ilffliiiilip 
 
 IBffl 
 
 mm 
 
Cornell Ituuemtg Stihtarg 
 
 Strata, New fork 
 
 BOUGHT WITH THE INCOME OF THE 
 
 SAGE ENDOWMENT FUND 
 
 THE GIFT OF_ 
 
 HENRY W. SAGE 
 
 1691 
 
arV17960 C ° rne " Univers ">' Ubr *n/ 
 'mES&SJSL .t!?e..Rractical man 
 
 olin.anx 
 
MATHEMATICS 
 
 FOR 
 
 THE PRACTICAL MAN 
 
 EXPLAINING SIMPLY AND QUICKLY 
 ALL THE ELEMENTS OF 
 
 ALGEBRA, GEOMETRY, TRIGONOMETRY, 
 LOGARITHMS, COORDINATE 
 GEOMETRY, CALCULUS 
 WITH ANSWERS TO PROBLErtS 
 
 BY 
 
 GEORGE HOWE, M.E. 
 
 ILLUSTRATED 
 
 SEVENTH THOUSAND 
 
 NEW YORK 
 
 D. VAN NOSTRAND COMPANY 
 
 25 Park Place 
 
 1918 
 
 ■•0 
 
Copyright, 191 1, by 
 D. VAN NOSTRAND COMPANY 
 
 Copyright, igig, by 
 D. VAN NOSTRAND COMPANY 
 
 Stanbope Ipnes 
 
 r. H. G1LSON COMPANY 
 BOSTON. U.S.A. 
 
Dedicated To 
 Mtavm Agrea, pf.S. 
 
 PRESIDENT OF THE UNIVERSITY OF TENNESSEE 
 "MY GOOD FRIEND AND GUIDE." 
 
Cornell University 
 Library 
 
 The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031266871 
 
PREFACE 
 
 In preparing this work the author has been prompted 
 by many reasons, the most important of which are: 
 
 The dearth of short but complete books covering the 
 fundamentals of mathematics. 
 
 The tendency of those elementary books which " begin 
 at the beginning " to treat the subject in a popular rather 
 than in a scientific manner. 
 
 Those who have had experience in lecturing to large 
 bodies of men in night classes know that they are com- 
 posed partly of practical engineers who have had consider- 
 able experience in the operation of machinery, but no 
 scientific training whatsoever; partly of men who have de- 
 voted some time to study through correspondence schools 
 and similar methods of instruction; partly of men who 
 have had a good education in some non-technical field of 
 work but, feeling a distinct calling to the engineering 
 profession, have sought special training from night lecture 
 courses; partly of commercial engineering salesmen, whose 
 preparation has been non-technical and who realize in this 
 fact a serious handicap whenever an important sale is to 
 be negotiated and they are brought into competition with 
 the skill of trained engineers; and finally, of young men 
 leaving high schools and academies anxious to become 
 engineers but who are unable to attend college for that 
 purpose. Therefore it is apparent that with this wide 
 
iv PREFACE 
 
 difference in the degree of preparation of its students any 
 course of study must begin with studies which are quite 
 familiar to a large number but which have been forgotten 
 or perhaps never undertaken by a large number of others. 
 
 And here lies the best hope of this textbook. It "begins 
 at the beginning," assumes no mathematical knowledge be- 
 yond arithmetic on the part of the student, has endeavored 
 to gather together in a concise and simple yet accurate and 
 scientific form those fundamental notions of mathematics 
 without which any studies in engineering are impossible, 
 omitting the usual diffuseness of elementary works, and 
 making no pretense at elaborate demonstrations, believing 
 that where there is much chaff the seed is easily lost. 
 
 I have therefore made it the policy of this book that 
 no technical difficulties will be waived, no obstacles cir- 
 cumscribed in the pursuit of any theory or any conception. 
 Straightforward discussion has been adopted; where 
 obstacles have been met, an attempt has been made to 
 strike at their very roots, and proceed no further until 
 they have been thoroughly unearthed. 
 
 With this introduction, I beg to submit this modest 
 attempt to the engineering world, being amply repaid if, 
 even in a small way, it may advance the general knowledge 
 of mathematics. 
 
 GEORGE HOWE. 
 
 New York, September, 1910. 
 
TABLE OF CONTENTS 
 
 Chapter Page 
 I. Fundamentals of Algebra. Addition and Subtrac- 
 tion i 
 
 II. Fundamentals of Algebra. Multiplication and Divi- 
 sion, I -. 7 
 
 III. Fundamentals of Algebra. Multiplication and Divi- 
 
 sion, II ■ 12 
 
 IV. Fundamentals of Algebra. Factoring 21 
 
 V. Fundamentals of Algebra. Involution and Evolu- 
 tion 25 
 
 VI. Fundamentals of Algebra. Simple Equations 29 
 
 VII. Fundamentals of Algebra. Simultaneous Equa- 
 tions 41 
 
 VIII. Fundamentals of Algebra. Quadratic Equations 48 
 
 DC. Fundamentals of Algebra. Variation 55 
 
 X. Some Elements of Geometry 61 
 
 XI. Elementary Principles of Trigonometry 75 
 
 XII. Logarithms ■ 85 
 
 XIII. Elementary Principles of Coordinate Geometry 95 
 
 XIV. Elementary Prlnclples of the Calculus no 
 
MATHEMATICS 
 
 CHAPTER I 
 
 FUNDAMENTALS OF ALGEBRA 
 
 Addition and Subtraction 
 
 As an introduction to this chapter on the fundamental 
 principles of algebra, I will say that it is absolutely 
 essential to an understanding of engineering that the 
 fundamental principles of algebra be thoroughly digested 
 and redigested, — in short, literally soaked into one's 
 mind and method of thought. 
 
 Algebra is a very simple science — extremely simple 
 if looked at from a common-sense standpoint. If not 
 seen thus, it can be made most intricate and, in fact, 
 incomprehensible. It is arithmetic simplified, — a short 
 cut to arithmetic. In arithmetic we would say, if one 
 hat costs 5 cents, 10 hats cost^ 50 cents. In algebra 
 we would say, if one a costs 5 cents, then 10 a cost 50 
 cents, a being used here to represent "hat." a is what 
 we term in algebra a symbol, and all quantities are 
 handled by means of such symbols, a is presumed 
 to represent one thing; b, another symbol, is presumed 
 
2 MATHEMATICS 
 
 to represent another thing, c another, d another, and 
 so on for any number of objects. The usefulness 
 and simplicity, therefore, of using symbols to repre- 
 sent objects is obvious. Suppose a merchant in the 
 furniture business to be taking stock. He would go 
 through his stock rooms and, seeing 10 chairs, he 
 would actually write down "10 chairs"; 5 tables, he 
 would actually write out "5 tables"; 4 beds, he would 
 actually write this out, and so on. Now, if he had at 
 the start agreed to represent chairs by the letter a, 
 tables by the letter b, beds by the letter c, and so on, 
 he would have been saved the necessity of writing 
 down the names of these articles each time, and could 
 have written 10 a, 5 b, and 4 c. 
 
 Definition of a Symbol. — A symbol is some letter by 
 which it is agreed to represent some object or thing. 
 
 When a problem is to be worked in algebra, the first 
 thing necessary is to make a choice of symbols, namely, 
 to assign certain letters to each of the different objects 
 concerned with the problem, — in other words, to get 
 up a code. When this code is once established it must 
 be rigorously maintained; that is, if, in the solution of 
 any problem or set of problems, it is once stipulated 
 that a shall represent a chair, then wherever a appears 
 it means a chair, and wherever the word chair would 
 be inserted an a must be placed — the code must not 
 be changed. 
 
FUNDAMENTALS OF ALGEBRA 3 
 
 Positivity and Negativity. — Now, in algebraic thought, 
 not only do we use symbols to represent various objects 
 and things, but we use the signs plus (+) or minus (— ) 
 before the symbols, to indicate what we call the positivity 
 or negativity of the object. 
 
 Addition and Subtraction. — Algebraically, if, in go- 
 ing over his stock and accounts, a merchant finds that 
 he has 4 tables in stock, and on glancing over his 
 books finds that he owes 3 tables, he would represent 
 the 4 tables in stock by such a form as +4 a, a 
 representing table; the 3 tables which he owes he would 
 represent by —3 a, the plus sign indicating that which 
 he has on hand and the minus sign that which he owes. 
 Grouping the quantities +4 a and —3a together, 
 in other words, striking a balance, one would get +a, 
 which represents the one table which he owns over and 
 above that which he owes. The plus sign, then, is taken 
 to indicate all things on hand, all quantities greater 
 than zero. The minus sign is taken to indicate all 
 those things which are owed, all things less than zero. 
 
 Suppose the following to be the inventory of a cer- 
 tain quantity of stock: +8 a, — 2 a, +6 b, —3 c, 
 +4 a, — 2 &, — 2 c, +5 c. Now, on grouping these 
 quantities together and striking a balance, it will be 
 seen that there are 8 of those things which are repre- 
 sented by a on hand; likewise 4 more, represented by 
 4 a, on hand; 2 are owed, namely, —2 a. Therefore, 
 
4 MATHEMATICS 
 
 on grouping +8 a, +4 a, and —2a together, +10 a 
 will be the result. Now, collecting those terms repre- 
 senting the objects which we have called b, we have 
 +6b and —2 b, giving as a result +46. Grouping 
 —3 c, — 2 c, and +5 c together^ will give o, because 
 +5 c represents 5 c's on hand, and —3c and —2c 
 represent that 5 c's are owed; therefore, these quan- 
 tities neutralize and strike a balance. Therefore, 
 
 + 8a — 2 a + 66 — 3 c + 4a — 2b — 2 c + 5 c 
 reduces to + 10 a + 4 b. 
 
 This process of gathering together and simplifying a 
 collection of terms having different signs is what we 
 call in algebra addition and subtraction. Nothing is 
 more simple, and yet nothing should be more thoroughly 
 understood before proceeding further. It is obviously 
 impossible to add one table to one chair and thereby 
 get two chairs, or one bojbk to one hat and get two 
 books; whereas it is perfectly possible to add one book 
 to another book and get' two books, one chair to an- 
 other chair and thereby get two chairs. 
 
 Rule. — Like symbols can be added and subtracted, and 
 only like symbols. 
 
 a + a will give 2 a; 30 + 50 will give 8a; a + b 
 will not give 2 a or 2 b, but will simply give a + b, 
 this being the simplest form in which the addition of 
 these two terms can be expressed. 
 
FUNDAMENTALS OF ALGEBRA 5 
 
 Coefficients. — In any term such as +8 a the plus 
 sign indicates that the object is on hand or greater than 
 zero, the 8 indicates the number of them on hand, it 
 is the numerical part of the term and is called the 
 coefficient, and the a indicates the nature of the ob- 
 ject, whether it is a chair or a book or a table that we 
 have represented by the symbol a. In the term +6 a, 
 the plus (+) sign indicates that the object is owned, or 
 greater than zero, the 6 indicates the number of objects 
 on hand, and the a their nature. If a man has $20 
 in his pocket and he owes $50, it is evident that if he 
 paid up as far as he could, he would still owe $30. If 
 we had represented $1 by the letter a, then the $20 in 
 his pocket would be represented by + 20 a, the $50 
 that he owed by — 50 a. On grouping these terms 
 together, which is the same process as the settling of 
 accounts, the result would be —30 a. 
 
 Algebraic Expressions. — An algebraic expression con- 
 sists of two or more terms; for instance, + a + b is 
 an algebraic expression; +a-|-2& + cisan algebraic 
 expression; + $a + $b + 6b + c is another alge- 
 braic expression, but this last one can be written more 
 simply, for the 5 b and 6 b can be grouped together in 
 one term, making n b, and the expression now becomes 
 + 3a + 11b + c, which is as simple as it can be 
 written. It is always advisable to group together into 
 the smallest number of terms any algebraic expression 
 
6 MATHEMATICS 
 
 wherever it is met in a problem, and thus simplify the 
 manipulation or handling of it. 
 
 When there is no sign before the first term of an 
 expression the plus (+) sign is intended. , 
 i To subtract one quantity from another, change the 
 sign and then group the quantities into one term, as just 
 explained. Thus: to subtract 4 a from + 12 a we 
 write — 4 a -+• i2j^wMclvsiniplifies into 4-8 a. Again, 
 subtracting i"a from 4- 6 a we would have — 2 a + 6 a, 
 which equals +4 a. 
 
 PROBLEMS 
 
 Simplify the following expressions : 
 
 1. ioa + 5b + 6c-8a~3d + b. 4 ^ V- &B 
 
 2. a — 6 + c — 10 a — 7, c + 2 &. 
 
 3. 10J + 3Z + 8& — 4 d — 6.z — 12 & + 5 a — 3d 
 
 + 8z — ioa + 8b — $a — 6z + iob. 
 
 4. 5« — 4 y + 3 ,z — 2 &.+ 4 y + * + z + a — j x 
 
 + 6y. 
 
 5. 3 6— 2 a + 5 c + 7 a — 10 6j— 8 c 4- 4 a— b + c. 
 
 6. — 2 » + a + 6 + ioy — 6 *— y— 7 a + 3 b + 2y. 
 
 7. 4X — y + z + x + 15Z — 3X + 6y^7y+ 12Z. 
 
 r r ■ ■ 
 
CHAPTER II 
 
 FUNDAMENTALS OF ALGEBRA 
 
 Multiplication and Division 
 
 We have seen how the use of algebra simplifies the 
 operations of addition and subtraction, but in multipli- 
 cation and division this simplification is far greater, and 
 the great weapon of thought which algebra is to become 
 to the student is now realized for the first time. If the 
 student of arithmetic is asked to multiply one foot by 
 one foot, his result is one square foot, the square foot 
 being very different from the foot. Now, ask hira to 
 multiply one chair by one table. How can he express 
 the result? What word can he use to signify the result? 
 Is there any conception in his mind as to the appear- 
 ance of the object which would be obtained by multi- 
 plying one chair by one table? In algebra all this is 
 simplified. If we represent a table by a, and a chair by 
 b, and we multiply a by b, we obtain the expression ab, 
 which represents in its entirety the multiplication of a 
 chair by a table. We need no word, no name by which 
 to call it; we simply use the form ab, and that carries 
 to our mind the notion of the thing which we call a 
 multiplied by the thing which we call b. And thus the 
 
 7 
 
8 MATHEMATICS 
 
 form is carried without any further thought being given 
 to it. 
 
 Exponents. — The multiplication of a by a may be 
 represented by aa. But here we have a further short 
 cut, namely, a 2 . This 2, called an exponent, indicates that 
 two a's have been multiplied by each other; a X a X a 
 would give us a 3 , the 3 indicating that three a's have 
 been multiplied by one another; and so on. The expo- 
 nent simply signifies the number of times the symbol has 
 been multiplied by itself. 
 
 Now, suppose a 2 were multiplied by a 3 ;j the result would 
 be a 5 , since a 2 signifies that 2 a's are multiplied together, 
 and a 3 indicates that 3 a's are multiplied together; then 
 multiplying these two expressions by each other simply 
 indicates that 5 a's are multiplied together, a 3 X a 7 
 would likewise give us a 10 , a 4 X a 4 would give us a 8 , 
 a 4 X a 4 X a 2 X a 3 would give us a 13 , and so on. 
 
 Rule. — The multiplication by each other of symbols 
 representing similar objects is accomplished by adding 
 their exponents. 
 
 Indentity of Symbols. — Now, in the foregoing it must 
 be clearly seen that the combined symbol ab is different 
 from either a or b ; ab must be handled as differently 
 from a or b as c would be handled; in other words, it is 
 an absolutely new symbol. Likewise a 2 is as different 
 from a as a square foot is from a linear foot, and a 3 is 
 as different from a 2 as one cubic foot is from one square 
 
FUNDAMENTALS OF ALGEBRA 9 
 
 foot, a 2 is a distinct symbol, a 3 is a distinct symbol, 
 and can only be grouped together with other a 3 's. For 
 example, if an algebraic expression such as this were met : 
 
 a 2 + a + ab + a 3 + 3 a 2 — 2 a — ab, 
 to simplify it we could group together the a 2 and the 
 +3 a 2 , giving +4 a 2 ; the +a and the — 2a give — a; 
 the +ab and the — ab neutralize each other; there is 
 only one term with the symbol a 3 . Therefore the 
 above expression simplified would be 4 a 2 — a + a 8 . 
 This is as simple as it can be expressed. Above all 
 things the most important is never to group unlike 
 symbols together by addition and subtraction. Re- 
 member fundamentally that a, b, ab, a 2 , a 3 , a* are all 
 separate and distinct symbols, each representing a 
 separate and distinct thing. 
 
 Suppose we have a X b X c. It gives us the term 
 abc. If we have a 2 X b we get a 2 b. If we have ab X ab, 
 we get a 2 b 2 - If we have 2 ab X 2 ab we get 4 a 2 b 2 ; 
 6 a 2 W X 3 c, we get 18 a 2 b 3 c; and so on. Whenever two 
 terms are multiplied by each other, the coefficients are 
 multiplied together, and the similar parts of the symbols 
 are multiplied together. 
 • Division. — Just as when in arithmetic we write 
 
 down -to mean 2 divided by 3, in algebra we writer 
 
 3 o 
 
 to mean a divided by b. a is called a numerator and 
 
 b a denominator, and the expression 7 is called a frac- 
 
 
 
IO MATHEMATICS 
 
 tion. If a 3 is multiplied by a 2 , we have seen that the 
 
 result is a 5 , obtained by adding the exponents 3 and 2. 
 
 If a 3 is divided by a 2 , the result is a, which is obtained 
 
 a?b 
 by subtracting 2 from 3. Therefore — would equal a, 
 
 the a in the denominator dividing into o 2 in the nu- 
 merator a times, and the b in the denominator cancel- 
 ing the b in the numerator. Division is then simply the 
 inverse of multiplication, which is patent. On simplify- 
 
 a A b 2 c 3 . a 2 b 
 
 ing such an expression as , we obtain — , and so on. 
 
 Or OCr C 
 
 Negative Exponents. — But there is a more scientific 
 and logical way of explaining division as the inverse 
 of multiplication, and it is thus : Suppose we have the 
 
 fraction—. This may be written a -2 , or the term b 2 
 
 a 2 
 
 may be written — ; that is, any term may be changed 
 
 from the numerator of a fraction to the denominator by 
 
 simply changing the sign of its exponent. For example, 
 
 a 5 
 
 -j may be written o 5 X a 2 . Multiplying these two 
 
 terms together, which is accomplished by adding their 
 exponents, would give us a 3 , 3 being the result of 
 the addition of 5 and —2. It is scarcely necessary, 
 therefore, to make a separate l'aw for division if one is 
 made for multiplication, when it is seen that division 
 simply changes the sign of the exponent. This should 
 
FUNDAMENTALS OF ALGEBRA II 
 
 be carefully considered and thought over by the pupil, 
 for it is of great importance. Take such an expression 
 
 as . Suppose all the symbols in the denominator 
 
 abc l 
 
 are placed in the numerator, then we have a^Wa^b^c, 
 or, simplifying, ab~ z <?, which may be further written 
 
 ft/** 
 
 — . The negative exponent is very serviceable, and it 
 b 6 
 
 is well to become thoroughly familiar with it. The fol- 
 lowing examples should be worked by the student. 
 
 PROBLEMS 
 
 Simplify the following : 
 i. 2aX $b X sab. 
 
 10 
 
 ii 
 
 1 2 a?bc X 4 c 2 b. 
 
 6 x X 5 y X 3 xy. 
 
 4 a?bc X 3 abc X 2 a 5 $ X 6 ft 2 . 
 
 aW 
 
 a 2 d? ' 
 a" 2 X & 3 X a 6 fi 2 c. 
 abc 2 X 6-?a-V X o 3 6 3 . 
 a'b^ch 
 
 10 a 2 6 X s/*" 1 ^-* X 7^ X io- J a. 
 o 2 a 4 
 
 5 a Wa 12 
 45 a 3 X 6^' 
 
CHAPTER III 
 FUNDAMENTALS OF ALGEBRA 
 
 Multiplication and Division (Continued). 
 
 Having illustrated and explained the principles of 
 multiplication and division of algebraic terms, we will 
 in this lecture inquire into the nature of these processes 
 as they apply to algebraic expressions^ Before doing 
 this, however, let us investigate a little further into the 
 principles of fractions. 
 
 Fractions. — We have said that the fraction 7 indi- 
 
 
 
 cated that a was divided by b, just as in arithme- 
 tic \ indicates that 1 is divided by 3. Suppose we 
 
 multiply the fraction -by 3, we obtain** , our procedure 
 
 3 3 
 
 being to multiply the numerator 1 by 3. Similarly, 
 
 if we had multiplied the fraction - by 3, our result 
 
 would have been ^~ • 
 
 
 Rule. — The multiplication of a fraction by any quan- 
 tity is accomplished by multiplying its numerator by 
 
 2 a? 
 that quantity; thus, —7- multiplied by 3 a would give 
 
 — 7-- Conversely, when we divide a fraction by a 
 
FUNDAMENTALS OF ALGEBRA 13 
 
 quantity, we multiply its denominator by that quantity. 
 
 Thus, the fraction - when divided by 2 & gives -^t- 
 2 b 2 
 
 Finally, should we multiply the numerator and the 
 denominator by the same quantity, it is obvious that 
 we do not change the value of the fraction, for we have 
 multiplied and divided it by the same thing. From 
 this it must not be deduced that adding the same 
 quantity to both the numerator and the denominator of 
 a fraction will not change its value. The beginner is 
 likely to make this mistake, and he is here warned 
 against it. Suppose we add to both the numerator and 
 the denominator of the fraction f the quantity 2 . We will 
 obtain §, which is different in value from ^, proving 
 that the addition or subtraction of the same quantity 
 from both numerator and denominator of any fraction 
 changes its value. The multiplication or division of 
 both the numerator and the denominator by the same 
 quantity does not alter the value of a fraction one whit. 
 Multiplying two fractions by each other is accom- 
 plished by multiplying their numerators together and 
 
 multiplying their denominators together. Thus, - X - 
 
 c 
 
 would give us — ■ 
 be 
 
 Suppose it is desired to add the fraction \ to the 
 fraction \ . Arithmetic teaches us that it is first neces- 
 sary to reduce both fractions to a common denominator, 
 
14 MATHEMATICS 
 
 which in this case is 6, viz. : 1 + 1 = 1, the numera- 
 tors being added if the denominators are of a common 
 
 value. Likewise, if it is desired to add - to - , we must 
 
 o a 
 
 reduce both of these fractions to a common denominator, 
 
 which in this case is bd. (The common denominator of 
 
 several denominators is a quantity into which any one of 
 
 these denominators may be divided; thus b will divide into 
 
 bd, d times, and a* will divide into b d, b times.) Our fractions 
 
 then become — + — . The denominators now having a 
 bd bd 
 
 common value, the fractions may be added by adding 
 the numerators, resulting in — ^— . Likewise, adding 
 
 the fractions - -\ 1 , we find that the common 
 
 3 2a sa 
 
 denominator in this case is 6 a. The first fraction 
 
 becomes - — , the second ^— and the third — , the 
 6 a 6 a 6 a 
 
 2 Or "A— "l b ~\~ 2 C 
 
 result being the fraction f ! . This process 
 
 on 
 
 will be taken up and explained in more detail later, but 
 the student should make an attempt to apprehend the 
 principles here stated and solve the problems given at 
 the end of this lecture. 
 
