc., by a curve, we have the representa- tion of the equation y = aj? + h. 1116. Empiric functions. In practice it happens daily that 484 ANALYTIC GEOMETRY observation or experiments furnish a series of corresponding values of two variables, without any algebraic equation to rep- resent the law which governs these variables. In this case, taking the values of one of the variables for ab- scissas and the corresponding values of the other variable for ordinates, and drawing a smooth curve through the points thus obtained, if the points are near enough together, this curve will represent with sufficient accuracy the law which governs these variables. Such a curve furnishes a picture of the observed phenomena; it may be used to find any intermediate points that were not directly observed; if it closely resembles some known curve, it may be expressed by an equation or formula known as empiric; any anomaly which breaks the continuity of the curve indicates an error in the observations or a peculiarity in the phenomena observed. STRAIGHT LINE 1117. The general equation of a straight line with reference to a rectangular coordinate system is y = ax + b. Let any straight line AB be situated in the plane of the rectangular axis Ox and Oy. From any point M on this line drop a per- pendicular MP to the axis Ox; it determines the coordinates MP = y and OP = x of the point M. Through the point C where AB Fig. 282 intersects the axis Oy, draw CD parallel to the axis Ox. In the right triangle CME we have (1055), ME = CE X tan BCD. Adding EP to the two members of this equation, we have, ME + EP = CE X tan BCD + EP- Noting, first, that ME + EP = y; second, that the angle BCD, which the line makes with CD or the axis Ox, is constant, and therefore its tangent, which may be represented by a, called an angular coefficient, or the slope, is also constant; third, that CE = x; fourth, that EP = OC is also constant and may be y M^ B c -"""l^ D -^ P STRAIGHT LINE 485 represented by b, called the ordinate at the origin, the preceding equation takes the form y = ax + b, which is the equation of a straight line, since it was established for any point in the line, and took into account the different signs which enter into the equation. Remark 1. When the straight line AB passes th r n xLgh_the origin 0, the ordinate at the origin OC = 6 = 0, and the equa- tion becomes : y = ax. Remark 2. When AB is parallel to Ox, the angle BCD is zero, then the tan BCD = a = (1027), and the equation be- comes : y = b. Remark 3. In the case where a = and 6 = 0, the equation becomes : which indicates that the line coincides with the a;-axis. Remark 4. If the line were parallel to the y-axis or coin- cided with it, its equation would be obtained by interchanging y and x in the last two equations given above. Thus, we would have X = b and a; = 0, wherein b is no longer the ordinate at the origin, but the abscissa at the origin. Remark 5. It is seen that the equation of a straight line is of the first degree (510). Conversely, any equation of the first degree between two variables is the equation of a straight line. This is why straight lines are called lines of the first degree. 1118. The equation of a straight line whose slope is given and passes through a point, the coordinates of which are x' and y'. For the point {x', y'), we have, y' = ax' + b, and b = y' — ax'. Substituting this value of b in the general equation, y = ax + b, we have, y - y' = a{x - x'). 486 ANALYTIC GEOMETRY 1119. The equation of a straight line passing through two given points (x', y', and x" , y"). a being the unknown slope of the line, for the point {x', y'), we have (1118), y — y' = a{x — x'). This equation should be satisfied by putting y = y" and x = x", which gives y" — y' = o- (^" — ^')- Eliminating a by division, we have, y — y' X — ocf y" — y' x" — x' If one of the points is on the a;-axis, and the other on the y-axis, that is, if we have x' = p, y' = 0, and y" = q, x" = 0, the equation becomes: y__ ^-P or^ I y _i q -p p q If one of the points is at the origin, if for instance, y" = x" = 0, we have the equation of a straight Une through the origin to a point (a;', y'). Thus, y — y' _ X — x' y _ X — y' — x' y' x' 1120. The intersection of two straight lines given by their equa- tions. Any two lines, straight or curved, being given by their equa- tions, by solving the system of two equations with x and y as the unknowns, which cease to be indeterminate variables, the values obtained are the coordinates of the points of intersection of the lines. Thus the point of intersection of two hues (520, 1117) is b' -b J ab' - a'b x = ; and y = ;- • a — a' a — a' Conversely, having a system of two equations involving two unknowns to solve, if -the two lines represented by the equa- tions are constructed, the coordinates of each point of intersec- tion will be a solution of the system (580). 1121. Two straight lines perpendicular to each other, making CIRCLE 48? two angles with the a;-axis whose difference is equal to 90°, the tangents of these angles give the relation in article (1044); from which it follows that aa' = 1 or aa' + 1 = 0. CIRCLE. 1122. The definition of a circle (665) may be expressed in polar coordinates. Thus, if we put (1100); p = OM + r, and make a = and r constant, we see that, no matter what the value of a, we always have p = r, an equation which is satisfied by any point in the circumfer- ence of a circle whose center is at the origin and whose radius is r. 1123. General equation of a circle, with respect to a system of rectangular coordinates (1099). Let M be any point in the circumference of a circle whose center is C and whose radius is r. Let MA = y and OA = x, the coordinates of the point M, and CB = q and OB = p, the coordinates of the center, which remain constant. In the right triangle CDM (730): or Ml? = y^ + q^-2qy; or CD' = x^ + p^ — 2 px. MD MD = y-q, CD = x-p, (728) Adding the equations of MD and CD and replacing MD' + CD" by r', or y' + x' — 2 qy — 2 px + q' + p' = r^, y' — 2qy = r^ — x'^ + 2px q^ -p\ from which y = q ± Vg^ + r' — x' + 2 px — q^ — pi'. (572) Such is the general equation of the circle in rectangular co- ordinates. When E is the origin and EC the a;-axis, we have q = and p = r, and the general equation becomes or y^ + x' = 2rx — x' — 2rx + r' = r', and y = ± V2 rx — x', 4$§ ANALYTIC GEOMETRY If the center of the circle is at the origin, we have g' = and p = 0, and the equation becomes if + x^ = r^ and 2/ = ± Vr^ — 7?. It is seen that in each of the three cases which we have just examined, two values of y correspond to each value of x\ which is as it should be, since the equation of the circle is of the second degree. Furthermore, in the last two cases the values of y are equal and opposite in sign, which indicates that the curve is symmetrical with respect to the a;-axis. 1124. Uraw a tangent to a circle at a point M taken on the circumference. Draw the radius OM, and the perpendicular .4J5 at the extremity of this radius is the required tangent. Proof. It suffices to prove that AB has only ^^T^ the point M in common with the circle, that is, that any point C on this line, other than M, is outside of the circle. Drawing OC, this line is oblique and greater than OM, which is a radius; therefore the point C is outside of the circle, and AB is the required tangent at the point M (954). 1125. Since AB is tangent to the circle, all its points except M are situated outside of the circle; therefore any straight line OC is greater than OM; therefore the radius OM, drawn to the point of contact, is perpendicular to the tangent (620), and consequently to the circumference (678). Thus, to draw a normal at a certain point in the circumference, it suffices to pig, 285 connect this point to the center. 1126. Draw a tangent to a circle through a point M taken outside of the circle (954). Draw MO. On this line as a diameter describe a circumference which cuts the given circumference in the points T and T', then connecting these points with M, we have TM and T'M as the required tangents. Proof. Drawing the radii OT and OT', each of the angles OTM and OT'M is a right angle, being inscribed in a semicircle (684), and the lines MT and MT', perpendicular to the radii OT and OT' at their extremities, are tangent to the circle (1124). MLLIPSB 489 ELLIPSE 1127. The ellipse is a curve such that the sum MF + MF', of the distances of any point M to two fixed points, foci, F and F', is a constant quantity. It is seen that an ellipse is defined by its equation in focal coordinates (1101). Designating the radius vectors of the points in the curve by the variables p and p', and the constant sum by 2 a, we have, p + p' = 2 a. 1128. As in the case of a circle (666), a portion of an ellipse is an arc, and the straight line which joins the extremities of the arc is a chord. On an ellipse, and, in general, on any curve, an arc of one degree is one such that the normals erected at its extremities form an angle with each other of one degree. The chord AA', which passes through the foci, is the major axis of the ellipse. The chord BB', which is the perpen- dicular bisector of the major axis, is the minor axis of the ellipse. The point of intersection of the two axes is the center of the ellipse. Any chord which passes through the center is a diameter of the ellipse. The extremities A, A', B, and B' of the axes are the vertices of the elhpse. 1129. The foci are equally distant: 1st. From the vertices, AF = A'F' and AF' = A'F; 2d. From the center, OF = OF'. 1st. The vertices A and A' are part of the ellipse, the sums of their radius vectors are each equal to the constant 2 a (1127), and consequently equal to each other; therefore AF + AF' or 2AF + FF' = A'F' + A'F or 2 A'F' + F'F. Subtracting FF' from both members, we have 2 AF = 2 A'F', and AF = A'F', and for the same reason AF' = A'F. 2d. Having OA = OA' and AF = A'F', we also have, OA - AF = OA' - A'F' or OF = OF'. 490 ANALYTIC GEOMETRY 1130. The constant sum 2 a of the radius vectors is equal to the ■major axis. Since the point A is part of the elHpse, we have, AF + AF' = 2 a. Replacing AF' by its equal A'F, we have, AF + A'F = 2 a = AA'. 1131. The equation of an ellipse when the major and minor axes are taken as the coordinates (1099, 1128). Let 2 a = AA', the major axis, and 2 c = FF', the distance between the foci. We always have 2 a> 2 c or a> c. In the right triangles MPF' and MPF, we have respectively, MF' or p'2 = MF + PF'^ and MF^ or p" = FF + PF. Since MP = y, PF'= OF' + 0P = c + X, or TF^ = c^ + 3? + 2cx, (727) and PF = 0F - OP = c - x, or TF =& -\-3? -2cx. (728) Substituting these values in the formulas for p^ and p'^, p'2 = if + x^ -\- c^ + 2 ex and p^ — y^ + x^ + c^ — 2 ex. (a) Subtracting these two equations, we have p'2 — p2 or (p' + p) (p' - p) = 4:cx; 4: ex 4 ex 2 ex from which p — p = , = -^— = —— • P + P 2, a a Adding this equation to p' + p=2a, 2 ex ex we obtain 2 p' = h 2 o, from which p' = ha, a a and therefore p'^ = — =- + a^ + 2 ex. (727) a' Putting this value of p" and the value in (a) equal to each other, and eliminating the denominator a^, aY + aV + aV + 2 a^cx = cV + a* + 2 o?cx. ELLIPSE 491 Canceling the term 2 a?cx and grouping the terms, ay + (a^ - c2) a;2 = a? {a? - c^), representing the constant {a? — c=) by W (1133), we have for the equation of the curve : ay + 5V = a'V or ^ + ^ = 1; 2/ = ±-Va2-a:2; (571) which shows that for every value of x there are two equal values of y opposite in sign, and consequently the curve is symmetrical with respect to the x-axis. In expressing the value of x in terms of y, it will be seen that for every value of y there are two equal values of x opposite in sign, and consequently the curve is also symmetrical about the 2/-axis (1138). Remabk. In the case where a = h = r the equation of the elUpse becomes 2/2 ^ -J.2 _ y.2^ which is nothing other than the equation of a circle (1123). Thus, the circle is a special case of the ellipse, in which the semi-axes are equal to the radius r. Therefore the properties of the ellipse are also those of the circle. 1132. The straight lines BF and BF', which join the extremities of the minor axis to the foci, are each equal to the semi-major axis a. These lines are equal since they cut off equal distances from the foot of the perpendicular BO (620) Furthermore, we have, BF + BF' or 2BF = 2a and BF = a. 1133. Having BF = a, OF = c, if the semi-minor axis OB is represented by b, the right triangle BOF gives (730): P = a" - c^ Thus, in the equation of the ellipse (1131), the constant quantity h is the semir-minor axis. 1134. The distance FF' = 2 c between the foci is called the 492 ANALYTIC GEOMETRY 2 c c focal distance, and the ratio - — = - of the focal distance to the ti a a major axis is called the eccentricity of the ellipse. Designating this eccentricity by e, we have, ■=^^/^ The eccentricity of the ellipse lies always between and !• at the limit the ellipse is a circle, and at the limit 1 the curve is flattened to a straight line joining the vertices and the foci. 1135. The foci and one of the axes of an ellipse being given to find the other axis (Fig. 287). 1st. A A' being the major axis, and F and F' the foci (1128), the perpendicular bisector BB' of AA' coincides with the minor axis; and if from one of the foci F as center and AO = a as radius, an arc is described, it will cut BB' in the points B and B', which are the extremities of the minor axis (1132). 2d. If the minor axis BB' and the foci F and F' are given, to find the major axis, lay off to the right and left of the point on FF', the distance BF = a. 1136. The axes A A' and BB' of an ellipse being given, to find the foci (Fig. 287). From one of the extremities B of the minor axis, with the semi-major axis for radius, describe an arc which cuts AA' in the points F and F', which are the foci of the ellipse (1132). 1137. The ellipse is the geometrical locus of all the points the sum of whose radius vectors is equal to the major axis 2 a (609, 1130). 1st. M being a point situated outside of the ellipse, we have MF + MF' > 2 a. Drawing CF, the point C being on the ellipse, we have CF + CF' = 2 a. Replacing CF by the greater quantity MC + MF, we have, MF + MC + CF' or MF + MF' > 2 a. 2d. The point M' being situated within the ellipse, we have, M'F + M'F' < 2 a. Becatise drawing CF, the point C being on the ellipse, we have, CF + CM' + M'F' = 2 a. Fig. 288 ELLIPSE 493 Replacing CF + CM' by a smaller quantity M'F, we have, M'F + M'F' <2a. Corollary. The converse statements of the above are also true. 1138. The major and minor axis both divide the ellipse into two equal and symmetrical parts. 1st. M being a point on the ellipse, its corresponding symmetrical point M' with respect to the major axis AA' (836) is also on the ellipse. This follows from the equation of the curve (1131); furthermore, the two equal right triangles, MPF and M'PF, giving MF' = M'F', we have, M'F + M'F' = MF + MF' = 2 a, and the point M' is on the ellipse (1137). From this it' follows, that if the part of the ellipse AMA' be turned about the axis AA', it would come into coincidence with the part AM' A'; therefore they are equal and symmetrical. 2d. The point M", symmetrical to M with respect to the minor axis BB', is also on the ellipse. This follows directly from the equation, and may also be proved as follows: Having OP = OP' as quantities each equal to QM = QM", and OF = OF', it follows that FP' = F'P and FP = F'P'; and since MP = M"P', the two equal right triangles MPF', M"P'F, give MF' = M"F, and the two other equal triangles MPF, M"P'F', give MF = M"F'; it follows that M"F + M"F' = MF' + MF = 2a; therefore M" is on the ellipse, and the ellipse is also divided into two equal and sym- metrical parts by the minor axis. 1139. The center of the ellipse divides all the diameters into two equal parts. The point M being on the ellipse, prolong- ing MO to M', making M'O = MO, and drawing MF, MF', M'F, and M'F', in the quadrilateral MFM'F', the diagonals cutting each other in two equal parts, the figure is a parallelogram (660), and we have, M'F + M'F' = MF + MF' = 2 a. 494 ANALYTIC GEOMETRY Therefore the point M' is on the ellipse (1137), and MM' is a diameter divided into two equal parts at the point 0. 1140. Any diameter MM', other than the major and minor axes, divides the ellipse into two equal parts hut not symmetrical with respect to that diameter (837). Bringing the part MBM' upon the part M'B'M by turning it about as a center until M coincides with M' and M' with M, and considering any diameter BB', after the change, the part OB will coincide with the part OB', since the angle BOM = B'OM', and since OB = OB', the point B will coincide with the point B'. The point B being any point, it is seen that all the points on the part MBM' fall upon the curve M'B'M; therefore any diameter divides the ellipse into two equal parts. 1141. From the equation of the ellipse (1131), we may deduce, that for any point M, ^' (a) (729) y a' or r 3^ {a + x) (a — x) a' or noting that a + x = A'P and a — x = AP, „2 62 r AP X A'P a' which shows that the ratio of the square of an ordinate to the prod- uct of the corresponding segments of the major axis is equal to the B V M /^^ ~7 y \^^ &{ E' " 1 ^ F h" fT" "^ c /A ) ■^ P Kg. 291 Fig. 292 ratio of the squares of the minor and major axes, and therefore this ratio is constant. For another point, we would have. y- and then AP'XA'P' a"' y' AP X A'P l/^' AP'XA'P'' ELLIPSE 495 Thus the squares of the ordinates are to each other as the prod- ucts of the corresponding segments of the major axis. From the equation of the elHpse, and by the same process of reasoning, the same properties are found for the abscissas and the corresponding segments of the minor axis: x" a^ 3? BQX B'Q BQ X B'Q ¥ x'^ BQ' X B'Q' 1142. Describing a circle on the major axis as diameter, and drawing any corresponding ordinates MP = y and CP = Y of the elUpse and of this circle (Fig. 292), we have, Y a ' Proof. The right triangle OPC gives OCf -UP = CP'' or a" - x^ = T'. Substituting in equation (a) of the preceding article, we have, if W y h Y^ a^ Y a ^ ' Describing a circle upon the minor axis, the same relation is found to hold, thus, MQ X a ... 1143. From the equation (a) of the preceding article, we may consider any ellipse having 2 a and 2 h for its axes, as being a projection of a circle of the diameter 2 a upon the plane of the ellipse, and from the equation (6) that any circle of the diameter 2 h may he considered as being the projection on its plane of different ellipses having a common minor axis 2 b. From these relations, diverse interesting consequences relative to the supplementary chords, to the conjugate diameters, to the circumscribed parallelograms, and to the area of the elhpse, may be deduced (11 2). Thus the elUpse ABA'B', which has AA' = 2a and BB' = 26 for its axes, is the projection of a circle aba'V , having 2 a for its diameter, and its plane making an angle Q, whose cosine is -, with the plane of the ellipse. 496 ANALYTIC GEOMETRY The diameter aa' being parallel to the plane of the ellipse its true length is projected upon AA', and for any point m the pro- jection of the perpendicular mp to aa' is MP = mp, cOkS = mp x -; a from which it follows that M is part of an ellipse whose major axis is AA' and minor axis is BB' = 66"' cos ^ = a"^ X-. Each of the elements mp of a circle having its surface multi- plied by cos for its projection, the projection of the entire circle is equal to the surface of the circle multiplied by cos 6. This is not only true of the projection of a circle, but also of the projection of any plane surface. Drawing any diameter de of the circle, and the two chords cd and ce, which are perpendicular to each other (684), the di- ameters mm', nn', which pass through the middle points i and h of these chords, are also perpendicular to each other, and each one divides all the chords, which are parallel to the other, into two equal parts. Moreover, the tangents m, n, wl, n' form a circumscribed square rstu. Projecting these lines which have just been discussed upon the plane of the el- lipse, and representing the different points by the same letters, written as capitals, the chords CD and CE, which start from the same point C in the curve and end at the extremities of the same diameter DE, are called supplementary chords; the diameters MM', NN', are parallel to these chords, and each divides into two equal parts all chords parallel to the other, which property gives them the name of conjugate diameters; moreover, the projection of the square rstu is a circumscribed parallelogram RSTU, the sides of which are parallel to the conjugate diameters MM' and NN' passing through the points of contact; finally, the ELLIPSE 497 square rstu being constant, the area of the circumscribed paral- lelogram RSTU is also constant and equal to 4 a^ cos 6 = 4 a^ - = 4a6. 1144. Thus, in an ellipse, any diameter MM' which divides a chord AA' into two equal parts, divides all the chords NN', BB' . . ., parallel to the first, in the same manner, and the two diameters MM' and NN' of the elHpse are said to be conjugate diameters when each divides into two equal parts the chords parallel to the other. 1145. A diameter MM' (Fig. 294) being given, find its conju- gate. Draw a chord CC parallel to MM', and, drawing the diameter NN' through the middle point D of the chord CC, we have the conjugate diameter of MM'. When the major axis of the ellipse (Fig. 295) is known, draw- ing a chord AB parallel to MM' through its extremity A and joining B to A', the diameter NN' parallel to BA' is the conju- gate of MM' (1143). This construction is more simple than the preceding one. 1146. An ellipse being given, to determine: first, its center; second, its axes; third, its foci. 1st. Drawing two parallel chords CC and DD', the straight line EE' which joins the middle points of these chords is a diam- eter, the middle point of which is the center of the ellipse. 2d. From the center 0, describe a circle with a radius suf- ficiently long to cut the ellipse in four points ; then the line which joins E and G and the line which joins G and E' are respectively par- allel to the major and minor axes, and these axes may be drawn. 3d. Having the axes, the foci are determined as in~ article (1136). To determine the center of an arc of an ellipse, inscribe two par- allel chords in the arc; draw a line through the middle points of 498 ANALYTIC GEOMETRY these chords, then this line having the direction of a diameter will pass through the center. Now by drawing in two new chords parallel to each other, and repeating • the first construction, the intersection of the two bisectors will give the center of the ellipse. In case the arc is long enough, so that the circle GEE' can cut it in two points G and E ov G and £", the major or minor axis may be drawn, and, erecting a perpendicular to this axis at the center, the second axis is obtained. 1147. From article (1142) an easy method of constructing cm ellipse by points may be deduced. Describing circles on the axes as diameters (Fig. 292), drawing any radius OC and through the points C and D drawing parallels to the axes, these parallels meet at a point M on the ellipse. Thus, MP being parallel to OP, we have. MP CP OD OC or -j; = It is evident that as many such points may be constructed as desired, and when enough have been determined and connected by a smooth curve we have an ellipse (1099). 1148. Another method by points (1147). AA' being the major axis of an ellipse, and F and F' the foci, I F-- TT 1 1 m \ 7 ~'d' / N, r Fig. 297 Fig. 298 with F and F' as centers and a radius equal to Am, which may vary from AF to AF' , describe two arcs; then from the same cen- ters with a radius equal to A'm describe arcs which cut the first arcs in the points D, D', C, and C , which are on the ellipse, because Am + A'm = AA' = 2 a (1137). In this manner as many points may be determined as is desired, and a smooth curve connecting them is the required ellipse. 1149. A method used by gardeners for constructing an ellipse (Fig. 298). AA' being the major axis, and F and F' the foci, fasten the ends ELLIPSE 499 of a cord at F and F', making the length of the cord FM + MF' equal to the major axis AA'; hold the cord taut with a pointed stick at M, and walk around making a mark in the soil with the stick. If the cord is held taut, the sum of the radius vectors FM and F'M is always constant and equal to the major axis AA', and we have an ellipse. The same method may be used on paper by substituting a pencil for the sharp stick (1137). 1150. Construction of an ellipse with a rule (Fig. 299). Marking three points C, E, and G, on the edge of a thin rule, such that CG = OA = a the semi-major axis, and EG = OB = b the semi-minor axis, from which CE = a — b; moving the rule in such a way that the point E remains constantly upon AA\ and C upon BB', G will follow the curve of the ellipse whose major and minor axes are respectively AA' and BB'. This method is used for constructing the intrados and extrados of arches which have the form of an ellipse. The point G follows the curve of an ellipse because, drawing CH parallel to OA, we have, GP^GE y ^b . GH GC °^ Va^ - x' a ' and 2/ = - Va^ - x\ (1131) If from the point C as center, CE = a — & for radius, an arc of a circle had been described, the point E would have been determined; then drawing CE and prolonging it to G, making EG = b, the point G upon the ellipse would have been found. 1151. The elliptic-compasses are con- structed according to the principle demon- strated in the preceding article, and permit the construction of an ellipse by a con- tinued motion. It consists of two slots or guides assembled in the form of a cross (Fig. 300) so that they may be made to coincide with the axes AA' and BB' of the ellipse; a rod CD carrying two slides E and G, which may be fastened at any two points. The slide E carries a pivoted foot, which fits in the slot AA' and G a point or pencil which traces the curve when the rod is moved. At the extremity of the rod is another pivoted foot, which is fixed and 500 ANALrtlC CfSOMETRY slides in the slot BB'. Having fixed the slides in such a manner that CG = OA and EG = OB, and placed the instrument so that the guides coincide with the axes of the ellipse, an ellipse is traced by turning the rod CD. The middle point of CE describes a circle about the center 0. Therefore if this point is joined to the center by a link, one of the feet C or £ may be left off. 1152. Instead of spacing the points on the rule in such a man- ner as was done in article (1150), we may take GC = 6 and GE = a (Fig. 301). Moving the rule so that C follows the major axis and E the minor axis, the point G will describe an ellipse. GP EQ GC GE °'' V^ y a? (1131) 1153. Draw a tangent to an ellipse through a point M taken on the curve. Fig. 302 Draw the radius vectors MF' and MF, prolonging the latter so that MC = MF'; draw CF', and the perpendicular TP dropped from the point M on CF' is tangent to the ellipse, that is, that any other point M' taken on TP lies outside the ellipse. Proof. Joining M'F, M'F', and M'C, the triangle MCF' being isosceles, the straight line MP is perpendicular to CF' at its middle point, and we have M'F' = M'C. In the triangle FCM', we have M'F + M'C or M'F + M'F' > CF or MF + MF' or 2 a; therefore the point M' is situated outside the eUipse (1137), and TP is the tangent at the point M. Remark 1. The tangent TP bisects the angles, formed by each radiu vector with the prolongation of the other. Proof. The triangle F'MC being isosceles, the perpendicular ELLIPSE 501 MP bisects the angle F'MC at the vertex and also its vertical angle FMC. Remark 2. The preceding method for drawing a tangent to an ellipse, and those which follow, except that in (1159), do not require that the ellipse be constructed. This is a great advantage where the ellipse is constructed by points; because, as soon as a point is found, its tangent may be drawn, and in this manner the curve is blocked out, making it possible to draw it in with a lesser number of points. 