Cornell University Library The original of tliis book 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/cu31924031187408 Cornell University Library arV1679 Elements of geometry & trigonometry 3 1924 031 187 408 olln.anx ELEMENTS OF GEOMETRY TRIGONOMETRY, APPLICATIONS m MENSUEATION". BY CHARLES DA VIES. LL. D. AUTHOR OF FIRST LESSONS IX ARITHMETIC, ELEMENTARY ALGEBRA, PEAOTICAL MATHEMATICS FOR PRACTICAL MEN, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE GEOMETRY, SHADES, SHADOWS, AND PERSPECTIVE, ANA- LYTICAL GEOMETRY, DIFFERENTIAL AND INTKORAL CALCULUS. NEW YORK: A. S. BARNES & Co., Ill & 113 WILLIAM STEBET, (COENEB OF JOHN STKEBT.) SOLD BY BOOKSELLBBS, GENEBALI.Y, TUBOUGnOUT THK UNITBD STATES. 1 8 ,8 . £utDied according to Act of Congress, In the year KighteeD Bmubed and tWtfoaa, Bl CHARLES DAVIES, lu the Clerk's Office of the District Court of the United States for the Sootbtuu District of New York. PREFACE Those who are conversant with the preparation of ele- mentary text-books, have ex])erienced the difficuUy of adapting ihem to the various wants which tliey are in- tended to supply. The instilulions of education are of all grades, from the college to the district school, and although there is a wide difference between the extremes, the level, in passing from one grade to the other, is scarcely broken. Each of these classes of seminaries requires text-books adapted to its own peculiar wants ; and if each held its proper place in its own class, the task of supplying suit- able works would not be difficult. An indifferent college is generally inferior, in the system and scope of its instruction,to the academy or high school; while the district school is often found to be superior to its neighboring academy. The Geometry of Legendre, enibracing a complete course of Geometrical science, is all that is desired in the colleges and higher seminaries; while the Practical Mathematics for Practical Men, recently published, is designed to meet the wants of those schools which are strictly elementary and practical in their systems of instruction. .^^ PBEFAOE. ]3ut still a large class of seminaries remained tiusup- plied with a suitable text-book on Elementary G eometi-y and Trigonometry : viz., those where the pupils are car- ried beyond the acquisition of facts and mere practical knowledge, but have not time to go through with a ful course of mathematical studies. It is foi such, that the following work is designed. It lias been the aim of the author to present the striliing and important truths of Geometry in a form more simple and concise than could be adopted in a complete treatise, and yet to preserve the exactness of rigorous reasoning. In this system of Geometry nothing has been taken foT granted, and nothing passed over without being fully de- monstrated. The Trigonometry, including the applications to the measurements of heights and distances, has been writ- ten upon the same plan and for the same objects: it embraces all the important theorems and all the striking examples. In order, however, to render the apphcations of Ge- ometry ta the mensuration of surfaces and solids complete in itself, a few rules have been given which are not de- monstrated. This forms an exception to the general plan of the work, but being added in the form of an appendix, it does not materially break its unity. That the work may be useful in advancing the interests of education, is the hope and ardent wish of the author. FlBHEILL LaKDIKG, May, 1851 CONTENTS. BOOK I. Page. DuriHiT OKS arid Remarks, ..... 9—16 Axioms, ........ 16 Properties of Polygons, ...... 17 — 37 BOOK II. Of the Circle, 38 Problems relating to the First and Second Books, - - 53- -68 BOOK III. Patios and Proportions, -.....• 69 — * BOOK IV. Measurement of Areas and Proportions of Figures, . - 82 — 10 Problems relating to the Fourth Book, • - - 109 — 113 Appendix — Regular Polygons, ... . 113 — 115 BOOK V. Of Planes and their Angles, ... . 116 — 188 BOOK VI. 01 Solids, 126—162 Appendix, - 163—164 8 CONTENTS. TRIGONOMETRY. Page. Ot LoGAiirriiJf3, ...--■ .1f Scales, Definitions, and Explanation of Tables, - ITieorems, , . . . . Examples, ---.-- Application to Heights and Distances, - 165- 176- 181- 189- . - 193- . '. 202- -176 -181 -189 -192 -201 -210 APPLICATIONS OF GEOMETRY. Me.vsueation of Sckfaces, ... 211 General Principles, ... 211- -213 Contents of Figures, .... 213- -239 Mensuration of Solids, . - • 239 General Principles, .... 239- -240 Solidities of Figures, . - . 240- -247 Mensueation of the Round Bodies, 248 To find the Surface of a Cylinder, 248- -249 To find the Solidity of a Cylinder, 249- -250 To find the Surface of a Cone, 250- -251 To find the Solidity of a Cone, 251- 252 To find the Surface of the Frustum of a Cone 253 To find the Solidity of the Frustum of a Cone 254 To find the Surface of a Sphere, 255 To find the Surface of a Spherical Zone, 255- -25e To find the Solidity of a Sphere, - 256- -257 To find the Solidity of a Spherical Segment, 258 To find the Solidity of a Spheroid, 259- -260 To find the Surface of a Cylindrical Ring, 260- -201 To find the Solidity of a Cylindrical Rin g. 261- -?6J ELEMENTARY GEOMETRY BOOK I. DEFINITIONS AND REMARKS. 1. Extension has three dimensions, length, brcadtt, and thickness. Geometry is the science which has for its object ; 1st. The measurement of extension ; and 2dly, To discover, by means of such measurement, the properties and relatione of geometrical figures. 2. A Point is that which has place, or position, bat not magnitude. 3. A Line is length, without breadth or thickness. 4. A Straight Line is one which lies in the same direction between any two of . its points. 6. A Curve Line is one which changes is direction at every point. The word line when used alone, will designate a straight line ; and the word curve, a curve line. 6. A Surface is that which has length and breadth, with- out height or thickness. T. A Plane Surface is that which lies even throughout its whole extent, and with which a straight line, laid in any direction, will exactly coincide in its whole length. 8. A Curved Surface has length and breadth without thick- ness, and like a curve line is constantly changing its direction. 9. A Solid or Bodt/ is that which has length, breadth, and tliickness. Length, breadth, and thi 'kness are called dimen- 10 GEOMETRY. Def ini tior s. . eions. Hence, a solid has three dimensions, a surface two, and a line one. A point has no dimensions, but position only 10. Geometry treats of lines, surfaces, and solids. 11. A Demonstration is a course of reasoning which estab- lishes a truth. 12. An Hypothesis is a supposition on which a demonstra- tion may be founded. 13. A Theorem is something to be proved by demonstration. 14. A Problem is something proposed to be done. 15. A Proposition is something proposed either to be done or demonstrated — and may be either a problem or a theorem. 16. A Corollary is an obvious consequence, deduced front something that has gone before. 17. A Scholium a aremark on one or more preceding propo- sitions. 18. An Asoiom is a self evident proposition. OF ANGLES. 19. An Angle is the portion of a plane included between (wo straight lines which meet at a common point. The two straight lines are called the sides of the angle, and the common point of intersection, the vertex. Thus, the part of the plane included C between AB and ^ C is called an angle : ^^ AB and A are its siies, and A its vertex. J^ j An angle is generally read, by placing the letter at the ver- tex in the middle. Thus, we say, the angle CAB. We may, however, say simply, the angle A. 20. One line is said to be perpendicular to another when it inclines no more to the one side than to the other BOOK I. 11 Definitions, B C D B C B C B C The two angles formed are tlien equal to each other. Thus, if the line DB is per- pendicular to AC, the angle DBA will be equal to DBC. 21. When two lines are perpendicular to each other, the angles which they form are called right angles. Thus, DBA and DBC are called right angles. 22. An acute angle is less than a right angle. Thus, DBC is an acute angle. 23. An obtuse angle is greater than a right angle. Thus, DBC is an obtuse angle. 24. The circumference of a circle is a curve line all the points of which are equally distant from a certain point within called the centre. Thus, if all the points of the curve AEB are equally distant from the centre C, this curve will be the circumference of a circle. 25. Any portion of- the circumference, as AED, is called an arc. 26. The diameter of a circle is a straight line passing through the centre and terminating at the circumference. Thus, ACB is a diameter. 27. One half of the circumference, as A CB is called a semicircumference ; and one quarter of the circumference, as AC, is called a quadrant 12 GEOMETRY. Definiti ons. 28. The circumference of a circle is used for the measure- ment of angles. For this purpose it is divided into 360 equal parts called degrees, each degree into 60 equal parts caUed minutes, and each minute into 60 equal parts called seconds. The degrees, minutes, and seconds are marked thus " ' " ; and 9° 18' 16", are read, 9 degrees 18 minutes and 16 seconds. 29. Let us suppose the circumference of a circle to be divided into 360 degrees, beginning at the point B. If through the point of division marked 40, we draw CJE, then, the angle ECB will be equal to 40 degrees. If CF were drawn through the point of division marked 80, the angle BCF would be equal to 80 degrees. or LINES. 30. Two straight lines are said to be parallel, when being produced either way, as far as we please, they will not meet each other. 31. Two curves are said to be parallel or concentric, when they are the same dis- tance from each other at every point. 32. Oblique lines are those which ap- proach each other, and meet if sufficiently produced. 33. Lines which are parallel to the horizon, or to the water level, are called horizontal lines. 34. Lines which are perpendicular to the horizon, or to the water level, are called vertical lines. B O K I. 13 De fin it ions. OF PLANE FIGURKS. 35. A Plane Figure is a portion of a plane terminated on all sides by linos, either straight or curved. 36. If the lines which bound a figure are straight, the f.paco which. they inclose is called a rectilineal figure, oi polygon The lines themselves, taken together, are called the perimtter of the polygon. Hence, the perimeter of a polygon is the sum of all its sides. 37. A polygon of three sides is called a triangle. 38. A polygon of four sides is called a quadrilateral. 39. A polygon of five sides is called a pentagon. 40. A polygon of six sides is called a hexagon. 41. A polygon of seven sides is called a heptagon. 42 A polygon of eight sides is called an octagon. 2 14 GEOMETRY. De fin it io ns. 43. A polygon of nine sides is called a nonagon. 44. A polygon of ten sides is called a decagon. 45. A polygon of twelve sides is called a dodecagon. 46. There are several kinds of triangles. First. An equilateral triangle, which has its three sides all equal. Second. An isosceles triangle, which has two of its sides equal. Third. A scalene triangle, which has its three sides all unequal. Fourth. A right angled triangle, which lias one right angle. In the right angled triangle ABC, the side AC, opposite the right angle, is called the hypothenuse. 47. The base of a triangle is the side on which it stands. Thus, AB is the base of the triangle ACB. The altitude of a triangle is a line drawn from the angle opposite the base and per-j4 pendicular to the base. Thus, CD is the altitude of the tri angle ACB. BOOK I. 15 Definitions. 48. There are three kinds of quadrilaterals. 1, The trapezium, which has none of ite sides parallel. 2. The trapesoid, ■which has only two of its sides parallel. cnx 8. The parallelogram, which has its opposite sides parallel. 7 49. There are four kinds of parallelograms ; 1. The rhomboid, which has no right / nngle. / 2. The rhombus, or lozenge, which is an equilateral rhomboid. 3. The rectangle, which is an equian- gular parallelogram. 4. The square, which is both equilat- eral and equiangular. 16 GEOMETRY. Of Axioms . 50. A Diagonal of a figure is a line which joins the vertices of two angles not adjacent. 51. The base of a figure is the side on which it is suppu?a3 (() stand ; and the altitude is a line drawn from the opposite side or angle> perpendicular to the base. AXIOMS. 1. Things which are equal to the same thing are equal to each other. 2. If equals be added to equals, the wholes will be equal. 3. If equals be taken from equals, the remainders will be equal. 4. If equals be added to unequals, the wholes will be un- equal. 5. If equals be taken from unequals, the remainders will be unequal. 6. Things which are double of equal things, are equal to each other. 7. Things which are halves of the same thing, are equal to each other. 8. The whole is greater tlian any of its parts 9. The whole is equal to the sum of all its parts. 10. All right angles are equal to each other. H. A straight line is the shortest distance between two points. 12. Magnitudes, which being applied to each other, soin- cide throughout their whole extent, are equal. BOOK I. 17 Of Angles. PROPERTIES OF POLYGONS. THEOREM I. Mesry diameter of a circle divides Jhe circumference into two equal parts. Let ADBE be the circumference of a circlo, and ACB a, diameter: then will the part ADB be equal to the part AEB. For, suppose the part AEB to be turn- el arcund AB, until it shall fall on the part ADB. The curve AEB will then exactly coincide with the curve ADB, or else there would be some point in the curve AEB or ADB, unequally distant from the centre C, which is contrary to the definition of a circumference (Def. 24). Hence, the two curves will bo equal (Ax. 13). Corollary 1. If two lines, AB, DE, be drawn through the centre C perpen-: dicular to each other, each will divide the circumference into two equal parts ; and the entire circumference will be divided into the equal quadrants DB, DA, AE, and EB. Cor. 2. Hence, a right angle, as DCB, is measured by one quadrant, or 90 degrees ; two right angles by a semicircumfer- ence, or 180 degrees ; and four right angles by the whole cir- cumference, or 360 degrees 2* 18 GEOM E TR y. Of Angl es. THEOREM II. Ij one straight line meet another straight line, the sum of the two adjacent angles mil be equal to two rigfA angles. Let the straight line CD meet the / Blraight line AB, at the point C; then will the angle BCB plus the angle DC A be equal to two right angles, A C B About the centre C, with any radius as CB, suj)pose, a semicircuniference to bo described. Then, the angle DCB will be measured by the arc BD, and the angle DC A by the arc AD. But the sum of the two arcs is equal to a somicir cumference : hence, the sum of the two angles is equal to two right angles (Th. i. Cor. 2). Cor. 1. If one of the angles, as DCB, is a right angle, the other angle, DC A will also be a right ansle. Cor. 2. Hence, all the angles which can be formed at any point C, by any number of lines, GD, OE, CF, &c., drawn on the same side of AB, are equal to two right angles : for, they will be measured by a semicircumference. Cor. 3. If DC meets two lines CB, OA, making DOB plus DCA equal to two right angles, ACB will form one straight line. Cor. 4. Hence, also, all the angles which can be formed round any point, as C, are equal to four right angles. For, tlie sum of all the arcs which measure them, is equal to the entire circumference, which is the measure of four right angles (Th. i. Cor. 2). A B B O K 1 . 19 Of Triangles. A V~7J o THEOREM III. If two straight lines intersect each other, the opposite or ver- tical angles which they form, are equal. Let the two straight lines AB and ^ CD intersect each other at the point E : then will the opposite angle AEC be equal to DEB, and AED=CEB. For, since the line AE meets the '^ line CD, the angle AEC+AED= two right angles. But Bince the line DE meets the line AB, we have DEB+AED=: two right angles. Taking away from these equals the com- mon angle AED, and there will remain the angle AEC equal to the angle DEB (Ax. 3). In the same manner we may prove that the angle AED is equal to the angle CEB. THEOREM IV. If two triangles have two sides and the included angle of the one, equal to two sides and the included angle of the other, each to each, the two triangles will he equal. Let the triangles ABC and DEF have the side AC equal to DF, CB to FE, and the angle C equal to the angle F : then will the triangle A CB be equal to the triangle DEF. For, suppose the side ^ C, of the ^ -B -D ^ triangle ACB, to be placed on DF, so that the extremity C shall fall on the extremity F: then, since the sides are equal, A will fall on D. But since the angle C is equal to the angle F, tiie line CB 20 GEOMETRY Of Triangles. will fall on FE ; and since CB is equal to FE, the extremity^ will fall on E ; and consequently the side AB will fall on the side DE {Ax. 11). Hence, the I'vo triangles will fill the same space, and consequently are equal (Ax. 12.). Sckolium, Two triangles are said to be equal, when being applied the one to the other they exactly coincide (Ax. 12). Hence, equal triangles have their like parts equal, each to each, since those parts coincide with each other. The converse of the proposition is also true, namely, that two triangles which have all the parts of the one equal to the corresponding parts of the other, each to each, are equal : for if applied the one to the other, the equal parts will coincide. THEOREM V. If two triangles have two angles and the included side of the one, equal to two angles and the included side of the other, each to each, the two triangles wiU be equal. Let the two triangles ABC and DEF have the angle A equal to the angle D, the angle B equal to the angle E, and the included side AB equal to the included side DE : then will the triangle ABC bt equal to the triangle DEF. For, let the side AB be placed on the side DE, the extrem- ity A on the extremity D ; and since the sides are equal, the point B will fall on the point E. Then since the angle A is equal tp the angle D, the side B O K 1 . 21 Of Triangles. AC will take the direction DF: and since the angle £ is equal to the angle E, the side BC will fall or the side EF: hence, the point C will be found at the same time on DF and EF, and therefore will fall at the intersection F: consequently, all the parts of the triangle ABC will coincide with the parts of the triangle DEF, and therefore, the two triangles are equal THEOREM VI. In an isosceles triangle the angles opposite the equal sides art equal to each other. Q Let ABC be an isosceles triangle, hav- ing the side AC equal to the side CB : then will the angle A be equal to the an- gle B. ^ A For, suppose the line CD to be drawn dividing the angle C into two equal parts. Then, the two triangles ACD and DCB, have two sides and the included angle of the one equal to two sides and the in- cluded angle of the other, each to each : that is, the side A C equal to BC, the side CD common, and the included angle ACD equal to the included angle DCB : hence the two trian- gles are equal (Th. iv) ; and hence, the angle A is equal to the angle B. Cor. 1. Hence, the line which bisects the vertical angle o/ an isosceles triangle, bisects the base. It is also perpendicu- lar to the base, since the angle CD A is equal to the angle CDB. Cor. 2. Hence, also, every equilateral triangle, must also be equiangular: that is, have all its ang es equal, each to each 22 GEOMETRY Of Triangles. THEOREM VII. Conversely. — If a triangle has two of itx angles equal, the sides opposite those angles mil also be equal. In the triangle ABC, let the angle A be equal to the angle B: then vifill the side BO be equal to the side AC. For. if the tvifo sides are not equal, one of them must be greater than the other. Suppose AC to be the greater side. Then take a part AD equal to BC Now, in the two triangles ADB and jIBC, we have the aide AT)=BC, by hypothesis ; the side aB common, and the aagle A equal to the angle B : hence, the two triangles have two sides and the included angle of the one equal to two sides and the included angle of the other, each to each : hence, the two triangles are equal (Th. iv), that is, a part ADB is equal to the whole ABC, which is impossible (Ax. 8) : conse- quently, the side AC cannot be greater than the side CB, and hence, the triangle is isosceles. Scholium 1. The method of reasoning pursued in the last theorem, is called the " reduclio ad absurdum," or a proof that leads to a known absurdity. Let us analyze this method of reasoning. We wished to prove that the two sides AC, CB were equal. We supposed them unequal, an! .AC the greater — that was an hypothesis (See Def. 12). We then reasoned on the hypothesis, and proved a part equal to the whole, which we know to be false (A.X. 8) Hence, we conclude that the hypothesis is untrue, because after a correct chain of reasoning it leads to a resuU which we know to be absurd BOOK I. 23 Of Tr angles. Scholium 2. Generally, — If the demonstration is based on known principles, previously proved, or admitted in the ax- ioms, the conclusion will always be true. But, if the demon- stration is based on an hypothesis, (as in the last theorem, that AC was the greater side), and the conclusion is contrary to wkat has been previously proved, or admitted in the axioms then, it follows, that the hypothesis cannot be true. The former is called a direct, and the latter an indirect demonstration. / D THEOREM VIII. If two triangles have the three sides of the one equal to the three sides of the other, each to each, tite three angles wiU aha be equal, each to each. Let the two triangles ABC, ABD, have the side AB equal to the side AB, the side AC equal to AD, and the side CB equal to DB: then will the corres- ponding angles also be equal, viz : the angle A will be equal to the angle A, the angle B to the angle B, and the angle C to the angle D. For, suppose the triangles to be joined by their longest equal sides AB, and the line CD to be drawn. Then, since the side .AC is equal to AD, by hypothesis, the triangle AD C will be isosceles ; and therefore, the angle A CD will be equal to the angle ADC (Th. vi). In like manner, in the triangle CBD, the side CB is equal to DB : hence, th£ angle BCD is t qual to the angle BDC. Now, by the addition of equals, we have 24 G E O M E T R T . ^__ Of Triangles. that is, the angle ACB=ABB. /hv Now, the two triangles ACB and ABB ^\~\ — y^ have two sides and the included angle of nI/^ the one equal to two sides and the in- " eluded angle of the other, each to each: hence, the remainicg angles will be equal (Th. iv) : consequently, the angle CAB is equal to BAB, and the angle CBA to the angle ABB. Sch. The angles of the two triangles which are equal to each other, are those which lie opposite the equal sides. THEOREM IX. If one side of a triangle is produced, the outward angle is greater than cither of the inward opposite angles. Let ABC be a triangle, having the side AB produced to B : then will the outward angle CBB be greater than either of the inward opposite angles .A or C. ^q For, suppose the side CB to be bisected at the point E. Draw AE, and produce it until EF is equal to AE, and then draw BF. Now, since the two triangles AEC and BEFhave AE= EF and EC=EB, and the included angle AEC equal to the included angle BEF (Th. iii), the two triangles will be equal in all respects (Th. iv) : hence, the angle EBFvrill be equal to the angle C. But the angle CBB is greater than the angle CBF, consequently it is greater than the angle C. In like manner, if CB be produced to G, and AB be bi sectcd, it may be proved that the outward angle ABG, or its equal CBB (Th. iii), is greater than the angle .4. B O K I . ■ 25 01 Trianglea. THEOREM X. The sum of any two sides of a triangle is sifeater than the third side. Let ABC ha a triangle : then will the sum of two of its sides, as AC, CB, be greater than the third side AB. For the straiett line ^B is the short- C B est distance between the two points A and B (Ax. xi): hence, A C+ CB is greater than AB. THEOEEM XI. The greater side of every triangle is opposite the greater angle : and conversely, the greater angle is opposite the greater side. First. In the triangle CAB, let the an- gle C be greater than the angle B : then, will the side AB be greater than the side AC. For, draw CD, making the angle BCD equal to the angle B. Then, the triangle CBD will be isosceles: hence, the side CD=DB (Th. vii.) But, by the last theorem ^C is less than AD -{-CD; that is, less than AD-^-DB, and consequently less than AB. Secondly. Let us suppose the side AB to be greater than A C ; then will the angle C be greater than the angle B. For if the angle C were equal to B, the triangle CA B would be isosceles, and the side AC would be equal to AB (Th. vii) , which would be contrary to the hypothesis. Again, if the angle C were less than B, then, by the first part of the theorem, the side AB would be less than A C, which is also contrary to the hypothesis Hence, since C 26 GEOMETRY. Of ParalUl Lines. cannoi bo equal to B, nor less than B, it follows that it miiBl be greater THEOREM XII. If a straight line intersect two parallel lines, the alternate angle? mil be equal. G If two parallel straight lines, AB CD, are intersected by a third line GH, the ^Z angles ABF and EFI) are called alternate -^ y^-'^'^^ angles. It is required to prove that these G /f" 1) angles are equal. ■" If they are unequal one of them must be ^eater than the other. Suppose EFD to be the greater angle. Now conceive FB to be drawn, maldng the angle EFB equal to the angle AEF, and meeting A E in B. Then, in the triangle FEB the outward angle FEA is greater than either of the inward angles B or EFB (Th. ix.) ; and therefore, EFB can never be equalto.A£Fsolongas J!B meets EB. But since we have supposed EFD to be greater than AEF, it follows that EFB could not be equal to AEF, if FB fell be- low FD. Therefore, if the angle EFB is equal to the angle AEF, FB cannot meet AB, nor fall below FD, and conse- quently must coincide with the parallel CD (Def. 30) : and hence, the alternate angles AEF and EFD are equal. Cor. If a line be perpendicular to one of two parallel lines, it will also be per- pendicular lo the other. B Q K X . 27 Of Parallel Lines. THEOREM XIII. Conversely, — If a line intersect two straight lines, making the alternate angles equal, those straight lines will he parallel. Let the line EF meet the lines AB, CD. making tha angle AEF equal to the b/ angle EFD: then will the lines AB and ^ / ^ CD be parallel. C /T"^-^.^ -O For, if they are not parallel, suppose G through the point F the line FQ to be drawn parallel to AB. Then, because of the parallels AB, FG, the alternate angles, AEF and EFG will be equal (Th. xii). But, by hypothesis, the angle AEF is equal to EFD : hence, the angle EFD is equal to the angle EFG (Ax. 1) ; that is, a part is equal to the whole, which is absurd (Ax. 8) : therefore, no line but CD can be parallel to AB. Cor. If two lines are perpendicular to the same line, they will be parallel to each other. THEOREM XIV. If a line cut two parallel lines, the outward angle is equal to the inward opposite angle on the same side ; and the two inward angles, on the same side, are equal to two right angles. Let the line EF cut the two parallels .4.B, CD : then will the outward angle y E GB be equal to the inward opposite an- J 7^ B glo EHD ; and the two inward angles, C /^ n BGH and GHD, will be equal to two ^ ii"ht anjiloK. 28 GEOMETRY- Of ParalNl Lines First. Since the lines AB, CD, are parallel, the angle AGH equal to the alternate angle GHD I'h. xii) ; but the angle AGH is equal to the opposite angle EGB : hence, the is equal to the alternate angle GHD E (Th. xii); but the angle AGH is equal ^ -~^- — 'jg angle EGB is equal to the angle EHD ^ (Ax. I). Secondly. Since the two adjacent angles EGB and BGH are equal to two right angles (Th. ii) ; and since the angle EGB has been proved equal to EHD, it follows that the sum of BGH plus GHD, is also equal to two right angles. Cor. 1, Conversely, if one straight line meets two other straight hnes, making the angles on the same side equal to each other, those lines will bs parallel. Cor. 2. If a line intersect two other lines, making the sum of the two inward angles equal to two right angles, those two Jines will be parallel. Cor. 3. If a line intersect two other lines, making the sum of the two inward angles less than two right angles, those lines will not be parallel, but will meet if sufficiently produced. THEOREM XV. All straight lines which are parallel to the same line, are paraJld to each other. liCt the lines AB and CD be each par- q all el to EF: then will they be parallel \o each other. For. let the line GI be drawn perpen- dicular to F.F : then will it also be per- pendicular to the parallels AB, CD (Th. sii Cor.). A B C D B I F B O K 1 . 29 Of Triangles. Then, since the lines AB and CD are perpendicular to the line GI, they will be parallel to each other (Th. xiii. Ccr), THEOREM XVI. Tj cne side of a triangle be produced, the outward angle will hi equal to the sumof the inward opposite angles. In the triangle ABC, let the side AB be produced to D : then will the outward >X ^ angle CBD be equal to the sum of the in- /^ \ _,,-' ward opposite angles A and C. f- ^^ j. For, conceive the line B£ to be drawn parallel to the side AC. Then, since BC meets the two pa- rallels A C, BE, the alternate angles A CB and CBE will be equal (Th. xii). And since the lino AD cuts the two parallels BE and AC the angles EBD and CAB are equal to each other (Th. xiv). Therefore, the inward angles C and A, of the triangle ABC, are equal to the angles CBE and EBD ; and consequently, the sum of the two angles, A and C, is equal to -the outward angle CBD (Ax. 1). THEOEEM XVII. In any triangle the sum of the three angles ij equal to two right angles. Let ABC be any triangle : then will the sum of the three angles C A+B-}-C=two right angles. For, let the side AB be produced lo D ^ Then, the outward angle CBD =4+ C (Th. xvi). 3* 30 GEOMETRY. Of Triangles. To each of these equals add the angle CBA, and we shall have CBD+ CBA=A+ C+B. But the sum of the two angles CBD and CBA, is equal to two right angles -* ■" I I'h ii) : hence A-\-B+C=\.v!0 right angles (Ax. 1). Cor. 1. If two angles of one triangle be equal to two angles of another triangle, the third angles will also be equal (Ax. 3). Cor. 2. If one angle of one triangle be equal to one angle of another triangle, the sum of the two remaining angles in each triangle, will also be equal (Ax. 3). Cor. 3. If one angle of a triangle be a right angle, the sura of the other two angles will be equal to a right angle ; and each angle singly, will be acute. Cor. 4. No triangle can have more than one right angle, nor more. than one obtuse angle ; otherwise, the sum of the three angles would exceed two right angles : hence, at least two angles of every triangle must be acute. THEOREM XVIII. I. A perpendicular is the shortest line that can be drawn from a given point to a given line. II. If any number of lines be drawn from the same point, those which are nearest the perpendicular are less titan those which are morejremote. Lot .A be a given point, and DE a straight line. Suppose AB to be drawn pcrpendiculai to DE, and suppose the oblique lines AC and AD also to be ^ B O K 1 . 31 Of Triang.os. drawn : Then, AB will be shorter than either of the oblique lines, and AC will be less than AD First. Since the angle B, in the triangle ACB, is a right angle, the angle G will be acute (Th. xvii. Cor. 3) : and sinca the greater side of every triangle is opposite the greater angle v^Tb. xi), the side AO will be greater than AB. Sieondly. Since the angle AOB is acute, the adjacent angle ACD will be obtuse (Th. ii) : consequently, the angle D is acute (Th. xvii. Cor. 3), and therefore less than the angle ACD. And since the greater side of every triangle is oppo- site the greater angle, it follows that AD is greater than AQ. Cor. A perpendicular is the shortest distance from a point to a line. THEOREM XIX. If two nglit angled triangles have the hypothenuse and a sida of the one equal to the hypothenuse and a side of the other, the remaiiing parts will also he equal, each to each. Let the two right angled triangles A D ABC and DEF, have the hypothe- nuse A C equal to DF, and the side AB equal to DE : then will the re- maining parts be equal, each to each. " ^ ^ For, if the side BC is equal to EF, the corresponding an- gles of the two triangles will be equal (Th. viii). If the sides are unequal, suppose BC to be the greater, and take apart, BG, equal to EF, and draw AG. Then, in the two triangles ABG and DEF. the angle B is oqual to the angle E, the side AB to the side DE, and the side BG to the side EF : hence, the two triangles are equal in all respects (Th. iv) and consequently, the side AG is equal to 32 GEOMETRY. Of Polygons. DF. But DF is equal to AC, by hypothesis; therefore, AO is equal to J.C (Ax. 1). But this is impossible (Th, sviii) ; hence, the sides BC and EF cannot be unequal ; con- Bequenily, the triangles are equal (Th. viii). THEOREM XX. The sum of the four angles of every quadrilateral is equal to four right angles. Let A DBD be a quadrilateral : then will AA-B+C+D—ioviT right angles. Let the diagonal DC he drawn dividing ihe quadrilateral AB, into two triangles, BDC, ADC. Then, because the sum of the three angles of each triangle is equal to two right angles (Th. x\'ii), it folio n-s that the sum of the angles of both triangles is equal to four right angles. But the sum of the angles of both triangles, make up the angles of the quadrilateral. Hence, the sum of the four angles of the quadrilateral is equal to four right angles. Cor. 1. If then three of the angles be right angles, the fourtli angle will also be a right angle. Cor. 2. If the sum of two of the four angles be equal to two right angles, the sum of the remaining two will also be equal to two right angles. Cor. 3. Since all the angles of a square or rectangle, are equal to each other (Def. 48), and their sum equal to four right angles, it follows that each angle is equal to one right anjiio. THEOREM XXI. Tfte svm of all the interior angles of any polygon is equal to twice as many right angles, wanting four, as the figure has sides B K I . , 33 Of Polygons liBt ABCDE be any polygon: then will ihe sum of its inward angles A+B+C+D^E E^ be equil to twice as many right angles, wanting four, as the figure has sides. For, from any point P, within the poly- A S gon, draw the lines PA, PB, PC, PD, PE, to each of the angles, dividing the polygon into as many triangles as the figure has sides. Now, the sum of the three angles of each of these triangles is equal to two right angles (Th. xvii) : hence, the sum of the angles of all the triangles is equal to twice as many right an- gles as the figure has sides. But the sum of all the angles about the point P is equal to four right angles (Th. ii. Cor. 4) ; and since this sum makes no part of the inward angles of the polygon, it must be sub- tracted from the sum of all the angles of the triangles, before found. Hence, the sum of the interior angles of the polygon is equal to twice as many right angles, ^canting four, as the figure has sides. Sch. This proposition is not applicable to polygons which have re-entrant angles. The reasoning is limited to polygons with salient angles, which may properly be named convex polygons. THEOREM XXII. Ff every side of a polygon be produced out, the sum of all the ottt ward angles thereby formed, will he equal to four right angles. GEOMETRY. Of Polygons. Let A, B, C, D, and E, be tlie outward angles of a polygon formed by producing all the sides. Then will A-\-B+ C+D+Jr;=four right angles. For, each interior angle, plus its exte- lior angle, as A-\-a, is equal to two right angles (Th. ii). But there are as many exterior as interior angles, and as many of each as there are sides of the polygon : hence, the sum of all the interior and exterior angles will be equal to twice as many right angles as the polygon has sides. But the sum of all the interior angles together with four right angles, is equal to twice as many right angles as the polygon tias sides (Th. xxi) : that is, equal to the sum of all the in- ward and outward angles taken together. From each of these equal sums take away the inward angles, and there will remain, the outward angles equal to four right angles (Ax. 3). THEOREM XXIII. The opposite sides and angles of every paraRelogram are equal, each to each : and a diagonal divides the parallelogram into two equal triangles. Let ABCD be any parallelogram, and DB a diagonal : then will the opposite sidos and angles be equal to each other, each to each, and the diagonal DB will divide the parallelogram into two equal triangles. For, since the figure is a parallelogram, the sides AB, DC aro parallel, as also the sides AD, BC. Now, since the B O O K i . 35 Of Parallelograms. parallels are cut by the diagonal DB, the alternate angles will be equal (Th. xii) : that is the angle ADB=DBC and BDC=ABD. Ilonce the two triangles ABB BBC, having two angles in the one equal to two angles in the other, will have their third angles equal (Th. xvii. Cor. 1), viz. the angle A equal to the angle C, and these are two of the opposite angles of tho parallelogram. Also, if to the equal angles ABB, BBC, we add the equals BBC, ABB, the suras will be equal (Ax. 2) : viz. the whole angle ABC to the whole angle ABC, and these are the other two opposite angles of the parallelogram. Again, since the two triangles ABB, BBC, have the side BB common, and the two adjacent angles in the one equal to the two adjacent angles in the other, each to each, the two triangles will be equal (Th. v) : hence, the diagonal divides the parallelogram into two equal triangles. Cor. 1. If one angle of a parallelogram be a right angle, each of the angles will also be a right angle, and the parallelo- gram will be a rectangle. Cor. 2. Hence, also, the sum of either two adjacent angles of a parallelogram, viH be equal tp two right angles. THEOEEM XXIV.' If the opposite sides of a quadrilateral, are equal, each to each, tkt equal sides will be parallel, and the figure mil be a pa- railelo'rrani. »t5 GEOMETRY. 01 Parallelograms. Let ABCD be a quadrilateral, having its opposite sides respectively equal, viz. AB=CD and AD^BC thrn will these sides be parallel, and the ^ li^ure will be a parallelogram. For, draw the ijiagonal ED. Then, the two triangles ABD, BDC, have all the sides of the one equal to all the sides ol tiie other, each to each : therefore, the two triangles are equal (Th. viii) ; hence, the angle ADB, opposite the side AB, ia o-jual to the angle DBC opposite the side DC ; therefore, the sides AD, BC, are parallel (Th. xiii). For a like reason DC it parallel to AB, and the figure ABCD is a parallelogram. THEOREM XXT. If two opposite sides of a quadrilateral are equal and parallel, th'! remaining sides will also be equal and parallel, and the fignre mil be a parallelogram. Let ABCD be a quadrilateral, having the sides AB, CD, equal and parallel: then will the figure be a parallelogram. For, draw the diagonal DB, dividing the quadrilateral into two triangles. Then, since AB is parallel to DC, the alternate angles, ABD and BDC are equal (Th. xii) : moreover, the side BD is common ; ripnce the two triangles have two sides and the included angle of ilie one, equal to two sides and the included angle of the other: the triangles are therefore equal, and consequently, AD is equal to BC, and the angle ADB to the angle DBC; and consequently, AD is also parallel to BC (Th xiii) Tliorcfore, the figure ABCD is a parallelogram. BOOK 1 Of Parallelograms. THEOREM XXVI. The two diagonals cf a parallelogram divide each ether into eouaJ parts, or mutually bisect each other. Let ABCD be a parallelogram, and A C, BD Its two diagonals intersecting at E. Then will AE=EC and BE=ED. A Comparing the two triangles AEI) and BEC, we find the side AD=BC (Th. xxiii), the angle ADE = EBC and EAD=ECB : hence, the two triangles are equal (Th. v) : therefore, AE, the side opposite ABE, is equal to EC, the side opposite EBC; and ED is equal to EB Sch. In the case of a rhombus (Def. 48), the sides AB, BC being equal, the trian- gles AEB and BEC have all the sides of the one equal to the corresponding sides of the other, and are therefore equal. -4 Whence it follows that the angles AEB and BE C are equal. Therefore, the diagonals oi a rhombus bisect each other at right aigles. GEOMETRY. BOOK II, OF THE CIKCLE DEFINITIONS. 1. Tub ciicumference of a circle is a curve line, all the points of which are equally distant from a certain point within called the centre. 2. The circle is the space bounded by this curve line. 3. Every straightline, CA, CD, CE, drawn from the centre to the circumference, is called a radius or semidiamster. Every line which, like AB, passes through the centre and terminates in the circumfe- rence, is called a diameter. 4. Any portion of the circumference, as EFG, is called an arc. 5. A straight line, as EG, joining the-^ extremities of an arc, is called a chord. 6 A segment is the surface or portion of a circle included between an arc and its chord. Thus EFG is a segment. BOOK II 39 Defirritions. 7. A sector is the part of the circle in- cluded bo'tween an arc and the two radii drawn through its extremities. Thus, CAB is a sector 8. A straight line is said to be in- scribed in a circle, when its extremities are in the circumference. Thus, the line AB is inscribed in a circle. 9. An inscribed angle is one which is formed by two chords that intersect each other in the circumference. Thus, BAC is an inscribed angle. 10. An inscribed triangle is one which has its three angular points in the circumference. Thus, ABC is an inscribed triangle. B A 11. Any polygon is said to be in- scribed in a circle when the vertices of all the anyles are in the circumference. The ciicle is then said to circumscribe the polygon. 40 GK JMETRY. Dofini t ions. 1 3 A secant is a line whicli meets the cinnimference in two points, and lies parlly within and partly without the circle. Thua AB is a secant. £ 13. A tangent is a line which has but one point in common with the cir- cumference. Thus, CMB is a tangent. 14. Two circles are said to touch each other internally, when one lies within the other, and their circumfe- rences have but one point in common. 15. Two circles are said to touch each other externally, when one lies without the other, and their circumfe- rences have but one point in common BOOK II. 41 Of the Circle. THEOREM I. Every chord is less than a diamcler. Let AD be any chord. Draw •ho radii CA, CD to its extremities. We shall then have, AD less than AC+CD (Book I. Th. x*). But AC-^CD is equal to the diameter AB : hence, the chord AD is less than the diameter. THEOREM II. If from the centre of a circle a line he drawn to the middle of a chord, I. It mil be perpendicular to the chord ; II. And it will bisect the arc of the chord. Let C be the centre of a circle, and AB any chord. Draw CD through D, the middle point of the chord, and produce it to E: then will CD be perpendicular to the chord, and the arc AE equal to EB. First. Draw the two radii CA, OB. Then the two triangles A CD, D CB, have the three sides of the one equal to the three sides of the 'Note. When reference is maae fron; one theorem to another, in l!ie Bame Book, the number of the theorem referrfid to is alone given- but when the theorem' referred to is found in a preceding Book, the number of the Book is also g'ven. 4* 42 GEOMETRY. Of the Circle. 