 Law of_Signs. — Like signs multiplied or divided give 
 + and unlike signs give — . Thus : 
 
 + 3 a X + 2 a gives + 6 a 2 , 
 also — 3 a X —20 gives + 6 a 2 , 
 
FUNDAMENTALS OF ALGEBRA 1 5 
 
 while + 3 a X — 2 a gives — § a 2 
 
 or — 3 a X + 20 gives — 6 a 2 ; 
 
 furthermore + 8 a 2 divided by + 2 a gives + 4 a, 
 
 and — 8 a 2 divided by — 2 a gives + 4 a 
 
 while — 8 a 2 divided by + 2 a gives — 4 a 
 
 or + 8 a 2 divided by — 2 a gives — 4 a. 
 
 Multiplication of an Algebraic Expression by a 
 Quantity. — As previously said, an algebraic expres- 
 sion consists of two or more terms. 3 a, 5 b, are terms, 
 but 3 a + S b is an algebraic expression. If the stock 
 of a merchant consists of 10 tables and 5 chairs, and he 
 doubles his stock, it is evident that he must double the 
 number of tables and also the number of chairs, namely, 
 increase it to 20 tables and 10 chairs. Likewise, when 
 an algebraic expression which consists of 3 a + 2 b is 
 doubled, or, what is the same thing, multiplied by 2, 
 each term must be doubled or multiplied by 2, result- 
 ing in the expression 6 a + 4 b. Similarly, when an 
 algebraic expression consisting of several terms is 
 divided by any number, each term must be divided by 
 that number. 
 
 Rule. — An algebraic expression must be treated as a 
 unit. Whenever it is multiplied or divided by any quan- 
 tity, each term of the expression must be multiplied or 
 divided by that quantity. For example: Multiplying 
 
16 MATHEMATICS 
 
 the expression 4 x + 3 y + 5 xy by the quantity 3 x will 
 give the following result : 12 a; 2 + 9 ary + 15 a^y, obtained 
 by multiplying each one of the separate terms by 3 a; 
 successively. 
 
 Division of an Algebraic Expression by a Quantity. — 
 Dividing the expression 6 a 3 + 2 a?b + 4 b 2 by 2 oft 
 
 would result in the expression = 2 t — \- a -\ , obtained 
 
 a 
 
 by dividing each term successively by 2 b. This rule 
 
 u -***« 
 
 must be remembered, as its importance cannot be over- 
 estimated. The numerator or denominator of a frac- 
 tion consisting of one or two or more terms must be 
 handled as a unit, this being one of the most important 
 applications of this rule. For example, in the fraction 
 
 or ■ , it is impossible to cancel out the a in 
 
 a a + 
 
 the numerator and denominator, for the reason that if 
 the numerator is divided by a, each term must be 
 divided by a, and the operation upon the one term o 
 without the same operation upon the term b would 
 
 be erroneous. If the fraction is multiplied by 3, 
 
 a 
 
 it becomes 2 2_. If the fraction is multi- 
 
 a a + b 
 
 2 2 (t ~~ 2 b 
 
 plied by - it becomes ■ ; and so on. Never 
 
 3 30 + 3* 
 
 forget that the numerator (or denominator) of a fraction 
 
 consisting of two or more terms is an algebraic expression 
 and must be handled as a unit. 
 
FUNDAMENTALS OF ALGEBRA 17 
 
 Multiplication of One Algebraic Expression by An- 
 other. — It is frequently desired to multiply an alge- 
 braic expression not only by a single term but by another 
 algebraic expression consisting of two or more terms, in 
 which case the first expression is multiplied throughout 
 by each term of the second expression. The terms which 
 result from this operation are then collected together 
 by addition and subtraction and the result expressed in 
 the simplest manner possible. Suppose it were desired 
 to multiply a + b by c + d. We would first multiply 
 a + b by c, which would give us ac + be. Then we 
 would multiply a + b by d, which would give us 
 ad + bd. Now, collecting the result of these two multi- 
 plications together, we obtain ac + be + ad + bd, viz. : 
 
 a + b 
 
 c + d 
 ac + be 
 
 ad + bd 
 ac + be + ad + bd 
 
 Again, let us multiply 
 
 2 a + b — 3 c 
 by a + 2 h — c 
 
 2 a 2 + ab — 2>ac 
 
 $ab + 2 b 2 — 6 be 
 
 — 2 ac — be + 3 c 2 
 
 and we have 
 
 2 a 2 + 5 ab - 5 ac + 2 b 2 - 7 be + 3 <?• 
 
1 8 MATHEMATICS 
 
 The Division of one Algebraic Expression by Another. 
 
 — This is somewhat more difficult to explain and under- 
 stand than the foregoing. In general it may be said 
 that, suppose we are required to divide the expression 
 6 a 2 + 17 ab + 12 b 2 by 3 a + 4 b, we would arrange 
 the expression in the following way : 
 
 6 a 2 + 17 ab + 12 b 2 | 3 a + 4b 
 6a 2 + Bab 2 a + 3 & 
 
 9 ab + 12 b 2 
 
 gab + 12 b 2 
 
 3 a 'will divide into 6 a 2 , 2 a times, and this is placed 
 in the quotient as shown. This 2 a is then multiplied 
 successively into each of the terms in the divisor, namely, 
 3 a + 4 b, and the result, namely, 6 a 2 + 8 ab, is placed 
 beneath the dividend, as shown. A line is then drawn 
 and this quantity subtracted from the dividend, leaving 
 gab. The +12 b 2 in the dividend is now carried. 
 Again, we observe that 3 a in the divisor will divide into 
 gab, +36 times, and we place this term in the divisor. 
 Multiplying 3 b by each of the terms of the divisor, as 
 before, will give us 9 ab + 12 b 2 ; and, subtracting this 
 as shown, nothing remains, the final result of the divi- 
 sion then being the expression 20 + 36. 
 
 This process should be studied and thoroughly under- 
 stood by the student. 
 
FUNDAMENTALS OF ALGEBRA 19 
 
 PROBLEMS 
 
 Solve the following problems : 
 
 1. Multiply the fraction = 2 — — by the quantity 3 x. 
 
 2. Divide the fraction — — by the quantity 3 a. 
 
 a 
 
 a 2 b 2 c 2 aW 
 
 3. Multiply the fraction — — by the fraction — — by 
 
 xy a 
 
 the fraction -r*« 
 
 
 4. Multiply the expression 43; + 33; + 22 by the 
 quantity 5 x. 
 
 5. Divide the expression 8 a?b + 4 a z b z — 2 ab 2 by the 
 quantity 2 ab. 
 
 6. Multiply the expression a + b by the expression 
 a — b. 
 
 7. Multiply the expression 2 a + b — c by the ex- 
 pression 3a — 2& + 4C. 
 
 8. Divide the expression a 2 — 2 ab + b 2 by a — b. 
 
 9. Divide the expression a 3 + 3 a 2 b + 3 a& 2 + 5 3 by 
 a + b. 
 
 10. Multiply the fraction ■ , by ;- 
 
 r J a — b a — b 
 
 11. Multiply the fraction -3— by by 
 
 ^ J c + d 2 a — c 
 
 12. Multiply the fraction by — - 2 by -• 
 
20 MATHEMATICS 
 
 13. Add together the fractions — — I (- -• 
 
 040 
 
 2 4. c 
 
 14. Add together the fractions — - — — + - • 
 
 3 a 2 2 a 6 
 
 15. Add together the fractions — ; 1 — ■ 1 
 
 40 2 c 6 
 
 16. Add together the fractions 
 
 17. Add together the fractions 
 
 2 a 46 
 
 a 2 
 
 a + b s a 
 
CHAPTER IV 
 
 FUNDAMENTALS OF ALGEBRA 
 
 Factoring 
 
 Definition of a Factor. — A factor of a quantity is 
 one of the two or more parts which when multiplied 
 together give the quantity. A factor is an integral 
 part of a quantity, and the ability to divide and sub- 
 divide a quantity, be it a single term or a whole expres- 
 sion, into those factors whose multiplication has created 
 it, is very valuable. 
 
 Factoring. — Suppose we take the number 6. Its 
 factors are readily detected as 2 and 3. Likewise the 
 factors of 10 are 5 and 2. The factors of 18 are 9 and 
 2; or, better still, 3X3X2. The factors of 30 are 
 3X2X5; and so on. The factors of the algebraic ex- 
 pression ab are readily detected as a and b, because 
 their multiplication created the term ab. The factors 
 of 6 abc are 3, 2, a, b and c. The factors of 25 ab are 5, 
 5, a and b, which are quite readily detected. 
 
 The factors of an expression consisting of two or more 
 terms, however, are not so readily seen and sometimes 
 require considerable ingenuity for their detection. Sup- 
 pose we have an algebraic expression in which all of 
 
22 MATHEMATICS 
 
 the terms have one or more common factors, — that is, 
 that one or more like factors appear in the make-up of 
 each term. It is often desirable in this case to remove 
 the common factors from the several terms, and in 
 order to do this without changing the value of any of 
 the terms, the common factor or factors are placed 
 outside of a parenthesis and the terms from which they 
 have been removed placed within the parenthesis in 
 their simplified form. Thus, in the algebraic expression 
 6 a 2 b + 3 a 3 , 3 a 2 is a common factor of both terms; 
 therefore we may write the expression, without chang- 
 ing its value, in the following manner: 3 a 2 (2 b + a). 
 The term 3 a 2 written outside of the parenthesis indi- 
 cates that it must be multiplied into each of the sepa- 
 rate terms within the parenthesis. Likewise, in the 
 expression 12 xy + 4 x 3 + 6 x 2 z + 8 xz, 2 x is a common 
 factor of each of the terms, and the expression may 
 be written 2 x (6 y + 2 x 2 + 3 xz + 4 z). It is often 
 desirable to factor in this simple manner. 
 
 Still further suppose we have a 2 + ab + ac + be; we 
 can take a out of the first two terms and c out of the 
 last two, thus: a (a + b) + c (a + b). Now we have 
 two separate terms and taking (a + b) out of each 
 we have (a + b) X (a + c). Likewise, in the expres- 
 sion 6 x 2 + 4 xy — 3 zx — 2 zy 
 we have 2 x (3 x + 2 y) — 2 (3 x + 2 y), 
 or, (3 x + 2 y) X (2 * — 2). 
 
FUNDAMENTALS OF ALGEBRA 23 
 
 Now, suppose we have the expression a 2 — 2 ab + b 2 . 
 We readily detect that this quantity is the result of 
 multiplying a — b by a — b; the first and last terms 
 are respectively the squares of a and b, while the 
 middle term is equal to twice the product of a and 
 b. Any expression where this is the case is a perfect 
 square; thus, 9 x 2 — 12 xy + 4 y 2 is the square of 
 3 x — 2 y, and may be written (3 x — 2 y) 2 . Remem- 
 bering these facts, a perfect square is readily detected. 
 
 The product of the sum and difference of two terms 
 such as (a + b) X (a — b) equals a 2 — b 2 ; or, briefly, 
 the product of the sum and difference of two numbers 
 is equal to the difference of their squares. 
 
 By trial it is often easy to discover the factors of 
 algebraic expressions; for example, 2 a 2 + 7 ab + 3 b 2 is 
 readily detected to be the product of 2 a + b and 
 a + 3&. 
 
 Factor the following : 
 
 1. 3oa?b. 
 
 2. 4&a i c. 
 
 3. 30x 2 y*z 3 . 
 
 4. 144 x 2 a 2 . 
 12 ab 2 (? 
 
 3 " Atfb 2 
 
 , 10 xy 2 
 
 6 - — 2 ' 
 2xry 
 
 7. 2 a 2 + ab — 2 ac — ic. 
 
24 24 MATHEMATICS 
 
 8. 3 x 2 + xy + 3 xc + cy. 
 
 9. 2X 2 + sxy + 2xz + $ yz. 
 
 10. a 2 — 2 ab + b 2 . 
 
 11. 4* 2 — i2ary + (jy 2 . 
 
 12. 81 a 2 + go ab + 25 b 2 . 
 
 13. 16 c 2 — 48 ca + 36 a 2 . 
 
 14. 4 afy + 5 *zy 2 — 10 xzy. 
 
 15. 30 ab + 15 a&c — 5 be. 
 
 16. 81 »y — 25 a 2 . 
 
 17. a 4 - 16 6 4 . 
 
 18. 144 afy 2 — 64 z 2 . 
 
 19. 4 a 2 — 8 ac + 4 c. 
 
 20. i6y 2 + 8xy + x 2 . 
 
 21. 6y 2 — $xy — 6X 2 . 
 
 22. 4 a 2 — 306 — 10 6 2 . 
 
 23. 6y 2 — i$xy +6X 2 . 
 
 24. 2 a 2 — 5 a& — 3 & 2 . 
 
 25. 2 a 2 + 9 a& + 10 b 2 . 
 
CHAPTER V 
 FUNDAMENTALS OF ALGEBRA 
 
 Involution and Evolution 
 
 We have in a previous chapter discussed the process 
 by which we can raise an algebraic term and even a 
 whole algebraic expression to any power desired, by 
 multiplying it by itself. Let us now investigate the 
 method of finding the square root and the cube root 
 of an algebraic expression, as we are frequently called 
 upon to do. 
 
 The square root of any term such as a 2 , a 4 , a 6 , 
 and so on, will be, respectively, ±a, ±a 2 , and 
 ±a 3 , obtained by dividing the exponents by 2. 
 The plus-or-minus sign (±) shows that either + a or 
 — a when squared would give us +0 2 . On taking the 
 square root, therefore, the plus-or-minus sign (±) is 
 always placed before the root. This is not the case in 
 the cube root, however. Likewise, the cube root of 
 such terms as a 3 , a 6 , a 9 , and so on, would be respec- 
 tively a, a 2 and a 3 , obtained by dividing the exponents 
 by 3. Similarly, the square root of 4 o 4 6 6 will be seen 
 to be ± 2 cPb 3 , obtained by taking the square root of 
 each factor of the term. And likewise the cube root 
 
 25 
 
26 MATHEMATICS 
 
 of — 27 a 9 6 6 will be — 3 aW. These facts are so self- 
 evident that it is scarcely necessary to dwell upon them. 
 However, the detection of the square and the cube root 
 of an algebraic expression consisting of several terms is 
 by no means so simple. 
 
 Square Root of an Algebraic Expression. — Suppose 
 we multiply the expression a + b by itself. We obtain 
 a 2 + 2 ab + b 2 . This is evidently the square of a + b. 
 Suppose then we are given this expression and asked to 
 determine its square root. We proceed in this manner : 
 Take the square root of the first term and isolate it, 
 calling it the trial root. The square root of a 2 is 
 a; therefore place a in the trial root. Now square 
 a and subtract this from the original expression, and 
 we have the remainder 2 ab + b 2 . For our trial divisor 
 we proceed as follows: Double the part of the root 
 already found, namely, a. This gives us 2 a. 20 will 
 go into 2 ab, the first term of the remainder, b times. 
 Add b to the trial root, and the same becomes a + b. 
 Now multiply the trial divisor by b, it gives us 2 ab + b 2 , 
 and subtracting this from our former remainder, we 
 have nothing left. The square root of our expression, 
 then, is seen to be a + b, viz. : 
 
 a 2 + 2 ab + b 2 \a + b 
 
 2 a + b 
 
 2\ab + b 2 
 2ab + b 2 
 
FUNDAMENTALS OF ALGEBRA 27 
 
 Likewise we see that the square root of 4 a 2 + 1 2 ab + 9 b 2 
 is 2 a + 3 b, viz. : 
 
 4jz 2 ,+ 12 ab + g b 2 | 2 g + 3 b 
 
 4 a 2 
 
 4 a + 3 & 
 
 12 aJ + 9 & 2 
 12 a& + 9 6 2 
 
 The Cube Root of an Algebraic Expression. — If we 
 
 multiply a + b by itself three times, in other words, 
 cube the expression, we obtain a 3 + 3 a 2 b + 3 ab 2 + b 3 . 
 It is evident, therefore, that if we had been given this 
 latter expression and asked to find its cube root, our 
 result should be a + b. In finding the cube root, a + b, 
 we proceed thus: We take the cube root of the first 
 term, namely, a, and place this in our trial root. Now 
 cube a, subtract the a 3 thus obtained from the original 
 expression, and we have as a remainder 3 a?b + 3 ab 2 + b 3 . 
 Now our trial divisor will consist as follows: Square 
 the part of the root already found and multiply same 
 by 3. This gives us 3 a 2 . Divide 3 a 2 into the first 
 term of the remainder, namely, 3 a 2 b, and it will go 
 b times, b then becomes the second term of the 
 root. Now add to the trial divisor three times the first 
 term of the root multiplied by the second term of the 
 root, which gives us 3 ab. Then add the second term 
 of the root square, namely, b 2 . Our full divisor now 
 becomes 3 a 2 + 3 ab + b 2 . Now multiply this full divisor 
 by b and subtract this from the former remainder, namely, 
 
28 
 
 MATHEMATICS 
 
 3 a?b + 3 ab 2 + W, and, having nothing left, we see that 
 the cube root of our original expression is a + b, viz. : 
 
 a 3 + sa 2 b + sab 2 + b s | a + b 
 
 3 a 2 + 3 ab + b 2 
 
 3 a 2 b + 3 a& 2 + < 
 3 a 2 b + 3 a& 2 + . 
 
 Likewise the cube root of 27 x 3 + 27 a; 2 + 9 x + 1 is 
 seen to be 3 x + 1, viz. : 
 
 2 , jx? + 2'jx 2 + gx + i I3X+1 
 27 x 3 
 2j x 2 + gx + 1 
 
 27 x 2 + 9 a; + 1 
 27 x 2 + gx + 1 
 
 Find the square root of the following expressions : 
 
 1. 16 x 2 + 24 xy + 9 y 2 . 
 
 2. 4 a 2 + 4 o& + Z> 2 . 
 
 3. 36x 2 + 24x3/ + 4y 2 - 
 
 4. 25 a 2 — 20 ab + 4 b 2 . 
 
 5. a 2 + 2 aft + 2 ac + 2 Sc + 6 2 + c 2 - 
 
 Find the cube root of the following expressions : 
 
 1. 8x 3 + 36x 2 )> + 54x;y 2 + 27 j/ 3 . 
 
 2. x 3 + 6x 2 y + 12 xy 2 + Sy 2 - 
 
 3. 27 a 3 + 81 a 2 b[+ 81 a& 2 + 27 W. 
 
CHAPTER VI 
 
 FUNDAMENTALS OF ALGEBRA 
 
 Simple Equations 
 
 An equation is the expression of the equality of two 
 things; thus, a = b signifies that whatever we call a is 
 equal to whatever we ca.ll b; for example, one pile of 
 money containing $100 in one shape or another is 
 equal to any other pile containing $100. It is evident 
 that if a quantity is added to or subtracted from one 
 side of an equation or equality, it must be added to or 
 subtracted from the other side of the equation or equal- 
 ity, in order to retain the equality of the two sides; 
 thus, if a = b, then a + c = b + c and a — c = b — c. 
 Similarly, if one side of an equation is multiplied 
 or divided by any quantity, the other side must be 
 multiplied or divided by the same quantity; thus, 
 
 if a = b, 
 
 then ac = be 
 
 j a b 
 
 and - = - . 
 
 c c 
 
 Similarly, if one side of an equation is squared, the 
 
 other side of the equation must be squared in order to 
 
 29 
 
30 MATHEMATICS 
 
 retain the equality. In general, whatever is done to 
 one side of an equation must also be done to the other 
 side in order to retain the equality of both sides. The 
 logic of this is self-evident. 
 
 Transposition. — Suppose we have the equation 
 a + b = c. Subtract b from both sides, and we have 
 a + b — b = c — b. On the left-hand side of the equa- 
 tion the -\-b and the —b will cancel out, leaving a, and 
 we have the result a = c — b. Compare this with our 
 original equation, and we will see that they are exactly 
 alike except for the fact that in the one b is on the 
 left-hand side of the equation, in the other b is on 
 the right-hand side of the equation; in one case its sign 
 is plus, in the other case its sign is minus. This indi- 
 cates that in order to change a term from one side of 
 an equation to the other side it is simply necessary to 
 change its sign; thus, 
 
 a — c + b = d 
 
 may be transposed into the equation 
 
 a = c — b + d, 
 
 or into the form a — d = c — b, 
 
 or into the form — d = c — a — b. 
 
 Any term may be transposed from one side of an equa- 
 tion to the other simply by changing its sign. 
 
 Adding or Subtracting Two Equations. — When two 
 equations are to be added to one another their corre- 
 
FUNDAMENTALS OF ALGEBRA 31 
 
 sponding sides are added to one another; thus, a + c = b 
 when added to a = d + e will give 2a + c = b + d + e. 
 Likewise 3 a + b = 2 c when subtracted^ from 
 ioa + 2b = 6c will yield 7 a + b = 4 c. 
 
 Multiplying or Dividing Two Equations by one An- 
 other. — When two equations are multiplied or divided 
 by one another their corresponding sides must be multi- 
 plied or divided by one another; thus, a = b multiplied by 
 c = d will give ac = bd, also a = b divided by c = d 
 
 .... a b 
 will give - = -■ 
 c a 
 
 Solution of an Equation. — Suppose we have such an 
 
 equation as 4 x + 10 = 2 x + 24, and it is desired that 
 
 this equation be solved for the value of x; that is, that 
 
 the value of the unknown quantity x be found. In 
 
 order to do this, the first process must always be 
 
 to group the terms containing x on one side of the 
 
 equation by themselves and all the other terms in the 
 
 equation on the other side of the equation. In this 
 
 case, grouping the terms containing the unknown 
 
 quantity x on the left-hand side of the equation we 
 
 " ave 4 x — 2 x = 24 — 10. 
 
 Now, collecting the like terms, this becomes 
 
 2 x = 14. 
 
 The next step is to divide the equation through by the 
 coefficient of x, namely, 2. Dividing the left-hand 
 
32 MATHEMATICS 
 
 side by 2, we have x. Dividing the right-hand side 
 by 2, we have 7. Our equation, therefore, has resolved 
 
 itself into 
 
 x = 7. 
 
 We therefore have the value of x. Substituting this 
 value in the original equation, namely, 
 4 x + 10 = 2 x + 24, 
 we see that the equation becomes 
 
 28 + 10 = 14 + 24, 
 or, 38 = 38, 
 
 which proves the result. 
 
 The process above described is the general method of 
 solving for an unknown quantity in a simple equation. 
 
 Let us now take the equation 
 
 2 ex + c = 40 — 5 x. 
 
 This equation contains two unknown quantities, namely, 
 c and x, either of which we may solve for. x is usually, 
 however, chosen to represent the unknown quantity, 
 whose value we wish to find, in an algebraic expression; 
 in fact, x, y and z are generally chosen to represent 
 unknown quantities. Let us solve for x in the above 
 equation. Again we group the two terms containing 
 x on one side of the equation by themselves and all 
 other terms on the other side, and we have 
 
 2 ex + 5 * = 40 — c. 
 