1154. Draw a normal to an ellipse at a point M (Fig. 302). Join M to the foci, then the bisector MN of the angle formed by the two radius vectors is normal to the ellipse, that is, per- pendicular to the tangent TM (678, 946). Proof. The angles CMF' and C'MF being equal, their halves are equal, and we have PMF' = TMF; since F'MN = NMF, adding these two equations, we obtain PMN = NMT; therefore MN is perpendicular to TM (614). Prolonging MN to FF', and projecting the point M on FF', the projection NQ of MN on FF' is called a subnormal. Since, when radius vectors FM and F'M are drawn from any point M, the angle of incidence formed with the tangent is equal to the angle of reflection (950), it follows that on an elliptic billiard table, a ball shot from one focus to any point on the cushion will pass through the other focus, then, after touching the cushion the second time, will pass through the first focus, and so on. The same is true of rays of heat or light which radiate from one focus of an elliptical mirror. Because of this reciprocal action of each focus they are called conjugate foci. 1155. MT being the tangent to the ellipse at the point M, drawn according to the construction in article (1153), and the center of the ellipse, in the triangle FF'C, the straight line OP bisects FF' and CF', and we have OP = -^ = a (699) ; which shows that the circle described on the axis AA', as diameter, passes through the point P, and is the geometrical locus of the projections P, P' , of the foci on the tangents (609, 715). Fig. 303 502 ANALYTIC GEOMETRY Describe a circle from the focus i^ as a center, with AA' = 2 o for a radius, then drawing any radius FC we have MC = MF'. Therefore an ellipse may be defined as a curve such that all its points are equally distant from the circumference of a circle (670) and a fixed point within the circle. From this definition a method of constructing the ellipse by points may be deduced (1147 to 1152). The circle described on the major axis AA' as diameter is often called the principal circle of the ellipse, and the one described from one of the foci as centers, with the major axis FC = AA' for radius, is called the directrix circle. From that which was said above, in order to draw a tangent to an ellipse at the point M (Fig. 303), describe a circle on A A' as \C/ Fig. 304 Fig. 305 diameter, and another having F as center and AA' for its radius; draw the radius FC passing through M, then CF' which will cut the circumference of the principal circle in P, and joining M to P, we have the required tangent. 1156. To draw a tangent to an ellipse parallel to a given straight line CD. From the focus F' draw F'G perpendicular to CD; from the other focus F with a radius FG = 2 a (1131), describe an arc which determines the point G; drawing FG, we have the point of contact M; the required tangent is now obtained by drawing a parallel to CD through M, or a perpendicular to F'G through M. To draw a tangent to an ellipse making any given angle with a given line, draw a tangent parallel to a line which makes the required angle with the given line (955). 1157. Draw a tangent to an ellipse through a point M taken outside of the ellipse. ELLIPSE 503 T-'^ From the point M as center, and with the distance from the point M to the nearest focus F' as radius, describe an arc; from the other focus F, with the major axis of the ellipse as radius, describe a second arc, which cuts the first in the points C and C" ; draw CF and C'F, which determine the points N and N', and drawing MN, MN', these lines are tangent to the ellipse at N and N'. Proof. From NF' + NF = CF, major axis, we have NF' = NC; since MF' = MC, the line MN is perpendicular to CF' at its middle point (621), and bisects the angle F'NC; therefore it is tangent to the ellipse at the point N (1153). 1158. Noting (Fig. 293) that the point of meeting of the tan- gent to the ellipse at M with the axis AA' is the projection of the point of meeting of the tangent to the circle at m with the diam- eter an', it follows (Fig. 306) that all eUipses having the same major axis AA', and the circle which has this major axis as its diam- eter, have the following property: namely, that the tangents drawn through the points M, N, ... where a plane perpendicular to the major axis cuts the ellipses and the circle, meet in the same point T on a prolongation of the major axis. This property reduces the difficulty of drawing a tangent to an ellipse to that of drawing one to a circle (1124, 1126). When the point through which the tangent is to be drawn is outside of the ellipse, it should be on the axis xx'. In the right triangle ONT, we have (705), OP:ON = ON:OT, andOr = ^, which determines the point T where the tangent to the ellipse at the point M meets the prolongation of the major axis AA'. Describing a circle on the minor axis BB', the tangent drawn to this circle at the point where it cuts ON and the tangent TM meet in the same point on the prolongation of the minor axis BB\ In this case the two points of contact lie in the same 504 ANALYTIC GEOMETRY plane perpendicular to the minor axis BB'. From this follows a method, analogous to the preceding one, for drawing a tangent to an ellipse at the point M. These two methods taken together give the points where the tangents to the ellipse meet the two axes; which may be used to verify the correctness of the con- struction of the ellipse. 1159. Another method of drawing a tangent to an ellipse at a point taken on the curve. Through the point M draw two chords MC and MD; through the points C and D draw two others CE ^ ■ and T)G parallel to MD and MC re- '^f^^EziIi:::^^ spectively, then the parallel MT to EG y^i ^ / \ drawn through M is the required tan- U X/ V. gent. ^v '/'~~^^2^^^') Drawing the diameter MM' , the chord ^v^^_j^^^^^^$>^ EG and all parallel to it are bisected; ,,^ MT is therefore parallel to the conjugate ^'e. 307 diameter of MM' (1141), which is still another method of determining the direc- tion of MT; but it is easier to construct EG than the conjugate diameter of MM'. Remark. The parallel M'T to EG or MT is also tangent to the ellipse. Thus, as is the case with the circle, tangents drawn at the extremities of the diameter of an ellipse are parallel (1143). 1160. Two ellipses are said to be similar when their axes are proportional, that is, when a : a' = & : 6' (1131). As is the case for two similar polygons (695) or circles (749, 750), if two ellipses are similar, the ratio of their axes is equal to the ratio of any homologous linear dimensions straight or curved. The surfaces of two similar ellipses are to each other as the squares of their axes (s : s' = a^ : a'^), and in general as the squares of any homologous linear dimensions. Two portions of similar ellipses whose perimeters are formed of homologous lines are also similar, and their surfaces are to each other as the surfaces of the ellipses. Similar ellipses have the same eccentricity e, since we have c:a = c':a' (1134). 1161. The length of an ellipse or an arc of an ellipse is not given exactly by any elementary geometrical construction (951); but, considering the ellipse or arc to be made up of a series of ELLIPSE 505 very short straight lines, the length is equal to the sum of these lines (1111). I being the length of a semi-ellipse, whose major and minor axes are respectively a and &, we have, '"«['- (i 'J- KH^f-K^ «-)"-■]■ in which e is the eccentricity of the ellipse (1134): «=^\/^=#=£¥^- (») When a = 6 = r-, we have e = 0, and therefore I = irr; which is as it should be, since the semi-ellipse becomes a semi-circle (752). e being put in the form y ^^ —^ , with the aid of logarithms, the value of e is easily computed; and letting o- rep- resent the sum of the quantities within the parentheses, we have, Z = Tra (1 — o); taking a = 1, l = 7T (1 - 0-). This gives the value of I with sufficient approximation, and is used in calculating the values in the fourth column of the fol- lowing table. Multiplying these tabular values by a expressed in feet or inches, we obtain I in feet or inches. Taking the axes of the ellipse as coordinate axes (Fig. 292), X being the abscissa MQ = OP, and y the ordinate MP, of any point M on the curve, and calling the angle corresponding to CO A, 9, we have, COS = - and sin 5 = t or w = & sin 6. For the point M, the value of the subnormal (1154) is: s =— cos 0. a The slope of the normal with reference to the a;-axis, designating the angle which the normal makes with the axis as a (1030), is: y b sin 6 a , „ tan a = - = — = - tan o. s W a b — COSP a It is useful to know this slope in constructing elliptical arches (1150). Example. Having a = 15 ft., and 6 = 10 ft., for a point in the curve whose abscissa is a; = 11.49060 ft., we have. 506 ANALYTIC GEOMETRY . 11.49060 ^naa^^ COS e = — — — = 0.76604. 15 From the table, the value of sin 6 which corresponds to this cos 6 is sin 6 = 0.64279. Therefore, we have y = IQ x 0.64279 = 6.4279 ft. In this manner any number of points may be determined and the curve drawn. The subnormal is s = ^ X 0.76604 = 5.10693, 15 and the slope of the normal is tan a = ^ X 0.83910 = 1.25865. Having tan a, the table (1071) gives a = 51° 32'. If the ratio - = cos 6 were not contained in the following table the table (1071) could be used. Having a = 15 ft., and 6 = 10 ft., I may be obtained as follows: Putting -= cos e = ^ = 0.66667. ° a 15 The angle 6 is constant for a given ellipse, and is equal to the angle COA (Fig. 292), wherein OP = OB = b (1143). When a = 1 we have b = cos 0. This is indicated in the sec- ond column of the table. Taking a = I, the table gives the length I of the perimeter of the semi-ellipse by interpolation (755): 2.64768 - 0.01823 n Smo ~ n SahI = 2-64768 - 00343 = 2.64425. 0.66913 — 0.65606 Therefore, in feet, we have, Z = 15 X 2.64425 = 39.66375 ft. Remark. If, instead of b, the semi-focal distance c (1134) had been given, we would have, c Va^ - V Vl - ^= Vl - cos^fl = sinft V a^ a a Then the table would give the value of I corresponding to sin S when a = 1. ELLIPSE 507 Designating the radius of curvature at the point whose abscissa is X, by p, we have, ^=ri' or, putting - = sin a or - = cos a, and - X -— cos B, ' '^ a a a a we have, a sin^ /8 sin a Designating the abscissa of the center of curvature by x', we have, c'x^ a* Table for the construction of the ellipse by points, for the deter- mination of the normal at any of these points and the calculation of the semi-perimeter. CB i perim. .2 i perim. 1 6 = cos e sin 9 I Differ- ^ b = cos B sin e I Differ- for a = 1 ences. for a = 1 ences. 0° 1.00000 0.00000 3.14159 0.00024 45° 0.70711 0.70711 2.70128 0.01767 1 0.99985 0.01745 3.14135 0.00072 46 0.69466 0.71934 2.68361 0.01787 2 0.99939 0.03490 3.14063 0.00120 47 0.68200 0.73135 2.66573 0.01806 3 0.99863 0.05234 3.13944 0.00167 48 0.66913 0.74314 2.64768 0.01823 4 0.99756 0.06976 3.13776 0.00215 49 0.65606 0.75471 2.62945 0.01838 5 0.99619 0.08716 3.13561 0.00262 50 0.64279 0.76604 2.61107 0.01852 6 0.99452 0.10453 3.13299 0.00310 51 0.62932 0.77715 2.59255 0.01865 7 0.99255 0.12187 3.12989 0.00357 52 0.61566 0.78801 2.57390 0.01876 8 0.99027 0.13917 3.12632 0.00404 53 0.60182 0.79864 2.55514 0.01885 9 0.98769 0.15643 3.12228 0.00451 54 0.58779 0.80902 2.53629 0.01894 10 0.98481 0.17365 3.11777 0.00498 55 0.57338 0.81915 2.51735 0.01899 11 0.98163 0.19081 3.11279 0.00544 56 0.55919 0.82904 2.49836 0.01904 12 0.97815 0.20791 3.10736 0.00590 57 0.54464 0.83867 2.47932 0.01907 13 0.97437 0.22495 3.10146 0.00635 58 0.52992 0.84805 2.46025 0.01908 14 0.97030 0.24192 3.09510 0.00680 59 0.51504 0.85717 2.44117 0.01906 15 0.96593 0.25882 3.08830 0.00726 60 0.50000 0.86603 2.42211 0.01904 16 0.96126 0.27564 3.08104 0.00771 61 0.48481 0.87462 2.40307 0.01898 17 0.956.30 0.29237 3.07333 0.00814 62 0.46947 0.88295 2.38409 0.01892 18 0.95106 0.30902 3.06519 0.00858 63 0.45399 0.89101 2.36517 0.01882 19 0.94552 0.32557 3.05661 0.00902 64 0.43837 0.89879 2.34625 0.01870 20 0.93969 0.34202 3.04759 0.00944 65 0.42262 0.90631 2.32766 0.01856 21 0.93358 0.35837 3.03815 0.00986 66 0.40674 0.91355 2.30909 0.01840 22 0.92718 0.37461 3.02829 0.01028 67 0.39073 0.92050 2.29069 0.01821 23 0.92050 0.39073 3.01801 0.01069 68 0.37461 0.92718 2.27248 0.01799 24 0.91355 0.40674 3.00732 0.01110 69 0.35837 0.93358 2.25449 0.01774 25 0.90631 0.42262 2.99822 0.01149 70 0.34202 0.93969 2.23675 0.01747 26 0.89879 0.43837 2.98473 0.01188 71 0.32557 0.94552 2.21928 0.01716 27 0.89101 0.45399 2.97285 0.01227 72 0.30902 0.95106 2.20212 0.01682 28 0.88295 0.46947 2.98058 0.01265 73 0.29237 0.95630 2.18530 0.01645 29 0.87462 0.48481 2.94793 0.01302 74 0.27.564 0.96126 2.16885 0.01604 30 0.86603 0.50000 2.93492 0.01337 75 0.25882 0.96593 2.15281 0.01560 31 0.85717 0.51504 2.92154 0.01373 76 0.24192 0.97030 2.13721 0.01510 32 0.84805 0.52992 2.90781 0.01408 77 0.22495 0.97437 2.12211 0.01456 33 0.83867 0.54464 2.89373 0.01441 78 0.20791 0.97815 2.10755 0.01398 34 0.82904 0.55919 2.87932 0.01474 79 0.19081 0.98163 2.09357 0.01335 35 0.81915 0.57338 2.86458 0.01506 80 0.17365 0.98481 2.08022 0.01265 36 0.80902 0.58779 2.84952 0.01538 81 0.15643 0.98769 2.06757 0.01189 37 0.79864 0.60182 2.83414 0.01567 82 0.13917 0.99027 2.05568 0.01106 38 0.78801 0.61566 2.81847 0.01596 83 0.12187 0.99255 2.04462 0.01015 39 0.77715 0.62932 2.80251 0.01623 84 0.10453 0.99452 2.03447 0.00915 40 0.76604 0.64279 2.78628 0.01651 85 0.08716 0.99619 2.02532 0.00803 41 0.75471 0.65606 2.76977 0.01677 86 0.06976 0.99756 2.01729 0.00678 42 0.74314 0.66913 2.75300 0.01701 87 0.05234 0.99863 2.01051 0.00535 43 0.73135 0.68200 2.73599 0.01724 88 0.03490 0.99939 2.00516 0.00366 44 0.71934 0.69466 2.71875 0.01747 89 0.01745 0.99985 2.00150 0.00150 45 0.70711 0.70711 2.70128 90 0.00000 1.00000 2.00000 508 ANALYTIC GEOMETRY Table of the perimeters, of ellipses, whose minor axes 2 b are all equal to 100. This second table is less rigorous in the decimal part, but gives the required results more directly. Major Axis. Perimeter. Major Axis. Perimeter. Major Axis. Perimeter. 2o 21 2a 21 2a 21 101 315.7478 350 762.0212 680 1400.0412 102 317.3364 360 780.9768 690 1419.6200 103 318.9249 370 799.9512 700 1439.2084 104 ,320.5135 380 819.0084 710 1458.8072 105 322.1021 390 838.0740 720 1478.4116 106 323.6907 400 857.1708 730 1498.0284 107 325.2792 410 876.2972 740 1517.6476 108 326.8678 420 895.4524 760 1537.-2756 109 328.4564 430 914.6324 760 1556.9120 110 330.0450 440 933.8376 770 1576.5548 120 346.2680 450 953.0668 780 1596.2048 130 362.7856 460 972.3192 790 1616.8624 140 379.5624 470 991.5944 800 1635.5248 ISO 396.5712 480 1010.8896 810 1655.1948 ISO 413.7792 490 1030.2064 820 1674.8704 1694.5504 170 431.1732 500 1049.5404 830 180 448.7276 510 1068.8901 840 1714.2392 190 466.4488 520 1088.2616 850 1733.9332 300 484.2652 530 1107.6492 860 1753.6321 210 502.2223 540 1127.0492 870 1773.3359 220 520.2924 550 1146.4672 880 1793.0446 230 538.4560 560 1165.8968 890 1812.7580 240 556.7612 570 1185.3452 900 1832.4772 250 575.0624 580 1204.8044 910 1852.2020 260 593.4832 590 1224.2776 920 1871.9300 270 611.9944 600 1243.7604 930 1891.6640 280 630.5401 610 1263.2568 940 1911.4004 290 649.1640 620 12S2.76S6 950 1931.1452 300 667.8392 630 1302.2852 960 1950.8916 310 686.5904 640 1321.8172 970 1970.6404 320 705.3808 650 1341.3571 980 1990.3943 330 724.2152 660 1360.9096 990 2010.1526 340 743.0984 670 1380.4708 1000 2029.9192 Example. For 2 a = 30 ft., and 2 b = 20 ft., making 2 6= 100, we have, 2a = 100^= 150. For this value of 2 a the table gives, 2 I = 396.5712. Therefore the value in feet is on 21= 396.5712 X ^ = 79.31424 feet, or I = 39.65712 feet, which is not greatly different from that obtained from the first table. 1162. Surface of the ellipse. Since we may consider an elhpse whose major axis is 2 a and minor axis 2 6, as being a projection ELLIPSE 509 of a circle whose diameter is 2 a, upon the plane of the ellipse; the angle between the plane of the circle and that of the ellipse being 6 and cos ^ = - (1143), the area S of the surface of the eUipse is, S = S' cos 6 tra? - = {■Tab), wherein S' is the area of the circle. For a = 3 ft., and 6 = 2 ft., we have, S = 3.1416 X 3 X 2 = 18.85 sq. ft. Thus we have S : S' = b : a. Therefore the surface of an ellipse is equivalent to that irr^ of a circle the radius of which is a mean proportional between the semi- major axis a and the semi-minor axis b, that is, r^ = a6 (753, 970). When the two foci of the ellipse approach each other until they coincide, the radius vectors of all points become equal to the semi-major axis which is equal to the semi-minor axis. The ellipse is then a circle having a = b = r for its radius, and therefore wr^ for its area. (See Part VI.) 1163. That portion of an ellipse included between two parallel chords is a segment. The area of a segment included between two chords parallel to either the major or minor axis. 1st. Describe a circle on the major axis AA' as diameter; then, after having deter- mined the area S' of the circular segment C'D'E'F' (760), the area of the segment of the ellipse CDEF is found from the pro- portion S :S' = b:a, from which S = S'X - • a Proof. Since the entire ellipse may be considered as being the projection of a circle (1162), we may also consider the segment of an ellipse as being the projection of the segment of a circle, and we have, S = S' cos 9= S'-- Fig. 308 510 ~ ANALYTIC GEOMETRY 2d. The chords Gil and IK, which bound the segment, being parallel to the major axis, describing a circle on the minor axis BB' as diameter, the area of the segment of the ellipse is given by the proportion GHIK or S : G'H'I'K' or |- = a : &, and S^S'r- When the parallel chords are perpendicular to the minor axis at its extremities, the segment becomes the ellipse, and that of the circle a circle of radius b, and we still have the ratio S -.S' = a:b. 1164. The ellipsoid of revolution is a solid generated by the revolution of an ellipse about one of its axes. 1165. The surface of an ellipsoid is not given by any elemen- tary algebraic expression. It may be computed by considering the generating ellipse as being made up of short straight lines, which generate cylinders, frustums, and cones of revolution; measuring all these lateral surfaces (906, 912, 908), and summing them, we have the approximate area of the ellipsoid. (See (1355) integral calculus.) 1166. The volume of an ellipsoid. When the ellipsoid has three unequal axes, that is, when a plane drawn through the center perpendicular to the major axis 2 a, does not determine a circle of diameter 2 b, as in the ellipsoid of revolution, but an ellipse having 2 b and 2 c for its axes, its volume is, F = - Trabc o For an ellipsoid of revolution, according as the ellipse turns upon its major or minor axis, it suffices to make c = 6 or c = a in the preceding formula, and we have respectively, V = ^7ra¥ or F = ^■^a'b. (See Part VI.) When a = b = r, that is, when the generating ellipse is a circle, we have, 7 = 1^3, which is as it should be, since the ellipsoid is a sphere of radius r (924). HYPERBOLA 511 HYPERBOLA 1167. The hyperbola is an open curve of two branches (Fig. 309), such that the difference MF' — MF between the distances of each of its points from two fixed points, called the foci F and F', is constant. It is seen that, like the ellipse (1127), the hyperbola is defined by its equation in focal coordinates (1101); designating the radius vectors of each point by the variables p and p' and the constant difference by 2 a, we have, p' — p = 2a. 1168. The straight line which passes through the foci F,F', of the hyperbola is the principal axis (Fig. 309). The segment AA' of the principal axis, intercepted by the cin^ve, is eaUed the transverse axis. The points A and A' are the vertices of the hyperbola. The perpendicular bisector of AA' is called the conjugate axis. 1169. The distances of the foci to the nearer vertices are equal, and therefore so are the distances from the foci to the center: AF = A'F' and FO = F'O. Proof. The vertices A and A' being on the hyperbola, we have, AF'-AF or AA' + A'F'-AF = A'F-A'F' or A'A+AF-A'F'. Canceling the quantity AA' common to both members of the equation, and trans- posing the like quantities to the same side of the equation, we have, 2 A'F' = 2AFotAF = A'F'; adding the quantity AA' to both members of this equation, we have A'F = AF', which shows that the distances from the foci to Fig. 309 the farther vertices are equal. Since AO = A'O, we have also FO = F'O. 1170. The constant difference 2 a of the radius vectors is equal to the transverse axis AA'. The point A being on the hyperbola, we have, AF' - AF or AA' + A'F' - AF == 2 a; 512 ANALYTIC GEOMETRY from which, noting that A'F' = AF (1169), AA' = 2 a. 1171. The equation of the hyperbola, taking the axes of the curve as coord/mate axes (1168). Let AA' = 2a and FF' = 2 c. We always have 2 a < 2 c or a < c. Since F'P = x + c and FP = x — c, the right triangles MPF' and MPF give respectively (730): p'2 = y^ + (x + cy and p' = y^ + (a; - cf; (o) developing (727, 728) and simplifying, p'^ ~ p^ = y' + x' + c' + 2cx — y^ — x^ — c' + 2cx = i ox; that is (729), (p' + P) (p' - P) = 4 ex, J , , 4 ex 4 ex 2 ca; and p + p = = — — = ; P — P 2 a a and, since p'- P= 2 a, adding these two equations, we have, and therefore, 2, z ex -^ ex p' = \- 2 a or p' = — ■ + a, a a p'2 = ^ + «=> + 2 ex. Putting this value of p'^ equal to that in equation (a), and ehm- inating the denominator a^, cfy^ + o^a^ + (^c^ + 2 c^cx = c^y? + o* + 2 a^cx. Canceling 2 a^cx, and transposing, ay + x2(a=' - (?) = a^a^ - c^). Representing the constant quantity (a^ — c^), which is neces- sarily negative, by — W (1186), we have for the equation of the hyperbola, xi^ x^ d^y"^ — Vx^ = — €?}?■ ox jT ^ = — 1, and ?/ = ± - Va;2 - a^. (571) HYPERBOLA 513 From this equation it follows that, like the ellipse (1131, 1138), the hyperbola is divided into two equal and symmetrical parts by each of its axes (839). This equation shows furthermore that X cannot be less than a, . and, according as x varies from ± a to ± (XI , y varies from to ± oo . Thus the curve is composed of two infinite branches. 1172. The distance 2 c = FF' between the foci is called the focal distance, and the ratio e of the focal distance to the trans- verse axis 2 a is called the eccentricity (1134). Thus we have, c e = - = a 1173. From the equation of the hyperbola (1171), we find for any point M (Fig. 309) : 7? — (^ or r {x + a) (a; — o) a^ (1141) Noting that a; + a = ± A'P and x — a = ±, AP, A'P X AP a^ This shows that the ratio of the square of an ordinate to the prod- uct of the corresponding segments of the principal axis is equal to the ratio of the square of the conjugate axis to the square of the transverse axis, and is therefore constant. For another point we would have, y'^ P A'P'X AP'~ a^' therefore, ■— =; A'P X AP A'P'X AP'' Fig. 310 Thus, the squares of the ordinates of two points are to each other as the products of the corresponding segments of the principal axis. 1174. The hyperbola is the geometrical locus of the points the difference of whose radius vectors is equal to the transverse axis 2 a of the curve (1137). 1st. The point M being situated between the two branches of the hyperbola, we have MF' - MF < 2 a. 514 ANALYTIC GEOMETRY Proof. Drawing CF', the point C is on the hyperbola; and we have, CF' - CF = 2a. Having CF' > MF' - MC (637), replacing CF' by this smaller quantity, MF' ~ MC - CF or MF' - MF < 2 a. 2d. The point M' not being between the two branches of the hyperbola, we have, M'F' - M'F > 2 a. Proof. Drawing CF, the point C is on the hyperbola, CF' - CF = 2 a; replacing the quantity CF by the smaller quantity M'F — M'C, we have, CF' - M'F + M'C or M'F' - M'F > 2 a. CoROLLAEY. The converse statements of the above are also true. 1175. The parts OM, OM', of the same straight line MM', included between the center and the branches of the hyperbola, are equal. Drawing MP perpendicular to Ox, and taldng PN = PM, the point N is on the hyperbola (1171). Drawing NQ perpendicular to Oy, and prolonging it until it meets MO at the point M'; since NM' is parallel to PO and PN = PM, we have MO = OM'. From this equation, and since OQ is parallel to MN, we have QM' = QN, and N is on the hyperbola, as is also its symmetrical point M'; therefore the point M', which gives OM' = OM, is situ- ated on the hyperbola, j^g 311 From this it is seen that the point may be considered as the center of the hyperbola, and straight lines, such as MM', as diameters. Straight lines which pass through the center and do not cut the hyperbola are called infinite diameters. Since any diameter cannot cut the hyperbola in more than two points, it cannot cut one of the branches in more than one point, and a chord in one of the branches does not meet the other. 1176. When the center is Joined to the middle i of a chord, HYPERBOLA 515 the diameter BB', which coincides with this line, bisects all chords EG, GH, etc., parallel to CD. The infinite diameter IK which connects the center to the middle e of the chord GC, bisects all chords HD parallel to GC (1144). As was the case with the ellipse, the two diameters BB' and IK, each of which bisects the chords parallel to the other, are called conjugate diameters. Having -a diameter of an hyperbola given, its conjugate is found in the same way as is that of the ellipse (1145, 1189). 1177. An hyperbola or an arc of an hyperbola being given, to find its center and its axes, operate as with an ellipse (1146). 1178. To trace an hyperbola by points. F and F' being the foci of an hyperbola, and A and A' the vertices, with F and F' as centers and A'M as radius, which Fig. 312 Fig. 313 may vary from AF to oo , describe arcs; then with the same centers F and F', with a radius equal to Am, describe arcs cutting each of the first in the points CD, which belong to one branch of the hyperbola, and CD', which belong to the other branch. Proof. Any of "these points gives CF'— CF = A'm — Am = AA' = 2a (1167). Varying the position of m on the prolongation of AF, as many points may be determined as are desired, and the smooth curve drawn through these points form the two branches of the hyper- bola. 1179. To trace an hyperbola by a continuous motion. Let (Fig. 313) F'E be a rule with a small hole at one end placed in line with one edge, and EDF be a string fastened at the other end of this same edge. Taking the length of this string EDF such that EF' - (ED + DF) = AA' = 2 a, fastening the ex- 516 ANALYTIC GEOMETRY tremity F' with a pivot at one focus and the end of the string p at the other focus, and turning the rule while' holding the string taut with a pencil D pressed tightly against the edge of the rule a branch of an hyperbola is traced. Proof. For any position D of the pencil, DF' DF = EF' - {ED + DF) = AA' = 2 a. The other branch of the hyperbola is traced in the same manner. 1180. To draw a tangent to an hyperbola through a point M taken on the curve (1153). Draw the radius vectors MF, MF'; take MC = MF, draw CF, and the per- pendicular MT, dropped from the point M on CF, is the required tangent; that is, that any point M', other than M, taken on this line, gives M'F' - M'F < AA' or 2 a. (1174) Proof. MT being perpendicular to CF at its middle point, the triangle MCF is Fig. 314 an isosceles triangle, and we have, F'C + CM' - M'F = F'C = MF' - MF = 2a. F'C + CM' > M'F'; M'F' - M'F <2a. But therefore Remark. The triangle MCF being isosceles, it is seen that the tangent bisects the angle included by the radius vectors. 1181. As in the ellipse (1159), the tangent to the hyperbola is parallel to the conjugate of the diameter drawn through the point of contact (1176); which gives a second method for drawing a tangent to an hyperbola. 1182. To draw a normal to an hyperbola through a point M (Fig. 314). The bisector MN of the angle FMC formed by the radius vector MF and the prolongation MC of the other radius vector, is the normal to the curve at the point M. Reasoning as in (1154), it may be proved that MN is perpendicular to MT at M. 1183. Two hyperbolas, and in general two curves, are said to be homofocal when they have the same foci. HYPERBOLA 517 An ellipse and an hyperbola, which are homofocal, cut each other at right angles. As bisector of the angle FMC, MT is both tangent to the ellipse and normal to the hyperbola, and, as bisector of the angle FMF', MN is both normal to the ellipse and tangent to the hyperbola, and MN and MT are perpendicular to each other whether we consider the ellipse (1154) or the hyperbola (1182). The method of determining the point T has been given (1158). 