3thcr, each to each : viz. A C equal to CB, being radii, AD equal to DB, by hypothesis, and CD common: hence, the corresponding angles are equal (Book I. Th, viii) : that is, the angle CD A equal to CDB, and the angle ACD equal to the angle DCB. But, since the angle CDA is equal to the angle CDB, the radius CE is perpendicular to the chord AB (Bk. I. Def. 20). Secondly. Since the angle ACE is equal to BCE, the arc AE will be equal to the arc EB, for equal angles must have equal measures (Bk. I. Def. 29). Hence, the radius drawn through the middle point of a chord, is perpendicular to the chord, and bisects the arc of the chord. Cor. Hence, a line which bisects a chord at right angles, bisects the arc of the chord, and passes through the centre of the circle. Also, a line drawn through the centre of the cir- cle and perpendicular to the chord, bisects it. THEOREM III. If more than two equal linescan he drawn from any point toithut a circle to the circumference, that point will be the centre. Lot D be any point within the circle ABC -"Then, if the three lines DA, T)B, and DC, drawn from the point D to the circumference, are equal, the jen- dicular on the line. Let A be the given point, and BD the given line From the point ^ as a centre, with b radius greater than the shortest distance to BD, describe an arc cut- ting BD in the points B and D. Then, with B and D as centres, and the same radius, describe two arcs intersecting each other at JS. Draw AFE, and it will be the perpendicular required. For, draw the equal radii AB, AD, BE and DE. Then, the two triangles EAB and EAD will have the sides of the one equal to the sides of the other, each to each ; hence, their corresponding angles will be equal (Bk. I. Th. viii), viz. the angle BAE to the angle DAE. Hence, the two triangles BAF and DAF will have two sides and the included angle of the one, equal to two sides and the included angle of the other, and therefore, the angle AFB will be equal to the angle A FD (Bk. I. Th. iv) : hence, AFE will be perpendicidar to BD. SECOND METHOD. When the given point A is nearly- opposite the extremity of the line. Dra w ^ C, to any point C of the lino BD. Bisect AC at P. Then, wiih P as a centre and PC as a ra- dius, doscribi; the semicircle CDA ; draw AD, and it will be perpendicular to CD since CDA is an angle in a semicircle (Bk. II. Th. x). B O K I I . 59 Prob ems. PROBLEM VIII. At a given point in a given line, to make an angle equal to a given angle. JiCt A be the given point, AE ilio given line, and IKL the given angle. ^ . From the vertex K, as a centre, ^ i A with any radius, describe the arc lli, terminating in the two sides of the angle : and draw the chord TL. . From the point A, as a centre, with a distance AE, equal to Kl, describe the arc BE ; then with E, as a centre, and a radius equal to the chord Ih, describe an arc cutting BE at Z); draw AB, and the angle EAB will be equal to the angle K. For, draw the chord BE. Then the two triangles IKh and EAB, having the three sides of the one equal to the three sides of the other, each to each, the angle EAB will be equal to the angle K (Bk. I. Th. viii). PROBLEM IX. Through a given point to draw a line that shall be parallel to a given line. Let A be the given point and p £ i?C the given line. 1 _,..--- \ With A&sa, centre, and any ra- \,--- J. „ dius crreater than the shortest dis- tance from .4 to BC, describe the indefinite arc BE. From the point E, as a centre, with the same radius, describe the arc AF: then, make EB equa to ^ F and draw AB, and it will be the required parallel. 60 GEOMETRY. Probl ems. For, since the arcs AF and ED are equal, the angles EAD and AEF, which they measure, are eijual : hence, the line AD is parallel to BC (Bk 1. Th xiii). ^ PROBLEM X. Two angles of a triangle being given or knovm, to find the third. Draw the indefinite line DEF. At any point, as E, make the angle DEC equal to one "^ E of the given angles, and then CEH equal to a second, by Prob. VlII ; then will the angle HEF be equal to the third angle of the triangle. For, the sum of the three angles of a triangle is equal to two right angles (Bk. I. Th. xvii) ; and the sum of the three angles on the same side of the line DE if equal to two nght angles (Bk. I. Th. ii. Cor. 2) : hence, if DEC and CEH art equal to two of the angles, the angle HEF will be eoual lo ihp remaining angle of the triangle PROBLEM XT. Three sides of a inangh being given, to aescnhe the rrtangie Let A, B, and C, be the gives sides. Draw DE, and. make it equal to ilic side A. From the point D. as a centre, with a radius equal to the -^ si>cond side B, describe an arc ■ o- BOOK II. 61 Problems. from £ as a centre, with the third side C, describe another arc intersecting the former in F: draw DF and FE: then will DEF be the required triangle. For, the three sides are respectively equal to the three lines .4,5, and C. PEOBLEM XII. The adjacent sides of a parallelogram, with the angle which tMy contain, being given, to describe the parallelosrom Let A and B be the given side.* ^ aiid C the given angle / Draw the line DE and make it /jZ equal to A At the point D mane A\ \ the angle EDF equal ro the angle C Make the side DF equal \o B Then describe iwo arcs, one from F as a centre, with a radius FG equal to DE, the other from £, as a centre, with a radius EG equal to DF Through the point G, the point of intersection, draw the lines EG and FG, and DEGF will be the required parallelogram For, ill the quadrilateral DFGE, the opposite sides DE and FG axe each equal to A : the opposite sides DF and EG are each equal to B, and the angle EDF is equal lo C. But, since the opposite sides are equal, they are also parallel (Bk. I. Th, xxiv), and therefore the figure is a parallelogram PROBLEM XIII, To describe a square on a gtvcv line 62 GEOMETRY, Problems. Let AB be the given line. At the point B draw £C perpendicu- lar to AB, by Problem VI, and then make it equal to AB. Tlien, with J. as a centre, and ra- dius equal to AB, describe an arc; and with C as a centre, and the same A B radius AB, describe another arc ; and through D, their point of intersection, draw AD and CD: then will ABCD be the required square. For, since the opposite sides are equal, the figure will be a parallelogram (Bk. I. Th. xxiv) : and since one of the angles is a right angle, the others will also be right angles (Bk. I. Th. xxiii. Cor. 1 ) ; and since the sides are all equal, the figure will be a square. PROBLEM XIV. To construct a rhombus, having given the length of one of the equal sides, and one of the angles. Let AB be equal to the given side, and E the given angle. ' "'^„ At B lay off an angle, ABC, equal to E, by Prob. VIIL and make BC equal to AB. Then, with A and C as centres, and a radius equal to AB, describe two arcs. Through D, their point of intersection, draw thr lines AD, CD: then will ABCD be the required ihombus. For, since the opposite sides are equal, they will be parallel (,Bk. I. Th. xxiv). But they are each equal to AB, and the BOOK II 63 Problems. ingle B is equal to the angle E : hence, ABCD is the re- quired rhombus. PROBLEM XV. To find tlte centre of a circle Draw any chor^ras AB, and bisect it fty Problem IV. Then, through F, the middle point, draw DCE, perpendicular to AB, by Problem VI. Then DCE will be a diameter of the circle (Bk. II. Th. ii. Cor.). Then bisect DE at C, and C will be the centre of the circle. PROBLEM XVI. To describe the circumference of a circle through, three git en points not in the same straight line. Let A, B, C, be the given points. Join these points by the straight lines AC AB, BC. Then, bisect any two of these straight lines, as AB, BC, by the perpendiculars OD, OP (Prob. iv) ; and the point O, where these per- pendiculars intersect each other, will be the centre of the circle. Then with O as a centre, and a radius equal to OA, de- scribe the circumfei'ence of a circle, and it will pass through the points A, B, and C. For, the two right angled triangles OAP and OBP have the side AP equal to the side BP, OP common, and the included 64 GEOMETRY. Problems. angles OPA and OPB equal, being right angles ; hence, the side OB is equal to OA (Bk. 1. Th iv). In like manner it may be showft. that OC is equal to OB. Hence, a circumference described with the radius OA, will pass through the points B and C Sch. This problem enables us to describe the circumference of a circle about a given triangle For, we may consider the vertices of the three angles as the three points through which the circumference is to pass. PROBLEM xvu. Through a given pmnt m the circumference of a circle, to drau) ■i tangent line to the circle. Let A be ttie given point j2. Through A, draw the radius ^C to the centre, and then draw DAE perpendicu- lar to AC, by Problem VI. Then will DAE be tangent to the circle at the point A (Bk. II. Th. v). PROBLEM XVIII. Through a given point without the circumference, to drav j tangent line to the circle. BOOK II, 65 Problems. Let C be the centre of the circle, and A the given point without the circle. Join A and the centre C, and on AC, as a diameter, describe a circumference. Through the poin s B and D where the two circumferences intersect each oiher, draw the lines. AB and AD: these lines will be tangent to the circle whose centre is C. For, since the angles ABC and ADC are each inscribed in a semicircle, they will be right angles (Bk. II. Th. x). Again, since the lines AB, AD, are each perpendicular to a radius at its extremity, they will be tangent to the circle (Bk. II. Th. v). PROBLEM XIX To inscribe a circle in a given, triangle. Let ABC be the given tri- angle. Bisect the angles A and B by the lines AG and BQ, meet- ing at the point 0. Prom O, let fall the perpendiculars OD, OE, OF, on the three sides of the triangle — these perpendiculars will be equal to each other. For, in the two right angled triangles DAG and FAG, wo li'ive the right angle D equal the right angle F, the angle FAG equal to DAO, and consequently, the third angles AOD and AGF are equal (Bk. I. Th. xvii. Cor 1). But the two triangles have a common side AG, hence, they are eqiiaC (Bk. I. Th. v), and consequently, GD is e'ual to OF 66 GEOMETRY. Problems. In a similar manner, it may be proved tliat OE and OD are equal : hence, tlie three per- pendiculars, OD, OF, and OE, are all equal. Now, if with O as a centre,^ and OP as a radius, we describe the circumljerence of a circle, it wil» pass through the points D and E. and since the sides of the triangle are perpendicular to the radii OF, OD, OE, they will be tangent to the circum- ference (Bk. II. Th. v). Hence, the circle will bo inscribed in the triangle. PROBLEM XX. To inscribe an equilateral triangle m a circle. Through the centre C draw any diam- eter, as ACB. From 5 as a centre, with a radius equal to BC, describe the arc DOE. Then, draw AD, AE, and DE, and DAE will be the required triangle. For, since the chords BD, BE, are each equal to the radius CB, the arcs BD, BE, are each equal to sixty degrees (Bk. II. Th. xix), and the arc DBE to one hundred and twenty degrees; hence, the angle DAE is equal to sixty degrees (Bk. II. Th. viii). Again, since the arc BD is equal to sixty degrees, and the arc BDA equal to one hundred and eighty degrees, it follows that DA will be equal to one hundred and twenty degrees : hence, the angle DEA is equal to sixty degrees, and conse- quently, the third angle ADE, is equal to sixty degrees. BOOK II. 67 Problems. Therefore, tlie tris-ngle ADE is equilateral (Bk, I, Th. vL Cor. 2). PROBLEM XXI. To inscribe a regular hexagon in a circle. Draw any radius, as .AC. Then ap' ply the radius AC around the circum- ference, and it will give the chords AD, /j DE, EF, FG, GH, and HA, which will be the sides of the regular hexagon. For, -^ ^^ ^ the side of a hexagon is equal to the radius (Bk. II. Th. xix). PEOBLEM XXII. To inscribe a square m a given circle. Let ABCD be the given circle. Draw the two diameters AC, BD, at righ angles to each other, and through the points A, B, C and D draw the lines AB, BC, CD, and DA: then will ABCD be the required square. For, the four right angled triangles, AOB, BOC, COD, and DOA are equal, since the sides AO, OB, OC, and OD are equal, being radii of the circle ; and the angles at are equal in each, being right angles : hence, the sides AB, BC, CD, and DA are equal (Bk. I. Th. iv). But each of the angles ABC, BCD, CDA, DAB, is a right angle, being an angle in a semicircle (Bk. II. Th x) : henco, the Igiure ABCD is a square (Bk. I. Def. 48) 68 GEOMETRY. Problems. Sck. If we bisect the arcs AB, BC, 'CD, DA, and join the points, we shall have a regular octagon in- scribed in the circle. If we again 5] bisect the arcs, and join the points of bisection, we shall have a regular polygon of sixteen sides. PROBLEM XXIII. To describe a square about a given circle. I'raw the diameters AB, DE, at _ „ H A r "\ V t ? J 3 / n right angles to each other. Through the extremities A and B draw FAG and HBl parallel to DE, and through E and D, draw FEH and GDI par- allel Xo AB: then will FGIHhe, the required square. For, since ACDG is a parallelogram, the opposite sides are equal (Bk. I. Th. xxiii) : and since the angle at C is a right angle all the other angles are right angles (Bk. I. Th. xxiii. Cor. 1): and as the same may be proved of each of the figures CI, CH and CF, it follows that all the angles, F, G, I, ai d //, are right angles, and that the sides GI, IH, HF, and FG, are equal, each being equal to the diameter of the circle. Hence the figure GIITF is a sauare (Bk. I. Def. 48). GEOMETRY. BOOK III. OF RATIOS AND PROPORTIONS. DEFINITIONS. 1. Ratio is the quotient arising from dividing one quantity by another quantity of the same kind. Thus, if the numbers 3 and 6 hare the same unit, the ratio of 3 to 6 will be expressed by And in general, if A and B represent quantities of the same kind, the ratio of A to B will be expre Bsed by £ A' 2. If there be four numbers, 2, 4, b, 16, laving such values that the second divided by the first is equal to the fourth di- vided by the third, the numbers are said to be in proportion. And in general, if there be four quantities A, B, C, and D, baring sii;h values that B D A=C' then, A is said to have the same ratio to B, that C has to D, or, the ratio of A to B is equal to the ratio of C to D When 70 G E M E T E Y . Of Ratios and Prop ort io ns. four quantities have this relation to each other, they are said to be in proportion. Hence, the proportion of four quantities results from an equahty of their ratios taken two and two. To express that the ratio of ^ to 5 is equal to the ratio of C to -Z>, we write the quantities thus : A :. B :: : D; and read, A is to B, as to B. The quantities which are compared together are called the terms of the proportion. The first and last terms are called the extremes, and the second and third terms, the means. Thus, A and B are the extremes, and B and the means. 3. Of four propoitional quantities, the first and third are called the antecedents, and the second and fourth the conse- quents ; and the last is said to be a fourth proportional to the other three taken in order. Thus, in the last proportion, A and C are the antecedents, and B and D the consequents. 4. Three quantities are in proportion when the first has the same ratio to the second, that the second has to the third ; and then the middle term is said to be a mean proportional betweeu the other two. For example, 3 : 6 : : 6 : 12 ; and 6 is a mean proportional between 3 and 12. 5. Quantities are said to be in proportion by znvcrsion , oi inversely, when the consequents are made the antecedents and the antecedents the consequents. Thus, if we have' the proportion 3 : 6 : : 8 : IS. ■ thu inverse proportion would be 6 .• 3 : : 16 : 8. BOOKIIl. 71 Of Ratios and Proportiors. 6. Quantities are said to be in proportion by altematicn, or alternately, when antecedent is compared with antecedent and consequent with, consequent. "thus, if we have the proportion 3 : 6 : : 8 : 16, the alternate proportion would be 3 : 8 : : 6 : 16. 7. Quantities are said to be in proportion by composition, when the sum of the antecedent and consequent is compared cither with antecedent or consequent. Thus, if we have the proportion 2 : 4 : : 8 T 16, the proportion by composition would be 2+4 : 4 :: 8+16 : 16; that is, 6 : 4 : : 24 : 16. 8. Quantities are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with the antecedent or consequent. Thus, if we have the proportion 3 : 9 : : 12 : 36, tlie proportion by division will be 9—3 : 9 :: 36—12 : 36; that is, 6 : 9 : : 24 : 36. 9. Equimultiples of two or more quantities are the products which arise from multiplying the quantities by the same number. Thus, if we have any two numbers, as 6 and 5, and multiply 72 GEOMETRY. Of Ratio8 and Proportions. them both by any number, as 9, the equimultiples will be 54 and 45 ; for 6x9=54 and 5x9=45. Also, mXA and mxB are equimultiples of A and B, the common multiplier being m. 10. Two variable quantities, A and B, are said to be re- ciprocally proportional, or inversely proportional, when one increases in the same ratio as the other diminishes. When this relation exists, either of them is equal to a constant quantity divided by the other. Thus, if we had any two numbers, as 2 and 4, so related to each other that if we divided one by any number we must multiply the other by the same number, one would increase in the same ratio as the other would diminish, and their product would not be changed. THEOREM I. If four quantities are in proportion, the product of the two ex- tremes will he equal to the product of the two means. If we have the proportion A : B :: C : D we have, by Def. 2, B_D A~C and by clearing the equation of fractions, we have BC=AD Sch. The general principle is verified in the proponion belwoen the numbers 2 : 10 : : 12 : 60 which gives 2x60=10x12=120 BOOKIII. 73 Of Ratios and Proportions. THEOREM II. If four quantities are so related to each other, that the product of two of them is equal to the product of the other two ; then two of them may he made the means, and the other two the extremes of a proportion. Let A, B, C, and D, have such values that BxC=AxD Divide both sides of the equation by A and we haro 4xC=D A Then divide both sides of the last equation by C, and we have B_D A~C hence, by JJef. 2, we have A : B :: C : D. Sch. The general truth may be verified by the numbers 2x18=9x4 which give 2 : 4. : : 9 : 18 THEOREM III. [f ihree quantities are in proportion, the product of the two extremes will he equal to the square cf the middle term. Let us suppose that we have A : B .: B : C Then, by Def, 2, we have B_C A~B and by clearing the equation of its fractions, we have 7 74 GEOMETRY. Of Ratios and Proportiona. Sch. The proposition may be verified by tbe nuinbers 3 : 6 : : 6 : 12 which give 3x12=6x6=36 THEOREM IV. 1/ four quantities are in proportion, tltey mil be tn proportton when taken alternately. Let A : B : : C : D Then, by Def. 2, we have B_D A~C Multiplying both members of this equation by — , we have B C_D A~B and consequently, A : C :: B : D. Sch. The theorem may be verified by the proportion 10 : 15 : : 20 : 30 for, we have, by alternation, 10 : 20 : : 15 : 30. THEOREM V. If there he two sets of proportions, having an antecedent and a consequent in the one, equal to an antecedent and a consequent in the other; then, the remaining terms wiU be proportional. If wo have A : B :. C . D, a.ni A : B : E : F; Hi en wo shall have BOOKlIl 75 Of Ratios and Propor ions. B D B ,F - = - and -=;g Hence, by Ax. 1, we have D_F C~E End consequently, C : D :: E : F. Sch. The proposition may be verified by the following proportions, 2 : 6 : : 8 : 24 and 2 : 6 : : 10 : 30 which give 8 : 24 : : 10 : 30. THEOREM VI. If four quantities are in proportion, they will be in proportion when taken inversely. If we have the proportion A i B :: C : D wo have, by Th. I, AxD=BxC, or BxC=AxD. Hence, we have, by Th. II, B : A :: D : C. Sch. The proposition may be verified by the proportion 7 : 14 :: 8 : 16; which, when taken inversely, gives 14 : 7 : : 16 : 8. THEOREM VII. If fimr quantities are in proportion, they will be in proportion by composition. 76 GEOMETRY. Of Ratios and Proportions. Let US suppose that we have A : B :: C : D wc shall then have AxD=BxC. To each of these equals, add BxD, and we hare \a+B)xD:^{C+D)xB; tnd by separating the factors by Th. II, we have A+B I B :: C+D : D. Sch. The proposition may be verified by the tollowing proportion, 9 : 27 : : 16 : 48. We shall have, by composition, 9+27 : 27 : : 16+48 : 48, that is, 36 : 27 : : 64 : 48, in which the ratio is three fourths. THEOREM VIII. If four quantities are in proportion, they will be in proportion by division. Let us suppose that we have A : B i: C : D; we shall then have AxD-BxC. From each of these equals let us subtract BxD, and »^e have {A-B)xD={C-D)xB; and by separating the factors by Th. II, we have, A-B : B :: C-D : D. Sch The proposition may be verified by the proportion, 24 • 8 : ; 48 : 16. B K I I I . 77 Of Ratios and Proport'ono We have, by division, 24-8 : 8 : : 48-16 : 16; owef to which it is necessary to raise a fixed number, in order te produce the first number. This fixed number is called the base of the system, and maj be any number except 1 : in the common system 10 is assumed as the base. 2. If we form those powers of 10, which are denoted by entire exponents, we shall have 10*' = 1 10^ = 10 , lO' — 1000 lO' = 100 , 10* = 10000, (fee. (fee. From the above table, it is plain, that 0, 1, 2, 3, 4, 7f I I I ' I ' I I ■— f 14. A scale of equal parts is formed by dividing a line of a given length into equal portions. If, for example, the line ab of a given length, say one inch, be divided into any number of equal parts, as 10, the scale thus formed, is called a scale of ten parts to the inch. The line ab, which is divided, is called the unit of the scale. This unit is laid off several times on the left of the divided line, and its extremities marked, 1, 2, 3, ifec. The unit of scales of equal parts, is, in general, either an inch, or an exaot part of an inch. If, for example, ab the unit INTRODUClION. 177 Description of InBtruments. of tlie scale, were half an inch, Ihe scale would be one of 10 parts to half an inch, or of 20 parts to the inch. If it'were required to take from the scale a line equal to two inches and six-tenths, place one foot of the dividers at 2 on the left, and extend the other to .6, which marks the sixth of the small divisions : the dividers will then embrace the required distance. DIAGONAL SCALE OP EttUAL PARTS. Hf c I M ( ; ; 1 p oa 1 / / M M / 08 1 1 1 19 III 07 1 j 1 n 06 1 \ 1 1 1 1 05 M "W n >l i 04 1 M / 1 / 03 / Ml 11 02 1 1 1 n ir OJ 1 M 1 .3.4,5.6. 7.,?, 9 A 15. This scale is thus constructed. Take ab for the unit of the scale, which may be one inch, j, -J- or f of an inch, in length. On ab describe the square abed. Divide the sides ab and dc each into ten equal parts. Draw af and the other nine parallels as in the figure. Produce ba to the left, and lay off the unit of the scale any convenient number of times, and mark the points 1, 2, 3, &o. Then, divide the line ad into ten equal parts, and through the points of division draw parallels to ab as in the figure. Now, the small divisions of the line ab are each one-tenth (.1) of ab ; they are therefore .1 of ad, or .1 of a^' or gh. If we consider the triangle adf, we see that the base df is 178 TRIGONOMETRY DoBcriptiou of Inatrumenta. one-tenth of ad, the unit of the scale. Since the distance froni a to the first horizontal line above ah, is one-tenth of the dis- tance ad, it follows that the distance measured on that line be- tween ad and af is one-tenth of df: but since one-tenth of a tenth is a hundredth, it follows that this distance is one-hun- dredth (.01) of the unit of the scale. A like distance measured on the second line will be two-hundre'dths (.02) of the unit of the scale ; on the third, .03 ; on the fourth, .04, &c. If it were required to take, in the dividers, the unit of the scale, and any number of tenths, place one foot of the diridera at 1, and extend the other to that figure between a and 6 which designates the tenths. If two or more units are required, the dividers must be placed on a point of division further to the left. When units, tenths, and hundredths, are required, place one foot of the dividers where the vertical line through the point which designates the units, intersects the line which designates the hundredths : then, extend the dividers to that line between ad and he which designates the tenths : the distance so deter- mined will be the one required. For example, to take oflF the distance 2.34, we place one foot of the dividers at I, and extend the other to e : and to take off the distance 2.58, we place one foot of the dividers at p and ex- tend the other to q. Remark I. If a line is so long that the whole of it cannot be taken from the scale, it must be divided, and the parts of it t.\ken from the scale in succession. Rbmark II. If a line be given upon the paper, its length tan be found by taking it in the dividers and applying it to the scale. INTEODXJCTION. 179 DoBCTiption of I ii«i trnmeut a. SCALE OF CH0KD3 16. If, with any radius, as AQ, we describe the quadrant QD, and then divide it into 90 equal parts, each part is called a degree. Through (7, and each point of division, let a chord be drawn, and let the lengths of these chords be accurately laid oif on a scale : such a scale is called a scale of chords. In the figure, the chords are drawn for every ten degrees. ■The scale of chords being once constructed, the radius of the circle from which the chords were obtained, is known ; for, the chord marked 60 is always equal to the radius of the circle. A scale of chords is generally laid down on the scales which belong to cases of mathematical instruments, and is marked cho. To lay off, at a given point of a line, with the scale of chords, an angle equal to a given angle. Let AB be the line, and A the given point. Take from the scale the chord of 60 'de- grees, and with this radius, and the point 4 as a centre, describe the arc BC. Then take from the scale 180 TRIGONOMETRY. Description of InstTuments. tlie chord of the given angle, say 30 degrees, and with ttis line as a radius, and ^ as a centre, describe an arc cutting BC in C. Through A and C draw the line AC, and BAC will be the re- quired angle. SEMICIKCULAE PKOTRACTOB. C A S Vl. This instrument is used to lay down, or protract angles. It may also be used to measure angles included between lines already drawn upon paper. It consists of a brass semicircle ABO divided to half degrees. The degrees are numbered from to 180, both ways; that is, from AtoB and from B to A. The divisions, in the figure, are only made to degrees. There is a small notch at the mid- dle of the diameter AB, which indicates the centre of tie pro- tractor. GUNTERS' SCALE. 18. This is a scale of two feet m length, on the faces of which a variety of scales is marked. The face oa which the T R I G Sr M E T R T. 181 Seflnitiona. divisions of inches are made, contains, however, all the scales necessary for laying down lines and angles. These are, the scale of equal parts, the diagonal scale of equal parts, and the scale of chords, all of which have been described. PLANE TRIGONOMETRY. DEFINITIONS AND EXPLANATION OP TABLES. 19. In every plane triangle there are six parts : three sides and three angles. These parts are so related to each other, that when one side and any two other parts are given, the remain- ing parts can be obtained, either by geometrical construction or by trigonometrical computation. 20. Plane Trigonometry explains the methods of computing the unknown parts of a plane triangle, when a sufficient cum- ber of the six parts is given. 21. For the purpose of trigonometrical calculation, the cir- cumference of the circle is supposed to be divided into 360 equal parts, called degrees ; each degree is supposed to be di- vided into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds. Degrees, minutes, and seconds, are designated respectively, 16 182 TRIGONOMETRY Definitions. by the characters ° ' ". For example, ten degrees, eighteen minutes, and fourteen seconds, would be written 10° 18' 14" If two lines be drawn through the centre of the circle, at right angles to each other, they will divide the circumference into four equal parts, of 90° eacK. Every right angle then, as EOA, is measured by an arc of 90° ; every acute angle, as BOA, by au arc less than 90° ; and every obtuse angle, aa FOA, by an arc greater than 90°. 22. The complement of an arc is what remains after subtracting the arc from 90°. Thus, the arc EB is the complement of AB. The sum of an arc and its complement is equal to 90°. 23. The supplement of an arc is what remains after subtracting the arc from 180°. Thus, GF\a. the sup- plement of the arc AEF. The sum of an arc and its sup- plement is equal to 180°. 24. The sine of an arc is the perpendicular let fall from one extremity of the arc on the diameter which passes through the other extremity. Thus, BD is the sine of the arc AB. 25. The cosine of an arc is the part of the diameter inter- cepted between the foot of the sine and centre. Thus, OD is the cosine of the arc AB, 26. The tangent of an arc is the line which touches it at one extremity, and is limited by a line drawn through the other extremity and the centre of the circle. Thus, AO is the tangent of the arc AB. TRIGOXOMETRr. 183 Definitions. 27. The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc, and limited by the tangent passing through the dther extremity. Thus, OC is the secant of the arc AB, 28. The four lines, £D, OD, AC, 00, depend for then values on the arc AB and the radius OA ; they are thus designated : sin AB for BD cos AB for OB tan ^5 for AO sec AB for 00 29. If ABE be equal to a quad- rant, or 90°, then HB will be the complement of AB. Let the lines JET and IB be drawn perpendicular to OK Then, JST, the tangent of UB, is called the cotangent of AB ; IB, the sine of^^, is equal to the cosine oi AB ; OT, the secant of KB, is called the cosecant of AB, In general, if A is any arc or angle, we have, cos A — sin (90°—^) cot A = tan (90°—^) cosec ^ = sec (90°— ^) 30. If we take an arc ABEF, greater than 90°, its Eine will be FH ; OH will be its cosine ; AQ \\s, tangent, and Q its secant. But FR is the sine of the arc GF, which is the supplement of AF, and OE is its cceine : hence, the sine of 184 TRIGOKOMETKT. Definitions. an arc is equal to the sine of its supplement ; and the cosine of an arc it equal to the cosine of its supplement.* Furthermore, AQ \s the tangent of the arc AF, and OQis its secant : GL is the tangent, and OL the secant of the sup- plemental arc GF. But since AQ is equal to GL, and OQ to OL, it follows that, the tangent of an arc is equal to the tangent of its supplement; and the secant of an arc is equal to the secant of its supplement.* Let us suppose, that in a circle of a given radius, the lengths of the sine, cosine, tangent, and cotangent, have been calculated for every minute or second of the quadrant, and arranged in a table ; such a table is called a table of sines and tangents. If the radius of the circle is 1, the table is called a table of natural sines. A table of natural sines, therefore, shows the values of the sines, cosines, tangents and cotangents of all the arcs of a quadrant, divided to minutes or seconds. If the sines, cosines, tangents, and secants are known for arcs less than 90°, those for arcs which are greater can be found from them. For if an arc is less than 90°, its supplement will be greater than 90°, and the values of these lines are the same for an arc and its supplement. Thus, if we know the sine of 20°, we also know the sine of its supplement IGO^ ; for the two are equal to each other. TABLE 01? LOGARITHMIC SINES. 31. In this table are arranged the logarithms of the nume- rical values of the sines, cosines, tangents, and cotangents of all * 'Diese relations arc between the numerical values of the tiigononietiical lines ) the algebraic Bigns, which they have iii the different qu.idranis, iue not considered. TEIGOWOMETRT. 185 Uses of the Tat>leB. the arcs of a quadrant, calculated to a radius of 10,000,000,000. The logaritlim of this radius is 10. lu the first and last hori- zontal lines of each page, are written the degrees whose sines, cosines, &o., are expressed on the page. The vertical columns on the left and right, are columns of minutes. CASE I. To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given arc or angle. 32. If the angle is less than 45°, look for the degrees in the first horizontal line of the different pages : then descend along the column of minutes, on the .left of the page, till you reach the number showing the minutes : then pass along the hori- zontal line till you come into the column designated, sine, cosine, tangent, or cotangent, as the case may be : the numbei BO indicated is the logarithm sought. Thus, on page 37, foi 19° 55' we find, sine 19° 55' 9.532312 cos 19° 55' 9.973215 tan 19° 65' 9.559097 cot 19° 55' 10.440903 33. If the angle is greater than 45°, search for the degrees along the bottom line of the different pages : then, ascend along the column of minutes on the right hand side of the page, till you reach the number expressing the minutes : then pass along the horizontal line into the column designated tang, cot, sine, or cosine, as the case may be: the number so pointed out is the logarithm required. 34. The column designated sine, at the top of the page, is 16* 18B TRIGONOMETRY. Uses of the Tables. designated by cosine at the bottom ; the one designated tang, by cotang, and the one designated cotang, by tang. The angle found by taking the degrees at the top of ths page and the minutes from the first vertical column on the left, is the complement of the angle found by taking the de- grees at the bottom of the page, and the minutes traced up in the right hand column to the same horizontal line. There- fore, sine, at the top of the page, should correspond with cosine, at the bottom ; cosine with sine, tang with cotang, and cotang with tang, as in the tables (Art. 11). If the angle is greater than 90°, we have only to subtract it from 180°, and take the sine, cosine, tangent or cotangent of the remainder. The column of the table next to the column of sines, and on the right of it, is designated by the letter D. This column is calculated in the following manner. Opening the table at any page, as 42, the sine of 24° is found to be 9.609313; that of 24° 01', 9.609597: their dif- ference is 284 ; this being divided by 60, the number of seconds in a minute, gives 4.73, which is entered in the column D. Now, supposing the increase of the logarithmic sine to be proportional to the increase of the arc, and it is nearly so for 60", it follows, that 4.73 is the increase of the sine for 1". Similarly, if the arc were 24° 20' the increase of the sine for 1", would be 4.65. The same remarks are applicable in respect of the column D, after the column cosine, and of the column B, between the tangents and cotangents. The column D between the columns tangents and cotangents, answers to both of these columns. TRIGONOMETRY. 197 Uses of the Tables. Now, if it were required to find tlie logarithmic sine of an arc expressed in degrees, minutes, and seconds, we have only to find the degrees and minutes as before ; then, multiply the corresponding tabular difference by the seconds, and add the pro- duct to the number first found, for the sine of the given arc Tius, if we wish the sine of 40° 26' The sine 40o 26' . . . Tabular difference 2.47 , . Number of seconds 28 . 28". . 9.811952 Product . . 69.16 to be added 69.16 Gives for the sine of 40° 26' 28" 9.812021. The decimal figures at the right are generally omitted in the final result ; but when they exceed five-tenths, the figure on the left of the decimal point is increased by 1 ; this gives the nearest approximate result. The tangent of an arc, in which there are seconds, is found in a manner entirely similar. In regard to the cosine and co- tangent, it must be remembered, that they increase while the arcs decrease, and decrease as the area are increased ; conse- quently, the proportional numbers found for the seconds, must be subtracted, not added. EXAMPLES. 1. To find the cosine of 3° 40' 40" The cosine of 3° 40' ... 9.999110 Tabular difference .13 , . . Number of seconds 40 . Product 5.20 to be subtracted 5.20 Gives for tte cosine of 3° 40' 40" . 9.999105 IS8 TRIGOM'OMETB.T. TJBeB of the Tables. 2. Find the tangent of 31° 28' 31" Ans. 9.884592. 3. Find the cotangent of 87° 57' 59" Ans. 8.550356. CASE 11. To find the degrees, minutes and seconds, answering to any given logarithmic sins, cosine, tangent or cotangent. 35. Search in the table, and in the proper column, and if the number be found, the degrees will be shown either at the top or bottom o*" the page, and the minutes in the side columns, either at the left or right. But, if the number cannot be found in the table, take from the table the degrees and minutes answering to the near- est less logarithm, the logarithm itself, and also the corres- ponding tabular diflference. Subtract the logarithm taken from the table from the given logarithm, annex two ciphers to the remainder, and then divide the remainder by the tabular dif- ference : the quotient will be seconds, and is to be connected with the degrees and minutes before found ; to be added for the sine and tangent, and subtracted for the cosine and co- tangent. EXAMPLES. 1. Find the arc answering to the sine 9.880054 Sine 49° 20', next less in the table 9.879963 Tabular difference . . . 1.81)91.00(50" Hence, the arc 49° 20' 50" corresponds to the giyon sine 0.880054. 2. Find the arc whose cotangent is , 10.008688 cot 44° 26', next less in the table 10.003591 Tabular difference . . . 4.21)97.00(23" TRIGONOMETRY. 189 Theorems. Hence, 44° 26'-23" = 44° 25' 37" is the arc answering to the given cotangent 10.008688. 3. Find the arc answering to tangent 9.9'79110. Ans. 43° 37' 21". 4. Find the arc answering to cosine 9.944599. Ans, 28° 19' 46". 36. "We shall now demonstrate the principal theorems of Plane Trigonometry, THEOREM I. The sides of a plane triangle are proportional to the sines of their opposite angles. Let AB C he a triangle ; then will CB : CA :: sin A : sin B. For, with ^ as a centre, and AD equal to the less side 5 C, as a radius, describe the arc DI: and with B as a centre and the equal radius BC, ■^ EI It F describe the arc CL: now JDE is the sine of the angle A, and CF is the sine of B, to the same radius AJ) or BC. But by similar triangles, AD : BE : : AG : CF. But AD being equal to BC, we have BO : sin ^ : : AC : sin B, or BC : AC : : em A : sin B. By comparing the sides AJB, ^C, in a similar manner, we should find, AB : AC : • &m : sin B. 19C TKIGONOMETRT. Theorems. THEOREM II. In any triangle, the sum, of the two aides containing either angle, is to their difference, as the tangent of half the sum of the two other'angles, to the tangent of half their difference. Let A CB be a triangle : then will AB + AC: AB-AC : : tan |(C + ^) : tan \{0~B). With ^ as a centre, and a radius AG the less of the two given sides, let the semicircle IFQE be de- scribed, meeting AB in /, and BA produced, in E. Then, BE will be the sum of the sides, and BI their difference. Draw C/ and -4i?'. Since CAE is an outward angle of the triangle ACB, it 'iS equal to the sum of the inward angles O and B (Bk. I, Tb. xvi.) But the angle CIE being at the circumference, is half the angle CAE at the centre (Bk. II, Th. viii. Cor 1) ; that is, half the sura of the angles G and B, or equaJ toKC + -S). The angle AFG = AGB, is also equal to ABC + BAF ; therefore, BAF = ACB - ABC. But, IGF= \{BAF) = :^{AGB — ABO), or \{G—B). With / and G as centres, and the common radius IG, let the arcs CD and IG be described, and draw the lines CE and IS perpendicular to IG. The perpendicular GE will pass through E, the extremity of the diameter IE, since the right angle ICE must be inscribed in a semicircle. But GE is the tangent of CIE = ^{G+B); and IH is the langert of IGB — \{G — B), to the common radius GI. TRIGONOMETRY. 191 Theorems. But since the lines CE and IH are parallel, the triangles BHI and BOE are similar, and give the proportion, BE : BI :: CE : IH, or Dy placing for BE and BI, CE and IH, their values, we have AB + AG: AB — AC:: tan \{C+B) : tan \[0 ~ B). THEOREM III, In any plane triangle, if a line is drawn from, the vertical angle perpendicular to the base, dividing it into two segments: then, the whole base, or sum of the segments, is to the sum of the two other sides, as the difference of those sides to the dif- ference of the segments. Let B AC be a triangle, and AB perpendicular to the base ; then will BC: CA + AB:: CA — AB: CD — BB For, Zs' = Ie5' + AO' (Bk. IV, Th. xii) ; and 'AC' = DCT + 'Aff by subtraction 'AG' — A^ = 'CB' — But since the difference of the squares of two lines is equivalent to the rectangle contained by their sum and difference (Davies' Legendre, Bk. IV, Prop, x,) we have, AG'~'aS'={AC-Is-AB) .{AC—AB) and W — BB'={CI>+I)B).{CI) — I)B) therefore, (CB + DB) . {CB—BB)=^{AC+AB) . {AO—AB) hence, CB -\- BB : AC + AB : : AC - ■ AB:CB — BB. 192 TRIGONOMETRY. Theorems. THEOREM IV. In any right-angled plane triangle, radius is to the tan- gent of either of the acute angles, as the side adjacent to the side opposite. Let CAB be the proposed triangle, _5 and denote the radius by R ; then will rr E: tan 0::A0 :AB. ^-^'^^ For, witb any radius as CD describe ^ the arc BH, and draw the tangent D G. From the similar triangles CDG and CAB we have CD -.BG \\ CA:AB; hence, i2 : tan C : : C^ : AB. By describing an arc with £ as a centre, we could show in the same manner that, B : tan B ::AB : AC. THEOREM V. In every right-angled plane triangle, radius is to the cosine of either of the acute angles, as the hypothenuse to the side adjacent. Let ABC be a triangle, right-angled, at B then will E : C0& A:: AC : AB. ^ For, from the point A as a centre, with ^^ any radius as AD, describe the arc DF, which will measure the angle A, and draw DH perpendicular to AB : then will AH be the cosine of A. The triangles ADE and ACB, being similar, we have AD : AE :: AC -.AB: that is, R: COS. A :: AC : AB. JEW TRIGONOMETRY. 1!)3 Applications. Remark. The relations between the. sides and angles of plane triangles, demonstrated in these five theorems, are suf- ficient to solve all the cases of Plane Trigonometry. Of the six parts •which make up a plane triangle, three must be given, and at least one of these a side, before the others can be de- termined. If the three angles are given, it is plain, that an indefi- nite number of similar triangles may be constructed, the angles of which shall be respectively equal to the angles that are given, and therefore, the sides could not be de- termined. Assuming, with this restriction, any three parts of a trian- gle as given, one of the four following cases will always be pre- sented. I. When two angles and a side are given, n. When two sides and an opposite angle are given. III. When two sides and the included angle are given. IV. When the three sides are given. CASE I. When two angles and a side are given. Add the given angles together and subtract their sum from 180 degrees. The remaining parts of the triangle can then be found by Theorem I. EXAMPLES. 