FUNDAMENTALS OF ALGEBRA 33 
 
 On the left-hand side of the equation we have two terms 
 
 containing xasa factor. Let us factor this expression 
 
 and we have 
 
 * (2 c + 5) = 40 - c. 
 
 Dividing through by the coefficient of x, which is the 
 parenthesis in this case, just as simple a coefficient to 
 handle as any other, and we have 
 
 40 — c 
 
 x = - aL — - 
 
 2c + s 
 
 This final result is the complete solution of the equa- 
 tion as to the value of x, for we have x isolated on 
 one side of the equation by itself, and its value on the 
 other side. In any equation containing any number of 
 unknown quantities represented by symbols, the complete 
 solution for the value of any one of the unknowns is 
 accomplished when we have isolated this unknown on one 
 side of the equation by itself. This is, therefore, the whole 
 object of our solution. 
 
 It is true that the value of x above shown still con- 
 tains an unknown quantity, c. Suppose the numerical 
 value of c were now given, we could immediately find 
 the corresponding numerical value of x; thus, suppose c 
 were equal to 2, we would have 
 
 40 — 2 
 
 x = , 
 
 4 + 5 
 
 or, 
 
 .-31 
 
34 MATHEMATICS 
 
 This is the numerical value of x, corresponding to the 
 numerical value 2 of c. If 4 had been assigned as 
 the numerical value of c we should have 
 
 XB= 4Q-4 = 3g. 
 
 8 + 5 13 
 
 Clearing of Fractions. — The above simple equations 
 contained no fractions. Suppose, however, that we are 
 asked to solve the equation 
 
 4226 
 
 Manifestly this equation cannot be treated at once in 
 the manner of the preceding example. The first step 
 in solving such an equation is the removal of all the 
 denominators of the fractions in the equation, this 
 step being called the Clearing of Fractions. 
 
 As previously seen, in order to add together the 
 fractions § and f we must reduce them to a common 
 denominator, 6 We then have t + f = f . Likewise, 
 in equations, before we can group or operate upon any 
 one of the terms we must reduce them to a common 
 denominator. The common denominator of several 
 denominators is any number into which any one of the 
 various denominators will divide, and the least common 
 denominator is the smallest such number. The product 
 of all the denominators — that is, multiplying them all 
 together — will always give a common denominator, but 
 
FUNDAMENTALS OF ALGEBRA 35 
 
 not always the least common denominator. The least 
 common denominator, being the smallest common de- 
 nominator, is always desirable in preference to a larger 
 number; but some ingenuity is needed frequently in 
 detecting it. The old rule of withdrawing all factors 
 common to at least two denominators and multiplying 
 them together, and then by what is left of the denomi- 
 nators, is probably the easiest and simplest way to pro- 
 ceed. Thus, suppose we have the denominators 6, 8, 
 9 and 4. 3 is common to both 6 and 9, leaving respec- 
 tively 2 and 3. 2 is common to 2, 8 and 4, leaving 
 respectively 1, 4 and 2, and still further common to 4 
 and 2. Finally, we have removed the common factors 
 3, 2 and 2, and we have left in the denominators 1, 2, 
 3 and 1. Multiplying all of these together we have 
 72, which is the Least Common Denominator of these 
 numbers, viz. : 
 
 3 1 6, 8, q, 4 
 
 2 I 2 , 8> 3,4 
 
 2 1 1. 4, 3, 2 
 i, 2,3, 1 
 
 3X2X2X1X2X3X1 = 7 2 - 
 
 Having determined the Least Common Denominator, 
 or any common denominator for that matter, the next 
 step is to multiply each denominator by such a quantity 
 as will change it into the Least Common Denominator. 
 If the denominator of a fraction is multiplied by any 
 
36 MATHEMATICS 
 
 quantity, as we have previously seen, the numerator 
 must be multiplied by that same quantity, or the value 
 of the fraction is changed. Therefore, in multiplying 
 the denominator of each fraction by a quantity, we 
 must also multiply the numerator. Returning to the 
 
 oc 6 
 equation which we had at the outset, namely, - -\ — = 
 
 4 2 
 
 "? OC c 
 
 * h ^ , we see that the common denominator here is 
 
 2 6 
 
 12. Our equation then becomes - \- — = — -\ 
 
 12 12 12 12 
 
 We have previously seen that the multiplication or 
 division of both sides of an equation by the same 
 quantity does not alter the value of the equation. 
 Therefore we can at once multiply both sides of this 
 equation by 12. Doing so, all the denominators dis- 
 appear. This is equivalent to merely canceling all the 
 denominators, and the equation is now changed to the 
 simple form 3 x + 36 = 18 x + 10. On transposition 
 
 this becomes 
 
 3 x — 18 x = 10 — 36, 
 
 or — 15 x = —26, 
 
 -26 
 
 or — x = 
 
 or +x = 
 
 J 5 
 + 26 
 
 IS 
 
 Again, let us now take the equation 
 2 x 10 a: 
 5* c 2 ~t' 
 
FUNDAMENTALS OF ALGEBRA 37 
 
 The least common denominator will at once be seen to 
 be 15 c 2 . Reducing all fractions to this common denomi- 
 nator we have 
 
 6 ex 150 1 _ $c 2 x 
 
 15 c 2 15 c 2 15 c 2 
 
 Canceling all denominators, we then have 
 
 6 ex + 150 = $c 2 x. 
 
 Transposing, we have 
 
 6 ex — 5 c 2 x = —150. 
 
 Taking a; as a common factor out of both of the terms 
 in which it appears, we have 
 
 * (6 c - 5 c 2 ) = - 150. 
 
 Dividing through by the parenthesis, we have 
 
 - ~ I 5° 
 6 c — s c 2 
 
 This is the value- of x. If some numerical value is 
 given to c, such as 2, for instance, we can then find 
 the corresponding numerical value of x by substituting 
 the numerical value of c in the above, and we have 
 
 — 150 —150 
 
 x = 2 — = |- =18.75. 
 
 12 — 20 — o 
 
 In this same manner all equations in which fractions 
 appear are solved. 
 
38 MATHEMATICS 
 
 PROBLEMS 
 
 Suppose we wish to make use of algebra in the solu- 
 tion of a simple problem usually worked arithmetically, 
 taking, for example, such a problem as this : A man pur- 
 chases a hat and coat for $15.00, and the coat costs 
 twice as much as the hat. How much did the hat cost? 
 We would proceed as follows : Let x equal the cost of 
 the hat. Since the coat cost twice as much as the hat, 
 then 2 x equals the cost of the coat, and x + 2 x = 15 
 is the equation representing the fact that the cost of 
 the coat plus the cost of the hat equals $15; therefore, 
 3 x = $15, from which x = $5; namely, the cost of the 
 hat was $5. 2 x then equals $10, the cost of the coat. 
 Thus many problems may be attacked. 
 
 Solve the following equations : 
 
 1. 6x — io + 4# + 3 = 2x + 20 — x + 15. 
 
 2. # + 5+3# + 6= — 10 a; + 25 + 8 #. 
 
 3. ex + 4 + x = ex + 8. Find the numerical value 
 of x if c = 3. 
 
 X . & X , 
 
 4- - + 3= T + 4- 
 
 3 5 2 3 
 
 6. - -\ = - + Find the numerical value of 
 
 c 4 c 3 12 c 
 
 x if c = 3. 
 
FUNDAMENTALS OF ALGEBRA 39 
 
 IOC ex. 8 xcx . 15 _. , ., 
 
 1 = * r -**■ . Find the numerical 
 
 3 C $C IO 2 C 
 
 
 value of x if c = 6. 
 
 
 * j j.? ■ 
 
 
 a + b 3 
 
 
 21, . 2 3 
 
 
 9. ' -f 3* 1 — x — J t- 
 a a — a 2 
 
 
 * , X 
 
 
 10. , , -r , — 10. 
 a + a — 
 
 
 11. Multiply a* + b = c* — b by 2 a - 
 
 - * = c + 10. 
 
 12. Multiply- +b=-byx = y + 3. 
 
 13. Divide a 2 — b 2 = c by a + b = c + 3. 
 
 14. Divide 2 a = 10 y by a = y + 2. 
 
 15. Add 2a + io = # + 3 — a" to 3 a — 7 = 20". 
 
 16. Add 4 ax + 2 y = — 10 x to 2 a# — 7 y = 5. 
 
 17. Add 15 z 2 + x = 5 to 3X = — 10 y + 7. 
 
 18. Subtract 2 a — d = 8 from 8 a + d = 12. 
 
 19. Subtract 3# + 7 = i5# 2 + y from 6 # + 5 = I8* 2 . 
 
 20. Subtract \- c = 7 from = 18. 
 
 ioy 5j> 
 
 21. Multiply!— ^-r - - = c by-^-= = i£±^ • 
 
 3« + o 3 c — a c 
 
 22. Solve the equation - = — 
 
 x x + 1 
 
 23. If a coat cost one-half as much as a gun and twice 
 as much as a hat, and all cost together $100, what is 
 the cost of each ? 
 
40 MATHEMATICS 
 
 24. The value of a horse is $15 more than twice the 
 value of a carriage, and the cost of both is $1000; what is 
 the cost of each ? 
 
 25. One-third of Anne's age is 5 years less than one- 
 half plus 2 years; what is her age? ■ «" 
 
 26. A merchant has 10 more chairs than tables in 
 stock. He sells four of each and adding up stock finds 
 that he now has twice as many chairs as tables. How 
 many of each did he have at first? 
 
CHAPTER VII 
 
 FUNDAMENTALS OF ALGEBRA 
 
 Simultaneous Equations 
 
 As seen in the previous chapter, when we have a 
 simple equation in which only one unknown quantity 
 appears, such, for instance, as x, we can, by alge- 
 braic processes, at once determine the numerical value 
 of this unknown quantity. Should another unknown 
 quantity, such as c, appear in this equation, in order 
 to determine the value of x some definite value must 
 be assigned to c. However, this is not always pos- 
 sible. An equation containing two unknown quanti-. 
 ties represents some manner of relation between these 
 quantities. If two separate and distinct equations repre- 
 senting two separate and distinct relations which exist 
 between the two unknown quantities can be found, 
 then the numerical values of the unknown quantities 
 become fixed, and either one can be determined without 
 knowing the corresponding value of the other. The 
 two separate equations are called simultaneous equa- 
 tions, since they represent simultaneous relations be- 
 tween the unknown quantity. The following is an 
 example: 
 
 41 
 
42 MATHEMATICS 
 
 x + y = 10. 
 x - y = 4. 
 
 The first equation represents one relation between a; 
 and y. The second equation represents another relation 
 subsisting between x and y. The solution for the numer- 
 ical value of x, or that of y, from these two equations, 
 consists in eliminating one of the unknowns, x or y as 
 the case may be, by adding or subtracting, dividing or 
 multiplying the equations by each other, as will be 
 seen in the following. Let us now find the value of 
 x in the first equation, and we see that this is 
 
 x = 10 — y. 
 Likewise in the second equation we have 
 
 x = 4 + y. 
 
 These two values of x may now be equated (things 
 
 equal to the same thing must be equal to each other), 
 
 and we have 
 
 10 - y = 4 + y, 
 
 or, — 2 y = 4 — 10, 
 
 - 2y = -6, 
 
 + 2y = +6, 
 
 V = 3- 
 
 Now, this is the value of y. In order to find the 
 value of x, we substitute this numerical value of 
 y in one of the equations containing both x and y, 
 
FUNDAMENTALS OF ALGEBRA 
 
 43 
 
 such as the first equation, x + y = 10. Substituting, 
 
 we have 
 
 x + 3 = 10. 
 
 Transposing, x = 10 — 3, 
 
 x = 7. 
 
 Here, then, we have found the values of both x and 
 y, the algebraic process having been made possible 
 by the fact that we had two equations connecting the 
 unknown quantities. 
 
 The simultaneous equations above given might have 
 been solved likewise by simply adding both equations 
 together, thus: 
 
 Adding 
 
 x + y = 10 
 
 and 
 
 x - y = 4, 
 
 we have 
 
 x -{- y + x — y = 14. 
 
 Here +y and 
 
 —y will cancel out, leaving 
 
 
 ix = 14, 
 
 
 x = 7. 
 
 Both of these processes are called elimination, the 
 principal object in solving simultaneous equations being 
 the elimination of unknown quantities until some equa- 
 tion is obtained in which only one unknown quantity 
 appears. 
 
 We have seen that by simply adding two equations 
 
44 MATHEMATICS 
 
 we have eliminated one of the unknowns. But sup- 
 pose the equations are of this type : 
 
 (i) ?,x + 2y = 12, 
 
 (2) x + y = 5. 
 
 Now we can proceed to solve these equations in one of 
 two ways: first, to find the value of x in each equation 
 and then equate these values of x, thus obtaining an 
 equation where only y appears as an unknown quan- 
 tity. But suppose we are trying to eliminate x from 
 these equations by addition; it will be seen that adding 
 will not eliminate *, nor even will subtraction eliminate 
 it. If, however, we multiply equation (2) by 3, it be- 
 comes 
 
 3^ + 33' = x 5- 
 
 Now, when this is subtracted from equation (1), thus : 
 3* + 2y = 12 
 
 3X + sy = 15 
 
 -y = -3 
 
 the terms in #, +3 x and + 30: respectively, will elimi- 
 nate, 3 y minus 231 leaves —y, and 12 — 15 leaves —3, 
 or -y = -3, 
 
 therefore + y = +3. 
 
 Just as in order to find the value of two unknowns 
 two distinct and separate equations are necessary to 
 express relations between these unknowns, likewise to 
 find the value of the unknowns in equations contain- 
 
FUNDAMENTALS OF ALGEBRA 45 
 
 ing three unknown quantities, three distinct and sepa- 
 rate equations are necessary. Thus, we may have the 
 equations 
 
 (1) x + y + z = 6, 
 
 (2) x — y + 2Z = 1, 
 
 (3) x + y - z = 4. 
 
 We now combine any two of these equations, for in- 
 stance the first and the second, with the idea of elimi- 
 nating one of the unknown quantities, as x. Subtracting 
 equation (2) from (1), we will have 
 
 (4) 2 y - z = 5. 
 
 Now taking any other two of the equations, such as the 
 second and the third, and subtracting one from the other, 
 with a view to eliminating x, and we have 
 
 (5) -27 + 3Z = -3. 
 
 We now have two equations containing two unknowns, 
 which we solve as before explained. For instance, add- 
 ing them, we have 
 
 2Z = 2, 
 Z = I. 
 
 Substituting this value of z in equation (4), we have 
 
 2 y - 1 = 5> 
 2y = 6, 
 
 y = 3- 
 
46 MATHEMATICS 
 
 Substituting both of these values of z and y in equation 
 (i), we have 
 
 x + 3 + i = 6, 
 
 X = 2. 
 
 Thus we see that with three unknowns three distinct 
 and separate equations connecting them are necessary in 
 order that their values may be found. Likewise with 
 four unknowns four distinct and separate equations 
 showing relations between them are necessary. In each 
 case where we have a larger number than two equations, 
 we combine the equations together two at a time, each 
 time ehminating one of the unknown quantities, and, 
 using the resultant equations, continue in the same 
 course until we have finally resolved into one final 
 equation containing only one unknown. To find the 
 value of the other unknowns we then work backward, 
 substituting the value of the one unknown found in an 
 equation containing two unknowns, and both of these in 
 an equation containing three unknowns, and so on. 
 
 The solution of simultaneous equations is very im- 
 portant, and the student should practice on this subject 
 until he is thoroughly familiar with every one of these 
 steps. 
 
 Solve the following problems : 
 i. 2X + y = 8, 
 
 2 y — x = 6. 
 
FUNDAMENTALS OF ALGEBRA 
 
 47 
 
 2. x + y = 7, 
 3* -y = 13. 
 
 3. 4x = y + 2, 
 
 x + y = 3. 
 
 4. Find the value of x, y and z in the following 
 
 equations : 
 
 x + y + z = 10, 
 
 2X + y — z = 9, 
 
 # + 2y + z = 12. 
 
 5. Find the value of x, y and z in the following 
 equations : 
 
 2# + 3y+2Z = 20, 
 
 x + sy + z = 13, 
 
 # + y + 2z = 13. 
 
 X 
 
 6. 
 
 + y = 10, 
 
 10. 
 
 - + = 100 x + a if a = 8, 
 4 3 a 
 
 2# , 
 
 — = y + 10. 
 
 5 
 
 3* + y = 15, 
 
 x = 6 + 7 y. 
 
 os y _ 
 
 a + & a — b 
 
 x + y - 5 
 
 3*-y + 6* = 8, 
 
 y— 10 + 4 y = # 
 
 .,fl = 6, 
 
 if 
 
 6 = 5- 
 
CHAPTER VIII 
 FUNDAMENTALS OF ALGEBRA 
 
 Quadratic Equations 
 
 Thus far we have handled equations where the 
 unknown whose value we were solving for entered the 
 equation in the first power. Suppose, however, that 
 the unknown entered the equation in the second power; 
 for instance, the unknown x enters the equation thus, 
 
 x 2 = 12 — 2# 2 . 
 In solving this equation in the usual manner we obtain 
 Sx 2 = 12, 
 x 2 = 4. 
 Taking the square root of both sides, 
 
 x = ±2. 
 We first obtained the value of x 2 and then took the 
 square root of this to find the value of x. The solu- 
 tion of such an equation is seen to be just as simple in 
 every respect as a simple equation where the unknown 
 did not appear as a square. But suppose that we have 
 such an equation as this : 
 
 4X 2 + Sx = 12. 
 We see that none of the processes thus far discussed 
 
 will do. We must therefore find some way of grouping 
 
 48 
 
FUNDAMENTALS OF ALGEBRA 49 
 
 x 2 and x together which will give us a single term in x 
 when we take the square root of both sides; this device 
 is called " Completing the square in x." 
 
 It consists as follows: Group together all terms in x 2 
 into a single term, likewise all terms containing x into 
 another single term. Place these on the left-hand side 
 of the equation and everything else on the right-hand 
 side of the equation. Now divide through by the 
 coefficient of x 2 . In the above equation this is 4. Hav- 
 ing done this, add to the right-hand side of the equa- 
 tion the square of one-half of the coefficient of x. If 
 this is added to one side of the equation it must like- 
 wise be added to the other side of the equation. Thus: 
 4.x 2 + 8x = 12. 
 
 Dividing through by the coefficient of x 2 , namely 4, we 
 
 have 
 
 x 2 + 2 x = 3. 
 
 Adding to both sides the square of one-half of the 
 
 coefficient of x, which is 2 in the term 2 x, 
 
 x 2 + 2X + 1 =3 + 1. 
 
 The left-hand side of this equation has now been made 
 into the perfect square of x + 1, and therefore may be 
 expressed thus: 
 
 (* + i) 2 = 4. 
 Now taking the square root of both sides we have 
 
 x + 1 = ±2. 
 
50 MATHEMATICS 
 
 Therefore, using the plus sign of 2, we have 
 
 x = 1. 
 Using the minus sign of 2 we have 
 
 x = -3. 
 The student will note that there must, in the nature of 
 the case, be two distinct and separate roots to a quad- 
 ratic equation, due to the plus and minus signs above 
 mentioned. 
 To recapitulate the preceding steps, we have : 
 
 (1) Group all the terms in x 2 and x on one side of the 
 equation alone, placing those in x 2 first. 
 
 (2) Divide through by the coefficient of x 2 . 
 
 (3) Add to both sides of the equation the square of 
 one-half of the coefficient of the x term. 
 
 (4) Take the square root of both sides (the left-hand 
 side being a perfect square). Then solve as for a simple 
 equation in x. 
 
 Example: Solve for * in the following equation: 
 4 x 2 = 56 — 20 x, 
 4 x 2 + 20 x = 56, 
 x 2 + 5 x = 14, 
 
 x 2 + 5X + ^ =14 + ^, 
 4 4 
 
 x? + $x + ^- = —, 
 4 4 
 
FUNDAMENTALS OF ALGEBRA 51 
 
 Taking the square root of both sides we have 
 
 2 2 
 
 2 2 
 
 a; = 2 or —7. 
 
 Example: Solve for * in the following equation: 
 2a^-4a; + 5=a5 2 + 2a;-io-3« 2 + 33, 
 2 a? — a; 2 + 33? — 4a; — 2 a; = 33 — 10 — 5, 
 4 a? — 6 a; = 18, 
 
 x 2 - 
 
 4 
 
 4 ' 
 
 
 
 x 2 - 
 
 2 
 
 18 
 = — i 
 
 4 
 
 
 
 x 2 - 
 
 2 16 
 
 l 8 + 
 
 4 
 
 £ 
 
 (.. 
 
 4/ 
 
 16 
 
 & 
 
 
 (,- 
 
 4/ 
 
 81 
 
 
 
 * - 
 
 .2 = 
 4 
 
 ±2, 
 4 
 
 
 
 re = 
 
 = ±2 
 4 
 
 4 
 
 
 
 * = +3 or - 1| ■ 
 
52 MATHEMATICS 
 
 Solving an Equation which Contains a Root. — Fre- 
 quently we meet with an equation which contains a 
 square or a cube root. In such cases it is necessary to 
 get rid of the square or cube root sign as quickly as 
 possible. To do this the root is usually placed on one 
 side of the equation by itself, and then both sides are 
 squared or cubed, as the case may be, thus : 
 
 Example: Solve the equation 
 
 V2 x + 6 + 5 a = 10. 
 Solving for the root, we have 
 
 V2 x + 6 = 10 — 5 a. 
 Now squaring both sides we have 
 
 2 x + 6 = 100 — 100 a + 25 a 2 , 
 or, 2 x = 25 a 2 — 100 a + 100 — 6, 
 
 _ 25 a 2 — 100 a + 94 
 
 2 
 
 In any event, our prime object is first to get the square- 
 root sign on one side of the equation by itself if possible, 
 so that it may be removed by squaring. 
 
 Or the equation may be of the type 
 
 2S + I = . 4 
 
 Vfl — X 
 
 Squaring both sides we have 
 
 21 1 *6 
 
 a — x 
 
FUNDAMENTALS OF ALGEBRA 53 
 
 Clearing fractions we have 
 
 — 4<i 2 x — 4<ix — x + 4a 3 + 4a 2 + a = 16, 
 
 — x (4 a 2 + 4 a + 1) = —4 a 3 — 4 a 2 — a + 16, 
 
 _ 4 a 3 + 4 a 2 + ^ — 16 
 4 a 2 + 4 a + 1 
 
 PROBLEMS 
 
 Solve the following equations for the value of x : 
 
 1. 5 a; 2 — 15 x = —10. 
 
 2. 3 x 2 + 4 x + 20 = 44. 
 
 3. 23? + 11 = x 2 + 4X + 7. 
 
 4. jc 2 + 4se = 2x + 2x 2 — 8. 
 
 5. 73: + 15 - x 2 = 32; + 18. 
 
 6. jc 4 + 2 x 2 = 24. 
 
 7. x 2 +^ + 6x 2 = 10. 
 
 a 
 
 1 * 4 x 2 , 2 a; 
 
 9. 14 + 6 3; = = 1- ■ — — 7. 
 
 7 2 a 
 
 
 3? 
 
 
 10. 
 
 ^ # — 2. 
 
 a + 6 ^ 
 
 
 II. 
 
 3 3? + 5 3; - 15 = 0. 
 
 
 12. 
 