1184. Hyperbolic mirrors (1154). A ray of light or heat emanating from the focus i^ of a hyperbolic mirror (Fig. 315) strikes any point M and is reflected in the direction MC and appears to come from the focus F'. As is seen, all the reflected rays, instead of meeting at the same point, as in the elliptical mirror, appear to come from the same point F', which is a virtual focus and not a conjugate focus. The space in front of the mirror in the angle DF'D' receives both the direct rays, from the source at F and those reflected by Fig. 315 Fig. 316 the mirror. Thus it is seen that when a large area is to be lighted, a hyperbolic mirror should be used. 1185. What was said in article (1155) concerning the ellipse holds good for the hyperbola. MT being the tangent drawn to the hyperbola at M, accord- ing to the construction of article (1180), and the center of the hyperbola, in the triangle FF'C the straight Une OP bisecting which shows that the FC and FF' F'C we have OP = —^ = a circle described upon AA' as diameter passes through the point P, and that it is the geometrical locus of the projections P, P', of the foci upon the tangents (1155) (Fig. 316). The circle described from one of the foci F' as center, with 518 ANALYTIC GEOMETRY AA' = 2 a as radius, has the property that when any radius F'C is prolonged to the hyperbola MC = MF. Therefore, an hyperbola may be defined as a curve such that all of its points are equally distant from the circumference of a circle and a fixed point outside of that circle. From this definition a method may be deduced for the construc- tion of the hyperbola by points, but it is quite complicated. The circle described on 4 A' as a diameter is called the principal circle of the hyperbola, and that described from one of the foci as center with the transverse axis AA' as radius is called the directrix circle. From that which has been said, in order to draw a tangent to an hyperbola at the point M, describe a circle on AA' as diameter, and another with F' as a center with A A' for a radius; draw F'M, then CF, which will intersect the cir- cumference of the principal circle at P, and connecting M to P we have the required tangent. 1186. Asymptotes. The branches of the hyperbola extend to infinity, and the diameters increase to a maximum angle with the principal axis, at which angle they extend from + oo to — oo Kg. 317 ' (1175). The two infinite diameters which meet the hyperbola at infinity are called the asymptotes. They are tangent to the branches at infinity. When the point of contact M (Fig. 316) moves along the curve, the point P describes the principal circle and the point C the directrix circle, whose center is at the focus F' (1185). Since the straight lines OP and F'C are always parallel (1185), the angles OPF and F'CF are always equal; and if one of the angles OPF becomes -a right angle, the other F'CF also becomes a right angle, and FC is tangent to the principal circle and also to the directrix circle. Then (Fig. 317) the tangent MP per- pendicular to FC at its middle point and the radius OP are in the same straight line; and since the point of contact is at the intersection of the two parallels OP and F'C, which is at infinity, the line OM is an asymptote. Therefore, to trace an asymptote, connect the center to the HYPERBOLA 519 point of contact P of the tangent to the principal circle drawn through F. The other tangent FP' drawn to the principal circle gives the other asymptote ON', and the tangents drawn from F' to the same circle determine the asymptotes ON, OM', of the second branch of the hyperbola; but, since the figure is symmetrical, the asymptotes of the second branch are prolonga- tions of those of the first. Therefore the hyperbola has two asymptotes. Erecting perpendiculars to AA' at A and A', and completing a rectangle whose vertices are on the asymptotes, the two right triangles OPF OAI having an acute angle common and the side OP = OA, being radii of the same circle, are equal, and 01 = OF = c. Therefore, to trace the asymptotes, from one of the vertices A' as center, with OF as radius, describe an arc which cuts the transverse axis in B and B' ; draw the rectangle / I'l"!'" on AA' and BB', and the diagonals of this rectangle are the asymptotes; they may be traced without constructing the rect- angle / 77"/'", by simply drawing parallels to A'B and to AB through the center 0. In the right triangle A' OB we have OF = A^ - A^ = & — a' = 6^ (1171). This is why BB' = 2 6 is taken as the length of the conjugate axis. 1187. An hyperbola is equilateral when the asymptotes are perpendicular to each other. Then the rectangle / ITT" (Fig. 317) is a square, and the two axes 2 a and 2 b are equal. 1188. Two hyperbolas are said to be con- jugate when, having the same asymptotes and equal focal distances, FF' = ff', the transverse axis of one is the conjugate axis of the other. From that which has been said, the points F, I, f, are on an arc of the same circle, whose center is and radius is OF = c. The transverse axis AA' = 2 a and the conjugate axis BB' = 2 6 of the hyperbola FF' are respectively the conjugate axis 2 b' and the transverse axis 2 a' of the conjugate hyperbola jf'. We have, a'2 = &2 = c^ _ a^ and b'^ = a' = c^ - b\ When one of the hyperbolas is equilateral (1187), its conjugate rig. 318 S20 ANALYTIC GEOMETRY is also. We have, a' = ¥ = a" = h'\ /.'2 _ -2 _ = c^ = 2 a^" = 2 62 thus the two hyperbolas are identical. 1189. When the asymptotes are traced (1186), to draw the conjugate to a given diameter LU (1176), through L, draw a par- allel LD to the farther asymptote; it cuts the other asymptote in E; take EG = EL, and GO is the required conjugate diameter. This construction is based upon the fact that each asymptote bisects the parallels to the other which are included between two conjugate diameters. Thus, the asymptote MM' bisects GL and all lines parallel to it and included between the conjugate diam- eters LU and GG'; it also bisects all parallels AB' , A'B', . . ., included between the other two conjugate diameters AA', BE'. 1190. To draw a tangent to an hyperbola through a point M exterior to the hyperbola (1157). From the point M as center, with a radius equal to the dis- tance MF to the nearer focus, describe an arc; from the other Pig. 319 Fig. 320 Kg. 321 focus F', with a radius 2 a = AA', describe another arc which cuts the first in the two points C and C'; draw FC and FC, and the perpendiculars MT, MT', dropped from the point M to the middle points of these chords, are tangents to the hyperbola at the points T and T'. The points of contact T and T' may be obtained directly, by drawing F'C and F'C' and prolonging these lines imtil they cut the hyperbola; because, if it was desired to draw a tangent at the point T where F'C meets the hyperbola, we would lay off TF on TF', thus determining the point C; then T would be on the hyperbola, and we would have TF' - TF = 2a = CF'; we HYPERBOLA • 521 would then draw FC, and the perpendicular dropped from the point T to the middle of FC would be the tangent (1180). This perpendicular coinciding with that which was drawn through M, the latter is also tangent to the hyperbola at the point T. In the same way it may be shown that MT' is tangent at T'. 1191. Taking the asymptotes Ox' and Oy' of the hyperbola as coordinate axes, the equation of the curve becomes (1171, 1186), ^Y = 4 ' which shows that the product of the coordinates, perpendicular or oblique, MQ = x' and MF = y' , is constant, and that the parallelogram OFMQ formed by the coordinates of any point and the asymptotes is also constant, since, designating the angle included by the asymptotes by 6, the base of this parallelogram is x', its altitude is y' sin 6 and the area of its surface is S = x'y' sm = — -. — sm 6. 4 When the hyperbola is equilateral, = 90° and sin = 1; that is, OFMQ becomes a rectangle (Fig. 321), & = x'y' = —^- • 1192. The area of an hyperbola. Making the constant quantity x'y' = ^1±^ = m\ (1191) the area A of the figure MM'F'F included by the arc MM' the asymptote and the two ordinates y' and y" is x" A = m^ sin L. —r > x' wherein x' = OF, x" = OF', and L. -= Napierian logarithm (407, 408, and 1796). When the hyperbola is equilateral (1187), we have sin = 1, and therefore, x" x' 522 ANALYTIC GEOMETRY If we take m as unity, ^ = L. ^, x' and in the case where the point M is at the vertex A of the hyper- bola, since a;' = 1 and x' = y', x'y' = m? = \, and A = Lo x". This property of the Napierian logarithms gives them the name, hyperbolic logarithms. 1193. According as an hyperbola revolves about its conjugate axis or its transverse axis (1168), it generates an un-parted hyper- boloid or a bi-parted hyperboloid. PAfeABOLA 1194. A parabola is an open-branched curve (Fig. 322), all points of which are equally distant from a fixed point or fociis F, and a fixed straight line or directrix OD. The parabola, like the ellipse and hyperbola, is defined in focal coordinates (1127, 1167). Designating the radius vectors of different points on the curve by the variables p and /, we have, P = P'. Two parabolas having the same focus are Kg. 322 said to be confocal (1183). 1195. The perpendicular Fx to the direc- trix drawn through the focus is the axis of the parabola. The point A, where the axis cuts the curve; is the vertex of the parabola. Twice the constant distance FO between the focus and the directrix is called the parameter of the parabola; it is represented by 2 p, and determines the parabola. The vertex, being part of the curve, bisects the distance FO, and we have, OA = AF = lp. 1196. The chord BB' drawn through the focus perpendicular to the axis is called the latus rectum and is equal to the parameter 2 p. From the definition of a parabola and the fact that parallels PARABOLA 523 comprehended between parallels are equal, we have FB = FB' = 0F = p and BB' = 2 p. 1197. The equation of the parabola referred to coordinate axes, when one coincides with the axis Ax and the other passes through the vertex A parallel to the directrix of the curve OD. In the right triangle MFP (730), p^=MF + FP'=y' + (x~l pj; also, (x + ipj; Putting these two values of p^ equal to each other, y^ + 3? + -^p^ — px — x" + -gp' + px. Simplifjdng, we have the equation of the curve, 2/' = 2 px, and (571) y =± V2 px. For every value of x there are two equal values of y opposite in sign, therefore the curve is symmetrical about its x-axis. Solving the equation for x, y^ 2p y' being necessarily positive (537), x is always positive, and the curve is situated entirely on one side of the y-axis. When X varies from to oc,y varies from to ± oo ; consequently the curve has one branch extending to infinity on both the + y and the — y side of the x-axis. If p is negative, the curve is open on the left side. 1198. The squares of the ordinates of the parabola are to each other as the corresponding abscissas (1141, 1173). From the equation of the parabola (1197), y^ = 2 px and y" = 2 px' and -^ = - • 2/" x' 1199. From the equation 2/^ = 2 px, we have, ^=2p, X 524 ANALYTIC GEOMETRY which shows that the ratio of the square of an ordinate to the corresponding abscissa is constant and equal to the parameter 2 p. For X = ^j we have y^ = p^ or y =^ p. Thus the ordinate which corresponds to the focus is equal to the distance from the focus to the directrix (1196). 1200. The parabola is the geometrical locus of the points equally distant from the focus and the directrix (1137, 1174). 1st. The point M being outside the parabola, we have MQ < MF. Proof. Prolonging QM, and drawing CF, we have, CF - CM < MF; replacing CF by its equal CQ, CQ - CM or MQ < MF. 2d. The point M' being inside the curve, we have M'Q > M'F; because, having M'C + CF > M'F, replacing CF by CQ, M'C + CQ or M'Q > M'F. Corollary. The converse statements of 1st and 2d are both true. 1201. The axis of the parabola divides the curve into two equal and symmetrical parts. C being any point in the curve (Fig. 323), '^'^ drawing the perpendicular CP to Ox, and '^ taking PB = PC, the point B symmetrical to C is on the parabola. Proof. Drawing BF, we have CF = BF (621); furthermore, since CF = CQ and CQ = BQ', we have BF = BQ'; which cannot be unless the point B is on the curve (1200); therefore the two parts of the curve are sym- metrical with respect to the axis and equal each to each (839). This was proved in article (1197). 1202. The ellipse being the geometrical locus of the points, such as M, which are equally distant from the focus F and the PARABOLA 525 directrix circle whose center is at the other focus F' (1155), the vertex A and the focus F remaining fixed, according as the vertex A', the focus F', and the center M'Q' (620), and therefore M'Q' < M'F. Remark 1. Since the triangle MFQ is isosceles, it follows that the tangent MT bisects the angle FMQ and the radius vectors. Remark 2. The angle QMT = MTF, being alternate interior angles; and QMT =■ TFM being base angles of an isosceles triangle. The triangle MTF being isosceles, it follows that in order to draw a tangent at the point M, lay off from the focus Kg. 332 prp ^ pj^ ^^^ (Jj.^^ MT. Having FT = FM = MQ = OP, and AO = AF, it follows that we also have AT = AP. Remark 3. Taking MB = MQ = MF = FT, the chord CD, which passes through F and B, is parallel to the tangent MT, and is bisected at the point B. From this we have a method for drawing a chord through F which is bisected by a given diameter MB (1204, 1207). Drawing through the extremity of this diameter a parallel to the chord, it will be tangent to the curve; which gives a third method for drawing a tangent to a parabola (1208). Remark 4. Having drawn the diameter MB, and the axis of 532 ANALYTIC GEOMETRY the parabola, as per (1206), drawing the tangent MT, the triangle MTF is isosceles, and the perpendicular bisector of its base determines the focus F at the intersection of this line with the axis, and the point Q at the intersection of this same line with the diameter MB determines the directrix. Tliis is a second method for determining the focus and directrix of a parabola (1206). Remark 5. Having AO = AF, the perpendicular erected at A to the axis AN of the parabola passes through the middle point of FQ (699), that is, at the point where the tangent cuts the line FQ, which is perpendicular to it; therefore the geometrical locus of the projection of the focus on the tangents is the per- pendicular erected at the vertex A (1155, 1185). 1215. To draw a normal to the parabola. The bisector MN of the angle FMB, which is included by one radius vector and the prolongation of the other, is normal to the curve at the point M. It may be proved that MN is perpendicular to the tangent MT, as was done in article (1154). Having FT = OP (1214, Rkmaek 2), we have FP = OT, and since AF = AO, we have AP = AT = x, and TP = 2 x. This being true, the point M is on the curve, and we have (1197), 2/^ = 2 px. Representing the subnormal PN by s, the right triangle TMN gives (705), y^ = s X TP = s x2x. Putting these two values of if equal to each other, 2 sx = 2 px, then s = p. Thus, for the parabola, the subnormal is constant and equal to the semi-parameter p = OF. This furnishes an easy method of drawing a normal or a tangent to the parabola at any given point M. 1216. Parabolic mirror, ear-trumpet, megaphone, etc. In a para- bolic mirror, all rays FAI (Fig. 332) emanated from the focus are reflected along lines MB, parallel to the axis. All rays paral- lel to the axis which strike the mirror from outside are reflected to- the focus. This property is utilized in ear-trumpets. The sound which enters the trumpet is reflected to the focus, and, the end being removed, the focus is brought inside the ear (Fig. 333). PARABOLA 633 The megaphone is sometimes made by combining an ellipsoid and a paraboloid (Fig. 334) so that they have a focus F in com- mon, the mouth being placed at the other focus F' of the ellipse. 1217. The path of a projectile would be a parabola were it not for the resistance of the air which modifies the curve. The cables on suspension bridges have a curvature which is very nearly parabolic, and in practice may be taken as such. 1218. To draw a tangent to a parabola parallel to a given straight rig. 333 Fig. 334 line CD (Fig. 332), follow the same course as for the elHpse (1156). Thus, draw FQ from the focus perpendicular to CD, and the per- pendicular bisector of FQ is the required tangent. To obtain the point of contact, draw QM parallel to the axis. It is seen that the tangent and its point of contact may be determined without constructing the parabola, when the axis, focus, and directrix are given. The problem is impossible when CD is parallel to the axis, because then the perpendicular FQ meets the directrix at infinity. 1219. To draw a tangent to a parabola through a point M out- side the curve. From the point M as center, and with MF as radius, describe an arc which cuts the directrix in the points D and D'; draw FD and FD'; then the perpendiculars to these lines, dropped from the point M, are tangents to the parabola at the points T and T', which are given directly by drawing parallels to the axis through D and D'. If a tangent to the curve was to be drawn at the point T, a perpendicular would be dropped from this point to the middle point of FD (1214); but this perpendicular would coincide with that which was drawn from the point M; because, the triangle MDF being isosceles, this perpendicular also passes through the middle point of FD. Pig. 335 534 ANALYTIC GEOMETRY 1220. As was the case with the ellipse and hyperbola, there is no method in elementary geometry by which the length of an arc of a parabola can be accurately determined. 1221. The surface of a parabolic segment ABCD, included between the vertex and the chord BD perpendicular to the axis, is equal to 2 - of the rectangle EDBG, whose altitude is BD and whose base is AC; thus we have (1329) (Fig- 336), Mg. 336 surface ABCD = | AC X BD, or From this, surface ABC = t;ACX BC. surface ABE = | surface ACBE = ^AC X BC. Noting that the segment BIKD, included between the two chords BD, IK, perpendicular to the axis, is the difference between two segments AILK and ABCD, we have. surface BIKD = :^AL X IK -ACXBD 2 {AL XIK-ACX BD). M being the point of contact of the tangent MT parallel to 2 ID (1214, Remark 3), the surface of the segments AIQD is ^ of the surface of the rectangle IDT'T, which has the same base ID and the same altitude MP as the segment; thus we have. surface AIQD = -MP X ID. The segment whose base is perpendicular to the axis is simply a special case of the general theorem. 1222. The solid generated by the revolution of a parabola about its axis is called a paraboloid. PARABOLA 535 1223. The surface of the paraboloid generated by the rotation of an arc AI upon the axis (Fig. 336). Take LR = 2 AF, and IS = SAF; draw SR; then take lU = IR, and draw UV parallel to SR; from which we have, IS :IR = lU or IR : IV. The surface s of the paraboloid is equal to the lateral surface of a right cylinder having IR for its diameter and IV for its g altitude, less ^ of the surface of a circle having AF for its radius; thus we have (753, 906, and 1340), s= TT- 772 .77-1 ttZp'. (a) o Representing the ordinate IL by y, since we have LR = 2 AF 3 = p, (1205), and IS = 3 AF = ^p, the right triangle ILR gives, IR=^ From the above proportion. Tv = ^J^^ t±^ _ ^iy' + p') IS 3 3p oP Since AF' = ^, 4 substituting these values in the formula (a), This expression permits the calculation of s without any geo- metrical construction when the values of p and y are known. Since, representing AL by x, y^ = 2 px (1197), s may also be expressed in terms of x, thus: 4: X + 2 V 2 s = 'r\/2px + fX 3 — - - 3 'rp'- 1224. The volume of a paraboloid generated by the rotation of the parabolic segment AIL about the axis, the base IL being per- 536 ANALYTIC GEOMETRY pendicular to the axis (Fig. 336), is equal to that of a right cylin- der having AL for its radius and 2 AF for its altitude. Repre- senting the volume by i> (907 and 1340), V = TT- AL^ ■ 2 AF. Making AL = x and 2 AF = p (1195), V = TTX^p. Replacing x^ by ^(1197), 4p CURVES OF THE SECOND DEGREE, OR CONIC SECTIONS 1225. A parabola may be considered as the limit of an ellipse when its major axis approaches infinity, the distance between one vertex and focus remaining constant (1202). The parabola may also be considered as the limit of the hj'per- bola. Placing the origin at the vertex of the ellipse, of the hyperbola and of the parabola, these three curves are represented by the general equation, y^ = 2px + qx', wherem V = — and o = - = ^ • a ^ a a^ According as 5 < 0, q > 0, or g = 0, the equation becomes, 1st. v^ = 2 — x 5 a;2 ellipse; 2d. y^ = 2 — x-\ — ^x^, hyperbola; 3d. 2/^ = 2 — x, parabola. Changing the origin to the vertex at the left, and thus chang- ing a; to a: — a in the general equation (1131), equation 1st is obtained. CURVES OF SECOND DEGREE 537 In a like manner, changing the origin to the vertex at the right, and thus changing a; to a; + a, the general equation of the hyper- bola (1171) becomes equation 2d. 1226. The ellipse, hyperbola, and parabola are called second- degree curves i because the equations of these curves are of the second degree (1131, 1171, 1197), and all equations of the second degree involving two variables represent these curves. 1227. The curve of intersection of any secant plane with a right cone of revolution (841) is of the second degree, unless the plane passes through the vertex. The section is an ellipse if the plane cuts all the elements of the cone; and if the plane is perpendicular to the axis, the section is a circle (843). The section is an hyperbola when the plane is parallel to two elements of the cone; one of the branches is on one nappe and the other branch on the other nappe of the cone. When the plane is parallel to only one element, it cuts only one nappe, and the section is a parabola. All planes which cut the elements of a cylinder of revolution determine an ellipse, which is as it should be, since a cylinder may be considered as a cone whose vertex is at infinity. Since the plane which determines the parabola or hyperbola is parallel to one or two elements, and therefore to the axis, it cannot cut the lateral surface except along an element (842), and therefore determines no curve. Any ellipse or parabola may be laid out upon the lateral sur- face of a given cone of revolution. The same is true of the hy- perbola when the angle between the asymptotes is less than the angle between the opposite elements of the cone. Because of these properties, the name conic sections is often given to curves of the second degree. 1228. The ellipse, the hyperbola, and the parabola are convex curves; that is, that a straight line cannot cut them in more than two points (648). This follows from the determination of the points common to a given straight line and an ellipse, hyperbola, or parabola. 1st. F and F' being the foci of the ellipse, if a point M of the given line MM' is on the curve, prolonging F'M to C, making MC = FM, the point C is on the directrix circle described from the focus F' as center (1155), and determining the point / sym- 538 ANALYTIC GEOMETRY metrical to F with respect to MM' (836), it is seen that M is the center of a circle tangent to the directrix circle and passing through the two points F and /. Then (964) describing a circle passing through F and / and cutting the directrix circle in any two points / and /', if from the point of intersection E of Ff and //' a tangent to the directrix circle is drawn and the point of contact C connected to the focus F', the line CF' cuts MM' in the required point M. Thus, to find the point M, describe the directrix circle, drop a perpendicular from F upon MM', draw an arbitrary circle through Kg. 337 Fig. 338 F and its symmetrical /, and from the intersection E of Ff and //' draw a tangent EC to the directrix circle, then draw CF', which cuts MM' in M. The second tangent EC drawn through E to the directrix circle determines in the same way a second point M' common to the straight line MM' and the ellipse. Since evidently there are as many common points as there are tangents to the directrix circle which pass through the point E, there are two, one, or none, according as the point E is outside of, upon, or inside of, the directrix circle. The line MM' is a secant in the first case, a tangent in the second, and does not meet the ellipse in the third. 2d. For the hyperbola the same course is followed. Thus, for the construction, from the focus F' as center describe the directrix circle (1185), determine the point / symmetrical to F with respect to MM', describe a circle passing through F and / and cutting the directrix circle in two points / and /', draw the chords fF and //', and through their point of intersection E draw the tangents EC and EC to the directrix circle; then con- LEMNISCATE, CISSOID, STROPHOID, LIMACON 539 Kig. 339 necting the points of contact C and C to the focus F', these lines cut the given line in the required points M and M'. As in the preceding case, MM' has two, one, or no points com- mon with the hyperbola, according as the point E is outside, upon, or within the directrix circle. 3d. For the parabola, taking / sym- metrical to the focus F with respect to MM', if the point M is on the curve, OD being the directrix, M is the center of a circle tangent to OD and passing through F and /. M may be deter- mined without drawing the circle (960). Thus, draw FE perpendicular to MM', take EC a mean proportional between EF and Ef, ovWf = EF x Ef, and the perpendicular drawn through C to OD determines the point M. M' is also the center of a circle tangent to OD at C and pass- ing through F and /. The tangent EC to this circle gives EC'' = EF x Ef = W, then EC = EC. Thus the same mean proportional laid off above and below E determines the two points M and M'. When MM' passes through the focus, / coincides with the focus F, the points C and C are obtained by erecting the per- pendicular FE to MM' and taking EC = EC = EF. When MM' is parallel to the axis and consequently perpen- dicular to the directrix, C being the intersection of MM' with OD, draw FC and erect its perpendicular bisector which will cut MM' in the point M equally distant from F and C or OD, and is therefore on the parabola. M is the only point common to MM' and the curve; because any other point is unequally distant from F and OD, since it is not on the perpendicular bisector of FC. If the point / is on OD, there is but one point in common, and MM' is tangent to the curve; and if / is on the other side of OD, there is no point in common, and MM' does not meet the curve. LEMNISCATE. CISSOID. STROPHOID. LIMACON 1229. Although these four curves are of no great practical import, they nevertheless deserve to be mentioned. 540 ANALYTIC GEOMETRY y M F Kg. 340 1st. The lemniscate is the locus oi the points M, such that the product MF x MF' of the radius vectors is equal to the square of half the focal distance FF'. Designating the constant FF' by -2 a, and MF and MF' by p and p', the equation of the curve in focal coordinates and in rectangular coor- dinates is respectively (1102), pp' = a? and y" = a V4 r* + a^ - {o? -\- aF). (1111) 2d. A circle of diameter OA and a tangent to the circle at the extremity of this diameter being given, laying off on any secant OC, which passes through 0, OM = CD, the locus of the positions of the point M is the cissoid of Diodes. Designating the diameter OA by a, the variable angle COx by a, and the variable distance OM by p, the equation of the curve in polar coordinates and rectangular coor- dinates is respectively, a sin^ a P = cos li and y V a — The curve has two symmetrical branches with respect to OA, and is included between Oy and AB, having AB for its asymptote. 