1. In a plane triangle ABO, there are given the angle A = 58° 07', the angle B ~ 22' 37', and the side AB = A 408 yards. Required the other parts. 17 1!M TRIGONOMETRY. Applications. GBOMETRICALLT. Draw an indefinite straight line A£, and from the scale of equal parts lay off AB equal to 408. Then at A lay off an angle equal to 68° 01', and at B an angle equal to 22° 37', and draw the lines AO and BC: then will ABC be the tri- angle required. The angle may be measured either with the protractor or the scale of chords (Arts. 16 and IT), and will be found equal to 99° 16'. The sides AC and BO may be measured by re- ferring them to the scale of equal parts (Art. 2). We shall find AC = 158.9 and BC = 351. yards. TRIIiONOMETRICALLT BT LOGARITHMS. To the angle . . A = 58° 07' Add the angle . B = 22° 37' Their sum = 80° 44' taken from . . . 180° 00' leaves C ... 99° 16' which, exceeding 00° we use its supplement 80° 44'. To find the side BC. As sin C 99° 16' ar. comp. . 0.005705 ': sin A 58° 01' : 9.928972 : : AB 408 . 2.610660 : BO 351.024 (after rejecting 10) 2.545337 Remark. The logarithm of the fourth term of a proportion is obtained by adding the logarithm of the second term to that of the third, and subtracting from their sum the logarithm of the first term. But to subtract the first term is the same as TRIGONOMETRY. 195 Application B. to add its aritlimetical complement and reject IC from the sura (Art. 13) : hence, the arithmetical complement of the first term added to the logarithms of the second and third terms, minus ten, will. give tlie logarithm of the fourth term. To find side AO. As sin G 99° 16' ar. comp. 0.005705 : sin B 22" 3Y' • • • 9.584968 : : AB 408 • • • 2.610660 : AC 158.976 • • • 2.201333 2. In a triangle ABC, there are given A = 38° 25', B = 67° 42', and AB = 400 : 'required the remaining parts. Ans. C= 83° 53', 5(7= 249.974, AO = 340.04. CASE II. When two sides and an opposite angle are given. In a plane triangle ABO, there are C" given AC = 216, CB = 117, the "^ angle A = 22" 37', to find the other _^^ parts. GEOMETRICALLT. Draw an indefinite right line ABB' : from any point as A, draw AO making BAO = 22° 37', and make AO = 210. With C as a centre, and a radius equal to 117, the other given side, describe the arc B'B; draw B'C and BC: then will either of the triangles ABC or AB'C, answer all the condi- tions of the question. 196 TEIGONOMETRY. Applications. TRIG ONOMETRIC ALLY. To find the angle B. As BO 117 . ar. comp. . .' T.931814 ^C 216 2.334454 : sin ^ 22° ST' 9.584968 sin B' 45° 13' 55", or ABQ 134° 46' 05" 9.851236 The ambiguity in this, and similar examples, ai-ises in con- sequence of the first proportion being true for either of the angles ABO, or AB'O, which are supplements of each other, and therefore have the same sine (Art. 30). As long as the two triangles exist, the ambiguity will continue. But if tha side CB, opposite the given angle, is greater than AO, the arc BB' will cut the line ABB', on the same side of the point A, in but one point, and then there will be only one triangle an- swering the conditions. If the side GB is equal to the perpendicular Cd, the arc BB' will be tangent to ABB', and in this case also there will be but one triangle. When CB is less than the perpen- dicular Gd, the arc BB' will not intersect the base ABB', and m that case, no triangle can be formed, or it will be impossible to fulfil the conditions of the problem. 2. Given two sides of a triangle 50 and 40 respectively, and the angle opposite the latter equal to 32° : required the re- maining parts of the triangle. Ans. If the angle opposite the side 60 is acute, it is equal to 41° 28' 69" ; the third angle is then equal to 106° 31' 01", and the third side to '72.368. If the angle opposite the side TRIGONOMETRY. 197 Applications. 50 is obtuse, it is equal to 138° 31' 01", tlie tliiid angle to d" 28' 59", and the remaining side to 12.430. CASE III. "Wlien the two sides and tlieir included angle are given. Let ABO be a triangle; AB, BC, the given sides, and B the given angle. Since B is known, we can find the sum of the two other angles : for A+ = 180° — B and 1{A + C)= i(180° - B) "We next find half the difference of the angles A and by Theorem ii., viz. BC + BA:BC-BA: -.tan ^{A + 0) : tanJ(^- C): in which we consider BO greater than BA, and therefore A ia greater than C; since the greater angle must be opposite the greater side. Having found half the difference of A and 0, by adding it to the half sum, ^{A + 0), we obtain the greater angle, and by subtracting it from hsilf the sum, we obtain the less. That is \{A + C) + \[A -0) = A, and ^A+0)-\[A-0)=C. Having found the angles A and O, the third side AG may- be found by the proportion. sin A : sm B :: BO : AO. EXAMPLES. 1. In the triangle ABO, let BO = 540, AB = 450, and the included angle 5 = 80° : required the remaining parts. 17* 198 rEIGOWOMETRT. Applications. GBOMETRICALLT. Draw an indefinite right line BQ and from any point, as B, lay off a distance BG = 540. At B make the angle CBA = 80° : draw BA and make the distance BA = 450 ; draw AC; then will ABO be the required triangle. TRIQONOMETRICALLT. BO + BA = 540 + 450 = 990 ; and BO — BA = 540 — 450 = 90. A+ 0= 180° —B = 180° —80° = 100°, and therefore, l(^ + C) = i(100°) = 50° To find i{A—0). As. BO + BA 990 . ar. comp. . 7.004365 BO—BA 90 ... . 1.954243 tan \{A + 0) 50° . . . . 10.076187 tanl(^— C) 6° 11' . . . 9.034795 Hence, 50° + 6° 11' == 56° 11' = A; and 50° — 6° 11' = 43° 49' = C. To find the third side AO. As sin 43° 49' . ar. comp. . . 0.159672 : sin B 80° ... . . 9.993351 :: AB 450 .... . 2.653213 : AO 640X82 ... . 2.806230 2. Given two sides of a plane triangle, 1686 and 960, and their included angle 128° 04' : required the other parts. Ans. Angles, 33° 34' 39"; 18° 21' 21"; side 2100. TRIGONOMEIRY. 199 Applications. CASE IV. Having given the three sides of a plane triangle, to find the angles. Let fall a perpendicular from the angle opposite the greater side, dividing the given triangle into two right-angled triangles : then find the difl'erence of the segments of the base by Theo- rem iii. Half this difference being added to half the base^ gives the greater segment ; and, being subtracted from half the base, gives the less segment. Then, since the greater segment belongs to the right-angled triangle having the greatest hypo- thenuse, we have the sides and right angle of two right-angled triangles, to find the acute angles. EXAMPLES. 1. The sides of a plane trian- gle being given; viz. BQ = 40, AC = 34 and AB = 25 : required the angles. -B GEOMETRICALLT. With the three given lines as sides construct a triangle as in Bk. IL Prob. xi. Then measure the angles of the Iriangk either with the protractor or scale of chords. TEIGONOMBIRICALLT. As BO : AG + AB : : AC - AB : CD - BD That is, 40 . : 59 _ 40 -f 13.275 Then, 9 69 X 9 And 40 — 13.275 40 — 26.6375 = CD = 13.3625 = BD. — 13.276 '■H)0 rEXGONOMETET. A,p plications. In the triangle DAC, to find the angle BAG. As AC 34 . . ar. comp. . 8.468521 DG 26.6375 .... 1.425493 sin B, 90° 10.000000 sin BAG 51° 34' 40" . . . 9.894014 In the triangle BAB, to find the angle BAB. As AB 25 ar. comp. . 8.602000 BB 13.3025 . . . 1.125887 sin B 90° ... . 10.000000 sin BAB 32° 18' 35" . . . 9.727947 Hence 90° — i)^C= 90° — 61° 34' 40" = 38° 26' 20" = Q and 90° — BAB = 90° — 32° 18' 35" = 57° 41' 26" = B and BAB + BAG = 51° 34' 40" + 32° 18' 35" = 83° 53' 15" = A. 2. In a triangle, in which the sides are 4-, 5 and 6, what are the angles 3 Avs. 41° 24' 35" ; 55° 46' 16" ; and 82° 49' 09". SOLUTION OF KIGHT-ANGLED TRIANGLES. The unknown parts of a rjght-angled triangle may be found by either of the four last cases : or, if two of tlie sides are given, by means of the property that the square of the hypo- thenuse is equivalent to the sum of the squares of the two other sides. Or the parts may be found by Theorems iv. and v. EXAMPLES. 1. In a right-angled triangle BAG, tnere are given the hypothenuse BG — 250, and the base AG = 240: re- C- quired the other parts. TRIGONOMETRY. 201 Applications. To find tlte angle B. As BQ 250 , ar comp, . 7.6020G0 AO 240 ... . 2.380211 sin A 90° ... . 10.000000 sin B 73° 44' 23" . . . 9.982271 But C = 90° — ^ = 90° — 73° 44' 23" = 16° 15' 37" : Or C may be found from the proportion. As CB 250 ar. comp. . 7.602060 2.380211 10.000000 CB 250 £ AO 240 R . cos G 16° 15' 37" 9.982271 To find side AB by Theorem iv. As R ar. comp. . 0.000000 : tan a 16° 15' 37" . . . 9.464889 :: AO 240 .... 2.380211 AB 70.0003 .... 1.845100 2. In a right-angled triangle BAO, there are given AC z= 384, and B = 53° 08' : required the remaining parts. Ans. AB = 287.96 ; B0= 479.979 ; C = 36° 52' DEFINITIONS. 1. A horizontal angle is one whose sides are horizontal ; ita plane is also horizontal. 2. An angle of elevation or depression, has one horizontal side, and the other oblique, but lying directly above or below the first. 202 TKIGOIJ-OMETET. Applications. APPLICATIOlf TO HEIGHTS AND DISTANCES. PKOBLBM I. T0 determine the horizontal distance to a 'point which is inac- cessible by reason of an intervening river. Let be the point. Measure along the bank of the river a hori- zontal base line AB, and select the stations A and B, in such a manner that each can be seen from the other, and the point C from both of them. Then measure the horizontal angles CAB and CBA, with an instrument adapted to that purpose. Let U3 suppose that we have found AB = 600 yards, CAB = 51° 35' and CBA = 64° 51'. The at gle G = 180" — (^ + ^) = To find the distance BG. .570 34'. As sin 51° 34' ar. comp. • 0.073649 sin A 51° 35' . • 9.926431 : : AB 600 ... • 2.778151 BG 600.11 yards. To find the distance AC. " 2.778231 Aa sin C 51° 34' ar. comp. • 0.073G49 ; sin B 64° 51' , 9.956744 • • • • AB 600 . ; . 2.778151 • AG 643.94 )'ards. . • 2.808544 T'K I G N M E T K T. 203 Applioations. PROBLEM ir. Tc determine the altitude of an inaccessible object above a given horisontal plane. riRST METHOD Suppose D to be the inaccessible object, and BC the horizontal plane _,,--'''' from which the altitude is to be n^-'"^ «^-^^* estimated: then, if we suppose DG \/ /' y to be a vertical line, it will repre- \ V'-'''' sent the required distance. \^' Measure any horizontal base line, as BA ; and at the ex- tremities B and A, measure the horizontal angles OBA and CAB. Measure also, the angle of elevation BBC. Then in the triangle CBA there will be known, two angles and the side AB ; the side B C can therefore be determined. Having found BC, we shall have, in the right-angled triangle BBC, the base BC and the angle at the base, to find the per- pendicular BC, which measures the altitude of the point D above the horizontal plane BC. Let us suppose that we have found BA = 780 yards-, the horizontal angle CBA = 41° 24', the horizontal angle CAB — 96° 28', and the angle of eleva- tion BBC=1Q° 4:3'. In the triangle BCA, to find the horizontal distance BC. The angle 5C^ = 7 80° - (41° 24' + 96° 28') = 42° 08'= 0. As sin (7 . . 42° 08' ar. comp. . O.lYSSeg : sin ^ . 96° 28' . . . . 9.99'7228 :: AB . 180 .... 2.892095 BC . 1155.29 .... 3.062692 204 TRIGONOMETBT. Applications. p In the riglit-anglod triangle DBO, to find DC. As R ar. com p. : tan DBG 10° 43' 0.000000 g.2'77043 7C 1155.29 3.062692 DQ 218.64 2.339'735 Remark I. It might, at first, appear that the solution which we have given, requires that the points B and A should be in the same horizontal plane; but it is entirely independent of such a supposition. For, the horizontal distance, which is represented by BA, is the same, whether the station A is on the same level with B, above it, or below it. The horizontal angles CAB and CBA are also the same, so long as the point C is in the verti- cal line D C. Therefore, if the horizontal line through A should cut the vertical line B C, at any point as E, above or below C, AB would still be the horizontal distance between B and A, and AE which is equal to AC, would be the horizontal dis- tance between A and C. If at A, we measure the angle of elevation of the point i), we shall know in the right-angled triangle DAE, the base vl.S', and the angle at the base ; from which the perpendicular DE can be determined. TEIGOIfOMETET. 205 Applications. Let US suppose that we had measured the angle of elevation' DAE, and found it equal to 20° 15'. First: In the triangle BAG, to find AG or its equal AE, As sin Q 42°. 08' ar. comp. • 0.173309 : sin JS 41° 24' . 9.820406 :: AB TSO , 2.892095 AG ■768.9. • 2.885870 In the r ght-angled triangle DAE, to find BE. As R ar. comp. , 0.000000 : tan A 20° 15' , 9.566932 :: AE 768.9 . 2.885870 : DE 283.66 • 2.452802 Now, since Z^C is less than SE, it follows that the station B is above the station A. That is, DE — D(f= 283.66 — 218.64 — 65.02 = EO, which expresses the vertical distance that the station B is ahove the station A. Remark II. It should be remembered, that the vertical dis tanoe which is obtained by the calculation, is estimated from a horizontal line passing through the eye at the time of ob- servation. Hence, the height of the instrument is to be added, in order to obtain the true result. SECOND METHOD. When the nature of the ground will admit of it, measure a base line AB in the direction of the object D. Then mea- sure with the instrument the angles of elevation at A and B. Then, since the outward angle DBO is equal to the sum 18 206 TRIGONOMETKY. Applications. of the ang.es A and ADB, it follows, that the angle ADB is equal to the difference of the angles of de- ration at A and B. Hence, we can find all the parts of the triangle ADB. Having found DB, and knowing the angle DBC, we can find the altitude DO, This method supposes that the stations A and B are on the same horizontal plane ; and therefore can only be used when the line AB is nearly horizontal. Let us suppose that we have measured the base line, and the two angles of elevation, and (AB = 975 yards, A = 15° 36', DBC= 27° 29'; required the altitude DC, First: ADB = DBG- A = 27° 29' - 16° 36' = 11° 53'. In the triangle ADB, to find BD. As sin 2) 11° 53' ar. comp. sin A 15° 36' ... AB 975 ... . DB 1273.3 0.686302 9.429623 2.989005 3.104930 As In the triangle DBG, to find DG. H ar. comp. sin B 'z1° 29 DB 1273.3 DG 587.61 0.000000 9.064163 3.104930 2.769093 TRIGONOMETRY. 207 Applications. PEOBLEM III. To determine the perpendicular distance of an object below a given horizontal plane. Suppose to be directly over the given object, and A the point through which the horizontal plane ^ is supposed to pass. Measure a horizontal base line AB, and at the stations A and B ^^ conceive the two horizontal lines AG, BC, to be drawn. The oblique lines from A and B to the object will be the hypothenusei of two right-angled triangles, of which AC, BC, are the bases. The perpendiculars of these triangles will be the dis- tances from the horizxintal lines AC, BC, to the object. If we turn the triangles about their bases AC, BC, until they become horizontal, the object, in the first case, will fall at C", and in the second at C". Measure the horizontal angles CAB, GBA, and also the angles of depression G'AC, C"BG. Le't us suppose that we have ■ f AB = 672 yards BAG = 12° 29' found i ABC = 39° 20' G'AC =21° 4:9' G"BG= 19° 10' First: In the triangle ABC, the horizontal angle AO.B =a 180° - {A+ B) = 180° - 111° 49' = 68° 11'. 208 TB.IGONOM iSTR'S Applications. To find the horizontal distance AQ. As sin C 68° 11' ar. comp. , . Bin B 39° 20' ... '.'. AB 672 ... 0.032275 9.801973 2.827369 • • AC 458.79 • 2.661617 To find the horizontal distance BG. As sin G 68° 11' . ar. comp. . 0.032276 sin A 72° 29' ... 9.979380 i : AB 672 .... 2.827369 BO 690.28 .... 2.839024 In the triangle CAC, to find CG: As B ar. comp. . 0.000000 tan C'AQ 27° 49' . . 9.722315 ; : AG 458.79 . . ; . 2.661617 GG' 242.06 . . 2.383932 In the triangle CBG", to find GG" As R . ar. comp. . . 0.000000 : tan G"BG 19° 10' . . . . 9.541061 .: BG 690.28 .... 2.839024 : GG" 239.93 .... 2.380085 TEiaONOMETRT, 209 Applications. Hence also, CQ' — CC" = 242.06 — 239.93 = 2.13 yards wliich is the height of the station A above station B. PKOBLBMS. 1. Wanting to know the distance between two inaccessible objects, which lie in a direct line from the bottom of a towei of 120 feet in height, the angles of depression are measjired, and are found to be, of the nearer 57°, of the more remote 25° 30' : required the distance between them. Ans. 173.656 feet. 2. In order to iind the distance between two trees A and B, which could not be directly measured because of a pool which occupied the intermediate space, the dis- tances of a third point C from each of them were measured, and also the included angle A CB : it was found that CB = 6 72 yards C/4 = 588 yards AOB = 55° 40' ; required the distance AB, Ans. 592.96T yards. 3. Being on a horizontal plane, and wanting to ascertain the height of a tower, standing on the top of an inaccessible hill, there were measured, the angle of elevation of the top of the hill 40°, and of the top of the tower 61° ; then mea- suring in a direct line 180 feet farther from the hill, the angle of elevation of the top of the tower was 33° 45' ; required the height of the tower. Ans. 83.998 feet. 18* 210 TRIGONOMETRY. Applications. 4. Wanting to know the horizon- tal distance between two inaccessi- ble objects E and W, th« following measurements were made, ( AB= 636 yards BAW=iQ'' 16' VIZ. ^ WAE = 51° 40' ABM! = 42" 22' £!BW= 71° 07' required the distance £!W. Ans. 939.527 yards. 5. Wanting to know the aorizontal distance between two inaccessible objects A and B, '"■ and not finding any station from which both of them could be seen, two points O and D, were chosen, at a distance from 3ach other, equal to 200 yards ; from the former of these points A could be seen, and from the latter £, and at each of the :3oints C and D a staff was set up. From C a distance OF was measured, not in the direction DO, equal to 200 yards, and from I) a distance DU equal to 200 yards, and the follow- big angles taken, fAFO = 83° 00' BDH = 54° 30' AGD= 53° 30' BDO = 156° 25' ACF— 54° 31' BED — 88° 30' Ans. AB = 345.467 yards. APPLICATIONS OF GEOMETRY. MENSURATION OF SURFACES. DEFINITIONS. 1 The area of any figure has already been defined to be the measure of its surface (Bk. IV. Def. 7). This measure is merely the number of squares which the figure contains. A square whose side is one inch, one foot, or one yard, &c., is called the measuring unit ; and the area or contents of a figure is expressed by the number of such squares which the figure contains. 2. In the questions involving decimals, the decimals are generally carried to four places, and then taken to the nearest figure. That is, if the fifth decimal figure is 5, or greater than 5, the fourth figure is increased by one. 3. Surveyors, in measuring land, generally use a chain called Gunter's chain. This chain is four rods, or 66 feet in length, and is divided into 100 links. 4. An acre is a surface equal in extent to 10 square chains; that is, equal to a rectangle of which one side is ten chains and the other side one chain. One quarter of an acre, is called a rood. Since the chain is 4 rods in length, 1 square chain contains 16 square rods ; and therefore, an acre, wriich is 10 square chains, contains 160 square rods, and a rood contains 40 square rods. The square r ids are called perches. 212 APPLICATIONS ' Mensuration of Surfaces. 5. Land is generally computed in acres, roods, and perches which are respectively designated by the letters A, R, P. When the linear dimensions of a survey are chains or links the area will be expressed in square chains or square links, and it is necessary to form a rule for reducing this area to acres, roods, and perches. For this purpose, let us form llie following TABLE. 1 square chain=100x 100 = 10000 square links. 1 acre=10 square chains = 100000 square links. 1 acre =4 roods = 160 perches. 1 square mile = 6400 square chains = 640 acres. 6. Now, when the linear dimensions are links, the area will be expressed in square links, and may be reduced to acres by dividing by 100000, the number of square links in an acre : that is, by pointing off five decimal places from the right hand. If the decimal part be then multiplied by 4, and five places ol decimals pointed off from the right hand, the figures to the left hand will express the roods. If the decimal part of tliis result be now multiplied by 40, and five places for decimals pointed off, as before, the figures to the left will express the perches. If one of the dimensions be in links, and the other in chains, the chains may be reduced to links by annexing twp ciphers or, the multiplication may be made without annexing the ci- phers, and the product reduced to acres and decimals of an acre, by pointing off three decimal places at the right hand. When both dimensions are in chains, the product is re- OF GEOMETRY. 213 Mensuration of Surfaces. (luced to acres by dividing by 10, or pointing off one decimal place. From which we conclude : that, I. If links he multiplied hy Units, the product is reduced to a/rei hy pointing off Jive decimal places from the right hand. II. If chains he multiplied hy links, the product is reduced to acres by pointing off three decimal places from the right hand. III. If chains he multiplied hy cjiains, the product is reduced to acres hy pointing off one decimal place from the right hand. 7. Since there are 16.5 feet in a rod, a square rod is equal to 16.5 X 16.5=372.25 square feet. If the last number be multiplied by 160, we shall have 272.25 X 160=43560 the square feet in an acre. Since there are 9 square feet in a square yard, if the last number be divided by 9, we obtain 4840=the number of square yards in an acre. PROBLEM I. To find the area of a square, a rectangle, a rhombus, or a parallelogram. RULE. Multiply the hase hy the perpendicular height and the product will he the area (Bk. IV. Th. viii). EXAMPLES. 1. Required the area of the square A B CD, each of whose sides is 36 feet D C A i 214 APPLICATIONS Mensuration of Surfaces. We multiply two sides of he square together, and the product is the area in square feet. Operation. 36x36=1296 ;?,/. fi. 2. How many acres, roods, and perches, in a square whose side is 35.35 chains? Ans. 124 ^. 1 ii. 1 P. 3. What is the area of a square whose side is 8 feet 4 inches ? ' Ans. 69 ft. 5' 4". 4. Wliat is the contents of a square field whose side is 46 rods? Ans. 13 ^. OiJ. 36 P. 5. What is the area of a square whose side is 4769 yards 7 Ans. 22743361 sq. yds 6. What is the area of the parallelo- gram ABCD, of which the base AD is 64 feet, and altitude DE, 36 feet ? D A E We multiply the base 64, by the perpendicular height 36, and the product is the re- qiiired area. Operation. 64x36=2304*9. /<. 7. What is the area of a parallelogram whose base is 12.85 yards, and altitude 8.5 ? Ans. 104,125 sq. yds. 8. What is the area of a parallelogram whose base is S.IS chains, and altitude 6 chains ? ' Ans. 5 A. \ R. P. 9. What is the area of a parallelogram whose base is 7 fee) 9 inches, and altitude 3 feet 6 inches ? Ans. 27 sq.ft. 1'6'. OF GEOMETRY. 215 Mensuration of Surfaces. ]0. To find the area of a ror.tangle ABCD, of which the base AB=A5 yards, and the altitude AD=::15 yards. Here Ave simply multiply the base by the altitude, and the product is the area. D ~B Operation 45x15 = 675 sq. yds. 11. What is the area of a rectangle whose base is 14 feet 6 inches, and breadth 4 feet 9 inches 1 Ans. 68 sq.ft. 10' 6". 12. Find the area of a rectangular board whose length is 1 12 feet, and breadth 9 inches. Ans. 84 sq. ft. 13. Required the area of a rhombus whose base is 10.51 and breadth 4.28 chains. Ans. 4 A. 1 R. 39.7 P+. 14. Required the area of a rectangle whose base is 13 feet 6 inches, and altitude 9 feet 3 inches. Ans-. 115 sq. ft. 7' 6" PROBLEM II. To find the area of a triangle, when the base and altitude are known. RULE. I. Multiply the base by tlie altitude, and half the product will he the area. II. Multiply the base by half the altitude and the product mill be the area (Bk. IV. Th. ix). EXAMPLES. 1 . Required the area of the triangle ABC, whose base AB is 10,75 foet, and altitude 7,25 feet. 15 216 AVPLICATIONS Mensuration of Surfaces. We first multiply the base by the altitude, and then di- vide Ihe product by 2. Operation. 10,75X7,25=77,9375 and 77,9375^2=38,96875 =:area. 2. What is the area of a triangle whose base is 18 feet 4 inches, and altitude 11 feet 10 inches? Ans. 108 sq. ft. 5' 8". 3. What is the area of a triangle whose base is 12.25 chains, and altitude 8.5 chains 1 Ans. 5 A. OR. 33 P. 4. What is the area of a triangle whose base is 20 feel, and altitude 10.25 feet. Ans. 102.5 sq. ft. 5. Find the area of a triangle whose base is 625 and alti- tude 520 feet. Ans. 162500 sq. ft. 6. Find the number of square yards in a triangle whose oase is 40 and altitude 30 feet. Ans. 66f sq. yds. 7. What is the area of a triangle whose base is 72.7 yards, and altitude 36.5 yards ? Ans. 1326,775 sq. yds PROBLEM III. To find the area of a triangle when the three sides are knov/n. RULE, I. Add tlie three sides together and take half their sum. II. From this half sum take each side separately. III. Multiply together the half sum and each of the thief remainders, and then extract the square root of the product, U)hick will he the required ana. OF GE ME r R Y. 217 Mensuration of Surfaces. " EXAMPLES. 1. Find the area of a triangle whose sides are 20, 30, aiitl 40 rods. 20 45 45 45 30 20 30 40 40 25 1st rem. 15 2d rem. 5 3d rem. 2)90 45 half sum i> Then, to obtain the product, we have 45X25X15X5 = 84375; from which we find area= •/84375 =290,4737 perches. 2. How many square yards of plastering are there in a tri- angle, whose sides are 30, 40, and 50 feet? Ans. 66|. 3. The sides of a triangular field are 49 chains, 50.25 chains, and 25.69 : what is its area? Ans. 61 A. 1 R. 39,68 P. 4. What is the area of an isosceles triangle, whose base is 20, and each of the equal sides 15 1 Ans. Ill 803. 5. How many acres are there in a triangle whose three sides are 380, 420 and 765 yards. Ans. 9 ^. ii. 38 P. 6. How many square yards in a triangle whose sides are 13, 14, and 15 feet. Ans. 9^. 7 What is the area of an equilateral triangle whose side is 25 feet ? -Ans. 270.6329 sq. ft. 8. What is the area of a triongle whose sides are 24, 36, and 48 yards 1 Ans 418.282 sq. yds. 1 17 ^18 APPLICATIONS M ensuration of Surfacea. __^ PROBLEM IV. To find tl'.e liypothenuse of a right angled triangle wlujn the base and perpendicular are known. RULE. I. Square each of the sides separately. II. Add the squares together. III. Extract the square root of the sum, which will be the hy- oothenuse of the triangle (Bk. IV. Th. xii). EXAMPLES. 1. In the right angled triangle ABC, we have, AB=30 feet, BC=40 feet, to find AC. We first square each side, and then take the sum, of which we extract the square root, which mves A Operation. 30*= 900 40'=1600 sum=2500 ^C=-v/2500=50 feet. 2. The wall of a building, on the brink of a river, is 120 feet high, and the breadth of the river 70 yards : what is tho length of a line which would reach from the top of the wall to tlie opposite edge of the river? Ans. 241.86 ft. 3. The side roofs of a house of which the eaves are of tho same height, form a right angle at the top. Now, the length of tho rafters on one side is 10 feet, and on the other 14 feet • what is the breadth of the house ? Ans. 1 7.204 ft. 4. What would be the width of the house, in the last ex- ample, if the rafters on each side were 10 feet? Ans. 14.142 ft. OF liEOMETR\ 219 Mensuration of Surfaces. 5. What would be the width, if the rafters on each side were 1 4 feot ? Ans. 19.7989 fi. PROBLEM V. When the hypothenuse and one side o[ a right angled tii ande are known, to find the other side Square the hypothenuse and also the other given side, and take their difference : extract the square root of this difference, and the result will he the required side (Bk. IV. Th. xii. Cor.). EXAMPLES. 1. In the right angled triangle ^5 C, there are given AC=50 feet, and AB—AQ feet, required the side EC. We first square the hypoth- enuse and the other side, after wJiich we take the difl^erence, and then extract the square root, which gives BC=-/900=30feet. 2 The height of a precipice on the brink of a river is 103 feet, and a line of 320 feet in length will just reach from the top of it to the opposite bank : required the breadth of the river. Ans. 302.9703 ft. 3. The h)rpothenuse oi a triangle is 53 yards, and tne per pendiciilar 45 yards : what is the base 1 Ans. 28 yds. 4. A ladder 60 feet iti length, will reach to a window 40 Operation 50^=2500 40^=1600 Diff.= 900 •SZO APPLICATIONS Mensur ation of Surf ac es. feet from the ground on one side of the street, and by turning it over to the other side, it will reach a window 50 feet from tho ground : required the breadth of the street. Ans. 77.8875 /r. PROBLEM VI. To find the area of a trapezoid. Multiply the sum of the parallel sides by the perpendicular diitance between them, and then divide the product by two : the quctient will be the area (Bk. IV. Th. x). EXAMPLES. B 1 . Required the area of the trapezoid / |\ ABCD, having given /- L \ A. E B AB=321.51 feet, DC=214.24 feet, and C£=171.16 feet Operation. We first find the sum of the sides, and then multiply it by the perpendicular height, after which, we divide the product bv 2, for the area. 321.51+214.24=535.75= simi of parallel sides. Then, 535.75 X 171.16=91698.97 , 91698.97 and, =45849.485 1 =the area. 2. What is the area of a trapezoid, the parallel sides of wliich, are 12.41 and 8.22 chains and the perpendicular dis- tance between them 5.15 chains ? Ans. 5 A.l R. 9.956 P. 3. Required the area of a trapezoid whose parallel sides OF GEOMETRY. 221 Mensuration of Surfaces. arc 25 feet 6 inches, and 18 feet 9 inches, and the perpen- dicular distance between them 10 feet and 5 inches. Ans. 230 sq. ft. 5' 7". 4. Required the area of a trapezoid whose parallel sides are 20.5 and 12.25, and the perpendicular distance' between them 10.75 yards. Ans. 176.03125 sq. ijds. 5. What is the area of a trapezoid whose parallel sides are 7.50 chains, and 12.25 chains, and the perpendicular height 15.40 chains 1 Ans. 15 A. R. 33.2 P PROBLEM VII. To find the area of a quadrilateral. RULE. Measure the four sides of the quadrilateral, and also one of the diagonals : the quadrilateral will thus be divided into two trian- gles, in both of which all the sides will be known. Then, find the areas of the triangles separately, and their sum will hs tht area of the quadrilateral. EXAMrLES. 1. Suppose that we have meas- ured the sides and diagonal A C, of the quadrilateral ABCD, and found ^5=40.05 chains; 00=29.87 chains, £0=26.27 chains AD =37.07 chains, and AC=55 chains : required the area of the quadrUatsral. ,„» Ans. 101 A. 1 R 15 P 19* 222 APPLICATIONS Mens I. ration of Surfac es ■ Remark. — Instead of measuring n Ale four sides of the quadrilateral, ^/ j n. we may let fall the perpendicu- ^^ i i X. lars Bb, Dg, on the diagonal AC. ^^^ ^1 /^^ Die area of the triangles may then ^^^^'/''^ be determined by measuring these -B perpendiculars and diagonal AC. The pendiculars are,Z)g -::i 18.95 chains, and Bb =17. 92 chains. 2. Required the area of a quadrilateral whose diagonal is C0.5, and two perpendiculars 24.5, and 30.1 feet. Ans. 2197.65 sq.ft. 3. What is the area of a quadrilateral whose diagonal is 108 feet 6 inches, and the perpendiculars 5C feet 3 inches, and 60 feet 9 inches ? Ans. 6347 sq.ft. 3'. 4. How many square yards of paving in a quadrilateral whose diagonal is 65 feet, and the two perpendiculars 28, and 33^ feet ? Ans. 222,V sq. yds. 5. Required the area of a quadrilateral whose diagonal is 42 feet, and the two perpendiculars 18, and 16 feet. Atis. 714 sq. ft. 6. What is the area of a quadrilateral in which the diago> nal is 320,75 chains, and the two perpendiculars 69.73 chains, and 130.27 chains ? Ans. 3207 A. 2 R. PROBLEM Till. To find the area of a regular polygon. RULE. Multiphj half the perimeter of the figure by the perpendicular let fall from the centre on one of the sides, and the product inH hr. the area (Bk. IV. Th. xxvi) OF GEUMETRY. 223 Mensuration of Surfaces. EXAMPLES. 1. Required the area of the regular pentagon ABODE, each of whose sides AB, BC, &c., is 25 feet, and the perpendicular OP, 17.2 feet. We first multiply one side by the number of sides and divide the product by 2 : this gives half the perimeter which we multiply by the perpen- dicular for the area. 25 X *i =62-5 =half the perim- eter. Then, 62.5x17.2 = 1075 sq. ft. z=the area. 2. The side of a regular pentagon is 20 yards, and the j)er- pendicular from the centre on one of the sides 13,76382 ; re- quired the area. Ans. 688.191 sq. yds. 3. The side of a regular hexagon is 14, and the perpen dicular from the csntre on one of the sides 12,1243556: re quired the area. Ans. 509.2239352 sq.ft. 4. Required the area of a regular hexagon whose side is 14 6, and perpendicular from the centre 12.64 feet. Ans. 553.632 sq ft. 5. Required the area of a heptagon whose side is 19,3^ Rnd perpendicular 20 feet. Ans. 1356.6 sq. ft. TliC following table shows the areas of the ten regulai 224 APPLICATIONS Mensuration of Surfaces. polygons when the side of each is equal to 1 : it also shows the length of the radius of the inscribed circle. Number of sides. Names. Areas. Radius of inscribed circle. 3 Triangle, 0.4330137 0.2886751 4 Square, i.ooooqoo 0.5000000 5 Pentagon, 1.7204774 0.6881910 6 Hexagon, 2.5980763 0.8660354 7 Heptagon, 3.6339124 1.0383617 8 Octagon, 4.8284371 1.2071068 9 Nonagon, 6.1818342 1.3737387 10 Decagon, 7.6942088 1.5388418 11 Undecagon, 9.3656404 1.2038437 , 12 ~ Dodecagon, 11.1961534 1.8660354 Now, since the areas of similar polygons are to each other as the squares described on their homologous sides (Bk. IV Th. xx), we have 1* : tabular area : : any side squared : area. Hence, to find the area of a regular polygon, we have the following I Square the side of ilie polygon. 11. Multiply the square so found, hy the tabular area set oppO' site the polygon of the same number of sides, and the product will be the area. EXAMPLES. 1. What is the area of a regular hexagon whose side is 201 20''=:400 and tabular area=2,5980763. Ilonce, 2.5980762 X 400= 1039.23048=the area. OF GEOMtTRX. 225 Mensuration of Surfaces. 2. What is the area of a pentagon whose side is 25 ? Ans. 1075.298375. I 3. What ia the area of a heptagon whose side is 30 feet '! Ans. 3370.52116. 4. What is the area of an octagon whose side is 10 feet ? Ans. 482.84271 sq. ft 5. The side of a nonagon is 50 : what is its area? Ans. 15454.5605 6. The side of an undecagon is 20 : what is its area? Ans. 3746.25616. 7. The side of a dodecagon is 40 : what is its area ? Ans. 17913.84384 PROBLEM IX. To find the area of a long and in-egular figure, bounded on one side by a straight line. RULE. I.' Divide the right line or base into any number of equa. varts, and measure the breadth of the figure at the points of di- vision, and also at the extremities of the base. II. Add together the intermediate breadths, and half the sum of the extreme ones. III. Multiply this sum by the base line, and divide the product by the number of equal pans of the base. EXAMPLES. 1 . The breadths of an irregu- ^^ lar figure, at five equidistant =16.4578=chord of the half arc. 16.4578x8-30 „. „„^^ .„„ = 33.8874 rz: arc ADB. 20 e30 APPLICATIONS __ Mensuration of Surfaces 2 What is the length of an arc the chord of which is 24 foot, and the radius of the circle 20 feet 1 Ans. 25.7309 ft. 3. The chord of an arc is 16 and the diameter of the circle 20 : what is the length of the arc ? Ans. 18.5178. 4. The chord of an arc is 50, and the chord of half the arc is 27 : what is the length of the arc ? Ans. 55 j. PROBLEM xiv. To find the area of a circle when the diameter and circtmi- ference are both known. RULE. Multiply the circumference hy hdf the radius and the prodnet will he the area (Bk. IV. Th. xxvii). EXAMPLES. 1. What is the area of a circle whose diameter is 10, and circumference 31.416? If the diameter be 10, the Operation. 31.416x2^ = 78.54; radius is 5, and half the ra- dius is 25: hence, the cir- cumference multiplied by 2^ which is the area, givos the area. 2. Find the area of a circle whose diameter is 7; and cir- cumference 21.9912 yards. Ans. 38.4846 yds. 3. How many square yards in a circle whose diameter is 3j feet, and circiunferenco 10.9956. Ans. 1.069016.' 4. What is the area of a circle whose diameter is 100, and circumference 314.16? An j. 7854 OF GEOMETRY. 231 Mensura tion of Surfaces. 5. What is the area of a circle whose diameter is 1, and ciicTimference 3.1416. Arts. 0.7854. 6. What is the area of a circle whose diameter is 40, and circumference 131.94721 Ans. 1319.472. PROBLEM xy. To find the area of a circle when the diameter only is known. RULft. Square ihe diameter, and then multiply by the decimal .7854 EXAMPLES. What is the area of a circle whose diameter is 5 ? We square the diameter, which gives us 25, and we then multiply this number and the decimal .7854 to- gether. Operation. .7854 5^= 25 39270 15708 area= 19.6350 2. What is the area of a circle whose diameter is 7 ? Ans. 38.4846. 3. What is the area of a circle whose diameter is 4,5 1 Ans. 15.90435. 4. What is the number of square yards in a circle whoso diameter is \\ yards ? Ans. 1.069016. 5. What is tlie area of a circle whose diameter is 8.75 feet? Ans. 60.1322 sq.ft. PROBLEM XVI. To find the area of a circle when the circumference only is known. 232 APPLICATIONS M ensuration of Surfa cea. RULE. Multiply the square of the circumference by the decimal .07958, and the product will he the area very nearly. EXAMPLES. 1. What is the area of a circle whose circumference is 3.1416? We first square the cir- cumference, and then multi- ply by the decimal .07958. Operation. 3.1416 =9,86965056 ,07958 area =.7854+ 2. What is the area of a circle whose circumference is £ l? Ans. 659.00198. 3. Suppose a wheel turns twice in tracking I65 feet, and that it turns just 200 times in going round a circular bowling- green : what'is the area in acres, roods, and perches ? Ans. iA.^R. 35.8 V. 4. How many square feet are there in a circle whose cir- cumference is 10.9956 yards? Ans. 86.5933. 5. How many perches are there in a circle whose circum- ference is 7 miles ? ^n*. 399300.608. PROBLEM XTII. Having given a circle, to find a square which shall have an equal area. RULE. I. The diameter X.S862:= side of an equivalent square II. The circumference X, 2821= side of an equivalent square. OF G E M E 'J R Y. 203 Mensur ation of Surfav-.es. EXAMPLES. 1. The diameter of a circle is 100 : what is the side of a B pare of equal area ? Ans. 88.C2. 2. The diameter of a circular fishpond is 20 feet, wliat would be the side of a square fishpond of an equal area 1 Ans. 17.724 ft. 3. A man has a circular meadow of which the diameter is 875 yards, and wishes to exchange it for a square one of equal size : what must be the side of the square ? Ans. 775.425. 4. The circumference of a circle is 200 : what is the side of a square of an equal area 1 Ans. 56.42. 5. The circumference of a round fishpond is 400 yards • what is the side of a square pond of equal area ? Ans. 112.84. 6. The circumference of a circular bowling-green is 412 yards : what is the side of a square one of equal area ? Ans. 116.2252 yds. 7. The circumference of a circular walk is 625 : what is the side of a square containing the same area 1 Ans. 176.3125. PROBLEM XVIII. Ilaring given the diameter or circumference of a circle, lo find the side of the inscribed square. RULE. I , The diameter X .7071 =:side of the inscribed square. II. The circumference X .2251 —side of the inscribed square, 20* 231 APPLICATIONS Mensuration of Surfaces. EXAMPLES. 1. The diameter AB of a circle is 400 : what is the value of AC, the side of the inscribed square ? Here, ,7071 X 400 =282.8400=^0. 2. The diameter of a circle is 412 feet: what is the side of the inscribed square ? Ans. 291.3252 ft. 3. If the diameter of a circle be 600 what is the side of the inscribed square ? Atis. 424.26. 4. The circumference of a circle is 312 feet: what is the side of the inscribed square ? Ans. 70.2312 ft. 5. The circumference of a circle is 819 yards : what is the side of the inscribed square ? Ans. 184.3569 yds. 6. The circumference of a circle is 715 : what is the side of the inscribed square 1 Ans. i 60.9465. 7. The circumference of a circular walk is 625 : what is the side of an inscribed square? Ans. 140.6875. PROBLEM \ix. To find the area of a circular sector. RULE. I. Find the length of the arc by Prollem XII. It, Multiply the arc by one half the radius, and the product will be the area. O F G E O M E T R Y . 23S Mensuration of Surfaces. GZAMFLES. 1. What is the area of the circular sector ACB, the arc AB containing 18°, and the radius CA being equal to 3 feet. J< First, .01745 X IS x3 = .94230=length AB. Then, .94230 X li=1.41345=area 2. What is the area of a sector of a circle in which the ra- dius is 20 and the arc one of 22 degrees 1 Ans. 76.7800. 3. Required the area of a sector whose radius is 25 and the arc of 147° 29'. Ans. 804.2448. 4. Required the area of a semicircle in which the radius is 13. Ans. 2G5.4143. 5. What is the area of a circular sector when the length of the arc is 650 feet and the radius 325 ? Ans. 105625 sq. ft. PROBLEM XX. To find the area of a segment of a circle. RULE. I. Find the area of the sector having the same arc wi'h the segment, hij the last Problem. II. Find the area of the triangle formed by the chord of the segment and the two radii through its extremities. III. If the segment is greater than the semicircle, add the ttca areas together; but if it is less, subtract them, and the result in eithei case, will be the area required. 236 APPLICATIC NS Mensuration of Surfaces. EXAMPLES. ] . What i3 the area of the seg- ment ADB, the chord ^5=24 feet and CAz=2Q feet. First, CP=-v/cI*-^P* = -/400— 144=16 Then, PD= CD- CP=20-16=4. And, ^I>=-v/a?+PZ)*=v'144+ 16 = 12,64911 . .lMl^H21±Z?i^25,7309. then. arc ADB-- Arc ADB=25,7309 half radius = 10 area sector ADBC=257,3090 area C^5=192 ^P=13 CP= 16 area C^5=193 65,309 =area of segment ADB 2. Find the area of the segment AFB, knowing the following lines, viz: AB=20.5; FP— 17.17; AF =20; FG=n-5; and 0^ = 11.64. Arc ^gp^^-gx8-^P 11.5x8-20 . 3 3 ~ and sector ^GP5C=24 X 11.64=279.36 : bul CP=PP—AC=17.17— 11.64=5.53: „, .^„ ABxCP 20.5x5.53 Then, area^CB= ^= =56.0825 OF GEOMETRY. 237 Mensuration o f Surfaces. Then, area of sector 4^50=279.36 do. of triangle ABC= 56.6825 gives area of segnient AFB=336.0i25 3 What is the area of a segment; the radius of tho circle being 10 and the chord of the arc 13 yards 1 Ans. 16.324 sq. yds. 4. Required the area of the segment of a circle whose chord is 16, and the diameter of the circle 20. Ans. 44.5903. 5. What is the area of a segment whose arc is a quadrant, the diameter of the circle being 18 ? Ans. 63.6174. 