 (* + 2) 2 + 2 (* + 2) 
 
 = —1 
 
 i3- 
 
 (3; - 3) 2 - 103; + 7 = 
 
 = 0. 
 
 14. 
 
 (* - a) 2 - (* + a) 2 = 
 
 x+a x+b _ n 
 
 = 3- 
 
 3: — a 
 
 -b 
 
54 54 MATHEMATICS 
 
 l6 3* + 7 X + 2 _ 12 
 
 2 6 x + I 
 
 „ a? — 2 a; + 3 + 2 a; 
 ' ax 8 
 
 18. S? - X ~ I = ^ + 6 . 
 
 4 
 
 o 64 
 
 19. 8 = 
 
 Vx + 
 
 20. Va; + a + 10 a = 15. 
 
 21 
 a 
 
 X =Vx~+7. 
 
 22. 3 * + 5 = 2 + V 3 a: + 4. 
 
CHAPTER IX 
 VARIATION 
 
 This is a subject of the utmost importance in the 
 mathematical education of the student of science. It 
 is one to which, unfortunately, too little attention is 
 paid in the average mathematical textbook. Indeed, 
 it is not infrequent to find a student with an excellent 
 mathematical training who has but vaguely grasped 
 the notions of variation, and still it is upon variation 
 that we depend for nearly every physical law! 
 
 Fundamentally, variation means nothing more than 
 finding the constants which connect two mutually 
 varying quantities. Let us, for .instance, take wheat 
 and money. We know in a general way that the more 
 money we have the more wheat we can purchase. 
 This is a variation between wheat and money. But we 
 can go no further in determining exactly how many 
 bushels of wheat a certain amount of money will buy 
 before we establish some definite constant relation 
 between wheat and money, namely, the price per 
 bushel of wheat. This price is called the Constant of 
 the variation. Likewise, whenever two quantities are 
 varying together, the movement of one depending 
 absolutely upon the movement of the other, it is im- 
 
 S5 
 
56 MATHEMATICS 
 
 possible to find out exactly what value of one corre- 
 sponds with a given value of the other at any time, 
 unless we know exactly what constant relation subsists 
 between the two. 
 
 Where one quantity, a, varies as another quantity, 
 namely, increases or decreases in value as another quan- 
 tity, b, we represent the fact in this manner : 
 
 a cc b. 
 
 Now, wherever we have such a relation we can immedi- 
 ately write 
 
 a = some constant X b, 
 
 a = KXb. 
 
 If we observe closely two corresponding values of a and 
 b, we can substitute them in this equation and find out 
 the value of this constant. This is the process which 
 the experimenter in a laboratory has resorted to in de- 
 ducing all the laws of science. 
 
 Experimentation in a laboratory will enable us to 
 determine, not one, but a long series of corresponding 
 values of two varying quantities. This series of values 
 will give us an idea of the nature of their variation. 
 We may then write down the variation as above shown, 
 and solve for the constant. This constant establishes 
 the relation between a and b at all times, and is there- 
 fore all-important. Thus, suppose the experimenter in 
 a laboratory observes that by suspending a weight of 
 
VARIATION 57 
 
 ioo pounds on a wire of a certain length and size it 
 stretched one-tenth of an inch. On suspending 200 
 pounds he observes that it stretches two-tenths of an 
 inch. On suspending 300 pounds he observes that it 
 stretches three-tenths of an inch, and so on. He at 
 once sees that there is a constant relation between the 
 elongation and the weight producing it. He then 
 writes : 
 
 Elongation <* weight. 
 
 Elongation = some constant X weight. 
 
 E = K X W. 
 
 Now this is an equation. Suppose we substitute one of 
 the sets of values of elongation and weight, namely, 
 
 .3 of an inch and 300 lbs. 
 We have .3 = K X 300. 
 
 Therefore K = .001. 
 
 Now, this is an absolute constant for the stretch of that 
 wire, and if at any time we wish to know how much a 
 certain weight, say 500 lbs., will stretch that wire, we 
 simply have to write down the equation; 
 
 E = K X W. 
 Substituting elong. = .001 X 500, 
 and we have elong. = .5 of an inch. 
 
 Thus, in general, the student will remember that where 
 two quantities vary as each other we can change this 
 variation, which cannot be handled mathematically, 
 
58 MATHEMATICS 
 
 into an equation which can be handled with absolute 
 definiteness and precision by simply inserting a con- 
 stant into the variation. 
 
 Inverse Variation. — Sometimes we have one quantity 
 increasing at the same rate that another decreases; 
 thus, the pressure on a certain amount of air increases 
 as its volume is decreased, and we write 
 
 i 
 
 V oc -, 
 P 
 
 then v = KX~- 
 
 P 
 
 Wherever one quantity increases as another decreases, 
 we call this an inverse variation, and we express it in the 
 manner above shown. Frequently one quantity varies 
 as the square or the cube or the fourth power of the 
 other ; for instance, the area of a square varies as the 
 square of its side, and we write 
 
 A « b\ 
 or, A = Kb 2 . 
 
 Again, one quantity may vary inversely as the square of 
 the other, as, for example, the intensity of light, which 
 varies inversely as the square of the distance from its 
 source, thus: 
 
 , i 
 
 A oc —, 
 
 A = Kj 2 - 
 
VARIATION 
 
 59 
 
 Grouping of Variations. — Sometimes we have a 
 quantity varying as one quantity and also varying as 
 another quantity. In such cases we may group these 
 two variations into a single variation. Thus, we say 
 that 
 
 a <x b, 
 
 also a oc c, 
 
 then a a 5 X c, 
 
 or, a = KXbXc. 
 
 This is obviously correct; for, suppose we say that the 
 weight which a beam will sustain in end-on compression 
 varies directly as its width, also directly as its depth, 
 we see at a glance that the weight will vary as the 
 cross-sectional area, which is the product of the width 
 by the depth. 
 Sometimes we have such variations as this : 
 
 a oc b, 
 
 also a oc -j 
 
 c 
 
 then a oc -• 
 
 c 
 
 This is practically the same as the previous case, with 
 the exception that instead of two direct variations we 
 have one direct and one inverse variation. 
 
 There is much interesting theory in variation, which, 
 however, is unimportant for our purposes and which 
 
60 MATHEMATICS 
 
 I will therefore omit. If the student thoroughly mas- 
 ters the principles above mentioned he will find them 
 of . inestimable value in comprehending the deduction 
 of scientific equations. 
 
 PROBLEMS 
 
 i. If a oc b and we have a set of values showing that 
 when a — 500, b = 10, what is the constant of this 
 variation ? 
 
 2. If a oc b 2 , and the constant of the variation is 
 2205, what is the value of b when = 5? 
 
 3. a oc b; also a « -. or, a oc -. If we find that 
 
 c' c 
 
 when a = 100, then 6 = 5 and c = 3, what is the con- 
 stant of this variation ? 
 
 4. a oc be. The constant of the variation equals 12. 
 What is the value of a when 6 = 2 and c = 8 ? 
 
 5. 0= KX-. If K = 15 and a = 6 and 6 = 2. 
 
 c 
 
 what is the value of c ? 
 
CHAPTER X 
 
 SOME ELEMENTS OF GEOMETRY 
 
 In this chapter I will attempt to explain briefly some 
 elementary notions of geometry which will materially 
 aid the student to a thorough understanding of many 
 physical theories. At the start let us accept the fol- 
 lowing axioms and definitions of terms which we will 
 employ. 
 
 Axioms and Definitions : 
 
 I. Geometry is the science of space. 
 
 II. There are only three fundamental directions or 
 dimensions in space, namely, length, breadth and depth. 
 
 III. A geometrical point has theoretically no dimen- 
 sions. 
 
 IV. A geometrical line has theoretically only one 
 dimension, — length. 
 
 V. A geometrical surface or plane has theoretically 
 only two dimensions, namely, length and breadth. 
 
 VI. A geometrical body occupies space and has three 
 dimensions, — length, breadth and depth. 
 
 VII. An angle is the opening or divergence between 
 
 two straight lines which cut or intersect each other; 
 
 thus, in Fig. i, 
 
 61 
 
62 
 
 MATHEMATICS 
 
 Fig i. 
 4- a is an angle between the lines AB and CD, and may 
 be expressed thus, 4- a or 4- BOD. 
 
 VIII. When two lines lying in the same surface or 
 plane are so drawn that they never approach or retreat 
 from each other, no matter how long they are actually 
 extended, they are said to be parallel; thus, in Fig. 2, 
 A B 
 
 Fig. 2. 
 the lines AB and CD are parallel. 
 
 IX. A definite portion of a surface or plane bounded 
 by lines is called a polygon; thus, Fig. 3 shows a polygon. 
 
 Fig. 3- 
 
SOME ELEMENTS OF GEOMETRY 
 
 63 
 
 X. A polygon bounded by three sides is called a 
 triangle (Fig. 4). 
 
 Fig. 4. 
 
 XI. A polygon bounded by four sides is called a 
 quadrangle (Fig. 5), and if the opposite sides are parallel, 
 a parallelogram (Fig. 6). 
 
 Fig. 5- 
 
 Fig. 6. 
 
 XII. When a line has revolved about a point until it 
 has swept through a com- 
 plete circle, or 360 , it 
 comes back to its original 
 position. When it has re- 
 volved one quarter of a 
 circle, or 90 , away from its 
 original position, it is said 
 to be at right angles or 
 perpendicular to its original 
 position; thus, the angle a (Fig. 7) is a right a. 
 
6 4 
 
 MATHEMATICS 
 
 between the lines AB and CD, which are perpendicular 
 to each other. 
 
 XIII. An angle less than a right angle is called an 
 acute angle. 
 
 XIV. An angle greater than a right angle is called an 
 obtuse angle. 
 
 XV. The addition of two right angles makes a straight 
 line. 
 
 XVI. Two angles which when placed side by side or 
 added together make a right angle, or 90 , are said to 
 be complements of each other; thus, 4- 30 and 4- 6o° are 
 complementary angles. 
 
 i XVII. Two angles which when added together form 
 180 , or a straight line, are said to be supplements of 
 each other; thus, £ 130 and 4- 50 are supplementary 
 angles. 
 
 XVIII. When one of the inside angles of a triangle is 
 a right angle, it is called a right-angle triangle (Fig. 8), 
 
 Fig. 8. 
 
 and the side AB opposite the right angle is called its 
 hypothenuse. 
 
SOME ELEMENTS OF GEOMETRY 
 
 65 
 
 XIX. A rectangle is a parallelogram whose angles 
 are all right angles (Fig. 9a), and a square is a rectangle 
 whose sides are all equal (Fig. 9). 
 
 Fig. 9. 
 
 Fig. 9a. 
 
 XX. A circle is a curved line, all points of which are 
 equally distant or equidistant from a fixed point called a 
 center (Fig. 10). 
 
 Fig. 10. 
 
 With these as- 
 sumptions we may 
 now proceed. Let Fl s- "• 
 
 us look at Fig. n. BM and CN are parallel lines cut 
 by the common transversal or intersecting line RS. • It 
 is seen at a glance that the 4- ROM and £ BOA, 
 called vertical angles, are equal; likewise 4- ROM and 
 
66 MATHEMATICS 
 
 4 RAN, called exterior interior angles, are equal; 
 likewise 4 BOA and 4 RAN, called opposite in- 
 terior angles, are equal. These facts are actually 
 proved by placing one on the other, when they will 
 coincide exactly. The 4 ROM and 4 BOR are 
 supplementary, as their sum forms the straight line 
 BM, or i8o°. Likewise 4 ROM and 4 MOS, or 
 4 NAS, are supplementary. 
 
 In general, we have this rule : When the corresponding 
 sides of any two angles are parallel to each other, the angles 
 are either equal or supplementary. 
 
 Triangles. — Let us now investigate some of the prop- 
 erties of the triangle ABC (Fig. 12). Through A 
 
 Fig. 12. 
 draw a line, MN, parallel to BC. At a glance we see 
 that the sum of the angles a, d, and e is equal to 180 , 
 or two right angles, — 
 
 4 a + 4d+ 4e = 180 . 
 But 4 c is equal to 4 d, and 4 b is equal to 4 e, as 
 previously seen; therefore we have 
 
 4a+ 4c+ 4b= 180 . 
 
SOME ELEMENTS OF GEOMETRY 67 
 
 This demonstration proves the fact that the sum of 
 all the inside or interior angles of any triangle is equal 
 to 180 , or, what is the same thing, two right angles. 
 Now, if the triangle is a right triangle and one of its 
 angles is itself a right angle, then the sum joJ the two 
 remaining angles must be equal to one right angle, or 90 . 
 This fact should be most carefully noted, as it is very 
 important. 
 
 When we have two triangles with all the angles of 
 the one equal to the corresponding angles of the other, 
 as in Fig. 13, they are called similar triangles. 
 
 Fig. 13- 
 
 When we have two triangles with all three sides of 
 the one equal to the corresponding sides of the other, 
 they are equal to each other (Fig. 14), for they may be 
 
 Fig. 14. 
 
 perfectly superposed on each other. In fact, the two 
 triangles are seen to be equal if two sides and the in- 
 
68 
 
 MATHEMATICS 
 
 eluded angle of the one are equal to two sides and the 
 included angle of the other; or, if one side and two 
 angles of the one are equal to one side and the corre- 
 sponding angles respectively of the other; or, if one side 
 and the angle opposite to it of the one are equal to one 
 side and the corresponding angle of the other. 
 
 Projections. — The projection of any given tract, such 
 as AB (Fig. 15), upon a line, such as MN, is that space, 
 
 tL 
 
 H 
 
 Fig. 15- 
 CD, on the line MN bounded by two lines drawn 
 from A and B respectively perpendicular to MN. 
 
 Rectangles and Parallelograms. — A line drawn be- 
 tween opposite corners of a parallelogram is called a 
 diagonal; thus, AC is a diagonal in Fig. 16. It is along 
 
 B C 
 
 Fig. 16. 
 
 this diagonal that a body would move if pulled in the 
 direction of AB by one force, and in the direction AD 
 by another, the two forces having the same relative 
 
SOME ELEMENTS OF GEOMETRY 
 
 6 9 
 
 magnitudes as the relative lengths of AB and AD. 
 This fact is only mentioned here as illustrative of one 
 of the principles of mechanics. 
 
 Fig. 17. 
 
 The area of a rectangle is equal to the product of the 
 length by the breadth; thus, in Fig. 17, 
 
 Area of ABDC = AB X AC. 
 
 This fact is so patent as not to need explanation. 
 
 Suppose we have a parallelogram (Fig. 18), however, 
 what is its area equal to ? 
 
 Fig. 18. 
 
 The perpendicular distance BF between the sides BC 
 and AD of a parallelogram is called its altitude. Ex- 
 tend the base AD and draw CE perpendicular to it. 
 
7° 
 
 MATHEMATICS 
 
 Now we have the rectangle BCEF, whose area we know 
 to be equal to BC X BF. But the triangles ABF and 
 DCE are equal (having 2 sides and 2 angles mutually- 
 equal), and we observe that the rectangle is nothing else 
 than the parallelogram with the triangle ABF chipped 
 off and the triangle DCE added on, and since these are 
 equal, the rectangle is equal to the parallelogram, which 
 then has the same area as it; or, 
 
 Area of parallelogram A BCD = BC X BF. 
 
 If, now, we consider the area of the triangle ABC 
 (Fig. 19), we see that by drawing the lines AD and CD 
 
 D 
 
 parallel to BC and AB respectively, we have the parallel- 
 ogram BA DC, and we observe that the triangles ABC 
 and A DC are equal. Therefore triangle ABC equals 
 one-half of the parallelogram, and since the area of this 
 is equal to BC X AH, then the 
 
 Area of the triangle ABC = %BCX AH, 
 which means that the area of a triangle is equal to 
 one-half of the product of the base by the altitude. 
 
SOME ELEMENTS OF GEOMETRY 
 
 71 
 
 Circles. — Comparison between the lengths of the 
 diameter and circumference of a circle (Fig. 20) made 
 with the utmost care shows that 
 the circumference is 3.1416 times as 
 long as the diameter. This con- 
 stant, 3.1416, is usually expressed 
 by the Greek letter pi (ir). There- 
 fore, the circumference of a circle is 
 equal to ir X the diameter. Fig. 20. 
 
 circum. = ird, 
 circum. = 2 irr 
 
 if r , the radius, is used instead of the diameter. 
 
 The area of a segment of a circle (Fig. 21), like the 
 area of a triangle, is equal to \ of the product of the 
 base by the altitude, or \a Y. r. 
 This comes from the fact that the 
 segment may be divided up into a 
 very large number of small segments 
 whose bases, being very small, have 
 very little curvature, and may there- 
 fore be considered as small triangles. Therefore, if we 
 consider the whole circle, where the length of the arc 
 is 2 wr, the area is 
 
 5 X 2 irr X r = irr 2 , 
 Area circle = irr 2 - 
 
7 2 
 
 MATHEMATICS 
 
 I will conclude this chapter by a discussion of. one of 
 the most important properties of the right-angle tri- 
 angle, namely, that the~square erected on its hypothe- 
 nuse is equal to the sum of the squares erected on 
 its other two sides ; that is, that in the triangle ABC 
 (Fig. 22) AC 2 = AB* + BC\ 
 
 /I 
 
 
 1 
 / 
 
 / 
 
 1 
 1 
 
 1 
 
 1 
 
 •', 
 
 \ 
 \ 
 \ 
 \ 
 
 \ 
 
 \ 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 Fig. 22. 
 
 To prove ANMC = BCRS + ABHK, 
 
 or length AC = length BC + lengths^S*. 
 
 This is a difficult problem and one of the most interest- 
 ing and historic ones that the whole realm of mathe- 
 matics can offer, therefore I will only suggest its solu- 
 
SOME ELEMENTS OF GEOMETRY 73 
 
 tion and leave a little reasoning for the student himself 
 to do. 
 
 triangle ARC = triangle BMC, 
 
 triangle ARC = %CRxBC 
 
 •= % of the square BCRS, 
 
 triangle BCM = %CMxCO 
 
 = \ of rectangle COFM . 
 Therefore 
 
 § of square 5Cii5 = \ of rectangle COFAf , 
 
 or BCRS = COFM. 
 
 Similarly for the other side 
 
 ABHK = AOFN. 
 But 
 
 COFM + AOFN = whole square ACMN. 
 Therefore ACMN = BCRS + ABHK. 
 AC 2 = BC 2 + AB\ 
 
 PROBLEMS 
 
 i. What is the area of a rectangle 8 ft. long by 12 
 ft. wide ? 
 
 2. What is the area of a triangle whose base is 20 ft. 
 and whose altitude is 18 ft.? 
 
 3. What is the area of a circle whose radius is 9 ft.? 
 
 4. What is the length of the hypothenuse of a right- 
 angle triangle if the other two sides are respectively 
 6 ft. and 9 ft.? 
 
74 MATHEMATICS 
 
 5. What is the circumference of a circle whose diame- 
 ter is 20 ft.? 
 
 6. The hypothenuse of a right-angle triangle is 25 ft. 
 and one side is 18 ft.; what is the other side ? 
 
 7. If the area of a circle is 600 sq. ft., what is its 
 diameter ? 
 
 8. The circumference of the earth is 25,000 miles; 
 what is its diameter in miles ? 
 
 9. The area of a triangle is 30 sq. ft. and its base is 
 8 ft. ; what is its altitude ? 
 
 10. The area of a parallelogram is 100 sq. feet and 
 its base is 25 ft.; what is its altitude? 
 
CHAPTER XI 
 
 ELEMENTARY PRINCIPLES OF TRIGONOMETRY 
 
 Trigonometry is the science of angles; its province is 
 to teach us how to measure and employ angles with the 
 same ease that we handle lengths and areas. 
 
 In a previous chapter we have defined an angle as the 
 opening or the divergence between two intersecting 
 lines, AB and CD (Fig. 23). The next question is, How 
 
 Fig. 23. 
 
 are we going to measure this angle ? We have already 
 seen that we can do this in one way by employing 
 degrees, a complete circle being 360 . But there are 
 many instances which the student will meet later on 
 where the use of degrees would be meaningless. It 
 is then that certain constants connected with the angle, 
 called its functions, must be resorted to. Suppose we 
 have the angle a shown in Fig. 24. Now let us choose 
 
 75 
 
76 
 
 MATHEMATICS 
 
 a point anywhere either on the line AB or CD ; for 
 instance, the point P. From P drop a line which will 
 
 Fig. 24. 
 
 be perpendicular to CD. This gives us a right-angle 
 triangle whose sides we may call a, b and c respectively. 
 We may now define the following functions of the 4- a : 
 
 a 
 sine a = -> 
 
 cosme a = -1 
 c 
 
 tangent a = -> 
 
 which means that the sme of an angle is obtained by 
 dividing the side opposite to it by thehypothenuse; the 
 cosine, by dividing the side adjacent to it by the hypo- 
 thenuse; and the tangent, by dividing the side opposite 
 by the side adjacent. 
 
 These values, sine, cosine and tangent, are therefore 
 nothing but ratios, — pure numbers, — and under no dr- 
 
ELEMENTARY PRINCIPLES OF TRIGONOMETRY 77 
 
 cumstances should be taken for anything else. This is 
 one of the greatest faults that I have to find with many 
 texts and handbooks in not insisting on this point. 
 
 Looking at Fig. 24, it is evident that no matter where 
 I choose P, the values of the sine, cosine and tangent 
 will be the same; for if I choose P farther out on the 
 line I will increase c, but at the same time a will increase 
 
 in the same proportion, the quotient of - being always 
 
 c 
 
 the same wherever P may be chosen. 
 Likewise - and - will always remain constant. The 
 
 CO 
 
 sine, cosine and tangent are therefore always fixed and 
 constant quantities for any given angle. I might have 
 remarked that if P had been chosen on the line CD and 
 the perpendicular drawn to AB, as shown by the dotted 
 lines (Fig. 24), the hypothenuse and adjacent side simply 
 exchange places, but the value of the sine, cosine and 
 tangent would remain the same. 
 
 Since these functions, namely, sine, cosine and tan- 
 gent, of any angle remain the same at all times, they 
 become very convenient handles for employing the 
 angle. The sines, cosines and tangents of all angles of 
 every size may be actually measured and computed 
 with great care once and for all time, and then arranged 
 in tabulated form, so that by referring to this table one 
 can immediately find the sine, cosine or tangent of any 
 angle; or, on the other hand,, if a certain value said to 
 
78 MATHEMATICS 
 
 be the sine, cosine or tangent of an unknown angle is 
 given, the angle that it corresponds to may be found 
 from the table. Such a table may be found at the end 
 of this book, giving the sines, cosines and tangents of 
 all angles taken 6 minutes apart. Some special compi- 
 lations'of these tables give the values for all angles taken 
 only one minute apart, and some even closer, say 10 sec- 
 onds apart. 
 
 On reference to the table, the sine of io° is .1736, the 
 cosine of io° is .9848, the sine of 24 36' is .4163, the co- 
 sine of 24 36' is .9092. In the table of sines and cosines 
 the decimal point is understood to be before every value, 
 for, if we refer back to our definition of sine and cosine, 
 we will see that these values can never be greater than 
 1; in fact, they will always be less than 1, since the 
 hypothenuse c is always the longest side of the right 
 angle and therefore a and b are always less than it. 
 