3d. A right angle yOx and a fixed point pjg, g^i A on one of its sides being given, draw any line AD through A, and from the point D at its intersection with Oy lay off DM = DN = DO; the locus of the points M and N is the strophoid. Designating the constant OA by a, the variable angle DAx by u., and the variable distance AM or AN by p, the equation of the curve in polar coordinates and rectangular coordinates is respectively, ^^i±^^^and2/=±xv/^^- cos a \ a — X The curve is symmetrical with respect to Ax. When the moving line occupies the position Ax, the two points M and N LEMNISCATE, CISSOID, STROPHOID, LIMACON 541 coincide in 0. When the ordinate OD becomes ± oo, the point N is Skt A and the point M at infinity; and since DM = DN, it is seen that in taking OB = OA, the perpendicular BE to Ax is asymptote to the two branches of the curve. The tangents to the curve at form angles of 45° with Ox, and are therefore perpendicular to each other. The perpendicu- lar to .Ax erected at A is also tangent to the curve. 4th. Through a point A on the circumfer- ence of a circle, draw any secant AD; starting from D, lay off on this secant a constant dis- tance DN = DM. The locus of the points M and N is the limagon of Pascal. Designating the constant DM = DN by a, the diameter AB by b, the variable angle DAx by VL, and the variable distances AM and AN by p, A being the origin, the equation of the curve in polar coordinates and rect- angular coordinates is respectively, p = b cosa ± a and (y^ + x^ - bxY = a' {y^ + x^). The curve is symmetrical with respect to Ax. Fig. 343 shows a special case where' a < b. When AD coincides with ,,vM' Fig. 342 Fig. 343 Fig. 344 Ax we have BC = BE = a, and one of the two branches starts from C and the other from E. The line AD turning comes into the position AD', which gives AD' = a; then the point N is at A, and AD' is tangent to the curve. Since beyond the position AD' we have AD" < a, the point N" is below Ax. For the posi- 542 ANALYTIC GEOMETRY tion M"'N"' perpendicular to Ax, we have AM'" = AN'" = a. The angle a varying from 0° to 90° in the direction BL, the point M generates the arc CMM'" and the point N the arc EN AN'" and these two arcs form half of the curve, a varying also from 0° to 90° in the direction BD^, the point M describes the arc CMjN'" and the point N the arc EN^AM'"; these arcs are symmetrical to the first two with respect to Ax, and therefore meet and form a smooth, continuous curve. Note, li a = 0, the equation becomes p = b cos u,, which is that of the circle AB. THE SPIRAL ARCHIMEDES 1230. The spiral of Archimedes is a plane curve, traced by a point which moves about a fixed point in such a manner that any two radius vectors are in the same ratio as the angles they make with the initial line Ox. Thus the spiral is defined by its equation in polar coordinates (1100). Designating the coordinates by p and a, p = aa + b, wherein : jo is the variable distance of the generating point from the pole or the radius vector; a is the variable angle which the radius vector makes with the axis Ox; a is the constant coefficient expressing the augmentation of p corresponding to the augmentation of a of one unit, of a degree for example; & is a constant which expresses the value of p when a = 0; thus b is the distance from the pole to the point in the axis Ox where the generating point starts. In Fig. 344 the point starts from the pole, therefore b = 0, and the equation of the curve is, 1231. Each arc of the curve described by the point during one revolution about the pole, is called a spire. The distance between any two consecutive spires, measured on the radius vector p, is constant, and is called the pitch. It rep- resents the distance which the generating point travels away THE SPIRAL ARCHIMEDES 543 from or toward the pole for each spire. Thus, for a = 360°, and corresponding to 1°, if we represent the pitch by p, we have, p = aX360 or a=g|Q- 1232. To construct the spiral of Archimedes (Fig. 344). Ox being the axis, the pole, assuming b = 0, that is, that the gen- erating point starts from the pole 0, lay off the pitch OA from on Ox; divide OA into a certain number of equal parts, 8 for example; from the point as center, with OA as radius, describe a circle, and divide it into the same number, 8, of equal parts. Drawing the radii to these points of division, and laying off on radius 1 the distance 1; on radius 2, the distance 2; on radius 3, the distance 3, etc., all the points thus determined lie upon the spiral. Proof. Any of these points B gives, O A 0B:0A = BOA : 360, and OB = ~X BOA, or p = ^xa=aa. (1230) To trace the second spire, prolong the radius vectors, and lay off the pitch OA upon each one, starting from the first spire. Thus, on 1 lay off OA from 1 ; on 2 lay off OA from 2, etc. 1233. To draw a tangent to the spiral at a point M taken on the curve (Fig. 345). The following construction is based upon the general principle: That the tangent to any curve generated by a point, whose motion has two components, is the diagonal of a parallelogram whose sides have the same directions as the two components of the motion and are equal to the distances passed through along the lines of these motions in the process of generation. For the spiral of Archimedes, the motion of the generating point M is composed of two components: one along a straight line OM, the other along a circle whose radius is OM, that is, along the perpendicular MC to the radius OM. Starting from M, lay off on MO the length MD equal to the pitch p of the spiral, and on the perpendicular MC lay off a length MC equal to the circumference 2 ir x OM of the circle whose radius is OM; then completing the parallelogram MDTC, which in this case is 544 ANALYTIC GEOMETRY a rectangle, and has p and 2 t x OM for its sides, the diagonal MT is tangent to the curve at the point M. Laying the length MC equal to the arc MB described with the radius MO, off on MC, and completing the parallelogram MOT'C, the diagonal MT' is also tangent to the curve, that is, MT' coincides with MT; and we have the following proportion: OM : MC = MB : MC, or OM : arc MB = p : 2ir x OM. 1234. To draw a normal to the spiral at the point M (Fig. 345), draw the tangent MT, and the perpendicular MN erected to MT at the. point M is the re- quired normal. 1235. The surface of a segment of a spiral OBB'O included by the radius vector OB and the arc BB'O subtended by it (Fig. 344) is equal to one-third of the product of the surface gf the circle whose radius is the radius vector OB = p and the ratio of this radius to the pitch OA = p. Thus, s being the surface, we have (753) Fig. 345 1 ^ /^^2 . , OS 1 p irp^ -.XOB x^=3VX-=3^- (a) From this it follows that the area of the surface included by the first spire is equal to one-third the surface of the circle whose radius is the pitch OA = p. Making p = p in equation (a), 1 -Trp^ («) The surface of the first two spires is ^ of the area of the circle O whose radius is the pitch p. Putting p = 2p in the general equation (a), .8 '^P'_8 , s = 3 p = 3-r Subtracting the area of the first spire from that of the two spires, we have the area of the second spire. 8,1, I'^V' THE SPIRAL ARCHIMEDES 545 Finally, to obtain the surface of the spiral S included between two radius vectors OB = p and OB' = p', take the difference between the segments which terminate at these radius vectors, thus: 1236. Volutes. Having traced the first spiral with b = (1232), if a second one is traced' with b = 1, for example (Fig. vN \ Vb\ X/^ X\ / y\6 J -SsX \ 7 / "* '\io -£?! ^v\ ck is /\ \ \p ^^^ i„^.— \ \^ 2> , , aA + bB , ^ bB + cC S = abBA + bcCB H = ab h be ^— ^ -I (a) E Let ai = E; ab = be = = — ,. which assumes the projection ai to be divided into n equal parts, and aA = y„, bB = j/i, cC = 2/2 . . . il = i'„ be the different ordinates of the curve. Substituting these expressions in the equation (a), n 2 n 2 n 2 Simplifying, we have, s = §{^ + y. + y. + ---y.-. + ^y which is simple and easy to apply. 1268. Thomas Simpson's formula. This formula gives the area of a plane curve Alia (Fig. 357) more accurately than the preceding one. The number n of divisions of ai being even, Thomas Simpson has shown that the area S of the curve is given approximately by the following expression: W 3 jj^[2/o + 2/» + 4 (2/1 + 2/8+ 2/6+ • • • +2/n-i) +2 (J/2+2/4 + 2/6+ . . . + 2/„_2)]. E — being the distance between two consecutive ordinates, it is seen that the approximate value of the area S of the curve is equal E to the product of a third of the distance - — between two consecutive ^ ' ' 3 ^ ordinates, and the sum (?/„ + j/„) of the two extreme ordinates, plus 4 times the sum of the odd ordinates {y^ + 2/s + ?/b + • • • y«-\}i P^''^ 2 times the sum of the even ordinates {y^ + 2/4 + 2/6 + • ' 'Vn-i)- For n = 8, we have, S = §-^[2/0 + 2/8+ 4 (2/1 + 2/8 + 2/6 + 2/7) + 2 (2/2 + 2/4 + 2/6)]- Remark. In case one or both extremities of the curve fall upon the base line ai, the ordinates at those points are put equal to and the above formulas used. If the curve is closed, draw a line through the middle and find the area on each side of the line. This may be done in one single MISCELLANEOUS CURVES 559 operation by taking the ordinates as the sums of the correspond- ing ordinates of the two parts of the curve and using the above formulas. Derivation of the preceding formula. It may be assumed without appreciable error that the arc ABC of the curve (Fig. 357) coincides with the arc of a parabola passing through the three points A, B, and C, and having its IC C B Vo y, C A| becomes constant a, for any value of the radius vector. Therefore we have, io = a = constant. 2d. The polar equation of a circle. If the center of the circle is taken as pole, designating the radius as R, we have, p = R = constant. If the center is not at the pole, as in Fig. 283 (1223), and if the coordinates of the center are designated by./3 and •*, and those of any point M on the circumference by p and . This property may be used for constructing a graph- ical table for giving approximate values of the sines of angles. 4th. The polar equation of the ellipse, the hyperbola, and the parabola. If, for the ellipse and hyperbola, the focus at the right is taken as the pole and in the parabola the focus is taken as pole (Fig. 290, eUipse, Fig. 310, hyperbola. Fig. 322, parabola), the three curves have the common equation. 1 + e cos CO 52 g wherein p = — > e = - ; a a a and b are the semi-axes of the ellipse and hyperbola, and 2 c is the focal distance. The ratio - gives the relations, - < 1 for the ellipse, a ^ ' - > 1 for the hyperbola, /J - = 1 for the parabola. 5th. Spiral of Archimedes (see 1230) is represented by the equation, p = a -\- b. (See 1339, rectification of the spiral of Archimedes.) Logarithmic spiral is represented by the equation, log p = ko) or f^" — I y = ax + b, '^' J,, ggj in which x and y are the variables, and a and b the constants. As we have seen (1117), this is the equation of a straight line AB, b is the ordinate OC at the origin, and a is the slope. For x = 0, y = b, and taking OC = b, the point C is on the straight line AB; making x = OP, and taking y = ax + b, the point M belongs also to the line. DIFFERENTIAL CALCULUS 567 which is then determined by the points C and M, and may be indefinitely prolonged. Example 2. Let S be the area of a rectangle, b and h its two dimensions, then (716), S = bh. Supposing the base b constant and the altitude h variable, this equation is one of a straight line passing through the origin. Taldng an abscissa equal to a value of h, and erecting an ordinate equal to bh, we have a second point on the line, which is then determined. Any ordinate of this line represents the area S of this rectangle whose altitude is the corresponding abscissa, that is, that this area S will contain the unit of surface as many times as the ordinate contains the unit of length. Example 3. y = az^. (1) y and x being variables and a a constant, this is the equation of a parabola (1197), which is constructed by assuming different values for x and calculating the corresponding values of y. From equation (1), a and putting - = 2 p, 2py = x'. The quantity p is the distance from the focus to the directrix, and -T is the distance from the vertex of the parabola to the focus and the directrix. Remark. The law of falling bodies, is of the same form as (1). Example 4. The fimction, y = aoi?, in which a is a constant, is an equation of parabolic curve of the 668 ELEMENTS OF CALCULUS third degree, which can be constructed by points, giving differ- ent values to x and solving for y. The curve which represents the following equation may be constructed in the same way: V being the volume of a sphere (920). Any ordinate of the curve would express the volume of the sphere whose radius is the abscissa R; that is, the sphere would contain as many \mits of volume as the ordinate contained units of length. The functions, y = x' — ax^ + bx — c and y = x^ + ax^ + x^ — hx^ — ex — d, which contain different powers of the independent variable, may also be represented by curves constructed by points (580). Example 5. A variable quantity may be a function of sev- eral other variables. Thus, a being a constant, V = axyz. Such a function may be plotted when the values of y and z, which correspond to different values of x, are known. If, for example, xyz = x', then V = ax', and the values of V are represented by the ordinates of a straight line passing through the origin, the values of x' being the ab- scissas. If we had y = axz^, according as we put xz^ = x' or xz^ = x'^, we would have, y = ax' or y = ax'^, which are the equation of a straight line and that of a parabola, respectively. These divers examples show that the value of any function may be represented by the ordinates of a curve, the abscissas of which represent the values of the independent variable. Conversely, any curve referred to two axes represents the law of the simultaneous variation of two variables x and y. DIFFERENTIAL CALCULUS 569 1273. The variation of functions. Increasing and decreasing functions. Example 1. Let an equation of the first degree, involving two variables, that is, an equation of a straight line, be given (1272), y = ax + b. (1) If the independent variable x = OP is increased by a quantity PP' = a, the function y = MP becomes y' = M'P', and we have, (2) y'= a{x + a) + b; subtracting (1) from (2), y' - y = aa or M'Q = a X PP'; dividing both members by a, y' - y M'Q which shows that the ratio of the increment y' — y oi the function y to that of the incre- ment a of the variable x, is in- dependent of these increments. Example 2. Given the function, y = ax", X becoming (x+a), and designating the new value of the func- tion by y', y' = a(x + ay = ax'' + 2 aax + ao?. Fig. 362 The increment of the function is y' y = 2 aax + aa? and = 2ax + aa. The ratio y y is not independent of a, as in the first example; but, according as a decreases, the term oa becomes smaller and smaller, and it is evident that a may become so small that the term aa may be neglected in comparison to the term 2 ax, and at the limit we have, y^^iy^2ax. 570 ELEMENTS OF CALCULUS 1/ — v Thus the ratio ~ has a determinate limit 2 ax, u This property is general for all algebraic relations involving two variables. A fimction y = f {x) is increasing or decreasing according as y increases or decreases when x increases. Thus, Fig. 362 repre- sents an increasing function, and Fig. 363 a decreasing function. The same function can be alternately increasing and decreasing. 1274. A differential quantity. Differential coefficient. Derivw- live. Object of differential calculus. When the increment a of the abscissa or variable x is small, it is designated by Ax, pro- nounced delta X, and may be considered as a fraction of a;; in the same way a small increment y — y' oi y is designated by Ay. Thus, y' -y ^^y a Ax When At/ and Ax decrease and become infinitely small, the Hmit is represented by dx and dy. In example 2 of the preced- ing article, the limit of the ratio of the increments of y and x is, -f-= 2ax and dy — 2 axdx, which shows that an infinitely small increment dy of the function or ordinate y is expressed algebraically by the product of the infinitely small increment dx of the variable abscissa x and the variable coefficient 2 ax. The quantities dy and dx, considered as being infinitely small, are called the differentials of y and x. The coefficient 2 ax by which the differential dx is multiplied to obtain the differential dy, is called the differential coefficient. The ratio -f- is called the derivative of y with respect to x, or the derivative of the function y with respect to the variable x; it is equal to the differential coefficient. In the preceding example the inverse ratio, dx 1 dy 2 ax is the derivative of x with respect to y; a; is then the function, DIFFERENTIAL CALCULUS 571 and y the variable. Ordinarily the derivative -^ is designated by 2/' or /'(a;); thus, ! = / = «.); which indicates that the derivative of the function y is taken with respect to the variable x. If the derivative of x with re- spect to y had been taken, we would have, I =^' = /'(.). The difference between two quantities must not be confused with the differential of a quantity. Thus, having y' — y = 1 aax + ax^, the differential of the fimction y is dy = 2 axdx. It is seen that the difference between two quantities, no matter how smaU, may be expressed in numbers, while the differential dy cannot. The differential of a quantity must be considered as an al- gebraic expression or symbol resulting from a calculation; but a dv derivative -p has a perfectly determinate value, and may be expressed in numbers. The chief purpose of differential calculus is the determination of the law which governs the increments of a function and those of the variable upon which it depends, that is, the value of the ratio -rr ■ dx 1275. Geometric interpretation of the derivative of a function. Let C be any curve referred to two rectangular coordinate axes, whose equation is, y = f{x). f(x) represents the calculations which must be made in con- structing the curve by points, by assuming different values of X and calculating the corresponding values of y. Let us con- sider the points M and M' of the curve C whose coordinates are respectively y^x and y'x'. It is seen that in going from M to M', 572 ELEMENTS OF CALCULUS the ordinate increases by the amount M'Q, which is positive or negative according as the function is increasing (Fig. 364) or decreasing (Fig. 365), and the ratio of the simultaneous incre- ments of the ordinates and abscissas is, M'Q y' PP' y X' — X Drawing the secant MM', the ratio , _ ' is the tangent of the angle •*, which is included by MM' and the a;-axis,- and if the point M' approaches M indefinitely, that is, if the increments y J '■A Q ^ ^ y 1 1 i< X, \ ^M Q M' ^ C 1 P' X. Fig. 364 Kg. 365 are indefinitely decreased, the tangent MM' will approach a limit where it is tangent to the curve and M' coincides with M. This corresponds to the limit, 2/' = ^ from the ratio ^7 — ^ • ax x' — X Thus the limit of the ratio of the simultaneous increments of a function y and the variable x is equal to the tangent of the slope of the curve C which represents the function. The determination of this limit of the ratio solves the general problem of tangents, which is an important application of differential calculus. DIFFERENTIALS AITO DERIVATIVES OF FUNDAMENTAL FUNCTIONS y = x'". y = log X, y = sin x. 1276. Let it be given to determine the derivative of the differential of 2/ = af . (1) 1st. Assume that m is whole and positive. Gi\'ing x an incre- ment Ax, a corresponding increment Aj/ follows for y, and equa- tion (1) becomes (564), DERIVATIVES OF FUNDAMENTAL FUNCTIONS 573 y + Ay = {x + Ax)'"= 3f+ maf-^Ax + ^^'"~^-' x""'' (Axf + • • • (2) Subtracting (1) from (2), Ay = mx'-'-'Ax + ^ ^^ ~ ^^ a;"-^ (Ax)^ + . . . Dividing both members bj' Ax, ^=mx-^+ '^^'^-^h -^Ax + ... Ax 1-2 Ay which shows that the value of the ratio -^ contains one term Ax mx"~* independent of the increment Ax, but that all the others contain Ax as a factor. Since Ax may be taken infinitely small, the terms which contain Ax as a factor become negligible when Ax and Ay are infinitely small, and we have as a Umit, lim^or ^ = mx'"-S (3) Ax dx ' ^ m — 1 or (1274) y' = /'(x) = mx' which shows that to obtain the derivative y' of the junction j/ = x"* it suffices to take the variable x with its exponent m for coefficient, and the same exponent less one m — 1 for an exponent. Thus, for y = a?, we have -^ or y' = 5 x*, and for i/ = x, we have y' = x" = 1. (553) From the equation (3), dy = mx'^~^dx, which shows that the differential dy of the function y is equal to the derivative of y with respect to x, multiplied by the differential dx of the variable x. 2d. Case where the exponent of x is a fraction. Given the func- tion t y = xi> in which p and q are whole positive numbers. Raising both members to the q power, we have (555), t/« = x". 574 ELEMENTS OF CALCULUS Taking the differential of each member (1st), qy^~Hy = px^^^dx. dy _ p a;""' Transposing, dx q 2/'~^ Having a;^"' = — and 2/'~' = — > (555) we have dy_p^y__ dx q X y''' and since r = of, dy_py^ dx q X Substituting P x'i for y, y' = /.(,) or ^ = 2x^^ dx q Thus the 1st rule applies in this case. From this equation, we have the equation of the differential, thus; ^ 3 p -1 dx. 1. y = V^ = -x\ ^ dx 1 2 1 , (554) , 7 dx and dy 2 ^ 3d. Negative exponent. Given y = x-^, in which m is a whole number. This may be written (518), 2/=^' (1) or - = a;". (2) y = x"" - (x + Axy, DERIVATIVES OF FUNDAMENTAL FUNCTIONS 575 Increasing x by Ax and y by At/, the equation (2) becomes, —^ = (x + Axy. (3) Subtracting (2) from (3), 1 1 y y + ^y K^^ or -^-^-=[(x+A.)--.'»], y^ + y^y 2/' + y^y Ay 1 _ — [(a: + Ax)" — a:"] Ax 2/^ + yAy Ax Making the increments infinitely small and approaching the limit, yAy is negligible, and from (1st) we have, dy 1 „ , -f- — = - maf-^; ax 2/ and / = -^ = — y^mx"^\ (4) Since the equation (1) gives y^ = —^> by replacing y^ by its value in (4), we have, y' = ^= - -L^a;m-i = _ ^3.».-i-2m ^ _ ;„aj-"-^ (5) Thus the rule given in 1st applies when the exponent of the variable x is negative. The relation (5) shows that the deriva- tive is negative. This is as it should be, because, according to equation (1), an increment Ax of the variable x corresponds to a diminution of y, that is, a negative increment of the function. The differential is deduced from (5), thus: dy or dx~"' = — mx~'^~^dx. 1277. Derivative and differential of y = log X. (1) x increasing by Ax, y increases by a corresponding quantity Ay, and we have, y + Ay = log (x + Ax), (2) 576 ELEMENTS OF CALCULUS Subtracting (1) from (2), t^y = log {x + Ax) - log X = log — = log { 1 H ) , (396) X \ X / Ax Ax Putting (3) x Ax 1 Ax = — or — = — ) m X m expression (3) becomes: log('l+-) , ^. logfl+iV Ax X X \ mj X m Taking the limit dx of Ax, which corresponds to m = oo, and representing the limiting value of (1 -\ 1 by e, we have dy __ log e ( dx ^ To obtain the value of e, expand ( 1 H — 1 by the binomial theorem of Newton (564): mj m 1-2 1 • 2 ■ 3 mr m(m — 1) (w — 2) ■ • • (m — n + 1) 1 1 _ 1-2-3- • n nf " n^' canceling the common factors in each term and dividing by m, V m/ 1 • 2 \ m/ 1 • 2 • 3 \ m/ \ m/ 1 2 and if m = oo. the terms — . — . . . become zero: m' m =(-^r l+l+T^+ , i . +■•• + 1. 2 1-2- 3 ' 1.2.3...n + . I + I + ^ 1 • 2 • 3 ■ • • n(n + 1) ^ 1 ■ 2 • 3 • • • n(n + 1) (n +2) DERIVATIVES OF FUNDAMENTAL FUNCTIONS 577 The terms containing n, having the common factors the limit of their sum, less the first term, is 1 1.2.3-n'™U+l'''(n+l)(n + 2)"^(n+l)(n+2)(w+3)+"J"^^^ The sum of the fractions placed in parentheses being smaller than the sum of the terms of the descending geometrical progres- sion, 1 1 1 n+1' (n + iy (n+ ly of which the first term and the constant multiplier are n+ I this sum having - for its limit, the sum of the terms within the parentheses of expression (4) is smaller than — • Therefore, the value of e may be calculated with any desired degree of approx- imation, and it is found that e = 2.7182818 and log e = 0.4342945. (407) The derivative oi y = log x is, therefore, / dy ,,, . log e 0.4342945 and the differential is , , dx ay = log e 1278. Derivative and differential of y = sin X. (1) Giving the increment Aj; to the arc x, the function y or the sin takes the corresponding increment Aj/, and (1) becomes, y + Ay = sin (x + Ax). (2) Subtracting (1) from (2), Ay = sin (x + Ax) — sin x. (3) Since (1276) sin p — smq=2cos-^(p + q) sin ^(P ~ 5'). (4) 678 ELEMENTS OF CALCULUS putting p = X + Ax and q = x, we have, p + q=2x+Ax, p-q=Ax, or ■^(p + q) = x+—< ^^p-q)^'^. The relation (4) applied to the difference (3) gives Ay = 2 cos (x + ~] sin i-^j • Dividing both members by Ax, Ay ^"°^(^ + T)^^^(f) Ax Ax Dividing both terms of the fraction in the second member by 2, ^^ cos(^+^)sin(^) Ax Ax ~2 The ratio of the sin 1— j to — having 1 for its limit (1277), we have, dy ^=cos(x+^). dx -^ being negligible, we have, dy dx and the differential is, dy = cos xdx. THEOREMS OF DIFFERENTIATION 1279. The derivative and differential of a constant quantity are zero. Given the functions, y = F (x), (1) y = Fix) + C, (2) which differ only by the constant C. THEOREMS OF DIFFERENTIATION 579 From (1), y + Ay = F {x + Ax); from (2), y + Ay = F{x + Ax) + C; both of these expressions give the same value for the increment Ay of the fmiction Ay = F (x + Ax) - F (x). Therefore both give, Ay _ F(x + Ax) - F(x) Ax Ax , dy ,. F {x + Ax) - F (x) _„ , and 2/' = ^ = hm -i ^^ ^ = F'(x). Thus the derivatives of the functions (1) and (2) are the same, as are also their differentials; thus both give, dy = F' (x) dx. The constant C disappears in the process of differentiation. 1280. The derivative and differential of the sum of several func- tions are respectively the sum of the derivatives and the sum of the differentials of the functions. Given the sum y = F(x)+F,{x)+F,{x) + ... (1) in which F{x), F^{x), F-^ix) . . ., designate different algebraic quantities expressed in terms of x; for example, F {x) = log X, F^ (x) = sin x, F^ (x) = x™ . . . If X is increased by the increment Ax, the quantity y increases by a corresponding increment Ay, and relation (1) becomes, y + Ay = F{x + Ax) + F^{x + Ax) + i^^ (x + Ax) + • • • (2) Subtracting (1) from (2), we have, Ay =\_F{x + Ax) - F (x)] + [i^i {x + Ax) - Fj, (x)] + [J5'j(x + Ax)-i^2(x)] + •■• dividing both members by Ax and equating the limits, dy ,. F{x + Ax)-F{x) , ,. F^{x ■]- Ax) - F^ix) ^ -T- = lim — ^^ r^ ^-^ + lim -^ — r f- • • • dx Ax Ax or y' = F' (x) + F\ (x) + F'^(x) +■■■ = -^+cosx + mx"-! + • • • 580 ELEMENTS OF CALCULUS which was to be proved. In the same manner the differential is obtained, dy = F' (x) dx + F\ {x) dx + F'^ {x)dx ->r ■■■ loff e = dx + cos xdx + ma;™"' dx + • • • 1281. The derivative of the 'product of several functions or vari- ables is equal to the sum of the products which arc obtained by mul- tiplying the derivative of each function by the product of the other variables. 1st. Given, the fimction y = wv; (1) deducing the derivative, y' = vu' + uv', {A) in which the variables u and v are the functions of x, such that, for example, u = log X, V = sin X. Increasing x by the increment Ax, u, v, and y take the corre- sponding increments Aw, Av, and Ay, and relation (1) becomes, y + Ay = (u + Am) (v + Av) = ux + vAu + uAv + AuAv. (2) Subtracting (1) from (2), Ay = vAu + uAv + AuAv; dividing by Ax, Aw Am Av Aw . Ax Ax Ax Ax ' and equating the limits, dy du , dv , , , /on or y' = vu' + uv'. The limit -t- dv oi — Av is negligible, since the factor dv is an infinitesimal, u' and v' designate the derivatives of u and v with respect to x, and the relation (3) is the required derivative. For u = log X, and v = sin x, we have (1277, 1278), dy . log e , , ^ = sm X — ^ 1- log X cos X. dx X THEOREMS OF DIFFERENTIATION 581 From the relation (3) the differential is deduced, dy — vu'dx + uv'dx = vdu + udv; which gives, in this case, dy = sin x — ^ dx + log x cos xdx, (4) and y = smx —^ — f- log x cos x. Thus, the derivative of the product of two variables is equal to the sum of the products obtained by multiplying each variable by the derivative of the other. 