6. Tho diameter of a circle is 100, and the chord of the sestraent 60 : what is the area of the segment ? Ans. 408, nearly. PROBLEM XXI. To find the area of an ellipse. Multiply the two axes together, and their product by the decimal ,7854, and the result will be the required area. EXAMPLES. 1 . Required the area of an ellipse, whose transverse axis .4.5=70 feet, and the conjugate axis Z)-E= 50 feet. 45 Xi?-E=70x 50=3500: Then, .7854 X 3500 = 2748.9 = area. 2. Required the area of an ellipse whose axes are 24 and 13. Ans. 339.2928 2118 APPLICATIONS Mensuration of Surfaces. 3. V\ hat is the area of an ellipse whose axes are 80 and 60 1. Ans. 3709.92. 4. What is the area of an ellipse whose axes are 50 and 451 Ans. 1707.13. FEOBLEM XXII. To find the area of a circular ring : that is, ths area in- cluded between the circumferences of two circles, having a common centre. RULE. I. Square the diameter of each ring, and subtract the square of the less from that of the greater. II. Multiply the difference of the squares by the decimal ,7854, and the product will be the area. EXAMPLES. 1. In the concentric circles having the common centre C, we have AB = \0 yds., and DE=G yards: what is the area of the space in- cluded between them ? JS.4*=ro'=100 DE ^= 6^= 36 Diflerence=64 Then, 63x.7854=50.2656=area. 2. What ia the area of the ring when the diameters of tlie circle are 20 and 10 ' Ans. 235.62. OF GEOMETRY. 2;J9 Mensuration of Solids. 3. If th,e diameters are 20 and 1-5, wliat will be *lie area in- cluded between the circumferences ? Ans. 137.445. 4. If the diameters are 16 and 10, what wiU be the area in- cluded between the circumferences ? Ans. 122.5224. 5 Two diameters are 21.75 and 9.5; required the area of the circidar ring. Ans. 300.6609. 6. If the two diameters are 4 and G, what is the area of the ring? j1»s. 15.708 [ENSURATION OF SOLIDS. DEFINITIONS. The mensuration of solids is divided into two parts. 1st, The mensuration of the surfaces of solids : and 2d, The mensuration of their solidities. We have already seen that the imit of measure for plane sturfaces, is a square whose side is the unit of length (Bk. IV Def. 7). 2. A curve line which is expressed by numbers is also re- ferred to an unit of length, and its numerical value is the num- ber of times which the line contains the unit. If then, we suppose the linear unit to be reduced to a straight line, and a square constructed on this line, tms square will be the unit of measure for curved surfaces. 3. The unit of solidity is a cube, whose edge is the unit in which the linear dimensions of the solid are expressed ; and 240 APPLICATIONS Mensuration of Solids. the face of this cube is the superficial unit in which the sur- face of the solid is estimated (Bk. VI. Th. xiii. Sch). 4. The following is a table of solid measure. 1 cubic foot =1728 cubic inches. 1 cubic yard =27 1 cubic rod =4492J 1 ale gallon =282 1 wine gallon=231 cubic feet, cubic feet, cubic inches, cubic inches. 1 bushel =2150,42 cubic inches. PROBLEM I. To find the surface of a right prism. Multiply the perimeter of the base hy the altitude and the pro- duct will be the convex surface : and to this add the area of ths bases, when the entire surface is required (Bk. VI. Th. i). EXAMPLES 1. Find the entu-e surface of the regular prism whose base is the reg- ular polygon ABCBE and altitude AF, when each side of the base is 20 feet and the altitude AF, 50 feet. .'^\ S D AB-ifBC-\-CB+BE+EA=zlOO; and ^f =50: then {AB-\-BC-\- CD+DE+EA) x .4F=convex surface OF GEOMETRY. 241 Mensu ration of Solids. which becomes, 100x50=5000 square feet; which is me convex surface. For the area of the end, we have AB X tabular number = area j4SCZ)i?, that is, 20^ X tabular number, or 400x1.720477=688.1908= Ihe area ABCDE. Then, convex surface = 5000 square feet. lower base G88.1908 square feet. upper base 688.1908 square feet. Entire surface 6376.3816 2. What is the surface of a cube, the length of each side being 20 feet 1 Ans. 2400 sq. ft. 3. Find the entire surface of a triangular prism, whose base is an equilateral triangle, having each of its sides equal to 18 inches, and altitude 20 feet. Ans. 91.949 sj. ft. 4. What is the convex surface of a regular octagonal prism, the side of whose base is 15 and altitude 12 feet? Ans. 1440 sq. ft. 5. What must be paid for lining a rectangular cistern with lead at 2d a pound, the thickness of the lead being such as to require 71b. for each square foot of surface ; the inner dimen- sions of the cistern being as follows : viz. the length 3 feet 2 inches, the breadth 2 feet 8 inches, and the depth 2 feet 6 inches? Ans. £2 3s. lOfrf. PROBLEM II. To and the solidity of a prism. RULE. Multiply the area of the hose by the perpendicular height, and the product will be the solidity. 21.2 APPLICATIONS Mensuration of Solids. EXAMPLES. 1 , What is the solidity of a reg- ular pentagonal prism whose altitude is 20, and each side of the base 15 feet? To find the area of the base we have by Problem VIII. page 178. kn> 15''=225: and 225x1.7204774=387.107415= the area of the base : hence, 387.107415 x20=7742.1483=solidity. 2. What is the solid contents of a cube whose side is 24 inches ? Ans. 13824 solid in. 3. How many cubic feet in a block of marble, of which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 2 feet 6 inches 1 Ans. 21J solid ft. 4. How many gallons of water, ale measure, will a cistern contain whose dimensions are the same as in the last ex- ample ? Ans. 1291^ 5. Required the solidity of a triangular prism whose alti tude is 10 feet, and the three sides of its triangular base 3, 4, and 5 feet. An». 60 solid ji. 6. What- is the solidity of a square prism whose height is 6 J feet, and each side of the base 1| foot ? Ans 95 solid Ji. OF GEOMETRY. 243 Mensuration of Solids. 7. What is the solidity of a prism whjse base is an equi- lateral triangle, each side of which is 4 feet, the height of the prism being 10 feet? Ans. 69.282 solid ft. 8. Wliat is the number of cubic or solid feel in a regular pentagonal prism of which the altitude is 15 feet and each side of the base 3.75 feet ? Ans. 362.913 PROBLEM III. To find the surface of a regular pyramid. RULE. Multiply the perimeter of the base by half the slant height, and the product will be the convex surface : to this add the area of the base, if the entire surface is required (Bk. VI. Th vi) EXAMPLES. 1. In the regular pentagonal pyramid S—ABCDE, the slant height SF is equal to 45, and each side of the base is 15 feet: required the convex sur- face, and also the entire surface. 15x5 = 75=perimeter of the base, 75x225 = 1687.5 square feet^area of convex surface. And 15 =225: then 225 X 1.7204774=:387.107415=:the area uf the base. Ilencc, convex surface =1687.5 areaof the base = 387.107415 Entire surface =2074.607415 square (eet. 244 APPLICA.TIOMS Mensuration of Solids. 2. What is the convex surface of a regular triangular pyra mid, the slant height being 20 feet, and each side of the base 3 feet ? Ans. 90 sq. ft. 3. What is the entire surface of a regular pyramid whose slant height is 15 feet, and the base a regular pentagon, of which each side is 25 feet? Ans. 2012.798 sq. ft. PROBLEM IV. To find the convex surface of the frustum of a regular pjramid. Multiply half the sum of the perimeters of the two bases by the slant height of the frustum, and the product will be the con- vex surface (Bk. VI. Th. vii). EXAMPLES. 1. In the frustum of the regular pen- tagonal pyramid each side of the lower base is 30, and each side of the upper base is 20 'feet, and the slant height fF is equal to 15 feet. What is the convex surface of the frustum ? Ans. 1 875 sq. ft. 2. Hovsf many square feet are there in the convex surface of the frustum of a square pyramid, whose slant height is 10 feet, each side of the lower base 3 feet 4 inches, and each side of the upper base 2 feet 2 inches ? Ans. 110. 3. What is the convex surface of the frustum of a hepta^o- nal pyramid whose slant height is 55 feet, each side of the lower base 8 feet, and each side of the upper base 4 feet ? Ans. 2310 sq. ft. OF GEOMBTRY, 24.i Mensuration of Solids. PROBLEM V. To find the soliditv of a pjTami J. / RULE. MuUiply tl.s area of the base by the altitude and divide 'he pro- duct by 3, the quotient will be the solidity (Bk. VI. Th. xvii). EXAMPLES. 1 What is the solidity of a pyramid tlie area of whose base is 215 square feet and the altitude 80=45 feet? First, 215x415=: 9675: then, 9675^ 3 = 3225 which is the solidity expressed in solid feet. 2. Kequired the solidity of a square pyramid, each side of its base being 30 and its altitude 25. Ans. 7500 solid ft. 3. How many solid yards are there in a triangidar pyramid whose altitude is 90 feet, and each side of its base 3 yards ? Ans. 38.97117. 4. How many solid feet in a triangular pyramid the altitude of which is 14 feet 6 inches, and the three sides of its base 5, 6 and 7 feet? Ans. 71.0353. 5. What is the solidity of a regular pentagonal pjrramid, its altitude being 12 feet, and eac-h side of its base 2 feet? Ans. 27.537G solid fi 21* ' 246 APPLICATIONS Mensuratio n of S olids. 6 How many solid feet in a regular hexagonal p)T:amid, whose altitude is 6.4 feet, and each side of the base 6 inches' Ans. 1.38564. 7. How many solid feet are contained in a hexagonal pyta- inid the height of which is 45 feet, and each side of the base 10 feet? , ^715.3897.1143. 8. The spire of a church is an octagonal pyramid, each side of the base being 5 feet 10 inches, and its perpendicular height 45 feet. Within is a cavity, or hollow part, each side of the base being 4 feet 11 inches, and its perpendicular height 41 feet: how many yards of stone does the spire contain? Ans. 32.197353, PROBLEM VI. To find the solidity of the frustum of a pyramid. Add, together the areas of the two bases of the frustum and a geometrical mean proportional between them ; and then multi' ply the sum hy the altitude, and take one-third the product f:T the solidity. EXAMPLES. 1 . What is the solidity of the frus- tum of a pentagonal pyramid the area of the lower base being 16 and of the upper base 9 square feet, the altitude being 7 feet ' F G E M E T R Y . 247 M ensuration of Solids. First, 16x9=144 : 11x611,^144=12, the mean Then, area of lower base =16 area of upper base = 9 mean of bases =12 ~37 height 7 3) 259 solidity = 86|- solid ft. 2. What is the number of. solid feet in a piece of timber whose bases are squares, each side of the lower base being 15 inches, and each side of the upper base being 6 inches the length being 24 feet ? Ans. 19.5. 3. Required the solidity of a regular pentagonal frustum, whose altitude is 6 feet, each side of the lower base 18 inches, and each side" of the upper base 6 inches. Ans. 9.Z\925 solid ft. 4. What is the contents of a regular hexagonal frustum, whose height is 6 feet, the side of the greater end 18 inches, and of the less end 13 inches ? Ans. 24.681724 cubic ft. 5. How many cubic feet in a square piece of timber, tho areas of the two ends being 504 and 372 inches, and its length 31i feet ? Ans. 95.447. 6. What is the solidity of a squared piece of timber, its length being 18 feet, each side of the greater base 18 inches, and eaih side of the smaller 12 inches ? Ans. 28.5 cubic ft. 7. What is tho solidity of the frustum of a regular hexago- nal pyramid, the side of the greater end being 3 feet, that of lliR loss 2 feet, and the height 12 feet? Ans. 197.453776 solid ft. 2i8 APPLICATIONS Mensuration of Solids. MEASURES OF THE THREE ROUND BODIES. PROBLEM I To find the surface of a cylinder. Multiply the circumference of the lose hy the altitude, and the product will be the convex surface ; and to this, add the areas of the two bases, when the entire surface is required (Bk. VI. Th. ii). EXAMPLES. 1. What is the entire surface of the cylinder in which AB, the diameter of the base, is 12 feet, and the altitude JSF 30 feet ? First, to find the circumference of the base, (Prob. X. page 180) : we have 3.1416 X 13=37.6992=circumference of the base. Then, 37.6992x30= 1130.9760= convex surface. Also, 12^=144: and 144x.7854=113.0976 = area of the base. Then. convex surface =1130.9760 lower base 113.0976 upper base 113.0976 Entire area =1357.1712 2. What is the convex surface of a cylinder, the Jiametef of whose base is 20, and the altitude 50 feet ? Ans. 3141.6 sg. ft OF GEOMETRY. 249 Mensuration of the Round Bodies. 3. Required the entire surface of a cylinder, whose altituJe is 20 feet, and the 'x.5236=90x4900x.5236=230907.6 cubic feet, which is iho solidity. 260 APPLICATIONS Menauratinn of Cylindrical Kings . 2. What is th.e solidity of a prolate spheriod, whose fixed axisis 100, and revolving axis 6 feet? Ans. 1884.96. 3. What is the solidity of an oblate spheroid, whose fixed axis is 60, and revolving axis 100 ? A7ts. 3141 GO. 4. What is the solidity of a prolate spheroid, whose axea are 40 and 50 1 Ans. 41888. 0. What is the solidity of an oblate spheroid, whose axes are 20 and 10 ? Ans. 2094.4. 6. What is the solidity of a prolate spheroid, whose axes are 55 and 33 ? Ans. 31361.022. 7. What is the solidity of an oblate spheroid, whose axes are 85 and 75 ? . Ans. OF CYLINDRICAL RINGS. A cji-lindrical ring is formed by bending a cylinder until the two ends meet each other. Thus, if a cylinder be bent round until the axis takes the position mon, a solid will be formed, which is called a cylin- drical ring. The line AB is called the outer, and cd the inner diameter. PROBLEM XII. To find the convex surface of a cylindrical ring. RULE. 1. To the thickness of the ring add the inner diameter. II. Multiply this sum by the thickness, and the product by 9.8696, the result will be the area. OF GEOMETRV 201 Mensuration of Cylindrical Rings. EXAMPLES. 1 . The thickness Ac, of a cyliudri- sal ring is 3 inches, and the inner 'liaineter cd, is 12 inches : what is llie convex surface ? Ac+cd=3 t-12 = 15: 15x3x9.8696zr:444.132 square inches=the surface. 2. The thickness of a cjlindrical ring is 4 inches, and the inner diameter 18 inches : what is the convex surface t Ans. 868.52 sg. in. 3. The thickness of a cylindrical ring is 3 inches, and the inner diameter 18 inches : what is the convex surface ? Ans. 394.784 sg. in. PROBLEM XIII. To find the solidity of a cylindrical ring. RULE. I To the thickness of a ring add the inner diameter II. Multiply this sum hy the square of half the thickness, and the product hy 9.8696, the result will he the required solidity. EXAMPLES. 1. What is the solidity of an anchor ring, whose inner di- ameler is 8 inches, and thickness in metal 3 inches? 8-l-3=]l: then, 11 x(ifx 9.8696=244.2726, which ex- presses the solidity in cubic inches. 2. The inner diameter of a cylindrical ring is 18 inches, and iho thickness 4 inches : what is the solidity of the ring ? Ans. 868.5248 cubic inches 262 APPLICATIONS Mensuration of Cylindrical Rings. 3. Required the solidity of a cylindrical ring whose thick- ness is 2 inches, and inner diameter 12 inches 1 Ans. 138.1744 cubic in 4. What is the solidity of a cylindrical ring, whose thick- ness is 4 inches, and inner diameter 16 inches? Ans. 789.568 cubic tn. 5. What is the solidity of a cylindrical ring, whose thick- ness is 8 inches, and inner diameter 20 inches 1 Ans. 6. What is the solidity of a cylindrical ring whose thick- ness is 5 inches, and inner diameter 18 inches ? Ans. A TABLE OF LOGARITHMS OF NUMBEES From 1 to 10,000 N. Log. N. Log. N. Log. N. Log. I oooooo 26 1 414973 5i 1.707570 76 1-880814 2 3oio3o 11 \ 43 1 364 52 1 •716003 77 1-886491 1-892085 3 477I2I 447158 53 1-724276 78 4 602060 29 I 4(12398 54 1-732394 79 1-897627 5 698970 3o I 477121 55 1 -740363 80 1-903000 6 778151 845098 3i 1 491362 56 1-748188 81 1-908485 I : 32 1 5o5i5o 57 1.755875 82 i-9i38i4 903090 33 I 5i85i4 58 1-763428 83 1-919078 9 954243 34 I 531479 59 1-770852 84 1-924279 10 r oooooo 35 1 544068 60 1-778151 85 1-929410 II I 041393 36 1 5563o3 61 1-785330 86 1-93449S 12 I 079IBI 1 13943 ll ] 568202. 62 1-792392 ll 1 -939519 1-944483 i3 I 579784 63 1 ■799341 1-806181 U I I46I28 39 1 591065 64 89 I -949390 |5 I 176091 40 1 602060 65 1-812913 90 1-954243 i6 I 204120 41 1 612784 66 1-819544 9' 1-959041 ■7 I 230449 255273 42 I 623249 67 1-826075 92 1-963788 i8 I 43 I 633468 68 1-832509 93 1-968483 19 I 278754 44 I 643453 69 1-838849 94' 1-973128 20 I 3oio3o 45 1 653213 70 1-845098 1-851258 95 1-977724 21 1 322219 342423 46 1 662758 71 96 1-982271 22 I % \ 672098 72 1-857333 97 1-986772 23 I 361728 681241 73 1-863323 98 1-991226 24 I 380211 49 > 690196 74 1-S692.32 1-875061 99 1-995635 j5 I 397940 5a 1 698970 75 100 2-000000 Remaek.— rln the following table, in the nine right- hand columns of each page, where the first or lead- ing figures change from 9's to O's, poinis or dots are introduced instead of the O's, to catch the eye, and to indicate that from thence the two figures of the Log- arithm- to be taken from the second column/ stand in the next line below. 2 A TABLE OF logahithms from 1 TO 10,000. N. 1 2 3 4 5 6 7 8 9 3891 i). ~ 43a 100 000000 0434 0868 i3oi 1734 2166 2598 3029 3461 lOI 4321 475 1 5i8i 5609 6o38 6466 6894 7321 7748 8174 428 loa 8600 9026 3259 9451 368o 9876 •3oo •724 1147 1570 1993 241 5 424 io3 012837 4100 4521 4940 535o 5779 6j?7 6616 f? 104 7033 745 1 7868 8284 8700 2841 9116 §252 9532 3664 9947 4075 8164 •36 1 ?l 416 io5 021189 i6o3 2016 2428 4486 413 io6 53o6 5715 6125 6533 6942 735o 7757 1812 8571 8978 408 '"2 lo8 9384 033424 Ull •195 •5oo 1004 1408 2216 2619 3o2i 404 4227 8223 4628 5029 543o 5830 6230 6629 7028 41)0 109 7426 7825 8620 9017 5414 3352 9811 3755 7664 •207 4148 8o53 •602 ii 396 no III 041393 5323 1787 5714 9606 2182 6io5 2576 6495 •38o l^ 4540 8442 3^ 113 9218 053078 ni •766 45i3 1538 1924 2309 2694 386 Ii3 3463 423o xt 5378 9185 5760 6142 6524 382 1 14 6905 7286 7666 8045 8426 9d63 3333 9942 3709 7443 •320 In ]i5 060698 4458 1075 1452 1829 2206 2582 2958 4o83 376 116 4832 5205 .5586 5953 I352 6325 6699 7071 7815 372 \\l 8186 8557 8928 9298 ••38 •407 •776 1145 i5i4 lif 071882 225o 2617 6276 3718 4o85 445 1 4816 5182 366 119 5547 5912 9543 3i44 6640 7004 7368 7731 1347 8094 8457 8819 363 120 079181 082785 ?1S! •266 •626 n 1707 2067 2426 35o 121 3861 4219 4934 5291 5647 6004 III 122 636o 6716 7071 7426 7781 8i36 8490 8845 9198 o552 3071 355 123 9905 093422 •258 •6ii •963 i3i5 1667 2018 2370 3721 35 1 124 3772 7257 0715 4122 447' 4820 5169 55i8 5866 62i5 6563 349 346 125 6910 100371 7604 7951 8298 8644 8990 2434 9335 9681 ••26 125 1059 i4o3 1747 2091 2777 3119 3452 343 III 38o4 4146 4487 4828 5169 8565 55io 585i 6191 653 1 6871 340 7210 7549 ,888 8227 8903 9241 9579 f.]t •253 338 129 iioSgo 0926 1263 1 599 1934 2270 26o5 2940 3609 335 i3o 113943 4277 4611 4944 5278 56ii 5943 6276 9585 6608 6940 333 i3i 7271 7603 7934 8265 85o5 1888 8926 9256 t4 6456 •245 33o l32 120574 0903 I23l i56o 22l6 2544 2871 3525 338 i33 3852 4178 45o4 483o 5i55 5481 S806 6i3i 6781 325 i34 7105 7429 o655 7753 8076 8399 8722 9045 9368 9690 ••13 323 i35 i3o334 0977 1298 1619 I'&l 2360 358o 2900 3219 54o3 321 r36 3539 3858 % 4496 4814 545i 5769 6nR6 3i8 III 6721 143015 7037 7671 7987 83o3 8618 8934 9249 9554 3i5 •194 •5o8 •822 ii35 I45o 1753 4885 2076 2389 2702 58i8 3i4 139 3327 6438 3630 3951 4263 4574 5196 55o7 3u 140 I46'i28 6748 9S35 7o58 7367 38i5 7985 8394 86o3 8911 309. 141 152288 9527 •142 •449 io53 1370 1676 1982 307 142 2594 2900 32o5 35io 4120 4434 4728 5o32 3o5 143 5336 5640 5943 6245 6549 6852 7154 7457 7759 8061 3o3 144 8362 8664 8965 9266 9567 9868 •l58 •469 11^ 1068 3oi 145 l5i368 1667 1967 2366 2554 3863 3i5i 3460 4o55 299 146 4353 465o 4947 5244 5541 5838 6i34 643o 6726 7022 397 \u 73i7 7613 8203 8497 1434 8793 9086 9380 9674 tl 295 170262 o555 1141 1726 4541 30J9 33ll 25o3 293 149 3i85 3478 638 1 3769 4060 435 1 4932 7825 5222 55ia 58o2 ¥ i5o 176091 6670 6959 7248 7535 8ii3 8401 8689 1558 i5i 8977 9264 9552 9§39 •126 •4i3 l^ •985 3839 1272 287 ■ 32 181844 2129 241 5 2700 2985 3270 4123 4407 285 i53 4691 4975! 5259 5542 5825 6108 6391 6674 6955 7239 283 l54 75?i 7803 8084 8366 8547 8928 9309' 9490 2010 3309 til] 5345 ••5i 281 1 55 190332 06l2 36^1 1171 1451 1730 2846 270 156 3i25 34o3 3959 4237 45i4 4792 5069 5633 278 \U 5899 6176 8o32 1070 6453 6729 7oo5 72S1 75d6' 7832 8107 8382 276 8607 9206 9481 0755 ••39 •303^ ^577 •85o 1 1 34 274 >59 201397 1943 2216 2488I 2761 3o33 33o5 3577 3848 372 N. I 2 3 i 4 ! 5 6 1 7 8 i 9 D. A TABLE OF LOUARITHMS FROM '. TO 10,000. i N. 1 I 1 3 3 1 4 5 6 1 7 8 1 9 D.n i6o 204120' 439 1 1 4663 4934 52o4 54751 5745 6016 7634! 7904' 8173 8441 8710 6286 6556! ■27, i6i 6826' 7096! 7365 95i5' 97831 "Di 8979 9347 260 i6l •3i9 ^386 •853: 1121 i388 1654 1921 207 43 14 4579 2(6 i63 212188. 2454I 2720 2986 3252 35i8 3783 4049 5638 5902 6166 643o 6694 164 4844 5109' 5373 6957 7221 26 '1 9585 9846 262 i55 7484! 7747| 8010': 8273 8536 8798 9060 9323 166 220108: 0370 o63i 0892 ii53, 1414; 1675 1936 2196 2456 261 \u 2716 2976! 3236 3496 6084 3755 4oi5 42741 4533 4792 5o5i III 5309' 5568 5826 6342 6600 6858: 71 i5 7372 7630 169 7887 8144 840a 8657 8913 9170 9426 9682 9938 •193 255 . 170 j3o449 0704 C/960 35o4 I2l5 1470 1724 1979 2234 2488 '2742 354 'T IViS 325o 3707 4011 4264 4517 4770 5o23 5276 253 172 5781 6o33 6285 6537 6789 7041 7292 7544' 7795 252 173 8046 ,8297 8548 8799 9049 ?^^? 9550 gSoo ••5o ^300 35o 'H ''tn "3X 1048 IW^ 1 546 2044 2293 2541 2790 249 175 3534 4o3o 4277 4525 4772 5019: 5266i 24S 176 55i3 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 \]l 7973 8219 8464 8709 8954 1395 9108 i638 9443 9687 Itl •176 245 250420 0664 0908 3338 Ii5i 1881 2125 2610 243 \lt 2853 3096 358o 3822 4064 43o6 4548 4790 5o3i 243, 255273 55i4 5755 83g8 0787 6237 6477 6718 6058 9355 7.98 7439 341 i8t 7679 iv,l 81 58 8637 8877 91 16 9594 9833 III 182 260071 o548 1025 1263 i5oi 1739 \y^ 2214 1 83 245 1 2688 2925 3162 3399 3636 3873 4109 4582 337 iSi 4818 5o54 5290 5525 5761 5996 6232 6467 6702 6937 235 1 85 7.72 7406 7641 7875 8IIO 8344 8578 8812 90461 9279 i377 1609 334 186 9D.3 9746 9980 •2l3 •446 •679 •912 1144 333 III 271842 2074 4389 23o6 2538 2770 3ooi 3233 3464 3696 3927 232 4i58 4620 485o 5o8i 53ii 5542 5772 6002 6232 23o 189 5462 6921 7.5i 7380 7609 7838 8067 8296 8525 lit 190 278754 281033 92 1 1 ■9439 1715 9667 9896 •123 •35 1 •578 •806 191 no! 1488 1942 2169 2396 2622 2849 3075 227 192 33oi 3527 3753 3979 42o5 443 1 4636 4882 5io7 5332 226 193 5557 5782 6007 6232 6456 6681 6905 7i3o 7354 7578 225 194 7802 8026 8249 8473 8696 8920 9143 9366 tl 9812 223 195 290035 0257 0480 0702 0925 1147 l36g i5gi 2o34 222 I9S 2256 2478 2699 2920 3i4i 3353 3584 38o4 4025 4246 221 "97 4466 4687 4907 5i27 5347 5567 5787 6007 6226 6446 220 198 6665 6884 7104 7323 7542 7761 7979 8193 8416 8635 2Iq "99 8853 9071 9289 9507 9725 9943 •161 •378 •595 •8i3 3l3 200 3oio3o 1247 1464 1681 .898 4059 2H4 233i 2547 2764 2980 317 201 3196 3412 3628 3844 4275 4491 6639 877S 4706 4921 5i35 216 202 5351 5566 5781 5995 62 1 1 6425 5854 7068 7282 2l5 3o3 x 9843 7924 8i37i 835i 8564 8991 9204 9417 2l3 904 ••56 •2681 •481 •693; '906 2812 3o23 iii8 i33o 1 542 213 Jo5 3in54 3867 1966 %ll 2389' 2600 323i 3445 3656 311 206 4078 4499! 4710 4920 5i3o 5340 555i 5760 210 207 208 5970 6180 fx 6599 6809 86^9 889§ 7018 7227 7436 7646 7854 208 8o63 8272 9106 93i4 9522 9730 9938 309 320146 o354 o562 0769 0977 1184 1391 3458 159S i8o5 2012 207 311 323219 2426 2633 2839I 3o46 3252 3655 3871 4077 206 , 311 4282 4488 4694 48oql 5io5| 53 10 55i6 5721 5926 6i3i 2o5 312 6336 6541 6745J 6956 7i55| 7359] 7563 8787, 8991 9194! 9398; 9601 7767 7972 8176 304 ||3 838o 8583 9805 •••8 •211 203 >M 330414 0617 0810 1022 12251 1427 i63o i832 2o34 2236 202 2l5 2438 2640 2842 3o44l 3346! 3447' 3649 4655 4856 5o57 52571 5458 5658 385o 4o5i 4253 202 1 216 4454 585o 6o59 6260' 201 217 6460 6660 6860 7060' 7260I 7459' 7659 7858 8o58i 8237 200 218 8456 8656 8S55 9034 9253. 9431 9650! 9849J "47 •246; 199 119 340444 C642 0841 Io39 1237 1435 i632 i83o! 2028 2225 198 N. i I 1 3 1 3 _i 1 i A 1 7 f 8 9 D. 1 A TABLE OF LUOAIilTHMS FROM 1 TO 10,000. N. 320 ' 1 ' 3 4 5 6 1 7 8 9 D. 342423 3620! 2817 3oi4 3212 3409 36o6! 38o2 3999 4196 \u 221 4392 4589I 4785 4981 5178 7ii5 5374 5570 5766 5963 6167 322 6353: 6549' 6744 6939 733o 73251 7720 7913; 8110 9860; ••54 195 223 83o5 85oo; 8694 8889 90S3, 9378 ^9472; 9666 '91 234 350248 0442 o636 0829 1023 I3l6 1410: i6o3 1796; 1989 .93 223 2i83 2375 3568 3761 2954i 3i47 3339' 3532 3724 3916 5643, 5834 193 ,226 4ii>S 43oi 4493 4685 4876; 5o58' 5260, 54J2 19J ?/22 6026 6217 6408 6599 6790 6981' 7172 7363 7354! 7744 191 7^35 8125 83i6 85o6 8696! 8886 9076 9266 9456! 9646 ii 22C) 9835 ••25 •3l5 •404 2482I 2671 4363 4501 X I161 i35o: i539 23o 361728 To 2io5 2294 3o48 3236' 3424 23 1 3612 3988 5862 4176 4739 4926 5ii3 53oi 188 232 5488 5675 6o4o 6236 6423 6610 tit •5i3 6983 7169 Is? 233 7355 7542 71291 791D 8101 8287 •143 8473 8845 9o3o 234 9216 9401 93871 9772 ??^ •328 •698 •883 183 235 371068 1253 14371 1622 1991 2175 236o 2544 2,28 184 236 3912 3096 4932 3280! 3464 3647 3«, 401 3 4198 4382 4565 184 i3i 4748 5ii5l 5298 5481 5664 5846 6029 6212 6394 i83 till till 6942 7124 8761 §943 73o6 7488 7670 7853 8o34 8216 183 J39 9124 93o6 9487 9668] 9B49 ••3o 181 240 380211 0392 0573 0754 0934 iii5 1296I 1476! i656 1837 181 241 2017 38i5 2197 23771 2557 2737 2917 3097 3377 3456 3636 180 242 5% 41741 4353 4533 4712 4891 5070 5249 5428 \]t 243 56o6 5964 6142 6321 6499I ^^77 6856 7034 7212 244 7390 7568 7746 7023 9320 9698 8101 8279J 84361 8634i 8811 8989 n8 245 9166 9343 9875 ••5i •328 •403 •582 •739 "i 246 390935 1112 1388 1464 1641 1817 1993 2169! 2345 3531 248 2697 4452 2873 3048 3224 3400I 3575 3731 3926; 4101 4277 176 4637 4802 4977 5i52; 5326 55oi 5676! 585o 6o35 ■75 249 6199 6374 6548 6732 6896 7071 8634 880S IH^ 7419; 7392 7766 174 25o 397940 8114 8287 8461 8981 9>54 9328 •383 To56 gSoi 173 231 9674 9847 ••20 •192 •365 ^538 •711 1228 m 233 401401 1573 1745 -1917 2089 1 2261 3433 26o3| 2777 2n 172 253 3l2I 3292 3464 3633 3807 56SS 4149. 4320: 4492 3858 6029I 6199 IT 234 4834 5oo5 6881 5346 5517 6370 171 255 6540 6710 705 1 8749 7231 7391 7561 773lj 790' 8070 170 256 8240 8410 8579 S918: 9087 9337 9436, 9595; 9764 169 257 9933 •102 3635 •440 •609! ^777 •946 1114 1283| 1431J i6g 1 25S 411620 178a 2124 2293 2461 2629' 2796; 2964I 3i32; 168 1 259 33oo 3467 38o3 3970 4137 58o8 43o3] 4472, 4639! 4806 ■67 260 414973 5i4o 5307 5474 5641 5974 6141 1 63o8; 6474 167 261 6641 6807 6973 l^t 7306! 7472 8964' 9 '29 7633 7804 7970 8i33 9460; 9625 9791 166 262 83oi 8467 8633 9295 1 65 263 9956 •121 •286 •431 •5i5 ^781 •945 2590 iiiO| 1373I 1439 i65 264 421604 1788 1933 l^] 2261; 2436 2734: 2018! 3082 164 265 3246 3410 3374 3901: 40631 422SI 4393 4355 4718 iC4 266 4S82 5045 5208 5371 5334 5*97; 3860J 6o23: 6186 6349 i63 26^ 65n 66-4 6836 ^. 7161, 7324 8783; 8944 7486 764s; 7811 7973 162 8i35 8297 8450 ••75 9I06 9268 9429 9591 162 269 9752 9914 •236 •39S •359 •730 •SSi .042 I203 161 370 43 1 364 i525 1846 2007J 2167 23j8 24S3 2649 2800 i6i 271 6i6J 3i3o 3290 3456 36 10 3770 3930 4050 j 4349 4409 56S5! 5844 6004 160 372 4739 4888 5048 5207! 5367 5526 i59 373 6333 6481 6640 6798 6957 7116 8701 72-5, 7433 7592 8859; 90'7 9<73 \lt 274 7751 7909 8067' 8226; 8384 8542 9648 9806' 9964 •122 275 9333 1066 •279 •437 .594, •752 1 53 276 440909 1334 i38i| 1538 1695 3950! 3io6' 3263 1853 2009I 2166! 2323 i57 l]l 2480 2637 lll\ 3419 3576; 3732I 3S89 .57 4045 4301 43i3' 4669' 4825 4981 5i37 52o3 544Q i56 '79 56o4 5760 5915 6071 6226, 63821 6537 6692 6848, 70o3 i55 N. / _l_ 3 4 1 5 1 6 7 8 9 "£." A TABLE OP LOOAUITIIMS FROM ] ro 10,000. s N. I 2 3 4 5 1 6 7 8 9 ^~ 280 447158 8706 IU\ 7468 7623 7778 7933 8088 8242 8397 8552 l55 aSi 9or5 9170 9324 9478 9633 9787 9941 ••95 l633 1 54 s:82 450249 1786 33i8 0433 0557 0711 0865 10181 1172 i326 '479 i54 283 1940 2093 2247 2400 2553] 2706 2859 3oi2J 3i65 1 53 284 3471 3624 3777 8980 4082 433d 4387 4540 4692 i53 1 285 4845 4997 5i5o 53o2 5454 56o6 5758 5910 6062 6214 l52 286 6366 65i8 6670 6821 8487 9995 7125 8638 7276 8789 7428 7^79 773' l52 288 7882 8o33 81S4 8336 8940 9091 9242 i5i 9392 9543 9694 9845 •146 •296 •447 194S •597 •748 i5i 289 460B98 1048 iigS i348 1499 1649 '799 2098 2248 l5o 290 462398 2548 2697 2847 2997 3 146 3296 4788 3445 foii 3744 1 5a 291 3893 4042 410.; 4340 4490 4639 6126 4936 5234 149 292 5383 5532 568o 5829 5977 6274 6423 6671 6719 148 293 6858 7016 8495 7164 8643 7312 7460 7608 7756 7904 9380 8o52 8200 294 8347 S? 8988 9085 9333 •99^ 9675 143 295 9823 9965 •116 •410 •557 •704 o85i 1 145 \U 296 ''m 1438 i585 1782 1878 2025 2171 23i8 2464 2610 III 2903 4362 3 049 45o8 3195 4653 3341 3487 3633 3779 3925 538i 4071 146 4216 4799 4944 6397 5090 5230 5525 146 299 5671 58i6 5962 6107 6252 6542 6687 6832 6976 145 3 00 mil 7266 S711 741 1 ''8855 7555 7700 7844 7989 8i33 8278 8422 145 3oI 9143 9287 9431 9575 97 '9 9863 144 3o3 480007 oi5i 0294 o582 0725 0869 1012 Ii56 1299 144 3o3 1443 i586 1729 1872 2016 2159 2302 2445 2588 2731 143 3o4 2874 3oi6 3i59 4585 3302 3445 3587 3730 3872 40i5 4167 143 3o5 4300 4442 4727 4869 Soil 5i53 5295 5437 5579 142 3o6 5721 5863 6oo5 6147 6289 643o 6572 6714 6855 6997 142 307 7i38 855.1 7380 7421 7563 7704 7845 7986 9h° 8.27 8269 8410 141 3o8 8692 8833 8974 9>I4 9255 9537 9677 9818 141 309 9958 491362 •"99 •239 •38o •520 •661 •801 •941 2341 io8r 1222 140 3io l502 1642 1782 1922 2062 2301 3481 2621 140 3ii 2760 2900 3 040 3i79 3319 3458 3507 tit 3737 3875 401 5 139 3l2 4! 55 4294 4433 4572 4711 485o 5128 5267 5406 139 3i3 5544 5683 5832 5960 8362 6338 65i5 6653 6791 'M 3i4 6o3o 83ii 7068 8448 7206 7344 8724 7621 8999 7759 7897 8o33 8.73 3i5 8586 9>37 9375 9:112 9550 i38 3i6 . 96S7 9S24 riti "99 •236 •374 •5ii •648 •785 •922 ■37 3i7 001039 1196 1470 1607 1744 1880 3017 2i54 23gl 3655 137 3i8 2427 2564 2700 2837 nil 3109 3346 3382 35 18 i36 319 . i'^V 3927 4o63 4199 4471 4607 4743 4878 -5014 136 320 5o5iDo 5386 5421 5537 5693 5828 8664 6099 6234 5370 136 321 63o5 6640 6776 6911 7046 71S1 7451 7586 7721 i35 323 7836 7991 8126 8260 8395 8530 8799 •143 8934 9068 135 323 0203 9«7 9471 9606 9740 9874 •••9 •277 •411 1 34 324 516543 0679 o8i3 0947 icSi I2l5 1 349 1482 1616 1730 134 32D 1 833 2017 2l5l 2284 2418 255i 2684 2818 2931 3o84 133 326 '3218 335i 34S4 36i7 3750 3883 4016 4149 4282 4414 i33 327 4548 4681 4813 4946 5079 521 I 5344 5476 3609 3741 i33 328 0874 6od6 6139 6271 6403 6535 6668 6800 6932 7064 132 3^9 7196 7328 7460 7592 8909 7724 7855 7987 8ri9 825i 8382 |32 33o 5iB5i4 8646 8777 9040 9171 93o3 9434 9566 9697 i3i 33r 9828 qoSgi ••90 •221 •353 •48,( •6i5 •743 •876 1007. i3i 335 3ju38 1269' 1400 .53o 1661 1792 1^23 2d53; 2i83 23i4 i3i 3)3 2444 2575,' 2705 2835 2966 3096 3225 3356 3486 36 16 iT»._ 334 3746 3876 4006 4i36 4266 4396 4525 4656 4785 4915, i3o 335 5o45 5i74 53o4 5434 5563 5693 5823 5951 6081 6210 119 336 6339 6469 6598 6727 6856 6985 7114 7243 7872 75oi 129 337 7630 7759' 7888 8016 8145 8274 S402 8531 3660 8788 129 338 8917 0045, 9174 9302 9430 9559 9687 9S15 9943 ••72 129 339 13o2ooj 6328i 0456 o584 0712 0840 0968 1096' 1223 i35i 128 N. I = 3 4 5_ 6 7 i 8 9 D. i A .TAULB OF LO&AUITHMS FaOM 1 TO 10,000. ~N." ■ 1 ^ 3 4 1 5 6 7 8 9 D. 340 531479 1607 1734 1862 1990 2117 2245 2372 25oo 2627 128 1 341 2734 2882 3009 3i36 3264 3391 35i8 3645 3772 3899 127 J42 4026 4i53 4280 4407 4534 4661 4787 4914 5o4i 5167 III 343 5294 65d8 7819 5421 5547 3674 69^7 8197 9452 58oo 5927 6o53 6180 63o6 6432 344 343 6685 7945 681 1 8071 7063 8322 844§ 8574 744' 7567 8699 8825 99^4 "79 lit 126 126 346 9076 9202 9327 '^ll 9703 9829 •204 125 347 348 5 'o329 0455 o58o 0705 0955 1080 22o3 2327 i2o5 i33o 1454 125 ■'579 <704 1829 1953 loiR 2452 2576 2701 i]5 349 382? 2950 3074 3'99 3323 3447 3571 4688 4812 3696 3820 3y44 124 350 544068 iX 43i6 44J10 4564 4936J 5o6o 5i83 124 35i 5307 6543 5555 5678 58o2 5925] 6049 6172 6296 6419 124 35: 6666 6789 6913 7o36 7159; 7282 hSg' 85i2 74o5' 7529 8635 8758 76.12 123 353 7773 7898 802i 8144 8267 8881 123 354 9003 9126 9249 0473 9371 9494 9616 9739 9861! 9984 •106 123 355 550228 o35i 0595 0717 8840 0962 1084 1206 |32S 122 356 i45o 1572 1694 1S16 1938 2o6o' 2181 23o3, 2425 2547 122 357 2668 2790 2911 3o33 3i55 3276J 3393 4480' 4610 35i9 3640 3762 121 358 3883 4004 4126 4247 4368 4731 1 4852 4973 121 35, 5094 5213 5336 5457 5578 67S5 5690' a82o 6905 7026 Bios; 8228 5940 6061 6182 121 360 5563o3 6423 6544 6664 7146 7267 6340' 4469 9548 9667 til 120 361 po7 8709 .7627 8829 7748 8948 7S68 7988 120 363 9068 9188 9308, 9428 9787 120 363 9907 ••26 •i46 •265 •385 •5o4' •62A •743', •863 •982 119 364 56i 101 1221 i34o 1439 1378 1698 i8ii 2887 3odB 1936 2o55 2174 119 365 2293 2412 a53i 2630 2769 3i25 3244 3362 119 366 3/,8i 36oo 3718 3837 3933 4074' 4 19* 43 111 4429 4548 367 4666 4784 4903 5021 5i39 5237' 5376 5494 56 12 5730 368 5848 5966 6084 6202 63 20 6437 6555 6673, 6791 6909 118 369 7026 83 1^ 7262 8436 7379 7497 7614' 7732 8788, 8905 7849' 7967 9023 9140 8084 118 3-0 568202 8554 8671- 9257 "7 37. 9374 9491 9608 9725 9842 9959 ^^76 •.53, .309 •426 "7 372 570343 0660 0776 0893 1010 1126 1243 1339' 1476 1592 2765 "7 373 1709 1823 1942 2038 2174 3336 2291 2407 3432 3568 252J 263g 116 374 J872 4o3i 2988 3io4 3220 3684 38oo 3915 116 375 4147 4263 4379 4494 45ro 4726 4R41 4937 5072 116 376 5i88 53o3 5419 6534 S65o| S765' 588o 5996 6. II 6226 n5 377 6341 6457 6572 6687 6802 6917; 7o32 8066; 8181 7147 7262 8295 8410 7377 8525 ii5 378 mi 8754 886S 7836 8983 795i ii5 379 9097 921a 9326 9441' 9555 9669 114 3ao 5797S4 580925 9°oS 1039 "12 •126 •241 •355 "469 1493; 160S •583 ^697 1722 i836 •81 T 114 38 1 Ii53 1267 i38i 1950 114 382 2063 2177 2291 2404 25i8 26J1' 2-451 2858, 2972 3o83 114 383 3199 3312 3/,25 3539 3652 3763, 3879 4896; 5009 3992' 4103 4218 Ii3 384 433. 4444 4557 4670 4783 5i22, 5235 5348 ii3 385 5461 5574 5686 5799 5912 6024 6137 6250 6362 6473 ii3 386 6587 6700 7823 8944 6812 6925 7037 7149 7262 8272, 8334 2496! Im 7599 8720 9833 112 387 77" 7935 8047 8160 112 388 8832 9o56 9167 r^ 93911 95o3 9610 9726 •330 •ai2 1843 1955 112 389 9930! ••61 12S7 •284 •5o7 •619 •953 112 390 ^91065 Wh '399 l5io 1621 1732 2843 3o66 III 391 3177 2399 35o8 aSio 2621 2732 2954 3064 3175 III 39a 3286 3397 .36i8 3729' 384e 4834| 4945 3950 4061 4171 4182 III ^393 4393 45o3 4614 4724 5827 5o55 5i65 5276 5386 no 394 5496 56o6 m 5937, 6047 7037' 7146 6157 6267 6377 7476 8372 6487 no 395 8791' 8900 9883 99Q> (100973 108 J 6927 7256 7366 8462 7586 no 396 79'4 8024 8i34 8243 8353 8681 no III 9009 9119 9328 9337' 9446 •319 •428 '537 1408 i5i7 i6a5 9336 9665 9774 109 •lOI •210 •646 •755 •864 109 399 II9II 12Q9 1734' i843i 1951 109 , ». ,0 ii;s|3|4;5|6 7 ■ 1 8 i 9 ^•i & lABLE OF LOOARITHMB FROM 1 TO 10,000. 1 ' N. I 2 3 1 4 5 6 2711 7 8 9 |D. 3o35i 108 400 602060 2169 2277 336i 2386 2494 2603 2819; 2928 401 3144 3253 3469 3577 4658 3686 3794 3902, 4010 4118; 108 1 403 4226 4334 4442 455o 4766 4874 4982! 5089 5197 108 4o3 53o5 54i3 5521 5628 5736 5844 5951 6059 6166 6274 108 404 638 1 6489 6596 6704 6919 7026 8098 9167 7i33 8205 7241 7348 8419 107 4o5 7455 7562 8633 1669 8740 9808 7777 8847 7884 8954 799' 83i2 107 406 8526 9061 9274 9381 9488 107 407 9594 9701 9914 ••21 •128 •234 •341 •447 •354] 10] 1617 106 2678 106 408 610660 0767 1829 0873 •0979 1086 1192 S254 i4o5 i5ii 409 1723 1^36 2041 2,148 23^0 2466 »572 410 612784 2890 2906 3 102 3207 33 13 3419 3525 363o 3736 106 411 3842 3947 4o53 4i59 52i3 4264 4370 4473 458 1 4686 4792 106 412 tin 5oo3 5io8 5319 5424 5529 5634 5740 5845 lo5 4i3 6o35 6160 6265 5370 6476 658 1 6686 SS84 6895 io5 414 41 5 7000 8048 . 7io5 8i53 7210 8257 73i5 8362 7420 8466 7525 8571 8676 8780 ^^ io5 io5 416 9093 9198 9302 9406 9611 9615 9719 9824 9928 •i32 104 418 620106 0240 0344 0448 o552 o656 0760 0864 0968 1072 104 1.176 1280 i384 1488 1592 1695 2732 m 1903 2007 2110 ■ 104 4ig 2214 _23i8 2421 2525 2628 3^7? 3o42 3i46 104 420 623249 3353 3456 3559 3663 3766 3869 4076 4179 io3 421 4282 • 4385 4488 459' 4695 ilf, 4901 5oo4 5io7 5210 io3 422 53i2 5415 55i8 5621 5724 5929 6o32 6i35 6238 io3 423 6340 6443 6546 6648 6751 6853 6956 7g58 8082 7161 7263 io3 424 7366 8389 7468 8491 7571 S593 lb^3 8695 7775 8797 9817 7878 7980 8i85 8287 102 425 8900 9002 9104 9206 9308 102 426 9410 9^i^ 9613 97i5 9919 ••21 •123 •224 •326 102 427 630*428 o53o o63i 0733 0835 0936 io38 ii39 2i53 1241 i342 103 428 1444 1045 1647 1748 1849 -195. 2052 2255 2356 101 429 2457 633468 2559 2660 2761 2862 2g63 3o64 3i65 3266 3367 101 43o 356g 4578 5584 3670 377. 3872 4880 3973 4074 5o8i 4175 3182 4276 4376 100 43 1 5484 4679 4779 57^D 4981 5283 5383 loo 432 5683 5886 5986 6087 6187 6287 6388 100 433 6488 6588 6688 6789 6889 6989 9088 7189 8190 9188 7290 9387 ICO 434 435 8489 ?5^9 IX Wo 9785 8888 7900 8988 8290 9287 99 99 435 9456 9586 9686 9885 9984 ••84 •i83 •283 •382 99 437 438 640481 o58i 0680 0779 0879 0978 1077 1177 1276 1375 99 1474 1573 1672 1771 1871 1970 2069 3o58 2168 2267 2366 99 439 2465 2563 2662 2761 2860 2959 3i56 3255 J354 99 98 Mo 643453 355 1 365o 3749 3847 .3946 4044 4143 4242 4340 44 1 4439 4537 4636 4734 4832 4931 5029 ■5,127 5226 5324 98 442 5422 5521 56i9 57.J 6698 5gi5 5913 6S94 601 1 6j 10 6208 63o6 98 443 6404 6502 6600 6796 6992 7089 V^l 7283 98 444 7383 7481 845a 2^79 lit 7774 1872 8848 Its 8067 8262 98 445 836o 855d 8750 9043 9140 9237 97 446 9335 943 » 9,53o 9627 9724 9821 9919 <»i6 •ii3 •210 97 447 65o3o8 o4o5 o5o2 0599 0696 0793 0987 io84 1181 97 448 1278 1375 1472 1569 1666 1762 I8D9 1956 2o53 2i5o 97 449 2246 2343 2440 2536 2633 2730 2826 2923 3888 3oi9 3ii6 '9I 45o 653213 3309 34o5 35o2 3598 3695 3791 4754 3984 4080 401 4177 5.38 4369 4465 4562 4658 485o 4946 5o42 96 4i2 5235 533i 5427 5523 56i9 6^73 58io 5906 6002 96 453 6098 6194 6290 6386 5482 %l 6769 7725 6864 6960 96 454 7o56 7i5j 7247 7343 8298 9230 7438 8393 7629 8584 7820 8774 8870 95 455 8011 8107 8202 8488 8679 95 456 8965 9060 9155 9346 9441 9536 9631 9726 9821 95 457 9916 660865 "n •106 •201 "296 •391 1339 •486 •58 1 •676 4771 95 458 0960 ]o55 11 DO 1245 1434 i529 1623 2^63 459 i8i3 1907 2002 2096 2191 2286 238o 2475 2569 ,95 N. I 2 3 1 4 5 1 6 7 8 9 D. : 23* A TABLE OF LOGARITHMS FROM 1 TO 10,000. fN. 1 460 I 2 3 4 5 6 7 I 8 9 94 662758 2852 2947 3o4l 3i33 323a 3324 3418 35i2 3607 4548 461 3701 4642 3795 3889 3983 4078 4172 4266 436o 4454 94 462 4736 4830 4924 5362 5oi8 5lI2 5206 tlfj l^ol 5393 633 1 5487 94 463 558i 5675 5769 5936 6o5o 6143 6424 94 464 465 65r8 7453 8386 6612 7546 8479 670? 7640 8572 S! 6892 7826 8739 6986 Mil lilt 7266 8199 7360 8293 9| 93 466 8045 9875 9o38 -9131 9224 93 : 46^ 93.7 9410 95o3 9596 9782 9967 ••60 •i53 93 J70246 0339 1265 0431 o524 0617 1 543 0710 0802 6895 0988 1080 9,^ 46g , "■'^ i358 I45i 1636 1728 1821 1913 2836 2oo5 93 470 672098 2190 2283 2373 2467 256o 2652 2744 3666 2929 92 1 471 302I 3ii3 32o5 3297 4218 3390 3482 3574 3758 3S5o 92 17= 3942 4861 4034 4126 43io 4402 4494 4586 4677 4769 92 473 4953 5870 6785 5045 5i37 bo53 5228 5320 5412 55o3 5595 5687 92 474 5778 5962 6145 6236 6328 6419 7333 8245 65u 6602 92 475 6694 6876 7789 8700 6968 7881 8791 7059 8o63 7242 7424 8336 7516 8427 9' 476 7607 85i8 tf. mi 81 54 9' 1 478 lul 9064 9155 9246 9337 9' 9428 9519 9610 9700 979" 9073 ••63 •i54 •243 91 479 4 So 68o336 0426 o5i7 0607 i5i3 0698 0789 0879 0970 1874 1060 ii5i 9' 681241 i332 1422 i6o3 1693 1784 1964 2867 2055 90 , 43i 2143 2235 2326 2416 25o5 2596 2686 2777 2957 90 i 4S2 3o47 3i37 3227 3317 3407 3497 3587 3677 3767 3857 .JO 4a3 ^2H 4037 4127 4217 4307 4396 4486 4576 4655 4756 90 i 4S4 4845 4o35 5831 5o3 3 5ii4 5204 5294 5383 5473 5563 5652 ^ 1 4S3 5742 592! 6Si5 6010 6100 6279 6363 6458 6547 486 6636 6726 6904 6994 7886 IZ 7261 8i53 7351 8242 7440 833 1 89 48-7 488 7529 7618 in 7796 i 89 8420 85o9 86S7 8776 8953 9841 9042 9i3i 9220 ?9 4S9 9309 939? 94S6 9575 9664 9753 9930 ••19 •107 89 490 690196 0285 0373 0462 o55o o&3g 0728 0816 0900 -W93 1877 ; 491 '°h 1170 1258 i347 1435 1524 1612 1700! 1780 ! ^'yi 1965 2053 2142 223o 23i8 2406 2494 2583 2671 2759 88 493 1847 2935 3o23 3iii 3199 3287 3375 3463 355i 3639 88 494 ^ri 38i5 3903 im 4078 4166 4234 4342 443o 4317 88 493 4603 4693 4781 4956 5§32 5o44 5i3i 5219 53o7 5394 88 495 5482 5569 5657 5744 6618 5919 6007 6^8 6182 6269 8? 497 6356 6444 6531 5706 6793 7665 8535 68S0 7055 7142 8o!4 87 498 7229 l%l 74o4 ^362 7578 8449 7752 8622 7839 8709 7926 87 499 8101 8275 X 8883 87 Sou 698970 9057 9>44 9231 9317 9404 9491 9578 975 1 87 5oi 9838 9924 ••11 ••98 •184 •271 •3d8 2444 •33 1 •617 u 5o2 '"W^ 0790 0877 0963 loSo Ii36 1222 1 309 1395 2258 1482 5o3 1634 26o3 1827 1913 ■im 20S6 2172 2344 85 5o4 243 1 25i7 2689 lUl 2947 3807 3o33 3119 32o5 86 5o5 3291 4i5i 3377 3463 3549 4408 3721 3893 4751 5A07 3979 4o65 86 5o6 4235 4332 4494 5350 4579 4663 4837 4922 86 lu 5oo8 5094 5179 5265 5436 5522 5593 'Si 86 5864 ^2"? 6o3d 6120 6206 6291 6376 6452 6547 85 5o9 6718 707570 8421 68o3 5888 8676 7059 7144 8931 73i5 8166 90i5 7400 8251 9100 7485 8336 9i85 85 5io 5ii 7655 85o6 lit: 8761 8846 9609 9694 85 85 012 9270 9355 9440 9524 0625 9863 9948 ••33 85 5i3 1807 0202 02S7 0371 0456J o54o 0710 v.n 0879 1723 85 5i4 1048 u32 IS17 i3oi| 1385 1470 1 554 84 5i5 1S92 2734 1976 2060 2144 2229 23i3 3^?^ 2481 2555 84 5i6 2600 2818 290a 2986' 3070 3i54 3323 3407 84 ^'1 51^ IZ 3575 3659 3743 3826 3910 im 4078 4162 4246 84 4414 4497 4581 5335 5418 4655: 4749 55o2 5d85 4916 5ooo 5oS4 84 519 5167 5j5i 5669 5753 5836 5920 84 D. N. I 2 3 4 5 1 6 7 « 9 A 'i'Ain.E 1 I 1 OP LOOAIUTHMS FKOM 1 TO 10,000. 9 N. ■--- 3 ; 4 1 5 6 7 8 1 9' 1 D. Sao 716003 6087; 6170' 6254 6337 ^421 6304 6588 6671! 6754, 83 521 6838 6921 7004 7088, 7171 7234 7338 8169 9000 7421 8253 9083 7504 8336 9165 7087 83 8419: 83 9348 83 553 533 7671 7754; 7337 8502 35851 8668 792c i 8oo3 8086 875JI 8834 8917 534 9331 9414] 9497 9580! 9663 9745 9828 99iJ 9994 •"77; 83 523 720159 0242, 0323 0407! 0490 0573 o655 0738 0821 0903 1 83 'J2b 09S6 1068 Ii5i 1233 J3i5 1398 1481 |563 1646 172S. 83 511 538 1811 1893 1975 2o58 2140 2222 23o5 2387 2469 3553! 83 2634 2716 3520 2881 2963 3045 3127 3948 3209 3291 3374! 82 ; 539 3456 3538 3702 3784 4604 3866 4o3o 4112 4194 t' 53o 724276 4358 4440 4522 4685 4767 4849 4931 5oi3 2" i 53 1 5095 5175 5258 5340 5423 53o3 5585 5657 6483 5748 583o §' 532 5912 8435 6075 5i56 6238 6320 6401 6564 6646 82 533 6727 6890 6972 7053 7'34 7216 8029 8841 7297 8110 8922 7379 8191 goo3 7460 81 534 535 7541 8354 Ei nil 7866 8678 94S9 •298 ^759 8273 9084 8i 81 536 9165 9246 f,ll 9408 9570 965i 9732 98.3 9893 8r 537 533 9974 730782 ••55 •217 •378 11S6 •459 •540 •621 •702 81 o863 0944 1024 iio5 1266 1 347 142S i5o8 81 539 1589 1669 1750 i83o 1911 1991 2073 2l52 2233 23,3 81 540 732394 2474 2555 2635 271 5 2796 3876 3956 3o37 3i!7 80 541 3 197 3278 3358 3438 35i8 3598 3679 4480 3759 3839 3919 80 54 a 3099 4800 4079 4880 4i6o 4240 4320 4400 4560 4640 4730 80 543 4960 5o4o 5l20 5200 5270 5359 5439 55i9 80 544 5599 5679 5759 5838 5918 5998 6o7§ 6i57 6237 63i7 80 545 6397 7io3 8781 9572 6476 6^56 6635 6715 7311 83o5 6795 Ml 6874 6954 7034 71.3 So 546 7272 7352 8146 7431 8225 7670 llt^ 7829 7908 79 547 8067 8463 8622 8701 79 54« 8860 8939 goi8 9097 9177 9968 9256 9335 9414 9493 19 1 549 9651 9731 9810 988Q ••47 •126 •205 •284 79 55o 74o363 0442 052l 0600 0678 1467 0757 o836 0915 0994 10731 79 55 1 Il52 I23o i3o9 i388 1546 1624 i7o3 1782 1860I 79 532 1939 2018 2096 2175 2254 2332 2411 24S9 2568 2647 i 79 553 2720 2804 2882 2961 3o39 3823 3ii8 3196 3980 3273 3353 343 1 -a 554 35io 3588 3667 3745 3902 4058 4i36 42i5 78 555 4293 4371 4449 4528 4606 4584 4762 4840 iigig 4997 78 555 5075 5i53 5231 5309 5387 5465 5543 5621 5699 5777 78 557 5855 5g33 601 1 6089 6167 6245 6323 6401 6479 6556 78 55!J 6634 6712 6790 6868 5945 7023 7101 V^M 7256 8o33 7334 8110 78 559 7412 n 7567 834J 7645 8498 7800 8576 7878 I** 56o 748188 8421 8653 8731 8S08 8885 77 56i 8963 9040 911^ 9195 9272 9350 9427 9504 9582 9659 77 .552 9736 9814 989 1 9968 ••43 •I23 •200 •277 •354 ^31 77 563 75o5o8 o5S6 0653 0740 0817 0894 0971 1048 II25 1202 77 564 1270 i356 1433 i5io 1 587 1664 1741 1818 1895 1972 77 565 2048 2125 2202 2279 2356 2433 25o9 2586 2663 2740 77 565 2816 2893 2970 37i6 45oi 3o47 3i23 3300 3277 3353 3430 35o6 77 567 3583 366o 38i3 3889 3966 4042 4119 4883 4195 4272 77 568 4348 4425 4578 4654 4730 4S07 4960 5o36 76 569 5lI2 5189 5265 5341 5417 5494 6256 5570 5646 5722 5799 76 570 755875 5951 6027 6788 6io3 6180 6332 6408 6484 656o 76 57. 