 Obviously, - and -> the values respectively of sine and 
 c c 
 
 cosine, being a smaller quantity divided by a larger, 
 can never be greater than 1. Not so with the tangent; 
 for angles between o° and 45 , a is less than b, 
 therefore - is less than 1 ; but for angles between 45° 
 
 and 90 , a is greater than b, and therefore - is greater 
 
 
 
 than 1. Thus, on reference to the table the tangent 
 of io° 24' is seen to be .1835, the tangent of 45 isi, 
 the tangent of 6o° 30' is 1.7675. 
 
ELEMENTARY PRINCIPLES OF TRIGONOMETRY 79 
 
 Now let us work backwards. Suppose we are given 
 •3437 as the sine of a certain angle, to find the angle. 
 On reference to the table we find that this is the sine 
 of 20°6', therefore this is the angle sought. Again, 
 suppose we have .8878 as the cosine of an angle, to find 
 the angle. On reference to the table we find that this 
 is the angle 27 24'. Likewise suppose we are given 
 3.5339 as the tangent of an angle, to find the angle. 
 The tables show that this is the angle 74 12'. 
 
 When an angle or value which is sought cannot be 
 found in the tables, we must prorate between the next 
 higher and lower values. This process is called inter- 
 polation, and is merely a question of proportion. It will 
 be explained in detail in the chapter on Logarithms. 
 
 Relation of Sine and Cosine. — On reference to Fig. 
 
 2 q we see that the sine a = - ; 
 
 c 
 
 but if we take 0, the other 
 acute angle of the right-angle 
 triangle, we see that cosine 
 
 Remembering; always the fundamental definition of 
 
 sine and cosine, namely, 
 
 Opposite side 
 
 sme = rr . 
 
 Hypothenuse 
 
 Adjacent side 
 
 cosine = =-* — 7: > 
 
 Hypothenuse 
 
80 MATHEMATICS 
 
 we see that the cosine jS is equal to the same thing as 
 the sine a, therefore 
 
 sine a = cosine /3. 
 
 Now, if we refer back to our geometry, we will re- 
 member that the sum of the three angles of a triangle 
 = 180 , or two right angles, and therefore in a right- 
 angle triangle 4 a- + 4 P = 90 , or 1 right angle. In 
 other words 4 a and 4 P are complementary angles. 
 We then have the following general law: "The sine of 
 an angle is equal to the cosine of its complement." Thus, 
 if we have a table of sines or cosines from o° to 90 , or 
 sines and cosines between o° and 45 , we make use of 
 this principle. If we are asked to find the sine of 68° 
 we may look for the cosine of (90 — 68°), or 22 ; or, if 
 we want the cosine of 68°, we may look for the sine of 
 (90 - 68°), or 22 . 
 
 Other Functions. — There are some other functions 
 of the angle which are seldom used, but which I will 
 mention here, namely, 
 
 Cotangent = -> 
 a 
 
 Secant = - > 
 
 
 Cosecant = -• 
 a 
 
 Other Relations of Sine and Cosine. — We have seen 
 
ELEMENTARY PRINCIPLES OF TRIGONOMETRY 8 1 
 
 7 
 
 that the sine a= - and the cosine a=-. Also from 
 c c 
 
 geometry 
 
 ffl 2 + tf = C 2. ( T ) 
 
 Dividing equation (i) by c 2 we have 
 a 2 , 5 2 
 
 But this is nothing but the square of the sine plus the 
 square of the cosine of £ a, therefore 
 
 (sine a) 2 + (cosine a) 2 = i . 
 Other relations whose proof is too intricate to enter 
 into now are sine 2 « = 2 sin a cos a, 
 
 cos 2 a = 1 — 2 Sin - * a, 
 
 or 
 
 COS 2 a 
 
 cos' a 
 
 2 r* - 
 
 Use of Trigonometry. — Trigonometry is invaluable 
 in triangulation of all kinds. When two sides or one 
 side and an acute angle of a right-angle triangle are 
 A s b 
 
 Fig. 26. 
 
 given, the other two sides can be easily found. Sup- 
 pose we wish to measure the distance BC across the 
 river in Fig. 26 ; we proceed as follows : First we lay off 
 
82 
 
 MATHEMATICS 
 
 and measure the distance AB along the shore; then by 
 means of a transit we sight perpendicularly across the 
 river and erect a flag at C; then we sight from A to B 
 and from A to C and determine the angle a. Now, as 
 before seen, we know that 
 
 tangent a = - ■ 
 b 
 
 Suppose b had been iooo ft. and £ a was 40 , then 
 
 tangent 40 = 
 
 IOOO 
 
 The tables show that the tangent of 40 is .8391; 
 
 then 
 
 .8391 = 
 
 1000 
 
 therefore a = 839.1 ft. 
 
 Thus we have found the distance across the river to be 
 839.1 ft. 
 
 Likewise in Fig. 27, suppose c = 300 and £ a= 36 , 
 to find a and b. We have 
 
 sine a = -j 
 c 
 
 or 
 
 sine 36 = 
 
 300 
 
ELEMENTARY PRINCIPLES OF TRIGONOMETRY 83 
 
 From the tables sine 36 = .5878. 
 
 
 .5878 = — . 
 300 
 
 
 a = .5878 X 300, 
 
 or 
 
 a = 176.34 ft. 
 
 Likewise 
 
 b 
 cosine a = - ■ 
 c 
 
 From table, 
 
 cosine 36' = • .8090, 
 
 therefore 
 
 .8090 = > 
 
 300 
 
 or 
 
 b =242.7 ft. 
 
 Now, if we had been told that a = 225 and b = 100, to 
 find £ a and c, we would have proceeded thus : 
 
 tangent a = - • 
 
 
 Therefore tangent a = ■ — a i 
 
 100 
 
 tangent a = 2.25 ft. 
 
 The tables show that this corresponds to the angle 
 
 66° 4'. 
 
 Therefore a = 66° 4'. 
 
 Now to find c we have 
 
 a 
 c 
 
 sin 66° 4' = ^. 
 c 
 
84 MATHEMATICS 
 
 From tables, sine 66° 4' = .9140, 
 
 22 c 
 therefore .9140 = — >*> 
 
 c 
 
 22 c 
 or c = ■ — 2 - = 248.5 ft. 
 
 .9140 
 
 And thus we may proceed, the use of a little judgment 
 being all that is necessary to the solution of the most 
 difficult problems of triangulation. 
 
 PROBLEMS 
 
 i. Find the sine, cosine and tangent of 32° 20'. 
 
 2. Find the sine, cosine and tangent of 8i° 24'. 
 
 3. What angle is it whose sine is .4320 ? 
 
 4. What angle is it whose cosine is .1836 ? 
 
 5. What angle is it whose tangent is .753 ? 
 
 6. What angle is it whose cosine is .8755 ? 
 
 In a right-angle triangle — 
 
 7. If a =300 ft. and t a. = 30 , what are c and b ? 
 
 8. If a = 500 ft. and b = 315 ft., what are %■ a and c ? 
 
 9. If c = 1250 ft. and £ a = 8o°, what are b and a ? 
 10. If b = 250 ft. and c = 530 ft., what are % a 
 
 and al 
 
CHAPTER XII 
 LOGARITHMS 
 
 I have inserted this chapter on logarithms because 
 I consider a knowledge of them very essential to the 
 education of any engineer. 
 
 Definition. — A logarithm is the power to which we 
 must raise a given base to produce a given number. 
 Thus, suppose we choose 10 as our base, we will say 
 that 2 is the logarithm of ioo, because we must raise 10 to 
 the second power — in other words, square it — in order 
 to produce ioo. Likewise 3 is the logarithm of 1000, 
 for we have to raise 10 to the third power (thus, io 3 ) to 
 produce 1000. The logarithm of 10,000 would then be 
 4, and the logarithm of 100,000 would be 5, and so on. 
 
 The base of the universally used Common System of 
 logarithms is 10; of the Naperian or Natural System, e 
 or 2.7. The latter is seldom used. 
 
 We see that the logarithms of such numbers as 100, 
 1000, 10,000, etc., are easily detected; but suppose we 
 have a number such as 300, then the difficulty of find- 
 ing its logarithm is apparent. We have seen that io 2 
 is 100, and io 3 equals 1000, therefore the number 300, 
 which lies between 100 and 1000, must have a logarithm 
 which lies between the logarithms of 100 and 1000, 
 
 8s 
 
86 MATHEMATICS 
 
 namely 2 and 3 respectively. Reference to a table of 
 logarithms at the end of this book, which we will ex- 
 plain later, shows that the logarithm of 300 is 2.4771, 
 which means that 10 raised to the 2.477iths power will 
 give 300. The whole number in a logarithm, for exam- 
 ple the 2 in the above case, is called the characteristic; 
 the decimal part of the logarithm, namely, .4771, is 
 called the mantissa. It will be hard for the student to 
 understand at first what is meant by raising 10 to a 
 fractional part of a power, but he should not worry 
 about this at the present time; as he studies more 
 deeply into mathematics the notion will dawn on him 
 more clearly. 
 
 We now see that every number has a logarithm, no 
 matter how large or how small it may be; every number 
 can be produced by raising 10 to some power, and this 
 power is what we call the logarithm of the number. 
 Mathematicians have carefully worked out and tabu- 
 lated the logarithm of every number, and by reference 
 to these tables we can find the logarithm corresponding 
 to any number, or vice versa. A short table of loga- 
 rithms is shown at the end of this book. 
 
 Now take the number 351.1400; we find its logarithm 
 is 2.545,479. Like all numbers which lie between 100 
 and 1000 its characteristic is 3. The numbers which 
 lie between 1000 and 10.000 have 3 as a characteristic; 
 between 10 and 100, 1 as a characteristic. We there- 
 
LOGARITHMS 87 
 
 fore have the rule that the characteristic is always one 
 less than the number of places to the left of the decimal 
 point. Thus, if we have the number 31875.12, we 
 immediately see that the characteristic of its logarithm 
 will be 4, because there are five places to the left of the 
 decimal point. Since it is so easy to detect the char- 
 acteristic, it is never put in logarithmic tables, the man- 
 tissa or decimal part being the only part that the tables 
 need include. 
 
 If one looked in a table for a logarithm of 125.60, he 
 would only find .09,899. This is only the mantissa of 
 the logarithm, and he would himself have to insert the 
 characteristic, which, being one less than the number of 
 places to the left of the decimal point, would in this 
 case be 2 ; therefore the logarithm of 125.6 is 2.09,899. 
 
 Furthermore, the mantissa of the logarithms of 
 3.4546, 34.546, 34546, 3454-6, etc., are all exactly the 
 same. The characteristic of the logarithm is the only 
 thing which the decimal point changes, thus : 
 
 log 3-4S46 = 0.538,398, 
 log 34-546 = 1.538,398, 
 log 34546 = 2.538,398, 
 log 3454-6 = 3-53 8 .398, 
 etc. 
 
 Therefore, in looking for the logarithm of a number, 
 first put down the characteristic on the basis of the 
 
88 MATHEMATICS 
 
 above rules, then look for the mantissa in a table, 
 neglecting the position of the decimal point altogether. 
 Thus, if we are looking for the logarithm of .9840, we 
 first write down the characteristic, which in this case 
 would be — 1 (there are no places to the left of the 
 decimal point in this case, therefore one less than none 
 is — 1). Now look in a table of logarithms for the 
 mantissa which corresponds to .9840, and we find this 
 to be .993,083; therefore 
 
 log .9840 = -1.993,083. 
 
 If the number had been 98.40 the logarithm would have 
 been +1.993,083. 
 
 When we have such a number as .084, the character- 
 istic of its logarithm would be — 2, there being one 
 less than no places at all to the left of its decimal point; 
 for, even if the decimal point were moved to the right 
 one place, you would still have no places to the left of 
 the decimal point; therefore 
 
 log .00,386 = - 3.586,587, 
 
 log 38.6 =1.586,587, 
 
 log 386 =2.586,587, 
 
 log 386,000 = 5.586,587. 
 
 Interpolation. — Suppose we are asked to find the 
 logarithm of 2468; immediately write down 3 as the 
 characteristic. Now, on reference to the logarithmic 
 
LOGARITHMS 89 
 
 table at the end of this book, we see that the loga- 
 rithms of 2460 and 2470 are given, but not 2468. Thus : 
 
 log 2460 = 3.3909, 
 
 log 2468 = ? 
 
 log 2470 = 3.3927. 
 
 We find that the total difference between the two 
 given logarithms, namely 3909 and 3927, is 16, the 
 total difference between the numbers corresponding to 
 these logarithms is 10, the difference between 2460 and 
 2468 is 8; therefore the logarithm to be found lies A 
 of the distance across the bridge between the two given 
 logarithms 3909 and 3927. The whole distance across 
 is 16. A of 16 is 12.8. Adding this to 3909 we have 
 3921.8; therefore 
 
 log of 2468 = 3.39,218. 
 
 Reference to column 8 in the interpolation columns to the 
 right of the table would have given this value at once. 
 
 Many elaborate tables of logarithms may be purchased 
 at small cost which make interpolation almost unneces- 
 sary for practical purposes. 
 
 Now let us work backwards and find the number if 
 we know its logarithm. Suppose we have given the 
 logarithm 3.6201. Referring to our table, we see that 
 the mantissa .6201 corresponds to the number 417; the 
 characteristic 3 tells us that there must be four places 
 to the left of the decimal point; therefore 
 3.6201 is the log of 4170.0. 
 
QO MATHEMATICS 
 
 Now, for interpolation we have the same principles 
 aforesaid. Let us find the number whose log is — 3.7304. 
 In the table we find that 
 
 log 7300 corresponds to the number 5370, 
 log 7304 corresponds to the number ? 
 log 7308 corresponds to the number 5380. 
 
 Therefore it is evident that 
 
 7304 corresponds to 5375. 
 
 Now the characteristic of our logarithm is — 3 ; from 
 this we know that there must be two zeros to the left 
 of the decimal point; therefore 
 
 —3.7304 is the log of the number .005375. 
 
 Likewise 
 
 — 2.7304 is the log of the number .05375, 
 .7304 is the log of the number 5.375, 
 4.7304 is the log of the number 53,750. 
 
 Use of the Logarithm. — Having thoroughly under- 
 stood the nature and meaning of a logarithm, let us 
 investigate its use mathematically. It changes multi- 
 plication and division into addition and subtraction; in- 
 volution and evolution into multiplication and division. 
 
 We have seen in algebra that 
 
 a 2 X a 5 = a 5+2 , or a 7 , 
 
 a s 
 and that — = a 8-3 , or a 5 . 
 
LOGARITHMS 91 
 
 In other words, multiplication or division of like symbols 
 was accomplished by adding or subtracting their ex- 
 ponents, as the case may be. Again, we have seen that 
 
 (a 2 ) 2 = a\ 
 or V& = a 2 . 
 
 In the first case a 2 squared gives a 4 , and in the second 
 case the cube root of a 6 is a 2 ; to raise a number to a 
 power you multiply its exponent by that power; to find 
 any root of it you divide its exponent by the exponent 
 of the root. Now, then, suppose we multiply 336 by 
 5380; we find that 
 
 log of 336 = io 2 - 6263 , 
 log of 5380 = io 37308 . 
 Then 336 X 5380 is the same thing as io 26263 X io 37308 - 
 But io 2-6263 X io 3,7308 = io 2 ' 5263+3 ' 7308 = io 6-2571 . 
 
 We have simply added the exponents, remembering that 
 these exponents are nothing but the logarithms of 336 
 and 5380 respectively. 
 
 Well, now, what number is io 62571 equal to? Look- 
 ing in a table of logarithms we see that the mantissa 
 .2571 corresponds to 1808; the characteristic 6 tells us 
 that there must be seven places to the left of the decimal; 
 
 therefore 
 
 io 62571 = 1,808,000. 
 
 If the student notes carefully the foregoing he will see 
 that in order to multiply 336 by 5380 we simply find 
 
92 MATHEMATICS 
 
 their logarithms, add them together, getting another 
 logarithm, and then find the number corresponding 
 to this logarithm. Any numbers may be multiplied 
 together in this simple manner; thus, if we multiply 
 217 X 4876 X 3.185 X .0438 X 890, we have 
 
 log 217 = 2.3365 
 
 log 4876 = 3.6880 
 
 log 3.185 = .5031 
 
 log .0438 = -2.6415* 
 
 log 890 = 2.9494 
 
 Adding we get 8.1185 
 
 We must now find the number corresponding to the 
 logarithm 8. n 85. Our tables show us that 
 
 8.1185 is the log of 131,380,000. 
 
 Therefore 131,380,000 is the result of the above multi- 
 plication. 
 
 To divide one number by another we subtract the 
 logarithm of the latter from the logarithm of the 
 former; thus, 3865 4- 735: 
 
 log 3865 = 3.5872 
 
 tog 735 = 2-8663 
 
 .7209 
 
 The tables show that .7209 is the logarithm of 5.259; 
 
 therefore 
 
 3865 ■*■ 735 = 5-259- 
 
 * The — 2 does not carry its negativity to the mantissa. 
 
LOGARITHMS 93 
 
 As explained above, if we wish to square a number, we 
 simply multiply its logarithm by 2 and then rind what 
 number the result is the logarithm of. If we had 
 wished to raise it to the third, fourth or higher power, 
 we would simply have multiplied by 3, 4 or higher power, 
 as the case may be. Thus, suppose we wish to cube 
 9879 ; we have 
 
 log 9897 = 3.9947 
 
 3 
 
 1 1. 9841 
 
 1 1. 9841 is the log of 964,000,000,000; 
 therefore 9879 cubed = 964,000,000,000. 
 
 Likewise, if we wish to find the square root, the cube 
 root, or fourth root or any root of a number, we simply 
 divide its logarithm by 2, 3, 4 or whatever the root may 
 be; thus, suppose we wish to find the square root of 
 36,850, we have 
 
 log 36,850 = 4-5664- 
 
 4.5664 -s- 2 = 2.2832. 
 
 2.2832 is the log. of 191.98; therefore the square root of 
 36,850 is 191.98. 
 
 The student should go over this chapter very care- 
 fully, so as to become thoroughly familiar with the 
 principles involved. 
 
94 MATHEMATICS 
 
 PROBLEMS 
 
 i. Find the logarithm of 3872. 
 
 2. Find the logarithm of 73.56. 
 
 3. Find the logarithm of .00988. 
 
 4. Find the logarithm of 41,267. 
 
 5. Find the number whose logarithm is 2.8236. 
 
 6. Find the number whose logarithm is 4.87175. 
 
 7. Find the number whose logarithm is —1.4385. 
 
 8. Find the number whose logarithm is —4.3821. 
 
 9. Find the number whose logarithm is 3.36175. 
 
 10. Multiply 2261 by 4335. 
 
 1 1 . Multiply 62 18 by 3998. 
 
 12. ' Multiply 231.9 by 478.8 by 7613 by .921. 
 
 13. Multiply .00983 by .0291. 
 
 14. Multiply .222 by .00054. 
 
 15. Divide 27,683 by 856. 
 
 16. Divide 4337 by 38.88. 
 
 17. Divide .9286 by 28.75. 
 
 18. Divide .0428 by 1.136. 
 
 19. Divide 3995 by .003,337. 
 
 20. Find the square of 4291. 
 
 21. Raise 22.91 to the fourth power. 
 
 22. Raise .0236 to the third power. 
 
 23. Find the square root of 302,060. 
 
 24. Find the cube root of 77.85. 
 
 25. Find the square root of .087,64. 
 
 26. Find the fifth root of 226,170,000. 
 
CHAPTER XIII 
 
 ELEMENTARY PRINCIPLES OF COORDINATE GEOMETRY 
 
 Coordinate Geometry may be called graphic algebra, 
 or equation drawing, in that it depicts algebraic equa- 
 tions not by means of symbols and terms but by means 
 of curves and lines. Nothing is more familiar to the 
 engineer, or in fact to any one, than to see the results of 
 machine tests or statistics and data of any kind shown 
 graphically by means of curves. The same analogy 
 exists between an algebraic equation and the curve 
 which graphically represents it as between the verbal 
 description of a landscape and its actual photograph; 
 the photograph tells at a glance more than could be 
 said in many thousands of words. Therefore the stu- 
 dent will realize how important it is that he master the 
 few fundamental principles of coordinate geometry which 
 we will discuss briefly in this chapter. 
 
 An Equation. — When discussing equations we remem- 
 ber that where we have an equation which contains 
 two unknown quantities, if we assign some numerical 
 value to one of them we may immediately find the cor- 
 responding numerical value of the other; for example, take 
 
 the equation 
 
 x = y + 4. 
 
 95 
 
96 MATHEMATICS 
 
 In this equation we have two unknown quantities, 
 namely, x and y; we cannot find the value of either 
 unless we know the value of the other. Let us say that 
 y = 1; then we see that we would get a corresponding 
 value, x = 5; for y = 2, x = 6; thus: 
 
 If y = 1, then a; = 5, 
 y = 2, a; = 6, 
 
 y = 3> * = 7, 
 
 y = 4, x = 8, 
 
 y = S, a; = 9, etc. 
 
 The equation then represents the relation in value 
 existing between * and y, and for any specific value of 
 x we can find the corresponding specific value of y. 
 Instead of writing down, as above, a list of such cor- 
 responding values, we may show them graphically thus : 
 Draw two lines perpendicular to each other; make one 
 of them the * line and the other the y line. These two 
 lines are called axes. Now draw parallel to these axes 
 equi-spaced lines forming cross-sections, as shown in 
 Fig. 28, and letter the intersections of these lines with 
 the axes 1, 2, 3, 4, 5, 6, etc., as shown. 
 
 Now let us plot the corresponding values, y = 1, 
 x = 5. This will be a point 1 space up on the y axis 
 and 5 spaces out on the x axis, and is denoted by letter 
 A in the figure. In plotting the corresponding values 
 y = 2, * = 6, we get the point B; the next set of values 
 
COORDINATE GEOMETRY 
 
 97 
 
 gives us the point C, the next D, and so on. Suppose 
 we draw a line through these points; this line, called 
 the curve of the equation, tells everything in a graphical 
 
 2nd quadrant 
 
 4th quadrant 
 
 Fig. 28. 
 
 way that the equation does algebraically. If this line 
 has been drawn accurately we can from it find out at 
 a glance what value of y corresponds to any given value 
 of x, and vice versa. For example, suppose we wish to 
 see what value of y corresponds to the value x = 6|; 
 
98 MATHEMATICS 
 
 we run our eyes along the x axis until we come to 6|, 
 then up until we strike the curve, then back upon the 
 y axis, where we note that y = z\. 
 
 Negative Values of x and y. — When we started at o 
 and counted i, 2, 3, 4, etc., to the right along the x 
 axis, we might just as well have counted to the left, 
 
 — 1, —2, —3, —4, etc. (Fig. 28), and likewise we 
 might have counted downwards along the y axis, — 1, 
 
 — 2, —3, —4, etc. The values, then, to the left of o 
 on the x axis and below o on the y axis are the negative 
 values of x and y. Still using the equation x = y + 4, 
 let us give the following values to y and note the cor- 
 responding values of x in the equation x = y + 4 : 
 
 If y = 0, 
 
 then 
 
 1 x = 4, 
 
 y = -1, 
 
 
 * = 3> 
 
 y = -2, 
 
 
 # = 2, 
 
 y= -3> 
 
 
 X= I, 
 
 y= -4, 
 
 
 X = 0, 
 
 y= -5, 
 
 
 a; = —1, 
 
 y = -6, 
 
 
 # = —2, 
 
 y= -7, 
 
 
 a; = -3. 
 