2d. Given the product, y = stv, (5) of three variables which are functions of x; we have, for example, s = a;", t = log x, V = sin x. Putting St = u, du = sdt + tds, the relation (5) becomes, y = uv; and from (1st) its derivative is, , _dy _ dv du dx dx dx Substituting for u and du, dy ^dv , V , ,, , ^, . .dv , dt ds '£^^*di^d^^^^^^ ^^^^ ^^^di'^^^'di^^^'dc' tt/Jy U/Jb ilJb titl' t*U/ U'.t' Designating -j- , 3- and -j- respectively by v', t' , and s', w' = ^s' + si)i' + tes' (6) ^ dx which gives that which was to be proved. Applying the formula (6) to the given example, we have, 2/'= -J- = a;" log X cos X + x" sin x-^ + log x sin xmx'""^ The differential of y is deduced from (6), dy = stv'dx + svt'dx + tvs'dx, and for the given example, we have, dy = x^log x cos xdx + x" sin x -^^ dx + log x sin xmx'"-'^dx. In the same manner it may be shown that this theorem applies to any number of factors. 582 ELEMENTS OF CALCULUS 3d. Special case where one of the factors is constant. Given the product, y = aa;"", in which the factor a is a constant. Applying the general rule for the differentiation of two factors (1st), -^ = amx'^-^ + = owia;'""'; and the differential is, dy = amx^^^dx. Thus in the differentiation of a product, all constant factors enter both the derivative and the differential as coefficient. 1282. Derivative and differential of a quotient or a fraction. Given the function, ;»=-:. . (1) in which u and v are functions of the same variable x, we have, for example, u = x"" and v = log x. From relation (1) we deduce (482), y = uv~^. Applying the rule for the differentiation of the product of two factors (1281, 1st) and taking the differentials. , , „ , vdu udv vdu — udv\ ,_, V ^du ~ uv 'dv = — 5 r- = 5 \ (2) V V V ^ To obtain the derivative of relation (1), u = yv. Taking the derivative of both members with respect to x, we have (1281, 1st), du _ dv dy dx dx dx' (3) du dv dy Tx^'^di^'^dx' and dy du y dv dx vdx v dx Substituting - for y, dy du u dv dx vdx v^ dx THEOREMS OF DIFFERENTIATION 683 or designating the derivatives of y, u, and v, with respect to x, by y', W, and v', , u' uv' vu' — uv' y = 2 = — 2 — which shows that ihe derivative of a quotient is equal to the product of the denominator by the derivative of the numerator less the pro- duct of the numerator by the derivative of the denominator, all being divided by the square of the denominator. Comparing the relations (2) and (4), it is seen that by replacing the word derivative by that of differential in the last rule, the rule for the differential of a quotient is obtained. 1283. Derivatives of a function of a function. When a function is not expressed directly by the independent variable x, as in the examples y = log (sin x), y = log (x™), y = sin (mx + c), it is said to be a function of a function. Such relations are written thus: y = Ff{x). In these examples the quantity within the parenthesis is itself a function of x; representing it by u, the preceding expressions may be written : y — log M or M == sin x; y = log w or w = a;""; y = sin u or u = m£ + c. The quantity y is called the principal function, u the subordi- nate function, and x the independent variable. It is easy to find an algebraic relation between these different quantities. Writing the identity Aa; ~ Am Ax which is true no matter what the simultaneous increments Ax, Am and Ay of the variables x, u and y may be. Equating the limits dx du dx 684 ELEMENTS OF CALCULUS -^ being the derivative of y with respect to u, and j- that of u with respect to x, it is seen that the derivative of a function of a function is equal to the product of the derivatives of the simple func- tions which compose it. Example 1. Find the derivative of y = log (sin a;). (1) Putting u = sin x, (2) the relation (1) becomes y = log u; and taking the derivative (1755), dy ^ loge du u Taking the derivative of u with respect to x (1278), the rela- tion (2) gives du -r = cos X. dij du Substituting for -^ and -r- in relation (a), dy log e -T" = — ^— cos x: ax u then substituting for u, dy log e log e ^,„,,» y'= -f- = -T-^ cos X = -^— ■ (1041) ax sin X tan x Taking the differential, dy = , dx. tana; Example 2. Find the derivative of y = cos x; (3) from (1053) y = sin (90° - x). (4) Putting M = 90° - X, (5) and substituting in (4), y = sin u. Taking the derivative of y with respect to the subordinate func- tion u (1278), dij -p = cos M = cos (90° — a;) = sin x. y = Tx= -''''''' THEOREMS OF DIFFERENTIATION 585 From the relation (5), taking the derivative of u with respect to a; (1276, 1279, 1288), du _ dx dif die Substituting for -j^ and -j- in relation (a), we have and the differential dy = — sin xdx. Example 3. Derivative of a radical of the second degree. y = Va' - x\ (6) Squaring, y'^ = o? — x^. Differentiating both members (1276, 1279, 1280), 2 ydy = — 2 xdx. Simplifying and transposing, — xdx xdx dy = y Va^ J , dy X and y>=-^ = - —== dx ya'—: which may be written dy — 2x y'=-r = dx 2 ^a^ _ x^ that is, the derivative of a radical of the second degree is obtained by dividing the derivative of the quantity under the radical by twice the radical. The same problem may be solved by aid of the theorem of a function of a function. Putting du u = a' — x^, du = — 2 xdx and j- = —2x. The relation (6) may be written from (1276) ^2/ _!..-*_ 1 I .. (553) - u- ? = du 2 2ui 2'^a^-x' 686 ELEMENTS OF CALCULUS Substituting for -p and -p in (a) dy - — r-^ X-2a; = - 2 •Va=' - x' X dx ^a'-x'' id J xdx dy z= . ^/a'-x' Example 4. Find the derivative of y= cosx from (1282) dy _ cos X cos x — (sin x X — sin a;) _ cos^ x + sin' x dx cos^a; cos^x Having cos' x + sin' x = 1 and cos' x = rr— — — 5— , (1041) 1 + tan' X ^ ^ we have ^ dx = — 5— = 1 + tan' X, cos' X and dy = (1 + tan'x) dx. 4th. For y = cotx write (1041) and from (1282) cosx ^ sm X dy (sin XX- ■ sin x) — sin'x cos X cos X — sin' x — cos' x - 1 dx sin'x sin'x = - (1 + cot' x), therefore y' = — (i + cot' x), and dy = =-^— = — (1 + cot' x) dx. sm' X Derivatives and differentials of inverse trigonometric functions. 5th. For y = sin-'x, which indicates that y is the arc or angle whose tangent is equal to X, and we may write X = siny, which gives (1278) -J^ = cos y; and 2/'=^ = J_ = dx cos y Vl — sin' y Vl — x' then dy = '^^ Vl-a? 590 ELEMENTS OF CALCULUS 6th. For y = cos-' x, write a; = cos y. From (1283, Example 2) da; -^ = — sin w, ,^ , dy 1 1 1_ then y' =-f- = - ■^— = - ^/, „ ^ = - 7^= dx sin 2/ VI — cos^ y VI — ■^ d^ and dj/ = 7th. For y = tan-' x, write a; = tan y. dx From (3) -^ = 1 + tan'' y; dy then 2/' = S = dx 1 + tan'' 2/ 1 + a;^ J dx and '^2/=j-qr^- 8th. For y = cot-» a;, write X = cot j/. Taking the derivative g = -(l + cot2 2/)=-(l + a?), dy -1 1287. Examples of derivatives of trigonometric functions. Example 1. Find the derivative of . _, V2 Rx - x" „. 2/ = sin ^ (1) Putting z = V2 Ea; - a;^, (2) 2' = 2 i?a; - a-!. (3) The relation (1) becomes 2/=sin-i|. (4) From (1283) V'^t^P,- (^^ THEOREMS OF DIFFERENTIATION 591 From (4) we deduce dy 1 R dz . L, 2^ R — X and from (2) (1284, Example 3), dz 2{R-x) R-x dx 2 'sliix - x^ sjRx - x^ Therefore the required derivative (A) is , dy R R-x R dx R-x V-Ra; - x' S/Rx - x^ Example 2. Find the derivative of . , V2 Ry - y^ u = sm-i ^ — ^ • (1) K Let y = F {x) = 2 ax (2) be given to find the derivative -j- of the function u with respect to X. Putting z= ^2Ry-f, (3) 2== = 2% -2/^. The relation (1) becomes u = sin~'^- (4) The theorem of a function of a function (1284): du _ du dz dy ,-■. dx dz dy dx The relations (4), (3) and (2) give the derivatives: du _ R dz R — y' dz R — y dy '\l2Ry-f' ^ = 2a. dx Therefore the relation (5) gives the required derivative : du 2aR dx V2 Ry - f 692 ELEMENTS OF CALCULUS « 1288. Derivatives and differentials of implicit functions. To apply the foregoing rules to the determination of the deriva- dv tive -p , commence by solving the equations for y, that is, reduc- ing them to the form y = /(a;). But often this method is laborious, and it may be simpler to have recourse to a general theorem which does not require the solution of the equation with respect to one of the variables. Let us assume that all the terms of an equation have been transposed to one side, and reduced to the form f{^,y) = o, (1) which indicates that a relation exists between the two variables X and y such that the simultaneous values of the two written in one member make that member equal to zero. Giving X an increment Ax, y takes a corresponding increment Ay, and the relation (1) becomes fix + Ax,y^ Ay) =0. (2) Subtractmg (1) from (2), fix + Ax,y + Ay) - f (x, y) = 0. Subtracting and adding the fimction f{x + Ax,y), in which y is considered as a constant and a; as a variable, we have f{x + Ax,y+ Ay) - fix + Ax,y) + f(x + Ax, y) - f ix, y) = 0. Dividing all the terms by Ax, fix + Ax,y + Ay) - / (a: + Ax, y) f jx + Ax, y)-f jx, y) ^^ Ax Ax Multiplying and dividing the first term by Ay, we have fix + Ax,y + Ay)-fix + Ax,y) Ay ^ fix + Ax,y)- fix,y) _^ ^g^ Ay Ax Ax , ' which is true, no matter what the simultaneous increments Ax and Ay may be THEOREMS OF DIFFERENTIATION 593 In taking the limits, it is to be noted: 1st. That lim -^ = f\ (x 4- Ax, y), representing by f'„ (x + Ax, y) the derivative with respect to y of the function /(x + Ax, j/), in which x + Ax is considered as a constant and y as a. variable (1274); 2d. That Hn./J^±^^^^li(^ = /,(x,,), that is, the derivative with respect to x of the given function / (x, y) in which y is considered as a constant and x as a variable. Then the limit of the relation (3) is f'yix,y)^ + f'^{x,y) = 0, and the required derivative rfy^ -rAx,y) ^ , y dx rAx,y) ^^^ Thus the derivative of an implicit function involving two vari- ables is equal to at least the derivative of the given function taken with respect to x, considering x as variable and y as constant, divided by the derivative of the same function with respect to y, considering X as constant and y as variable. Remark. The quantities f'^ and /'„ are called partial deriv- atives of the function / (x, y). Example 1. Find the derivative of the implicit function (1131) ay + Vx" - a^W = 0. (5) We have - f'J.x, 2/) = - 2 b^x and f'y (x, y) = 2 a^y; ,, , , dy —2b'x — ¥x therefore w' = -2. = — = — ^ '^ dx 2 a^y a'y Remark. The same result is obtained by taking the differen- tials of the different terms of the relation (5). Thus, we have 2 a^ydy + 2 Vxdx = 0; . . , dy — b^x transposmg, 2/. = _| = _^. 694 ELEMENTS OF CALCULUS Example 2. Find the derivative of the function 2/^ = 2 px. Write / (x, y) = y^ - 2 px = 0, then - /', (x, y) = 2p and /', (x, y) =2y, and 2/' = J = |^ = £. " dx 2y y Example 3. Find the derivative of (y - qY + {x- pY = r*. Having / {x, y) = {y - qf + (x - p)2 _ r^ = 0, we have f'xix, y) = r^{x - pY = r.ix" -2px+p')=2x-2p = 2(x-p), f'vix,y) = f'y{y-qy = 2(y-q), and , ^ ^ _ - f':c {x, y) _ -2{x-p) ^ -{x-p) dx f'y{x,y) 2 iy-q) y-q ' 1289. Compound functions . Let us consider a function of two variables u and v which we will designate by y = F{u,v). (1) The quantities u and v being the functions of x, it is required to dy find the derivative y' = ~^- ^ dx Giving X an increment Ax, the other variables, u, v and y, take the corresponding increments Aw, Av and Ay, and the rela- tion (1) becomes y + Ay = F {u + Au, V + Av). (2) Subtracting (1) from (2), Ay = F (u + Au, V + Av) - F (u, v). (3) Adding and subtracting the following mixed function in the second member of (3), F iu,v + Av), we obtain + Av), V). _ {F (u + Au, V + Av)— F (u, V + ^~\ +F (u,v + Av)-F {u, v) With reference to the mixed function, it must be observed that u is to be considered as a constant and v as a variable. This THEOREMS OF DIFFERENTIATION 595 being true, if all the terms of the last relation are divided by Ax, we have % _ F {u + Am, V + Ay) — F {u, V + Ay) Ax Ax F {u,v + Av) — F {u, v) + Ax Finally, if the common factors Am and Av are introduced into the two general terms of the last expression, we have Ay _ F (u + All, V + Av) — F (u,v + Av) Am Ax Am Aa; F {u,v + Av) — F (u, v) Av Av Ax The limits of these ratios are Ay , Am , Av Aa; ^ ' Ax Ax which may be written y' = F\ (m, v) u' + F\ (m, v) v', (5) designating the derivative of F (u, v + Av) by F\, neglecting Av and considering w as a constant and m as a variable ; likewise the derivative of F (m, v) is F'y when v is the variable, and the rela- tion (5) may be written y' = F'y + F\v, (6) Thus, the derivative of a compound function of two variables u and V is equal to the sum of the products obtained by multiplying each partial derivative by the derivative of the corresponding vari- able taken with respect to the independent variable x. Remark. This theorem is of general apphcation. Thus the function y = F{u, V, z) gives y' = F'„m' 4 F'^v' + F',z'. Example 1. Find the derivative of y .= a;^"^. Putting a; = M and sin a; = f, we have y = u\ Applying theorem (6), that is, y' = F'y + F'y, 696 ELEMENTS OF CALCULUS we have successively log e ' y' = sin aix""^-' + a;™^!-^^ cos x. loge Example 2. Find the derivative of y = 3f, (a) which may be written in the form y = u" by putting u = x and v = ". Applying theorem (6), log e \ log e/ Remark. This derivative may also be found as follows. From the given function (a), taking the logarithms, we have logy = X log X, and the derivative gives log e log e , , y' = a;— 2 — h log a; = log e + log x, y X Example 3. Find the derivative of the compound function y = uv which is y' = F\u' - F\v'. {A) As a special case, take y = X log X. Putting u = X and v = log x, the theorem (^4.) gives y'= log X H ^ X = log e + log x. This result may also be obtained by applying the theorem relative to the product of two functions (1281). Example 4. Application of the theorem of compound functions to the determination of an implicit function. THEOREMS OF DIFFERENTIATION 597 The theorem of (1288) may be deduced from the general theorem of compound functions. Let the implicit function F{x,y) = (A) be given. Comparing this with y = F (u, v) and putting u = x and v = y, the latter gives y'= F'ji' + F\v'. (B) From the relation F (x, y) = 0, it is seen that the derivative of the two members should be zero. Then (B) gives = F'^' + F'yy'; but the derivative x' is equal to one, and the above expression reduces to = 2^',+ F\y', -F' from which y' = F' TANGENTS. 1290. We saw in article (1275) that the limit -f^ of the ratio ax of the increment of the function y to that of the variable x, was equal to the slope of the tangent to the curve which represents the function. From this property it is easy to deduce a method of drawing a tangent to a curve whose equation is given and determine the equation of the tangent, y' and x' being the coordinates of the point of contact of a tangent to any curve, the equation of any line which passes through this point is (1118) y - y' ==. a{x - x'). In order that this line be tangent to the curve, the coefficient a must be equal to the derivative -j- of the equation of the curve taken at the point of contact. From this it follows that the general equation of a tangent to any curve is We will now apply this equation in some examples. 1291. Tangent to a circle. 698 ELEMENTS OF CALCULUS The equation of a circle referred to its center being (1123) 2/2 -|- a;2 = j^^ applying the rule for implicit functions (1288) we have dy _ — X dx y For the point of contact (x', y'), which is given, the derivative is dy _ — xf dx y' Therefore the equation of a tangent to the circle' at this point, dii upon substituting for -^ in (a) of the preceding article, becomes — x' y - y'= -TT (^ - ^0- Eliminating the denominator and reducing, yy' — 2/'^ = — xx' + x''^ or yy' + xx' = y'^ + x'^ = r'. Thus the sum yy' + xx^ = r^ = constant. 1292. Tangent to an ellipse. Theequationof an ellipse referred to its principal axes being(1131) d?y'^ + V^x^ = xW, from (1288) ^ = ^^, dx ^:^^^%=-^ -^2Ry- y\ (1) The equation of a tangent at the point M is y - ^ = m{x - a), (A) a and j3 being the coordinates of the point of contact and in the dv slope of the tangent. We know that m = -^ is the derivative of equation (1) of the curve. To find this derivative, put z = V2 Ry - y^, then z' = 2 Ry - f, (2) and equation (1) may be written X = sin~' D ~ 2- (^) K Taking the derivatives of all the terms with respect to the independent variable a; (1298) , z' , z' 1= — , —z'= V R2 V _ R^ V R^ substituting for z^, 1 = ^' {"p — )' and z'=^^^- (4) y In equation (2), taking the derivative with respect to x, R-y V2%- y The relations (4) and (5) give R-y R-y z'= , _ " .y'. (5) y', y yl2R - f , , , /2R-y and 3/'= 1/ = m. 602 ELEMENTS OP CALCULUS Therefore the equation (A) of the tangent to a cycloid is j2R-y, y-P = \/-\^i^-<^)- (6) For the highest point or the vertex of the cycloid, we have y = 2R, and the value of the coefRcient m is ^\/^ J 2R-2R - m=y 2^ =0. Thus the tangent is parallel to the a;-axis or the base of the cycloid. Remark. If the point of contact is placed at the height of the center of the generating circle, we have y = R, and the coefficient becomes which shows that the angle between the tangent and the a;-axis is 45°. At the origin and at the end of the cycloid, we have y = 0, and the coefficient for each of these values is ./R m= y-Q =. (a) If for 0) = 2 IT we have p = a, the preceding equation (a) gives a = K2w, and K = -pr-- Equation (a) becomes p = -^r-w. (6) Z'jr THEOREMS OF DIFFERENTIATION 603 The general expression of the slope of the tangent with respect to the radius vector, as given by equation (2), has the value /, d<0 2 TT 2 Trp tan o = p-r- = p — = — - . dp a a This value is for the curve traced in (1233). For p = 0, tan = 0; therefore, at the origin the spiral is tangent to the polar axis. 2d. The spiral not starting at the pole has the equation of the form. P = b + Ko>. (A) For 0) = 0, jo = 6. If for each revolution of the spiral the radius vector increases by an amount a, the above equation will hold for (1) = 2 57 and p = b + a, and we have b + a = b + K2Tr and K = TT- • Then the equation of the spiral is and we have, as in the first example, » dot 2-ir tan = p-r = p dp a For p = b y^e have m = and tan e = - 2 ff. a Thus the first element of the spiral is no longer tangent to the polar axis as in the preceding case. If we make 6 = 0, the spiral passes through the pole, and we have tan ^ = 0. Example 2. Tangent to a logarithmic spiral (1270). The equation of the logarithmic spiral is log p = An>{^ c,A^ \ ,^ x/ >^' >V ■v a: Fig. 369 makes with the axis Ox. Thus the curves C^ and Cs concave at M with respect to Ox and C and C-^ are convex to the same line Ox. The concavity and convexity constitute the direction of bend- ing of a curve. Let us express analytically the distinctive char- acter of the direction of bending with respect to the a;-axis. Fig. 370 For the curves C and Cj, the function 2/ = / (x) being increasing (1273), their tangents make acute angles with the X-axis and their slopes or angular coefficients are positive. Constructing the curves C and C , representing their first deriva- tives ^• !=''«' the ordinates of both of these curves will be positive, but they will have a characteristic difference due to the opposite direc- 608 ELEMENTS OF CALCULUS tions of bending of the curves C and Cj; thus the ordinates of the curve C will be increasing the same as the corresponding function, while the ordinates of C^' will be decreasing. It is seen, in fact, that x increasing . the tangent makes greater and greater acute angles with the a;-axis, the slopes increase, and the function -f- = f (a;) = y', which is represented by the curve Ci, is also increasing. In the same way it is seen that in- creasing X, the tangent to the curve Cj makes smaller and smaller acute angles with the x-axis; therefore the slopes diminish, and the function -^ = f (a;) = y', which is represented by the curve Ci, is decreasing. Now constructing the curves C" and C/', representing the second derivatives of the original functions 1/ = /(x), we have curves whose equations have the form that is, the ordinates y" of which are equal to the slopes of the tangents to the curves C and C'l, it is easily seen that /" (x) is positive and increasing in the case of the curve C Thus the curve C which is convex to the x-axis corresponds to the curve C" whose ordinates are positive, and the curve C^ concave to the x-axis corresponds to the curve Cj" whose ordi- nates are negative. As is shown in Fig. 370, this property applies also to the curves C^ and d; and in general, we may say that any curve whose equation is of the form y = fix) is convex or concave to the x-axis according as y" = f" (x) is posi- tive or negative. POINT OF INFLECTION. 1302. In general, the second derivative of a curve for the point of inflection is zero or equal to ±00. 1st. General Case. '^Yhen a curve AMB changes its direction of bending, the point M where this change takes place is called a point of inflection. Drawing a tangent to the curve at the point M, the two- elements Mm and Mn which are situated just POINT OF INFLECTION 609 before and just after the point of inflection lie on opposite sides of the tangent MT; y = / ix), y' = /' (X) = dx y" = /" (x) M dx being respectively the equations of the required curve AMB, B ■B' H. -^ Fig. 371 and of the first and second derivative functions; if there is a point of inflection M, we obtain for this point y" = /" {x) = 0) which indicates that the point M" of the second derivative curve is on the x-axis.- This is evident a 'priori, because, the portion AM being con- cave to Ox, the corresponding curve A"M" of the second deriva- tive has negative ordinates (1301), and the portion MB being convex to Ox, the corresponding curve M"B" of its second deri- vative has positive ordinates; from this it follows that the con- tinuous curve A"M"B" must cut the axis at M". The same is true of the curves A^M^B^ and AJtit^B^. 2d. Special Case. Given, two curves AMB and AJiliBi whose points of inflection M and Mj correspond to the tangents MT and M^Ti which are parallel to the j/-axis. Constructing the first and second derivative curves, it is easily seen that the points M" and Af/', which correspond to the points of inflection M 610 ELEMENTS OF CALCULUS y A l» 1 A' 0, r Oz 1 r I f". Kg. 372 and Mj, are situated at in- finity ; that is, the second de- rivatives for the points M and Ml are 2/"=/"(z)=±oo. Thus, for the points of in- flection of a curve whose equa- tion is y = f{x), we have y" = r'{x) =0, or y" = /" (a;) = ± oo . Exception. It is possible for a curve whose equation is y = fix) to give y" fix) =±00 without having a point of inflection. For example, the equation of a circle is j (y - qy + (x - pY - J^ = 0, or / (x, y) = 0. From (1291, Example 3), dy _p — X ~ dx y — q and therefore (1286, 1299), M M' Fig. 373 2/"= /" (x) {y - if For X = OP = p — r and y = q, that is, for the point M, we have — r y and for x = p + r and y = q, that is, for M', we have Thus the second derivatives for the points M and M' are — co and -I- 00 ; nevertheless, they are not points of inflection, but there is a change in the direction of bending with respect to the a;-axis. TAYLOR'S THEOREM Example. Given a sine curve whose equation is y = sin X. y From (1282, 1287) y' = /' (a;) = cos x, y" = fix) = — sin x. The value 2/"= fix) = corresponding to a; = 0, tt, 2 ir, 3 T, . . . n TT, since for these values of x we have y = 0,it follows that all these points of inflection 0, M, M^, M^, . 611 M' W^ Fig. 374 i-a . . are situated on the a;-axis, and furthermore, the corresponding points 0^, M" , M", M" .... on the curve representing the function y" = f" (x) are also on the axis. TAYLOR'S THEOREM 1303. Preliminary theorem. If in a function y=-f{x), (1) x is replaced hy x -{■ h, it follows that y takes the value y' and relation (1) becomes y' = f{x + h). (2) dv' The first derivative -^ dx of y' with respect to x, considering x as a variable and /i as a constant, is equal to the first derivative dy' -TT of y' with respect to h, constant. Thus, we have dy' -TT of y' with respect to h, considering /i as a variable and a; as a dy^^dy^^ dx dh In fact, putting x + h = x', relation (2) becomes 2/'=/('x'), and or dy' dx' 7 = /'(A dy' d(x-{-h) = /' (x + h). (3) 612 ELEMENTS OF CALCULUS Assuming h constant and x variable, d{x + h) = dx, and expression (3) may be written %-n^^K). (4) Now supposing x constant and h variable, d{x + h) = dh, and relation (3) becomes f = /'(x + ^). (5) Equating expressions (4) and (5), dy^^dl^ dx dh 1304. Taylor's theorem. Suppose that the expansion of the function y' = f(x + h) (1) with respect to the successive powers of h be given, y' = y + Ah + Bh^ + Ch^ + D¥+ ■ ■ ■ ■ 2) It is evident that the polynomial which expresses the value of y' contains an infinite number of terms, in which the exponent of h increases indefinitely from the first term where it is zero. The coefficients A, B, C, D, . . . . , are unknown functions of the variable x, which are to be determined. Taking the derivative of y' with respect to h in equation (2), we have (1276) % = A + 2Bh + 3Ch' + iDh^ + . • - (3) ah In the same equation (2) the derivative of y' with respect to x considering h constant, is dy' dy dA , , dB ,. , dC , , , ,.. dx dx dx dx dx The first members of equations (3) and (4) being equal (1303), equating the second members, we have A + 2Bh+SCh'+4J)h'+ • • ■ = ^ + ^h+ ^W + ^¥+ • • • (5) dx dx dx ax TAYLOR'S THEOREM 613 Putting the terms of the same order equal to each other, we have A-=^, B = — , c = —, D = — dx 2dx' 3 da; ' 4 dx ' ' Replacing A by its value in the expression of B, then B by its new value in C, etc., we have A =^, dx d^ R _ dx _ d?"y 1 "Ydx"!^ T2' d^ _d^J^_d?y 1 D = 3dxl-2 dx' 1 • 2 • 3 d^ dx' 1 d*y 1 4 dx 1 • 2 ■ 3 dxM • 2 • 3 • 4 Substituting these values oi A, B,C, D, . . . . , in the series (2), we have dy cPy fe^ d'y fe' d^y fe^ ^ ^ ^ dx ^ dxM • 2 ^ dx' 1 . 2 • 3 ^ dxH • 2 ■ 3 • 4 which may be written in the form f{x + h)=fix) + r (X) h + r (x) ^ + /"' (x) j-|^ + /"(^)rT2^4+--- («) which is Taylor's theorem for expanding a function with the aid of its successive derivatives. 1305. Madaurin's theorem or a special case of Taylor's theorer.i. If in the function y'=f{x + h) (A) and in its expansion (1304) y' = f(x) + r (X) h + /" (X) ■— +r w i7^ + • • • (1) X is made equal to and h = x, the function (A) becomes (desig- nating y' by y) y = f(.x), and its expansion takes the form 2/ = /(x) =/(0)+/'(0)x + /"(O) 3^2 + r (0) X7^ + • ■ • (2) 614 ELEMENTS OF CALCULUS which is known as Maclaurin's theorem, and in which / (0), /' (0), /"(O), . . . . , are values of the function y and its successive de- rivatives when X = 0. 1306. Application of Taylor's and Maclaurin's theorems to the expansion of the sine and cosine in terms of the arc. 1st. Expand y' = {x + a)". From this relation we deduce successively (1276, 1305) /(x)= J/=x"', f (x)= moif~^, f"lx)=m(m- l)3f-^, f" (x)= m (m - 1) (m - 2) x"-', Substituting these values of /(x), /'(x), fix), ... .in formula (6) (1304), and noting that h is replaced by a, we have (x + a)™ = x™ + max"-! + "^ \"^ ~ ^' a^x""-^ m (m - 1) (m - 2) + 1-2.3 "'' + which is nothing other than Newton's binomial theorem (564). 