6636 6712 6864 6940 7016 7992 7168 7244 7320 76 572 7396 8i5d 7472 7548 83o6 7624 8382 7700 7775 8533 8609 8685 9366 9441 8oo3 8079 -6 573 8230 8458 8761 8836 ~6 574 . 8912 8988 9o6v 9139 ■9214 9290 9517 9592 76 57'- 9668 9743 9819 9894 9970 ••45 •121I "196 •272 •347 '^ 570 760422 0498 0573 0649 0724 1^. 0875 0950 10231 IIOI 75 t^l 1 176 I25l 1326 1402 '477 2228 1627 2378 1702 1778 1853 75 1028 2679 2oo3 2078 2i53 23o3 2453 2539 2604 3278 3353 75 579 2754 2829 2904 2978 3o53 3128 32o3 75 N. I 3 3 4 5 6 7 a 1 ,' D. 10 A TABLE OF ~»Al,Ui« ia X nviMM. m. KV ^\fy\f^ «/• N. 38o I 3 3 4 , 5 6 7 8 9 I». 763428 35o3 3578' 3653 3727 38o2 3877 3952 4027 4101 7f ^5 58i 4176 4331 4325 4400 4475 455o 4624 4699 4774 4848 582 4923 4998 5072 5i47 5221 5296 5370 5443 55 20 5594 6338 73 583 5669 641 3 5743 58i8 5966 6041 6785 6ii5 6190 6264 74 584 6487 6562 6710 6859 7007 7082 74 585 585 7i55 Ms 723o 7972 8712 7304 8046 7379 8120 7453 8194 8934 0410 8268 7601 8342 8416 7749 8490 92J0 7823 8564 74 74 589 8786 8860 9008 9082 91 56 93o3 74 9377 7701 1 5 945i 0189 9525 0263 life 9745 0484 9820 o557 tf. 9968 0705 ••42 0778 74 74 5go 770852 0925 0999 1734 1073 1146 1220 1293 1357 1440 i5i4 H 5,, 1587 l66i 1808 1881 1955 2028 2102 2175 2248 H 5,2 3322 2395 2468 3542 25i5 2688 2762 2835 2908 2981 73 5,3 3o55 3i28 3201 3274 3348 3421 3494 3567 3640 3713 ''I 594 3786 4517 5246 386o 3933 4006 4079 41 52 4225 .4398 4371 4444 "ll 5,5 4590 4553 4736 4809 5538 4882 4955 5o28 5ioo 5.73 ■'^ 5^6 5319 5392 5465 56io 5683 5736 5829 5902 73 5974 6047 6120 6193 6265 6338 6411 5483 6556 6639 73 6701 6774 6846 6919 6992 7064 "^'Jl 7309 7282 8006 8730 7334 8079 8802 73 600 7427 778i5i 7499 8224 ?2^ 7644 8368 8441 m 7862 8585 ^i 72 73 601 8874 8947 9019 9091 9163 9236 9308 9380 9453 9524 72 602 9596 9669 9741 9813 9885 "29 •101 'i''? •245 72 5o3 7803I7 0389 o46i o533 o6o5 1396 v,n 0831 0893 0965 72 604 1037 1109 I181 1253 1324 1S40 1612 i584 72 6o5 .755 1827 1899 1971 2042 21 14 2186 2358 2329 2401 72 6o5 2473 3189 2544 2616 2688 4189 283 1 2902 44o3 3046 3ii7 72 607 3260 3332 34o3 3546 36i8 3761 3832 71 60S 3904 X 4046 4118 4261 4332 4475 4546 71 609 4617 4760 483i 4902 4974 5o45 5ii6 5.87 5259 7< 610 785330 5401 5472 5543 56i5 5686 5757 5828 5899 5970 T 611 6041 6II2 6i83 6254 6325 6396 6457 6538 6609 5680 71 612 6751 6822 6893 6964 7035 7,55 7177 7248 7319 1^ 71 6i3 7460 8168 7531 7602 7673 838i ll^i 7815 8522 7885 ^ 8027 8098 7' bi4 8239 83io 8593 8734 8804 7> 6i5 8875 958 1 8946 9016 9087 9157 9228 9399 9369 9440 95io T 616 9651 9722 9792 9863 9933 •••4 ••74 •144 •2l5 7° 6.7 bi8 790285 o356 0426 0496 0567 0637 0707 0778 1480 0848 0918 70 0988 io5g 1129 "99 1269 1 340 1410 i55o 1620 70 619 1 691 1761 i83i 1901 I971 2041 2111 3l8l 2232 2322 70 620 792392 2462 2532 2602 2672 2742 2S12 2883 2932 3032 70 621 3092 3162 323i 33oi 3371 3441 35ii 358i 3651 3731 70 622 3790 4488 386o 3930 4000 4070 41 39 4209 4279 4349 4418 70 623 4558 4637 4697 4767 48361 4906 5043 5ii5 70 624 5i85 5254 5324 5393 6088 5463 5532 56o2 5741 58ii 70 625 588o |S 6019 6713 61 58 622- 6297 6366 6435 65o5 69 626 6574 6782 6852 6921 7060 7129 7.98 ^ 628 7258 7960 8o3t 74o« 8858 7545 8236 7614 83o5 837^ S^3 7821 85i3 ilr. 69 629 865 1 8726 87?o ^ ^ 9065 9i34| 93o3 9372 69 63o 799341 9409 S 30029 ooot 9478 0167 til 9754 9833; 9892 ^8 ^ 63 J o3o5 0373 0442 o5n| o38o 69 532 0717 0786 o854 0023 1609 2295 0992 1061 1129 i8i5 1198 1356 i8§4! 1953 25581 2637 1335 69 633 1404 1472 i54i 1678 1747 2021 69 6)4 J089 SI 58 3225 2363 2433 25oo 371P ^ 635 2774 2842 2910 3594 3979 3o47 3ii6 3i84 3253,' 3321 3339 635 3457 3525 3662 3730 3708 4480 3867 4546 3^5 4l 7016! 75i8| 8019' 85 20' 9020' 939519 940018 o5i6' 1014 iSii sooS 25o4 3aoa 3.195 3989 -L 3S67 39201 3973 4396 4449 1 4302 5o3o 5558 5980 6o33 6D07 6359 7033, 7083 7558! 7611 8o83| 8i35 8607! 8659 9130 9183 9653 9706 0176' 0228 0697, 0749 1218 1270 1738 1790 2258 »3io 2777! 2829 3296 3348 38 1 4; 3865 433,1] :43a3 4848; 4890 5364 54i5 5879 5931 6394 6445 6908, 6959 7422, 7473 7935; 7986 8447 8498 8959^ 9010 9470, 9521 9981 "32 0491 o542 1000 io5i 1509' i56o 2017] 2068 2524 2375 3o3l! 3o82 3538, 3589 4044! 4094 4549' 4599 5o54 5io4 5558. 56o8 6061! 6111 6554 6614 7066 711 4568, 7618 8069 81 19 8570 S620 907c 9130 9569 961 0068 oil o566 0616 io54 iii4 i5&i 1611 2038 2107 2354 26o3 2653 3o4t) 3009! 3148, 3544 359.I' 36j3' 4o38 4o88| 413-! 6o85 6612 7i38 7663 8188 8712 9235 9758 0280 0801 l322 1842 2362 2881 3399 3917 4434 4951 5467 5982 6497 7011 7524 8037 8549 906 9572 "83 0592 1 102 i6io 2118 2626 3i33 363n 4145 465o 5i54 5558 6162 6655 7167 7668 8169 867a 91J0 9661 0l6( o655 ii63 1660 2157 4o26i 407( 4535! 46oi 5o83! 5i35 56ii! 5664 6i38. 6664 7190 7716 8240 8764 9287 9810 o332 o853 1374 619 6717 7243 7768 8293 8816 9340 9862 o384 0906 1426 1946 »4i4 2466 2933. 2985 345i!.33o3 3969' 402 4486; 4538 5oo3. 5o54 55 18. 5570 6034! 608S 6548 6600 70621 7114 7576, 7627 80S8 8140 8601 8652 9112! 9163 9623 9674 •i34| " 0643', 0694 Ii53j i2o4 1661 1712 2169 2220 2677 2727 3i83j 3234 3690 3740 4195 4246 4700 53o5 5709 4751 5255 5759 6212 6262 6715, 6755 7217! 7267 7718, 7769 8219 8269 8720 8770 9220 9270 9719 9769 02 1 8 0267 0716 0763 12l3 1710 2307 1253 1760 2256 2702 2702 319S 3347 3692 3742 4186 4235 4 ! 5 4i32 4660 5189 5716 6243 6770 7295 7820 8345 8869 9392 9914 0436 0958 1478 25i8 4184 4713 5241 5769 6296 6822 7348 7873 8397 8921 9444 9967 0489 1010 i53o 2o5o 1 ^^1° 3o37] 3089 3555 3607 4072 4589 5io5 5621 6i37 665i 7i65 7678 Vio3 92i5 4124 4641 5i57 5673 6188 6702 7216 7730 8242 8754 9266 9725, 9776 •236, "sS? 0745 0796 :254 i3o5 1753; i8i4 2271' 2322 277S1 2829 3285 3333 3791, 3341 4296 4347 4801 4853 33o6; 5356 5So9 5S6o 63i3 6363 681 3| 6865 7317! 7367 7819 7S69 8320 8370 8820 8870: 9320 9369' 9819 9869' o3i7| U357: 081 5 o855; I3i3 i352 1800' 1839 23o6' 2355 2801 28S1 32971 3346 3791! 3841 4235, 4335 4237J 4290 4766: 4819 5294 5347 5822 6349 6875 740Q 7,923 8450 8973 9496 "19 o54l 1062 i582 2102 2622 3i4o 3658 4176 4693 52' It &240 6754 7268 7781 8293 88o5 9317 9827 •338 0847 i356 i865 2372 33^ 38g2 4397 4902 5406 3910 6413 6916 741S 79'9 8420 8920 5875 6401 6927 7453 OD02 9026 9049 ••71 0093 I114 l634 2i54 2674 3192 3710 4228 4744 5261 5776 629 68o5 7832 8345 8837 9368 9879 •389 0898 1407 1915 2423 2930 3437 3943 4448 4953 5457 5960 6463 6966 7468 7969 8470 8970 9419 9469 9918 9968 04171 0467 0915 1412 1909 24o5 2901 3396 3890 4334 8 0964 1462 1938 2455 2950 3443 3939 4433 9 I -^J A TABLE 0? tOGAKITHMS FKOM 1 TO 10,000. 15 N. 1 1 , 2 3 1 4 5 1 6 7 8 9 D. 88o 944483 4332 458 1 463i 4680 4729I 4779 4828, 4877: 49271 49 88i 4976' 5o23 5'469 33 1 3 3074 5i24 5173 5222; 5272, 5321 5370; 5419 49 882 5567 56i6 5665 57i5 6764! 58i3 5862 j 3912 49 883 3961 6010 6059 6108, 6157; 6207! 6256! 63o5 66oo" 6649I 6698', 6747; 6796 6354 64o3 49 884 6452 65oi 655i 6843, 6894 7336 7383 49 883 6943 7^3 8462 7041 7090! 7140 7189 7238, 7287 49 886 887 888 7434 7532 8022 7581! 763o 8070 8119 til llf, nil S313I lili 49 49 85 1 1 856o S609 8657 8706 87,55, 8804! 8853 49 889 8902 949390 8931 8999 9048 9097 9146 'M 9244 9202^ 9341 9780 9829 49 8go 9439 9488 9535 9585 9634 9731 49 S91 987B 9926 9975 0462 ••24 ••73i •121 ,•170 •219 •267 •3i6' 49 1 ^9? q5o365 o4i4 o5ii o56o 0608 o657 0706 0754 o8o3 49 893 o85i 1^1 0949 1435 S 1046 IP93 i58o 1143 1 192 1240 1289 1775 49 IH 1338 i532 1629 1677 1726 it ^9^ 1823 1872, 1920 2I5! 2017 2066 2114 2i63 2211 2260 896 23o8 2356 24o5 25o2 235o 3o^? 2647 2696 2744 48 897 898 2792 2841 2889 3373 2933 2986 3o34 3i3i 3i8o 3223 48 3276 3325 3421 3470 35i8 3566 36i5 36631 3711 48 899 3760 33o8 3856 3903 4387 3953 4001 4049 4098 4146 4194 48 900 954243 4291 4339 4433 4484 4532 458o 4628 4677 5o62 5iio' 3i38 48 901 4725 4773 4821 4869 4918 i'o 4966 3oi4 48 902 5207 5255 53o3 535 1 5447 5493 5343 5692, 5640 48 903 5688 5736 5784 5832 6928 5976 6024 6072 6120 48 904 6168 6216 6265 63 13 636 1 6409 6888 6457 65o5 6533 660 1 48 905 6649 6697 6745 6793 6840 6936 6984 7032 7080 48 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7539 48 lU t^ 7655 8.34 3793 8181 773i 8229 7799 8277 7847 8325 m 7942 8421 7990 8o33 .8468 85 1 6 48 48 9"9 8364 8612 8659 8707 8753 88o3 883o 8898 8946 8994 48 910 959041 9089 9137 9185 9232 928P 9328 9375 9423 947' 48 911 9518 9566 9614 9661 V^st 9757 9804 9852 9900 •376 9947 48 9IJ ,9995 .••42 ••90 •i38 •233 •280 •328 •423 48 913 960471 o5i8 o566 o6i3 0661 0709 0756 0804 o85i 0899 48 9U 0946 0994 1 041 1089 1 563 Ii36 1 184 I23l \',^ i326 i374 47 9.5 142 1 1469 1943 i5i6 161 1 1 658 1706 i8oi 1848 47 916 1895 1990 2038 2083 2l32 2i3o 2227 2273 2322 47 9'2 2-369 2843 2417 2464 25 1 1 2359 2606 2653 2701 2748 2795 47 918 2890 2937 2985 3o32 3079 3126 3174 3221 3268 47 919 33i6 3363 3410 3457 35o4 3552 3599 3646 3693 3741 47 920 • 963788 3835 3882 3929 3977 4024 4071 4118 41 65 4212 47 921 426a 4307 4354 4401 4448 4495 4542 4390 463j 3io8 4684 47 922 473 1 4778 4823 4872 4919 4966 5oi3 5o6i 5i33 47 923 5202 5249 5296 3343 5390 5437 5484 553i 3378 5625 47 924 5672 5719 5766 58i3 3860 5907 6376 3954 6001 6048 6095 47 923 6142 6189 665S 6236 6283 6329 6423 6470 63i7 6364 47 926 661 1 6705 6732 6799 6845 6892 ^7^ 6986 7o33 47 927 7080 7": 7173 7220 -7267 73i4 7361 7434 7501 47 928 7548 7642 Slog 7688 8(36 III! 7782 7829 P^ 7922 7969 47 929 8016 8002 8249 8296 3390 8856 8436 47 980 968483 833o 8376 8623 8670 8716 8763 8810 47 93. 8950 8996 9043 9090 9i36 9183 mi .9276 9323 47 93J Hl^ 9463 qoaq 9536 9602 9649 9742 9789 983; 47 933 9882 9928, 9975 ••21 ••63 •ii4 •161 •207 •234 •3oo % 934 970347 0393 o858 0440 0486 o333 0379 0626 0672 0719 Ii83 0765 ^H 0812 0904 1369 0951 0997 1044 1090 1 354 1187 I22t 46 936 1276 1322 I4i5 1461 i5o8 1601 1647 169^ 2157 46 tl 1740 1786 i832 1879 1925 2.383 1971 2018 2064 :iio 46 2203 2249 2295 2342 2434 2481 252- 2373 2610 46 939 N. 2666 2712 2708 2804 283i 2897 2943 2989 3o35: 3o82 46 : I 2 3 |- 4 3 6 7 8 1 9 ~D.' 10 A TABLE or LOGARITHMS FROM 1 TO 10,000. N. I 2 3 4 5 -6 7 8 9 46 940 973128 3174 3636 3220 3265 33i3 3359 34o5 345i 3497 3969 3543 94i 3590 4o5i 3682 3728 3774 3820 3866 3913 4.003 46 943 'Ml 4143 4189 4233 4281 4327 4788 4374 4834 4420 4466 46 943 45i2 4604 465o 4696; 4742 5i56 5202 4880 ^?o5 46 944 fBl 5oi8 5064 Silo 5248 5294 5340 5386 46 945 5478 lii 6854 5524 5570 56i6 5662 5707 5753 im 5845 46 946 5891 5983 6029 6488 6075 6121 6167 6626 6212 63o4 46 947 948 6330 6442 6533 6579 6671 6717 6763 46 6808 6900 ,358 6946 6992 7037 7495 7083 7129 7175 7220 46 949 7266 7312 74o3 7449 754. 7586 7632 854^ 7678 8i35 8591 46 960 95 1 977724 8r8i 8226 7815 8272 7861 83i7 W>3 8409 8043 85oo 46 45 952 8637 9093 8683 8728 8774 8819 9275 8865 8911 8956 9002 9047 46 953 9i38 9184 923o 9321 9366 9412 9457 95o3 46 954 9548 9594 9639 9685 9730 9776 9821 9867 X 995s 46 953 980003 0049 o5o3 0094 0140 018D 023l 0276 o322 0412 45 955 0458 o549 0594 0640; o685 073o 0776 0821 0867 45 \$l 0912 l366 0957 ioo3 1048 1093 1139 1184! 1229 i547 1592 1637 i683 1273 l320 45 1411 i456 i5oi 1728 .773 45 959 1819I 1864 1909 1954 2362 2407 2000 2045 2090, 2i35 2181 2226 45 960 982271 23i6 2452 1 2497 2543 2588 2633 2678 45 961 2723 2^69 2814 2859 2904 2949 2994 3356' 3401 3446 3o4o 3o83 3i3o 45 962 3175] 3220 3265" 33 10 3491 3536 358i 45 ^63 3626 3671 3716I 3762 3807! 3852 3897 3942 4392 3987 4o32 45 964 4077 4122 4167' 4212 4257' 43o2 4347 4437 4482 45 965 4527 4572 4617' 4662 4707 1 4752 4797 484a 4887 4932 5382 45 966 4977 3022 5067' 5lI2 5i57 5202 5247 5292 5337 45 967 0426 547 1 55i6! 556i 5606; 5651 5596 5741 5786 5830 45 968 5875; 5920 5963 6010 6o33 6100 6:44 6189 6234 6279 45 969 63J4 6369! 64i3 6458 65o3l 6548 6093 6637 6682 6727 45 970 98677J 6817] 6861 6906 6951; 6996 7040 7398 7443! 748S 7o85 7i3o 7175 45 97' 7219| 7264; 7309I 7353 7532 8024 8470 7622 806S 85.4 43 912 973 7666 77 1 1 7756 7800 81 13 8157I 8202 8247 7845; 7890 7934 82911 8336 838i 8737! 8782 8826 l%l 43 45 974 8559' 8604] 8648; 8693 .90051 9049' 9094 9i38 8871 8916 8960 45 975 9183 9227 1 9272 93 16 94o5 45 976 945o' 9494! 9539 9583 9628; 9672 9717 9761 9806 9850 44 978 98q51 9939I 9983 ••28 "72, 'in •161 •206 •25o •294 0738 44 ^S? o383 0428 0472 o5i6 o36i o6o5 o65o tr^ 44 979 0827 0S71 0916 1359 0960 1004 1049 1093 1536 1 182 44 9S0 991226 1270 i3i5 i4o3 14481 149a i53o 1625 44 981 1669 n.3 1758 1802 1846; 1890 2288; 2333 i 1933 2377 "979 2023' 2067 44 9S2 2111 2 1 56 2200 2244 2421 2465 25o9 44 983 2554 2598! 2642 3o39 3o83 2686 2730 2774 2819 2863 2907; 2931 44 984 34?6 3127 3568 3172, 3216 3260 33o4 3348 3392 3833 44 985 3480; 3524 36i3| 3657 3701 3745 37S9 44 986 3877 3921 436i 3965 4009 4o53. 409- 4493 4537 4i4i 4iS3 4229 4273 44 $1 4317 44o5 4449 458 1 4623 4660 47i3 44 4757 4801 4845 4889 4933 5372 nu 5o21 3o65 5io8 5i52 44 989 5196 5240I 5284 595635 5679 5723 5328 5460 55o4 5547 55oi 44 991 5767 58ii 5854 5898 5942 6380 5986; 6o3o 44 991 6074 6117 6161 62o5 6249 '6^?^ 6337 6424 5468 44 992 65i2 6555 6599 6949' 6993! 7037 73861 74J0; 7474 6643 6687 6774 6818 6862 6906 7299 7343 44 993 7080 7124 7168 7212 7235 44 994 75.7 7561 tt 7648 7692 8129 7735 8172 mt 44 995 7823 7867' 79>o 8259 83o3 8347 86q5 8739* 87S2 7904 8390 7998 8o85 44 90 8434 S477 8521 8354 85o8 8652 44 997 998 8S26 8869 93o5 8913 9348 8956 9392 9000 9043 9087 44 9i3i] 9174' 921S 9261 9435 9479 9913 9522 44 999 .N. 9565 9609I 9652 9696 9739 9783 9826 9870 9957 43 1 ' 1 2 3 4 5 6 7 8 !d. A TABLE OF LOGAEITHMIC SINES AND TANGENTS FOR EVERT DEGREE AND MINUTE OF THE QUADRANT. fiEMAEK. The minutes in the left-hand column of each page, increasing do-wnwards, belong to the de- grees at the top ; and those increasing upwards, in the right-hand column, belong to Ae degrees below. 18 (0 SEORBEB.) & TABLE OF lOGAItrniMlC M. Sine 0-000000 D. Cosine ! D. " Tarig." D. Cotaiig. 10-000000' O-3PO0OO Infinite. 60 I 6-463726 5017-17 000000 1 -00 6-453'726 5oi7 11 13-536274 59 235244 58 1 764756 940847 2934 85 000000; • 00 764755 940847 2934 3 2082 3i 000000! -00 2082 3i 059153 57 4 7-065786 162696 I6i5 11 000000. • 00 7-055786 i6i5 17 12-934214 56 5 l3l9 000000 •00 162696 i3i9 1 837304 55 758122 54 6 241877 11 15 11 9 •999999, -01 241878 1113 1 308824 966 366816 852 999999^ •01 308825 99^ 591175! 53 633 183; 52 8 54 999999 -01 366817 8d2 54 9 417968 762 63 -01 417970 463727 762 63 582o3o 5i 10 463725 689 88 -01 689 88 536273 5o 12-494880 49 457091 48 II ;-5o5ii8 629 81 9-999998 999997 -or : 7;D05l20 629 81 12 542906 579 36 • oi '542909 579 33 1 3 577668 536 41 999997 ■01 577672 536 42 422328, 47 14 609853 499 38 999996 -01 609857 499 39 390143, 45 1-5 639816 467 667845; 438 14 999996 -of 639820 43? i5 36oi8c 45 i6 81 999995 -01 667849 82 332i5i 44 \l 694173, 4i3 72 999995 -01 694179 4i3 ll 3o582i 43 718997 391 35 999994 -01 719004 391 l^'^l 42 19 742477; 3-1 27 999993 • 01 742484 371 28 41 20 754754 353 i5 999993 -01 764761 35i 36 235239 40 21 7-785943, 336 806.146 321 72 9-999992 -01 '■^% 336 73 12-214049I 39 22 75 999991 •01 321 .76 193843 38 23 8j545i| 3o8 o5 999990 999989 999988 ■01 825460 3o8 06 174540 37 24 843934' 295 % -02 843944 295 49 i56o56 36 25 86 1 662 ■ 283 -02 861674 283 90 138325 35 26 878695 273 2] 999988 -02 878708 273 18 121292 34 11 29 ,895085 263 999987 -02 895099 263 25 104901 33 910879 253 926119I 245 ?? 999986 999985| 999983 -02 -02 910804 926134 254 245 01 40 089106 073856 32 3i 3o 940842; 237 33 -02 940858 237 35 069142 3o 3i 7-955082 229 80 9-999982 • 02 ■'■^^SiS" 229 81 1 2 ■ 044900 It 32 968870 982233 222 73 999981 -02 968889 982253 222 75 o3iiii 33 216 08 999980 -02 216 10 017747 27 34 995198 8-007787 209 203 81 999979 • 02 995219 12 83 004781 26 35 90 999977 -02 8 007809 020045 ?3 11-992191 23 36 020021 198 3i 999976 • 02 198 24 u 031919 043501 193 02 999975 -02 o3 1 945 043527 \tl o5 23 I §8 01 999973 -02 o3 955473 22 39 054781 i83 25 999972 -02 05480Q i83 27 945191 21 40 o65p6 178 72 999971 -02 o658o6 178 74 934194 20 41 8-076300 086965 174 41 9-999969 999968 -02 8 07653 I ■74 44 11-923469 9 1 3oo3 \l 42 170 166 3i -02 086997 170 34 43 097183 ll 999966 -02 097217 166 42 902783 892757 883o37 17 44 107167 116926 162 999964 .03 107202 162 68 16 45 \lt 08 999963 -o3 1 16963 \U 10 i5 46 126471 66 999961 -o3 126310 68 8-;3490 14 ii i358io l52 38 999>;.D9 9999581 •o3 1 35851 l52 41 864149 |3 144953 149 24 oJ 144996 149 146 27 855004 12 49 1 53907 146 22 999956 -03 153952 27 846048 II 5o 162681 143 33 999954, -03 i627i7 143 36 837273 10 5i 8-171280 140 54 9-999952, -o3 S-171328 140 57 11 82S672 I 52 179713 137 85 999950: 999948 -03 i88o36 137-90 135-32 820237 53 187985 190102 i35 29 -03 81 1964 8o3844 I 54 l32 80 999946 -03 196156 132-84 55 204070; i3o 41 999944 .o3 204126 i3o-44 796874 788047 5 55 211895 128 10 999942 • 04 211953 219641 128-14 4 U 219581 125 87 999940 ■ 04 125-90 780359 772805 3 227134 123 ^4 999938 ■ 04 227195 123-76 121-58 2 5, 234557 121 999936, -04 234621 765379 6o 241855 119-63 999934I -04 241921 119-67 758079 Cosiiie 1). Siuo |8D° Cotang. D. Tang. M. SINffiS AND TANGENTS . (1 DEGREE.) 19 M. Sine D. Cosine D. 1 Tang. : X>- Cotang. 8-24i855 119.63 9-999934 • 04 8-241921 1 H9-67 11-768079 60 I 249033 117-68 999932 .04 249102 ; i'7-72 115-84 760898 5I 3 266094 I1D.80 999929 .04 256i65 743835 3 263042 113.98 999927 999925 .04 263ii5 114-02 736885 u 4 2698B1 112-21 .04 269956 112-25 730044 5 276614; iio-5o :83243: 108-83 999922 .04 276691 110-64 723309 55 6 999920 .04 283323 108-87 716677 64 I 289773 lO-J-21 999918 .04 289866 107.26 710144! 53 1 296207 100-65 999915 .04 296292 302634 105.70 703708 5j 9 302546 104.13 999913 .04 104.18 697366 5i 10 308794 102-66 999910 .04 308884 102-70 691116 5o , '1 8-3i4904 101-22 9.999907 .04 8-3i5o46 101-26 672886 1? 12 321027 327016 99-82 999905 .04 321122 99-87 i3 98-47 999902 .04 327114 98-61 47 ■4 332924 97-14 999899 .o5 333025 97-19 96-90 666976 46 i5 333753 93-86 999897 ■o5 338856 661144 45 i6 344504 94-60 999894 :o5 344&10 94-65 660390 44 \l 35oi8i 93-38 999891 •o5 350289 36i43o 93-43 649711 43 355783' 92.19 999888 .o5 92-24 644 io5 42 19 36i3i5 01-03 999885 .06 91-08 6o-o5 88-85 638570 41 20 366777 999882 .o5 J66895 633io5 40 21 8.372171 9.999879 .o5 8-372292 11-627708 ^ 22 '& 87.72 999876 •o5 377622 382889 87-77 622378 23 86.67 999873 .o5 86.72 617111 37 24 387962 85.64 999870 .05 388092 393234 85.70 611908 36 25 393101 84.64 999867 .o5 84-70 606766 35 26 398179 83.66 999864 .o5 398315 83.71 601686 34 2^ 4o3i99 82.71 999861 .05 4o3338 82-76 81-82 596662 33 408161 tu 999858 •o5 4o83o4 691696 586787 32 ?9 4i3o68 999854 •o5 4i32i3 80-91 3i 3o 417919 79.96 999851 •06 418068 80-02 581932 3o 3i 8.422717 ]U 9.999848 ■06 8-422860 427618 70.14 Ii.577i3i It 32 427462 999844 •06 78.30 572382 33 432156 Pi 999841 .06 4323i5 77-45 567686 27 ¥. 436800 999838 .06 436062 441 56o •^^.63 563c38 26 35 441394 73-77 999834 •06 75.83 568440 26 36 445941 74-99 999831 •06 446110 73.06 553B50 24 ll 45o44o 74-22 999827 •06 45o6i3 74-28 549387 23 454893 73.46 999823 .06 455070 469481 73.52 544930 540519 22 39 459301 463665 72.73 999820 •06 72-79 21 40 72.00 9998161 .06 463849 72.06 536i5i 20 41 8.467985 71.29 9.999812 • 05 8-468172 71.35 11-531828 \l 42 472263 70.60 .06 472454 70.66 r?i 6? '65 627646 43 476498 480693 69.91 .06 476693 480892 485o5o 523307 17 44. tilt 999801 .06 51-9108 16 45 484848 999797 .07 614960 i5 46 488963 i-!' 999793 .07 480170 493260 68-01 5io83o 14 • % 493040 999790 .07 67-38 506760 i3 497078 5oioSo 66.69 999786 .07 497293 66-76 502707 12 49 66-08 999782 .07 501298 66-i5 498702 11 5a 5o5o45 65-48 999778 .07 506267 65-55 494733 10 1 5i 8.508974 512867 64-89 9-999774 .07 8-609200 5169^1 64-96 11-490800 I 52 64-3i 999769 999765 .07 64-39 486902 53 516726 52o55i 63-75 .07 63-82 483o39 7 54 63.19 999761 -07 620790 63-26 479210 473414 6 55 524343 62.64 999767 ■07 524586 62-72 5 55 528102 62.11 999753 .07 528349 62-18 471661 467920 4 u 531823 61.58 999748 -07 532080 61 -65 3 535523 61.06 999744 -07 535779 61 -i3 464221 3 ^ 539186 60-55 999740 .07 539447 60-62 460553 1 00 542819 60-04 999735 .07 543084 60-12 466916 1 CbsiQe 1 D. Sine ! S8° Cotang. D. T-ing 1 1 20 (2 DEGREES.) A TABLE OF LOG iSIlftMIC M. Sine D. Cosine D. Tang. ' D. Cotang. 8-542819 60-04 9-999735 .07 8-S43084 60-12 II -455916 60 I 546422 59-55 999731 -07 546691 59-62 453309 U 2 549995 553539 5g-o6 58-58 999726 -07 550268 59-14 58-66 449732 3 999722 ■08 553817 446183 57 4 557054 58-11 999717 -08 557336 58-19 57-73 442664 56 5 56o54o 57-65 999713 .08 560828 439172 55 6 563999 567431 57-19 999708 •08 564291 57-27 & 54 I 56-74 56 -io 999704 999699 -08 567727 56-82 53 570836 •08 571137I 56-38 428863 52 9 574214 55-87 999604 .08 574520 55-95 420480 5i 10 II 8.5^^392 55.44 55-02 999689 9-999685 .08 .08 s.UtU 55-52 55- ic 422123 11-418792 5o 1? 12 584193 54-60 999680 .08 584514 54-68 415486 i3 587469 54-19 999675 .08 587795 591051 594283 54-27 4i22o5 ^1 U i5 590721 593948 53-79 53-39 ^^11 .08 .08 53-87 53-47 53-08 408949 405717 46 45 i6 597152 53-00 999660 -08 597492 4o25oS 44 \l 6oo333 52-61 999655 -08 6oo6''7 52-70 52-32 399323 43 603489 52-23 99yD3o -08 6o3839 606978 396161 42 >9 60662J 51-86 999645 09 51-04 51-58 363022 11-3861?! 41 20 609734 8-612823 5i-49 999640 ■09 610094 8-6131S9 40 21 5i-i5 9 -99963 5 -09 5l-2I 39 33 22 613891 50-76 999629 • 09 616262 50-85 383738 23 618937 00-41 999624 -09 6i93i3 5o-5o 380687 ^,^ 24 621962 5o.o6 999619 -09 622343 5o-i5 377657 374548 36 25 624965 49- -72 49-38 999614 -09 625352 49-8i 35 26 627948 999608 -09I 628340 49-47 371660J 34 2^' 6309 n 49-04 48-71 999603 -09' 63i3o£ 49-13 48-80 368692! 33 633854 999597 -09 634256 355744 362816 32 29 636776 639680 48-39 999592 999586 -09 637184 48-48 3i 3o 48-06 -09 640093 -09] 8 •642982 ■09] 645853 48-16 11-357018 So 3i 8-642563 47-75 9-999081 47-84 29 28 32 64542 8 47-43 999075 47-53 354147 33 648274 47-12 999570 ■09! 648704 47-22 46-91 46-61 351296 27 34 65i 102 46-82 999564 -09 65 1 537 348463 26 3d 603911 46-52 999558 -10 654352 343548; 25 1 36 656702 46-22 99955^ -10 657149 65992? 46-31 34285 I 24 ll 659475 45-92 999547 ■10 46-02 340072 23 662230 45-63 999041 -10 662689 665433 45.73 337311 22 39 664968 45-35 TO335 -10 43-44 334557 21 40 667689 8-67o3q3 673080 45 06 999529 -10 66816c 45-26 331840 20 4i 44-79 9-999524 -10 8-670870 44-88 11-329130 \t 42 44- 5i 999518 ■ 10 673553 44-61 326437 43 675751 44-24 999512 -10 676239 44-34 323761 \l 44 678405 681043 43-97 999506 -10 67«nnn 44-17 321100 45 43-70 999500 -10 681544 43-80 3 18455 i5 46 683665 43-44 999493 -10 684172 686784 43-54 3 1 5828 14 % 686272 688863 43-18 999487 -10 43-28 3i32i6 i3 42-92 42-67 999481 • 10 689381 43 -o3 3io5i9 12 49 691438 999475 999469 9-999463 •10 691953 694029 42-77 3oSo37 II 5o 8-696543 42-42 • 10 42-52 305471 10 5i 42-17 •11 8-697081 42-28 1 1 -302919 3oo3S3 t 52 699073 701589 41-92 41-68 999456 •II 699617 42 -o3 53 999450 •II 702 l3q 704646 41-79 297861 1 7 295354. 6 54 704090 41-44 999443 -II 41-OD 55 706577 41-21 999437 -11 707140 41-32 292860 5 56 709049 40-97 999431 -11 709618 41-08 2903S2 4 u 711507 40-74 999424 •n 712083 40-83 287917 3 713952 716383 4o-5i 999418 -11 714534 40-62 285465 3 5, 40-20 40 05 999411 -11 716972 40-40 283028 I 6a 718800 999404 ■ 11 719396 40-17 280604 M. Cosine D. Sine 87°l Cotang. D. Tang. SINES AND TANGENTS . (3 DEGREES. ) 21 11. Sine D. ■ Cosi&e D. Tang. D. Colang. 1 8-718800 40-06 9-9tf94o4 •11 »^7i.9396 40-17 11-280604'! 60 t 721204 39-84 999398 ■ II 721806 39-95 278194: 5g 275796 58 2 723595 3g-52 999391 999384 -n 724204 39-74 3 728337 39-41 • II 7J5588 39-52 273412! 57 4 3?-9? 999378 -II 728959 73i3i7 39-30 271041! 56 5 , 730688 99937' -II lut 268683, 55 6 1 733027 38-77 999364 -12 733663 266337! 54 7 ■ 735354 38-57 999357 • 12 735996 38-68 264004! 53 8 737667 38-36 999350 -12 7383.7 740626 38-48 261683, 5s 9 739969 38-i6 999343 -12 38-27 259374 1 5i 10 742259 37-96 999336 -12 742922 38-07 257078 5o II 8-744536 37-76 37-56 9-999329 • 12 8 •740207 37-68 11-254793 49 12 746802 999-322 -12 747479 252521 48 l3 749055 37-37 9993i5 -12 749740 37-49 250260 % 14 751297 36-98 999308 -12 7^1989 37-29 24801 1 i5 753528 99g3oI • 12 754227 756453 37-10 345773 45 l5 755747 36-79 999204 999286 -12 36-92 243547 44 \l 757955 36-61 -12 758668 36-73 24i332 43 76oi5i 36 -'43 999279 -12 760872 36-55 239128 236935 42 19 762337 36-24 999272 -12 763o65 36-35 41 20 764511 36-06 999265 -12 765246 36-iS 234754 11-232583 40- 21 8.766675 35-88 9-999257 -12 8^7674i7 76957^ 36-00 39 22 768828 35.70 999250 -i3 35-83 230422; 38 1 23 770970 35.53 999242 -i3 w 35-65 228273 11 24 773101 33.35 999235 -i3 35-48 226134 25 775223 35-18 999227 -i3 775995 35-31 224oo5 35 26 777333 35-01 999220 -i3 778114 780222 35-14 231886 34 11 Win 34-84 999212 -i3 34-97 ■ 219778 217680 215592 33 34-67 99g2o5 -i3 783320 34-80 32 ?9 7836o5 34-51 999197 -i3 784408 34-64 3i 3o « ''l^^ll 34-3i 999189 -i3 786486 34-47 2i35i4 3o 3i 8-787736 34-18 9-9ggl8l -i3 8-788554 34-3i 11-211446 29 32 789787 34-02 999174 999166 -13 790613 34- 15 209387 28 33 33-86 -i3 792662 33-99 33-83 20733^ 27 34 793859 33-70 999158 -i3 794701 205299 2^ 35 7g588i 33-54 999 I 5o -i3 796731 33-68 203269 25 36 797894 33-39 33-23 999U2 -i3 798752 800763 33-52 201248 24 u itiZ 999134 -i3 33-37 199237 23 33-08 ■999126 -i3 802765 33-22 ig7235j 22 195242 21 39 803876 l^-9l 999118 -i3 804758 33-07 40 8o5852 32-78 999110 -i3 806742 32-g2 193258 20 41 8-807819 32-63 9-999102 -i3 8-808717 32-78 u-191283 189317 19 42 809777 32-49 999094 -14 810683 32-62 18 43 811726 32-34 999086 -14 812641 32-48 187359 n 44 813667 32-19 32 -o5 999077 ■14 814589 32-33 I854II 16 45 815599 999069 -14 816529 32-19 32-o5 183471 i5 46 817522 31-91 9ggo6i •14 818461 181539 14 tl ^'9436 3i-77 99go53 -14 820384 31-91 179616] i3 821343 3i-63 999044 •14 822298 31-77 177702 12 P 823240 3i-49 999036 ■14 824205 31-63 175795 11 5o 825i3o 31-35 999027 ■14 826103 3i-5o •73897 11-172008 10 5i 8-827011 828884 3l-22 9.999019 •14 8-827992 3i-36 t 52 3i-o8 999010 ■1-4 829874 3i-23 170136 53 830749 3o-g5 990002 •14 831748 3i-io 168253 I 54 832607 3o-82 998993 •14 8336i3 30-96 166387 55 834456 30-69 998984 -14 835471 3o-83' 164029 i 56 S36297 838i3o 3o-56 998976 •14 837321 30-70 162679 4 57 3o-43 998967 -i5 839163 3o-57 160S37 3 58 839955 3o-3o 99S958 -i5 840998 3o-45 159002 2 ^ 841774 30-17 ggSgSo •i5 842825 30-32 167175 I 5o 843SBD 3o-oo 998941 •i5 844544 3o-i9 155356 Cosima D Sine S6°l Cotang. D. . Tang. M. 22 (4 DBORKSS .) A TABLE OF LJGARITIiMIC M. 1 Sine 1 d; Cosine D. Tang. ■ D. T5tai(g. 60 ■8-843585 { 3o-o5 9.998941 ■ i5 8-844644 30.19 11.1^5356 I 845381 847183 848971 29-92 998932 ■ i5 846455 30.07 J 53545 59 1 29-80 998923 .i5 84?26o 29.95 29.82 j5i74o 58 3 29-67 998914 .i5 85oo57 149943 1481 54 57 4 85075 1 29-55 998817 .i5 85 1846 29.70 56 5 852525 29-43 -15 853628 29.58 146372 55 6 854291 29-31 .i5 855403 29.46 144597 54 I 856049 29-19 998878 .i5 857171 858,32 29.35 142829 3i .41068 52 857801 29-07 28-96 28-84 998869 .i5 29.23 9 859546 998860 ■l5 860686 29.11 .39314 01 10 861283 99885. •i5 862433 29.00 28.88 .37567 5o II 8-863014 28.- 73 9.998841 •15 8.864173 II -135827 § 12 864738 28-61 998832 .i5 865906 28.77 .34094 i3 866455 28 -So 998623 .16 867632 28.66 .33368 47 14 868i65 28.39 998813 .16 869351 28.54 .30649' 46 1 i5 869868 28-28 998804 .16 871064 28-43 128936 ' 45 i6 871565 28-17 ffl .16 872770 28-32 127280 44 \l 873255 28-06 .16 874469 28.21 12553. 43 874938 l]t 998776 • 16 876162 28. 1 .23838 42 "} 876615 998766 .16 877849 28.00 .22l5l 41 20 878285 27-73 998757 ■ 16 879529 27.89 1 2047 1 40 21 8-879949 27.^3 9.998747 .16 8.88i202 27.79 27-68 II 118798 ll 22 881607 883258 27-52 998738 .16 882869 117.31 23 27-42 998728 • 16 884530 27-58 ..5470 l2 24 884903 886542 27.31 998718 •16 886185 27-47 .138.5 36 25 2-7-21 998708 .16 887833 27.37 .12167 35 26 888174 27-11 ^86?9 .16 889476 27-27 iio524 34 11 889801 27-00 •16 89.1.2 27.17 108888 33 891421 26-90 26-80 998679 • 16 892742 27.07 I07258 32 =9 893035 998669 ■17 894366 26.97 105634 3i 36 894643 26-70 998659 •17 8.897?96 26.87 104016 3o 3i 8-896246 26.60 9.998649 -n 26.77 II.IO3404 ll 32 897842 26-51 998639 •17 899203 26.67 100797 33 899432 26-41 998629 •■7 900803 26.58 099197 27 34 '^IV^ 26-31 ^8619 ■n 902398 903987 905570 26.48 097602 26 35 26-22 998609 •>7 26.38 0960.3 25 36 904169 26-12 998599 -17 26.29 094430 24 ll 905736 26-03 » •17 907147 26-20 092853 23 l^&tl 25-93 25-84 •17 908719 26-10 09.281 22 39 998568 ■17 910285 26-01 0§o7.5 2. 40 910404 25-75 998558 •17 911846 25-92 25-83 088.54 20 41 8-911949 913488 25-66 9-998548 •17 8.913401 I. -086599 \l 42 S5-56 998537 ■17 914951 25.74 o85o49 o835o5 43 9l5o22 25-47 9,8521 998516 ■M 916495 918034 2555 \l 44 9i655o 25-38 25.55 081966 45 918073 25-29 998506 •i8j 919568 '^''2 080432 i5 45 9195911 921103 25-20 25-12 998495 998485 ■ 18 921096 .18 922619 25.38 25.30 .« 14 i3 922610 25-o3 998474 .18 924136 25.21 *075864 12 49 924112 24-94 24-86 998464 .18 92564c ■18 927156 25. .2 07435. II 5o 925609 998453 25.03 072844 10 5i 8-927100 928587 93oof3 24-77 24-69 9-998442 ■18 8-928658 24.95 34.86 11.071342 I 5a 998431 •18 93oi55 ^6^53 53 24-60 998421 ■18 931647 24.78 I 54 931 544 24.52 998410 ■18 9331 34 24.70 24.61 066866 55 93301 5 24-43 & •18 934616 055384 5 56 934481 24-35 ■18 93609" 937565 939032 24.53 06390- 4 u 935942 932io8 24.27 24-19 -i .18 .18 24-45 24.37 06243: 05^4! 3 1 5, 938850 24.11 998355 .18 940494 24.30 I 6o 940J96 14.03 998344 .18 941952 34.21 ' Cosine D. __Siae_ J5 = Cotaug. D. Tang. _«• SINKS AND TANGENTS (5 DEGREES-) 23 M. Bine D. CoBino B. Tang- T>. Cotang. ' 8 940295 941733 24 -03 9-998344 •19 8-941962 24 21 1 1 •068048 5o I 23-94 23-67 998333 .19 943404 24 i3 055596 u 2 943174 998322 ■ 19 944852 24 o5 o55i48 3 944606 23-79 998311 ■ 19 946195 947734 23 97 o537o5 ll 4 946034 23.71 23-63 998300 •19 23 tl 062266 5 947456 948874 950287 951696 998289 •19 949168 23 o5o832 55 6 23-55 Sii .19 950697 33 1^ 049403 54 7 23-48 • 19 992021 23 66 047979 046539 53 8 23-40 998255 • 19 953441 23 60 52 9 953100 23.32 998243 .19 954856 23 5i 045144 5i 10 9544*9 23-25 998232 •19 9S6267 8-957674 23 44 043733 5o II 8.955»94 95J284 958670 960032 23-17 9.998220 •9 23 37 11-042326 1? 12 i3 23-10 23-02 998209 998'97 998166 •19 •19 ■9S9075 961825 23 23 l^ 040925 039527 036134 U 22-95 22-88 22-80 . -19 23 14 i5 i6 961429 964801 998174 998163 -19 • 19 963255 964639 2.3 23 07 00 038745 oJ535i 45 44 3 964,170 22-73 998151 -19 966019 22 ?? 033981 43 965334 22-66 998130 998128 -20 95^7^6 22 86 032606 42 >9 966893 22-59 -20 22 79 o3i234 41 20 968249 22-52 998116 -20 970133 22 7' 029867 11-028604 40 21 8-969600 22-44 9-998104 -20 8-071406 972855 22 65 ll 22 970947 22-38 ogsX • 20 22 57 027145 23 972289 22.31 -20 974209 22 5i 026791 37 24 973628 22-24 998068 -20 975550 22 44 024440 35 1 25 974962 22-17 998056 -20 976906 22 37 023094 35 26 976293 22-10 998044' -20 978248 22 3o 021762 34 u 977619 981573 8.982883 S2-D3 998032: -20 979586 980921 22 23 020414 33 21-97 998020I -20 22 '7 019079 32 ?9 21-90 998008 -20 982201 22 10 017749 or6423 3i 3o 21-83 997906 9-997985 ••20 983577 22 04 3o 3i 2I.-77 •20 8-984899 21 97 ii-oiSioi 29 28 32 9841 89 21-70 997972 997959 -20 986217 21 11 013783 33 985491 21-63 •20 987532 988842 21 012468 ll 34 986789 988083 21-57 997947 997935 •20 21 78 011158 35 21 -5o •21 990149 21 7' 00985 1 006549 25 36 989374 21-44 997922 •21 991451 21 65 24 ll 990660 21-38 997910 •21 99? 7^0 21 58 oopSo 003955 23 991943 21-3l 997897 •21 994045 21 52 22 39 993222 21-25 9978§5 •21 995337 21 46 004663 21 40 8 ■995768 21-19 997872 -21 996624 21 40 003376 20 4i 21-12 9-997860 -21 8-997908 21 34 11^002092 \l 42 997036 998299 21-06 997847 -Ji 999188 21 27 00081 2 43 2 1 - 00 997835 -21 9-000465 21 21 Io^g99535 996262 17 44 999560 20-94 20-87 997822 •21 001738 21 i5 16 45 9-000816 997809 •21 oo3oo7 21 It 996993 i5 45 ooao6g oo33i8 20-82 997797 997784 •SI 004272 21 996728 14 s 20-76 •21 005534 20 97 994466 i3 004563 20-7(1 997771 •21 006792 20 11 993208 12 f9 oo58o5 20-64 9977D8 •21 008047 009298 20 991953 II 5o 007044 9-008278 20-58 997745 -21 20 80 10 5i 20-52 9-997732 ■21 9.010646 20 74 988210 1 52 009610 20-46 997719 -21 011790 oi3o3i 20 68 53 010737 20-40 997706 •21 20 62 986969 7 54 01 1962 20-34 997693 •22 014268 20 56 986732 5 55 oi3i82 20-29 20-23 997680 •22 oi55o2 20 5i 984498 5 56 014400 997667 •22 016732 20 45 983268 4 ^5^ 0i56i3 26-17 997664 -22 017969 019183 20 40 982041 3 016824 20-12 997641 -22 20 33 . 980817 2 59 oi8o3i 20-06 997628 -22 020403 20 28 979597 978380 I 6o 019235 20-00 997614 •22 021620 20 23 M. Cosine D- Sine !84° Cotang. D. Tailg. 24 (6 DEGREBe .) A TABLE OF LOOAKITHMIC M. Sine D. Cosine D. Tung. D- Cotang- I )-Di9233 20-00 9-997614 -22 9-021620 20-23 10-978380 60 1 020435 19-95 19-89 997601 • 22 022834 20-17 ^7^956' P 3 021632 W588 -22 024044 20-11 3 012825 19-84 997574 • 22 025331 20-06 972343; 55 4 024016 19-78 997561 -23 036455 30-00 5 025203 19-73 997547 -22 027655 028852 19.95 6 026386 19-67 997534 •23 19.79 971148! 34 2 027567 028744 19-62 19-57 997520 997507 -23 -23 o3oo46 o3i237 9M 3J 52 9 029918 19.51 997493 997480 -23 032425 19.74 9^2?''^ S| 10 031089 19-47 •23 033609 19.69 966391 5o II 9-032257 19-41 9-997466 -23 9-034791 19.64 10-965209 1? u 033421 19-36 997452 -23 035969 .9-58 964031 l3 034582 19-30 997439 -23 037144 19-53 962856 52 U 035741 036896 19-25 997425 •23 0383 16 19-48 961684 46 i5 19-20 997411 •23 039485 '9-£ 96051 5 45 i6 o33o48 19-15 997397 •23 04065 1 19-38 955349 958187 44 ;? o3gi97 19-10 W383 •23 o4i8i3 19-33 43 o4o342 19-05 18-99 997365 •23 042973 19-28 957027 43 '9 041485 997353 •23 044 i3o 19-23 955870 41 ao 042625 18-54 18-89 997341 •23 045284 19-18 954716 40 21 9-043762 044895 9-997327 •24 9.046434 19-13 10-953566 It 22 18-84 9973 1 3 •24 047582 19-08 95241 8 23 046026 18-79 18-73 697299 ■24 048727 049869 o5iooS b52i44 19-03 •18-98 951273 il 24 047154 048279 049400 997283 •24 95oi3i 36 25 26 18-70 i8-65 997271 997257 •24 •24 18-93 18-89 If^^ 35 34 11 o5o5ig o5i635 i8-6o 997242 ■24 053277 18-84 946733 33 18-53 997228 ■24 o54407 18-79 945593 32 29 052749 053859 i8-5o 997214 ■24 055535 18-74 944463 3i 36 i8-45 997>9? 9-997185 -24 056659 •I'l? 943341 3o 3i 9-054966 iS-4i -24 9-057781 ig-65 10-942219 ll 32 o56o7i i8-36 997170 -24 038900 >fl9 941 100 33 057172 058271 i8-3i 997136 •24 060016 18-53 930984 938S70 ll 34 18-27 997141 •24 061 i3o l8-5i 35 059367 l8-22 997 '27 •24 062240 18-46 937760 25 36 060460 18-17 997112 ■24 063348 18-42 936652 34 ll o6i55i ,8-i3 997083 •24 064453 18-37 935547 33 062639 18-08 •25 065556 18-33 934444 33 39 053724 064806 i8-o4 997068 •25 066555 18-28 933343 31 40 17-99 997053 •25 9-068846 i8^24 93224S 30 41 9-065885 17-04 9.997039 •25 i8^i9 io-93ii54 \l 42 066962 17-00 997024 •25 069933 l8^13 930062 43 o68o36 17-86 997009 9061)94 •25 071027 i8-io 928973 \l 44 069107 17-81 •25 072113 18-06 9278S7 45 070176 '7-71 996979 •25 073197 074278 l8-02 926803 i5 46 071242 17-72 996964 •23 >7-97 925722 14 S 072306 17-68 996949 •25 075355 17-93 •7-89 924644 i3 073366 17-63 996934 ■25 076432 923568 12 49 074424 17-59 596919 •25 0775o5 17-84 922495 II 00 075480 17-33 9-c,96?89 -25 078576 17^80 921424 10 5i 0-076533 17-30 -23 9-079644 17-75 10-930336 t 52 077583 078631 17-46 996874 -25 080710 17-72 919290 918227 53 17-42 996858 -25 081773 oSc833 \Ul 7 54 079676 080719 17-38 996843 -25 917167 916109 6 55 17-33 996828 -25 083891 WM 5 56 081739 17-29 17-25 996812 -26 084947 9i5o53 4 u 082707 o8383a 996797 • 26 086000 ii-5i 914000 3 17-21 996782 -26 087050 088098 17-47. 17-43 912950' 2 59 084864 17-17 i7-i3 996766 • 26 911.02 I 6a 085894 996751 -26 089144 17-38 910836 Ooaine D. iSiuo 1 83° Gotani;. D. Tansr- M, BINKS AND TAN&ENT6 . (7 t. CUItlillSS. 26 ran Bino r ). Cosine D. Tting" 0. Cotiing. ' 9-085894 17 i3 9-996751 •26 9-089144 17-38 10-9108561 60 I 086922 17 09 996735 •26 09018-7 091228 17-34 908772 58 3 087947 17 088970 17 04 996720 • 26 17-30 3 00 996704 •26 092266 17-27 907784 57 4 089990 16 96 996688 •26 093302 17-22 906698, 5o 5 091008 16 92 996673 -26 094336 17-19 906664! 55 6 092024 16 88 996657 -26 095367 17-10 904633 54 n 093037 16 84 996641 -26 096896 17-11 9o36o5 53 8 094047 16 80 996625 •26 097422 17-07 902678! 5a 9 095056 *T6 76 9966 1 •26 09S446 17-03 16-95 901554! 5i 10 096062 16 73 996594 •26 099468 000532 10-899613 so II 9 '097065 16 098066 16 68 9.99657S •27 9-100487 49 12 65 996662 -27 ioi5o4 16-91 898496; 48 i3 099065 16 61 996546 •27 102619 i6-g7 896468I.46 14 100062 16 57 996530 ■27 103532 16-84 i5 ioio56 16 53 99651 4 •27 104542 16-80 895468 45 i6 1*12048 16 s 996498 •27 io555o 16-76 894450 44 \l io3o37 16 996482 ■27 106556 16-72 893444 43 104025 16 41 996465 •27 107559 io856o 16-69 i6-65 892441 1 42 '9 loSoio 16 33 ti^ •27 891440 4i 20 10599^ 16 34 •27 109559 16-61 8904411 40 10-889444 39 888449 38 21 9-106973 16 3o 9-996417 •27 9-iio556 i6-58 22 107951 16 27 996400 ■27 1 1 1 55 1 16-54 23 108927 16 23 996)84 •27 112543 i6-5o 887457: 37 886467 36 24 109901 16 '9 996368 •27 113533 16-46 25 110873 16 ■16 996351 -27 114621 16-43 885479! 35 884493: 34 26 11 1842 16 12 996335 ■27 11 5507 16-89 11 112809 '6 08 996818 ■■U 116491 i6-36 883509 33 .113774 16 o5 99j3d2 118462 16-32 882528; 32 29 1 14737 16 n5^9S i5 9-H6656 i5 01 996235 • 28 16-29 16-2! 881648; 3i 3o 97 996269 .28 119429 880671 3o 3i 94 9-996252 .28 9 - 1 20404 l6-'22 10-879696 29 32 . 117613 i5 118567, i5 ^7 996235 -28 '2 '377 122848 16-18 87S623 28 33 -9962 19 -28 i6-i5 877662 27 34 119519 i5 83 996202 ■23 128817 i6-ii 876683 26 35 120469 1 5 80 996185' -28 124284 16-07 876716 25 36 121417 i5 76 -996168 ■ 28 126249 16-04 87.4751 24 37 33 122362 l5 73 ■ 996151 •28 12621 1 i5-oi 873789 23 ,872828 22 i233o6 ID tt 996134 ■28 127172 ■ >5-97 39 124248 1 5 996117 -28 128180 i5-94 871870 21 40 125187 i5 62 996100 •23 1 29087 16-91 870918 20 10-S69959' 19 86c|Oo6| 1 8 86So56j 17 1 41 9-126125 i5 59 9-996088 ■29 9-i3oo4i .5-^7 42 127060 1 5 56 996066 • 29 180994 13-84 43 127993 i5 52 996049 •29 131944 182893 138839 i5-8i 44 128925 i5 129804 i5 tl 996082 .29 15-77 867107 16 45 9960161 •29 15-74 866161 i5 46 130781 i5 42 995998 •29 134784 .5-71 866216 14 47 131706 i5 39 996980 -29 135726 '^^■' ,864274 13 48 132630 i5 33 996963 ■29 186667 15.64 863333 12 49 i3355i i5 32 996946 •29 187605 i5-6i 862896 861458 11 5o 134470 i5 9-135387 i5 It 996928 ■ 29 138542 15-58 10 5i 9-995911 •29 9-189476 15-55 10-860624 9 52 i363o3 1 5 22 995894 ■29 140409 i5-5i 869591 8 53 137216 i5 '9 995376 • 29 i4i34o 15.48 858660 I 54 i3Bi28 i5 16 995359 -29 142269 i5-45 867731 55 189037 i5 12 996841 -29 143196 i5-42 856804 5 56 139944 i5 09 996823 -29 144121 15.39 855879 854956 4 ll i4o85o i5 06 996806 -29 146044 i5-35 3 141754 i5 o3 995-88; • 29 ■45n66 146885 15-32 854034' 2 59 142655 1 5 00 995771,' • 29 16-29 853ii5, I 60 143555 14 96 995768' 1?2 147803 15-26 8521 97!; _Co6inb P Sine !!■ ■»o Co tano^. _P.._ Taiig. M. 26 (8 DEGREES.) A TABLE OF LOOARITHMIO M. 1 Sine . D Cosine D. Tang. J). Colang^ 6e' '9.143555 14 96 9-995753 • 3o 9-147803 i5-26 10-852107 851282 1 144453 14 93 995735 -3o 148718 i5-23 5? 2 i4534q 14 ll 995717 -3o 149632 l5-20 85o368 3 146243 14 995699 -3o i5o544 15-17 849456 U 4 147136 14 84 995681 -3o i5i454 i5.i4 848546 5 148026 14 81 995664 -3o 152363 i5.ii 847637 55 6 14801 5 149802 14 78 993646 -3o 153269 i5-o8 846731 845826 54 I 14 75 995628 -3o 154174 1 5-05 53 150686 14 72 995610 -3o 1 55077 15-02 844923 53 9 i5i569 14 69 995591 -3o .55978 14-99 844022 5i 10 1 52451 14 66 995573 -3o 156877 14-96 843123 5o II 9-i5333o 14 63 0-995555 -3o 9-157775 14-93 10-842225 it 12 i542o8 14 60 995537 -3o 158671 14-50 14-87 841329 840435 i3 i55o83 14 57 995519 -3o 159565 47 14 150957 14 54 995501 -3i 160457 14-84 83^53 46 i5 i5683o 14 5i 995482 -3i I6i347 14-81 45 16 157700 158569 14 48 995464 -3i 162236 14-79 837764 44 [I 14 45 995446 -31 i63i23 . 14-76 836877 43 159433 14 42 995427 ■3i 164008 14-73 835992 42 ■9 i6o3oi 14 39 995409 • 31 164892 14-70 835io8 41 20 161 164 14 36 995390 •3i 165774 14-67 . 834226 4D 21 9-162025 14 33 9-995372 • 31 9- 166654 14-64 10-833346 ll 22 162885 14 3o 995353 -3i 167532 14-61 832468 23 163743 14 27 995334 -3i 168409 14-58 83i59i '2 24 164600 U 24 995316 -3i 169284 14-55 83o7i6 36 25 165454 14 22 995297 993278 -3i 170157 14-53 829843 828971 35 26 166307 14 '9 -3i 171029 i4-5o 34 '■'i 167159 168008 14 16 995260 -3i 171899 14-47 828101 33 14 i3 995241 -32 172767 14-44 827233 32 29 168856 14 10 995222 -32 173634 14-42 826356 3i 3o 169702 14 07 995203 -32 174499 14-39 82530I 3o 3i 9-170547 14 o5 9-995184 -32 9-175362 14-36 10-824638 It 32 171389 14 02 995165 -32 176224 14-33 823776 33 172230 i3 99 995146 -32 177084 i4-3i - 822916 27 34 173070 i3 96 995 12T 995108 -32 177942 14-28 822o58 26 35 173908 i3 94 -32 178799 179653 14-25 821201 25 36 174744 i3 tl 995089 -32 14-23 820345 24 ll 175578 i3 995070 -32 i8o5o8 14-20 819492 818640 23 176411 i3 86 g95o5 1 -32 i8i36o 1417 22 39 177242 178072 i3 83 995o32 -32 182211 i4-i5 817789 21 40 i3 80 99501 3 -32 l83o59 14-12 816941 20 41 9-178900 i3 77 9-994993 •32 9-183907 14-09 10-816093 \l it \lllf, i3 74 994974 -32 184752 14-07 815248 43 i3 i 994953 •32 185557 186439 14-04 8i44o3 n 44 181374 i3 994935 •32 14-02 8i356i 16' 45 182196 i3 994916 994896 •33 187280 18S120 13-99 812720 i5 46 i83oi6 i3 64 .33 13-96 811880 i4 % 183834 i3 61 994877 • 33 188958 i3^^ 811042 l3 1 8465 1 i3 ll 994867 9^4838 •33 189794 190629 i3^9i 810206 12 49 18546& i3 •33 8^538 II 5o 186280 i3 53 994818 •33 191462 lO 5i 9. 1 8709 J i3 5i 9.994798 .33 9-192294 13^84 10-807706 I 5j .89^10 190325 i3 48 994779 994759 .33 193124 i3^8i 806876 53 i3 46 .33 193953 \l^ 806047 I 54 i3 43 994739 • 33 194780 8o5220 55 i3 41 994719 .33 195606 .3^74 804394 5 55 191130 i3 38 994680 .33 196430 13.66 803570 4 u 191933 i3 36 .33 197253 198074 802747 3 192734 i3 33 994660 .33 801926 3 59 193534 i3 3o 994640 -33 198894 13-64 801106 I 60 194332 i3 18 994620 •33 199713 i3-0i 800287 Oosino D. Siuo ' iio Cotanff. D. Tanjr. ft SINUS AND TAKOENTS . (9 DKGIIEKS. > 21 M, B'^ . D. Cosine D. Tang. D. Cotang. 1 60 9-i9433s 13-28 9-994620 -33 9-1997.3 200520 201345 i3-6i 10-800287 I 199 Uq 195925 13-26 994600 -33 i3-59 m^ 5? 2 13-23 994580 -33 i3*56 3 195719 l3-3I 994560 •34 202.59 13-54 7978 ir 57 4 i3-i8 994540 -34 202971 13-52 79^51? 55 5 i3-!6 994519 •34 203782 i3-49 6 I 9909 I li i3 99i499 -34 204592 13-47 795^08 54 ; \ 199879 i3-ii 994479 ■34 205400 13^45 794600 1 53 j 20C666 i3.o8 994459 994438 -34 206207 2070.3 i3-42 793793 792907 792183 52 9 r>oi45i l3-o6 •34 i3-4o 5i 10 302234 i3-o4 994418 ,34 207817 9.208619 i3-38 5o II 5 203017 i3-oi 9.994397 994077 •34 13-35 10.791381. ^ 12 203797 204577 205354 12-99 •34 209420 i3-33 789780 788982 I3 12-96 994357 •34 !''.3?20 i3.3i 47 14 12-94 994336 •34 nioiS 13-28 46 i5 2o6i3i 12-92 12-89 9943.6 •34 2fi8i5 i3-26 78SI85 45 i6 206906 994295 ■34 212611 13-24 786595 44 ;? 207679 12-87 994274 994254 -35 2.34o5 13-21 43 208452 12-85 -35 214198 214989 i3-i9 786802 42 '9 209223 12-82 994233 •35 13.17 i3.i5 78601. 