 The point y = o, x = 4 is seen to be on the x axis at 
 the point 4. The point y= — i, * = 3 is at point E, 
 that is, 1 below the x axis and 3 to the right of the y 
 axis. The points y = — 2, x = 2 and y = —3, a; = 1 
 are seen to be respectively points F and G. Point 
 
COORDINATE GEOMETRY 99 
 
 y = — 4, x = o is zero along the x axis, and is there- 
 fore at —4 on the y axis. Point y — — 5, x = — 1 
 is seen to be 5 below o on the y axis and 1 to 
 the left of o along the x axis (both x and y are now 
 negative), namely, at the point H. Point y = — 6, 
 a; = — 2 is at /, and so on. 
 
 The student will note that all points in the first 
 quadrant have positive values for both x and y, all 
 points in the second quadrant have positive values for 
 y (being all above o so far as the y axis is concerned), 
 but negative values for x (being to the left of o), all 
 points in the third quadrant have negative values 
 for both x and y, while all points in the fourth quadrant 
 have positive values of x and negative values of y. 
 
 Coordinates. — The corresponding x and y values of 
 a point are called its coordinates, the vertical or y value 
 is called its ordinate, while the horizontal or x value is 
 called the abscissa; thus at point A, x = 5, y = 1, here 
 5 is called the abscissa, while 1 is called the ordinate of 
 point A. Likewise at point G, where y = — 3, x = 1, 
 here —3 is the ordinate and 1 the abscissa of G. 
 
 Straight Lines. — The student has no doubt observed 
 that all points plotted in the equation x = y + 4 have 
 fallen on a straight line, and this will always be the case 
 where both of the unknowns (in this case x and y) enter 
 the equation only in the first power; but the line will 
 not be a straight one if either x or y or both of them 
 
IOO MATHEMATICS 
 
 enter the equation as a square or as a higher power; thus, 
 x? = y + 4 will not plot out a straight line because we 
 have x 2 in the equation. Whenever both of the un- 
 knowns in the equation which we happen to be plotting 
 (be they x and y, a and b, x and a, etc.) enter the 
 equation in the first power, the equation is called a 
 linear equation, and it will always plot a straight line; 
 thus, ^x -\- $y — 20 is a linear equation, and if plotted 
 will give a straight line. 
 
 Conic Sections. — If either or both of the unknown 
 quantities enter into the equation in the second power, 
 and no higher power, the equation will always represent 
 one of the following curves: a circle or an ellipse, a 
 parabola or an hyperbola. These curves are called the 
 conic sections. A typical equation of a circle is x 2 + y 2 
 = ^; a typical equation of a parabola is y 2 = \qx\ a 
 typical equation of a hyperbola is x 2 — y 2 = r 2 , or, 
 also, xy = c 2 . 
 
 It is noted in every one of these equations that we 
 have the second power of x or y, except in the equation 
 xy = c 2 , one of the equations of the hyperbola. In this 
 equation, however, although both x and y are in the 
 first power, they are multiplied by each other, which 
 practically makes a second power. 
 
 I have said that any equation containing x or y in 
 the second power, and in no higher power, represents 
 one of the curves of the conic sections whose type forms 
 
COORDINATE GEOMETRY 
 
 IOI 
 
 we have just given. But sometimes the equations do 
 not correspond to these types exactly and require some 
 manipulation to bring them into the type form. 
 
 Let us take the equation of a circle, namely, m? + y 2 = 
 S 2 , and plot it as shown in Fig. 29. 
 
 Fig. 29. 
 
 We see that it is a circle with its center at the inter- 
 section of the coordinate axes. Now take the equa- 
 tion (x - 2) 2 + (y - 3) 2 = 5 2 - Plotting this, Fig. 30, 
 we see that it is the same circle with its center at the 
 point whose coordinates are 2 and 3. This equation 
 and the first equation of the circle are identical in form, 
 
102 
 
 MATHEMATICS 
 
 but frequently it is difficult at a glance to discover this 
 identity, therefore much ingenuity is frequently required 
 in detecting same. 
 
 Fig. 3°- 
 
 In plotting the equation of a hyperbola, xy = 25 
 (Fig. 31), we recognize this as a curve which is met 
 with very frequently in engineering practice, and a 
 knowledge of its general laws is of great value. 
 
 Similarly, in plotting a parabola (Fig. 32), y 2 = 4 *, 
 we see another familiar curve. 
 
 In this brief chapter we can only call attention to the 
 conic sections, as their study is of academic more than 
 
1 10 
 
 10 
 
 Fig. 31. 
 
 16* 
 
 H 
 
 
 
 
 
 40 
 
 
 fc*^-"" 
 
 
 
 5 
 
 
 
 i 
 
 
 
 
 
 5 
 
 10 
 
 % 
 
 Fig. 32. 
 
 103 
 
104 MATHEMATICS 
 
 of pure engineering interest. However, as the student 
 progresses in his knowledge of mathematics, I would 
 suggest that he take up the subject in detail as one 
 which will offer much fascination. 
 
 Other Curves. — All other equations containing un- 
 known quantities which enter in h'gher powers than the 
 second power, represent a large variety of curves called 
 cubic curves. 
 
 The student may find the curve corresponding to 
 engineering laws whose equations he will hereafter 
 study. The main point of the whole discussion of this 
 chapter is to teach him the methods of plotting, and if 
 successful in this one point, this is as far as we shall go 
 at the present time. 
 
 Intersection of Curves and Straight Lines. — When 
 studying simultaneous equations we saw that if we had 
 two equations showing the relation between two un- 
 known quantities, such for instance as the equations 
 
 x + y = 7, 
 x -y = 3, 
 
 we could eHminate one of the unknown quantities in 
 these equations and obtain the values of x and y which 
 will satisfy both equations; thus, in the above equations, 
 eliminating y, we have 
 
 2X = 10, 
 
 * = 5- 
 
COORDINATE GEOMETRY 
 
 *°5 
 
 Substituting this value of x in one of the equations, we 
 have y — 2 
 
 Now each one of the above equations represents a 
 straight line, and each line can be plotted as shown in 
 Fig- 33- 
 
 Fig. 33- 
 
 Their point of intersection is obviously a point on 
 both lines. The coordinates of this point, then, x = 5 
 and y = 2, should satisfy both equations, and we have 
 already seen this. Therefore, in general, where we 
 
106 MATHEMATICS 
 
 have two equations each showing a relation in value 
 between the two unknown quantities, x and y, by com- 
 bining these equations, namely, eliminating one of the 
 unknown quantities and solving for the other, our 
 result will be the point or points of intersection of both 
 curves represented by the equations. Thus, if we add 
 the equations of two circles, 
 
 x 2 + y 2 = 4 2 , 
 (* - 2) 2 + y 2 = s 2 , 
 
 and if the student plots these equations separately and 
 then combines them, eliminating one of the unknown 
 quantities and solving for the other, his results will be 
 the points of intersection of both curves. 
 
 Plotting of Data. — When plotting mathematically 
 with absolute accuracy the curve of an equation, what- 
 ever scale we use along one axis we must employ along 
 the other axis. But, for practical results in plotting 
 curves which show the relative values of several vary- 
 ing quantities during a test or which show the opera- 
 tion of machines under certain conditions, we depart 
 from mathematical accuracy in the curve for the sake 
 of convenience and choose such scales of value along 
 each axis as we may deem appropriate. Thus, suppose 
 we were plotting the characteristic curve of a shunt 
 dynamo which had given the following sets of values 
 from no load to full load operation : 
 
COORDINATE GEOMETRY 
 
 107 
 
 VOLTS 
 122 
 I20 
 Il8 
 Il6 
 114 
 III 
 107 
 
 AMPERES 
 o 
 
 5 
 10 
 
 15 
 
 19 
 
 22 
 
 25 
 
 V 
 
 
 
 
 
 130 
 
 - 
 
 
 
 
 120 
 
 ~ 
 
 
 
 
 110 
 
 
 
 
 
 100 
 
 
 
 
 
 90 
 
 
 
 
 
 80 
 
 CO 
 
 > 
 
 - 
 
 ' 
 
 
 
 60 
 
 * 
 
 
 
 
 50 
 
 - 
 
 
 
 
 40 
 
 
 
 
 
 30 
 
 - 
 
 
 
 
 20 
 
 
 
 
 
 10 
 
 
 » 1 
 
 . 1 1 1 1 r 
 
 
 O 
 
 
 3 10 
 
 15 20 25 30 35 40 
 AMPERES 
 
 X 
 
 Fig- 34- 
 
 We plot this curve for convenience in a manner as 
 shown in Fig. 34. Along the volts axis we choose a 
 scale which is compressed to within one-half of the 
 
108 MATHEMATICS 
 
 space that we choose for the amperes along the ampere 
 axis. However, we might have chosen this entirely at 
 our own discretion and the curve would have had the 
 same significance to an engineer. 
 
 PROBLEMS 
 
 Plot the curves and lines corresponding to the follow- 
 ing equations : 
 
 i. x = 7,y + 10. 
 
 2. 2x + sy = 15. 
 
 3. x- 2y = 4. 
 
 4. 10 y + $x = -8. 
 
 5. x 2 + y 2 = 36. 
 
 6. x 2 = 16 y. 
 
 7. x 2 — y 2 = 16. 
 
 8. 3 x 2 + (y - 2) 2 = 25. 
 
 Find the intersections of the following curves and 
 lines : 
 
 1. sx + y = 10, 
 4# — y = 6. 
 
 2. x 2 + y 2 = 81, 
 
 x — y = 10. 
 
 3. ary = 40, 
 3X + y = 5. 
 
COORDINATE GEOMETRY 109 
 
 Plot the following volt-ampere curve : 
 
 VOLTS AMPERES 
 
 55° O 
 
 548 20 
 
 545 39 
 
 54i 55 
 
 536 79 
 
 529 91 
 
 521 102 
 
 510 US 
 
CHAPTER XIV 
 
 ELEMENTARY PRINCIPLES OF THE CALCULUS 
 
 It is not my aim in this short chapter to do more 
 than point out and explain a few of the fundamental 
 ideas of the calculus which may be of value to a 
 practical working knowledge of engineering. To the 
 advanced student no study can offer more intellectual 
 and to some extent practical interest than the ad- 
 vanced theories of calculus, but it must be admitted 
 that very little beyond the fundamental principles ever 
 enter into the work of the practical engineer. 
 
 In a general sense the study of calculus covers an inves- 
 tigation into the innermost properties of variable quan- 
 tities, that is quantities which have variable values as 
 against those which have absolutely constant, perpetual 
 and absolutely fixed values. (In previous chapters we 
 have seen what was meant by a constant quantity and 
 what was meant by a variable quantity in an equation.) 
 By the innermost properties of a variable quantity we mean 
 finding out in the minutest detail just how this quantity 
 originated; what infinitesimal (that is, exceedingly small) 
 parts go to make it up; how it increases or diminishes 
 with reference to other quantities; what its rate of 
 increasing or diminishing may be; what its greatest 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS III 
 
 and least values are; what is the smallest particle into 
 which it may be divided; and what is the result of 
 adding all of the smallest particles together. All of 
 the processes of the calculus therefore are either analy- 
 sis or synthesis, that is, either tearing up a quantity 
 into its smallest parts or building up and adding 
 together these smallest parts to make the quantity. 
 We call the analysis, or tearing apart, differentiation; 
 we call the synthesis, or building up, integration. 
 
 DIFFERENTIATION 
 
 Suppose we take the straight line (Fig. 35) of length 
 x. If we divide it into a large number of parts, greater 
 than a million or a billion or any number of which we 
 
 /dx 
 
 t- ^- =1 
 
 Fig. 35- 
 
 have any conception, we say that each part is infinitesi- 
 mally small, — that is, it is small beyond conceivable 
 length. We represent such inconceivably small lengths 
 by an expression Ax or 8x. Likewise, if we have a 
 surface and divide it into infinitely small parts, and if 
 we call a the area of the surface, the small infinitesimal 
 portion of that surface we represent by Aa or ha. 
 These quantities, namely, Sx and Sa, are called the 
 differential of x and a respectively. 
 
112 
 
 MATHEMATICS 
 
 We have seen that the differential of a line of the 
 length x is Sx. Now suppose we have a square each of 
 whose sides is *, as shown in Fig. 36. The area of that 
 square is then x 2 . Suppose now we increase the length 
 
 -dX THICKNESS 
 
 dX THICKNESSs 
 
 Fig. 36. 
 
 of each side by an infinitesimally small amount, Sx, 
 making the length of each side x + 5a;. If we complete 
 a square with this new length as its side, the new square 
 will obviously be larger than the old square by a very 
 small amount. The actual area of the new square will 
 be equal to the area of the old square + the additions 
 to it. The area of the old square was equal to x 2 . 
 The addition consists of two fine strips each x long by 
 Sx wide and a small square having Sx as the length of 
 its side. The area of the addition then is 
 
 (x X Sx) + (x X Sx) + (Sx X Sx) = additional area. 
 
 (The student should note this very carefully.) There- 
 fore the addition equals 
 
 2xSx + (Sx) 2 = additional area. 
 Now the smaller Sx becomes, the smaller in, more rapid 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 1 13 
 
 proportion does 8x 2 , which is the area of the small 
 square, become. Likewise the smaller 8x is, the thinner 
 do the strips whose areas are x 5x become; but the strips 
 do not diminish in value as fast as the small square 
 diminishes, and, in fact, the small square vanishes so 
 rapidly in comparison with the strips that even when 
 the strips are of appreciable size the area of the small 
 square is inappreciable, and we may say practically that 
 by increasing the length of the side x of the square 
 shown in Fig. 36 by the length 8x we increase its area 
 by the quantity 2 x 8x. 
 
 Again, if we reduce the side x of the square by the 
 length 8x, we reduce the area of the square by the 
 amount 2 x 8x. This infinitesimal quantity, out of a 
 very large number of which the square consists or may 
 be considered as made up of, is equal to the differential of 
 the square, namely, the differential of x 2 . We thus see 
 that the differential of the quantity x 2 is equal to 2 x 8x. 
 Likewise, if we had considered the case of a cube instead 
 of a square, we would have found that the differential of 
 the cube X s would have been 3 x 2 8x. Likewise, by more 
 elaborate investigations we find that the differential of 
 x 4 = 4 X s 8x. Summarizing, then, the foregoing results 
 we have * differential of * = 8x, 
 
 differential of x 2 = 2 x 8x, 
 
 differential of x 3 = 3 x 2 8x, 
 
 differential of as 4 = 4 z 3 &*• 
 
114 MATHEMATICS 
 
 From these we see that there is a very simple and defi- 
 nite law by which we can at once find the differential of 
 any power of x. 
 
 Law. — Reduce the power of x by one, multiply by 
 Sx and place before the whole a coefficient which is the 
 same number as the power of x which we are differen- 
 tiating; thus, if we differentiate x 5 we get 5 ic 4 dx; also, 
 if we differentiate X s we get 6 « 5 8x. 
 
 I will repeat here that it is necessary for the student 
 to get a clear conception of what is meant by differen- 
 tiation ; and I also repeat that in differentiating any 
 quantity our object is to find out and get the value of 
 the very small parts of which it is constructed (the rate 
 of growth). Thus we have seen that a line is con- 
 structed of small lengths Sx all placed together; that a 
 square grows or evolves by placing fine strips one next 
 the other; that a cube is built up of thin surfaces placed 
 one over the other; and so on. 
 
 Differentiation Similar to Acceleration. — We have just 
 said that finding the value of the differential, or one of 
 the smallest particles whose gradual addition to a quan- 
 tity makes the quantity, is the same as finding out the 
 rate of growth, and this is what we understood by the 
 ordinary term acceleration. Now we can begin to see 
 concretely just what we are aiming at in the term 
 differential. The student should stop right here, think 
 over all that has gone before and weigh each word of 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS IIS 
 
 what we are saying with extreme care, for if he under- 
 stands that the differentiation of a quantity gives us 
 the rate of growth or acceleration of that quantity he 
 has mastered the most important idea, in fact the key- 
 note idea of all the calculus; I repeat, the keynote idea. 
 Before going further let us stop for a little illustration. 
 
 Example. — If a train is running at a constant speed 
 of ten miles an hour, the speed is constant, unvarying 
 and therefore has no rate of change, since it does not 
 change at all. If we call x the speed of the train, there- 
 fore x would be a constant quantity, and if we put it in an 
 equation it would have a constant value and be called a 
 constant. In algebra we have seen that we do not 
 usually designate a constant or known quantity by the 
 symbol *, but rather by the symbols a, k, etc. 
 
 Now on the other hand suppose the speed of the train 
 was changing; say in the first hour it made ten miles, in 
 the second hour eleven miles, in the third hour twelve 
 miles, in the fourth hour thirteen miles, etc. It is evi- 
 dent that the speed is increasing one mile per hour each 
 hour. This increase of speed we have always called the 
 acceleration or rate of growth of the speed. Now if we 
 designated the speed of the train by the symbol x, we 
 see that x would be a variable quantity and would have 
 a different value for every hour, every minute, every 
 second, every instant that the train was running. The 
 speed x would constantly at every instant have added 
 
Ii6 MATHEMATICS 
 
 to it a little more speed, namely 8x, and if we can find 
 the value of this small quantity 5x for each instant of 
 time we would have the differential of speed *, or in 
 other words the acceleration of the speed x. Now let us 
 repeat, x would have to be a variable quantity in order 
 to have any differential at all, and if it is a variable quan- 
 tity and has a differential, then that differential is the 
 rate of growth or acceleration with which the value of 
 that quantity x is increasing or diminishing as the case 
 may be. We now see the significance of the term 
 differential. 
 
 One more illustration. We all know that if a ball 
 is thrown straight up in the air it starts up with great 
 speed and gradually stops and begins to fall. Then as 
 it falls it continues to increase its speed of falling until 
 it strikes the earth with the same speed that it was 
 thrown up with. Now we know that the force of 
 gravity has been pulling on that ball from the time that 
 it left our hands and has accelerated its speed back- 
 wards until it came to a stop in the air, and then 
 speeded it to the earth. This instantaneous change in 
 the speed of the ball we have called the acceleration of 
 gravity, and is the rate of change of the speed of the ball. 
 Prom careful observation we find this to be 32 ft. per 
 second per second. A little further on we will learn 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 117 
 
 how to express the concrete value of 8x in simple 
 form. 
 
 Differentiation of Constants. — Now let us remember 
 that a constant quantity, since it has no rate of change, 
 cannot be differentiated; therefore its differential is zero. 
 If, however, a variable quantity such as x is multiplied by 
 a constant quantity such as 6, making the quantity 6 x, 
 of course this does not prevent you from differentiating 
 the variable part, namely x; but of course the constant 
 quantity remains unchanged; thus the differential of 
 6 = 0. 
 
 But the differential of 3 a; = 3 5a:, 
 
 the differential of. 4 x 2 = 4 times 2 x 8x = 8 x Sx, 
 the differential of 2 x 3 = 2 times 3 x 2 8x = 6 x 2 8x, 
 and so on. 
 
 Differential of a Sum or Difference. — We have seen 
 how to find the djffierential of a single term. Let us now 
 take up an algebraic expression consisting of several 
 terms with positive or negative signs before them; for 
 
 example 
 
 x 2 — 2 a; + 6 + 3 x 4 . 
 
 In differentiating such an expression it is obvious that 
 we must differentiate each term separately, for each 
 term is separate and distinct from the other terms, and 
 therefore its differential or rate of growth will be dis- 
 tinct and separate from the differential of the other 
 terms; thus 
 
u8 
 
 MATHEMATICS 
 
 The differential of (x 2 — 2 x + 6 + 3 ce 4 ) 
 
 = 2XSX — 2 8X + I2X 3 8X. 
 
 We need scarcely say that if we differentiate one side 
 of an algebraic equation we must also differentiate the 
 other side; for we have already seen that whatever oper- 
 ation is performed to one side of an equation must be 
 performed to the other side in order to retain the 
 equality. Thus if we differentiate 
 
 x 2 + 4 = 6 x — 10, 
 we get 2 x 8x + o = 6 5x — o, 
 
 or 2 x 8x = 6 8x. 
 
 Differentiation of a Product. — In Fig. 37 we have 
 a rectangle whose sides are x and y and whose area is 
 
 dx 
 
 Rg- 37- 
 
 therefore equal to the product xy. Now increase its 
 sides by a small amount and we have the old area 
 added to by two thin strips and a small rectangle, 
 thus: 
 
 New area = Old area + y8x + 8ydx + x 8y. 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 119 
 
 8y 8x is negligibly small; therefore we see that the 
 differential of the original area xy = x 8y + y 5x. This 
 can be generalized for every case and we have the law 
 Law. — " The differential of the product of two vari- 
 ables is equal to the first multiplied by the differential of 
 the second plus the second multiplied by the differential 
 of the first." Thus, 
 
 Differential x 2 y = x 2 Sy + 2 yx Sx. 
 This law holds for any number of variables. 
 
 Differential xyz = xy 8z + xz 5y + yz Sx. 
 Differential of a Fraction. — If we are asked to differ- 
 entiate the fraction - we first write it in the form xy -1 , 
 
 y 
 
 using the negative exponent; now on differentiating we 
 
 have 
 
 Differential xy~ x = — xy" 2 By + y _1 8x 
 
 _ x8y 8x 
 
 9 
 
 y y 
 
 Reducing to a common denominator we have 
 
 T\-a 4-- 1 -1 * x8y . y8x 
 Differential xy * or - = 1- + ■*-— 
 
 y y* -ir 
 
 _ y 8x — x by 
 
 f 
 
 Law. — The differential of a fraction is then seen to 
 be equal to the differential of the numerator times the 
 denominator, minus the differential of the denominator 
 
120 MATHEMATICS 
 
 times the numerator, all divided by the square of the 
 
 denominator. 
 
 Differential of One Quantity with Respect to Another. 
 
 — Thus far we have considered the differential of a 
 
 variable with respect to itself, that is, we have considered 
 
 its rate of development in so far as it was itself alone 
 
 concerned. Suppose however we have two variable 
 
 quantities dependent on each other, that is, as one 
 
 changes the other changes, and we are asked to find the 
 
 rate of change of the one with respect to the other, 
 
 that is, to find the rate of change of one knowing the 
 
 rate of change of the other. At a glance we see that 
 
 this should be a very simple process, for if we know the 
 
 relation which subsists between two variable quantities, 
 
 this relation being expressed in the form of an equation 
 
 between the two quantities, we should readily be able 
 
 to tell the relation which will hold between similar 
 
 deductions from these quantities. Let us for instance 
 
 take the equation 
 
 x = y + 2. 
 
 Here we have the two variables x and y tied together 
 by an equation which establishes a relation between 
 them. As we have previously seen, if we give any 
 definite value to y we will find a corresponding value 
 for x. Referring to our chapter on coordinate geometry 
 we see that this is the equation of the line shown in 
 Fig. 38. 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 121 
 
 y 
 
 Fig. 38. 
 