2d. Expansion of sine x as a function of arc x. From the function 2/ = sin X we deduce successively (1278, 1283) / (x) = sin X, f" (x) = sin x /' (x) = cos X, /^ (x) = cos X, /" (x) = — sin X, /" (x) = — sin x, /'" (x) = — cos X, ^"{x) = — cos X, Making arc x = 0° in these expressions, and using the notation of Maclaurin's theorem (1305), we have fix) = / (0) = sin X = sin 0° = 0, fix) = /'(O) = cos X = cos 0° = 1, fix) = /"(0)= - sin X = - sin 0° = 0, /'"(x) =/"'(0)= - cosx = - cos 0° = - 1, /i^(x) =f">^(0)= sin X = sin 0° = 0, MAXIMA AND MINIMA 615 Substituting these values of f{x), fix), f" (x) ... in form- ula (2) of (1305), and noting that the odd terms are equal to zero, we have a;' ^ -»■' sin a; = a; — - — - — - + 1-2-3 1-2-3-4-5 l-2.3-4-5.,6-7 3d. Expansion of the cos x as a function of the arc x. From the function y = cos X we deduce successively f(x) = cos X, f*^ (a;) = cos x, f'{x) = - sin X, f (x) = - sin x, f"{x) = — cos X, p"^ (x) = — cos X, f"'{x) = sin X, r"(^) = sin x, Making arc x = 0°, and using the notation of Maclaurin's theo- rem, these expressions become (1305) fix) = / (0) = cos X = cos 0° = 1, fix) = /'(O) = - sin a; = - sin 0° = - 0, fix) = /"(O) = - cos X = - cos 0° = - 1, f"'{x) =/"'(0) = sina; = sinO° = 0, f" i^) =f^ (0) = cos X = cos 0° = 1, Substituting these values of fix), fix), fix) ... in Maclau- rin's formula (1305), and noting that the even terms equal zero, we have: , x^ x* a;' X* cosx=l——— +————— — +- 1-2 ' 1.2-3-4 1-2-3-4-5-6 1- 2- 3-4- 5-6- 7-8 MAXIMA AND MINIMA 1307. Maxima and minima of functions. Let the curve C represent the function y = fK-i)- If for a value OP = x of the abscissa, the- corresponding value MP = y oi the ordinate is greater than the vaiues^'^he ordi- nates m'-p' and m"p" , corresponding to the abscissas Op' and Qp" one of which comes just before and the other just after OP = x, the function or the ordinate y = MP is said to be a maximum. 616 ELEMENTS OF CALCULUS In the same way the ordinate M^P^ being smaller than the ones infinitely near it, the-ordinate or the function y which it represents, is said to be a minimum. Thus, in general, a function is a maxi- mum or a minimum according as a particular value is greater or smaller than the values infinitely near the point in question. As shown in Fig. 375: 1st. A function may have several maximum values and several minimum values; 2d. A minimum ilf,P, may be greater than a maximum MjPj,- 3d. A maximum or a minimum may be positive or negative. A function may have relative maximum and minimum values, and at the same time have an absolute maximum and an absolute minimum value. In order to obtain a clear conception of the behavior of a function when it passes through maximum and minimum values, construct the curves C, C„ and Cj, representing the given function and its first and second derivatives (1299), y'= fix) and y" = f"{x). At first the function y = f{^) is increasing, that is, when the abscissa Op' is increased, the ordi- nate m'p' increases also, and this is true until the point M is reached, where the function takes a maximum value y = MP. Up to this point the slope remained positive, that is. y' = r'.-)- dy dx remains positive, but diminishes continuously until at M it is equal to_^frr&i • The tangent to the curve C at M is parallel to the~a?^xis. Starting at M the function y becomes decreasing, that is, when the abscissa Op", for example, is increased, the ordinate m"p" MAXIMA AND MINIMA 617 decreases; this goes on until at M^ the function reaches a mini- mum. From M to M^ the slope or first derivative is negative. It goes on increasing up to the point of inflection between M and Af j, and from this point it de- creases continuously un- til it reaches M^, where it becomes zero, since the tangent to the curve at Ml is parallel to the a;-axis. In the same way, be- tween Ml and M^, the function is increasing, — ^ ^^ and the first derivative is positive, becoming zero at M2, which is another maximum. Thus for all maximum or minimum values of the function the first derivative is zero. y = / (a;), 0; that is, the points M', M^', M./ . . . which correspond to the points M, Ml, M2, . . . are situated on the axis Oia;i. To distinguish a maximum from a minimum we have recourse to the curve Cj which represents the second derivative. It is seen that the ordinate of the curve Cj or the second derivative, which corresponds to the maximum MP, is negative, while the second derivative, which corresponds to the minimum MJP^, is positive. It may be demonstrated that this is always the case. Thus, when the function, y = f{x) is increasing, the first derivative for the part m', for example, is positive, and at M is equal to zero. Since a quantity which is positive tends towards zero, it is decreasing, as is indicated by the portion AM' of the curve C^, and therefore. 618 ELEMENTS OP CALCULUS is a decreasing function. This established, as we see in Fig. 376, when a function is decreasing, the derivative of this function is negative; therefore, the second derivative M"P" is negative when the original function reaches a maximum value. In the same manner it may be demonstrated that the second derivative of a function corresponding to a minimum value of that function, is positive. Since it is simply the sign of the second derivative which dis- tinguishes between maximum and minimum values of a given function, if it happens that the second derivative is zero, it can have no sign, and could not indicate whether the corresponding value of the function were a maximum or a minimum. In this case it is necessary to have recourse to the 3d and 4th derivatives, as shown below. We have seen (1304) that a function y = f{x + h) may be written in the form, The increment of the function may be written: +rix) f(_x+h)-fix)=nx)h+f"{x)^+rix)j^ h* ;+■ 1-2-3-4 If for a certain value of x the functions /' (x) and /" (x) are zero at the same time (Fig. 377), this last relation is reduced to f{x + h)-fix)=f (x) 1 ■2-3 + r(.x) + ■~iK Mg. 377 1 •2-3-4 and since when the increment h of the vari- xg able X is very small, the terms of the second member which follow the first term are neeU- gible in comparison with it, and we have, f{x + h)-fix) = rix)^^^- (1) Therefore, if the increment f{x + h)—f (x) of the function is zero, which corresponds to a maximum or a minimum, we have, MAXIMA AND MINIMA 619 which requires that /"'(^) = 0; since the increment h of the abscissa, although very small, is not zero. Thus we see that the maximum or minimum of a function cor- responds to f"'{x) = 0. It now remains to determine when we have a maximum and when a minimum. Noting that before a maximum the incre- ment f{x'+h) — f {x) is positive and before a minimum it is negative, from the relation (1) f"'{x) has the same sign as this increment, since h and therefore h^ is always positive. Since a positive function /'" (a;) which approaches zero is decreasing, and the derivative of a decreasing function is negative, it follows that l"{x) is negative for a maximum value of the function (Fig. 377). For the same reason, if the increment f{x + h) — / (x) is nega- tive, /'" (x) - — jr — - wUl be negative, and therefore /'" (x) will be L ' ^ ' o negative. Since a negative function which approaches zero is increasing, and the derivative of an increasing function is posi- tive, it follows that /"■ (x) is positive for a minimum value of the function. There is a maximum or a m,inimum when the third derivative f" {x) is zero, and it is a maximum or a minimum according as the fourth derivative /'^ (x) is negative or positive. In general, when several successive derivatives are equal to zero, there is neither maximum nor minimum if the first derivative after the one which is not equal to zero is of an odd order; but if it is of an even order, there is a maximum or minimum, according as it is negative or positive. 1308. A function y oi a single variable x being given in the form y = f(x), (1) to find the maximum or minimum of this function, take the first derivative of y with respect to x and put it equal to zero, thus: f = /' (^) = 0. (2) 620 ELEMENTS OF CALCULUS This equation solved for x gives the value of x corresponding to the maximum or minimum. Then find the second derivative, y" = f"{x), (3) and according as this derivative is negative or positive, there is a maximum or a minimum. The value of x deduced from equation (2), substituted in equation (1), gives a maximum or minimum value of y. If the second derivative y" is zero, take the third and fourth derivatives, 2/'"=/'"(x), (4) t^-rix); ^ (5) put f"{x) = 0, and solve for x and substitute in (1), which will give the maximum or minimum value of y according as y^ is negative or positive. If the fourth derivative were also zero, we would take the fifth and sixth, and so on. 1309. Applications of the preceding rule. Example 1. The product y of two variables x and z, whose sum c is constant, is a maximum when the two factors are equal (583). Accordingly, we have, X + z = c {a) y = xz (b) From (a) z = c — X. Substituting this value in (b), y = ex — x^. (1) Taking the first derivative and putting it equal to zero (1276, 1280), g = /'(x) = c-2a;=0. (2) Solving for x, we obtain the value corresponding to the maxi- mum or minimum, c ^ = 2- Taking the second derivative (1279), %-r'ix) = -'^. MAXIMA AND MINIMA 621 This derivative being negative, x = ^ corresponds to a maximum and not to a minimum. Substituting this value in (a), we find c '=2 Thus we have a maximum when the two factors are equal, c Example 2. Of all cylinders having the same volume V, de- termine which has the minimum total surface S. r being the radius of the base and h the altitude of the cylin- der, we have, S = 2-n-r^ + 2 7rrh, (a) V and F = TT r'h, h = — = • (b) Substituting this value of h in (a), we obtain an expression in- volving only two variables S and r, S=27rr2 + — =27rr2 + 2Fr-'. (1) r Taking the first derivative and putting it equal to zero, ^ =:f'(r) = 4:7rr-2Vr-^=0. (2) Solving for r, we obtain the value of r corresponding to the maximum or minimum. Taking the second derivative, ^ = /" (r) = 4 ,r + 4 7r-' = 4 ^ + ^ . This derivative being positive, »• = V ^5~ corresponds to a mini- mum and not to a maximum. Substituting this value of r in (1), we obtain the minimum value of S in terms of V; but the dimension h being of more importance, substituting in (3) the value of V given in (6), we have, r = V — — or r^ = -2" and h = 2r=2y ^ 622 ELEMENTS OF CALCULUS Thus *S is a minimum when the altitude of the cylinder is twice the radius of the base, and we have y = 2 IT r = —— . 4 Example 3. The mean temperature in a chimney correspond- ing to the maximum draft, according to the old theory of P^clet, is expressed by the formula wherein Qi is the weight of air passed through the chimney per second; 1.3 is the weight of a cubic meter of air at 0° and 860 milli- meter pressure; D is one side of the minimum interior section, taken as square; a = 0.00367 is the temperature coefficient of air; M is a constant for any one class of chimneys; t' is the mean temperature of the air in the chimney; t is the temperature of the outside air. M 1.3 D' y ^ being a constant quantity for any one chimney, Q^ will be a max- / t' - t / t' — t imum when 1.3 D^ y j- — ; — -^, or y —. — - — -fr^ is a maximum. V (1 + aty V (1 + aty Representing this radical by y and the variable t' by x, we have, or 2/' + 2 axy^ + aVy^ — x + t = 0. Taking the first derivative (1288) and putting it equal to zero, dy _ — 2 g?/" — 2 ay a: + 1 _ dx 2y + i axy + 2 a^x^y This being true only when ~2ay^ -2 a^fx + 1 = 0, or - 2 a?/^ (1 + ax) + 1 = 0. Substituting the value of y^ given in (1), we have MAXIMA AND MINIMA 623 from which we deduce successively, 2 a -— — = 1, , 1 + ax 2 ax — 2at = 1 + ax,' ax = 1 + 2at, x= - + 2t. a If we assume the temperature t of the outside air to be zero, we have 1310. Special cases of maxima and minima. 1st. When a function has a value equal to infinity or zero, this value cannot properly be considered as a maximum or a minimum. The parabola whose equation is (1197) 2/2= 2px, giving y = for x = 0, and y = ± co for a; = oo , the function varies continuously from + ob to — oo , and has neither maximum nor minimum. The derivative of the preceding function being ax V putting it equal to zero. /'(^) = f = o, we have y = ± , values which correspond to a; = oo . Thus the points at which the tangents are parallel to the x-axis are at infinity. For x = 0, we have y = 0, and therefore, /'(a;) = ^ = oo. y Thus the ^/-axis is tangent to the curve. If the logarithmic curve, y = log X, is given: Taking the derivative (1281), dy_,,, log e _ 0.4342945 . dx^'^^^x X ' 624 ELEMENTS OF CALCULUS putting this derivative equal to zero, „ , , 0.4342945 f (x)= z 0; from this x = cc , and therefore, y = log x = oo ; moreover, since for a; = 0, we have y = log = — oo , the f imction varies con- tinuously from +00 to — 00, and >l nevertheless has no maximum nor ^ minimum. 2d. Another peculiarity of maxima and minima. Point of retrogression. When a curve has two branches AM and MB, having a common tangent parallel to the ^/-axis (Fig. 378), the point M necessarily cor- responds to a maximum or a mini- Pi 378 mum. At this point M the slope of the tangent is y y] p .^ r' J v X Oj f A Xj dy dx = /'(a;) = rtoo, which is not zero. Fig. 379 The point M is called the point of retrogression. A point of retrogression M (Fig. 379) may correspond to a tangent whose slope is not parallel to the y-a.ids, that is, a value of dy dx 3d. A curve may give a value of zero for the first derivative, and still have neither maximum nor mini- mum. This is the case when the curve (Fig. 380) has a tangent at a point of in- flection which is parallel to the a;-axis; be- cause for this point, /' (x) = 0. This case may be recognized from the fact that, starting from the point M, the curve is convex or concave to the a;-axis, according as the second derivative is positive or negative (1301). It may also be noted that in the case where M is a point of inflection the first deriva- tive does not change its sign, since the tangent to the curve at Fig. 380 MAXIMA AND MINIMA 625 that point and beyond does not change the direction of its slope with reference to the a;-axis ; except that it is zero at the point of inflection. Example of curves which have a maximum, a minimum, and a point of inflection. Given the equation a;' - 3 a; + 1 Fig. 381 y = x" - 6x + 1 (1) of a curve referred to a system of coordi- nate axes Ox and Oy. Taking the first and second derivative, we have, y' = 3x'-3, y" = 6x. For the point of inflection M the second derivative is equal to zero (1302). y" = 6x = 0, and a; = 0. It is seen that the point of inflection is situated on the y-axis. To determine the ordinate, make a; = in equation (1), which gives 3/ = 1. To obtain the coordinates of the points Mi and M^ correspond- ing to the minimum and maximum, put the first derivative equal to zero, 3a;2- 3 = 0; then, a; = ± 1. Therefore, equation (1) gives, 2/ = 1 - 3 + 1 = - 1, 2/=-l + 3 + l=+3. Thus the points M^ and M^ have the coordinates (a; = + 1 (x= — 1 1311. A study of quantities which have an indeterminate form. Let us consider a quotient of two functions of the same vari- able x, y = W:- (1) Giving X the value a, we have, <^(x) 626 ELEMENTS OF CALCULUS Putting u = F (x), (2) V = (x). (3) The relation (1) may be written, y = -- (4) , Giving an increment Aa; to the variable x, the variables u, v and y take corresponding increments, and relation (4) becomes w + Am . dividing both terms of the fraction by Aa;, u + Am y + ^y^ -n^ - (s) Aa; If for the value x = a, the functions (2) and (3) become zero, it follows that relation (5) has the limit Am _'Ki_F'u . y''Av~F'v' Ax that is, the value of the given quotient will be given by the quo- tient of the derivatives of both the terms, in which x = a. Example 1. Find the value of X"- 1 ^ = ¥^ ' for a; = 1. The direct calculation gives the indeterminate form, To make certain that the value is really indeterminate, replace the two terms by their derivatives, and in the new quotient put a; = 1. naf~* y = -y- = n, which is the required value. MAXIMA AND MINIMA 627 Example 2. Calculate _ ax*— ah" ax — aP J for the particular value x = \?. The direct calculation gives, Taking the derivatives of both the terms, and putting x = 6^ we obtain the rekl value, 3aa^ 3a6* „,, a a Example 3. Calculate the following expression for x = 30°: 1 ^ — sm a; y = J- {A) sin a; - - Sin 30° = ^ J consequently the value of the expression takes the indeterminate form, Taking the derivatives of both terms of {A), — cos x y = = — 1. cos X It may be noted that the given expression reduces to the con- stant value — 1 for all values of x. Thus, 1 - — sm a; 2/ = 1^ ~Js1^ZjI^-x. sm a; — ^ sm ^— t, Example 4. Referring to the form ^, let the function 00' y = - (a) be given, u and v being functions of x. It is required to calcu- late the value of y where a particular value, given to x gives u = ao and V = CO ; such that 00 " 00 628 ELEMENTS OF CALCULUS The relation (o) may be written y = r (b) Since v and u become infinite for a particular value x = a, the reciprocals - and - are equal to zero. Therefore, we may con- * sider y in relation (6) as having the form y = -ior the particular value X = a; and applying the above rule, that is, substituting the first derivatives for the terms of the quotient (6), the re- quired value is obtained, _ v' _v? v' ^~ r~~^ u'' U V? v' or -= "2 -7- V ir w CanceUing the common factor - , we have, ,. u u' lim - = -T • V V fit Thus we calculate the value of i/ = - as in the first example, by substituting the derivatives of the terms in the given expres- sion and putting x = a. Example 5. Find the value of the function log X y=x ior X = 00 . The direct calculation gives 00 y = — '^ 00 Taking the derivatives of the terms of the fraction separately, and making a; = oo , we obtain the real value, a; 00 ' If, giving y=\^x' MAXIMA AND MINIMA 629 the value for a; = oo is desired, replacing both terms by their de- rivatives and putting x = , the real value is obtained, _ 1 _ X _ 00 log e log e log e X Example 6. Find the value of y = tan x (a) cos x ' for x = 90°. The direct calciilation gives 1 Q>) 2/= 00 - --=00 — 00. The relation (a) may be written sin x 1 sin X — 1 ^ cosx cos x cos X For a; = 90°, this becomes 1-1 y = o' Substituting the derivatives for the terms of the fraction (6), and making x = 90°, we have, _ cos X _ cos 90° _ _ y " - sin 90° ~ ~^1~ " 31 - "■ Remark. This value, x = 90, corresponds to a maximum of the given function. sin X — 1 y = • cos X Thus taking the derivative, , cos^x — (sin x — Vj {— sin x) y' — , " cos^ x cos^ X + sin^ x — sin a; I — sin x or w'= = s ^ cos^ X cos^ X The maximum corresponds to 1 — sin a; = 0; then sin X = 1, and x = 90°. For all other values of x the function y is negative. 630 ELEMENTS OF CALCULUS RADII OF CURVATURE 1312. The equation of a curve MM'D of the form 2/ - / (a;) being given to find the value of the radius of curvature (1239). Let M and M' be two points on the curve, MA and M'B the tangents to the curve at these points, and MC and M'C the normals at the same points. Decreasing the arc MM', at the limit the chord MM' coincides with the tangent to the curve at M; and the triangle MCM', whose vertex C is the center of curvature, is a right tri- angle, and we have tanC = MM' 'MC and MC: MM' tan C The angle C included by the two normals, and the angle /3 included by the tangents, are equal, having their sides perpen- dicular to each other; and we have tan C = tan /3, and therefore. MC = MM' tan/3 (1) The angle a' being an exterior angle of the triangle AEB, we have ft = a - a, and (1046) tan a' — tan a i' — i tan^S = 1 + tan a tan a' 1+ ii' (2) designating the trigonometric tangents by i and i'. Since at the Umit the slope of the tangents differs only by a differential di, we have, i'= i + di; and substituting this value in (2), •i -1- di — i tan j8 = di (3) \+ i{i + di) 1 + 1=' + idi Furthermore, the right triangle MM'Q gives MM' = ^MQ" + WQ^ = "^{dxY + {dyf = dx y 1 + (^Y - (4) RADII OF CURVATURE 63l Substituting the values (3) and (4) for tan /8 and MM' in (1), we have, dx y/l + /^Y(l + i^ + idi) MC = di Noting that idi in the numerator may be neglected in com- parison with 1 + i^, dividing both terms of the fraction by dx and designating the radius of curvature MC by p, we have P - (i ' (1 + ^^) di dx Having « = tan a = ^ = f (x) and — = — = /" (x), the above relation may be written, (1 + u' jxmHi + [f (x)f) (1 + [f (x)Y)^ ... P- fix) - fix) ■ ^^^ If the sign of the numerator is always taken as plus +, p will have the same sign as /" (x), and consequently will be positive or negative according as the curve is concave to the positive 2/-ordinates or the negative 2/-ordinates. 1. Application to the parabola. The equation of curvature being (1197) 2/' = 2 px, we have successively, dx ' ^ ' y i^ = [/'(x)P = ^;. dy yfx °'^ ^*=p- Differentiating this last relation (1281), di . .dy „ yTx'^'Tx-''' di , -2 n. 2/^ + ^- = 0; , di ,„, . - «? -P' and T- or /" (x) = = — r~ " dx ' ^ ^ y f 632 ELEMENTS OF CALCULUS These values substituted in formula (5) for the radius of curva- ture give, T indicates that p has a sign opposite to that of y. For 2/ = 0, _ (P!)f _ 2! = J9. Thus at the vertex of the parabola the radius of curvature is twice the distance from the vertex to the focus (1195). 2. Application to the circle. From the equation of the circle (1123) y^+ x^= r^, we deduce successively (1288), dy , , — X ■2 ^' -x = yi, — dx = idy + ydi, ^_l/_i ,dy\_-{\ + i') dx or • y\ dx) y y f Substituting these values of /' (x) and /" {x) in the general formula (5), we have. P= /'l+-')^ 3 ^ W ^ ^ (y' + x^)h^ = T(2/- + .^^ = T V^m:^== Tr. -(2/Hx^) -(2/2 + x^)(j/^)f Thus the radius of curvature is constant and equal to the radius of a circle. 3. Application to the sine wave (1296, Fig. 367). The equation of the curve is y = smx, or y =^ R sin x, RADII OF CURVATURE 633 and /' (x) = R cos x, f" (x) = - R sin x. The formula (5) for the radius of curvature gives, _ (1 + -R' cos^a;)^ — 22 sin a; For x = 0, iroT 180°, 2 tt or 360°, (1 + R')i that is, at the points 0, B, D - . ., there is an inflection or change in curvature. n For a; = 2 ' ~o~ ' • • • *^^ radius of curvature has the value P = —n = T pj which is the radius of curvature in A, C, . . . 4. Application to the ellipse. From the equation of the ellipse, ay+ 6V= a'b^, we deduce successively, '. ^ ■' =* a^y » aY ff'M = yff = - a'b'y + a'b^xy' ^ -^ y^ ^ «V ^^2/ y ' „ -b\ ¥3? Substituting these values of y' and 2/" in the general formula (5) for the radius of curvature, we obtain, \, "^ aYI (ay + bV)^ '' ~ _ /_&^_^\ a^'b^Cft^a^ - ay) " Va'2/ a*yy For a = 6 = r, the formula gives p = r, which is as it should be, since the curve is then a circle. (j2 For X = and y = h, p = t- ' which is the radius of curvature W of the minor vertices of the axis. For w = and x = a, p = —, a which is the radius of curvature of the vertices of the major axis. INTEGRAL CALCULUS INTRODUCTION 1313. The object of integral calculus. Integration. Integral. Integral calculus can be used to find a function 2/ = / (a;) whose derivative y' = /' (x) is given; or to find a function y = fix) whose differential or differential coefficient dy = /' (x) dx is given. As is seen, integral calculus is the inverse of differential calculus. Thus the fundamental functions (1276, 1277, 1278, 1283) 2/ = a;"", y = log x, y = sin x, y = cos x, having respectively the derivatives and differentials log e y' = ma;"'-', y' = — 2_ , y' — cos x, y' = — sin x; loff 6 dy — mx^~'^ dxj dy = — ^— dx, dy = cos xdxj dy = — sin xdx, X if one of these derivatives or differentials is given, the above table gives the fundamental function from which it is derived. However, since the same derivative, for example, y' = mx"'~^, or the same differential, dy = mx'^~'-dx, corresponds to two functions, namely, 2/ = / (x) = x™ and y = f(x)+C (1) C being a constant (1279), which can be determined, the result of an integration is always written in the form y = fix) + C, 634 INTEGRAL CALCULUS 635 Fig. 383 which signifies tHat if the curve C (Fig. 383), whose equation is y = f{^), satisfies the conditions, the same will be true of all other curves C, whose ordinate at any point A gives, AP = MP ± MA. The length MA is the constant C in relation (1). It is to be noted that the three curves have the same slope at the points A, M, and A, since /' (x) is the same for each; that is, the tangents at these points are parallels. In practice, the constant C ceases to be arbitrary as soon as one point on the curve is known, or, which is the same thing, as soon as a system of values of x and y are known ; because, substituting these values in equation (1) we may solve for C. The process of finding the function of a differential equation y = fix)+C dy — f (x) dx is called integration, and the function is the integral of the differ- ential dy. 1314. Geometrical interpretation of an integral. Sign of integra- tion. Limits of an integral. Definite integral. Indefinite integral. y' = r (X) = dy dx' The first derivative, being given, we have dy = f {x) dx, and wish to find the original function Suppose the problem to be solved, and let the curve AMM'B represent the function. Considering the two points M and M', which approach infinitely near each other; at the limit, the increment M'Q' of the ordinate MP is the differential dy of this : V~^ ordinate MP = y ; and the increment PP' of the abscissa OP is the differential dx of 636 ELEMENTS OF CALCULUS the abscissa OP = x; and it is seen that in order to pass from the ordinate of a point A on the curve to another point B, the sum of a certain number of increments M'Q', M"Q", . . . must be added to the ordinate at the point A. Since at the hmit the arc MM' coincides with the chord MM' or with tangent to the curve at M, the figure MM'Q' is a right triangle, and we have, M'Q' - MQ' tan (M'MQ'), or dy = dx-^ = dxf {x) = y'dx. The element M'M" gives, M"Q" = dy, = dx,^J and since we have the same for each element of the curve AB, it is seen that the quantity BC which is to be added to the ordinate at the point A in order to obtain that at/ the point B, is equal to the sum of the differentials dy, dy^, . . . that is, My = "Sii/dx, wherein ^dy represents the sum of all the quantities analogous to dy' and 'Sy'dx the sum of all the products analogous to y'dx. This sum is the required integral of dy, and is written J dy = J y'dx, which is read, integral of dy equal to integral of y'dx. To indicate that this sum or integral is to be taken from the point A to the point B, designating the abscissa at A by a and that at B by b, we write. dy^ I y'dx. which is read, integral between the limits a and b of dy equal to the integral between the limits a and b of y'dx, and signifies that the integral of the differential quantity of the form dy = /' (x) dx is the sum of the increments dy of the function y, made between the limits a and b corresponding to two ordinates or particular finite values of the function y. One of these limits can be zero INTEGRAL CALCULUS 637 or negative ; that is what happens when the point A is on the 2/-axis or at the left of it; in each case the integral is written, dy = \ y'dx, and I dy = \ y'dx. The hmit a being negative, the hmit h can also be zero or nega- tive. An integral taken between two limits is called a definite inte- gral, and an integral under the general form / dy is called an indefinite integral. 1315. The calculation of a definite integral whose limits are given. Let y= fx^dx (1) ^ ^/ be given. Then from (1276, 1313), 2/ = I + C. (2) Q 'o p n « Now let it be required to calculate this in- ^^' ^^ tegral between the limits corresponding to the points A and B, whose coordinates are lx = a = OP „(x = b = 0R ^\y=a'=AP' \y = h'=BR' To calculate the integral / x^dx between the limits correspond- ing to the points A and B, amounts to finding the length BQ which must be added to AP in order to obtain BR. From the relation (2) we have, AP = y = ^ + C and BR = y =^^ + C and BR- AP = ^-^= \ 3?dx. Thus the required result is obtained by substituting successively in the indefinite integral (1) the values of x which correspond to the ■ limits of the integral and taking the algebraic difference of these two results. 1316. A definite integral may be represented geometrically by the area of a curve. 