41 20 209992 12-80 994212 .35 2.5780 784220 40 21 9-210760 12-78 9-994.91 .35 9-216568 l3.12 10.783432 3a 22 211026 12-75 994 r7i •35 217356 218.42 13.10 782644 38 23 2I22QI 2i3o55 12-73 994150 -35 i3.cS 781868 37 24 12-71 994129 994. OS -35 2.8926 i3.c5 781074 31^ 25 2i38i8 12-68 -35 219710 i3-c3 ■)8o290 35 26 214579 215338 12-66 994087 -35 220492 i3-(i 77o5o8 77S728 34 11 12-64 994066 .35 221272 2220D2 I2-(^ 33 216097 216854 12>6l 994045 -35 12-J7 777948 32 p 12-59 994024 -35 222830 12-1)4 777170 3» 3o 217609 9-218363 12-57 9g4oo3 -35 2236o6 12 -J2 776394 3o 3i 12-5D 9-993981 •35 9-2243S2 12-50 12-88 10.775618 ll 32 219116 12-53 993960 -35 225i56 774844 33 219868 12-5o 993939 9939. S -35 225929 12 86 774071 27 34 220618 12-48 -35 226700 12 84 773300 36 35 23i367 222115 12-46 993896 -36 227471 12 81 772629 25 36 12-44 993875 • 36 228239 12-79 771761 24 ll 222861 12-42 993854 ■36 229007 12-77 770993 23 3236o6 12-39 993832 •36 229773 12-76 770227 22 3g 224349 12-37 993811 -36 23o539 11-73 21 40 225oo2 9-225833 12-35 993789 9-993768 -36 23.3o2 12-71 10.767935 20 41 12-33 ■-36 9-232065 12 -6q ;? 42 226573 12-31 .993746 ■36 232826 d 767174 43 227311 228048 12-28 993720 -36 233586 766414 \i 44 12-26 993703 -36 234345 lJ-62 765655 45 228784 12-24 99368. 36 235 io3 I2-6o 764897 r5 46 229518 12-22 993660 -36 235859 12-58 764141 14 % 230252 12-20 993638 ■36 2366.4 2-56 763386 i3 230984 12-18 9936.6 -36 237368 238. 20 :2-54 762632 12 P 231714 12-16 993594 ■37 12-52 761880 5o S32444 12-14 993572 •37 238872 12-5o 761128 10 ■5i 9-233172 12-12 9-993550 •37 9-239622 12-48 10.760378 t 53 233899 234625 12-09 993528 i^ 240371 12-46 759629 ■J58882 52 12-07 993506 •37 241118 12-44 7 54 335349 236073 12-05 993484 •37 241865 12-42 768135 6 55 I2-03 993462 ■37 242610 .2-40 757390 5 56 3367,5 12-01 993440 •37 243354 12-38 766646 4 U 2375i5 238235 11-99 9934.8 •37 244097 12-36 755903 3 11-97 993396 •37 , 244839 12-34 755.61 3 52 238953 239670 1.-95 993374 .3i 245579 12-32 754421 1 6*) 1.-93 99335. ■37 346319 13-3o 753681 1 Ooaiue D. Sice 80° Cotms- D. ..?■««■ 28 (10 UEGKEEB.) A lAULE OF LOGAUITHMIC M. Sine D. Couine D- Tang. D. Cotang. 9-239670 11-93 9.993351 37 9.246319 12-30 10-7536S1 60 1 240386 11-91 993329 3t 247057 12-28 762943 5q 2 241101 11 -69 993307 37 248530 12-26 762206 58 3 241814 11-87 993285 37 12-24 761470 22 ' 4 242526 11-85 993262 37 249264 12-22 750736 56 5 243237 11-83 993240 ll 249998 260730 12-20 760002 55 b 243947 H-81 993217 12-18 mi 54 I 244656 11-79 993195 38 261461 12-17 53 245363 11-77 993172 38 262191 12-15 747809 53 9 246060 246775 11-75 993149 38 262920 12-13 747080 746352 5i 10 11-73 993127 38 253648 12-11 5o II 0-247478 248181 11-71 9-993104 ■88 9-254374 12-09 10-746625 S 12 11-69 993081 38 255 100 12:07 744900 i3 248883 \]:tl 993059 38 255824 13 o5 744176 47 14 249583 993o3o 38 256547 12-03 743453 46 i5 250282 11-63 9^013 38 267269 12-01 74.731 45 i6 250980 11-61 992990 38 268710 12-00 742010 44 \l 251677 \i:ll 992967 38 11-98 741290 43 252373 992944 38 269429 11-96 740671 42 '9 253067 11-56 992921 38 260146 11-94 739854 41 20 253761 11-54 99289S 38 260863 11-92 739137 40 21 9-254453 11-52 9.992875 38 9-261578 \\X io-738422| 39 1 22 255 144 11 5o 992852 38 262292 737708 38 23 255834 11.48 992829I 39 263oo5 11-87 736995 37 24 256523 11-46 992806 ?9 263717 11-85 736283 35 25 237211 11-44 992783 ^9 264428 11-83 735572 35 26 257898 258583 11-42 992759] ?9 265 1 38 11-81 734862 34 11 ■11-41 9927361 39 255847 ;;:7^^ 734153 33 259268 11-39 9927131 ^9 266555 733445 32 ? 25g95i 11.37 992690 ^9 267261 11-76 732739 732033 3i 3o 260633 11-35 992666! 39- 267967 39 9.268671 11-74 3o 3i 9-26i3i4 11-33 0-99i043: 11-72 10-731329 29 28 32 261994 ii-3i 992619 39 269375 11-70 730626 33 262673 11 -3o 992596 39 270077 11-69 729923 27 34 263301 11-28 992672 ^9 270779 \\:U 729221 728021 35 35 264027 264703 265377 11-26 992549 992D25 ?9 271479 272178 35 36 11-24 ?9 11-64 727822 34 ll 11-22 992501 39 272876 11-62 727124 23 26605 1 11-20 992478 40 273573 11-60 726427 32 3q 266723 1119 992454 40 274269 11-58 725731 21 40 267395 9 - 268065 268734 \\:\l 992430 40 274964 11.57 723035 20 41 9-992406 40 9.275558 11.55 10724342 \l 42 1113 992382 40 276351 11.53 723649 43 269402 11-11 992359 40 277043 •11 -51 722967 >7 41 270069 270735 U-IO 99233! 40 277734 278424 iiSo 722266 i5 45 Ii-o8 992311 40 11-48 721576I 1 5 46 271400 11-06 992287 40 2791 i3 11-47 720887; 14 ii 272064 11 -o5 992263 40 279801 380488 11-45 7201991 i3 iH 272726 273388 1103 992239' 40 11-43 718826 11 ^9 11.01 992214 40 281174 n-41 5o 274049 10.98 992190 9.992166 40 28i85S 11-40 71S142 10 5i " iX 40 9-282543 11-38 10-717458 9 5: 10.96 992142 40 383235 11-36 716775 8 53 276024 10. 94 992117 41 383907 u-35 716093 I ?i 276681 10. 92 992oq3 992069I 41 ■(8458S 11-33 716412 55 277337 10.89 41 280368 I1-3I 714732 5 56 278644 992044, 41 SI 11.30 714053 4 u 10-87 10-86 992020; 41 11-28 713376 3 279297 279948 260599 951996 41 287301 11-25 712699 3 1 59 10-84 99'97'i 41 ^m 11. 25 7 1202 J 1 1 6o 10.82 991947 ^-^ 11. 23 7I134S M. Cosine U. Sine I ?1 Cotang. D. Taw'. SINES AND TANGENTS. (11 DEOiiEES.) '29 M. Sine 9 280399 281248 U. Cosine | D. Tatig. D. Cotang. 10 82 9-99'9i7i -41 9-238652 11-23 10-711348 60 • 10 81 991922^ -41 289326 11-22 710674 ^2 2 281897 10 79 991897! •41 289999 11-20 -71 0001 58 3 282544 10 ]l 991873; -41 29067 1 11-18 'M 57 4 283190 283836 10 991848 -41 291342 11-17 56 5 10 74 991S23 •41 292013 II-15 707987 55 6 284480 10 72 991799 •41 292682 11-14 707318 54 I 285 1 24 10 7' 99' 774 •42 293350 11-12 7o665o 53 285766 10 69 991749 •42 294017 ll^II 7o5g83 7053 1 6 52 9 286408 10 67 991724 •42 294684 ii^og 5i 10 287048 10 66 991699 •42 295349 9-296013 Il^OJ 11 ■06 704651 5o II '•M 10 64 9-991674 •42 10-703987 703323 4o 12 TO 63 991649 •42 296677 11^04 48 |1 288964 10 6i 991624 ■42 297339 298001 ii-o3 702661 47 14 289600 10 5? 991599 •42 11-01 701999 701338 46 i5 290236 10 58 99'574 ■42 , 298662 11-00 45 i6 290870 10 56 991549 ■42 ■299322 10-98 700678 44 \l 291504 \ 10 54 991524 ■42 299980 10-96 700020 43 292137 292768 10 53 991498 •42 3oo638 . 10-95 699362 42 19 10 5i 991473 •42 801295 ' 10-93 698705 41 20 293399 10 5d 991448 •42 801951 10-92 698049 10-697393 696739 40 21 9-294029 2g4658 10 48 9-991422 •42 9-302607 10-90 lO-Sg 39 38 22 10 46 991397 •42 3o326i 23 2g5286 10 45 991372 •43 303914 304567 10-87 696086 37 24 295913 296539 10 43 991346 •43 10-86 695433 36 25 10 42 991321 •43 3o52i8 10-84 694782 35 26 S97164 10 40 991295 •43 3o586g 10-83 6g4i3i 34 11 297788 298412 10 39 991270 •43 3o65ig 307168 10-81 6g348i 33 10 37 991244 •43 10-80 692832 32 ?9 299034 10 36 991218 ■43 307815 10-78 692185 3i 3o 299655 10 34 991193 •43 3o8463 10-77 6gi537 3o 3i 9-300276 10 32 9-991167 •43 g-309109 10-75 10-690891 =2 32 300S95 10 3i 991 i4i •43 309754 10-74 690246 689602 28 33 3oi5i4 10 29 991115 ■43 310398 10-73 27 34 302i32 10 28 991090 ■43 311042 10-71 688g58 6883 1 5 26 35 3o2748 10 26 991064 .43 3ii685 10-70 25 36 3o3364 10 25 991038 •43 312327 10-68 687673 24 ^3 3o3g79 30439.3 10 23 991012 •43 312967 10-67 687033 23 38 10 22 990986 990960 •43 3i36o8 10-65 686392 685753 22 39 3o5207 10 20 .43 314247 10-64 21 40 3o58i9 10 '9 990934! •44 3 1 4885 10-62 6851 i5 20 41 9 -306430 10 17 9-9909081 990882: •44 9-3i5523 io-6i 10-634477 19 18 42 307041 10 16 •44 3i6i59 316795 317430 318064 10-60 683841 43 307650 10 14 990855; 990829 990803 •44 10-58 683205 17 44 308259- 10 i3 ■44 10-57 682570 16 45 30S867 10 1 1 •44 10-55 681936 68i3o3 i5 46 309474 3iooSo 10 10 990777 1 •44 318697 10-54 14 47 10 08 990750, •44 3i932g 10-53 680671 13 48 3 10685 10 07 990724; ■44 3igg6i 320592 io-5i 680039 i: |9 311289 311893 10 o5 990697, ■44 io-5o 679408 678778 .11 5o 10 04 990671 •44 321222 10-48 10 5i 9-312495 10 o3 9-990644 -44 9-32i85i 10-47 10-678149 ? 52 3 1 3097 313698 10 01 990618 -44 322479 10-45 677521 53 10 00 990391 ■44 323io6 10-44 676894 7 5i 314297 9 93 990565 •44 323733 10-43 676267 6 55 314897 9 97 990533, •44 324358 10-41 675642 5 56 30495 9 96 99o5_i 1 .45 3249S3 10-40 675017 4 Pu 316092 316689 9 94 990435 •45 325607 10-39 674393 3 58 9 93 990453, .45 326231 ■°'?I 673769 2 59 317284 9 91 990431, •45 326853 10-36 673147 I 6a 317879 9 90 990404 .45 327475 10-35 672325 CoBine L). Sine 780 Cotiinif. D. Tiuur- sr. 30 (12 DEOKEES.) A TABLE OF LOGARITHMIC a. 6 . , Sine D. GoaSlic D. . '-Taiig^ D. Ootaiig. 60 9-3r787g 318473 ■ ^8 9-99040/ .45 9-327474 10-35 10-672526 I 990378 .45 328093 10-33 671905 si 2 319066 9-87 99035 1 •45 828715 10-32 671285 3 319658 9-86 990324 .45 329334 io-3o 670666 11 4 320249 9.84 990207 .45 329953 330370 10^29 10^28 67004^ 669430 66S8i3 5 320840 9-83 990270 .45 ss 6 321430 9-82 990243 .45 331187 IO-56 54 ? 322019 9-8o 990215 .45 33i8o3 lo^jS 668197 667582 53 322607 9-79 990188 •45 332418 10^24 5j 9 323194 9-77 990161 •45 333o33 I0^23 666967 61 666354J 30 ic 32378a 9-324366 9-76 990134 ■ 40 333646 10^2I II 9-75 9-990107 -46 9-334259 10-20 13-665741! 49 665129' 48 ij 324950 325534 9-73 990079 -46 ' 334871 10-19 i3 9-72 990052 .46 335482 10-17 664518, 47 U 326ri7 9-70 990025 .46 336093 10-16 668907: 45 i5 326700 P^ 989997 .46 336702 10-13 653298; 45 662689' 44 i6- 327281 989970 •46 337311 10-13 :? 327862 328442 9-56 989942 •46 337919 338527 10-12 662081! 43 9-55 . 989915 •46 lO-II 661473, 42 19 329021 9.64 989887 •46 339133 lo-ro 660867 41 20 329599 9-62 989860 •46 339789 10-08 660261. 40 91 9-330176 9-61 9-989832 •46 9-340344 10-07 10-659655' 39 659002; 38 658448 37 ;2 33oi53 9-60 989804 •46 340948 341352 10-06 23 33i3;9 331903 9-58 989777 •46 10-04 24 9-57 99 362356 8 988163 .50 374193 374766 9 39 38 626807 41 20 362889 8 88 988133 • 50 9 626244 40 21 9-363422 8 87 9-988103 .50 9-375319 9 37 10-624681 It 22 363954 8 85 988073 -50 375881 9 35 624119 623558 23 364485 8 84 988043 -50 376442 9 34 37 24 36501 6 8 S3 988013 -50 377003 9 33 622997 36 25 365546 8 82 987983 -50 377563 37S122 9 32 622437 621S78 35 26 366075 8 81 987953 -50 9 3 1 34 11 366604 8 80 987922 987892 -50 378681 9 3o 621319 33 367i3i 8 79 -50 379239 9 l^ 620761 32 29 367659 368 1 85 8 77 987862 -50 379797 38o354 9 6202o3 3i 3o 8 76 987832 •51 9 57 619646 3o 3i 9-3687U 8 75 9.987801 •51 9-380910 9 26 10-619090 618534 2g 32 369236 8 74 987771 -51 381466 9 25 28 33 369761 8 73 987740 -51 382020 9 24 617980 ^I 34 370285 8 72 987679 .51 382675 9 23 617425 616871 26 35 370808 8 7' -51 383129 9 22 25 36 371330 8 70 987649 -51 383682 9 21 6i63i8 24 ll 371852 8 69 987618 .51 384234 9 20 616766 23 372373 8 tl 987588 •51 384786 9 [t 6i52i4 22 39 372894 8 987557 •51 385337 385888 9 614663 21 40 3734r4 8 65 987026 •51 9 17 6i4|i2 20 41 9-373933 8 64 9-987496 .5. 9-386438 9 i5 IO-613562 '2 42 374452 8 63 987465 •51 387536 388084 9 14 6i3oi3 18 43 374970 375487 8 62 987434 ■51 9 i3 *12454 \l 44 8 61 987403 •52 9 12 611916 611369 45 376003 8 60 987372 .52 388631 9 11 i5 46 376519 377035 8 5^ 987341 •52 389118 9 10 610822 14 47 8 987310 .52 389724 9 2 610276 i3 48 378063 8 57 > Z'S ■ 52 390270 9 609730 12 49 8 56 •52 390815 9 07 609185 608640 11 5o 378077 8 54 987211 9-987186 •52 391360 9 06 10 5i 9-379089 8 53 •52 9-391903 9 o5 10-608097 607553 9 52 lltll 8 52 987155 -52 392447 9 04 8 53 8 5i 987124 ' -52 392989 393531 9 o3 607011 ] 54 380624 8 5o 987092 . -52 9 02 606469 5 55 38ii34 8 49 987061 ; -52 394073 9 01 606927 5 56 381643 8 48 987030 ' -52 394614 00 6o5386 4 u 382152 8 S 986998 -52 395x54 a 9-3 604846 3 382661 8 986967 1 -52 395694 8 98 604306 2 59 383 168 8 45 986936 •52 396233 8 97 603767 I 6o_ 383675 8 •44 986904 1 ^52 396771 8-96 603229 Cosine D. Sine \7e° Cotanjf. 1>. X-^'ig.... __M._ 25* 32 (14 DEGREES.) A lABLE OP LOGARITHMIC M. Sine D. Cosine B. Tang. D. Cotang. 9-383675 8.44 9-986904 ■52 9.396771 S.gb io^6o3229 60 I 384182 8-43 986873 ■53 397309 S-gb 602691 39 6031 54 58 3 384687 8-42 986841 •53 397846 398383 8.^5 3 38DI92 8-41 986809 ■ 53 8^94 60 1 6 1 7 57 4 385607 8-40 986778 •53 398919 8^93 601081 56 5 386201 8-39 8-38 986746 • 53 399453 8^92 600545 55 5 386704 986714 •53 399990 8-91 600010 54 I 3S7207 8-37 9B6683! ■53 . 4oo52/ i-g° 599476 598942 33 387709 8-36 9S665ii •53 4oio58 8^89 53 9 388210 8-35 986619 •53 401591 8^88 598409 5i IC 3887 1 1 8-34 986587 •53 402124 8^87 597876 5o II 9'389aii 8-33 9-986555 •53 9^402656 8^86 10^597344 it 12 3897 1 1 8-32 986523 •53 403187 403718 8^85 596813 i3 390210 8.3i 986491 986459 •53 8^84 596282 ^1 14 390708 8-3o •53 404249 8^83 595751 45 i5 391206 8-28 986427 •53 404778 8^82 595222 45 i6 391703 8-27 986395 •53 4o53o8 8^8i 594692 44 n 392199 8-26 986363 •54 405835 8^8o 594164 43 i8 392695 8-25 , 986331 •54 405364 8^7o 593636 42 '9 393191 8-24 986299 •54 406892 8^7§ 593108 41 20 393685 8-23 986266 •54 407419 8-77 8^76 592581 40 21 9 -394 179 394673 8-22 9-986234 •54 9^407943 10-592055 U 22 8-21 986202 •54 408471 8^75 591529 23 395166 8-20 986169 •54 408097 409321 8-74 591003 ij 24 395653 8-19 986137 •54 ^■'^ 590479 589953 36 25 396150 8-i8 9S6104I •54 410045 8.73 35 26 396641 8-17 986072, •54 4io569 8^72 34 ^^ 397132 8-17 986039; •54 .411092 8^7i 33 39762! 8-i5 986007 •54 4ii6i5 8^70 588385 32 29 39S1 1 1 8-i5 985974 •54 412137 8^69 8^68 587863 3i 3o 398600 8-14 985942 •54 412658 587342 3o 3i 9-399088 8-13 9-985909 985876 • 55 9^4i3i79 8^67 I0.58682I ll 32 399575 8-12 • 55 413699 8^66 5S630I 33 400062 8-11 985843 • 55 414219 8^65 585781 27 34 400549 8-10 9S5811 •55 41473s 8^64 585262' 26 35 401035 8-09 8-08 985778 •55 415257 8-64 584743i 25 36 4oi520 985745 •55 415775 8-63 534223 24 ll 4o2oo5 8-07 985712 • 55 416293 8^62 583707 23 402489 8-o6 985679 ■ 55 416S10 8^6i 533190 22 39 402972 8-o5 985646 • 55 417326 8^6o 582674 21 40 403455 8-04 o856i3 .55 417842 9^4i8358 8 -So 582158 20 41 9-403938 8-o3 9-985580 .55 8^5S io^58i642 19 18 42 404420 8-02 985547 • 55 418873 s'-sl 58.127 43 4o4boi 8-01 985514 •55 419387 58o6.3 \l 44 405382 8-00 985480 •55 419901 8^55 580099 5795S5 45. 4o5862 7-99 7-98 9S5447 • 55 4204:5 8^55 i5 46 406341 985414 • 56 420927 8^54 l]Ul 14 % 406820 T97 7-96 9853S0 • 56 421440 8-53 i3 407399 985347 • 56 42ig52 8^52 57S048 11 49 407777 408354 7-95 985314 ■56 422463 8^5i 577537 II 5o 7-94 985280 •56 422974 8^5o 577026 10 5i 9 408731 7-94 9-985247 •56 9^423484 S^4o 8^48 io^5765i6 i 32 409307 7-93 985213 •56 423993 424303 576007 53 409682 7.92 983180 ■56 8^48 itx n 54 410137 7-9' 98J146 ■56 425ou 8-47 6 55 410633 ]-i 9851.3 ■56 425519 8^45 5744SI 5 56 4 1 1 1 06 983079I ■56 426027 S^45 573973 i tl 411579 983043 ■ 56 426534 844 573466 3 412052 7.87 985011 ■56 427041 8-43 572959 572453 3 5q 412524 7-86 984978 ■56 4aSo52 Oolang. 843 I 6b 412996 7-85 984944 56 8-42 57.948 L_.. . Oosiiio D. Sine " rsoj I>. Tang. M. 1 SrNRS AXn TAN OK NTS (15 DKOItKES. ) 33 •^r Sine D. Cosine | 1). 1 I'an?- D. Cotang 5o 9.4I2996 85 9-984944' ■57 1 9-428052 8-42 10-57194S I 413467 413938 84 984910, •?7 428557 8-41 571443 It 2 83 984876; ■37 42906: 8-40 57093s 3 414408 83 984842, 'V 429366 8-39 570434 II 4 414878 82 984K08 ■57 43oo'fo 8-38 569730 5 415347 4i58i5 81 984774 ■37 430573 8-38- 569427 568925 55 6 So 984740 'V 431073 8-37 8-36 54 I 416283 72 78 084706 17 431377 568423 53 416751 984672 984637 984603 •57 432079 432580 8-35 567921 52 Q 417217 ]l 'V 8-34 567420 566920 5i 10 417684 9-4ioi5o 1^ 433080 8-33 5o II 75 9.984569 ■57 9-433580 8-32 .0-566420 tt 11 4i86i5 74 984533 v 434080 8-32 565920 i3 419079 73 984500 ■^J 434579 8-3i 565421 47 14 419544 73 984466 :U 435078 8-3o 564922 46 i5 420007 V 98443a 435376 8-29 8.28 8-28 564424 45 |6 420470 420933 42i3g5 421857 4223iS V 70 9843S ■58 • 58 436073 ' 436570 437067 437563 563927 563430 44 43 984328 ■ 58 8-27 562033 42 >9 984294 ■ 58 8-26 562437 41 20 67 9842D9 ■ 58 438059 8-25 561941 40 21 9-422778 67 9-984224 ■ 58 9-438554 8-24 10-561446 39 38 22 423238 66 984190 984155 ■ 58 439048 8-23 560932 23 423697 65 ■58 439543 S-23 560437 u 24 424106 64 984120 ■58 44oo36 8-22 559964 25 424615 63 984085 ■58 440029 8-21 559471 558978 35 26 425073 425530 62 9S4050 ■ 58 441022 8-20 34 27 6i 984015 ■58 44i5i4 8-19 558486 33 28 425987 60 983981 ■58 442006 8-i| 557994 537303 32 29 426443 60 983946 ■58 442497 442988 8-18 3i 3o 426899 9-427354 So 9S2'i •58 8-17 8-16 557012 3o 3i 58 9-983875 ■ 58 9-443479 443968 10-556521 29 32 427809 428s63 57 983840 19 8-i5 556032I 28 33 56 9838o5 ■ 59 444458 8-i5 5555421 27 34 428717 55 933779 ■?9 444947 8-14 555o53; 26 35 429170 54 98373D ■59 445435 8-i3 554565; 25 36 429623 53 9837CO ■?9 445923 8-12 554077' 24 553589' 23 \L 430075 52 983664 i' 446411 8-12 430527 430978 52 983629 .59 446898 8-11 553102 22 39 5i ^fM ■39 447384 8-10 532616 21 40 41 431429 9-431879 5o 49 983558 9-983523 1' ■59 447870 9-448356 8-09 8-09 552 i3o 10-531644 20 42 432329 49 983487 i9 448841 8-08 53ii5cJ 43 432778 48 983452 19 449326 8-07 550674 17 44 433226 47 983416 1' 449810 8-06 550190 16 45 433675 46 983381 ■ 59 450294 8-06 549706 i5 46 434122 45 983345 19 450777 8-03 549223 14 % 434569 44 983309 ■ 59 451260 8-04 548740 i3 43501 6 44 9^^']^ ■60 451743 8-o3 548257 12 49 435462 43 983238 ■ 60 452225 8-02 547773 II 5c 435go8 9-436353 42 983202 ■60 452706 8-02 547294 10 5i 41 9-983166 ■60 9-453187 453668 8-01 10-546813 g 52 436798 40 983i3o •60 8-00 546332 53 437242 40 9^°?! -60 454148 7-99 545852 I 54 437686 438129 39 983o58 ■60 454628 7-99 545372 55 38 983022 ■ 60 455107 7 -98 544893 5 56 433572 37 982986 • 60 455586 7-97 544414 4 \l 439014 36 982950 •60 456064 7-96 543936 3 439456 36 982914 •60 456542 7-96 543458 3 59 44o3?8 35 982878 ■60 457019 7-95 542981 542504 I 60 34 982843 • 60 457496 7-94 Cosine D. Sine 1 r4o Cotang. D. Tang. M. 34 (16 DEGREES.) A TABLE OF LOGARITHMIC r-HT Sine D. Coaino D. Tang. D. Cotang. 9 -440333 7-34 9-982842 ■ 60 9-457496 7-94 10-542504 60 I 440778 7-33 982805 • 60 457973 7-93 542027 ?2 2 441218 7-32 982769 ■ 61 458449 458925 7-93 54i55i 58 3 441658 7-31 982733 •61 7-92 541075 57 4 442096 442535 7-3i - 982696 •61 459400 7-91 540600 56 9 7-30 982660 •61 459875 7.90 540125 55 6 442973 III 982624 •61 460349 539651 54 I 443410 982587 •61 460823 539177 538703 53 443847 7-37 982551 •61 461297 7-88 52 9 444284 7-27 982514 •61 461770 7-88 538230 5i 10 444720 7-25 982477 •61 462242 i,:tl X 537758 5o 11 9-445i55 7-25 0-982441 •61 9-462714 10-537286 ii 12 445590 7-24 982404 •61 463 186 7-85 ' 536814 i3 446025 •^-23 982367 •61 463658 7-85 536342 "I 14 446459 446893 •1-23 982331 •61 464129 7-84 53587, 45 i5 7-22 llllf, •61 464599 7-83 535401 45 i6 447326 7-21 ■61 465069 7-83 534931 44 \l 447759 448191 7-20 982220 ■62 465539 466008 7-82 534461 43 7-20 982183 •62 7.81 533993 42 >9 448623 7-10 7-i8 982146 ■62 466476 7.80 533D24 41 20 449054 982109 -52 466945 7-80 533od5 40 21 9-449485 7-17 9-982072 •62 9-457413 ?:?? 10-532587 3^ 22 449915 45o345 7-16 9S2035 ■62 467880 468347 533120 23 7-6 98.998 -63 7.78 53i553 ll 24 450775 7-i5 981961 ■62 468814 ]:U 531186 36 25 45i204 7-14 981924 •62 469280 53o73o 35 26 45i632 7-i3 981 S86 ■62 469745 7-75 530254 34 11 452060 7-.3 981849 •62 470211 7.75 529789 33 452488 7-12 981812 •62 470676 7-74 529324 538859 33 ?9 452915 453342 7-11 981774 -52 471141 7.73 3i 3o 7-10 981737 -63 471605 7.73 528305 10-527932 3o 3i 9-453768 7-10 9-981699 •63 9-472068 T72 =8 32 454194 ]-2 981662 -63 472532 7.71 527468 2S 33 454619 981625 -63 472995 473457 7.71 527005 27 34 455o44 7-07 981587 ■63 ni 535543 26 35 455469 455893 7-07 981549 •63 473919 474381 526081 25 35 7-o5 981512 -63 v^ 525619! 24 1 ll 4563 16 7 -05 981474 ■63 474842 525 1 58 23 456739 7.04 981436 •63 475303 7-67 524697 534237 32 39 457162 7-04 981399 •53 475763 7.67 21 40 457534 9 -458006 7 -03 981351 ■63 476223 7-65 523777 20 41 7 -02 9-981323 •53 9-476683 7-65 io^5333i7 19 42 » 458427 7-01 981285 •63 477' 42 7-65 522858 18 43 458848 7.01 9S1247 ■63 477601 478059 7-64 522399 \l 44 459268 7*00 981209 •53 7-63 521941 45 459688 t'^ 981171 •63 478517 478975 7-63 521483 i5 46 460108 981133 • 54 7-62 52I035 14 % 460527 460945 461354 6-98 981095 981057 .54 479431 7^6i 52o568 i3 tu •54 479889 480345 7-5i 520I 1 1 13 49 981019 •64 7-5o 519655 11 5o 461782 6-95 980981 •54 480801 7-59 ■o^iii;?? 10 5i 9-462199 6-95 9-980942 • 64 9-481257 I 52 462616 5-94 98X5 •64 481712 518288 53' 463o32 6.93 .64 482167 7-57 517833 I 54 463448 6.93 980827 •64 482621 V.ll l:^it 55 463S64 5-92 980789 ■64 4S3075 5 56 464279 6-91 980750 •64 483529 7.55 516471 4 u 454694 6-90 980J12 980673 ■64 483982 7^55 5i5oi8 3 465 108 6-QO -64 484435 7^54 5i5565 2 59 465522 980635 -64 484887 7^53 5i5ii3 I 6o 455935 980396 -64 4S5339 7^53 5i465i Ji. Oosiuo D. Siuo rso Cotaug. D. Tang. SINES AND TANGENTS. (l? DEGREES. ) 35 1^ Sine D. Cosine D. Tang. D. Cotang. __ 9-465035 466348 5.88 Q- 980596 980558 -64 9-485339 7-55 !0-5i465i 50 , I 6 88 .64 486791 •52 614209 613768 ^ 3 466761 6 'd 980519 ■65 486242 •5i 3 467173 467535 6 980480 -65 486693 5i 5i3307 ^I 4 6 85 980442 -65 487143 5o 61.2857 56 5 467996 6 85 980403 -65 487693 488043 49 612407 65 6 468407 6 84 980364 -65 49 511967 54 7 468817 6 83 980325 -65 488492 48 5ii5o8 53 8 469227 6 83 980286 -65 488941 489390 489838 47 611069 52 9 469631 470040 6 82 980247 980208 •65 47 5io5io 5i 10 6 8i • 65 46 5ioi52 60 II 9-470455 6 80 9-980169 ■ 65 9-490286 46 10-609714 it 12 470863 6 80 980130 • 65 490733 ,45 609267 508820 i3 471271 6 7^^ 980091 980002 -65 491180 44 47 U 471679 6 • 65 491627 44 593373 46 i5 472086 6 78 980012 • 65 492073 43 607927 45 i6 472492 6 77 979973 ■ 65 492619 492965 43 507481 44 ;? 472898 6 76 979895 979855 • 66 42 507035 43 473304 6 76 •66 493410 41 506690 42 '9 473710 6 75 •66 493354 40 5o5i46 41 30 4741 i5 6 74 979816 •66 494299 9-494743 40 506701 40 21 9-474519 474923 475327 6 74 9-97977,5 .66 40 10-605267 39 22 6 73 979787 • 66 495186 39 504814 38 ■ 23 6 72 -& ■ 66 495630 38 604870 37 24 475730 6 72 ■ 66 496073 37 508927 35 25 476133 6 71 979618 • 66 496615 37 5o3435 35 26 476536 6 70 979579 • 66 496957 497399 36 5o3o43 34 2? 476938 477340 6 69 979539 • 66 35 602601 33 2S 6 '^2 979499 979459 • 65 497841 498282 35 602169 32 ^' 477741 478142 6 68 .66 34 601718 3i 3o 6 67 979420 • 66 498722 34 501278 3o 3i 9-478542 6 67 9-979380 • 66 9-499163 33 io-5oo837 29 32 478942 479342 6 66 979340 • 66 499603 33 600397 499958 499019 23 33 6 65 979800 • 67 5ooo42 32 ^2 34 479741 480140 6 65 979260 • 67 500481 31 25 3i 6 64 979220 • 67 600920 5oi359 ' 31 499080 498641 25 36 480539 6 63 979180 • 57 30 24 ^^ 480937 6 63 . 979"4o .67 izm 30 498203 23 481334 6 62 979100 .67 29 28 497765 22 3, 481731 6 61 979069 • 67 502672 497328 21 40 482128 6 61 979019 9-978979 ■67 5o3i09 28 496S91 10-495454 20 4i 9-482525 6 60 -67 9 -503545 27 \l 42 482921 4833 1 5 6 59 978898 .67 503982 27 496018 43 6 ll -67 5o44i8 25 496682 17 44 483712 5 978858 -67 604354 25 495145 i5 45 484107 6 57 9788.7 -67 606289 25 494711 i5 46 484501 6 57 9787-7 • 67 606724 24 494276 i4 tl 484895 6 56 978786 ■''I 5o6i59 506593 24 493841 j3 486289 6 55 Sit -68 23 493407 12 49 485682 6 55 -68 607027 22 492973 492540 11 Dq 486075 6 54 978615 • 68 607460 7 12 10 5i 9-486467 6 53 9-978574 ■ 68 9-507893 5o8326 21 10-492107 I 53 486860 6 53 978533 -68 21 491674 53 487261 6 52 978498 -68 608769 20 491241 7 54 487643 6 5i 978402 -68 609191 '9 490809 5 55 488034 6 5i 978411 •68 609622 '2 490378 489946 439615 5 56 488424 6 5o 978870 • 68 610064 18 4 ll 488814 6 5o 978320 ■68 5io485 18 3 489204 6 t 978288 ■58 510916 \l 489084 488564 3 5, ffl 6 978247 978206 -68 5ii346 I 6a 6 48 -63 611776 7-16 483224 Coaine D. Sine rao Cotiuig. D. Tang. M. S6 (18 UEGliEEft.) A rABLE OF LOOARITHMIO Sine D. Cosine ) !)• Tang. D. Cotaug. 9-489982 49037 1 6-48 9-978206' -68 9-511776 7-i6 10-488224 60 I 6-48 978165 • 68 012206 7-16 487794 1 490759 6-47 978124 ■68 512635 7-i5 487360 3 491147 4gi53D 6.46 978083 .69 5i3o64 7-14 486936 48650J 4 6-46 978042 .69 613493 7-14 66 5 6 491922 492308 6-45 6-44 978001 977960 977918 977877 .69 •69 61392, 5i434g 7-i3 7-13 486079 485651 65 54 I 492695 6-44 .69 5i477i 7.12 485223 53 493081 6-43 .69 5i5204 7-12 484796 62 9 493466 6-42 977835 .69 5i563, 7-II 484369 fi i 10 493851 6-42 977794 9-977752 .69 5i5o57 7-10 63 II 9-494236 6-4r .69 9-516484 7-10 I0-4835i6 % 12 494621 6.41 9777" •69 616910 617335 7-09 483090 i3 495005 6-40 977669 .69 ■fo8 482666 "Z i5 495388 495772 6.39 6.39 6-38 977628 I77586 .69 ■ 69 518185 482239 48i8i5 46 43 i6 496134 977544 ■70 5i86io 7-07 481390 44 ■7 496537 6.37 »775o3 •70 619034 7-06 480966 43 iS 496919 497301 6-37 977461 •70 5X9458 7-06 480D42 42 '9 6-36 977419 •70 519882 7-o5 4801 ^8 41 20 497682 6-36 9-977335 •70 52o3o5 7 -06 479696 40 ,21 9-498064 6-35 •70 9-620728 7 -04 10-479272 39 •'22 498444 6-34 977293 ■70 52ii5i 7-03 47S849 38 !3 498825 6-34 9772DI •70 621573 7-03 478427 ^I 14 499204 6-33 977269 •70 621995 7 -03 478005 35 ID 499584 6-32 97T67 •70 622417 7-02 477583 35 16 499963 5oo342 6-32 977126 •70 522838 7-02 477162 34 n 6.3i 977083 •70 52325, 7-01 476741 33 !8 500721 6-31 977041 •70 52368o 7-01 476320 32 '9 501099 6-3o 976999 976967 •70 6241 00 7-00 475900 3i !o 501476 6-29 •70 524520 6-99 475480 3o li 9-5oi854 6-29 6-28 9-976914 •70 9-52493g 52535g 525778 626197 6.90 10-475061 ^i 12 50223l -71 6.9g 474641 28 13 502607 6.28 •71 6-98 474222 'I 14 502984 5o336o 6.27 976787 -7' 6.97 4738o3 26 i5 6-26 976745 -7' 626616 6-97 473383 25 !6 503735 6.26 976702 -7' 627033 6.95 472967 24 i? 5o4iio 6-25 976660 •71 627461 6.96 472549 23 IS 504485 6-25 976617 -71 62786S 628285 6.93 472132 22 I9 504860 6-24 976574 •71 6-96 471715 21 40 5o5234 6-23 976532 -71 628702 ^94 471298 10-470881 20 il 9-5o5&o8 6-23 9-976480 976446 •71 g-529iig f-9^ '2 12 505981 5o6354 6-22 -7' 52g536 5-93 470466 18 43 6-22 976404 976361 676318 ■T 529950 53o366 6-93 47O05o M 44 606727 6-21 ■v 692 469634 45 607099 6-20 ■71 530781 6-91 469219 i5 i5 507471 6-20 976275 •71 63iig6 691 468804 14 S 507843 608214 6-19 976282 ■72 53i6ii 6-90 4683S9 i3 6-19 6-18 9761,89! •72 532025 6-90 12 49 5o8585 976146' •72 53243g 632853 5-89 II 5o 508906 ;■ 509326 6-18 976103 • 72 6.8g 467147 10 5i 6-17 6-16 9-976060 •72 9-533265 6.88 10.466734 I .52 609696 976017 •72 533679 5-88 466321 53 5ioo65 6-i5 975974 •72 534092 6.87 466908 7 54 610434 6-i5 970930 •72 534004 6.87 465496 465o84 a 55 5io8o3 6-15 976887 1 -72 634gi6 535328 6-86 5 56 611172 6.14 9758441 -72 5-86 464672 4 u 5ii540 6.i3 976800 ■72 53573g 6-85 464261 3 5iigo7 612275 6.i3 976767 •72 536i5o 6-S5 463850 2 59 6-12 975714 975670 •72 536561 6.84 463439 463028 I 60 512642 6-12 •72 636g72 6-84 Cosine D. Sine ilio Cotaug. D. Tiuig. M. SINKS AND TANGENTS. (19 DEUREES. 37 ■M. ■ me ■ D. Cosine D. Timg. D. Co tang. r~i 0. 9. 5,1 2642 6-12 9-975670 •73 9-536972 537382 6-84 10-463028; 60 1 I 5 I 3009 6 11 975627 73 6 83 462618 U 2 5i3373 6 n 975583 73 537792 538202 6 83 462208 3 513741 6 10 975530 73 6 82 461798 57 4 514107 6 09 975452 73 538611 6 82 461389 56 5 514472 li °9 73 539020 6 81 460980 55 6 5i4837 6 08 075408 73 539429 6 81 460071 54 I 5l5202 6 o3 9-^5365 73 539837 540245 6 80 460163 53 5i5566 6 07 975321 73 6 80 459755 5j 9 10 5i5g3o 516294 9-5I6657 6 6 2 & 73 73 540653 541061 6 6 79 459347 458939 10-458532 5i So 11 6 q5 '■& 73 9-541468 6 i^ 13 517020 6 o5 73 541875 542281 6 78 458125 i3 517382 6 o4 97510I 73 6 77 457719 47 U i5 IX^ 6 6 o4 ■o3 975057 975oi3 73 73 542688 543094 6 6 77 76 457312 456906 4565oi 46 45 i6 518468 6 o3 974969 974925 74 543499 6 76 44 17 518829 6 02 74 543905 544310 6 75 456095 43 1.8 519190 6 01 974S80 74 6 75 455600 42 '9 519551 6 01 974836 74 544715 6 74 455285 41 20 519911 6 00 974792 74 545119 6 74 454881 40 21 9-520271 6 00 9-91iliS 74 9-545524 6 73 10-454476 39 38 22 .520631 5 99 974703 74 545928 546331 6 73 454072 23 520990 5 99 974659 74 6 72 453669 453265 37 24 521349 5 98 974614 74 546735 6 72 36 25 521707 5 98 974570 74 547138 6 T 452862 35 26 522066 5 97 ^74525 74 547540 6 7' 462460 34 ^^ 522424 5 96 974481 74 547943 548345 6 70 462067 45 1 655 33 522781 5 96 974436 74 6 70 32 29 523i38 5 95 974391 74 548747 6 69 401253 3i ; 3o 523495 5 95 974347 75 549149 6 69 45o85i 3o 3i 9-523852 5 94 9-974302 75 9-549550 6 68 10-450460 ^2 32 524208 5 94 974257 75 549961 55o352 6 63 45oo4q 28 33 524564 5 93 974212 75 6 67 449648 27 34 524920 5 93 974167 75 550762 6 67 449248 448848 26 35 525275 5 92 974122 75 55ii52 6 66 25 36 525630 5 9' 974077 75 55 1 552 6 66 448448 24 ll 525984 526339 5 9> 974032 75 551962 552351 6 65 448048 23 5 90 973987 •P 6 65 447649 22 39 526693 5 2° 973942 75 552730 6 65 447260 21 40 527046 5 §9 975897 9-973S52 75 553149 9-553548 6 64 446861 20 41 9-527400 5 ^9 75 6 64 10-446462 19 42 527753 528105 5 8a 973807 75 - 553946 6 63 446064 18 43 5 88 973761 75 554344 6 63 445656 17 44 528458 5 87 973716 76 554741 6 62 440269 i5 45 528810 5 87 97367' 76 555 139 6 62 444861 i5 46 529161 5 86 973625 76 555535 6 61 444464 14 tl 529513 5 86 973580 76 555g33 556320 6 61 444067 .3 529864 5 85 973535 76 6 60 443671 12 49 53o2r5 5 85 973489 76 556723 6 60 443275 II 5o 53o565 5 84 973444 76 557121 6 59 442879 10-442483 10 5i 9-53o9i5 5 84 9 -973398 973352 76 9-557617 6 59 I 5a 531265 5 83 76 557913 5583o8 6 U 442087 53 53i6i4 5 82 973307 76 6 441692 I 54 531963 5323i2 5 82 973261 76 558702 6 58 441298 55 5 81 973216 76 559097 6 57 440903 i5 56 532661 5 81 973169' 76 559491 559885 6 S 440009 4401 1 5 4 5t 533009 5 80 973124 76 6 56 1 5S 533357 5 80 973078 973032 76 560279 560673 6 56 ^9721 1 59 533704 5 79 77 6 55 - 439327 438934 • ! 66 534062 5 78 972986 77 661066 6.55 Cosiiio ]). Sino 70° Cotaiipf. U Tauff. M._ 38 (20 DliOItEES.) A TABIE OF 1,00AKITUMI0 Sine Q. 534032 D. Cosine D. Taiig. 1). Cotang. 60 5.78 9-972986 •77 9.561066 6-55 10.438934 I 534399 5.77 972940 972894 -77 561459 6.54 438341 si 3 D34743 5-77 •77 56i85i 6.54 438149 ' 3 535002 53543s tv> 972848 ■77 562244 6-53 437755 437364 57 4 972802 •77 562636 6-53 56 5 535783 5-;6 972755 •77 563028 6-53 » 55 6 536129 5-75 972663 •77 563419 6-52 54 I 536474 5-74 •77 563811 6-53 436189 435798 53 5368.8 5-^4 972617 •77 564202 6-5i 5] 9 537163 5-^3 972570 •77 564592 9.56537; 6-51 435408 5i 10 537507 5-^3 972524 •77 6-50 435017 5o II 9-537851 538194 538538 5.72 9-972478 '^l 6-5o 10-434627 i^ 12 5-72 972431 565763 6-49 434237 i3 5-71 972385 -78 566i53 6-49 433847 433458 ii 14 538880 5.^^ 972338 -78 566542 6-49 6-48 ID 539223 5.^o 972291 .78 566932 567320 433068 45 |6 539565 5-70 972245 .78 6-48 432680 44 ;? 539907 5-69 972198 972i5i ■^t 568486 6-47 432291 43 540249 5-6? -78 6-47 431902 43i5i4 43 "9 540590 540931 972105 .78 6-46 41 20 5-68 972058 .78 568873 646 431127 40 21 <)-54i272 5-67 9-972011 •78 9-569261 6-45 10-430739 It 22 54i6i3 5-67 971964 -78 569648 6-45 43o353 23 541953 5-66 971917 971870 -78 570035 6-45 429965 429578 ll 24 542293 542633 5-66 -78 570422 6-44 25 5-65 971823 ■78 571581 6-44 439191 438805 35 26 542971 543310 5. 65 971776 •78 6-43 34 ^2 5-64 971729 •79 6-43 438419 428033 33 543649 5-64 971682 •79 572352 6-42 32 29 543987 544325 5-63 971635 •79 6-42 427648 3i 3o 5-63 971588 •79 572738 5-42 427262 3o 3i 0-544663 5-62 9971540 ■79 9.573123 6-41 10.426877 It 32 545000 5-62 971493 •79 573507 6-41 426493 33 545338 5-61 971446 •79 573892 6-40 426108 37 34 545674 5-61 971398 971351 •79 574276 6-40 425724 36 35 54601 1 5.60 •79 574660 6-39 425340 35 36 546347 5-60 97i3o3 •79 575044 6.39 424056 434573 34 ll 546683 5-59 971256 •79 575427 6.3o 33 547019 5.59 5-5S 971208 •79 573810 434190 32 39 V7354 971161 •79 576193 6-38 433807 31 40 547689 9-548024 5-58 971113 •79 576376 6-37 423424 30 41 5.57 9-971066 -80 '■PX 6.37 10-423041 ;g 42 548359 548693 5^5^ 971018 -80 6. 35 422659 43 970970 -80 577-23 6.36 421896 17 44 549027 5-56 970922 970874 -80 578104 6-36 i& 45 549360 5-55 -80 5784S6 6.35 42i5i4 • 5 46 549693 5.55 970827 970779 ■80 578867 079248 6-35 421 i33 14 ii 550026 5-54 •80 6. 34 420752 i3 55o359 5-54 970731 -80 379629 5.34 420371 13 49 550692 5-53 970683 ■So 580009 6. 34 4I999< 419611 II 5o 551024 5.53 970635 ■80 5So3S9 6. 33 10 5i 9-551356 5-52 9-970586 ■ 80 9.580769 6-J3 10-419231 4iS85i ? 52 551687 552018 5-02 970538 ■80 58 1149 5Si328 6-32 53 5.53 970490 -So 6-32 418472 I 54 552349 5-5i 97044a ■80 5S1907 6-33 418093 55 552680 5-5i 970394 •80 582286 6-3i 417714 5 56 553010 5-5o 970345 ■81 582655 6-3i 417535 4 u 553341 5-5o 970297 -Si 583043 6-3o 416957 3 553670 5-49 970249 -81 583432 6-3o 416378 3 59 554000 5-49 5-48 970200 .81 583SOO 6-29 416200 1 60 554329 970152 -81 584177 6-29 415833 Ooaiue D. Siue 6!)° Cotanir. 1). Tim?. 1 M. ) BlMfia AND TAS0ENT3. {2i DEUKBliS. OS Siue U, Cosiiia D. .81 Tang. D. Cotang. ?• 554329 554653 5-48 9-970152 9-584177 6^29 iu-4i5823 60 I 5.48 970103 .81 584553 6-29 415445 5? 1 554987 5-47 970055 .81 584932 6.28 4i5o68 3 5553 1 5 5-47 5.46 970006 969957 • 81 585309 6^28 414691 57 4 555643 ■ 81 585686 6.27 414314 56 5 555971 5.46 &l ■ 81 586062 6.27 413938 4i356i 55 6 556299 5-45 ■ 81 586439 6^27 54 7 556626 5-45 96981 1 ■ 81 58681 D 6-26 4i3i85 53 8 556953 5.44 969762 ■ Si 587190 6^26 412810 53 9 557280 5-44 969714 •81 587566 6^25 412434 5i 10 557606 5.43 969665 .81 587941 9.5883i6 6^25 412069 5o 11 9-557932 558258 5-43 9-969616 .82 6^25 10-411684 :i 12 5.43 969567 969518 .82 588691 6^24 411309 i3 558583 5-42 .82 589066 6^24 410934 47 14 . 558909 5-42 969469 ■82 589440 6^23 4io56o 46 i5 559234 5-41 969420 .82 589814 6^23 410186 45 i6 559558 5.41 969370 •82 590188 6^23 409812 44 \l 559883 5.40 969321 •82 590662 6-22 409438 •43 560207 5-40 969272 •82 590935 6^22 409065 42 '9 56o53i 5.39 969223 ■ 82 591308 6-22 408692 41 20 56o855 5.39 969173 •82 591681 6^21 408319 40 21 9-561178 5-38 9-969124 .82 9-592054 6^21 10-407946 407674 It 22 56i5oi 5-38 969075 .82 592426 6-20 23 561824 ^/l^ 969025 968976 •82 592798 6^20 407202 37 24 562146 5.37 5.3i -82 593170 6^19 406829 36 25 562468 968926 -83 593542 6-19 406468 35 26 562790 5-36 968871 •83 593914 6-i8 406086 34 11 563112 5-36 96882-7 •83 594285 6-i8 405715 33 563433 5-35 968777 968728 •83 594656 6^i8 405344 32 '9 563755 5-35 •83 696027 595398 6M7 404973 3i 3o 564075 5.34 968678 •83 6^17 404602 3o 3i 9-564396 5.34 9-968628 •83 9-596768 6^.n 6^16 10^404232 29 32 564716 5.33 968678 •83 596138 403862 28 33 565036 5-33 968628 •83 696608 6-i6 403492 27 34 565356 5.32 968479 •83 596878 6^i6 4o3i22 26 35 565676 5.32 968429 •83 697247 6^i5 402753 25 36 'X 5-3i 968379 •83 697616 6^i5 402384 24 u 5-3i 968329 968278 •83 stlhi 6-15 4o2oi5 23 566632 5-3i •83 6^i4 401646 22 39 566951 5-3o 968238 •84 598722 6^14 401278 21 40 567269 5.3o 968:78 •84 699091 9-509459 6^i3 400909 10 -40064 1 20 4i 9-567587 5-29 9-968128 •S' 6^i3 19 iS 42 567904 568222 ^'2 96S07O ■84 599827 6^i3 400173 43 5-28 96S027 ■84 600194 6.12 399806 17 44 568530 5-28 96797? ■84 600662 6-12 399438 16 45 568856 5-28 967027 •84 600929 6^ii 399071 i5 46 569172 5.27 967876 ■ 84 601296 6-11 398704 14 s 569488 ^'I 967826 •84 601662 6-11 398338 i3 569804 5-26 967775 ■84 602039 6-10 397971 12 49 570120 5-26 967725 ■84 602390 6-10 397606 5o 570435 5-25 967674 • 84 602761 6-10 397239 10-396873 10 5i 9-570751 5-25 9-967624 •84 9-6o3i27 6-09 t 52 571066 5-24 967673 ■ 84 603493 6o3858 6-09 396607 53 571380 5-24 967622 .85 6-09 396142 7 5-i 571695 5-23 967471 ■85 604223 6-o8 395777 6 55 572009 572323 5-23 967421 ■85 604588 6-08 390412 5 56 5-23 967370 ■85 604953 6o53i7 6-07 896047 4 u 572636 5-22 967310 967268 • 85 6-07 394683 3 672950 5-22 ■85 6o5682 6'C7 394318 2 5, 573263 5-21 967217 ■85 606046 6-06 393954 393590 TilD^. 1 6o 5,3575 5-21 967166 ■85 606410 6-o6 Ji7_ Cosinfi D. Sine SS° CotaiifT. D. iO (22 DB0BKE8.) A TABLE OF lOOABITUMIO M. Sine D. Cosino 1 1). Taiig. 1>. Cotang. 60 i 9-573575 5738§8 5-li 9-957166 .85 Q. 606410 6-o6 10-893590 I 5-20 967115 .85 606778 6-o6 ^''H'i'l 59 58 2 574200 5-20 967064 .85 607137 6-o5 392868 3 574512 5-ig 967013 .85 607500 6-o5 392500 ll 4 574824 5-19 966961 .85 607863 608225 6-04 392187 56 5 573136 l\l 066910 966839 966808 .85 6-04 391773 55 6 575447 575758 .85 6o8588 6-04 391412 54 I 5.18 .85 608960 609312 6-o3 391060 53 576069 5.17 966756 .86 6-o3 390688 5a 9 576379 576689 5-17 5.i6 066705 .86 609674 6-o3 390826 3«9?64 5i 10 966653 .86 6ioo36 6-02 5a II 9-576999 577309 577618 5.16 9-966602 .86 9-610397 610709 6-02 10-889603 ii 12 5.16 966550 .86 6-02 389241 i3 5-i5 966499 .86 &11120 6-01 388880 "I 14 577927 578236 5-i5 966447 .86 61 1 480 6-01 338320 46 i5 5-14 966895 .86 611841 6-OI 388159 45 l6 578545 5-14 966344 -86 612201 6-00 387799 387439 44 :r 578853 5-i3 966292 .86 612061 6-00 43 579162 5-i3 966240 .86 612921 6-00 387079 42 >9 579470 5-i3 966188 .86 618281 5.99 386719 41 20 V58oo85 5-12 966186 .86 6 1 3641 1^ 386339 40 21 5-12 9-966085 -87 9614000 10-386000 ^ 22 580392 5ii 966033 •87 614359 5.98 385641 23 580699 5.11 963981 •87 61471a 5.98 385282 ?i 24 58100D 5.11 963928 965876 -87 61 5077 5-97 384923 36 25 58i3i2 5-10 •87 615435 5-97 384365 35 26 58i6i8 5-10 9&5824 -87 615793 ri 384207 34 11 581924 5.09 965772 .87 616131 383849 33 582229 5-09 963720 -87 6i65o9 5-96 383491 383i33 32 29 582533 5-0? 963668 •87 616867 5-96 81 3o 582840 963613 •87 617224 5.95 3S2776 3o 3i 9-583i45 5.08 9-965563 -87 9-617582 ^95 10-382418 29 28 32 583449 5.07 965311 -87 618295 6i8652 5.95 382061 33 583754 5-07 965433 .87 5-94 381705 2 34 584058 5.06 965406 il 5.94 38i348 35 584361 5-06 965353 619008 5-94 380992 38o636 23 36 584665 5-06 965301 .88 619864 5.93 24 u 584968 5-o5 965248 .88 619721 5.93 380279 23 585272 5-05 965195 .88 620076 620482 5-93 379924 22 3g 585D74 5-o4 963143 .88 5.92 379568 21 40 585877 5-o4 965090 9-965037 .88 620787 5.92 879218 20 41 9-586179 586482 5-03 .88 9-621142 5.92 10-378858 10 42 5-03 964984 .«3 621497 621852 5.91 3785o3 43 586783 5-o3 96498 1 964879 .88 5.9. 878148 \l U 587080 5-02 .88 622207 5-90 377793 377439 45 587386 5-02 964826 • 88 622561 5.90 i5 46 587688 5-01 964773 .88 622915 5.90 377083 14 % lllt^ 5-01 964719 964666 .88 623269 628623 3-6781 i3 5-01 .89 5.89 376877 12 49 588590 5-00 964613 .89 628976 624880 5.^ 376024 11 5o 588890 5-00 964360 -85 875670 10 5i 9-589190 589489 4-99 9-964507 .89 9-624683 5.88 10-875317 t 52 4-99 964454 -89 625o36 5.88 374964 374612 53 589789 964400 -89 623388 5.87 I 54 590088 964347 .89 625741 ^^7 874269 55 590887 4.98 964294 -89 626093 5.87 5.86 878907 5 56 590686 4-97 964240 -89 626445 378055 4 u 590984 4-97 964187 -89 626797 5.86 878203 3 591282 ni 964133 -89 627149 5.86 372861 2 59 591680 964080 -8g 627301 5.85 l]li'^ I 60 591878 4-96 964026 -89 627802 5.85 l^osino D. Sine 87° Cotiiug. D. Tang. M. 1 SINES AND TANOEN're. (23 DEGHEBS. ) 41 M. Sine D. -1 Cosine | JD. 1 Tang. D. Cotaiif,-. 60 9-591878 4-96 g-964026 -89 9-627832 5 -83 10-37214^ 1 592176 4 95 963972; .89' 628203 5 -85 371797 09 5H 2 092473 4 95 963919' -89' 628554 80 371446 3 592770 4 95 • 963S6Di' -90! 628905 5 84 37iog5 ^"f 4 593067 4 94 9638111 -9o| 629253 5 84 370-743 370394 56 5 593363 4 94 963757! -go 629606 5 83 55 6 593659 4 93 963704' -90 629g56 5 83 370044 54 I 593955 4 93 96365o' -gc 63o3o6 5 83 3696g4 53 594251 4 93 963596J -go 63o656 5 83 36^995 52 9 594547 4 92 963542 -go 63ioo5 5 82 5i 10 594842 4 9= 963488 -gc 63i355 5 82 368645 5o 11 5.595137 4 9' 9.963434 -go g-63i704 5 82 10-368296 it 13 595432 4 9' 963379 -go 632053 5 81. 367947 ■ 3 595727 4 9' 963320 -go 632401 5 8i 367099 367250 47 14 59602 1 4 90 g6327i .90 632700 5 81 45 i5 596315 4 963217 -go g63i63 -go 633098 5 80 366go2 366553 45 i6 596609 596903 4 og 633447 5 80 44 \l 4 89 g63io8 -gi 633795 5 80 3662O0 43 597196 4 963054 -gi 634143 5 79 365857 42 '9 597490 4 88 962ggg' -91 962g45; -91 6344go 5 79 365510 41 20 5977«3 9.598075 4 86 634838 5 7? 365i62 40 21 4 87 g-g628go' -gi 9-635i85 5 7S 10.364815 39 22 593368 4 87 g62836' -gi 635532 5 78 364468 38 S3 59S660 4 87 962781 •91 635879 5 78 3641 2 1 37 24 598952 4 86 962727 -g. 636226 5 77 363774 .36 25 599244 4 86 g62672 •gi 636572 5 77 363428 35 26 599536 4 85 g626i7 -gl 636gig 637265 5 77 363o8i 34 21 599827 4 85 962062; -gi 5 77 362733 33 28 6001 iS 4 85 962508, -gi 63761 1 5 76 3623Sg 32 20 600409 4 84 g62453| -gi 637g56 6383o2 5 76 362044 3i 30 600700 4 84 g623gS ■g2 5 76 36 1 698 3o 3i 9.600990 4 84 g-g62343 ■g2 9-638647 5 7^ I0-36i3o3 29 32 601280 4 83 g62288 ■92 638992 5 75 361008 28 33 601570 4 83 g62233 ■g2 639337 5 75 36o663 27 34 601860 4 82 962178 •g2 639682 5 74 36o3iS 26 35 602 1 5o 4 82 962123' -92 640027 5 74 3ogg73 25 36 602439 602728 4 82 962067 -92 64037 1 5 74 35g62g 24 ll 4 81 962012 •g2 640716 5 73 339284 358940 338696 23 603017 4 81 96ig57 ■g2 641060 5 73 22 39 6o33o5 4 8i 961902 -92 641404 5 73 21 40 603594 4 80 g6i846j -g2 641747 72 358253 20 41 9-603882 4 80 9-''MT)'-! -92 g6i73o, •g2 9-642091 64?434 5 72 10-337909 \l 4r 604170 4 79 5 72 337566 43 604457 604745 4 79 g6l68o| -g2 642777 5 72 357223 \l 44 4 7? 78 g6i624 -93 643 i 20 5 7' 336880 45 6o5o32 4 ■ g6io6g' ■g3; 643463 961 5i3 -93 643806 5 7' 336537 i5 46 6o53i9 4 78 5 71 356194 14 % 6o56o6 4 78 961458 -g3 644148 5 70 333802 i3 6o58g2 4 77 g6i4o2 -93 644490 961346, -93; 644832 5 70 3335io 12 49 606179 4 77 5 70 335i68 11 5o 606465 4 76 96i2go'i ■t)3\ 640174 5 6g 354826 10 5i 9-606751 4 -6 gg6i23o -gS g-6455i6 .5 6g 10-354484 9 53 607036 4 76 g6ii7g. -gil 645807 5 6g 354143 8 53 607322 4 75 ■ 961123 ■g3 ■ 646 igg 5 ''g 333801 7 54 607607 4 75 961067 •93 646540 5 68 333460 6 55 607892 4 Ji 961011 •93 646881 5 68 353119 302778 5 56 608177 4 74 960955 ■93 647222 5 68 ( u 608461 4 74 g6o8g9 •93 647562 5 67 302438 3 608745 4 73 g6o843, ■g4 647go3 648243 648583 5 67 302097 301757 351417 2 Po 609029 609313 4 4 73 73 g6o786| -94 960730, -94 5 5 ^6^ I 'mT Cosine D. Sine go: _C_otttiig._ a._ - Tujig- 12 (24 DEGREES.) A TABLE OF LOGARITHMIC Siuo U. Coajue I). Ti'.ng- D. Cotang. 60 9-6o93i3 4 73 9-960730 .94 9-648583 5-66 10-351417 1 b^sU 4 72 960674 ■94 648923 5-66 351077 ll 1 4 72 960618 •94 649263 5.66 350787 350398 35oo58 3 610164 4 72 960061 ■94 649602 5-65 \l 4 610447 4 71 96o5o5 -94 649942 5-55 56 5 610729 4 71 960448 •94 65o28i 5-55 349380 55 6 611012 4 70 96o3g2 •94 65o62o 5-55 54 7 61 1294 4 70 960335 ■94 660969 5-54 349041 348703 53 8 611576 4 70 960279 •94 65i297 65 1 636 5-64 53 9 6ii858 4 69 960222 •94 5-64 348364 5i IC 612140 4 69 960165 ■94 651974 5-63 348026 5o 11 9-612421 4 69 9-960109 -95 9-652312 5-63 10-347688 % 13 612702 4 68 960032 -95 652650 5-63 347350 i3 612983 4 68 939995 .^5 652988 5-63 347013 47 14 613264 4 67 959938 ■95 653326 5-63 346674 45 i5 613545 4 67 959882 ■ li 653663 5-62 345337 45 i6 6i3825 4 tl 959825 .95 654000 5-62 346000 44 17 6i4io5 4 959768 •95 654337 5.61 345663 43 i» 614385 4 66 9D9711 -95 554674 5.61 345326 43 '9 614665 4 66 959654 •95 655011 5.61 3449B9 41 20 614944 4 65 959596 9-959539 -^5 655348 5.5i 344552 40 21 9-6i5223 4 65 -95 9 •655684 5.60 10-344316 ll 22 6k-')02 4 65 959482 -95 656o2o 5.60 343980 23 615781 4 64 959425 -95 656356 5.60 343544 37 24 616060 4 64 959368 •95 556692 5-59 343308 36 25 6i6338 4 64 959310 -96 557038 5.59 342972 35 26 616616 4 63 959253 ■96 657364 5.59 342636 34 u 616894 617172 4 4 63 62 959195 959138 .?6 -96 65I0T4 5-59 343301 341966 33 32 ?9 617450 4 62 959081 -96 658369 5-58 34163 1 3i 3o 9-618004 4 62 959023 -96 658-04 5-58 341396 3o 3i 4 61 9-958965 -^6 9-659039 559375 5-58 10-340961 2^ 3! 618281 4 61 958908 -96 5.57 340637 33 6i8558 4 61 958850 -96 ■559708 5-57 340393 339958 37 34 618834 4 60 958792 -96 660042 5-57 35 35 619110 4 60 958734 .96 660375 5-57 339634 35 36 6ig386 4 60 958677 -96 6607 1 5.55 III 34 ll 619662 4 59 908619 -96 661043 5-56 33 619938 4 59 958561 .96 661877 5-56 33 3, 6202 l3 4 9585o3 •97 661710 5-55 33S390 337o?7 10-337624 31 40 620488 4 908440 -97 662043 5.55 20 41 9-620763 4 58 9-958387 ■97 9 •662376 5-55 \l 43 621038 4 ^1 958329 ■97 662709 5-54 337291 43 62i3i3 4 57 958271 ■97 663043 5-54 336908 336625 17 44 621587 4 ll 958213 ■97 663375 5-54 i5 45 621861 4 958 1 54 ■97 663707 5-54 335393 i5 45 622135 4 55 958o3S •97 664039 5.53 335901 14 ii 622409 4 56 •97 664371 5.53 335539 i3 622682 4 55 957979 •97 664703 5-53 335297 334965 334634 13 49 622956 4 55 957921 •97 665o33 5-53 11 5o 623229 4 55 907863 •97 665366 5-52 10 5i 9'6235oa 4 54 . 