 Let us take any point P on this line. Its coordi- 
 nates are y and x respectively. Now choose another 
 point Pi a short distance away from P on the same 
 line. The abscissa of this new point will be a little 
 longer than that of the old point, and will equal x + 5x, 
 while the ordinate y of the old point has been increased 
 by Sy, making the ordinate of the new point y + dy. 
 From Fig. 38 we see that 
 
 tana = 
 
 _ §1 
 
 Sx 
 
122 
 
 MATHEMATICS 
 
 Therefore, if we know the tangent a and know either 8y 
 or 8x we can find the other. 
 
 In this example our equation represents a straight" 
 line, but the same would be true for any curve repre- 
 sented by any equation between x and y no matter how 
 complicated; thus Fig. 39 shows the relation between 
 
 1 Fig- 39- 
 
 8x and Sy at one point of the curve (a circle) whose 
 equation is x 2 + y 2 = 25. For every other point of the 
 
 circle tan a or — will have a different value. 5x and 
 8x 
 
 Sy while shown quite large in the figure for demon- 
 stration's sake are inconceivably small in reality; there- 
 fore the line AB in the figure is really a tangent of the 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 123 
 
 curve, and £ a is the angle which it makes with the 
 x axis. For every point on the curve this angle will be 
 different. 
 
 Mediate Differentiation. — Summarizing the forego- 
 ing we see that if we know any two of the three un- 
 knowns in equation tan a = -**• we can find the third. 
 
 Some textbooks represent tan a or -^ by y x and, — by x y . 
 
 This is a convenient notation and we will use it here. 
 
 Therefore we have 
 
 5a: tan a = Sy, 
 
 Sy . 
 — — = Sx, 
 tana 
 
 or Sy = Sx y x , 
 
 Sx = Sy x y . 
 
 This shows us that if we differentiate the quantity 
 3 x 2 as to x we obtain 6 * 5a;, but if we had wished to 
 differentiate it with respect to y we would first have to 
 differentiate it with respect to x and then multiply by 
 x v thus: 
 
 Differentiation of 3 x 2 as to y = 6 x Sy x r 
 Likewise if we have 4 y 3 and we wish to differentiate it 
 with respect to x we have 
 
 Differential of 4 y 3 as to x = 12 y 2 Sx y x .' 
 
 This is called mediate differentiation and is resorted to 
 primarily because we can differentiate a power with 
 
124 MATHEMATICS 
 
 respect to itself readily, but not with respect to some 
 other variable. 
 
 Law. — To differentiate any expression containing 
 x as to y, first differentiate it as to x and then multiply 
 by x v ty or vice versa. 
 
 We need this principle if we find the differential of 
 several terms some containing x and some y; thus if we 
 differentiate the equation 2 x 2 = y 3 — 10 with respect 
 
 to x we get 
 
 4 x Sx = 3 y 2 y x 8x + o, 
 
 or 4 * = 3 y%, 
 
 AX 
 
 or y x = — - > 
 
 if 
 
 4X 
 or tan a = • 3 — - ■ 
 
 3 3* 
 
 From this we see that by differentiating the original 
 equation of the curve we got finally an equation giving 
 the value tan a in terms of x and y, and if we fill out the 
 exact numerical values of x and y for any particular point 
 of the curve we will immediately be able to determine 
 the slant of the tangent of the curve at this point, as we 
 will numerically have the value of tangent a, and a is 
 the angle that the tangent makes with the x axis. 
 
 In just the same manner that we have proceeded 
 here we can proceed to find the direction of the tangent 
 
 of any curve whose equation we know. The differ- 
 
 Sy 
 ential of y as to *, namely -f- or y x , must be kept in 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 125 
 
 mind as the rate of change of y with respect to x, and 
 nothing so vividly portrays this fact as the inclination 
 of the tangent to the curve which shows the bend of 
 the curve at every point. 
 
 Differentials of Other Functions. — By elaborate pro- 
 cesses which cannot be mentioned here we find that the 
 
 Differential of the sine x as to x = cosine x Sx. 
 Differential of the cosine x as to x = — sin xSx. 
 
 Differential of the log x as to x = - Sx. 
 
 x 
 
 Differential of the sine y as to x = cosine y y x Sx. 
 Differential of the cosine y as to x = — sine y y x Sx. 
 
 Differential of the log y as to x = -y x Sx. 
 
 Maxima and Minima. — Referring back to the circle, 
 Fig. 39, once more, we see that 
 
 x 2 + y 2 = 25. 
 
 Differentiating this equation with reference to x we 
 
 have 
 
 2 x Sx + 2 y y x Sx = o, 
 
 or 2 x + 2 y y x = o, 
 
 x 
 
 y* = - -■ 
 
 Therefore tan a = 
 
 y 
 
 Now when tan a = o it is evident that the tangent to 
 the curve is parallel to the * axis. At this point y is 
 
126 MATHEMATICS 
 
 either a maximum or a minimum which can be readily 
 determined on reference to the curve. 
 
 x 
 o = — > 
 
 y 
 
 X = o. 
 Therefore x = o when y is maximum and in this particular 
 curve also minimum. 
 
 Law. — If we want to find the maximum or mini- 
 mum value of x in any equation containing x and. y, 
 we differentiate the equation with reference to y and 
 solve for the value of x v ; this we make equal to o and 
 then we solve for the value of y in the resulting 
 equation. 
 
 Example. — Find the maximum or minimum value 
 of x in the equation 
 
 y 2 — 14 x. 
 
 Differentiating with respect to y we have 
 
 2 y Sy = 14 x v Sy, 
 
 _ 2y 
 Xv ~ 14 ' 
 
 Equating this to o we have 
 
 2 y 
 
 14 
 
 or ~y = o. 
 
 In other words, we find that x has its minimum value 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 127 
 
 when y = o. We can readily see that this is actually 
 the case in Fig. 40, which shows the curve (a parabola). 
 
 Fig. 40. 
 
 INTEGRATION 
 
 Integration is the exact opposite of differentiation. 
 In differentiation we divide a body into its constituent 
 parts, in integration we add these constituent parts 
 together to produce the body. 
 
 Integration is indicated by the sign J ; thus, if we 
 wished to integrate Sx we would write 
 
 fsx. 
 
 Since integration is the opposite of differentiation, if we 
 are given a quantity and asked to integrate it, our 
 
128 
 
 MATHEMATICS 
 
 answer would be that quantity which differentiated 
 will give us our original quantity. For example, we 
 detect Sx as the derivative of x; therefore the integral 
 JSx = x. Likewise, we detect 4 x 3 8x as the differen- 
 tial of x 4 ; therefore the integral J4 a? dx = x\ 
 
 Fig. 41. 
 
 If we consider the line AB (Fig. 35) to be made up of 
 small parts 8x, we could sum up these parts thus : 
 
 Sx + Sx + Sx + dx + 8x + Sx 
 
 for millions of parts. But integration enables us to ex- 
 press this more simply and j Sx means the summation 
 of every single part Sx which goes to make up the line 
 AB, no matter how many parts there may be or how 
 small each part. But x is the whole length of the line 
 of indefinite length. To sum up any portion of the line 
 between the points or limits x = 1 and x = 4, we 
 
 would write 
 
 x = 4 
 
 Sx = (*) 
 
 f. 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 120. 
 
 Now these are definite integrals because they indicate 
 exactly between what limits or points we wish to find 
 the length of the line. This is true for all integrals. 
 Where no limits of integration are shown the integral 
 will yield only a general result, but when limits are 
 stated between which summation is to be made, then we 
 have a definite integral whose precise value we may 
 ascertain. 
 
 Refer back to the expression (x) ; in order to 
 
 x = 1 
 
 solve this, substitute inside of the parenthesis the 
 value of x for the upper limit of x, namely, 4, and sub- 
 stitute and subtract the value of x at the lower limit, 
 namely, 1 ; we then get 
 
 x = 4 
 (x) = (4 - 1) = 3. 
 
 x = 1 
 
 Thus 3 is the length of the line between 1 and 4. Or, 
 to give another illustration, suppose the solution of 
 some integral had given us 
 
 x = 2 
 then 
 
 & - 1) = ( 3 2 - 1) - (2 2 - 1) ='S- 
 
 X = 2 
 
 Here we simply substituted for x in the parenthesis its 
 upper limit, then subtracted from the quantity thus 
 
130 MATHEMATICS 
 
 obtained another quantity, which is had by substituting 
 the lower limit of x. 
 
 By higher mathematics and the theories of series we 
 prove that the integral of any power of a variable as to 
 itself is obtained by increasing the exponent by one and 
 dividing by the new exponent, thus : 
 
 /■ 
 
 / 
 
 x?bx = — . 
 3 
 
 4 3 s 8x = *— • 
 
 On close inspection this is seen to be the inverse of the 
 law of differentiation, which says to decrease the expo- 
 nent by one and multiply by the old exponent. 
 
 So many and so complex are the laws of nature and 
 so few and so limited the present conceptions of man 
 that only a few type forms of integrals may be actually 
 integrated. If the quantity under the integral sign by 
 some manipulation or device is brought into a form 
 where it is recognized as the differential of another 
 quantity, then integrating it will give that quantity. 
 
 The Integral of an Expression. — The integral of an 
 algebraic expression consisting of several terms is equal 
 to the sum of the integrals of each of the separate terms; 
 
 J x 2 Sx + 2 x 8x + 3 Sx 
 is the same thing as 
 
 fx 2 bx + §2 x bx + J3 5*. 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 1 31 
 
 The most common integrals to be met with practically 
 are: 
 
 (1) The integrals of some power of the variable 
 whose solution we have just explained 
 
 (2) The integrals of the sine and cosine, which are 
 
 J cosine x Sx = sine x, 
 f sine x Sx = — cosine *. 
 
 (3) The integral of the reciprocal, which is 
 
 /; 
 
 5x = log, *.* 
 
 x 
 
 Areas. — Up to the present we have considered only 
 the integration of a quantity with respect to itself. 
 Suppose now we integrate one quantity with respect to 
 another. 
 
 In Fig. 41 we have the curve PPi, which is the graphical 
 representation of some equation containing x and y. 
 If we wish to find the area which lies between the curve 
 and the x axis and between the two vertical lines 
 drawn at distances x =a and x = b respectively, we 
 divide the space up by vertical lines drawn Sx distance 
 apart. Now we would have a large number of small 
 strips each Sx wide and all having different heights, 
 namely, y h y 2 , y 3 , yi, etc. 
 
 The enumeration of all these areas would then be 
 yi Sx + y-2, Sx + y 3 Sx + yi Sx, etc. 
 
 * Log« means natural logarithm or logarithm to the Napierian base e 
 which is equal to 2.718 as distinguished from ordinary logarithms to the 
 base 10. In fact wherever log appears in this chapter it means log e . 
 
132 MATHEMATICS 
 
 Now calculus enables us to say 
 
 ySx. 
 
 x=a 
 
 J n x=b 
 y Sx cannot be readily solved. If 
 x=a 
 
 it were jx 8x we have seen that the result would be 
 
 x 2 
 
 — ; but this is not the case with fy Sx. We must then 
 
 2 J 
 
 find some way to replace y in this integral by some ex- 
 pression containing x. It is here then that we have to 
 resort to the equation of the curve PP\. From this 
 equation we find the value of y in terms of a;; we then 
 substitute this value of y in the integral fy 5x, and then 
 having an integral of x as to itself we can readily solve 
 it. Now, if the equation of the curve PPi is a complex 
 one this process becomes very difficult and sometimes 
 impossible. 
 
 A simple case of the above is the hyperbola xy = 10 
 (Fig. 42). If we wish to get the value of the shaded 
 area we have 
 
 Jr*:c= 12 ft. 
 y Sx. 
 1=5 ft. 
 
 From the equation of this curve we have 
 
 xy = 10, 
 10 
 
 y = 
 
 X 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 133 
 V 
 
 Fig. 42. 
 
 Therefore, substituting we have 
 
 Jr»i=i2 IO 
 — 
 x = 5 X 
 
 8x. 
 
 Area =10 (log* x) 
 
 x = 12 
 
 x = 5 
 
 = I0(l0g e I2 - l0ge5) 
 = 10(2.4817 — I.6077). 
 
 Area = 8.740 sq. ft. 
 
 Beyond this brief gist of the principles of calculus we 
 can go no further in this chapter. The student may not 
 understand the theories herein treated of at first — in 
 fact, it will take him, as it has taken every student, 
 
134 MATHEMATICS 
 
 many months before the true conceptions of calculus 
 dawn on him clearly. And, moreover, it is not essential 
 that he know calculus at all to follow the ordinary 
 engineering discussions. It is only where a student 
 wishes to obtain the deepest insight into the science that 
 he needs calculus, and to such a student I hope this 
 chapter will be of service as a brief preliminary to the 
 difficulties and complexities of that subject. 
 
 PROBLEMS 
 
 i. Differentiate 2 x 3 as to x. 
 
 2. Differentiate 12 # 2 as to x. 
 
 3. Differentiate 8 x 5 as to x. 
 
 4. Differentiate 3 x 2 + 4 x + 10 = 5 x 3 as to x. 
 
 5. Differentiate 4 y 2 — 3 x as to y. 
 
 6. Differentiate 14 yV as to y. 
 
 x 2 
 1. Differentiate — as to x. 
 
 y 
 
 8. Differentiate 2 y 2 — 4 qx as to y. 
 
 Find y x in the following equations: 
 
 9. x 2 + 2 y 2 = 100. 
 
 10. x* + y = 5. 
 
 11. x 2 — y 2 = 25. 
 
 12. $xy = 12. 
 
 13. What angle does the tangent line to the circle 
 x 2 + y 2 = g make with the x axis at the point where 
 x = 2? 
 
ELEMENTARY PRINCIPLES OF THE CALCULUS 135 
 
 14. What is the minimum value of y in the equation 
 x 2 = 153;? 
 
 15. Solve f 2 3? Sx. 
 
 16. Solve J 5 x 2 Sx. 
 
 17. Solve J 10 ax 5x + 5 x 2 Sx + 3 5*. 
 
 18. Solve J 3 sine x Sx. 
 
 19. Solve J 2 cosine x Sx. 
 
 1=2 
 
 Jr*x=i» 
 y 5* if a;y = 4. 
 
 22. Differentiate 10 sine x as to x. 
 
 23. Differentiate cosine a; sine x as to *. 
 
 24. Differentiate log x as to x. 
 
 y z 
 
 25. Differentiate ^ as to #. 
 
 ar 
 
 The following tables are reproduced from Ames and 
 Bliss's " Manual of Experimental Physics " by permis- 
 sion of the American Book Company. 
 
136 
 
 LOGARITHMS IOO TO IOOO 
 
 123 
 
 4 6 6 
 
 789 
 
 10 
 
 00000043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 Use preceding Table 
 
 11 
 12 
 13 
 
 0414 0453 
 0792 0828 
 "39 "73 
 
 0492 
 0864 
 1206 
 
 0531 0569 
 
 0899 
 
 0934 
 
 1239 1271 
 
 0607 
 0969 
 1303 
 
 0645 
 1004 
 1335 
 
 0682 
 1038 
 1367 
 
 0719 
 1072 
 1399 
 
 o755 
 1 106 
 1430 
 
 48 11 
 3710 
 36 10 
 
 19 23 
 17 21 
 16 19 
 
 3° 34 
 28 31 
 26 29 
 
 14 
 15 
 16 
 
 1461 
 1761 
 2041 
 
 1492 
 1790 
 2068 
 
 1523 
 1818 
 2095 
 
 1553 
 1847 
 
 2122 
 
 1584 
 1875 
 2148 
 
 1614 
 I903 
 2175 
 
 1644 
 
 1931 
 220I 
 
 1673 
 
 1959 
 2227 
 
 1703 
 I987 
 2253 
 
 1732 
 2014 
 
 2279 
 
 369 
 368 
 35 8 
 
 1 15 18 
 14 17 
 13 16 
 
 24 27 
 
 > 22 25 
 ■ 21 24 
 
 17 
 18 
 19 
 
 2304 
 
 2553 
 2788 
 
 2330 
 2577 
 2810 
 
 2355 
 2601 
 
 2833 
 
 2380 
 2625 
 2856 
 
 2405 
 2648 
 2878 
 
 2430 
 2672 
 2900 
 
 2455 
 2695 
 2923 
 
 2480 
 2718 
 2945 
 
 2504 
 2742 
 2967 
 
 2529 
 2765 
 2989 
 
 257 
 
 2 S 7 
 247 
 
 12 15 
 12 14 
 11 13 
 
 20 22 
 19 21 
 
 20 
 
 3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 246 
 
 11 13 
 
 21 
 22 
 23 
 24 
 25 
 26 
 
 3222 
 
 3424 
 3617 
 
 3243 
 3444 
 3636 
 
 3263 
 3404 
 3655 
 
 3284 
 3483 
 3674 
 
 3304 
 3502 
 3692 
 
 3324 
 3522 
 
 37" 
 
 3345 
 3541 
 3729 
 
 3365 
 3500 
 
 3747 
 
 3385 
 3579 
 3766 
 
 3404 
 
 3598 
 3784 
 
 246 
 2 4 
 
 246 
 
 10 12 
 10 12 
 9 " 
 
 16 18 
 15 17 
 15 17 
 
 3802 
 3979 
 
 3820 
 3997 
 
 4150 4166 
 
 3838 
 4014 
 4183 
 
 3856 
 
 3874 
 
 4031 4048 
 4200 4216 
 
 3892 
 4065 
 4232 
 
 3909 3927 
 
 4082 
 
 4099 
 
 4249 4265 
 
 3945 
 4116 
 4281 
 
 39°2 
 
 4133 
 
 4298 
 
 2 4 5 
 2 3 5 
 2 3 5 
 
 9 
 9 10 
 8 
 
 14 16 
 14 15 
 13 15 
 
 27 
 28 
 
 29 
 
 4314 4330 
 
 4472 
 
 4487 
 
 4624 4639 
 
 4346 
 4502 
 4654 
 
 4362 
 
 45i8 
 
 4378 
 4533 
 
 4669 4683 
 
 4393 
 4548 
 4698 
 
 4409 4425 
 4564 4579 
 4713 4728 
 
 4440 
 
 4594 
 4742 
 
 4456 
 4609 
 4757 
 
 235 
 235 
 
 3 4 
 
 8 9 
 8 9 
 7 9 
 
 13 *4 
 12 14 
 12 13 
 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 
 4771 
 
 4786 
 
 4800 
 
 4814 4829 
 
 4843 
 
 4857 4871 
 
 4886 
 
 4900 
 
 3 4 
 
 7 9 
 
 4914 4928 
 
 5051 
 
 5185 
 
 5065 
 5198 
 
 4942 
 5079 
 5211 
 
 4955 
 5092 
 5224 
 
 4969 
 5105 
 
 5237 
 
 4983 
 5119 
 5250 
 
 4997 
 5132 
 
 5263 
 
 501 1 
 
 5145 
 5276 
 
 5024 
 
 5159 
 
 5289 
 
 5038 
 5172 
 5302 
 
 3 4 
 3 4 
 3 4 
 
 11 12 
 11 12 
 10 12 
 
 5315 
 5441 
 5563 
 
 5328 
 
 5453 
 5575 
 
 5340 
 5465 
 
 5587 
 
 5353 
 5478 
 5599 
 
 5366 
 5490 
 5611 
 
 5378 
 5502 
 5623 
 
 5391 
 
 5403 
 
 5514 5527 
 
 5635 
 
 5647 
 
 54i6 
 5539 
 5658 
 
 5428 
 
 5551 
 5670 
 
 3 4 
 2 4 
 2 4 
 
 6 8 
 6 7 
 6 7 
 
 10 11 
 10 11 
 10 11 
 
 5682 
 5798 
 
 59" 
 
 5694 
 5809 
 5922 
 
 5705 
 5821 
 5933 
 
 5717 
 5832 
 
 5729 
 5843 
 
 5944 5955 
 
 5740 
 5855 
 5966 
 
 5752 5763 
 58665877 
 5977 5988 
 
 5775 
 
 5999 
 
 5786 
 
 5899 
 6010 
 
 9 10 
 9 10 
 9 
 
 6021 
 
 6031 
 
 6042 
 
 6053 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 9 10 
 
 41 
 42 
 43 
 
 6128 
 6232 
 6335 
 
 6138 
 6243 
 6345 
 
 6149 
 6253 
 6355 
 
 6160 6170 
 
 6263 
 6365 
 
 6274 
 6375 
 
 6180 
 6284 
 6385 
 
 6191 
 
 6201 
 
 6294 6304 
 
 6395 
 
 6405 
 
 6212 
 6314 
 
 6415 
 
 6222 
 6325 
 6425 
 
 5 6 
 5 6 
 5 6 
 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 51 
 52 
 53 
 