638 ELEMENTS OF CALCULUS Constructing the curves C and C" which represent respectively Q C the function and its first derivative, y = fix) (1) ^-|-n»), and from which dy = f (x) dx = y'dx. Fig. 386 Since in integrating this last expression jg, we obtain the original function (1), we have, j dy = j y'dx ov y = j y'dx. (2) The infinitesimal increment dx of the variable x being repre- sented geometrically by PP' = P^P^, and y' by the ordinate MiPi,the product y'dx is represented by the trapezoid MjPJ'iM^, since at the limit MJ^^ = M^P^, and it follows that the incre- ment dy = M'Q' of the ordinate y = MP of the curve C is rep- resented by the area M^PJP^M^. Since any other increment of the ordinate is likewise represented by a corresponding area, it follows that in passing from the ordinate at the point A to the ordinate at the point B, sum-total BD of all the increments of y will be represented by the sum of the corresponding areas, that is, by the area A^A^B^'B^. Thus, y'dx = 4i4,'5/B„ wherein a and h are the limits of the integral, that is, they deter- mine the ordinates which bound the area. Summing up, it is seen that the calculation of a definite inte- gral may always be reduced to the determination of the area of a curve included between two ordinates which correspond to the limits of the integral, thus representing the first derivative of the required function y = f(x) = J y'dx. RULES FOR INTEGRATION 639 RULES FOR INTEGRATION 1317. Integrals of simple functions. There is no general method of integration. Analogy serves as the rule. Thus the function y = x'" having the derivative (1280), (1) ^ = 2/' = mx""-S (2) and the differential, dy = mx'"~^dx, (3) if one of the expressions (2) or (3) were given to find the original function, the answer would be, and we would write. fdy=f> mx"-! dx = of + C, that is, the exponent m — 1 is increased by one unit and the quantity divided by the new exponent and dx; thus, ,j--^ I dy = j mx ^^dx ov y = ■ ^ = a;"; ^-'^ then the arbitrary constant C is added so as to obtain a general expression of the function whose derivative is mx"""'. Therefore we have. / afdx = ^Vt + C- n + 1 This rule does not apply in the case where n = —1. Thus we would have, /.-..=/§= -£;f^ + C = |Vc = l + C = oo + (7, or, if we had by analogy (1281), dx dy--- h-f- 'dx _ log X ^ log e (1) 640 ELEMENTS OF CALCULUS Table of Integrals and Their Corresponding Differentials da;»+» = (n + 1) x'^dx, (1280) fx^dx = ^—^ + C d\ogx = ^^^dx, (1281) r^^dx = \ogx + C. (2) ^ log a; ^ d^ ^ rd£ ^ log a: _|_ ^ ,g. log ex' J X log e ' ^ da^=^^^a''dx, (1289) fa^rfx = J^^a"' + C. (4) log e J log a ^ ' d sin a; =cos xrfa;, (1282) I cos, xdx = sinx ■{■ C. (5) rf cos a; = — sin xdx, (1287) I sin a;rfa; = — cos a; + C. (6) rftana;= — ^-={l + ta,n^x)dx, I — 5— = tan a; + C (7) cos^x ('1290) "^ ^ ^ d cot X = ^^ , (1290) (J^ = - cot a; + C. (8) Sin ic / sm ic d sec a; = — 5— dx, / - — 5 — da; = sec x + C. (9) COS^X J COS"* X ^ ' J cosx C- cosx , , ^ .^„. a CSC X = ^-5—' / — =—5 — da; = esc x + C. (10) sin^ X ,/ sin-* X ^ sin-i X = , (1290) / , = sin-^x + C. (11) Vi - x^ J vn^ d cos-' X = ~^^ , (1290) r 7^^ = cos-i x + C. (12) d tan-»x = j-^ , (1290) /j^ = tan"' x + C. (13) d cot-' X = -=^, (1290) r^^;:^ = cot-' x+C. (14) J. ~r X^ 1/ J- I *^ dor i dec dsec-'x= — , ) / — , = sec-' x + C. (15) 3;Vx2-1 J XVX^ _ 1 d CSC-' X = — ,. . ) / — . = CSC"' x+C. (16) xyy? — 1 »^ xVx^-l dl = ^, (1280) f^ = + -+C. (17) X ar J x^ X d^/x = -^ (1280) fe = 2V^ + C. (18) 2 Vx «^ Vx , F'(x)dx /,„„_v rF'(x)dx „ / „ . ■ , ^ ,,_, RULES FOR INTEGRATION 641 1318. The integral of the sum of several differentials of the same variable x is equal to the sum of the integrals tvhich compose this sum. Thus, the algebraic sum, y = u + V - z, (1) in which u, v and z are any functions of the same variable x, giving (1284), d{u + V — z) = du + dv — dz. Integrating both members, we have, I d(u + v - z)= I du+ I dv - j dz + C, or y = u + v-z + C, C being the sum of the constants which must be added to each particular integral. Example 1. Integrating the differential expression, dy = x'^dx + afdx — x'^dx, we obtain (1317), j.'t+i x""*"* a;*""*"' ^ ^ m+ 1 "*" n + 1 ~ p + 1 "^ ^° Example 2. Integrating, dy = dx + cos xdx, we obtain (1317), y = I dx + j cos xdx = log x + sin x + C. 1319. All constant factors in a differential expression appear in the coefficient of the integral of this expression. Thus, the function, y = af(x), in which a is a constant, giving (1285, 3d) dy = af'{x)dx. Integrating this function, we have af'{x)dx = af{x) + C. As example we have (1317) /° y=f5xMx = ^ + C. 642 ELEMENTS OF CALCULUS PRINCIPAL THEOREMS OF INTEGRATION 1320. Considering the constant coefficient, the integrals of certain functions (1317) may be deduced directly by making these constants appear as multipliers or divisors. Example 1. The differentials dy = — and dy = dx, differing only by the constant coefficient log e, their integrals differ also by this same coefficient; thus (1317), ""loge /dx X dx = log a; + C, X dx _ log X ^ X log e Remark. If the logarithms are taken in the Napierian system (408), since loge e = 1, we would have, ^dx f = loge X + C. Example 2. a and b being constant coefficients, we have (479, 1317, 1318), Uax + bx'y dx= \ a^xHx + / 2 abxMx + / b'^xl^dx _ aV 2 abx* 6V - "3" + ~4~ + T~ + ^• 1321. Integration by changing the independent variable or by substitution. A differential function which is not immediately iiitegrable sometimes becomes so by changing the independent variable. Example 1. Let it be required to integrate dy = {ax + bxTdx. . (1) The second member may be expanded by Newton's binomial theorem (530), and each term separately integrated; but it is simpler to operate in the following manner: Putting ax + bx = z, or {a + b)x = z, z 1 we have x = — r-r and dx = — — ^ dz, a + b a + b THEOREMS OF INTEGRATION 643 Substituting these values oi ax + bx and dx in relation (1), we have and integrating both members (1317, 1319), 1 ^+' ^ y a-Vbm+l^ ' ' then substituting ax +' hx for z, we have, l_{px + hxr^ y~a + b m+1 +^- Example 2. Find the integral y = I , -dx = I , dx = I — , dx. (1') X /xV Putting - = z, then dx = adz, and I - 1 = z^; and substituting in (1'), w= f—£=dz = a^ f-i£L= = a^sm-^z+C=a^sm-^- + C. (1317) Example 3. Find the integral /I Sin 2/ tana;da;= / dx. (1") t/ COS X Putting cos X = z, then dz = — sin a;rfa; or sin xdx = — dz, and substituting in (1"), r-dz^-logz^^^^hgco^^^^ ^ J z log e log e Taking the logarithms in the Napierian system (408), log^e = 1, and therefore y = — loge cos X + C. Example 4. A being a constant, integrate Putting z-b dz, ax + = z, x = and dx = — , 'a a 644 ELEMENTS OF CALCULUS and substituting in (1'"); A(z- hfdz A IzHz 2 hzdz b^dz\ dy = A/fds _ 2bzdz b^dz\ a'\ 2^ 2? «' / or dy = :^J^-2 hz-^dz + hH-H^ ; then integrating both members (1317, 1318, 1320), ^ a'Vloge -1 ^ -2^^ ^ ~ a^Vloge^ 2 22^/''"^' and replacing z by its value ax + b, ^ A / log (ax + b) 2b _ 6^ \ ^ a»\ log e "^ ax + 6 2 (oa; + 6)7 "^ *"• Example 5. Find the integral y = j Va^ — xMx. (a) 2 being taken as the first auxiliary variable, put X = a sin 2; (a') from (1756) dx = a cos 2 dz and x^ = a? siv? z, and therefore Va^ — a? = "^/a^ — d^ sin^ 2 = a Vl — sin^ 2 = a cos 2. (1041) Substituting these values in (a), y = j a^ cos^ zdz= a^ I cos^ 2 d2. (6) Having (1047) o 02 1 J 2 1 + cos 22 cos 2 2 = 2 cos^ 2—1, and cos^ 2 = ^ . the relation (6) may be written n+cos22 _ ,Cdz fcos 22 c?z , , Tcos 2 2 , 2/ = ^+a-J-^-d2. In order to integrate the second term of this last relation, put 2 2 = M, then 2 = ^ and dz = —, THEOREMS OF INTEGRATION 646 and then we have Since the relation (a') gives sin 2 = - and z = sin"' - , a a and from (1041, 1047) we have sin 2 z = 2 sin z cos z, and cos z = V 1 — sin^ z = \ \ 5 = . y of' a now substituting these values in the last expression for y, a' . _, X a? „x y/a? — x^ 2/=2^^^ a + T^a-^— '■ simplif jdng and adding the constant C, we have 2/ = ^ sin-i - + I V^^^^ + C "2 a 2 This formula finds application in (1328) for determining the area of the circle and the ellipse. Example 6. Find the integral y = f 'sf^'+l^dx , (a) wherein p is a constant. Putting Vp^ + x^ = z — X, (b) wherein z is an auxiliary variable, the relation (a) becomes y = j (z — x) dx = I zdx — j xdx = I zdx (a') From the relation (6) we deduce successively, f + x^= z^ -2zx + x\ p' = z^ — 2zx, (c) 2" — r)2 X = ;;— ^- 2z z =x+ •^f + a;2. (572) 2^ = 2 x' + p^ + 2 a; Vp" + x^. 646 ELEMENTS OF CALCULUS Differentiating the equation (c), we obtain (1276, 1279, 1280, 1281) = 2 zdz — 2 zdx — 2 xdz, from whicli dx = (z — x) dz = \ 2z ) _ {^ + p') dz z z 2z^ Substituting this value of dx'va. I z dx of relation (a'), we have Ldx^ fil+fUl = f^_^ + r§^ = ^ + ^ 1^^ (1320) J J 2z J 2 J 2 z 4i 2 log e ' Now substituting for z and z'^, This value of j zdx substituted in relation (a') gives the in- tegral upon adding the constant C; thus, , = /v^^.z.= ^ + |VPT^+|^l2i(^±^^+c. id) This formula will be used in (1338) for the rectification of a parabola, and in (1389) for the rectification of the spiral of Archimedes. 1322. Integration by parts. Integrating the expression dy = udv, in which u and v are functions of x, we obtain, y = I udv — uv — I vdu. In fact, differentiating the expression y = uv, we have (1281) dy = d (uv) = vdu + udv, from which, udv = d (uv) —vdu; and integrating both members, y = j udv = uv — j vdu. (A) THIiis THEOREMS OF INTEGRATION 64? as the integral of the product udv is transformed to an algebraic" difference one term of which is the product uv of the variables (functions of x), and the other / vdu, although of the same form as the given integral, may be simpler. Example 1. Find the integral y = f logxdx. Putting log X = u, we have du = i^^^ ; C1277) X and putting dx = dv, we have x = v. Then from formula (A), y = I log xdx = X log X - j X — = a; log x- I - x log e, or I logxdx = X (log a; - log e) + C = x log - + C. (396) Example 2. Find the integral y = j xsin xdx. Putting X = u, dx = du, and sin x dx = dv, v = j sin x dx = — cos x. (1317) Then from formula (A), j udv = uv — j vdu, y = I xsin xdx = - xcosx— I -cos xdx = -a;cosa; + sin a; + C. Example 3. Find the integral y = I x'a^dx. Putting x^ = u and a'" = v, we have 2xdx=du and ^^ a'dx = dv. (1285) log e 648 ELEMENTS OF CALCULUS Then from formula (A) , j udv = uv — \ vdu, y=i'x^d'dx=x'a''- \d'2xdx. (B) To calculate / a°2 x dx, put 2 X = u, then 2 dx = du and a^dx = dv, then r^^ w' = v. (1317) log a ^ ' Substituting once more in formula (A), fa^2xdx = 2x^^a^- f2^^a^dx J log a J log a log a log a log a log a \ log a/ Now substituting this integral in formula (B), 2/= fx^a^dx=xV-2^^a^(x-'^^) + C. J log o \ log a/ Example 4. Find the integral (1321) y = j Va^ — x'dx. Putting u = Va^ — x^ and x = v and differentiating, these relations give (1283) du = ' dx, and dx = dv. Va^ — x^ Therefore, from formula (A), j udv = uv — j vdu, v= I Va^ — x^dx = X Va^ — a;^ — / dx. (a) Multiplying and dividing the first member of this equation by Va^ — x^, f y/^^T^'dx = r f ~ "^^ dx = f , "' cZa; - (-1=^= dx. THEOREMS OF INTEGRATION 649 or, from (1794, Example 2), /; dx = a^sin"^ -, Va^ - x^ a Va^ - x'dx= a^ sin-> - - / , dx. (b) a J Va^-a:^ ^ ^ Adding the equations (a) and {b), we have, 2 f y/a? - x'dx = a^ sin-^- + x Va^ - x^; then the required integral is (1321) Va^ — x^dx = 7^- sin"' - + - Va^ — x' + C. 1323. Examples of integrals involving logarithmic functions. Example 1. Find the integral Replacing —^ ^ by the sum of two fractions; thus, putting ' ^ +^ (2) a^ — oc^ a + X a — X and reducing to a common denominator, 1 a:(g-^) + a(^ + g) a^ — 3^ €? — 1^ (3) The quantities A and 5 in the preceding relations are indeter- minate quantities, to which values may be assigned such that the two numerators of relation (3) be equal. Thus, putting A = B, aiA + B) = 1, A^B = ^. 2a Substituting these values of A and B in expression (2), we have a^ — x' 2 a\a + X a — xj ' and the given integral (1) becomes = C ^^ - C ^ ( ^^ \ ^^ \ ^ ~ J a^ — 3? J2a\a + x a — x/ °'" y^J 2a{a + x)^ J 2a(a-xy ^*^ 650 ELEMENTS OF-t!ALCULVS Putting a + X = u and a — x = v, (5) we have dx = du and — dx =+ dv. Relation (4) becomes, / du r - dv ^ logM _ log!) 2au J 2 av 2 a log e 2 a log e " Now replacing u and v by their values (5), and observing that the difference of two logarithms is equal to the logarithm of a quotient, 1 1 /a + x\ _ Example 2. Find the integral r dx ^ J x^-a^' Following the same method as in the first example, we obtain 2/=2^^"s(fT^)+^- Example 3. Find the integral r dz_ ~ J , loi z log e _x (1) Put a+^^ = ^, (2) z z X being an auxiliary variable. From (2) az + log e = a; (3) d^=^and.= ^-^^i^. (4) a a Relation (1) may be written dx rl^ fdx^^ rd_x/x- log e\ " J X J ax J ax\ a / z „_ f'^^ C\ogedx_x \ogx (gv Finally, by replacing x by its value (3), we obtain the required integral, y= -i [(az + log e) - log {az + log e)] + C. THEOREMS OF INTEGRATION 651 Example 4. Find the integral y r dz ' — I ' J log e z Following the same method as in the third example, we find 2/ = -2 [(az - log e) + log (az - log e)] + C. Example 5. Find the integral '^b'- r dz Referring to first example (1323), make the following substitu- tions in relation (1): a = 1 and x = — =— > (£) then, the above relation (A) may be written, Proceeding as in the first example in article (1323), we have, 1-x^ 2\l+x^ 1-xJ' and replacing x by its value (B), 1-x' 2r ^ loge^^ loge ' , , . . ^ ^ ^ ' and substitutmg m (C), / dz r dz 2(1+1^) -^2(1- 1^)' These integrals are the same as those in the third and fourth examples, considering a = 1, and we can write the result in the form y= 2^{z + log e) - log {z + log e)] + \i{z- log e) + log (z - log e)] + C. Simplifying, y = z - -^logiz + \oge) +2 (^ - log e) + C. 652 ELEMENTS OF CALCULUS 1324. Integrals of trigonometric functions obtained in the form of logarithmic functions. (1) Example 1. Find the integral r dx ^ J sinx Putting cos X = z, we have sin X = Vl — cos^ X = Vl - - z^ Taking the derivatives (1283, 3d), 2zdz cosxdx = dx = — zdz — dz cos X Vl - z^ Vl -2== Substituting in (1) the values of dx and sin x in terms of z, f -dz _ r di y~ J ll-z') J 1- Referring to the first example (1323), and considering a=l and a; = z, we obtain Changing the signs, " ^ = -/r^-2i^[iog(i-.)-iog(i + .)L ^ = 21^ ^°s (rrl) + ^- Replacing z by its value cos x, From (1048, 3d), tan 1 • /l — cos X 2^=Vl + cos X then iogtanix = llog([-j-^), therefore, (2) may be written, y = , log tan^a; + C. ^ loge ^ 2 THEOREMS OF INTEGRATION 653 Example 2. Find the integral /dx : cos X Putting sin x = z, and following the same course as in the preceding example, /dx 1 , /I + sin x\ , „ ^^ = 2krei°Hr^^i^J + ^' Remark. Generalization of the two preceding examples. The two following general integrals may be solved with the aid of the two preceding examples. /dx _ — cos X m — 2 r dx ,., sin"" a; (m — 1) sin™"' a; m—lj sin^^^a;' /dx _ sin a; m — 2 T dx _ .„. cos^x (m — l^cos^^'a; m—lj cos'"~*a; (m — 1) cos' For m = 2, the latter gives dx sin X I cos, X cos X = tanx, dx > cos a; which conforms with the result given in the table (1317). For m = 3, formula {B) gives /dx _ sin X 1 r dx „ cos^ X 2 cos^ x 2 J cos x Substituting the value foimd in the second example for / ■ / dx _ sinx 1 , /I + sin x \ „ cos^x ~ 2 cos^x 4 log e U — sin x/ Example 3. Find the integral /dx cos X fr,\ Putting sin x = z, we have, cos x = Vl - sin^ x or cos x = Vl - Taking the differentials, d sin X = dz or cos xdx = dz , dz , dz and ax = = , ^ • cosx Vl — 2^ z'. 654 ELEMENTS OF CALCULUS Substituting in relation (2), . Jg VI - a' frf2_log2. ^ r dz VI - a' _ r^_: loge therefore relation (1) gives / dx _ log sin X _ tan X log e Example 4. Find the integral r dx J cot a; COS iC Writing cot x = -. and putting cos x = z, and following a SlU X course analogous to that in the third example, we obtain /dx _ log cos X ^ cot X log e Example 5. Find the integral " J sin a; cos a; ^ ' This may be written (1069) r 2dx ^ r^dx_^ J 2 sm a; cos a; J sm 2 a; Putting 2x = z, X = -^j and 2dx = dz. Substituting in (2) the values of 2 a; and 2 dx in terms of z, we obtain (1324, Example 1) V = I -. — = log tan -^ = log tan x, '^ J smz ^ 2 ^ ' therefore 2/ = / -■ = log tan x + C. J smx cos X ° INTEGRATION BY SERIES Example 1. Find the integral dx -/r INTEGRATION BY SERIES 655 Referring to the table (1317, 13), we should write y = tan~'a;. Expanding (1 + x')~'^ according to the binomial theorem, j-p^ =dx(l + ar=)-i ^dxil-sf+x^-xn ■■■■). Integrating these different terms, y = tan-^x = a;— -a;^ + -a^ — -x'+ •••• o 5 7 Example 2. In the same way for dx /dx we should write, Expanding, y = sin ^x. U x-) ^+2+2.4+2.4-6 Multiplying by dx and integrating, sin-a; = :. + 273 + 27475 + 2T4T6T7 + --" Example 3. Given y — dx Veos^ x + 1. Expanding by the binomial theorem, Vcos'' a; + 1 = (cos^ x + 1)^ _^^g ^ ^^ 1 J_ ^ l_ _J- 5_ _J_ ^ 2 cos a; 8 cos'a; 16 cos^ x 128 cos' x Multiplying all the terms of the second member by dx, and inte- grating each term, we obtain, C , r dx ri dx , , „ y = I cos xdx + / 7^ / H — 5 — I- • • • • + 0. ^ J J 2 cos a; J 8 cos^a; Referring to the examples of number (1324), each term of this series is easily integrated. APPLICATIONS OF INTEGRAL CALCULUS QUADRATURE OF CURVES 1325. General solution of the quadrature of curves. Given the equation y = f{x) of a curve C, to find the area included between the ordinates AA' and BB', Y and X being the coordinates of '^ the point A, and Y' and X' those of the point B. Considering an element MPP'M' of this area included between the ordinates MP and M'P', y and x being the coordinates of the point M, at the limit those of the point M' will he y + dy and x + dx, and the element MPP'M' will be a trapezoid whose area we will designate by dS; then (723) Fig. 387 dS = y + (y + dy) dx. (1) This being established, we can easily conceive the entire sur- face AA'B'B as being divided into infinitely small trapezoids; then the total area S will be equal to the sum Sd/S or I dS of the areas of all the elementary trapezoids, and we have ■=Jds=Jy±^'y'^ S- ■dx. (2) Since dy in expressions (1) and (2) is negligible at the limit, the first one becomes, dS = ydx, and the second, S =fdS =Jydx. Calculating this integral in terms of x, and integrating between the limits x = X and x = X', we have (1314, 1315), nx' nx' 656 QUADRATURE OF CURVES 657 The same integral calculated in terms of y between the limits Y and Y', is S = r ydx. (3) From the equation of the curve y = fix), we deduce, dy in terms of x or dx in terms of y; which permits us to calculate the integral (3) in terms of one of the variables X or y. 1326. Example 1. The area of a right triangle. Given a straight line OB whose equation y is (1117) y = ax, (1) to calculate the area COC included between ol^x A' c' B' ^ the origin and the ordinate CC. Fig. 388 Let OC = b, and CC = h. The general formula (1325) is S = I ydx. Replacing y by its value in (1), and integrating (1317, 1319), S = faxdx=^+C. (2) To obtain the required area COC', take this integral between the hmits x = and x = b. Since for a; = and x = b we have respectively, and therefore, S= I ''^dy = -^ + C. Since for j/ = 0, »S = 0, = + C or = 0, therefore S>= f-dy = ^, and the required area is phi, h" '-11"' = ^ '^' QUADRATURE OF CURVES 659 Substituting the coordinates of the point C in relation (1), h = ab and a = v ; now substituting this value in (2), the required area is 2h~ 2' 1327. Example 2. The area of a trapezoid. To obtain the area of the trapezoid AA'B'B (Fig. 378), it suffices to calculate the integral S=Jydx (1325) between the Umits a; = X and x = X', X and X' being the ab- scissas at the extreme points A and B. The area of the trape- zoid is also equal to the difference between the areas of the triangles BOB' and AOA', that is (1326), Jrx' rx' nx or ,S = ^-^' = |(X'==-Z=) = |(X'-^Z)(Z'-Z). Since the equation of the line OB, y = ax, gives respectively for the points A and B, Y = aX and Y' = aX', by addition we have, Y + Y' = a(X + X') and (X + X') = L±J!. . Substituting this value of X -I- X' in the above formula for S, Y + Y' S^^^^(X'-X), which is the same expression given in (723) for the area of a trapezoid having Y and Y' for bases and X' — X for altitude. 1328. Example 3. Area of an ellipse and of a circle. The equation of an ellipse referred to its principal axes is (1131) = ^ V^ 660 ELEMENTS OP CALCULUS The general formula for areas (1325), S =j ydx, applied to the ellipse gives (1321, Example 5), S= f-^'^f^^dx J a Fig. 389 h ^ a 2 sin' -»- + - |V^^^ + C. a a 2 Taking this integral for a quarter of an ellipse, that is, between the hmits x = and a; = a, for a; = we have S = 0, there- fore C = 0, and for a; = a we have ~ ah . , , irab ,S = -2-sin'l=— , therefore for a quarter of an ellipse, Jo a S-- 2 ZS J '"'^ r — ar da; = — r- > 4 and for the total surface (1162), S = wab. When a = b = r, the ellipse becomes a circle of radius r, and we have (753, 1162) S ^■xt'. 1329. Example 4. The area of a segment of a parabola. The equation of a parabola referred to its vertex being y' = 2 px, ■ (1) the general formula for areas (1325), S=J ydx, gives S=f^/2px^dx= V2p^+C = |('\/2px)a;+C=|a;2/ + C. 2 Designating the coordinates of a point M by Y and X, the area of the segment MOP is obtained by taking the preceding inte- QUADRATURE OF CURVES 661 gral between the limits a; = and x = X. For x = 0, S = 0, and we have C = 0; therefore, the required area is (1221) ^ = fJ^xUx = lxY. We can integrate S = j ydx with respect to the variable y. Thus from relation (1) 2ydy = 2pdx and dx = -dy- This value of dx substituted in the general formula, gives S= p^dy=f^C. J p 3p Taking this integral between the limits y = and y = Y; since for ?/ = 0, S = and C = 0, the required area is Jo V 3p or, since ]P = 2 pX, 2pXY 2 1330. Example 5. The area of a sine wave. The equation of this curve being y = sin X, the general formula for areas (1325), S= j ydx, gives (1317) S = j sin x dx = — cos x + C. y /A\ E To obtain the area ^f of a segment OAP, take this integral be- 3 a; == and x = OP = ^, which give S =-\ + C and 1 the value of L is positive, and for a; < 1 the value of L is negative. Remark. From formula (3), ior x = ao, L = ao, which corre- sponds to the graph of the curve, since from the point A the curve extends to infinity in the direction of the positive y-axis. For a; = 0, L = - 00 , since the curve extends to infinity in the direction of the negative y-ends. Thus ior x = the formula (3) gives L = loge--log(loge+loge)+-logGoge - log e) + C. The last term gives - log (0) = — oo ; id therefore, L — — cc. Example 4. Rectification of a cycloid. "With the aid of the formula (1338), and the derivative of the equation of the cycloid (1297), R~y ' (2) dx V dx y y the problem is solved as shown below. To simplify the calculations the origin is changed to the ver- tex B (Fig. 349, 1243) of the cycloid (as was done in 1336). Then the ordinate y becomes {2 R — y), which, substituted in the derivative (2), gives dx \ 2 y dx y 2R-y' thus ^=V^IZl and (^Y=^^:^^. dy y y \dyf y Substituting this value in (1), L= '^J2R^= \l2R-2'^, L = 2 V2% + C or L = 2 V2%. RECTIFICATION OF CURVES 673 The constant C = 0, since the value y = corresponds to the vertex B of the curve, the origin of the axes. For y = 2 R, we have, L = 2 V^R^ = 4 ii. The total length of the curve, 2L = SR = iD, that is, the length of the cycloid is equal to four times the di- ameter of the generating circle. The base of the curve is equal to 2 iri2 = 3.1416 D. RECTIFICATION OF CURVES EXPRESSED IN POLAR COORDINATES 1339. General formula for rectification. Referring to the form- ula (1338) for the length of the differential arc, and substituting polar coordinates, we have, dL = y/idpy+Qximf = do,\/Wj+p\ wherein p and <« are the coordinates of the point, and L the length of the arc. ■ L-fdps/l+pitJ+C. (B) Example 1. Rectification of the logarithmic spiral. We have, log/»=4«>) . For o> = wehavep=l, p = 6-*" I ^ ■' For (0 = — 00 we have p = 0, and , d + C, C =- H; therefore relation (3) becomes L = Hb^" - F = H(b^- -1). From equation (1), p = b^"; therefore L in terms of the radius vector is L = i?(p-1). (4) For (0 = — 00 , we have p = and L = — H. Therefore, starting from the polar axis which corresponds to m = 0, the spiral makes an infinite number of turns before arriving at the RECTIFICATION OF CURVES 676 pole. The length of this portion of the spiral included between the pole and the origin (for which p = 1) is negative and has the value — H. Remark. From relation (4), L + H = Hp, that is, the length of the logarithmic spiral, measured from the pole to any point on the curve, is proportional to the radius vector which ends at that point. This property, which has long been known, may be used in graphically representing a system of logarithms.* Example 2. The rectification of the spiral of Archimedes (1230). Taking the equation of the curve in the form P = K<^, (1) the formula for rectification is From(l), %=K, therefore relation (2) may be written, L = fd, H we put — ^^ = 10 ir= 10 units, and if the base of the logarithmic system is 10 and the angular measure of the logarithm of this base is 2 ir, we have, 1 = log 10 =: ^ 2 TT, and -=k Substituting in equation (2), l°g'> = 2^ which gives for 0. = 0, P = t, U =Z 277, p = 10, L + IT= R, The spiral will have 1 at the origin and 10 at the end, and the points 2, 3 ... 9, 10, wiU divide it into equal arcs. 676 ELEMENTS OF CALCULUS Putting Wo>='+ 1 = 3-0), (4) we have ia^ + 1 = z^ — 2 z^, -.9 i (5) 2z d. Now >-X^ + ^) = ^^'"°'^ and I /t z^dz=7r'\/p — +C = -^7r'\Jp(2x + py + C. (a) 2 Since for x = 0, 5 = 0, = ^^p' + C and C = - ^^p\ o o To obtain the surface of a paraboloid included between tlie vertex and a section whose abscissa is X (Fig. 389), take the pre- ceding integral between the limits a; = and x = X, which is done simply by replacing a; by X and C by its value, in expres- sion (a); thus, /S = 27r^^/ dx V2 a; + p = Itt Vp (2 Z + pY -%vf = 2^[Vp(2Z + p)='-p^]. CUBATURE OF SOLIDS OP REVOLUTION 679 CUBATURE OF SOLIDS OF REVOLUTION 1341. General formula for the volume of a solid of revolution. Let y = f {x) be the equation of a meridian curve (Fig. 395) of a solid of revolution about the axis Ox. Consider this solid V as being made up of infinitely thin slices included ' between planes per- pendicular to the axis Ox. Since any one of these slices, that gen- erated by MPP'M' for example, at thp limit may be considered as the frustum of a cone, the radii of whose bases are MP = y and M'P' = y + dy, and whose altitude is PP' = dx, the volume dV of this sUce is (913), dV=^7rlf + iy + dyf + 2/ (2/ + dyy\ dx; or, neglecting dy in comparison with y, dV = n'^ iy^ + y^ + y^) dx = iry'^dx. Therefore, the volume V corresponding to the meridian AB is expressed by the indefinite integral y = ^Jfdx. (1) 1342. Example 1. The volume of a cone, generated by a right triangle OBP turning about the axis Ox which coincides with the side OP- The equation of the meridian being (1117) y = ax, substituting this value of y in the general equation (1) of the preceding article this equation becomes. =^.fa HHx^ ^ + ^ = § '^'^'^'^ + ^ = ^ '"^''^ + ^• Since for a; = 0, we have 7 = 0, = + C and C = 0. Taking the integral between the limits x = 0, which corresponds to y = 0, and x = h, which corresponds io y = r, and since C = 0, the re- quired volume is = ^ TaH'dx^^^r'h. (909) 680 ELEMENTS OP CALCULUS Example 2. The volume of an ellipsoid of revolution. The equation of the meridian is ^ (1131) and y^ = -{a^ - r"). Kg. 397 - ^i Substituting this value of y^ in equation (1) of the preceding article, V = ir I -^(a^— af)dx = IT f —^dx — tt / -^xHx b" a? a" 3 Since for a; = 0, V = 0, and substituting these values in the above integral C = 0, taking the integral between the limits a; = and a; = a, we obtain for half the volume of the ellipsoid, V = vVa — T -2 -^ = o '^"^» and for the whole volume, O (a) If the generating ellipse turned about its minor axis, we would have, (1166) which result is obtained by substituting b for a and a for 6 in formula (a), or by taking from the equation of the eUipse a'x^ + b^y^ — aW the following value of y'^, f=^,{b-^-o?), and substituting in the general formula. CENTER OF GRAVITY 1343. The moment and center of gravity of a figure. In order to calculate the center of gravity of a body from its geometrical form, we must assume that the body is composed of strictly homogeneous material. CENTER OF GRAVITY 681 A figure (line, surface or volume) may be considered as being composed of infinitesimal elements. The product of one of these elements and its distance from a plane is called the moment of this element with respect to this plane. The moments of two elements on opposite sides of the plane have opposite signs. The moment of a figure or a system of ele- ments is the algebraic sum of the moments of the different ele- ments which compose the figure or system. The center of gravity of a system of elements (lines, surfaces, or volumes) is a point, such that, if all the elements were concen- trated in it, the product of the sum of all the elements and the distance of the point from a certain plane, would be equal to the algebraic sum of the moments of the different " elements with respect to the same plane. 