9-957804 •97 9^665697 5-52 io-3343o3 I 5a 523774 4 54 957746 .93 666029 5.5a 333971 333640 53 524047 4 54 907687 .98 666360 5-5i I 54 624319 4 53 907628 .^8 666691 5-5i 333309 55 624591 4 53 907570 .93 66702 1 5-5i 332979 332648 5 56 624863 4 53 95751 1 -98 667352 5.5i 4 u 635135 4 5j 957452 -98 667682 56Soi3 5-5o 3323i8 3 6a54o6 4 52 957393 9573^5 957276 .98 5-5o 331987 33 1657 33i3a3 1 ^ 625677 625948 (.'nsine 4 4 53 5i .98 668343 66S673 5-5o 5-5o I M. "] .1. Sine 030 (.'otan^r. D. T«nff. SINES AND TANOJSNTS. (25 D£OAE£S. ) 43 M. Sino 1). Cosine D. Tang. D. Cotang. 9-635948 4 5i 9.957276 .98 9.668673 5-5o io-33i327 330998 60 1 626219 4 5i 95721-7 -98 669003 5-49 It 3 616490 4 31 957158J .98 669332 5-49 330668 3 636760 4 5o 957099 .98 669661 5-49 330339 57 4 627030 4 5o 957040 -98 669991 670320 5-48 330009 56 5 627300 4 5o 956981 -98 5-48 329680 55 6 627570 4 49 956021 ■99 6-70649 5-48 3293511 54 1 I 62J840 628109 4 49 956862; -99 t]:V2 5-48 329023 53 4 49 956803 1 -99 5.47 338694 52 9 628378 4 48 956744 -99 671634 5.47 328366 5i 10 628647 4 48 956684 •99 671963 5-47 338037 5o II 9-628916 4 47 q. 956625 ■99 9.673391 5.47 ,0.337709 tl 13 629185 4 47 956566 ■99 673619 5.46 337381 i3 629453 4 tl 9565o6 •99 672947 5.46 327053 ^l i.'l 639721 4 956447 ■99 673374 5.46 326726 326398 46 i5 639989 4 46 956387 ■99 673602 5-46 45 |6 630257 4 46 g56327 •99 673929 5.45 326071 44 17 63o534 4 46 956268 ■99 674257 5.45 335743 43 18 630793 4 45 956208,1-00 6-}4584 5.45 325416 42 ■9 6310D9 4 45 956148 I -00 674910 5.44 325090 41 20 63i336 4 45 956089 I -00 675337 5.44 324763 40 21 9-63i593 63 1859 632125 4 44 9.956029 1 -00 n. 675564 ' 6-i5890 5.44 ,0.324436 1% 22 4 44 955969 1. 00 5-44 3241 10 38 23 4 44 955909 1.00 676316 5.43 323784 \1 24 632392 632658 4 43 955849 1.00 676543 5.43 323457 36 25 4 43 955789 1.00 676869 5.43 333i3i 35 26 632923 4 43 955729 1.00 677194 5.43 322806 34 11 633189 4 42 955669 l.QO 677530 5.42 332480 33 633454 4 42 955609 1. 00 677846 678171 5-42 322154 32 29 633719 4 42 955548 |.00 5-42 321829 3i 36 633984 4 41 955488 I -00 678496 5.42 33i5o4 3o 3i 9-634249 4 41 9.955438 I-OI 9.678831 5-41 10.321179 29 32 6345i4 4 40 955368 I-Ol 679146 5-41 330834 28 33 634778 4 40 955307 I-OI 679471 5.41 32o53g 27 34 635o42 4 40 955347 I -01 679795 5.41 320205 26 35 6353o6 4 39 955186 1. 01 680120 5-40 319880 25 36 635570 4 39 955126 I-OI 680444 5.40 319556 24 . 37 635834 4 39 38 955o65 1. 01 680768 5.40 319232 318908 318584 33 3S 636097 4 g55oo5 1. 01 681092 5.40 33 3, 636360 4 38 954944 1. 01 681416 5.39 31 40 636623 4 38 954883 |.0I 681740 5.39 318260 20 41 9-636886 4 37 9.954833 1. 01 Q. 683063 5.39 10.317937 \l 43 637148 4 37 954762 1. 01 ^ 683387 5.39 317613 43 63741 1 4 37 954701 |.0I 682710 5.38 317290 '2 44 637673 ■ 4 37 954640 1. 01 683o33 5.38 316967 16 45 637935 4 36 954579 I .c: 683356 5-38 316644 i5 46 638197 638458 4 36 954518 1.02 683679 5-38 3i632i 14 % 4 36 954457 1.02 6S4001 5.37 315999 i3 638730 4 35 954396 954335 1-03 684324 5.37 315676 12 49 638981 4 35 1-02 684646 5-37 315354 II 5o 639242 4 35 954274 1-02 684968 5-37 3i5o33 10 5i 9 -639503 4 34 9.954318 1.02 9-685390 5-36 io-3i47io I 52 639764 4 34 954153 1-03 685612 5-36 314388 53 640024 4 34 954090 1-02 680934 5.36 314066 7 54 640384 4 33 954029 953968 1-02 686255 5.36 313745 6 55 640544 4 33 1.03 686577 5.35 3 1 3423 5 56 640804 4 33 953906 1.02 686898 5.35 3i3i02 4 5^ 641064 4 32 953845 1.03 687219 5.35 312781 3 641324 4 32 953783 1-03 687540 5.35 312460 3 59 641 584 4 32 953722 I-03 687861 5.34 312139 I 60 641843 4-3i 953660 1.03 688182 5.34 3n8i8 Cosine D. Sine 040 , Cot-aiifr. D. Tang. "mT 44 (20 DEGREES.) A TABLE OF LOGARITHMIC M. Siue U. Oosiue D. Tang. D. Cottvng. 9-641842 4 3i Q.-,95366o -o3 9-688182 5-34 io-3u8i8 60 I 642101 4 3i 953599 -o3 688502 5 34 3 1 1498 % 2 642360 4 3i 953537 -o3 688823 5 34 3I1I77 3 642618 4 3o 953475 -o3 689143 5 33 3 1 0867 i! 4 642877 4 3o 953413 -o3 689463 5 33 3 10537 56 5 643i35 4 3o 953352 -o3 689783 5 33 310217 55 6 643393 4 3o 953290 -o3 690103 5 33 309897 54 I 643650 4 29 953228 -o3 690423 5 33 » 53 643908 4 29 953166 -o3 690742 5 32 52 9 644165 4 '^ q53io4 -o3 691062 5 32 308938 5i 10 644423 4 953042 -OJ 691881 .5 32 308619 5o II 9-644680 4 28 9-952980 -04 9-691700 5 3l io-3o83oo H 12 644936 4 28 952918 -04 692019 5 3i 307081 48 i3 645193 6454S0 4 27 952855 -04 692338 5 3i 30766a ^1 14 4 27 & -04 692656 5 3i 307344 46 i3 645706 4 V, -04 692975 5 3i 307025 306707 45 i6 645962 4 952669 -04 693293 5 3o 44 \l 646218 4 26 952606 -04 693612 5 3o 3o6388 43 646474 4 26 952544 -04 698930 5 3o 306070 305752 42 '9 646729 4 25 952481 -04 694248 5 3o 41 20 646984 4 25 952419 .04 694566 5 29 3o5434 40 21 9-647240 4 25 9.952306 • 04 9-694883 5 29 io-3o5ii7 '^ 22 647494 4 24 952294 952231 -04 6g520i 5 29 304799 3044S2 23 647749 4 24 -04 695518 5 29 ^1 24 648004 4 24 952168 -o5 6g5836 5 ^§ 304164 36 25 648a 58 4 24 902106 -o5 6g6i53 5 3o3847 35 26 648312 4 23 952043 -o5 696470 5 28 3o353o 34 '7 64H766 4 23 951980 •o5 696787 5 28 3o32i3 33 28 649020 4 23 951017 951854 ■ o5 697103 5 28 302897 32 29 649274 4 22 -o5 697420 5 27 3o25«o 3i 3o 649027 4 22 951791 -o5 9-698053 5 27 302264 3o 3i 9-649781 4 22 9.951728 -05 5 27 10.301947 2? 32 65oo34 4 22 ^ 95i665 •o5 698869 698685 5 ^I 3oi63i 33 600287 4 21 951602 -o5 5 3oi3i5 ^Z 34 600539 4 21 95 1 539 -o5 699001 5 26 30^4 35 600792 4 21 951476 -o5 699316 5 26 25 36 65 I 044 4 20 951412 -o5 699632 5 26 3oo368 =4 ll 651297 4 20 95 1 349 .06 699947 5 26 3ooo53 23 65i349 4 20 951286 -06 700263 5 25 299737 22 39 65 1 800 4 19 95x232 -06 700578 5 25 299422 21 40 6o2o52 4 '9 951.59 -06 700893 5 25 290107 10.298792 20 41 g-6523o4 4 19 9.951096 -06 Q. 701 208 5 24 \l 42 652055 4 18 95io32 -06 70i523 5 24 298477 43 65a8o6 4 18 950968 -06 701837 5 24 39S163 \l 44 653o57 4 18 950905 -06 -702152 5 24 .297848 45 6533o8 4 18 950841 .06 702466 5 24 297534 i5 46 653558 4 17 950778 -06 702780 5 23 297220 296005 14 % 6538o8 4 17 950714 95o65o -06 703098 5 23 i3 654059 4 :z -06 703409 703723 5 23 296091 12 ^9 654309 4 95o586 -06 5 23 296277 290964 11 5o 654558 4 16 95o522 ■07 704036 5 22 10 5i 9 -654808 4 16 9.950458 -07 9-704350 5 22 10-295650 Q 52 53 655058 655301 655556 4 4 16 i5 950394 95o3jo -07 ■07 704663 704977 5 5 22 22 295337 295023 7 54 4 i5 900266 -07 705290 5 22 294710! 6 55 6558o5 4 i5 q5o202 -07 ■705603 5 21 294397! 5 294084 4 56 656o54 4 i4 90OI38 1 -07 705916 5 21 % 656302 4 14 900074 ■ 07 706228 5 21 293772 3 656501 4 14 950010 -07 706541 5 21 293459 293146 2 59 656799 4 i3 949045|i 949881 |_i -07 706854 5 21 I 60 607047 4-i3 -07 707166 5-20 292834 1 Cosiuo D. Sino l( 33° Cotauff. D. Tanff. M. SINES AND TANGENTS. (27 DEGREES. ) 4a M. Sine D. Cosine D. Tang. 5-20 Cotang. 10-292834 60 9 .6570 47 4m3 9-949881 1-07 9-707166 I 657295 4-i3 949816]! -07 707478 5-20 292522 59 3 637542 4-12 9497521-07 707790 708102 5-20 292210 58 3 657700 4-12 949688'! -08 5-20 291808 291586 57 i 658o37 4-12 949623:1-08 708414 5- 19 56 5 658284 4-12 g49558|i-o8 708726 5.19 291274 55 6 658531 4-11 949494 I -08 709037 5-19 29og63 54 I 608778 4-11 949429 1.08 7og34g 5-19 2go65i 53 65go25 4-u 949364 1.08 70g66o 5-19 5.18 2go34o 5i 9 659271 4-10 949300 1-08 709971 710282 200029 289718 5i •.3 659517 4-10 ■ 949235 1-08 5-18 5o II 9-659763 4-10 9-9^9170 i-o8 9-710593 5.18 10-289407 289096 288785 S 13 660009 . 660255 4-09 949105 i-o8 710904 5-i8 i3 4-09 949040 I -08 711215 5-18 tl r4 66o5oi 4-09 g48g75 I -08 711625 5-17 288475 15 660746 4-09 948910 i-o8 7II836 5.17 288164 45 i6 660991 661236 4-o8 948845 i-o8 712146 5.17 287854 44 \l 4-08 948780 1-09 712456 5-17 287544 43 661481 4-o8 948715 1-09 712766 5-16 287234 42 '9 661726 4-07 g4865o 1-09 713076 5-16 286924 41 20 661970 4-07 948584 1-09 713386 5-16 286614 40 21 9-662214 4-07 9-948519 1-09 9-713696 5-16 10-286304 39 38 22 662459 662703 4-07 4-o6 948454 I- 09 714005 5.16 285gg5 23 948388 i-og 714314 5-i5 285686 37 24 662946 4-06 948323 1-09 714624 5-i5 235376 36 25 663190 6634M 4-06 948257 1-09 714933 5-i5 285067 35 26 4-o5 . 948192 1-09 716242 5-i5 284768 34 ^^ 663677 4-o5 948126 i-og 7i555i 5-14 28444g 33 663920 4-o5 948060 1-09 7i586o 5-14 284140 32 29 664i63 4-o5 947995 I-IO 716168 .5-14 283832 3i 3o 664406 4-04 947929 I-IO 716477 9-716785 5-14 283523 3o 3i 9-664648 4-04 9-947863 I-IO 5-14 10-283215 11 32 664891 665i33 4-04 947797 947731 I-IO 717093 5-i3 282907 282599 33 4-o3 I-IO 717401 5-i3 27 34 665315 4-o3 947665 1-10 717709 718017 5-i3 2S2291 26 35 665617 4-o3 947600 I-IO 5-i3 281983 25 36 665859 4-02 947533 l-IO 718325 5-i3 2S1670 24 u 666100 4-02 947467 1-10 718633 5-12 281367 23 666342 4-02 947401 1 -10 718940 5-12 281060 22 3g 666583 4-02 947335 l-IO 719248 5-12 280762 21 40 666824 4-01 947269 I-IO 719555 5-12 280445 20 4i 9-667065 4-01 9-947203 I-IO 9-719862 5-12 io-28oi38 19 18 >7 42 43 6673o5 667546 4-01 4-01 947136 947070 i-ii 1-11 720169 720476 720783 5-11 5-II 279831 279624 44 667786 4-00 947004 i-ii 5-n 279217 16 45 668027 4-00 946937 I-U 721089 5-II 278911 l5 46 668267 4-00 946871 l-II 72l3g6 5-II 27S604 14 % 6685o6 3-99 946804 I-II 721702 5-10 27829S i3 668746 3-99 946738 1-11 722009 7223i5 5-10 277901 12 49 668986 3-99 g4667 1 l-II 5-10 2776^5 II 5o 669225 3-99 946604 I-II 722621 5-10 277379 10-277073 10 5i 9-669464 3-98 9 946538 I-Il 9-722927 5-10 t 52 669703 3-g8 946471 I-II 723232 5-og 276768 53 669942 3-93 946404 l-II 723538 5-Q9 276462 1 54 670181 3-97 946337 l-II 723844 5-09 276156 6 55 670419 3-97 946270 1-12 724149 5-09 275351 5 56 670653 3-97 946203 I-I2 724454 5-09 5.0S 270646 4 U t]l^. 3-97 946 1 36 I-I2 725o65 275241 3 3-96 946069 1-13 5.08 274935 3 =9 671372 3-96 946002 I-I2 725369 5.08 274631 I 60 671609 3-96 g45g35 I-I2 726674 5-08 274336 Cosine D. Sine 62° Co tang. D. Tang. M. 46 (28 DEGREES.) A TABLE OF LOGARITHMIC m: Sine 1 D. Cosine | D. Tang. 1 D. Cotang. 9-67i6o9| 3-96 9.945935 1. 1 2 9-725674! 5-08 10-274326 Jo 1 671841 3.93 945868,1.12 725979 5.08 274021 ll 1 = 672084 3.95 945800; 1-12 726284 5-07 273716 1 3 672321 3.95 945733,1-12 726588 5-07 273412 ?2 4 67255? 3.95 945666' I -12 726892 5-07 273108 56 5 672795 673032 3.94 945598;! -12 945531 1-12 727197 5.07 272803 55 6 3.94 727501 5-07 272499 54 I 673268 3-94 9454641-13 727805 728109 5-o6 272195; 53 673505 3.94 945396' I- i3 ?-°^ 271891 271588 52 9 673741 3.93 9453281.13 728412 5-o5 3i 10 67397; 3.93 94526i!i.i3 728716 5-o5 271284 3o II 9-674213 3.93 9-946193 i-i3 9.729020 1 5-06 10-270980 il 12 674448 3.92 945i25'i.i3 729323 : 5-o5 270677 i3 674684 3.92 945o58i.i3 729626 ^°^ 270374: 47 14 674919 675155 3.92 944990: i.i3 729929 730233 1 5-o5 2700711 46 i5 3.92 9449221.13 944854' I -i3 5-o5 269767 45 |6 675390 3.91 73o535 5-o5 269455 44 11 675624 3-91 944786' I -i3 73o838 5-04 269162 43 675859 3-91 944718 i-i3 731141 5-04 268859 41 268556! 41 '9 67609^ 3.91 944650' i-i3 731444 5-04 20 676328 3 '90 944582|i-i4 731746 5-04 268254: 40 21 9 676562 3-90 9-9445i4:i-i4 9-732048 5 -04 10-267952' 39 22 676795 677030 3.9a 944446: 1-14 73235i 5-o3 267649' 33 23 It 944377 I -14 732653 5-o3 267347; 37 24 677264 944309 1-14 732955 5-o3 2670431 36 23 '& 3-89 944241,1-14 733257 5-o3 266743 35 26 I'M 944172,1-14 733558 5-o3 266442 34 ^^ 677964 944104 1-14 733860 5-02 266140! 33 678197 3-88 944035 1. 14 734162 5-02 265838 32 29 6-;843o 3.88 943967 1. 1 4 734463 5-02 265537! 3i 3o 678663 3-88 943899I1.14 9-943830 1.14 734764 5-02 255236 3o 3i 9-678895 3.87 9-735o65 5-02 10-264934' 29 264633] 28 32 679128 3.87 943761 1.14 735367 5-02 33 679360 3.87 943693 Vi 5 735668 5-01 264332' 27 34 679592 3.87 943624 1.1 5 735969 5.0, 26403 1 ; 26 35 679824 3-86 943555 1. 15 736269 5-01 263731 25 36 6Soo56 3-86 943486 1. 1 5 736570 5-01 253430 a ll 680288 3-86 9434n!i.i5 736871 5-01 263129 23 68o5i9 3.85 9433481-15 737171 5-00 262S29I 22 39 680750 3.80 943279' I -i5 737471 5-00 262529' 21 40 680982 3-85 943210 1-15 73777' 5-O0 262229! 20 41 9-68i2i3 3-85 9-943i4i'i-i5 9-738071 5-00 10-261929; ID 261629: 18 42 681443 3.84 943072 1-15 738371 5-00 43 681674 3-84 943oo3 i-i5 738671 4-99 261329 '7 261029' 16 44 681905 3-84 942934' 1-15 738971 4-99 45 682135 3.84 9428641-15 739271 4-99 260729I l5 46 682365 3-83 942795:1-16 739570 4-99 260430, 14 % 682595 3-83 942726 I -16 739870 740169 740468 4-99 26oi3o i3 682825 3-83 9426561-16 4-98 259831, 12 49 683o55 3.83 942587 1-16 259532' 11 30 683284 3-82 942517! 1-16 9-942448; I -16 740767 g. 741 066 4-98 250233 10 10-258934! 9 258635! 8 5i 9-683514 3-82 4-98 52 683743 3-82 9423781-16 741365 4-9S 53 683972 3.82 9423o8 i-i6 741664 4-9S 258336! 7 258o38; 6 54 684201 3.81 942239 i-i6 741962 4-97 55 68w3o 3.81 942169:1-16 74:261 4-97 257739] 5 56 684658 3.81 942099 1-16 742559 742858 4-97 257441 1 4 u 684887 3.80 942029I1.16 4-97 257142 3 6851 i5 3.80 941959 i-i6 941889:1.17 743155 4-97 256844 a 59 685343 3-80 743454 4-97 4-96 256546 I 60 685571 3.80 941819 1-17 743752 266248 Cosine D. Sino 161° Cotang. D. Tang. M^. BINE8 AND TANGENTS. (29 DliOREBS. ) 47 M. Sine 1). Cosine D. Tang. JJ. Colang. 9-685571 3-8o 9-941819 1-17 9-743752 4 96 ,10-356248 60 I 685799 3 79 94174911-17 744o5o 4 96 sSSgSo 59 3 686027 3 79 941679 I -17 744348 4 96 255652 58 3 686254 3 79 941609 1-17 744645 4 96 255355 ^7 4 686482 3 ]t 94i539li.i7 744943 4 96 255o57 56 5 686709 686936 3 94I469I-I7 • 745240 4 95 254760 55 6 3 78 94.398l1.17 745538 4 95 254462 54 I 687163 3 78 941328 I -17 745835 4 95 254165 53 687389 3 78 941258 I -17 746132 4 9^ 253868 52 9 687616 3 77 941 187 I-I7 746429 4 95 253571 5i 10 687843 3 77 941117 9-941046 1. 17 746726 4 95 253274 5o II r. 688069 688295 3 77 I -18' 9-747023 4 94 '°tlUJ 49 48 12 3 77 940975 I -18 747319 4 94 i3 688521 3 76 940005 i-iS 747616 4 94 252384 ^1 14 688747 3 76 940834 1-18 747913 748200 7485o5 4 94 252087 46 i5 68K972 3 76 940763 i-i8 4 94 251791 45 i6 689198 3 76 940693 i-i8 4 93 251495 44 \l 689423 3 75 940622 i-i8 748801 4 9^ aSngo 260903 43 689648 3 75 94o55i 1-18 749097 4 93 42 '9 689873 3 75 940480 1-18 749393 4 9? 260607 41 20 690098 3 75 940409 i-iS 749689 4 93 25o3ii 40 21 9-690323 3 74 9-940338 I-I8 9-749983 4 93 io-i5ooi5 It 22 690548 3 74 940267 i-i8 750281 4 92 249719 23 690772 3 74 940196 i-i8 750576 4 92 249424 37 24 690996 3 74 940125 1-19 750872 4 92 249128 36 25 691220 3 73 940054 1-19 761167 4 92 248833 35 26 691444 3 ,3 939982 1-19 751462 4 92 248538 34 27 691668 3 73 93991 1 1-19 751757 4 92 248243 33 23 691892 3 73 000040 1-19 7D2002 4 9' 247948 32 29 692115 3 72 « I -19 752347 4 91 247653 3i 3o 692339 3 V I-I9 752642 4 91 247358 3o ■ 3i 9-692562 3 72 g-939625 1-19 9-752937 4 91 10-247063 29 32 692785 3 71 939554 I -19 753231 4 91 246769 28 33 693008 3 7' 939482 1-19 753026 4 9' 246474 ^I 34 693231 3 7< 939410 I-I9 753820 4 90 246180 26 35 693453 3 71 939339 1-19 7.54115 4 90 245885 25 36 693676 3 70 939267 1-20 754409 754703 4 90 245591 24 37 693898 3 70 939195 1-20 4 90 240297 23 38 694120 3 70 939123 1-20 754997 4 90 246003 22 39 694342 3 70 939052 93S980 1-20 750291 4 ? 244709 21 40 694564 3 69 1-20 755585 4 244410 20 41 •9-694786 3 69 9-93S908 938836 1-20 9-755S78 4 ^9 10-244122 19 42 695007 3 69 1-20 7564^5 4 ^9 243828 43 6g5229 3 6? 938763 1-20 4 ^9 243535 '2 44 695450 3 68 938691 1-20 .7,56759 4 ^9 243241 16 45 695671 3 68 938619 1-20 707052 4 ll 242948 i5 46 695892 3 68 938547 1-20 7.57345 4 242655 14 % 6g6ii3 3 68 938475 1-20 757638 4 88 242362 i3 696334 3 67 938402 I-2I 757931 4 88 242069 12 49 696554 3 67 938330 I-2I 768224 4 88 241776 241483 II 5o 696775 3 67 938258 1. 21 758517 4 88 10 5i 9-696995 3 67 9-938i85 I-2I 9-758810 4 88 10-241190 Q 8 52 697215 3 66 9381 i3 1-21 759102 4 87 240898 53 697435 3 66 938040 1-21 759395 4 i^ 24o6o5 I 54 697654 3 66 937967 937895 I-2I 759687 4 l^ 24o3i3 55 697874 3 66 I-2I 7-59979 4 87 24002 1 5 56 698094 3 65 937822 I-2I 760272 4 S"? 239728 i 57 69831 3 3 65 937749 I-2I 760564 4 ?I 239436 3 58 698532 3 65 937676 I -21 760856 4 86 239144 238852 2 59 698751 3 65 937604 1-21 761148 4 86 I 60 698970 3 64 937531 I -21 761439 4-86 238561 Coaipe n. Sine G0° Ootimg. D. Taiig^ M. 18 (30 DEOAEES.) A rABLK OF LOGARITHMIC M. Sino D. Cosine | D. Tang. D. Cotiuig- 1 10-238561 60 o 9 698970 3-54 9-937531 1^21 9.761439 4.86 I 699189 3-64 9374581-22 761731 4.86 238269' io 237977 1 58 1 699407 3-64 937385|i-22 937312 1-22 762023 4.86 3 699626 3-64 762314 4.86 237685 57 i 699844 3-63 9372381-22 762606 4.85 237394 56 S 700062 3-63 937i65|i-22 t'^. 4-85 237103: 55 6 700280 3-63 93709211-22 4.85 2368 1 2 54 I 700498 3-63 937019' I -22 ibM-.q 4.85 236521 53 700716 3-63 936946'! -22 9368721-22 763770 4.85 236230 52 9 700933 3-62 764061 4.85 235939 5i 10 70.. 5i 3-62 9367991.22 9-936725 1-22 764352 4.84 235648 5o II 9.701368 3-6J 9.764643 4-84 10-235357 % 12 701585 3.62 936652 1-23 764933 4-84 235067 i3 701802 3-61 9365781-23 765224 4-84 234776 47 14 702019 3-6i 9355o5i-23 765514 4.84 234485 46 i3 702236 3-6i 936431 ]i -23 7658o5 4-84 234195 45 i6 702452 3-6i 936357 1-23 766095 4-84 333905 44 \l 702669 3-60 936284 1-23 766385 4-83 2336i5 43 702883 3-60 9362ioli-23 766675 4-83 233325 42 19 7o3ioi 3-6o 936i36 1-23 ■jbbgbj 4-83 333o35 41 20 703317 9.703533 3 -60 936062 1-23 767255 4-83 232745 4D 21 3-59 9-935988 1-23 9.767545 4-83 10-332455; 39 1 22 703749 3-59 935914 935840 1-23 767834 768124 4-83 232166 38 23 703964 3-59 1-23 4-82 231876 37 24 704179 704395 3-59 935766 1-24 76S413 4-82 33 1 587 36 20 3-5? 935692 1-24 768703 4-82 331297 35 26 704610 935618 1-24 768992 4-82 23lQ08 34 ^? 704825 3-58 935543 1-24 ■ 769281 4-82 230719 33 7o5o4o 3-58 935469 1-24 769570 4-83 33o43o 33 29 705354 3-58 93539D 1-24 769860 4-81 33oi4o 3i 3a 705469 9-705683 3-57 935320 1-24 770148 4-8i 22oS52 3o 3i 3-57 9-935246 1-24 9.770437 4-81 10-229563 28 32 705898 3-57 935171 1-24 770726 4.81 229274 228985 33 706 r 1 2 3-57 935097 1-24 771015 4. Si 27 34 706326 3-56 935022 1-24 77i3o3 4-Si 228697 36 35 » 3-56 934948 934873 1-24 ??:^o 4.81 228408 35 36 3-56 1-24 4.80 328120 34 U 706967 3-56 934798 1-25 772168 4.80 327832 33 707180 3-55 934723 934649 1-25 7T457 4. So 337543 32 39 707393 3-55 1-25 772745 4-80 237355 236967 31 40 707606 3.55 934574 1-25 773033 4-8o 20 41 9-707819 708032 3-55 9-934499 1-25 9.773321 480 10-326679 \l 42 3-54 934424 1-25 773608 4-79 226392 43 708245 3-54 934349 1-25 7738o6 774184 4-79 226104 17 ii 708458 3-54 934274 1-25 4-79 3258i6 i6 45 708670 708882 3.54 934199 934123 1-25 774471 4-79 235539 i5 46 3-53 1-25 774759 775046 4-79 235341 14 % 709094 3-53 934048 1-25 4-79 324954 224667 i3 709306 3-53 933073 1.25 775333 i]% 12 P 709518 3-53 933898 1-26 775621 224379 II 5o 709730 3-53 933822 1.26 775908 4-78 234093 10 5i 9-709941 3.52 9-933747 1-26 9-776195 776482 4-78 «o-2238o5 I 52 710153 3-52 933671 1-26 4-78 3335i8 53 54 710364 710575 710786 3-52 3-52 933596 933520 1-26 1-26 V'-i'o^ 4-78 4.78 323231 223945 323658 I 55 3-5i 933445 1-26 777342 4-78 5 56 710997 711208 3-5i 933369 933293 1-26 777628 4-77 322372 323085 4 % 3-5i 1-26 777915 778201 4-77 3 711419 3-5i 933217 1-26 -4-77 221799 2 59 7116J9 3-5o 933141 1-26 778487 4-77 32l5i2! 1 6o 711839 3-5o 9330661-26 77S774 4-77 32I226| 1 Coaiue 1 D. Sine 50° GoVxDg. 1). - Tansf. 1 M. SIKE8 AND TAXGE.VTS. (31 DROHKES. ) 4 M. Sine D. 1 Cosiiio 1 D. 1 Tung. D. 1 COt!l.lJf. 60 9-7II839 3-5o 9-933o66 1-26 , 9-778774 4-77 10-221226 1 712030 3 5o 93299o|i •37 779060 4-77 4-76 220940! 59 1 712260 3 5o 932914,1 37 779346 23o654| 58 i 712469 3 49 932838 1 37 779632 4-76 22o368| 57 4 712679 3 49 932762 1 37 7799.8 4 --,6 220082 56 5 712880 71309S 3 49 932683 1 37 780203 4-76 2.9797! 55 6 3 49 932609 1 933333 1 37 7804S9 4-76 2193.1 34 I 7i33o8 3 45 37 780773 4-76 2.0225 53 2.8940 52 7135.7 3 48 932437,1 37 781060 4-76 9 713726 3 48 933380' 1 37 781346 4-75 2.8654 5i 10 713935 3 48 932304! 1 27 78.63. 4-75 2183691 5o 11 9'7i4i44 3 48 9-933228|i 27 9-781916 4-75 10-2.8084, 49 13 714352 3 47 932i5i,i 3^ 78220. 4-7= 217799 48 i3 714561 3 47 933075,1 782486 4-75 2.7314 ii i4 714769 714978 7i5iS6 3 47 931998 1 38 78377.1 4-75 317339 i5 3 47 93i93i'i 38 783o56 4-75 3.6944 45 i6 3 47 931 845; 1 28 78334. 4-75 3.6659 44 n 71 5394 3 46 9317681 931691 1 28 783626 4-74 216374 43 iS 7 1 56o2 3 46 28 7839.0 4-74 216090 42 19 715809 3 46 931614,1 28 784.95 4-74 2i58o5 4. 20 716017 3 46 93i537|i 38 784479 4-74 3.5321 40 21 9-716224 3 45 9o3i46o 1 38 9-784764 4-74 10-2153.36 ^8 22 716432 3 45 93.383 1 28 7S3048 4-74 ■ 3.4953 38 23 716639 3 45 93.306 I 28 783333 4-73 214668 ll 24 716846 3 45 93.329 I 29 7856.-6 4-73 2.4384 35 717053 3 45 93ii53 1 39 783900 4-73 214.00 35 36 717259 3 44 93.073 1 39 786.84 4-73 2.38.6 34 11 717466 3 44 9309981 29 78646S 4-73 2.3532 33 717673 3 44 930921 1 39 786733 4-73 2.3248 32 29 717379 7i8o85 3 44 930843.1- 29 787035 4-73 212964 3. 3o 3 43 9307661- 29 7873.9 4-72 2.26S. 3o 3i 9-718291 3 43 9-930688 1- 29 9-7S7603 4-72 .0-2.2397 29 28 32 718497 718703 3 43 ~ 93061 1 I - 29 787886 788.70 4-73 2.2.14 33 3 43 93o533 .- 29 4-73 2..830 V 34 718909 3 43 93o456;l- 39 788453 4-72 21.547 26 35 719114 3 42 93o378I- 29 788735 4-73 2.1264 25 36 719320 3 42 93o3oo 1- 3o 7890.9 4-72 2.0981 24 ll 719525 719730 3 42 930223. 1 3o 789302 4-71 2 . 0698 33 3 42 93o.45|i 3o 789385 4-71 2 . 04 . 5 33 3q 719935 3 41 9300671 1 3o 789868 4-71 210.32 31 4o 720140 3 41 939980 I 3o 790.5. 4-71. 209849 20 . 4i 9-720345 3 41 9-9290.11 9398331 3o 9 - 790433 4-7. .0-209567 19 ] 18 ; 42 720549 3 41 3o 790716 4-71 209284 43 720754 3 40 929753,1 3o 790999 79128. 4-71 209001 208719 17 : 44 720958 3 4c 929677 I 3o 4-71 16 , 45 721162 3 40 929599 1 3o 79.563 4-70 308437 'i5 : 46 721366 3 40 92952. 3o 791846 4-70 308.54 14 : % 721570 3 40 929442 3o 792.28 4-70 207872 i3 i 721774 3 39 929364 3i 7934.0 4-70 207590 12 ' 49 721978 3 39 9292S6 3. 792692 4-70 20730S II 5o 722181 3 39 929207 3i 792974 4-70 207026 10 5i 9-722385 3 39 9-929.391 3. 9-793256 4-70 10 206744 I; '52 722588 3 39 38 929050 1 928972 1 9388931 3i 793538 4-69 206462 .53 722791 3 3. 793819 4-69 206181 7 54 722994 3 38 3i 794101 4-69 205899 6 55 723197 3 38 928815 1 3. 794383 4-69 2o56i7 5 ■ 56 723400 3 38 928736 1 3. 794664 4-69 205336 4 57 7236o3 3 37 928657 1 3. 794945 4-69 2o5o55 3 58 • 7238o5 3 37 928578;! 3i ]$Vol 4-69 4-6-8 204773 . ,3 ' 59 724007 3 37 9284991 3i 20449a 'I : 6o 724210 3 37 928420I 1 3i 795789 4-68 204211 • '0 ' CoBiiie D. Si..e 1 S80 Ootanw. r>. Taiig. U.] 60 (32 DEGREES.) A table! of logarithmic M.' •.Sine i.. Cosine | I). Tang. D. Cotang. 60 9-724210 3.37 9-928420'! -32 9-795789 4-68 lo- 20421 1 I 724412 3.37 928342 1-32 7960-70 4-68 203930 5§ 2 724614 3-35 928263 1-32 7963D! 4-68 203649 203368 3 724816 3-36 928183 1-32 796532 4-68 P. 4 725017 3-36 928104 1-32 796913 4-68 2030S7 56 5 725219 3-35 928025]! -32 797'94 4-68 202806 55 6 725420 3-35 vX'y-3i 797475 4-58 202525; 54 I 725622 3-35 797755 4-58 202245! 53 725S23 3-35 927781 1-32 798036 4-57 201964, 52 9 726024 3-35 9277081-32 927629 1-32 9-937549'! -32 7983.6 4-67 201684 5i 10 726225 3-35 79S596 4-67 201 iOi 5o II 9.726426 3.34 9.798877 4-67 I0.20II23 ii 13 726626 3-34 927470'! -33 799>57 4-67 2J0843 i3 726827 3.34 927390'! -33 799437 4-67 200563 47 14 727027 727228 3-34 9273io|l-33 7997 > 7 4-67 200283 46 i5 3.34 927231 1-33 799997 800277 4-66 200003 45 l6 727428 3-33 927151 1-33 4-55 199723 44 \l 727628 3-33 927071,1-33 800557 4-66 199443 43 72782S 728027 3-33 926991 |! -33 8oo836 4-66 199154 198884 42 '9 3-33 926911;! -33 801116 4-56 41 20 72S227 3-33 926S3i,!-33 801396 4-56 198604 40 21 9.728427 3-32 9-926751 1-33 9-801675 4-56 10-198335 3o 22 728626 3-32 076671! 1-33 801955 4-56 198045 38 33 728825 3-32 925591'! -33 802234 4-65 197766 ll 24 729024 3-32 9265ii!i-34 8o25i3 4-65 197487 25 729223 3-31 925431,1-34 802792 4-65 197208; 35 26 729422 3.3i 926351 1-34 803072 4-65 196928I 34 'I 729621 3.3i 926270:1-34 8o335i 4-55 196649I 33 729S20 3.3i 926190;! -34 8o353o 4-55 196370 32 =9 730018 3-3o 9261 loll -34 80390S 4-65 196092 3i 3o 730216 3.3o 926029^ 1-34 804187 4-65 195813 3o 3i 9 -7304 1 5 3-30 9-925949 1-34 9-804466 4-64 10.195534 It 32 7306 1 3 3-3o 925863 !-34 804745 4-64 195255 33 730811 3.30 925783 1-34 8o5o23 4-64 194977 27 34 731009 3-29 925707 1-34 8o53o2 4-64 194698 26 35 731206 3-29 925626 1-34 8o558o 4-54 194420 25 36 731404 3-29 925545 1-35 8o5859 4-64 194141 24 u 731602 3-29 925465 1.35 806137 4-54 193853 23 731799 3-29 925384 1-35 8o54!5 4-63 193585 32 39 731996 3-2S 9253o3ii-35 806693 4-63 193307 2! 40 732193 3-28 925222JI-35 80697! 4-63 193029 30 41 9.732390 3-a8 9>925!4i 1-35 9-807249 4-63 10-19275! \t 42 •732587 3-28 925060 1-35 807527 4-63 192473 43 732784 3-28 924979l'-35 807805 4-63 192195 \l 44 732980 3.27 924897 1-35 808083 4-53 191917 45 733177 3.27 924816 1.35 8oS36i 4-53 191639 :5 45 733373 ^=^ 924735 1.36 8o8638 4-53 191362 14 % 733569 3-27 924654 1-36 808916 4-52 191084 13 733763 3-27 924572 1-36 809193 4.62 190S07 12 49 733961 3-26 924491 1-36 809471 4-63 190529 II 5o 734107 9-734353 3-26 924409 9924328 1-35 809748 4-63 190252 10-189975 10 5i 3-26 1-36 9.810025 4-63 ? 52 734549 3-25 924246 1-36 8!03o2 4.62 189698 53 734744 3-25 924164 I-35 8io58o 4-53 189420 I 54 •J34939 73di35 3-25 924083 1.36 810857 4-63 189143 55 3-25 92400! 1.36 8ii!34 4-6! 188866 5 56 735330 3-25 933919 I.35J 811410 4-6i 188590 4 u 735525 3-25 923837!!. 36 8! 1687 4-6i 188313 3 735719 3-24 923755:1 .37 811954 4'5i !88o36 1 5, 735914 3-24 923573 1-37 81224! 4-61 \i]in I 6o 736109 3-24 923591 1-37 812317 4-6i Cosino D. Sine BT" Ckitanpf. D. Tang. U. SINES AKV TANGENTS. (33 DEGREES. ) PI fMT ijine D. ^ Cosine D. Tang. D. Cotang. '■f(.ll^ 3 24 9-923591 1.37 9-812617 4-61 10-1874-83 "60" I 3 24 9.23509 1-37 812794 4-6i rS-7 2o6 59 3 73649S 3 24 923427 1-37 813070 4-6i 186930 38 3 7366g2 3 23 923345 -37 813347 4-60 186663 57 4 736886 3 23 923263; -37 8i3623 4-6o ■86377 66 5 737080 3 23 923181 -37 813899 4-6o 18610T 55 1 6 737274 3 23 923098 -37 814175 4-60 186825 54 I 737467 3 23 923016 •37 814452 4-60 185548 53 1 737661 3 22 922933 922851 -37 814728 4-60 186272 52 i 9 737855 3 22 ■M 8i5oo4 4-60 184996 ^ 1 10 738048 3 22 922768 816279 9-815555 4-60 184721 5i 11/ 9-738241 3 22 9-922686 -38 4-59 10-184445 ''g 13 V738434 3 22 922603 -38 8i583i 4-69 184160 183893 48 i3 738627 8 21 922620 .38 816107 4.59 47 U 738820 3 21 922438 -38 8i6382 4-59 i836i8 46 l5 739013 3 21 922355 -38 8i6658 4-59 i«3342 45 i6 739206 3 21 922272 -38 816933 4-59 183067 44 17 739398 3 21 922189 -38 817209 4-69 1S2791 43 18 739590 739783 3 20 922106 -38 817484 4-59 182616 42 '9 3 20 922023 -38 817759 818035 4-69 182241 41 20 739975 3 20 921940 -38 4-58 181966 40 31 9-740167 3 20 9-921857 -39 9-8i83io 4-58 10- 181690 39 38 22 740359 3 20 921774 -39 81 8585 4-58 181416 33 74o55o 3 '9 921691 -39 818860 4-58 181 i4o ll 34 740742 3 '9 921607 -39 819135 4-58 i8o865 23 740934 3 '9 921524 .39 819410 4-58 180690 35 26 741 1 25 3 '9 921441 -39 819684 4-58 i8o3i6 34 27 74i3i6 3 ]t 921357 -39 819969 4-58 1 8004 1 33 28 741 5o8 3 921374 .39 826234 4-58 179766 32 29 741699 3 18 921190 1 .39 820608 4-57 179492 3i 3o 741889 3 18 921107 1 .39 820783 4-57 179217 3o 3i 9-742080 3 18 9-921023 1 .39 9-821067 4-57 10-178943 29 28 32 742271 3 18 920939 I -40 821332 4-57 178668 33 742462 3 17 920856 1 -40 821606 4-57 178394 27 34 742652 3 '7 920773 1 .40 821880 4-57 178120 25 35 742842 3 >7 920688 1 .40 822154 4-57 177846 25 36 743o33 3 >7 920604 1 -40 822429 822703 4-57 177671 24 37 743223 3 17 920520 .40 4-67 177297 177023 23 38 743413 3 16 920436 -40 822977 4-55 32 39 743602 3 16 920352 -40 823260 4-56 176760 21 40 743792 3 16 920268 -40 823524 4-56 176476 20 41 9-743982 3 16 9-920184 -40 9-823798 4-56 10-176202 :i 42 744171 3 i5 920099 -40 824072 4-56 176928 43 744361 3 15 920015 .40 82J345 4-56 175655 17 44 744550 3 i5 919031 •41 824619 824893 4-56 175381 16 45 744739 3 i5 919S46 -41 4-55 176107 i5 46 744928 3 i5 919762 -41 835)66 4-56 174834 14 s 745117 3 i5 919677! •41 '' 825439 825713 4-55 174661 i3 7453o6 3 14 919593: -41 4-55 174287 12 49 745494 3 14 919508, -41 826986 4-55 174014 I' 5o 7456S3 3 14 919424! •41 826269 4-55 173741 10 5i 9-745871 3 14 9919339' -41 9-826532 4-55 10-173468 9 173195 8 53 746059 3 14 919254 -41 826806 4-55 53 746248 3 i3 919169! .41 "827078 4-55 172922 7 54 746436 4 i3 91908D 1-41 827351 4-55 172649 6 55 746624 3 i3 ?;c5| 1-/.I 827624 4-55 172376] 5 56 746812 3 i3 1-42 827897 4-54 172103 i 57 746999 747187 3 i3 9i8S3o 1-42 . 828170 4-54 171830 3 58 3 12 918745 1-42 828443 4-54 171668 3 59 747374 3 13 918659' 1-42 828715 4-54 171285 I ; 60 747562 3 12 918574 1-42 828987 4-54 171013 IZ", Coaiaa D. Sine OGo Cotimg. D. Tan^ M. »3 (34 degrees) a table of logarithmic sr Sino D. Cosina D. Taiig. D. Cotang. 1 j 9-747563 3-13 9-9185741 9184891 -42 9-828987 4-54 io-i7ioi3 60 1 747749 3-12 -42 829260 4.54 170740 170468 u 1 747936 748123 3-13 91 8404! I ■42 829532 4-54 3 3-II 9i83i8li ■42 829805 4-54 170195 ?I 4 748310 3-11 9182331 ■42 830077 4-54 169923 56 5 ]tini 3-II 918147 ' -42 83o349 4.53 169651 55 6 3-11 918062,1 ■42 83o62i 4.53 169379 54 I 748870 749056 3-II 3-10 917976 1 917891 1 -43 ■43 830893 83ii65 4-53 4-53 169107 168835 53 53 9 749243 3-10 917805 1 ■43 831437 4-53 168563 5i la 749429 9-7495i5 3-10 9177191 •43 831709 4-53 168291 So II 3-10 9-917634 1 •43 9.831981 4.53 io^i68oi9 % 13 749801 3-10 917548 1 .43 832253 4-53 167747 i3 749987 3-09 917463 1 -43 832525 4-53 167475 47 14 750172 3-09 917376 -43 f^'796 4-53 167204 45 l5 75o358 3-09 917290 1 -43 833o68 4-52 i6„932 45 i6 750543 3-09 917204I -43 833339 4-52 1 6666 1 44 ;? 750739 730914 12 917118: 917032 -44 -44 833611 833882 4-52 4-52 166389 166118 43 42 19 751284 3o8 916946 1 916659 9-916773 -44 834154 4-52 165846 41 30 3-o8 ■44 834425 453 163575 40 31 Q-75r46g 3-o8 ■44 9-834696 452 io^i653o4 l§ 33 ^ 751654 751839 752023 3-o3 916687 ■44 834967 453 i65o33 33 3-08 916600 ■44 835238 452 164762 37 34 3-07 9i65i4 •44 835509 432 164491 35 35 752208 752392 3.07 916427 •44 835780 45i 164220 35 36 3-07 916341 •44 836o5i 4-5i 163949 163678 34 11 752376 3-07 916254 •44 836322 45i 33 752760 3-07 916167 •43 836393 4-5i 163407 32 39 752944 3-o5 916081 ■45 836864 4^31 I63i36 3i 36 753128 3-o5 915994 ■ 45 837.34 45i 162866 3o 3i 0-753312 3-06 9-915907 -45 9 -837403 431 10-162595 11 33 753495 3-o5 915820 ■45 837673 431 162323 33 753679 3-o5 915733 •45 837946 45i 162034 '1 34 753862 3-o5 915646 ■45 833-2.6 45i 161784 26 35 754046 3-o5 915559 ■45 838487 4^5o .6.3.3 25 36 754229 754412 3-05 915472 ■45 838737 4-5o 16.243 24 u 3-o5 915385 ■45 839027 45o 160973 23 754595 3o5 915297 -45 '& 4-5o 160703 23 39 754778 3-04 915210 -45 45o 160432 21 40 754960 3-04 9i5i23 .46 839838 45o 160162 20 41 9-755143 3 -04 9-915035 -46 9^840108 4^5o 10- 159892 [t 755326 3-04 914948 914860 • 46 840378 4^30 159622 43 755508 3-04 •46 840647 4^50 159353 \l 44 755690 3-04 914773, 914680! ■46 840917 4-49 139083 I388i3 45 755872 3-o3 •46 841.87 4-49 i5 46 756o54 3-o3 914598! •46 84.457 841726 4.49 158543 14 s 756236 3-o3 9i45iO| •46 4.49 158274 i3 756418 3o3 914422 • 46 84.996 4.49 1 58004 12 49 756600 3o3 914334' ■ 46 842266 4.49 157734 11 50 756782 3-03 914246 ■47 842335 449 157465 10 5i 9-756963 3-03 a:.9'4i58 •47 9 842805 449 10-157193 156926 § 52 757144 3oj 914070, 9.3894 •47 843074 449 53 757326 3-02 •47 843343 449 156657 I 54 757507 737688 3-02 ■47 8436.3 4.40 4. 48 156388 55 3-01 913806' ■47 843S82 i56ii8 5 56 ]l& 3-01 913718' •47 844131 448 155849 u 3-01 91 3630 ■47 S44420 4^48 i5558o 758330 3-01 913541 •47 844689 84495s 4-48 l553ii 5, 758411 3-01 913453 ■47 4.48 1 55041 66 75359. 3-01 913365,1-47 845227 448 154773 CoBiiie D. Sine ISflo Cgtaiig. D. T"*'*- . M. SINES AWD TANQENTa . (35 DEGRBES ) 53 nr Sine D. Cosine D. J Tang. ID. Cotang. 9-75859, 3-OI 9-91336; 1-47! 9-8452271 4-48 10-154773 60 I 7D877J 3-00 9i327f '•'''^ 1 845496 4-48 164604 5? 2 75895! 3-00 9i3i8- I -481 845764I 4-48 154236 53 3 759I3J 3-00 913099 i-4f - 84603: 4-48 153967 153698 1 53430 57 i 7593 1 J 3-00 9i3oic I-4E 846302 4.48 56 5 759492 3-00 912922 912833 I.4E 84657c 4-47 55 6 759672 2-99 1-46 846839 4-47 i53i6i 54 I 75985J 2-99 912744 I-4S 84710- 847376 4-47 162898 53 76003 1 2.99 912655 1-48 4.47 162624 52 9 760211 2-99 912566 1-48 847644 4-47 152356 5i 10 760390 2-90 912477 1-48 84-7913 4-47 1520S7 5o II 9.760569 2-98 9-912388 1-48 9-848181 4-47 io-i5i8i9 49 12 760748 2-98 912299 1-49 848449 4-47 i5i55i 48 i3 760927 2-98 912210 1-49 848717 848986 4-47 i5i283 47 14 761106 2-98 912121 1-49 4-47 i5ioi4 46 i5 761285 2.98 9i2o3i 1-49 849254 4-47 160746 45 i6 761464 2.98 91 1942 1-49 849522 4-47 4-46 160478 44 \l 761642 2-97 911853 1-49 850^8 l5o2IO 43 761821 2-97 91 1763 1-49 4-46 149942 42 ■9 761999 2-97 911674 1-49 85o325 4-45 149673 41 20 9-7623^ 2-97 911684 1-49 850693 9-85o86i 4-46 149407 40 21 2-9? 9-9II495 1-49 4-46 I0-I4gl39 148871 39 22 762534 91 1405 1-49 85II29 4-46 38 23 762712 762889 -2.96 9Ii3i5 1-56 851396 4-45 148604 37 24 2-96 911226 1-50 85i664 4-46 148336 36 25 763067 763245 2-96 9III36 i-5o 85 193 1 4.46 148069 35 26 2-96 91 1046 i-5o 8524^? 4-46 147801 34 27 763422 2-96 910956 i-5o 4-46 147334 33 28 763600 2-95 910866 i-5o 852733 4-43 147267 32 ?9 1 ''1VV 2.95 910776 910686 i-5o 853001 4-45 146999 146732! 3i 3o 763954 2-95 i.5o 853268 4-45 3o 3i 1 9-764131 2-96 9-9io596!i-5o 9-853535 4-45 10-1464631' 29 32 764Joa 2-95 9io5o6]i-5o 853802 4-45 146198' 28 145931 27 33 764485 2-94 910415 i.-5o 854069 854336 4-45 34 764662 2-94 910325 i-5i 4-45 145664 26 35 76433s 2-94 910235 i-5i 854603 4-45 145397! 25 i45i3o! 24 36 765oi5 2-94 910144 i-5i 854870 4-45 -1 765191 2-94 910054 i-5i 855i37 4-45 1448631 23 33 765367 2-94 909963 i-5i 855404 4-45 144596; 22 39 7-65544 2-93 909873 i-5i 855671 4-44 144329 21 4o ■'^^2'? 2-93 909782 I -Si 855g38 4-44 144062 20 4i 9-765896 2.93 9-909691 i-5i 9-836204 4-44 10-143796 10 143529 18 42 766072 2-93 9O9601 i-5i 856471 4-44 43 766247 2-93 909610 i-5i 866737 4-44 143263 17 44 766423 2-93 9094191-51 909328 1-52 867004 4-44 142996 142730 16 45 766698 2-92 867270 4.44 i5 46 766774 2-92 909237 1-52 867537 4-44 142463 14 ^2 766949 2-92 909146 1-52 857803 4-44 142197 i3 48 767124 2-92 909o55|i -52 858069 4-44 12 49 767300 2-92 908964] I -52 908873:1-52 858336 4.44 141664 II 5o 767475 2-gi 858602 4-43 141398J 10 10-141132I 140866 8 5i 9-767649 2-91 9-908781 1-52 9-858868 -4-43 5j 767824 2-91 908690 1-52 869134 4-43 53 ]tlV,l 2-91 908599 1-52 869400 4-43 140600 7 54 2.91 908507 1-52 9084161-53 869666 4-43 i4o334 6 55 7683481 2-90 859932 4-43 140068 5 56 76S522J 2-90 9o8324!i-53 8601981 4-43 139802 4 u 768697 2-90 903233 1-53 S60464' 4-43 139536 3 768871 2-90 90S141 1-53 860730I 4-43 139270 2 5, 769043 2-90 908049 1-53 860996s, .-.43 139005 1 1 60 769219. Cosine | 2 90 907958 1-53 S612JJJ Cotanff 1 (h-43 138739 D. Sine 54^ !A__ _TMg. Mf 64 (30 Dt "1 GREES.) A TABLE OF LOQABITHMIC M. " Sine 5- Cceiiio D. Tang. D. Cotang. 9.769219 2-90 9-907958 1-53 9-861261 4-43 10-138739 138473 60 I 769393 2 89 90-|§66 1 ■53 86i527 4 43 U I 769566 2 89 907774 1 .53 861792 862o58 4 42 138208 3 769740 2 89 907682 1 .53 4 42 137942 57 i 769913 2 89 907590 1 ■53 862323 4 42 137677 56 5 770087 2 ll 907498 1 ■53 862689 4 42 137411 55 6 770260 2 907406 I •53 862854 4 42 137146 54 7 770433 2 88 907314 I -54 863119 863385 4 42 1 36881 53 8 770606 2 88 907222 ■54 4 42 i365i5 52 9 770779 2 88 9071291 •54 853650 4 42 136330 5i 10 770952 2 88 907037|. •54 853915 4 42 i36o83 30 II 9-771125 2 88 •54 9-864180 4 42 lo- 135820 ii 12 771298 2 87 906852 1 ■54 S64445 4 42 135555 i3 771470 2 87 906760 1 •54 864710 4 42 133290 47 i4 771643 2 87 9066671 -54 864975 4 4. i35o25 46 i5 771815 2 87 9055751 906482 1 -54 865240 4 4> 134760 45 i6 771987 2 87 -54 8655o5 4 41 134495 44 \l 772159 2 87 9063891 .55 865770 4 41 i3423o 43 77233. 2 86 906296 1 .55 866035 4 41 133965 42 "9 7725o3 2 86 906204 I .55 866300 4 41 133700 41 20 772675 2 86 906111 1 .55 865564 4 41 .33436 f 21 9-772847 2 86 9-906018 1 .55 9-866829 4 41 .0-.33I7. 39 22 773018 2 86 905925,1 905832 1 .55 867094 867358 4 41 .32906 38 23 773190 2 86 ■ 55 4 41 132642 37 24 773361 2 85 905739, -55 867623 4 41 132377 36 23 773533 2 85 905643, ■55 867887 4 4. i32ii3 35 26 773704 2 85 905552 -55 868i52 4 40 13184S 34 27 773875 2 85 905459 '1 -55 8584 1 6 4 40 I3i584 33 28 774046 2 85 9053661 -56! 868680 4 40 i3i32o 32 29 774317 3 85 903272 ■56 868945 4 40 i3io55 3i 3o 774388 2 84 905179 9-9o5o85 ■56 869209 9-869473 4 40 130794 3o 3i 9-774538 2 8/. ■56 4 40 io-i3o527 29 28 32 774729 774899 2 84 904992 1 904898 -56 869737 4 40 .30263 33 2 84 -56 870001 4 40 129999 27 34 775070 2 84 904804 -56 870265 4 40 129735 26 35 773240 2 84 904711 904617 ■ 55 870529 4 40 12947' 25 36 775410 2 83 -55 870793 4 40 129207 24 ll 775580 2 83 904523 ■ 55 871037 4 40 128943 23 775750 2 83 904439 904335 •57 871321 4 40 128679 12S415 22 ^5 775920 2 83 •57 871085 4 40 21 40 776090 2 83 904241 -57 87.849 4 39 i3Si5i 20 4' 9-776259 2 83 9-904147 ■57 9-872112 4 39 10-127S88 19 42 776429 2 fri 904053 •57 872376 4 39 127624 18 43 776598 2 82 903959 -57 872640 4 39 127360 17 4.1 776768 2 82 903864 ■57 872903 4 39 .27097 16. 45 7769-7 2 8a 903770 .57 873.67 4 39 126833 i5 46 777106 2 82 ■ 903676 •57 873430 4 39 125570 14 i u 777275 777444 2 2 81 81 9o358i 903487 •57 sW 4 4 39 39 ■ 263o6| .3 126043' .2 49 777613 2 81 903392 ■58 874230 4 39 125780 11 5o 777781 2 81 903298 •53 874484 4 39 1255.6 10 5i 9-777950 778119 2 81 9 -903203 • 58 9-874747 4 39 io^is5253 9 5j 2 81 903 108 • 58 875010 4 ll 124990 8 i3 ■'■'?'!] 2 80 9o3oi4 • 58 875273 4 124727 7 !)i 778455 2 80 902919 902824 • 58 875536 4 38 124464 6 55 778624 2 80 • 58 875800 4 38 124200 5 ^6 mm 2 80 902729 .58 876063 4 38 123937 123674 4 ^7 2 80 902534 • 58 876326 4 38 3 7791 2h 2 80 902539 -59 . 876589 4 38 123411 2 5y 779295 779463 2 79 902444 • 59 876851 4 38 1 23 1 49 1 6o 2-79 902349 .59 877114 4 38 122886 U IiT Cosine n. Sine 53°l Cotang. D. Tan?. SINES AND TANGENTS. (37 DEGREES. 1 SS M. Sine D. ■ GiasiBe D". Timg. D. Coti.hg. 9-779463 2 •79 9.9Q§349i-59 9-877114 4-38 10-122886 60 I 77963 1 2 •79 9022511 1.59 877377 4-38 122623 5n 1 779798 3 79 902 1 58', 1. 59 877640 4-38 I2236o 58 3 779966 2 79 902063 1.59 877903 878165 4-38 122097 121835 57 4 780133 3 9010671-59 4-33 56 5 780300 3 78 901872I1-59 878428 4-38 121672 55 5 780467 2 78 901176 901681 1.59 878691 4-38 121309 54 I 780634 2 78 1.59 878953 4-37 121047 53 7S0801 2 ■J^ 90i585 1.59 879216 4-37 120784 53 9 780968 2 ■'^ 901490 ..59 879478 4-37 120522 5i 10 781134 2 78 901394 I -60 879741 4-37 120259 5o II n. 781301 2 77 9-901298 I -60 9 - 88ooo3 4-37 I J. 1 19997 tt 12 781468 2 77 901202 I -60 880265 4-37 119735 i3 18; 634 2 77 901 106 I -60 880528 4-37 119472 47 14 781800 2 77 901010 1-60 880790 88io52 4-37 IIC,2I0 118948 46 i5 781966 3 77 900914 I -60 4-37 45 i6 782132 2 77 900818 I -60 88i3i4 4-37 11 8686 44 \l 782298 2 76 900723 1-60 881576 4-37 118424 43 782464 ■ 2 76 900626 I -60 881839 4-37 118161 42 '9 782630 2 76 900529 900433 1-60 882101 4-37 117899 117637 41 20 782796 2 76 i-6i 882363 4-36 40 21 9.782961 2 76 9-900337 1-61 9-882625 4-36 10-117375 It 22 783127 2 76 900240 1-61 882887 883148 4-36 117113 23 783292 783458 3 75 900144 1-61 4-36 1 1 6852 ll 24 3 75 900047 1-61 883410 4-36 116590 25 783623 2 75 1-61 883672 4-36 116328 35 26 783788 2 75 899854 I -61 883934 4-36 116066 34 11 7S3953 3 ^? 899757 i-6i 884196 4-36 ii58o4 33 Y84118 2 75. 899660 1-61 884457 4-36 115543 32 29 784282 3 74 899564 1.61 884719 4-36 115281 3i 3o 784447 2 74 899467 1-62 884980 4-36 Il5p20 3o 3i 9-784612 2 74 9-899370 1-62 9-885242 4-36 10-114758 20 32 784776 2 74 899273 1-62 8855o3 4-36 1 14497 28 33 784941 2 74 899176 1-62 885765 4-36 1 14235 27 34 785io5 2 74 899078 1-62 886026 4-36 118974 26 35 785269 2 73- 898981 898884 1-62 8862S8 4-36 113712 25 36 785433 2 73 1-62 886549 4-35 ii345i 24 ll 785597 2 73 898787 1-62 886810 4-35 113190 23 785761 2 73 898689 1-62 887072 4-35 1 1 2928 22 39 785925 S 73 898592 1-62 887333 4-35 1 1 2667 21 40 786089 3 73 898494 1-63 887594 9.887855 4-35 1 1 3406 20 41 9-786252 2 72 9-898397 1-63 4-35 10112145 :? 42 786416 2 72 898299 1-63 888116 4-35 H1884 43 786579 3 72 898202 1-63 888377 4-35 111623 >7 M 786742 2 72 8981041-63 888639 4-35 iii36i 16 45 786906 3 72 898006 1-63 888900 4-35 lllIOO i5 46 787069 3 72 897908 1-63 889160 4-35 1 1 0840 14 47 787232 2 71 897810 1-63 889421 4-35 110579 iio3i8 i3 48 787395 3 71 897712 1-63 889682 4-35 12 49 7875D7 2 7' 897614 1-63 889943 4-35 1 10057 11 5o 787720 3 71 897516 1-63 890204 4-34 10-109535 10 5i g-787883 2 71 9-897418 1-64 9-890465 4-34 9 52 788045 2 7' 897320 1-64 890725 4.34 109275 8 53 788208 2 71 897222 1-64 890986 4.34 109014 108753 7 54 788370 2 70 897123 1-64 891247 4-34- 6 55 788532 2 70 897025 1-64 891507 - 891768 4-34 108493 ioS:32 5 56 788694 788856 2 70 896926 896828 1-64 4-34 4 ll 3 70 1-64 89J028 4.34 107973 3 789018 2 70 896729 1-64 892289 4.34 107711 1 59 7891S0 3 70 896631 1-64 892549 4-34 1 0745 1 1 60 7893^2 2-69 896533 1-54 892810 4-34 107190 Oosii'D D. Sine 52° Cotfui?. D. Tang. M. 27* se (38 DEGREES.) A TABLE 07 LOGARITHMIC M. Sine D. Cosine D. Tang. D. Cctaug. 9-789342 2-69 9-896532 1-64 9-892810 4-34 10-107190 106930 60 I 789504 2-69 896433 1-65 893070 4-34 u 3 789665 2-69 896335 |i-65 893331 4-34 106669 3 789827 2-69 896236 1-65 893591 893851 4-34 106409 ?z 4 789988 2-69 896137 1-65 4-34 106149 56 5 790149 2-69 2-68 896038 1-65 8941 1 1 4-34 105889 55 6 7903 10 895939 li-65 894371 4-34 1 05629 105368 54 I 790471 2-63 895840! I -65 894632 4-33 53 790632 2-68 895741 1-65 894892 4-33 io5io8 02 9 790793 790954 2-68 895641 1-65 890152 4-33 1 04848 5i 10 2-68 895542 1-65 895412 4-33 104588 5o II 9-791H5 2-68 9-895443 1-66 9-895672 4-33 io-io432f ^ 12 791275 2-67 895343 1-66 895932 4-33 104068 i3 791436 2-67 895244 1-66 896192 896452 4-33 io38o8 ^I 14 791596 791757 2-67 895145 1-66 4-33 103548 46 |5 2-67 895045 1-66 896712 4-33 103288 45 i6 79'9'7 2-67 894945 1-66 896971 4-33 103029 44 n 792077 2-67 894846 1-66 897231 4-33 102769 43 i8 792237 2-66 894746 1-66 897491 4-33 . 102509 42 ■9 792397 792557 2-66 894646 1-66 897751 4-33 102249 41 20 2-66 894546 1-66 898010 4-33 101990 10-101730 40 21 9-792716 2-66 9-894446 1-67 9-898270 4-33 u 22 792876 2-66 894346 1-67 898530 4-33 101470 23 793o35 2-66 894246 1-67 898789 4-33 101211 u 24 793195 2-65 894146 1-67 899049 899308 4-32 100951 a5 793314 2-65 894046; I -67 4-32 100692 100432 35 26 793514 2-65 893946:1.67 8938461-67 899568 4-32 34 ^1 793673 2-65 899827 4-32 100173 33 28 793832 2-65 89374511-67 900086 4-32 099914 33 29 793991 2-65 893645 1-67 900346 4-32 099654 3i 3o 794100 2-64 893544 1-67 900605 4-32 099395 10-099136 09S876 3o 3i 9-794308 2-64 9-893444 1-68 9 900864 4-32 It 32 794467 2-64 893343 1-68 901124 4-32 33 794626 2-64 893243 I -68 901383 4-32 09% 17 11 34 794784 2-64 893142 1-68 901642 4-32 098358 35 79 ',942 2-64 893041 1-68 901901 4-32 09S099 25 36 795101 2-64 892940 892839 1-68 902160 4-32 097840 24 ll 795259 a-63 1-68 902419 4-32 097581 23 795417 2-63 89263? 