1344. The center of gravity of a straight line. First, the center of gravity is on the line, because, if we suppose it to be outside the line and pass a plane through it leaving the line entirely on one side of the plane, the product of the sum of all the elements and the distance of the center of gravity from the plane will be zero, while the moment of the line with respect to the same plane will evidently not be zero. The center of gravity is at the middle of the line, because, with respect to any plane passing through the middle, the product of the sum of all the elements and the distance from the point to the plane will be zero, and since the middle point divides the line into two symmetrical parts opposite in sign, the moment of the total line is also zero. Remark. By an analogous course of reasoning, we have in general : 1st. That all systems of geometrical lines, surfaces or volumes possessing a geometrical center have their center of gravity at the geometrical center. 2d. That any system composed of elements symmetrical in pairs with respect to a line or a plane (836, 839) has its center of gravity on this line or plane. 1345. Center of gravity of any plane curve AB. Drawing the coordinate axes Ox and Oy in the plane of the curve, the ' Fig. sss 682 ELEMENTS OF CALCULUS required center of gravity G 'will be determined when its co- ordinates X and Y are known, y being the ordinate of a point M, the moment of the element MM' = dL with respect to Ox is dL hf)' or, since -^ may be neglected in comparison with y, we have ydL. The algebraic sum of all the elementary moments, that is, the moment of the curve, is therefore, ^ydL =j ydL, and since this moment is equal to LY, L being the length of the curve, we have, n f ydL LY= jydL and Y = r ' (1) With respect to Oy, we have, // xdL XdL and X=^ (2) Remark. When the curve is given by its equation, y = f (x). From (1338) we have dL = -^JdyfTWf = dxsjl + (^Y . and ^ = /^^\/^^- and these values are substituted in equations (1) and (2). When the integrals resulting from these substitutions are too complicated, or the functions (1) and (2) are unknown, an approx- imate result may be obtained by using Thomas Simpson's form- ula (1333) for the calculation of the integrals \ ydL and I x dL. To do this, divide the curve into an even number n. of equal parts; from the points of division drop perpendiculars upon Ox\ CENTER OF GRAVITY 683 measure these perpendiculars i/„, 2/1, 2/27 •• • Vn, and making — = S, we have / y ^^ = 3 [2/o+2/«+4(2/iH-i/8+ +2/^1)+ 2(1/2+2/4+ 1- 2/^2)]. 1346. Center of gravity of an arc of a circle. The moment of the element MM' with respect to the axis OX (Fig. 399), which in this case is taken as the 2/-axis, is MM' X. ID OT X dL, and the moment of the arc is ^xdL = I xdL; but since MM' xID = PP' x r, or X dL = r dy, the moment of the arc is also, 'Srdy = r%dy = re, wherein c is the chord AB which is equal to S dy. The distance X from the center of gravity G to the center 0, designating the length of the arc L by a, is /xdL re L a X = (1) The arc being of n degrees, we have (758), 2'jrrn and 360 . n c „ . n These values of a and c substituted in relation (1) give n 360 rsm- Z = TTO For n = 180°, for example, we have sin ^ = sin 90° = 1, and therefore, n 2 _ 360 r _ 2^^ ^^2, ~ ISOtt" tt 22 11^' 684 ELEMENTS OF CALCULUS Thus the center of gravity of a semicircle is very approximately 7 — of a radius from the center. 1347. Center of gravity of plane surfaces, and in general of any surfaces or solids. General solution. Let m be an element dS of the surface bounded by any plane curve ABC, and y the distance of this element from the axis Ox, drawn in the plane of this surface. The product y dS is the moment of this element m, and the moment of the en- tire surface is (1343) fydS y /^ X. ^B I G ( X c i X Fig. 400 SY=%ydS= fydS and Y = S and with respect to the axis Oy we have, / SX = I xdS and X = / xdS ■S (1) (2) If the surface was not plane, instead of using two axes Ox and Oy in one plane, we would use three planes perpendicular to each other, and for each of these planes we would have a formula an- alogous to formula (1); which would make it possible to deter- mine the coordinate X, Y, and Z of the center of gravity with respect to the three planes. For solids we operate in the same manner, using the same formula (1), replacing the elements of surface dS by elements of volume dV. Whenever integrals (1) and (2) are obtained which are too com- pHcated, the formula of Thomas Simpson may be used (1333). Thus, choosing the axes Ox and Oy tangent to the surface, divide the projection I of the surface on the axis Ox into an even number n of equal parts - = 8; through these points of division draw perpendicu- lars to Ox; measure the portions y„, y^, y^, ■ . . yn, of these perpendiculars intercepted by the curve, and then from (1333), Fig. 401 . CENTER OF GRAVITY 686 8 S = 3 [2/0 + J/„ + 4 (j/i + 2/3 + . . . + y„_,)+2 (y.+y, + . . +y„_,)l Considering an infinitesimal element mm', limited by two paral- lels to Oy, the surface of this element is dS = ydx, taking y as the length mm' intercepted by the curve. Therefore, the moment of this element with respect to Oy is xdS = xydx, and that of the total surface / xydx iiuui wincu A. = To calculate / xy dx, put SX = I xy dx, from which X = S xy = 2/' and then we have approximately, /S ■ ' y'dx=-:g[y'o + y'„ + 4:(y'i + y',+... + y'^,) + 2(y', + y',+...+y',^,)l in which, y^' = yoXo = Po X = 0, Vi = Vt^i = yA 2// = 2/33:2 = 2 2/28, Vs = ViXi = 3 2/8 8, Vn = Vn^n = ny„^. Substituting these values, we have, Jx2/(ia; = -g|n2/» + 4[2/i + 32/3+---+ {n - 1) 2/„-i] + 2[2 2/2 + 42/4 +•••+ (n - 2)2/„_J|, and .^_ 8{w2/„+4[2/i+32/8+--- + (w-l)2/,^i] + 2[22/2+42/4+--+(n-2)2/„^]| ^o+2/»+4 (2/1+2/8+ • • •+J/«-i) + 2 (2/2+2/4+ ■ • •+2/»-2) Operating in the same manner for the axis Oy, the distance Y of the center of gravity from the axis Ox is obtained; but when the elements have been determined as in the above operation, it is simpler to operate as follows, z being the distance from the middle, that is, the center of gravity of the element mm' = dS ,686 ELEMENTS OF CALCULUS = ydx, to the axis Ox, the moment of this element with respect to the axis Ox is zdS = zy dx, and the moment of the total surface ■with respect to the same axis is r J I zydx SY = I zy dx, from which Y ■■ S Putting zy = y' , we have SY =Jy'dx=^[yo' + 2// + 4 (yi' + y,' + ■■■ + y'^^) + 2{y\ + y\+:- + y',^)]; in which 2/o' = y^o, Z|j, Zj, 22, . . , z„ being the distances from the middle points of the heights 2/o, Vx, 2/2; •• • or y^ to the axis Ox. Substituting these values, we have, 2/0 + 2/„ + 4 (2/1 + 2/s +• • •) + 2 (1/2 + J/4 + • • •) 1348. Center of gravity of the surface of a triangle. Through the vertex A draw an axis Ox parallel to the base BC. Then the surface of an infinitesimal element mm', parallel to the base, is dS = mm' X dy, and its moment is yds = yX mm' X dy. The two similar triangles Amm' and ABC give mm' V , , iy -1— = f and mm'= -f • b h h The elementary moment is ydS = ^dy, and the total moment -Jly'dy SY= /^2/^% = ^ + C. CENTER OF GRAVITY 687 Taking this integral between the limits y = and y = h, and making bh ^ we have for the total moment of the triangle ABC, Thus the center of gravity G lies on a line parallel to the base BC at a distance equal to one-third the altitude from it. In the same manner all three sides can be taken as bases, and the three parallels to the three sides intersect in a point G which is meeting- point of the three medians and the center of gravity. 1349. Center of gravity of a segment of a parabola, limited by a straight line AB perpendicular to the principal axis Ox, the equation of the parabola being (1197) 2/^=2 -px. The center of gravity being on the axis Ox, it is only necessary to determine its abscissa OG = X'. The surface of an element mm' included between two parallels infinitely near each other and parallel to the axis Oy, is dS = mm' dx = 2ydx, and its moment is xdS = 2 xydx, and therefore the moment of a parabolic segment is SX' = / 2 xy dx =J 2 x ■\/2px dx =J^ '^2px^dx = ^ Vlpa;^ C. Designating the coordinates of a point 4 by X and Y, and taking this integral between the limits x = and x = X, the constant C = 0, and we have, SX' = I V2^Xt = I V2^X^ = i YX\ 5 5 o Since in (1329) S = ^YX, t^^' 3 we have X' = j = r ^' Iyx 688 ELEMENTS OF CALCULUS 1350. Center of gravity of a zone AA'B'B. Since the figure is symmetrical, the center of gravity G hes upon the radius OC perpendicular to the planes AA' and BB' of the bases; and its distance OG = X from the center is all that remains to be determined. Take OC as the a;-axis, and let Oy be the trace of a plane perpendicular to Ox. Reasoning as in the two preceding articles, the surface of an element mm' of a zone in- cluded between two planes infinitely near each other and parallel to the plane Oy, is (915) dS = 2-n-R dx, and its moment with respect to Oy is xdS = 2 irRx dx, therefore the moment of the zone is SX = C2 -kRx dx = wRx' + C Taking this integral between the limits x = x' and a; = a;", we have SX = ,rR(x"' - x"); and since S = 2 irRH = 2irR (x" - x'). we have .R(x'--x-) ^l_ ^ 2wR{x"-x') 2^^ ^^^' which shows that the center of gravity G is at the middle of the height H of the zone. 1351. The center of gravity of the lateral surface of right cone. This center of gravity is situated upon the axis OP of the cone, and we have only to determine the value of OG = X' Taking OP as the axis of x, and the moments with respect to a plane Oy passing through the vertex perpendicular to Ox, desig- nating the slant height OA of the cone by I, for the expression of the surface of the ele- ment mm' included between two parallel planes perpendicular to the axis Ox, we have (912) dy\ dS .4 + 1) dl. CENTER OF GRAVITY 689 Neglecting -^ , dS -=2Trydl; then the moment of this element with respect to the plane Oy is xdS = 2 Tryx dl. Since we have ~ = cos a, dl = ^^ dl cos a ■ and since t = tan a, y=x tan a. Substituting these values, we have, ,„ 2 IT tana , , xdS = x' dx. cos a The moment of the lateral surface of the cone is, therefore, cos u. J COS a 3 Designating the coordinates of the point A by Z and Y, and taking the preceding integral between the limits x = and X = X; since the constant C = 0, we have for the moment of the lateral surface of the given cone, „, „ 2 IT tan a X' cos o 3 From (908), S = wYl, or, since Y = X tan a and I cos a ^^^rtena COS a and we have, 2 w tan a X^ ™ cos a 3 2-^ A = 7 = ^A. IT tan a ,,„ 6 x^ COS a Thus the center of gravity of the lateral surface of a cone is at 2 ^ the altitude as measured from the vertex. This is analogous to the position of the center of gravity of the surface of a tri- angle (1348). 1352. The center of gravity of any solid. Using three refer- 690 ELEMENTS OF CALCULUS ence planes perpendicular to each other, and operating with each as indicated in (1347), we obtain the three equations, VX and X ■■ / xdV fxdV, r fydv VY = JydV, and. Y ^ VZ = JzdV, and Z = V X A y X — ' G iV" ^ \ A, Z J' D I '-¥j Fig. 406 y In practice, when the integrals cannot be solved or are very comphcated, the formula of Thomas Simpson is used (1333). Thus, three planes perpendicular to each other and tangent to the solid are chosen. Let Ox and Oz be the intersections of two of these planes with that of the paper to determine the distance X of the center of gravity of the solid from the plane Oz. Draw a plane A„ tangent to the solid and parallel to the plane Oz; divide the portion I intercepted on Ox by the two planes Oz and An into an even number n of equal parts — = 8; through these points of division draw planes perpendicular to Ox; meas- ure the areas A^, A^, A^, ... ^„ of the sections determined by these planes and by 0^ and A,,; the areas A^j and A^ may be zero. Then from (1337) we have, F = |[4o + ^„ + 4 (4i + ^3 + • • • ) + 2 (^2 + A, + • . • )]• The volume of an element mm' of the solid, determined by two planes infinitely near each other and parallel to the axis Oz, is dY = A dx, A being the area of the section mm', and dx the thickness. The moment of the element mm' with respect to Oz is therefore, xdV = Axdx, CENTER OF GEAVIT'i and that of the total volume, VX= I Ax dx and X To calculate / Ax dx, put / Axdx 691 (1) Ax = y, and we have approximately, / Axdx = 3 [2/0 + 2/« + 4 (j/i + j/8 + . . . ) + 2 (j/2 + 1/4 + in which formula y^ = AaXa = A„xO = 0, Vn = A„y„ = ^„n8. Substituting these values, we obtain. )L /g3 Axdx = 3 [n.^„ + 4 (Ai + 3 yls + • • • ) + 2 (24, + 4 ^4 + ■ • . )]• then substituting the value of V in (1), we have 8M„+ 4(^ + 3^8 + •■•) +2(2^ + 4^+ ■•■)] X = ^0 + 4« + 4 (Ai + ^3 + • • • ) + 2 (^2 + ^4 + • • • ) In the same way we can find Z and Y, but if the centers of gravity of the sections A^, A^, A^ . . . are easily determined it is convenient to have recourse to the method in (1347) for obtain- ing Y. 1353. Center of gravity of any pyramid SABC. Any section of the pyramid made by a plane parallel to the base, has its center of gravity on a straight line Sg which joins the vertex and the center of gravity of the base. From this it follows that any element mm' included between two planes infinitely hear each other and parallel to the base has its center of gravity on the line Sb and therefore the cen- ter of gravity of the pyramid is also on this hne. This established, it remains to find the distance SG. Through the vertex S draw a plane parallel to the base ABC. Let Ox be the intersection of this plane with that of the paper. 692 ELEMENTS OF CALCULUS b being the base of the element mm', which at the limit may be supposed to be a prism, its volume is dV = bdy, and its moment with respect to the plane Ox is ydV = ybdy, and therefore the moment of the pyramid is YV = j ybdy. (1) V = 5^«- b B y' JJ2> » = !»■■ B and H being the base and the altitude of the pyramid, we have (891) ^ Furthermore, since Substituting these values of V and b in (1), we obtain, \BHY = ^Jfdy=^,t^C. Taking this integral between the limits 2/ = and y = H, we obtain the moment of the given pyramid, 1 B H* BH' and SBIP^B 4:BH 4 Therefore, the center of gravity lies upon the line Sg at a 3 distance Y = -H from the plane Ox, and we have SG = \sg. 1354. Center of gravity of solids of revolution. The general formulas of (1352) apply also to solids of revolution. But since solids of revolution are symmetrical with respect to the axis of revolution Ox, the center of gravity always lies upon this axis, and we have simply to determine its distance OG = X' from a certain plane perpendicular to Ox, which is expressed by a single equation. Fig. 408 CENTER OF GRAVITY 693 Thus, r f. IxdV FZ' = J xdV and X' = '^ ■ The volume of an element mm' included between two planes infinitely near each other and perpendicular to the plane Ox, being (1341) dV = Try' dx, the volume V = tt I y' dx. Furthermore, the moment of the element dV being xdV = Try'xdx, the total moment of the solid is "^ I u oc dx VX' = Tr fy^xdx and X' = -^ (1) n j y'dx When the value of V is known, it may be substituted in the denominator of (1), leaving the integral in the numerator to be calculated. However, the two integrals are so analogous that the value of one is easily deduced from the value of the other, and it is scarcely worth while to substitute the value V in the denominator. Example 1. Center of gravity of a paraboloid of revolution. The equation of the meridian curve- or generatrix OA being (1197) f = 2 px, substituting this value of y"^ in equation (1), and taking the inte- grals between the limits x = and a; = Z, we have, 2p r x'dx |Z3 Y' - '^" = - = -X xax ^ A" Example 2. Center of gravity of a right cone. The equation of the generatrix OA being that of a straight hne (1117) y = ax, 694 ELEMENTS OF CALCULUS ■ substituting this value of y in equation (1), and taking the inte- grals between the limits x = 0, and x = X, r^ 1 ■yf ^0 '± " XT- ^ - — Tvf -J ~4^' a? I 3?dx =;X^ Jo o which is the same as obtained in (1353) for the pyramid, and should be compared with that given for the lateral surface of the cone (1351). Example 3. Center of gravity of a spherical segment AA'BB' (Fig. 404). The equation of the generatrix AB being (1123) y^ = r^ — x^, substituting in the general equation (1) and taking ,the integrals between the limits x = x' and x = x", X' = r'dx- j x'dx r'ix" - x') - ~ + ^ '^{x"^-x") -~(x"*-x") r\x" -x') -I {x"^ - a;'^) For the hemi-sphere the limits are x = Q and a; = r, and we have 1 4 1 4 1 4 y, 2_ 4^ 4^^ 3 Thus the center of gravity of a hemi-sphere is at a distance from the center equal to f of the radius. RADIUS OF GYRATION AND MOMENT OF INERTU. j.;13:5B;' Thecproduct mr^ of a material element and the square of its distance from the axis of rotation is called the moment of inertia of the element with respect to that axis, and the sum "Smr^ MOMENT OF INERTIA 695 of the moments of inertia of all the material elements of a body with respect to an axis is the moment of inertia of the body with respect to that axis. The radius of gyration is a value E of r such that if the whole mass of the body was concentrated at that distance from the axis of rotation, the moment of inertia and consequently the kinetic energy of the body would remain unchanged for any given angular velocity. Since the bodies are supposed to be homogeneous, we may substitute the volume u of the elements for the mass m, and we have for the moment of inertia, Swr" = ig^Sw = UR' and ii^ = ^ , _/« or E' U ' wherein u is the volume of an element, U the total volume of the body, r the distance of an element from the axis of revolution, and R the radius of gyration. Example 1. Find the radius of gyration of a very small rod, which rotates about an axis Oy, one end of the rod being upon the axis. Let AB = 1 be the length of the rod, and s the area of its cross-section; then m being an element of the rod, whose „ length is dl, the volume of this element is u = sdl, and its moment of inertia, ux' — S3?dl. Since dl = —. — > A Sm a I>lg_ 4()g the moment of inertia of the element may be written U3? = -. — x'dx. sin a Therefore the general expression for the moment of inertia of the rod is %U7? =UE!' = ^^ f^dx =-^ ^+ C. smoj sma 6 696 ELEMENTS OF CALCULUS Taking this integral between the Umits a; = and x = BC the constant C = 0, and we obtain for the given rod AB, sin a 3 and noting that U = Is = s —. — , sin a s B^ ^.^sino^^l^ -^BC ^ sina Example 2. Find the radius of gyration of right circular cyl- inder turning about its axis. Let p be the radius of the cyhnder and I its length. The volume of an element included between two cylindrical surfaces having the same axis as the cylinder is u= [■ir(x + dxy — irx^ I, wherein u is the volume, x the radius of the inner cylinder, and X + dx that of the outer one. Simplifying and neglecting the infinitesimal of the second order tt {dxY I, we have, u = 2-irlxdx. The moment of inertia of this element is ux' = 2 trlx^ dx, and therefore the moment of inertia of the cylinder is UR:' = 2 ttZ fx^dx =27rlj+C. (1) Taking this integral between the limits x = and x = p, we have for the given cylinder, Substituting npH for U, we obtain. Trip* 1 ^ 2TrpH 2f- Example 3. Find the radius of gyration of a hollow cylinder, the exterior radius being p and the interior p' . MOMENT OF INERTIA 697 Take the integral (1) of Example 2, between the limits x = p' and X = p, which gives from which U = {irp' — Trp'^) Z, ^ 2^{p'-p")l 2^ ^'^ ^• Example 4. Find the radius of gyration of right circular cone turning about its axis. Let h be the altitude of the cone, and p the radius of its base. Taking the axis of the cone as the x-axis, the volume of an element included between two planes perpendicular to this axis is u = iry^dx, and its moment of inertia 2^2/ =-^'^y'dx. „. dx h , h , Since -r- = - , dx = -dy, dy p p "' and we may write, ^ ^2/^ = 7r~ V^^V- Therefore the general expression for the moment of inertia of a right circular cone is Taking this integral between the limits y = and y = p, we obtain for the cone in question, UR^^'^P*; and since '^ ~ 3 '^P'^' we have R - j^^ - JqP ■ 1357. Radius of gyration of any geometrical body. Referring to a system of three coordinate axes; let one of the axes be the axis of rotation 0, perpendicular to the plane of the paper; then u being the volume of an element situated at a distance r = Va;^ + y^ 698 ELEMENTS OF CALCULUS from the axis, its moment of inertia is wr* = U3? + uy^, and therefore the moment of inertia of the body is Pig. 410 Each of the two sums Swx^ and Swj/^ which make up the value of UR^ are calculated sepa- rately. Considering an infinitely thin slice of the body included between two planes perpendicular to the a;-axis, A being the area of the section, the volume of the slice is Adx, and since each element of the slice gives the same value for ux' we have for the whole slice %ux^ = AxHx, and consequently for the whole body S' "Skux^ — I AxHx. The degree of accuracy of this calculation depends evidently upon the section A, which may be constant or a variable follow- ing a certain law with respect to x, or vary in any manner. Considering the body as composed of infinitely thin slices per- pendicular to the 2/-axis, B being the area of the variable section, we have %mf = jByHy. Substituting these values in relation (1), we obtain ^ P \AxHx+ \ByHy UR^ = / Ax'dx + J ByHy, whence E" = ~ . Example 1. Find the radius of gyration of a rectangular paral- lelopiped turning about one of its edges. Let the edge c be the axis of rotation, and a . and b coincide with the axes x and y. First the /-\—/-\- sections A and B are constant, since A — be and B = ac, and we have. x^dx + ac j y^dy = bc-^+ ac-^- ng, a 411 V Y b T!l V "■ A^ V — C MOMENT OF INERTIA. 699 Since C/ = Sit = a6c, we have i abc (a" + 6^) i2'= r = ^ (a' + &")• Example 2. Find the radius of gyration of a right cylinder, whose base ABC is semi-parabolic, revolving about an axis A parallel to the axis of '^' the cylinder. Using the axes of the parabola Bx and By as co- ordinate axes, designating AB by a, AC by b, and the distance of an element from the axis of rotation by r, we have the relation r^ = {a — xY + y^- From this it follows that Sitr* = Sm (a - xY + %x\f, or VW = {^A {a - xfdx + r ByHy. «/o «/o The radius of gyration being independent of the length of the cylinder, we may assume the length to be 1. Therefore, for any section A or B, the equation of BC being 2/^ = 2 px, we have ^ = 2/ = "^2 px and B = a — x= a — ■^• Substituting these values in the above integrals, Ca (fl - xydx = V2^ Cx^ia^ -2ax + x^) dx Jo t/O /2 1 4 ? , 2 ^\ 16 -;r— 3 16 , 3 therefore C/E^ = Wd^"' + U"^' = IB "^ (I '^^ + ^') " Since, furthermore, we have X, r« i 2 , — t 2 , A rfa; = V2p / x dx = :^V2pa = ^ab, then ii;^ = g(|a^ + 6^)- 700 ELEMENTS OF CALCULUS Remark. When the integrals / Ax'dx and / Bifdy cannot be obtained algebraically, or when they are too complicated, the formula of Thomas Simpson may be used (1333). Thus, to calculate approximately / AxMx, divide the maxi- mum value of X into an even number n of equal parts 8 = - J through the points of division and at the extremities of I, draw planes perpendicular to the a;-axis; determine the areas A^^, A^, A^, . . . A„oi the sections made by the planes, and putting j/o = AoXo^ = Ao X = 0, Vi = ^la^i' = A^S\ 2/2 = A^x^^ = A^iS', 2/8 = AaXi^ = As 9 S', y„ = A„x„^ = A^n'Si^ we have approximately, J Ax'dx = - [2/„ + 4 (2/1 + 2/8 + - • •+ 2/„_.i) +2 (y^ + y^ +... + y^_^)] = ^[n"4„ + 4 (4i + 94s + 25^+. . .) +2(4^+16A4 + 364e + . . .)]. In the same way / By^dy is calculated, and dividing the sum of the results by C/ = Sm = I Adx, which may also be deter- mined by the formula of Thomas Simpson (1337), we obtain K' with sufficient approximation for all practical purposes. MOMENT OF INERTIA OF PLANE SURFACES 1358. Moment of inertia of plane surfaces with respect to an axis drawn in the plane of the surface (1356). 1st. The section being a rectangle, or in general a parallelogram, whose base is b and altitude h, if the base b is parallel to the neutral line Gx for any element, we have i = b dv v^, wherein the moment of inertia is i, the area of ^.* ^^^ the element is b dv, and its distance from the axis of rotation is v. m mf \ h G X MOMENT OF INERTIA 701 Therefore, the moment of inertia / of the section is / = bfv'dv = ~+C. (a) Taking the integral between the hmits and ~,C being for V = 0, we have for the moment of inertia /' of the part above the neutral axis Gx, 3 [2/ 24' h Taking the same integral between the limits — - and 0, we have ' for the moment of inertia /" of the part below the neutral axis Gx, r Therefore, = _ ^ /_ ^Y= — 3 V 2/ 24 ' r = 1" and 1 = 1' + I" = 2^ = %. 24 12 The same value is obtained when the integral is taken directly h h between the limits — - and ^ '• h I = b J f'^'=2i--^W^=12- ^131^)- 2d. The section being a hollow rectangle symmetrical about its axis, the moment of inertia / is the difference between the moments of inertia of two %^\4[ rectangles, one having the dimensions b and "^j h, and the other b' and h'; then from 1st I I I < — a — » /I h 'J \ j^W_Vhf^^ bW - b'h'" ^'s- *" lJ} 12 12 12 Fig. 415 jf 5' = 5^ tiiat is, if the web which joins the heads can be neglected, we have simply h{h^- h'^ 12 3d. Moment of inertia of a parallelogram ABCD with respect to one of its diagonals AC taken as axis. 702 ELEMENTS OF CALCULUS Kg. 416 Calling /' the moment of inertia of the triangle ABC with respect to its base AC = 6, and noting that mm' -.b = {h — v):h, , bQi-v) , b we have mm = — ^^-r ~ T ^' and therefore mm' dv = bdv — yvdv, h and r^ij\^dv-l£^dv = ^f-^-^^^-^- For the parallelogram ABCD (1st), 4th. The moment of inertia of a circle being the same for the axes OV and OU, we have / = fv^d<»= fu^ cio) or 7 = \f(v' + w') do>, wherein du> is the area of an element. Making v' + v? = r^ (733), and taking the element concentric to the circle, we have, Fig. 417 and then rfo) = 2 Trr dr, 7 = i f2Trr^dr. Taking this integral between the Umits and the exterior radius R, j_'tR* 4 5th. For a hollow circular section, whose exterior and interior radii are respectively R and R' (2d and 4th), we have, 4 4 4^ / 6th. Moment of inertia of an elliptical section having 2 a for its major axis and 2 b for its minor axis. MOMENT OF INERTIA 703 Describing a circle upon the major axis as diameter, the elements mm' and nn', taken at the same distance v from the axis, one in the circle and the other in the elHpse (1142), give mm' : nn' = b : a and mm' — -nn', a ' and we have d= j v^do) + / Pd = 0, and since j v d", a>" . . . (Fig, 422), included between the ordinates 2/0 and y^, y^ and 2/4 . . ., that is, between the successive even ordinates. The position of the center of gravity is often very difficult to determine; however, it is always near the middle ordinate, which it approaches as 8 is indefinitely decreased. To simplify the calculations, the following hypothesis, which is very near the truth, will be adopted. Giving the values which were found for s (1268, Figs. 358 and 359), to 0)', 0)", o'" . . ., and noting that v' = 8, v" = 3 8, v'" = 5 8 . . ., we may put. rig. 422 "V = 5(2/0 + 41/1+ 2/2) 8 = 3 (2/0 + iyi + yi), 0,'V' = |(2/,+ 42/8+ 2/4)3 8= - (3 2/2 + 4 X 32/3 + 32/0, <-"V" = |(2/4 + 42/5+ 2/.)58 = |(52/4 + 4 X 52/5 + 5y,), mv = g(2/„_!! + 42/,^! + 2/») (w - 1)8 = g [(n - 1) 2/„-2 + 4 (n - 1) 2/„-i + (n - 1) 2/„]- Adding these equations, for formula (2) we obtain ■S,u,v= n7 = -3{2/o + (n-l)2/»+4[2/i + 32/8+52/6+--- + (n-l)2/n-iJ + 2[22/2 + 42/4 + 62/6 +■ ■ •+ (n - 2)2/„_J|. (2') To calculate the formula (3), each element -