1-68 902679 902938 4-32 097321 23 39 795575 2-63 1-68 4-32 097062 21 40 795733 2-63 892536 1-68 903197 9-903455 4-3i 096S03 30 41 9-795891 2-63 9-892435 1-69 4-3i 10-096545 l§ 42 796049 2-63 S92334 1-69 903714 4-3i 096286 43 796206 2-63 892233 1-69 903973 4,-31 096027 \l 44 796364 2-62 892132 1-69 90433a •4-3i 095768 45 796521 2-62 892030 1-69 904491 904780 4-3i 095509 i5 46 796679 2-62 891929 1-69 4-31 og525o 14 % 796836 2-62 891726 1-69 9o5oo8 4-3i 094992 094733 i3 796993 797150 2-62 1-69 903267 4-3i 12 49 2-6l 891624 1-69 905526 4-3i 094474 II DO 797307 2-61 891523 1-70 905784 4-3i 094216 10 5i 9-797464 2-61 9-891421 1-70 9-906043 4-3i '°-°olXl t 52 797621 2-6l 891319 1-70 906302 4-3i 53 797777 261 891217 1-70 906560 4-3i 093440 7 54 797934 2-61 891115 1-70 906810 4-3i 093181 6 55 798091 2-61 891013 1-70 907077 4-3i 092923 5 56 798247 2-6l 890911 1-70 907336 4-3i 092664 i u 798403 2-60 890809 1-70 907594 907852 90S111 4-3i 093406 3 798560 2-6o Zlol 1-70 4-3i 092148 a 59 798716 798872 >-6o 1-70 4-3o 091889 I J±. 2-6o 890503 1-70 90S369 4-3o 091631 i Cosine D. Sine 51° Cotaiig. D. Tang. M. SINKS AND TANGENTS. (39 CSOllEES. 1 51 M. Sine U. Cosine | D. Tang. D. Cotang. ; 1 9-798872 z-6o 9-89o3o3ji-70 9-9o83t9 908628 4-3o 10-091631 60 I 799028 2 60 890400'! -71 4-3o 091372 It J 799184 2 60 890298'! -71 908886 4-3o 091114 3 799339 799495 799651 2 ^9 8go!95;i-7i 909144 4-3o 090856 57 4 2 59 800093:1-71 909402 4-3o 090598, 36 5 2 59 909660 4-3o 090340 55 6 799806 2 59 909918 4-3o 090082 089823 54 I 799962 2 59 8897831-71 910177 4-3o 33 8001 17 2 =2 889682 1-7! 910435 4-3o 089565 52 9 806272 2 58 889579 1. 71 910693 910951 4.30 089307 5i 10 800427 2 58 889477 1.71 4-3o 080049 5o II 9 -800582 2 58 9-889374 1-72 9 -91 1 209 4-30 10-08879! 088533 it 12 800737 800892 2 58 88927! 1-72 91 1467 4-3o i3 2 58 889168 1-72 911724 4-3o 088276 ii 14 801047 2- 58 889064 1-72 888961 1-72 91 1982 4.30 088018 46 i5 801201 2- 58 912240 4-3o 087760 087502 45 i6 8oi356 2- 57 888858 1-72 912498 912756 4-3o 44 \l 8oi5ii 2- 57 8887551.72 4-30 087244 43 8oi665 2- 37 88865! 1-72 9i3oi'4 4-29 086986 42 >9 801819 2 57 8885481.72 913271 4-29 086729 4! 20 801973 2 57 888444 1.73 913529 4-29 086471 40 21 9-802128 2 57 9-88834! 1-73 9-913787 4-29 !0-o862i3 It 22 802282 2 56 888237 1-73 914044 4-29 085956 23 802436 2 56 888134 i.:73 914302 4-29 085698 37 24 802389 802743 2 56 888o3o 1-73 914360 4-29 085440 36 25 2- 56 887926 887822 1-73 914817 4-29 o85i83 35 26 802897 8o3o5o 2 56 1-73 916075 4-29 084925 34 =7 2 55 887718 1-73 915332 4-29 084668 33 28 8o32o4 2 56 887614 1-73 915590 4-29 084410 ■32 ?9 803357 2 55 887510 1-73 9 '5847 4-29 084153 3! 3o 8o35ii 2- 55 887406 1-74 916104 4-29 083896 3o 3i 9-8o3664 2- 55 9-887302 1-74 9-916362 4-29 10-083638 29 1 32 8o38i7 2- 55 887198 '•74, 916619 4-29 08338! 28 33 803970 2 55 887093 8S6989 886885 1-74 916877 4-29 o83i23 27 34 804123 2 55 1-74 917134 4-29 082866 26 35 804276 2 54 1-74 917391 4-29 0S2609 25 36 804428 2 54 886780 1-74 917648 4-29 082352 24 ll 8o458i 2 54 886676 1-74 l\zl 4-29 4-28 0B2095 081837 23 804734 2 54 886571 1-74 22 3g 804886 2 54 886466 1-74 918420 4-28 o8i58o 21 4o 8o5o39 2 54 886362 1-75 918677 4-28 081323 20 4l 9-8o5i9i 2 54 Q-886257 1-75 9-918934 4-28 10-081066 IQ 18 42 8o5343 2 53 886152 1-75 9I9I9I 4-28 080809 43 805495 2 53 886047 1.75 919448 4-28 o8o552 \l 44 8o5647 2 53 886942 835837 1.75 919705 4-28 080295 o8oo38 45 & 2 53 1-75 919962 4-28 i5 46 2 53 885732 1-75 920219 4-28 079781 14 47 806103 2 53 885627 1-75 920476 4-28 079624 i3 48 806254 2 53 885522 1.75 920733 4-28 079267 12 49 806406 2 52 885416 1-75 920990 4-28 079010 078753 II 5o 806557 2 52 8853 II 1-76 921247 4-28 10 5i 9-806709 2 52 9-8832o5 1-76 9-92i5o3 4-28 10-078497 t 52 806S60 2 52 885100 1-76 921760 4-28 078240 53 807011 2 52 884994 884889 884783 1-^6 922017 4-28 077983 7 54 8o7i63 2 52 1-76 922274 4-28 077726 6 55 807314 2 52 1-76 922530 4-28 077470 5 56 807465 2 5i 884677 1-76 922187 4-28 077213 4 tl 807615 2 5i 884572 1-^6 923044 4-28 076966 3 807766 2 5i 884466 1-^6 Q233oO 4-28 076700 2 5, llZ] 2 -5i 884360 i.-;6 923557 4-27 076443 1 66 2 ■5x 884254 1-77 9238r3 4.27 076187 Cosine D. Sine 60° Cotang. D. Tang. IM. 58 (4C DEOSBES.) A TABLX OF LOGARITHMIC M. o Sine TJ. ' COBine m^ '■mg. D. Cotang, 9-808067 3-5l 9-884254 1-77 9-9S38.J3 4-27 10-076187 6a I 808218 2-5l 884148 '■77 924070 4 -37 075930 u 3 8o8368 3-5l 884042 ••77 924327 4 37 075673 3 8o85ig . 2-5o 883936 883829 883723 '•77 924583 4 27 075417 11 4 803669 2-5o '•77 924840 4 •37 075160 5 808819 2.5o '•77 925353 4 •37 074904 55 5 808969 3 -50 883617 '•77 4 37 074648 54 I 809119 3.5o 8835IO '■77 92586^ 4 • 27 074391 074135 53 809269 2-50 883404 1.76 4 -27 53 9 809419 2-49 883397 936133 4 ■27 073878 5i 10 809569 3'49 883191 9-883o84 1.78 926378 4 •27 073533 5o II 'tl^l 2-49 1.78 0-926634 4 •27 10-073355 1? 13 3-49 883977 1.78 936890 4 27 0731 IC i3 810017 3-49 883871 1.78 927147 4 27 072853 47 14 810167 3.49 3.48 883764 1-78 927403 4 27 073597 45 l5 8io3i6 883657 1.78 937659 927915 4 27 072341 45 l6 810465 3-48 883550 1-78 4 27 072085 44 \l 810614 3-48 882443 1.78 938171 4 27 071829 071573 43 810763 3-48 882336 '•79 928427 4 27 43 •9 810912 3-48 882229 '•79 928683 4 27 071317 41 30 811061 2-48 882121 '•79 938940 4 27 071060 40 31 9-8II2IO 3-48 9-882014 '•79 9-929196 4 27 10 070804 ll 32 8ii358 2-47 881907 '•79 929452 4 27 070548 33 8ii5o7 3-47 881799 '•79 929708 4 37 070393 070035 U 34 8ii655 3-47 881692 '•79 929964 4 36 35 811804 2-47 881 584 '•79 980220 4 36 069780 35 26 81 1952 2-47 881477 '•79 930475 4 36 069525 34 11 812100 2-47 881369 '■79 93073 1 4 36 069269 069013 068757 33 812248 2-47 2-46 881261 1-80 930987 4 36 33 29 812396 881 i53 1-80 931243 4 36 3i 3o 812544 3-46 88 1 046 i-8o 931499 9-931753 4 36 o685oi 3o 3i 9-812692 3-46 9-880938 1-80 4 36 10-068245 It 32 812840 2-46 88o83o 1-80 932010 4 36 067990 067734 33 812988 2-46 880733 I -80 932266 4 26 37 34 8i3i35 3-46 880613 I -80 932522 4 36 067478 26 35 8i3283 3-46 88o5o5 1-80 932778 4 36 067223 25 36 8i343o 2-45 880397 I -80 933 o33 4 26 066967 34 ll 8.3578 2-45 880289 1-81 933289 933545 4 36 066711 23 8t3725 2.45 880180 i-8i 4 26 066455 22 39 813872 2-45 880072 I -81 933800 4 26 066200 31 40 814019 2.45 879963 1-81 934056 4 36 065944 30 41 9-8i4i66 2-45 9-879855 i-8i 9934311 4 26 100656.S9 \l 42 8r43i3 2-45 879746 i-8i 934567 4 26 063433 43 8 1 4460 2-44 879637 1-81 934833 4 26 065 177 17 44 814607 2-44 879529 1-81 935078 4 26 064922 16 45 814753 2-44 879420 1-81 935333 4 36 064667 i5 46 814900 2-44 879311 1-81 935589 4 35 064411 I4 % 8i5o46 2-44 879202 1-82 935844 4 36 0641 56 i3 8i5i93 815339 2-44 870003 1-S2 9361 -10 4 25 063900 12 49 2-44 8jMs 1-82 956355 4 26 053645 II 56 8i54Sd 2-43 878875 1-82 936610 4 25 063390 io-o63i34 10 5i 9-8i563i 3.43 9-878766 1-82 9-936866 4 25 g 53 815778 3.43 87S556 i-Ss 937121 4 25 062879 53 815924 2-43 87S547 i-8a 937376 4 25 062624 7 062368 6 54 816069 816215 2-43 87843s 1-82 937632 4 25 55 2-43 878328 1-82 937887 4 25 062113 5 56 8i636i 3-43 878219 1-83 938142 4 25 o6i858 4 u 8i65o7 2-42 878109 1-83 938398 938653 4- 35 061602 3 8i6653 2.43 8T7999 1-83 4- 35 061347 3 59 816798 a -43 877§oo 877780 1-83 938908 4- 25 061092 060837 I 66 816943 2.4a 1-83 939163 4- 35 M. Cosine D. Sine 40° Co tang. D. _ Tang. SINES AND TANGENTS. • (41 DECaEES. ) 59 M. Sine D. Cosine D. Tar^. D. ~Oota-ag. c 9.81694;) 2 42 9-877780 1-83 9.939163 4-25 10.060837 60 I 817088 2 42 877670 1-83 939418 4-25 060682 It 2 817233 2 42 877560 -83 939673 4-25 060327 3 817379 2 42 877450 -83 939926 4-2.5 060072 57 4 817524 2 41 877340 -83 940183 4-25 059817 56 5 817668 2 41 877230 1-84 940438 4-23 359662 55 6 817813 2 41 877120 -84 940694 4-35 069306 54 I 817958 8i8io3 2 41 877010 876899 876789 1-84 940949 4-25 05905 1 058796 53 2 41 -84 941204 4-25 52 9 818247 2 41 -84 941458 4-25 058542 5i 10 818392 9.818536 2 41 876678 -84 941714 4-20 068286 5o II 2 40 q- 876568 -84 9-941968 4-25 10.058032 49 12 ^ 818681 2 40 876457 •84 942223 4-25 057777 48 i3 818825 2 40 876347 ■84 942478 4-25 057622 47 i4 818969 819113 2 40 876236 -85 942733 4-25 057267 46 • 5 2 40 876125 -85 942988 4-25 067012 45 i6 819257 2 40 876014 -85 943243 4-25 066767 44 \l 819401 2 40 875904 -85 943498 943752 4-25 o565o2 43 819545 2 39 l]llt] -85 4-25 056248 42 '9- 819689 2 39 -85 944007 4-25 055993 41 20 819832 2 39 875571 -85 944262 4-25 055738 40 21 9-819976 2 39 9-875450 -85 9-944517 4-25 10-055483 39 38 22 820120 2 39 875348 1 -85 944771 4-24 055229 23 820263 2 39 875237 1 -85 945026 4-24 054974 ll 24 820406 2 3? 875126 1 -86 945281 4-24 054719 25 82o55o 2 38 875014 1 -86 945535 4-24 054465 35 26 820693 820836 2 38 874903 1 -86 945790 4-24 054210 34 ^3 2 38 874680 1 •86 946045 4-24 053955 33 28 820979 2 38 -86 946299 946554 4-24 053701 32 29 821122 2 38 S74568 1 ■86 4-24 053446 3i 3o 821265 2 38 874456|i -86 946808 4-24 053192 3o 3i 9-821407 2 38 9-8743441 -86 9-947063 4-24 10-052937 It 32 82i55o 2 38 874232 I •87 947318 4-24 052682 33 821693 2 37 874121 1 -87 947572 4-24 052428 27 34 821835 2 37 874009 -87 947826 4-24 052174 26 35 82.977 2 37 873896; -87 948081 4-24 061919 25 36 822120 2 37 873784 -87 948336 4-24 o5i664 24 37 S22262 2 37 873672 ■87 948590 4-24 o5i4io 23 38 822404 2 37 873560 -87 948844 4-24 o5ii56 22 39 822546 2 37 873448 -87 l^ll 4-24 060901 21 40 822688 2 36 873335 •87 4-24 o5o647 ■20 41 9.822830 2 36 9-873223 •87 9-949607 4-24 10.050393 o5oi38 \l 42 822972 2 36 873110 -88 949862 4-24 43 823114 2 36 872998 -88 960116 4-24 049884 \l 44 823255 2 36 872885 -88 960370 4-24 049630 45 823397 2 36 872772 -88 930626 4-24 049376 i5 46 823539 2 36 872659 -88 900879 4-24 049121 048867 14 ii 823680 2 35 872547 ■88 95ii33 4-24 i3 823821 2 35 872434 -88 951 388 4-24 048612 12 49 823963 2 35 872321 -88 951642 4-24 048358 II 5o 824104 2 35 872208 -88 951896 4-24 048104 10 5i 9-824245 3 35 9-872093 871981 871868 -89 9-952100 4-24 10-047860 I 52 824386 2 35 -8g 962406 4-24 047696 53 824527 S24668 2 35 -89 902659 962913 4-24 047341 7 ^A 2 34 871755 -89 4-24 047087 6 II 824808 2 34 871641 -89 953167 4-23 046833 5 56 824949 2 34 871528 1-89 953421 4-23 046679 046325 4 u 825090 825230 2 34 871414 1-89 953675 4-23 3 2 34 871301 1-89 ffS 4-23 046071 2 59 825371 2 34 871187 -89 4-23 045817 I 60 825511 2-34 871073 1-90 954437 4-23 043563 M- Cosine D. Sine 18° Cotang. D. Tang. 60 (42 deuhees.) a TABLE OF LOGARITHMIC Ml Sine D. Cosiue 1 D. j Tang. D. \ Cotang. 9-82551 1 2-34 9-871073 I -90 9-954437 4-23 10 -045563 60 I 825651 2-33 870960'! -90 954691 4-23 045309 o45o55 u 3 825791 825931 2-33 870846!! -90 954945 4-23 3 2-33 8707324-90 955200 4-23 044800 57 4 82607 ' 2-33 870618 1 -9c 955454 4-23 044546 56 5 82621 1 2-33 87o5o4 1-90 955707 4-23 044293 55 6 826351 2-33 87039011-90 955961 4-23 o44o3ol 54 1 I 826491 8266 j I 2-33 870276 1-90 956215 4-23 043783 53 2-33 870161 1-90 956469 956723 4-23 043531 5j 9 826770 2-32 870047 I -91 4-23 043277 5i 10 826910 2-32 869933 1-9! 956977 4-23 o43o23 5o II 9-827049 2-32 9-869818 1-9! 9-957231 4-23 10-04276? 0425i5 it 12 827189 827328 2-32 869704 1-9! 957485 4-23 i3 2-32 S69589 1-9! 957739 957993 4-33 042261 47 14 827467 2-32 869474 1-91 4-23 042007 45 i5 827606 2-32 869360 1-91 958246 4-23 041754 45 i6 827745 2-32 869245 1.9! 958500 4-23 041 5oo 44 n 827884 828023 2-3l 869I30II.91 958754 4-23 041246 43 i8 2-3l . 869015 1.92 959008 4-23 040992 040738 42 19 8:Si62 2-3l 868900 1.92 959262 4-23 41 JO 828301 2-3l 868780 1.93 959516 4-23 040484 40 21 9 -.828439 2-3l 9 - 868670 1-92 9-909769 4-23 10 -04023 I It 22 828578 2-3l 868555 1-92 960023 4-23 039977 23 828716 2-3l 868440 1-92 960277 4-23 039733; 37 24 828855 2-3o 868324 1.92 96053 1 4-23 039469I 36 23 82.8993 829131 2-3o 863200 S68093 1 -92 960784 4-23 039216 35 26 2-3o 1-92 961038 4-23 038962 34 ^2 829269 2-3o 867978 867862 .-93 961291 4-23 038709 33 28 829407 2-3o 1.93 961545 4-23 038453 32 29 829545 2-3o 867747 1-93 961799 962052 4-23 03830 I 3i 3o ' 8296S3 2-3o 867631 1.93 4-23 037948 3o 3i 9.829821 2-29 9-8675i5 1.93 9-962306 4-23 10-037694 29 28 32 829939 2-29 867283 1.93 962560 4-23 037440 33 830097 830234 2-29 1-93 962813 4-23 037187 11 34 2-29 867167 1-93 963067 4-23 036933 35 83o372 2-29 867051 1-93 953330 4-23 o3668o 25 36 83o5o9 2-29 866935 866819 866703 1-94 963374 4-23 o36426 24 u 830646 2-29 1-94 963S27 4-23 o35i73 23 830784 lit 1-94 9640S1 4-23 035919 035655 23 39 830921 866586 1-94 964335 4-23 21 40 S3io58 2-28 866470 1-94 964588 4-22 o354i2 20 41 9-83II95 2-28 9-866353 1-94 9-964842 4-22 io-o35i58 \t 42 83i332 2-28 866237 1-94 965095 4-22 o349o5 43 831469 2-2S 866120 1-94 965349 4-2'S 03465 1 \l 44 83i6o6 2-28 866004 1-95 965602 4-22 034398 45 831742 831879 832015 2-28 865887 1-95 965855 4-32 034145 iS 46 2-28 865770 .-95 966105 4-22 033S91 033638 14 47 2-27 865653 1-95 965352 4-22 i3 4b 832152 2-27 865536 i-?5 966615 4-22 033334 12 ',9 832288 2-27 855419 1-95 966859 967123, 4-22 o33i3i II 5o 832425 2-27 865302 ..95 4-22 032877 10 5i 9-832561 2-27 9-865i85 1-95 9-967376, 4-22 10-032624 n 52 832697 2-27 865o6S 1-95 967629' 4-22 032371 8 53 832833 2-27 2-56 864950 864833 1.95 96iS83j 953i36 4-22 032117 7 54 833969 833io5 1-96 4-23 o3i364' 5 55 2-26 864716 I -lb 96S389' 958543! 4-22 o3i6iT 5 56 833241 2-26 864598, 1.96, 4-22 o3i357 4 u 8J3377 2-26 3644SI 1-96: 968896 4-32 o3iio4 3 833515 2-26 864363! 1.96] 969149I 969403' 4-22 o3o33i 2 59 833648 2-26 8642451-96 4-22 o3o597 I 6o 833783 2-26 864127 1-96 9696 55| 4-22 o3o344 1 Cosine D. Fiuo 41°. Cotansr. 1 U. Tanff-.. U. SINES AND TAKOENTS. (43 DEGBIiKS. ) 61 Sine D. Cosine | D. Tan?. D. Cotanff. __ 60 9-833783 2-26 9-854i27;i-96 9-969655 4-22 io-o3o344 I 833919 2 2D 864010 1-96 969909 4-22 oSoooi 59 1 834054 2 2; 863892,1-97 970162 4-22 029838; 58 1 3 834189 834325 2 25 863774li-97 970415 4-22 029584 57 < 2 25 863656! I- 97 970669 4-22 039331 55 5 834460 2 35 863538 1-97 970922 4-22 029078 55 6 834095 8347JO 2 25 8634191-97 971175 4-23 028825 54 7 2 25 863301 1.97 971429 4-22 028571 53 8 834865 2 25 863i83;i.97 971682 4-22 0283i8 5j 9 834999 835i34 2 24 863a54i-97 862946 I -$8 971935 4-22 028065 Si 10 2 24 972188 4-32 027812 5o II Q. 835269 835403 2 24 9-852827i.$8 9-972441 4-22 10-027559 027306 it 12 2 24 862709 1.98 972694 4-22 i3 835538 2 24 863590 1.98 972948 4-32 027052 47 14 835672 3 24 862471 1-98 973201 4-23 026799 45 |5 835807 2 24 862353:1-98 973454 4-22 025546 45 i6 835g4i 2 24 862234|i-98 973707 4-23 026293 44 n 836075 2 23 862115:1-98 973960 4-22 026040 43 i8 83&209 836343 2 23 861996 1-98 85i877'i-98 974213 4-22 025787 42 '9 2 23 974466 4-22 025534 41 JO 835477 2 33 851758,1-99 974719 •9-974973 4-32 025281 40 31 9-836611 2 23 9-86i638'i.99 4-23 10-025027 3q 024774 38 22 836145 836378 2 23 85i5i9i-99 975225 4-22 J3 2 23 86i4oo!i-99 975479 4-22 024521 37 .1.1 837012 3 22 86i28o'i-99 975732 4-22 024268 35 ' 25 3o7!46 i 32 86ii6ili-99 975985 4-22 024015 35 35 837279 S' 22 861041 1-99 976338 4-22 023762 34 27 8374! 2 3 22 860922.1-99 860802 1-99 976491 4-22 033509 33 28 837546 3 22 976744 4-22 023255 32 29 837679 2 22 8606832-00 976997 4-22 023oo3 3i 3o 837812 2- 32 85o552l3-oo 977350 4-22 022750 ■30 3i 9-837945 2 32 9-8604422-00 9-9775o3 4-33 10-022497 It 32 838078 2 21 86o322|2-oo 977756 4-22 032244 33 838211 2 21 860202 j2- 00 978009 4-33 031991 27 34 833344 2 21 860082,2-00 978262 4-22 02173S1 25 35 833477 2 31 859952|2-oo 97851 5 4-32 021485 25 35 838610 2 31 8598422-00 978768 4-22 021232 24 37 838742 2 21 85972i|2-oi 979031 4-22 020979 23 38 838875 3 21 859601 i2 -01 979274 4-23 020726 22 3, 839007 2 21 859480 2-01 979527 4-22 020473 21 4c 839140 2 20 85936o'2-oi 979780 4-22 020220 20 41 9-839272 2 30 9'859239'2-oi 9-9'^oo33 4-33 10-019967 \l 42 839404 2 30 859ii9'2-oi 980286 4-22 019714 43 839336 2 20 858998=7-01 980538 4-22 019462 '2 44 839668 2 20 858377 2-01 980791 4-21 019209 16 45 839S00 3 30 858755'2-(t2 98 1044 4-31 018956 i5 46 839932 2 20 858635 2-03 981397 4-31 018703 14 tl 840064 2 '9 8585i4'2-o2 981500 4-21 018450 i3 840196 2 '9 858393,2-02 981803 4-21 O18197 12 49 840328 2 '9 858272 2-02 982055 4-21 017944 II 5o 840459 2 19 858i5i',2-o2 982309 4-21 017691 10-017438 10 5l 9-840591 2 19 9'858o29'2-o3 %579n8'2-02 9-982552 4-21 t 52 840722 2 ■9 982814 4-21 017186 53 840854 2 ■9 857786 2-02 983067 4-21 016933 I 54 840985 2 857665;3.o3 983330 4-21 016680 6 55 841116 2 10 8075432-03 983573 4-21 016427 5 56 841247 2 18 857432 2-o3 983825 4-21 016174 4 u 841378 2 18 837300,2-03 984079 4-21 015921 3 84t5o9 2 -18 857178 2-o3 984331 4-21 oi566q oi54ic 2 59 ' 841640 2 -18 857055 2-o3 984584 4-21 I ~. 841771 2 •18 856934 2-63 984837 4-21 oifi63 a . Cofliiia D. Siui 40° Gotang. D. Tansr. IM.J ss (44 DEOREES.) A TABLE OF LOGARITHMIO M.' Siae D. Cosine D. 1 Tang. 1 v. Cotang. 60 h 55 54 53 5J 5i I 9-841771 841902 3-18 2-18 9-856034 2-03 856812 2-o3 9-984837 935090 4-21 4-21 io-oi5i63 OI49IO 3 842033 2-18 856690 2 - o4 985343 4-21 014657 3 8-«i63 2-17 856568 2-04 985596 4-21 014404 4 842294 2-17 856446 2-o4 985848 4-21 oi4i52 5 842424 2-17 856323 2 -04 986101 4-21 013899 6 842555 2-17 855201 2-04 985354 4-21 013646 I 842685 2-17 856078 2-04 986607 4-21 013393 8428i5 2-17 855956 855833 2-04 986860 4-21 Oi3i4o 9 842946 2-17 2-04 987112 4-21 012888 10 843076 2-17 2-l5 85571 1 2-05 987355 4-21 OI363S 5o II 9-843206 9-855588 2.o5 9-987618 4-31 10.0I3382 t 12 843336 2-l6 855455 2-o5 987871 4-21 012129 i3 843465 2-l6 855342 2-o5 988123 4-21 011877 il 14 843595 2-l5 855219 2-o5 988376 4-21 011634 45 i5 843725 2-l5 855096 2-o5 988629 4-21 01 1371 i6 843855 2-l5 854973 854850 2-o5 988882 4-21 011118 44 \l 843984 2-l5 2-o5 989134 4-21 oio8fi5 43 844114 2-l5 854727 2-05 989387 4-21 oio5i3 42 '9 844243 2-l5 8545o3 2-06 989640 4-21 oio35o 41 20 844372 2-l5 854480 2-06 989893 4-21 01 01 07 io 21 9 -844502 2-l5 9-854356 2-06 9-990145 4-21 10-009855 3q 38 22 844631 2-l5 854233 2-o5 990398 990651 4-21 009602 23 844760 2-l5 854109 2-06 4-21 009349 ll 35 24 844889 845oi8 2-l5 853g86 2-05 990903 4-21 009097 008844 25 2-l5 853862 2-06 991156 4-21 25 845 147 2-l5 853738 2-o5 991409 4-21 008591 34 ^? 845276 214 853614 2-07 991662 4-21 oo8338 33 845405 2-14 853490 2-07 99'9'4 4-31 008086 32 29 845533 2-14 853366 2-07 992167 4-21 007833 3i 3o 845662 2-14 853242 2-07 992420 4-21 007580 3o 3i 9-845790 214 9-853ii8 2-07 9-992672 4-21 10-007328 It 32 845919 2-14 802994 852869 2-07: 992925 4-21 007075 33 846047 2-14 2-07 993178 4-21 006822 H 34 8461-75 2-14 852745 2-07 993430 4-21 006570 26 35 846304 214 852620 2-07 993683 4-31 006317 35 36 846432 2-l3 852496 2-o3 993935 4-21 006064 24 ll 846560 2-l3 852371 2-o8 994189 4-21 oo5Si I 23 846688 2-l3 852247 2-oS 994441 4-31 oo555o oo53o6 23 39 846816 !-l3 852122 2-08 994694 4-21 21 40 846944 2-l3 851997 9-351872 2-08 994947 4-21 oo5o53 20 4i 9-847071 2-l3 2-08 9-995199 995452 4-21 10-004801 10 18 42 847199 2-l3 851747 2-03 4-21 004546 43 847327 2-l3 85i522 2-o8 995705 4-21 C0429J \l 44 847454 212 851497 2-09 995957 4-21 004043 45 847582 2-12 85i372]2-09 996210 4-21 003790 003537 003285 13 45 li^iii 212 85i246 2-og 996453 4-21 14 47 2-12 S5ll2II2-09 996715 4-21 i3 48 847964 2-12 850096' 2- 09 8508702-09 996968 4 21 oo3o32 13 49 848091 2-12 997221 4-21 002779 II 5o 8482 iS 2-12 8507452-09I 997473 4-21 00252- 10 5i 9-848345 2-12 9-85o6i9 2-09 85o493 2-10 9-997736 4-21 10-00227; t 52 84847J 2-II 99U-'9 4-21 002021 53 848599 2-11 85o368 2-io 998231 4-21 ooi76< ooiSii I 54 848726 2-11 850242 2-10 998484 4-21 55 848852 2-11 85oii6 2-10 998737 4-21 00126; 5 55 848979 2-II 8499^0 2 - 1 849S64!2-io 998989 4-21 OOIOII i ll 849106 2-H 999242 4-21 000758 3 849232 211 84973s 2-10 849611,2-10 999495 4-21 O0O3O5 3 59 849359 849485 2-n 999748 4-21 000253 I ti 2-11 849485 , 2 • 1 1 1 • 000000 4-31 10-000000 Cosine D. Sine |45o| Cotang. B. Tang. M- The JV'cetional Series of Standard SchoolSooks, ORTHOGRAPHY IM READiC. N'ATIOFAL SEHIES OP READERS AND SPELLERS, BY PAEKER & WATSOK The National Primer $25 National First Reader 38 National Second Reader fi3 National Third Reader 95 National Fourth Reader 1 50 National Fifth Reader 1 88 National Elementary Speller 25 National Pronouncing Speller 45 This unrivaled series has acquired for itself during a very fevr years of publication, a reputation and circulation never hefore attained by a series of Bchool readers in the same space of time. No contemporary books can be at all compared with them. The average annual increase in circulation exceeds 100,000 volumes. We challenge rival publishers to show such a record. The salient features of these works whichi have combined to render them so populat may he briefly recapitulated as follows ; 1. THE "WORD METHOD SYSTEM— This famous progressive method for young children originatad and was copyrighted with these books. It constitutes a process by which the beginner with words of one letter is gradually introduced to additional lists formed by prefixing or affixing single letters, and is thus led almost insensibly to th« mastery of the more difficult constructions. This is justly regarded as one of the most striking modern improvements in methods of teaching. 2. TREATMENT OF PEONUITGIATTOlir— The wants of the youngest scholara In this department are not overlooked. It may he said that from the first lesson the student by this method need never be at a loss for a prompt and accurate render- ing of every word encountered. 3. ARTICULATION AND ORTHOEPY are recognized as of primary im. portance. _ 3 iOver.y TAe JVatt07tal Series of Standard SchcolSooks* ORTHOGRAPHY AND READING-Continued. 4. PUNCTUATIOlf is inculcated ty a series of interesting reading lessons, the simple perusal of which suffices to fix its principles indelibly upon the mind. 5. ELOCUTION. Each of thchighcr Readers (3d, 4th and 5th) contains elaborate, Bcholarly, and thoroughly practical treatises on elocution. This feature alone has secured for the series many of its warmest fiends. 6. THE SELEOTIOUS are the croTvTiing glory of ihe series. Without exception It may be said that no volumes of the same size and character contain a collection so diversified, judicious, and artistic as this. It embraces the choicest gems of Englisli literature, so arranged as to aflford the reader ample exercise in every department cf style. So acceptable has the taste of the authors in this department proved, not only to the educational public but to the reading community at large, that thousands of copies of the Fourth and Fifth Beaders have found their way into public and private libraries throughout the country, where they arc in constant use as manuals of liter- ature, for reference as well as perusal. 7. AKEANG^EMEKT, The exercises arc so arranged as to present constantly al- ternating practice in the different styles of composition, while observing a definite plan of progression or gradation throughout the whole. In the higher books the ar- ticles are placed in formal sections and classified topically, thus concentratiug tiie in- terest and inculcating a principle of association likely to prove valuable in subsequent general reading. 8. UOTES AMD BIO&EAPHIGAL SKETCHES. These are full and adequate to every want. The biographical sketches present in pleasing style the history of ©very autlior laid under contribution. 9. ILLUSTEATIONS. These are plentiful, almost profuse, and of the highest character of art They are found in every volume of the seiies as far as and iucluding the Third Reader. 10. THE &EADATI01I is perfect. Each volume ovcrLips its companion pre- ceding or following in the series, so that the scholar, in passing from one to another, is barely conscious, save by the presence of the new book, of the transition, 11. THE PRICE is reasonable. The books were not fri»tmcd to the minimum of size in order that the publishers might be able to denominate them " the cheapest in the market," but were made large enough to cover and suffice for the grade indi- cated by the respective numbers. Tlius the child is not compelled to go over his First licader twice, or bo driven into the Second before he is prepared for it^ The compe- tent teachers who compiled the series made each volume just what it should be, leav- ing it for their brethren who should use the books to dedde what constitutes true citeapness* A glanco over the books will satisfy any one that the same amount of matter is nowhere furnished at a price more reasonable. Besides which another con- sideration enters into the question of relative economy, namely, the 12. BIITDIXG-. By the use of a material and process known only to themselves, fn common with all the publications of this house, the National Readers are warranted t« out-last any with which they may be compared — ^tho ratio of relative durability bfi- ing In their favor as two to one. 4 T/ie JVationaZ Series of Standard School-Sooks. SCHOOL-ROOM CARDS, To Accompany the National Headers. Eureka Alphabet Tablet *i so Presents the alphabet upon the Word Method SyEtem, byT7hicli the child will learn the alphabet in nine days, and make no Binall progress in reading and spelling in the same time. National School Tablets, lo Nos *7 50 Embrace reading and conversational exercises, object and moral Ics- Bons, form, color, &c. A complete set of these large and elegantly illus- trated Cards will embellish the school-room more than any other article of furniture. I '■ . ■■! ■ . — ■ '■ I- -. — I. — . READING. Fowle's Bible Reader $100 The narrative portions of the Bible, chronologically and topically ar- ranged, judiciously combined with selections from the Psalms, Proverbs, and other portions which inculcate important moral lessons or the groat truths of Christianity. The embarrassment and difficulty of reading the Bible itself, by course, as a class exercise, are obviated, and its UKe made feasible, by tlus means. North Carolina First Reader 50 North Carolina Second Reader ..... 75 North Carolina Third Reader 1 00 Pi'epared expressly for the schools of this State, byC. II. Wiley, Super- inte'ndent of Common Schools, and F, M. Hubbard, Professor of Lltera- ature in the State University. Parker's Rhetorical Reader 1 00 Designed to familiarize Readers with the pauses and other marks in general use, and lead them to the practice of modulation and inflection of the voice. Introductory Lessons in Reading and Elo- cution '75 Of similar character to the foregoing, for less advanced clasces. High School Literature 1 60 Admirable selections from a long list of the world's best writers, for e:^- ereise in reading, oratory, and composition. Speeches, dialogues, and model letters represent the latter department. 5 T^ie JVational Series of Standard SchoolSooki. ORT HOGRAP HY. SMITH'S SERIES Supplies a speller for every class in graded schools, and comprises the most com- plete and excellent treatise on English Orthography and its companion branches extant. 1. Smith's Little Speller $20 First Itound in the Ladder of Learning. 2. Smith's Juvenile Definer 45 Lessons composed of familiar words grouped with reference to siiuilar signification or use, and correctly spelled, accented, and defined, 3. Smith's Grammar-School Speller .... 50 Familiar words, grouped with reference to the sameness of sound of syl- lables differently spelled. Also definitions, complete rules for spelling and formation of derivatives, and exercises in false orthography. 4 Smith's Speller and Definer's Manual • 90 A complete School Dictionary containing 14,000 words, with various other useful matter in the way of Kales and Exercises. 5- Smith's Hand-Book of Etymology • • • i 25 The first and only Etymology to recognize the Anglo-Sacson our Tfwther tongue; containing also full lists of derivatives from the Latin, Greek, Oaelic, Swedish, Norman, &c., &c ; being, in fact, a complete etymology of the language for schools. Sherwood's Writing Speller 15 Sherwood's Speller and Defmer 15 Sherwood's Speller and Pronouncer ... 15 The Writing Speller consists of properly ruled and numbered blanks to receive tho words dictated by the teacher, with space for remarks and corrections. The other volumes may be used for the dictation or ordinary class exercises. Price's English Speller *15 A complete spelling-book for all grades, containing more matter than " Webster," manufactured in superior style, and sold at a lower price — consequently the cheapest speller extant Northend's Dictation Exercises ..... 63 Embracing valuable information on a thousand topics, commnnicated in such a manner as at once to relieve the exercise of spelling of its nsna] tedium, and combine it with instruction of a general character calculated to profit and amuse. Wright's Analytical Orthography .... 25 This standard work is popular, because it teacher the elementary sounds in a plaim and philosophical manner, and presents orthography and or- thoepy in an easy, uniform system of analysis or parsing. Fowle's False Orthography 45 Exercises for correction. Page's Normal Chart *3 75 Tho elementary sounds of the language for tho school-room waU& The JVational Series of Standard SchoolSooks. ENGLISH GRAMMAR. CLARK'S DIAGRAM SYSTEM. Clark's First Lessons in Grammar . . . 50 Clark's English Grammar i oo Clark's Key to English Grammar .... 60 Clark's Analysis of the English Language . 60 Clark's Grammatical Chart ^ oo The theory and practice of teaching grammar in American schoola is meeting with a thorough revolution from the use of this system. While the old methods offer proficiency to the pupil only after much weary plodding and dull memorizing, this affords from the inception the ad- vantage of pracUcal Object Teachinff^ addressing the eye by means of il- lustrative figures ; furnishes association to the memory, its most power- ful aid, and diverts the pupil by taxing Ids ingenuity. Teachers who are using Clark's Grammar uniformly testify that they and their pupils find it the most interesting study of the school course. Like all great and radical improvements, the system naturally met. at first with much unreasonable opposition. It has not only outlived the greater part of this opposition, but finds many of its warmest admirers among those who could not at first tolerate so radical an innovation. All it wants is an impartial trial, to convince Uie most skeptical of its merit'. No one who has fairly and intelligently tested it in the school-room has ever been known to go back to the old method. A great success is al- ready established, and it is easy to prophecy that the day is not far dis- tant when it will be the only system of teaching English Grammar. As the System is copyrighted, no other text-books can appropriate this ob- vious and great Improvement. Welch's Analysis of the English Sentence . i lo Eemarkahle for its new and simple classification, its method of treat- ing connectives, its explanations of the idioms and constructive laws of the language, &c, ETYMOLOGY. Smith's Complete Etymology, i 25 Containing the Anglo-Saxon, Trench, Dutch, German, Welsh, Danish, Gothic SwediBli, Gaelic, Italian, Latin, and Greek Boots, and the EnghBh vmrds derived therefrom accurately spelled, accented, and defined. The Topical Lexicon, • ■ • i 50 i This irork is a School Dictionary, an Etymology, a compilation of syn- onyms and a manual of general information. It differs from the ordinary lexicon in being arranged by topics instead of the letters of the alphabet^ thus realizing the apparent paradox of a " Readable Dictionary. ' An unusually valnable school-book. The JVaiionai Series of Standard SchootSook*. GEOGRAPHY. THE NATIONAL GEOGRAPHICAL SYSTEM. I. Monteith's First Lessons in Geography, % 35 II. Monteith's Introduction to the Manual, • 65 Hi. Monteith's New Manual of Geography, • i oo IV. Monteith's Physical & Intermediate Geog. i 75 V. McNally's System of Geography, • • • l 88 The only complete course of geographical InBtraction. Its circnlation. is almost unirerBal — its merits patent. A feir of the elements of ite popu- larity arc found la the following points of excellence. 1. PRACTICAL OBJECT TEACHING. The infant scholar is first mtrodnced to a'fiicture whence he may derive notions of the shape of the earth, the phenomena of day and night, the distribution of land and water, and the great natural divi^ons, which mere words would fail entirely to convey to the untutored mind. Other pic- tures follow on the same plan, and the child*s mind is called upon to grasp no idea without the aid of a pictorial illustration. Carried on to the higher books, this system culminates in No. 4, where such matters as climates, ocean currents, the winds, pecu- liarities of the carth''s crust, clouds and rain, are pictorially explained and rendered apparent to the most obtuse. The illustrations nsed for this purpose belong to the liighest grade of art. 2, OLEAEf BEAUTIFUL, AND OORSECT MAPS. In the lower numbers tlie maps avoid unneccssai-y detail, while respectively progressive, and affording the pnpn new matter for acG[uisLtion each time he approaches in the constantly enlarging circle the point of coiucidenco with previous lessons in the more elementary hoots. In No. 4, the maps embrace many new and striking features. One of the most effiictive of these is the new plan for displaying on each map the relative sizes cf countries not represented, thus ohviating much confutsion which has arisen from the necessity of .prescuting maps in the same atlas drawn on different scales. The maps of No. 5 have long been celebrated for their superior beauty and completeness. Tiiis is the only school-book in which the attempt to make a compHtQ atlas oteo clAar and diititict, h:i8 been successful. The map coloring throughout tho series is also notice- able. Delicate and subdued tints take the place of the startling glare of inhormoniouB colors which too frpquontly in such treatises dazzle tlie eyes, distract the attention, and servo to overwhelm tho names of towns and tho natural features of the landscape. 8 The J^atio9ial Series of Standard Schoot'SookS^ GEOGRAPHY-Continued. 3. THE VARIETY OF MAP ESEEOISE, Starting each time from a different basis, the pupil in many Instances approaches the same fact no less than six times, thus indelihly impressing it upon hta memory. At the same time this system is not allowed to become wetirisorae— the extent of exercise on each subject being graduated by its relatire importance or difficulty of acQuisition. 4. THE OHAEAOTER AND AERANGEMENT OF THE DESCEIPTIVE TEXT. The cream of tho science has been carefully culled, unimportant matter re- jected, elaboration aroidedf and a brief and concise manner of presentation cultivated. The orderly consideration of topics has contributed greatly to simplicity. Due atten- tion is paid to the facts in history and astronomy which are inseparably connected with, and important to the proper understanding of geography— and such only are admitted on any terms. In a. word, the National System teaches geography as a science, pure, simple, and exhaustive, 5. ALWAYS UP TO THE TIMES. The authors of these books, editorially speaking, never sleep. No change occurs in the boundaries of countries, or of coun- ties, no new discovery is laade, or railroad built, that is not at once noted and re- corded, and the next edition of each volume carries to every school-room the new or- der of things. « 6. STJPERIOR 0EADATIOU. This is the only series which famishes an avail- able volume for every possible class in graded schools. It is not contemplated that a pupil must necessarily go through every volume in succession to attain proficiency. On the contrary, two will suffice, hut three are advised ; and if the course will admit, the whole series should he pursued. At all events, the books are at hand for selection, and every teacher, of every grade, can find bmong tbem one ecsactly suited to his class. The best combination for those who wish to abridge the course consists of Nos. 1, 3, and 5, or where children are somewhat advanced in other studies when they com- mence geography, Nos. 2, 3, and 5. Where but two books are admissible, Nos. 2 and 4, or Nos. 3 and 5, are recommended. 7. FOSM OP THE VOLUMES AND MECHANICAL EXECUTION. The maps and text are no longer unnaturally divorced in accordance with the time-hon- ored practice of making text-books on this subject as inconvenient and expensive as possible. On the contrary, all map questions are to be found on the page opposite the map itself, and each hook is complete in one volume. The mechanical execution is unrivalled. Paper and printing are everything that could be desired, and the bind- ing is — A. S. Barnes and Company's. Ripley's Map Drawing ^i 25 This system adopts the circle as its basis, abandoning tbe processes by triangiilation, the square, parallels, and meridians, &o., which have been proved not feasible or natural in the development of this science. Sue- cess seems to indicate that the circle " has it." National Outline Maps For the school-roora walls. In preparation. 9 The JVaiional Series of Standard SchoolSooki. MATHEMATICS. BA¥iii' MTioiAi eonaii. • ARITHMETIC. 1. Davies' Primary Arithmetic $35 2. Davies' Intellectual Arithmetic 40 3. Davies' Elements of Written Arithmetic 50 4. Davies' Practical Arithmetic 1 00 Key to Practical Arithmetic *1 00 5. Davies' University Arithmetic 1 50 Key to University Arithmetic *1 50 ALGEBRA. 1. Davies' New Elementary Algebra 1 35 Key to Elementary Algebra *1 35 2. Davies' University Algebra , . . 1 60 Key to University Algebra *1 60 3. Davies' Bourdon's Algebra 2 35 Key to Bourdon's Algebra *3 35 GEOMETRY. 1. Davies' Elementary Geometry and Trigonometry . . . 1 40 2. Davies' Legendre's Geometry 3 35 3. Davies' Analytical Geometry and Calculus 3 50 4. Davies' Descriptive Geometry 3 75 MENSURATION. 1. Davies' Practical Mathematics and Mensuration . . . 1 40 2. Davies' Surveying and Navigation 3 50 3. Davies' Shades, Shadows, and Perspective 3 To MATHEMATICAL SCIENCE. Davies' Grammar of Arithmetic * 50 Davies' Outlines of Mathematical Science *1 00 Davies' Logic and Utility of Mathematics *1 50 Davies & Peck's Dictionary of Mathematics *3 75 10 The JVatlonal Series of Standard Schoot-Sooks. DAVIES' NATIONAL OOUESE of MATHEMATICS. ITS RECORD. In claiming for this series the first place among American text-books, of whatever class, the Publishers appeal to the magnificent record which its volumes have earned during the thirty-Jive years of Dr. Charles Davies* mathematical labors. The unre- mitting exertions of a life-time have placed the Tnoderh series on the same proud emi- nence among competitors that each of its predecessors has successively enjoyed in a course of constantly improved editions, now rounded to their perfect fruition— for it seems indeed that this science is susceptible of no further demonstration. During the period alluded to, many authors and editors in this department have started into public notice, and by borrowing ideas and processes original with Dr. Davies, have enjoyed a brief popularity, but are now almost unknown. Many of the series of to-day, built upon a similar basis, and described as "modern books," are destined to a similar fate ; while the most far-seeing eye will find it difficult to fix the time, on the basis of any data afforded by their past history, when these books will cease to increase and prosper, and fix a still firmer hold on the affection of every educated American. One cause of this unparalleled popularity is found in the fact that the enterprise of the author did not cease with the original completion of his books. Alw^ays a practi- cal teacher, he has incorporated in his text-books from time to time the advantages of every improvement in methods of teaching, and every advance in science. During all the years in which he has been laboring, he constantly submitted his own theories and those of others to the practical test of the class-room — approving, rejecting, or modifying them as the experience thus obtained might suggest. In this way he has been able to produce an almost perfect series of class-books, in which every depart- ment of mathematics has received minute and exhaustive attention. Nor has he yet retired from the field. Still in tlie prime of life, and enjoying a ripe experience which no other living mathematiJiau or teacher can emulate, bis pen is ever ready to carry on the good work, as the progress of science may demand. Wit- ness his recent exposition of the " Metric System," which received the official en- dorsement of Congress, by its Committee on Uniform Weights and Measures. Davtes' System 13 tue acknowledged Natio^jal Standaud foe tub United SxiTEB, for the following reasons : — 1st. It is the basis of instruction in the great national schools at West Point and Annapolis. 2d. It has received the quasi endorsement of the National Gongi'ess. 3d. It is exclusively used in tfae public schools of the National Capital. 4th. The officials of the Government use it as authority in all cases involving mathe- matical questions, 6th. Our great soldiers and sailors commanding the national armies and navies were educated in this system. So have been a majority 6f eminent scientists in this country. All these refer to " Davies" as authority. 6lh. A larger number of American citizens have received their education from this limn from any other series. Tth. The series has a larger circulation throughout the whole country than any other, being extansively used in every State in the Union. 11 50 T^te JVaiional Series of Standard SchoolSooks. MATHEMATICS-Continued. ARITHMETICAL EXAMPLES. Reuck's Examples in Denominate Numbers % Reuck's Examples in Arithmetic i 00 These volumes di£fer from tlie ordinary arithmetic in their peculiarly pracUcal character. They are composed mainly of examples, and afford the most severe and thorough discipline for the mind. While a book vhieh should contain a complete treatise of theory and practice would be too cumbersome for cvery-day use, the insufficiency of ^ocft'col examples has been a source of complaint, HIG-HER MATHEMATICS. Church's Elements of Calculus 2 50 Church's Analytical Geometry 2 50 Church's Descriptive Geometry, with Shades, Shadows, and Perspective 4 50 These volumes constitute the "West Point Course" in their several departments. Courtenay's Elements of Calculus • • ■ • 3 25 A work especially popular at the South. Hackley's Trigonometry 3 00 With applications to navigation and surveying, nautical and practical geometry and geodesy, and loganthmic, trigonometrical, and nautical tables. THE METRIC SYSTEM. The International System of Uniform Weights and Measures must hereafter be taught in all common-schools. Professor Charles Davies is tJie official exponent of the systeri, as indicated by the following resolutions, adopted by the Committee of the House of Kepresentatives, on a ^* Uniform System of Coinage, Weights, and Measures,** February 2, 1867 :— Uefiffloed, That this committee has observed with gratification the efforts made by the editors and publishers of several mathematical works, designed for the use of com- luon-BOhools and other Institutions of learning, to introduce the Metric System of Weights and Measures, as authorized by Congress, Into the system of instruction of the youth of the United States, in its various departments ; and, in order to extend further the knowledge of its advantages, alike in public education and in general use by the people, B« it further re&olved,^ That Professor Charles Dav^es, LL.D., of the State of New Vurk, he requested to confer with superintendents of public instruction, and teacheri of school;?, and others int«rested in a reform of the present incongruous system, and, by lectures and addresses, to promote its general introductiou and use. The official version of the Metric System, as prepared by Dr. Davies, may be found In the Written, Practical, and University Arithmetics of the Mathematical Series, ano la also published sepitrately, price postpaid, pxa cetita, 12 The SVaiionat Series of Standard Schoot-Sooks. HISTORY.