 6435 
 6532 
 6628 
 
 6444 
 6542 
 6637 
 
 6454 
 6551 
 6646 
 
 6464 
 6561 
 
 6474 
 6571 
 
 6656 6665 
 
 6484 
 6580 
 6675 
 
 6493 
 
 6503 
 
 6590 6599 
 6684 6693 
 
 6513 
 6609 
 6702 
 
 6522 
 6618 
 6712 
 
 5 6 
 5 6 
 5 6 
 
 6721 
 6812 
 6902 
 
 6730 
 6821 
 691 1 
 
 6739 
 6830 
 6920 
 
 6749 6758 
 6839 6848 
 
 6928 
 
 6937 
 
 6767 
 
 6857 
 6946 
 
 6776 6785 
 
 6866 
 6955 
 
 6875 
 6964 
 
 6794 
 6884 
 6972 
 
 6803 
 6893 
 6981 
 
 5 5 
 4 5 
 
 4 5 
 
 6990 6998 
 
 7007 
 
 70l6 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 7076 7084 
 7160 7168 
 
 7243 
 
 7251 
 
 7093 
 7177 
 7259 
 
 7IOI 
 
 7185 
 7267 
 
 7IIO 
 7193 
 7275 
 
 7"8 
 7202 
 7284 
 
 7126 7135 
 7210 7218 
 
 7292 
 
 7300 
 
 7143 
 7226 
 7308 
 
 7152 
 7235 
 7316 
 
 4 5 
 4 5 
 4 5 
 
 7 
 
 7 7 
 6 7 
 
 64 
 
 7324 7332 
 
 7340 
 
 7348 
 
 7356 
 
 7364 
 
 7372 
 
 7380 
 
 7388 
 
 7396 
 
 3 4 5 
 
 6 6 
 
LOGARITHMS 100 TO 1000 
 
 137 
 
 123 
 
 4 5 6 
 
 55 
 
 7404 
 
 7412 
 
 7419 
 
 7427 
 
 7435 
 
 7443 
 
 7451 
 
 7459 
 
 7466 
 
 7474 
 
 56 
 57 
 58 
 
 7482 
 7559 
 7634 
 
 7490 
 7566 
 7642 
 
 7497 
 7574 
 7649 
 
 7505 
 7582 
 7657 
 
 7513 
 7589 
 7664 
 
 7520 
 
 7597 
 7672 
 
 7528 
 
 7536 
 
 7604 7612 
 7679 7686 
 
 7543 
 7619 
 7694 
 
 7551 
 7627 
 7701 
 
 4 5 
 4 S 
 4 4 
 
 59 
 60 
 61 
 62 
 63 
 64 
 
 65 
 
 77097716 
 
 7782 
 7853 
 
 7789 
 7860 
 
 7723 
 
 7796 
 7868 
 
 7731 
 7803 
 
 7875 
 
 7738 
 7810 
 7882 
 
 7745 
 7818 
 7889 
 
 7752 
 7825 
 
 7760 
 7832 
 
 7896 7903 
 
 7767 
 
 7839 
 7910 
 
 7774 
 7846 
 7917 
 
 4 4 
 4 4 
 4 4 
 
 7924 
 
 7993 
 8062 
 
 7931 
 8000 
 
 7938 
 8007 
 8075 
 
 7945 
 
 7952 
 
 8014 8021 
 8082 8089 
 
 7959 
 8028 
 8096 
 
 7966 
 8035 
 8102 
 
 7973 
 8041 
 8109 
 
 7980 
 8048 
 8116 
 
 7987 
 8055 
 8122 
 
 3 4 
 3 4 
 3 4 
 
 8l2g 
 
 8136 
 
 8142 
 
 8149 8156 
 
 8l62 
 
 8169 8176 
 
 8182 
 
 8189 
 
 66 
 67 
 68 
 
 8i95 
 8261 
 
 8325 
 
 8202 
 8267 
 8331 
 
 8209 
 8274 
 8338 
 
 8215 
 
 8222 
 
 8280 8287 
 83448351 
 
 8228 
 8293 
 8357 
 
 8235 
 
 8241 
 
 8299 8306 
 
 8363 
 
 8370 
 
 8248 
 8312 
 8376 
 
 8254 
 
 8319 
 8382 
 
 3 4 
 3 4 
 
 3 4 
 
 69 
 70 
 71 
 
 8451 
 8513 
 
 8395 
 8457 
 8519 
 
 8401 
 8463 
 8525 
 
 8407 
 
 8414 
 
 8470 8476 
 
 8531 
 
 8537 
 
 8420 
 8482 
 8543 
 
 8426 
 8488 
 8549 
 
 8432 
 8494 
 8555 
 
 8439 
 8500 
 8561 
 
 8445 
 8506 
 
 8567 
 
 3 4 
 3 4 
 3 4 
 
 72 
 73 
 74 
 75 
 
 8573 
 8633 
 8692 
 
 8579 
 8639 
 8698 
 
 8585 
 8645 
 8704 
 
 8591 
 8651 
 
 8597 
 8657 
 
 8710 8716 
 
 8603 
 8663 
 8722 
 
 8609 
 8669 
 8727 
 
 8615 
 8675 
 8733 
 
 8621 
 8681 
 8739 
 
 8627 
 8686 
 8745 
 
 3 4 
 3 4 
 3 4 
 
 8751 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 3 3 
 
 76 
 77 
 78 
 
 8808 
 8865 
 
 |2I 
 
 8814 
 8871 
 8927 
 
 8820 
 8876 
 8932 
 
 8825 
 8882 
 8938 
 
 8831 
 8887 
 8943 
 
 8837 
 8893 
 8949 
 
 8842 
 8899 
 8954 
 
 8960 
 
 8854 
 89IO 
 8965 
 
 8859 
 
 8915 
 8971 
 
 3 3 
 3 3 
 3 3 
 
 79 
 80 
 81 
 
 8976 
 9031 
 
 9085 
 
 2 
 
 9036 
 9090 
 
 8987 
 9042 
 9096 
 
 8993 
 9047 
 9101 
 
 9°53 
 9106 
 
 9004 
 9058 
 9112 
 
 9009 
 9063 
 9117 
 
 9° r 5 
 9069 
 9122 
 
 9020 
 9074 
 9128 
 
 9025 
 9°79 
 9133 
 
 3 3 
 
 3 3 
 3 3 
 
 82 
 83 
 84 
 
 9138 
 9191 
 
 9 2 43 
 
 9*43 
 9196 
 9248 
 
 9149 
 9201 
 9253 
 
 9 I 54 9 I 59 
 
 9206 
 9258 
 
 9212 
 9263 
 
 9i65 
 9217 
 9269 
 
 9170 9175 
 
 9222 
 9274 
 
 9227 
 9279 
 
 9180 
 9232 
 9284 
 
 9186 
 9238 
 9289 
 
 3 3 
 3 3 
 3 3 
 
 85 
 
 9294 
 
 9299 
 
 9304 
 
 9309 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 3 3 
 
 86 
 87 
 88 
 
 9345 
 
 9350 
 
 9395 9400 
 
 9445 
 
 945o 
 
 9355 
 9405 
 9455 
 
 9360 9365 
 9410 9415 
 9460 9465 
 
 9370 
 9420 
 9469 
 
 9375 
 9425 
 
 9380 
 9430 
 
 9474 9479 
 
 9385 
 9435 
 9484 
 
 9390 
 9440 
 9489 
 
 3 3 
 
 2 3 
 2 3 
 
 89 
 90 
 91 
 92 
 93 
 94 
 9b 
 96 
 97 
 98 
 
 99 
 
 9494 9499 
 
 9542 
 
 9547 
 
 9590 9595 
 
 9504 
 9552 
 9600 
 
 95099513 
 
 9557 
 9605 
 
 9562 
 9609 
 
 95i8 
 9566 
 9614 
 
 9523 
 957i 
 
 9528 
 9576 
 
 9619 9624 
 
 9533 
 9581 
 9628 
 
 9538 
 9586 
 9633 
 
 9638 
 9685 
 9731 
 
 9643 
 9689 
 9736 
 
 9647 
 9694 
 9741 
 
 9652 
 
 9657 
 
 9699 9703 
 
 9745 
 
 975o 
 
 9661 
 9708 
 9754 
 
 96669671 
 
 9713 
 
 9717 
 
 9759 9763 
 
 9675 
 9722 
 9768 
 
 9680 
 9727 
 9773 
 
 9777 9782 
 
 9786 
 
 979i 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9823 
 9868 
 9912 
 
 9956 
 
 9827 
 9872 
 9917 
 
 9961 
 
 9832 
 9877 
 9921 
 
 9965 
 
 9836 9841 
 9886 
 9926 9930 
 
 9969 
 
 9974 
 
 9845 
 9890 
 
 9934 
 9978 
 
 9850 9854 
 9894 9899 
 9939 9943 
 
 9983 
 
 9987 
 
 9859 
 9903 
 9948 
 
 9991 
 
 9863 
 ggoS 
 9952 
 
 9956 
 
 2 3 
 
 2 3 
 
 2 3 
 
 2 3 
 
138 
 
 NATUKAL SINES 
 
 
 0' 
 
 6 
 
 12' 
 
 18 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 0° 
 
 oooo 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 OI57 
 
 369 
 
 12 15 
 
 1 
 
 2 
 3 
 
 0175 
 0349 
 0523 
 
 0192 
 0366 
 0541 
 
 0209 
 0384 
 0558 
 
 0227 
 0401 
 0576 
 
 0244 
 0419 
 0593 
 
 0262 
 0436 
 0610 
 
 0279 
 0454 
 0628 
 
 O297 
 0471 
 0645 
 
 0314 
 0488 
 0663 
 
 0332 
 0506 
 0680 
 
 369 
 
 369 
 369 
 
 12 15 
 12 15 
 12 15 
 
 4 
 S 
 6 
 
 0698 
 0872 
 
 1045 
 
 0715 
 0889 
 1063 
 
 0732 
 0906 
 1080 
 
 0750 
 0924 
 1097 
 
 0767 
 0941 
 1115 
 
 0785 
 0958 
 1132 
 
 0802 
 0976 
 1 149 
 
 0819 
 O993 
 H67 
 
 0837 
 
 IOII 
 
 1184 
 
 0854 
 1028 
 1 201 
 
 369 
 369 
 369 
 
 12 15 
 12 14 
 12 14 
 
 7 
 8 
 
 9 
 
 1219 
 
 1392 
 1564 
 
 1236 
 1409 
 1582 
 
 1253 
 1426 
 
 1599 
 
 1271 
 1444 
 1616 
 
 1288 
 1461 
 1633 
 
 1305 
 1478 
 1650 
 
 1323 
 1495 
 1668 
 
 1340 
 I513 
 1685 
 
 1357 
 1530 
 1702 
 
 1374 
 1547 
 1719 
 
 369 
 369 
 369 
 
 12 14 
 12 14 
 12 14 
 
 10 
 
 1736 
 
 1754 
 
 1771 
 
 1788 
 
 1805 
 
 1822 
 
 1840 
 
 I857 
 
 1874 
 
 1891 
 
 369 
 
 12 14 
 
 11 
 12 
 13 
 
 1908 
 2079 
 2250 
 
 1925 
 2096 
 2267 
 
 1942 
 2113 
 
 2284 
 
 1959 
 2130 
 2300 
 
 1977 
 2147 
 2317 
 
 1994 
 2164 
 
 2334 
 
 201 1 
 2181 
 2351 
 
 2028 
 2198 
 2368 
 
 2045 
 2215 
 
 2385 
 
 2062 
 2232 
 2402 
 
 369 
 
 369 
 368 
 
 11 14 
 11 14 
 11 14 
 
 14 
 15 
 16 
 
 2419 
 
 2588 
 2756 
 
 2436 
 2605 
 2773 
 
 2453 
 2622 
 2790 
 
 2470 
 2639 
 2807 
 
 2487 
 2656 
 2823 
 
 2504 
 2672 
 2840 
 
 2521 
 2689 
 2857 
 
 2538 
 2706 
 2874 
 
 2554 
 2723 
 2890 
 
 2571 
 2740 
 2907 
 
 368 
 368 
 368 
 
 11 14 
 11 14 
 11 14 
 
 17 
 18 
 19 
 
 2924 
 3090 
 
 3256 
 
 2940 
 3107 
 3272 
 
 2957 
 3123 
 3289 
 
 2974 
 3140 
 3305 
 
 2990 
 
 3156 
 3322 
 
 3007 
 3173 
 3338 
 
 3024 
 3190 
 3355 
 
 3040 
 3206 
 3371 
 
 3057 
 3223 
 3387 
 
 3074 
 3239 
 3404 
 
 368 
 368 
 
 3 5 8 
 
 11 14 
 11 14 
 11 14 
 
 20 
 
 3420 
 
 3437 
 
 3453 
 
 3469 
 
 3486 
 
 3502 
 
 3518 
 
 3535 
 
 3551 
 
 3567 
 
 3 5 8 
 
 11 14 
 
 21 
 22 
 23 
 
 3584 
 3746 
 3907 
 
 3600 
 3762 
 39 2 3 
 
 3616 
 3778 
 3939 
 
 3633 
 3795 
 3955 
 
 3649 
 3811 
 3971 
 
 3665 
 3827 
 398.7 
 
 3681 
 
 3843 
 4003 
 
 3697 
 
 3859 
 4019 
 
 3714 
 3875 
 4035 
 
 3730 
 3891 
 4051 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 11 14 
 11 14 
 11 14 
 
 24 
 25 
 26 
 
 4067 
 
 4226 
 
 4384 
 
 4083 
 4242 
 4399 
 
 4099 
 4258 
 4415 
 
 4"5 
 
 4274 
 
 4431 
 
 4131 
 4289 
 4446 
 
 4147 
 43Q5 
 4462 
 
 4163 
 4321 
 4478 
 
 4179 
 4337 
 4493 
 
 4195 
 4352 
 4509 
 
 4210 
 4368 
 4524 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 11 13 
 11 13 
 10 13 
 
 27 
 28 
 29 
 
 4540 
 4695 
 4848 
 
 4555 
 4710 
 4863 
 
 4571 
 4726 
 4879 
 
 4586 
 
 4741 
 4894 
 
 4602 
 4756 
 4909 
 
 4617 
 4772 
 4924 
 
 4633 
 4787 
 4939 
 
 4648 
 4802 
 4955 
 
 4664 
 4818 
 4970 
 
 4679 
 4833 
 4985 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 10 13 
 10 13 
 
 30 
 
 5000 
 
 5oi5 
 
 5030 
 
 5045 
 
 5060 
 
 5075 
 
 5090 
 
 5105 
 
 5120 
 
 5135 
 
 3 5 8 
 
 10 13 
 
 31 
 
 32 
 33 
 
 5150 
 5299 
 5446 
 
 5i65 
 5314 
 5461 
 
 5180 
 5329 
 5476 
 
 5195 
 5344 
 5490 
 
 5210 
 
 5358 
 5505 
 
 5225 
 5373 
 5519 
 
 5240 
 5388 
 5534 
 
 5255 
 5402 
 5548 
 
 5270 
 5417 
 5563 
 
 5284 
 5432 
 5577 
 
 2 5 7 
 2 5 7 
 2 5 7 
 
 10 12 
 10 12 
 10 12 
 
 34 
 35 
 36 
 
 5592 
 5736 
 
 5878 
 
 5606 
 575o 
 5892 
 
 5621 
 5764 
 5906 
 
 5635 
 ,5779 
 5920 
 
 5650 
 5793 
 5934 
 
 5664 
 5807 
 5948 
 
 5678 
 5821 
 5962 
 
 5693 
 5835 
 5976 
 
 5707 
 5850 
 
 5990 
 
 572i 
 5864 
 6004 
 
 2 5 7 
 2 5 7 
 2 5 7 
 
 10 12 
 
 10 12 
 
 9 12 
 
 37 
 38 
 39 
 
 6018 
 6157 
 
 6293 
 
 6032 
 6170 
 6307 
 
 6046 
 6184 
 6320 
 
 6060 
 6198 
 6334 
 
 6074 
 6211 
 6347 
 
 6088 
 6225 
 6361 
 
 6101 
 6239 
 6374 
 
 6115 
 6252 
 6388 
 
 6129 
 6266 
 6401 
 
 6143 
 6280 
 6414 
 
 2 5 7 
 2 5 7 
 247 
 
 9 12 
 9 11 
 9 11 
 
 40 
 
 6428 
 
 6441 
 
 6455 
 
 6468 
 
 6481 
 
 6494 
 
 6508 
 
 6521 
 
 6534 
 
 6547 
 
 247 
 
 9 11 
 
 41 
 
 42 
 43 
 
 6561 
 
 6691 
 6820 
 
 6574 
 6704 
 
 6833 
 
 6587 
 6717 
 6845 
 
 6600 
 6730 
 6858 
 
 6613 
 
 6743 
 6871 
 
 6626 
 6756 
 6884 
 
 6639 
 6769 
 6896 
 
 6652 
 6782 
 6909 
 
 6665 
 6794 
 6921 
 
 6678 
 6807 
 6934 
 
 247 
 246 
 246 
 
 9 11 
 9 11 
 8 11 
 
 44 
 
 6947 
 
 6959 
 
 6972 
 
 6984 
 
 6997 
 
 7009 
 
 7022 
 
 A. 
 
 7034 
 
 7046 
 
 7059 
 
 246 
 
 8 10 
 
NATURAL SINES 
 
 139 
 
 
 0' 
 
 6' 
 
 12' 
 
 18 
 
 24' 
 
 30' 
 
 [36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 45° 
 
 7071 
 
 7083 
 
 7096 
 
 7108 
 
 7120 
 
 7133 
 
 7145 
 
 7157 
 
 7169 
 
 7181 
 
 246 
 
 8 10 
 
 46 
 47 
 48 
 
 7193 
 73i4 
 743i 
 
 7206 
 7325 
 7443 
 
 7218 
 7337 
 7455 
 
 7230 
 
 7349 
 7466 
 
 7242 
 736i 
 7478 
 
 7254 
 7373 
 7490 
 
 7266 
 7385 
 7501 
 
 7278 
 7396 
 7513 
 
 7290 
 7408 
 7524 
 
 7302 
 7420 
 7536 
 
 246 
 246 
 246 
 
 8 10 
 8 10 
 8 10 
 
 49 
 50 
 51 
 
 7547 
 7660 
 
 777i 
 
 7558 
 7672 
 7782 
 
 757o 
 7683 
 7793 
 
 758i 
 7694 
 7804 
 
 7593 
 7705 
 7815 
 
 7604 
 7716 
 7826 
 
 7615 
 
 7727 
 7837 
 
 7627 
 7738 
 7848 
 
 7638 
 7749 
 7859 
 
 7649 
 7760 
 7869 
 
 246 
 246 
 245 
 
 8 9 
 7 9 
 7 9 
 
 52 
 53 
 54 
 
 7880 
 7986 
 8090 
 
 7891 
 
 7997 
 8100 
 
 7902 
 8007 
 
 8111 
 
 7912 
 8018 
 8121 
 
 7923 
 8028 
 8131 
 
 7934 
 8039 
 8141 
 
 7944 
 8049 
 8151 
 
 7955 
 8059 
 8161 
 
 7965 
 8070 
 8171 
 
 7976 
 8080 
 8181 
 
 245 
 235 
 2 3 5 
 
 7 9 
 7 9 
 7 8 
 
 55 
 
 8192 
 
 8202 
 
 8211 
 
 8221 
 
 8231 
 
 8241 
 
 8251 
 
 8261 
 
 8271 
 
 8281 
 
 2 3 5 
 
 7 8 
 
 56 
 57 
 68 
 
 8290 
 8387 
 8480 
 
 8300 
 8396 
 8490 
 
 8310 
 8406 
 8499 
 
 8320 
 8415 
 8508 
 
 8329 
 8425 
 8517 
 
 8339 
 8434 
 8526 
 
 8348 
 8443 
 8536 
 
 8358 
 8453 
 8545 
 
 8368 
 8462 
 8554 
 
 8377 
 8471 
 8563 
 
 2 3 5 
 2 3 5 
 2 3 5 
 
 6 8 
 6 8 
 6 8 
 
 59 
 60 
 61 
 
 8572 
 8660 
 8746 
 
 8581 
 8669 
 
 8755 
 
 8590 
 8678 
 8763 
 
 8599 
 8686 
 8771 
 
 8607 
 8695 
 8780 
 
 8616 
 8704 
 8788 
 
 8625 
 8712 
 8796 
 
 8634 
 8721 
 8805 
 
 8643 
 8729 
 8813 
 
 8652 
 8738 
 8821 
 
 1 3 4 
 1 3 4 
 1 3 4 
 
 6 7 
 6 7 
 6 7 
 
 62 
 63 
 64 
 
 8829 
 8910 
 8988 
 
 8838 
 8918 
 8996 
 
 8846 
 8926 
 9003 
 
 8854 
 8934 
 901 1 
 
 8862 
 8942 
 9018 
 
 8870 
 8949 
 9026 
 
 8878 
 8957 
 9033 
 
 8886 
 8965 
 9041 
 
 8894 
 
 8973 
 9048 
 
 8902 
 8980 
 9056 
 
 1 3 4 
 1 3 4 
 13 4 
 
 5 7 
 5 6 
 5 6 
 
 65 
 
 9063 
 
 9070 
 
 9078 
 
 9085 
 
 9092 
 
 9100 
 
 9107 
 
 9114 
 
 9121 
 
 9128 
 
 124 
 
 5 ° 
 
 66 
 67 
 68 
 
 9135 
 9205 
 9272 
 
 9143 
 9212 
 9278 
 
 9150 
 9219 
 9285 
 
 9157 
 9225 
 9291 
 
 9164 
 9232 
 9298 
 
 9171 
 9239 
 9304 
 
 9178 
 
 9245 
 9311 
 
 9184 
 9252 
 9317 
 
 9191 
 9259 
 9323 
 
 9198 
 9265 
 9330 
 
 123 
 123 
 123 
 
 5 6 
 4 6 
 4 5 
 
 69 
 70 
 71 
 
 9336 
 9397 
 9455 
 
 9342 
 9403 
 9461 
 
 9348 
 9409 
 9466 
 
 9354 
 9415 
 9472 
 
 9361 
 9421 
 9478 
 
 9367 
 9426 
 9483 
 
 9373 
 9432 
 9489 
 
 9379 
 9438 
 9494 
 
 9385 
 9444 
 9500 
 
 9391 
 9449 
 9505 
 
 123 
 123 
 123 
 
 4 5 
 4 5 
 4 5 
 
 72 
 73 
 74 
 
 951 1 
 
 9563 
 9613 
 
 9516 
 9568 
 9617 
 
 9521 
 
 9573 
 9622 
 
 9527 
 9578 
 9627 
 
 9532 
 9583 
 9632 
 
 9537 
 9588 
 9636 
 
 9542 
 9593 
 9641 
 
 9548 
 9598 
 9646 
 
 9553 
 9603 
 9650 
 
 9558 
 9608 
 9655 
 
 123 
 122 
 122 
 
 4 4 
 3 4 
 3 4 
 
 75 
 
 9659 
 
 9664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9686 
 
 9690 
 
 9694 
 
 9699 
 
 112 
 
 3 4 
 
 76 
 77 
 78 
 
 9703 
 9744 
 9781 
 
 9707 
 9748 
 9785 
 
 9711 
 
 9751 
 9789 
 
 9715 
 9755 
 9792 
 
 9720 
 
 9759 
 9796 
 
 9724 
 9763 
 9799 
 
 9728 
 9767 
 
 9803 
 
 9732 
 9770 
 9806 
 
 9736 
 9774 
 9810 
 
 9740 
 9778 
 9813 
 
 112 
 112 
 112 
 
 3 3 
 3 3 
 2 3 
 
 79 
 80 
 81 
 
 9816 
 9848 
 9877 
 
 9820 
 
 9851 
 9880 
 
 9823 
 9854 
 9882 
 
 9826 
 
 9857 
 9885 
 
 9829 
 9860 
 9888 
 
 9833 
 9863 
 9890 
 
 9836 
 9866 
 9893 
 
 9839 
 9869 
 
 9895 
 
 9842 
 9871 
 9898 
 
 9845 
 9874 
 9900 
 
 112 
 
 Oil 
 Oil 
 
 2 3 
 2 2 
 2 2 
 
 82 
 83 
 84 
 
 9903 
 9925 
 9945 
 
 9905 
 9928 
 9947 
 
 9907 
 993° 
 9949 
 
 9910 
 9932 
 995i 
 
 9912 
 
 9934 
 9952 
 
 9914 
 9936 
 9954 
 
 9917 
 9938 
 9956 
 
 9919 
 9940 
 9957 
 
 9921 
 9942 
 9959 
 
 9923 
 9943 
 9960 
 
 Oil 
 Oil 
 Oil 
 
 2 2 
 1 2 
 1 1 
 
 85 
 
 9962 
 
 9963 
 
 9965 
 
 9966 
 
 9968 
 
 9969 
 
 9971 
 
 9972 
 
 9973 
 
 9974 
 
 001 
 
 1 1 
 
 86 
 87 
 88 
 
 9976 
 9986 
 9994 
 
 9977 
 9987 
 9995 
 
 9978 
 9988 
 9995 
 
 9979 
 9989 
 9996 
 
 9980 
 9990 
 9996 
 
 9981 
 999° 
 9997 
 
 9982 
 9991 
 9997 
 
 9983 
 9992 
 9997 
 
 9984 
 
 9993 
 9998 
 
 99»5 
 9993 
 9998 
 
 001 
 000 
 000 
 
 1 1 
 1 1 
 
 
 89 
 
 9998 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 1.000 
 
 nearly 
 
 1. 000 
 
 nearly 
 
 1.000 
 
 nearly 
 
 1.000 
 
 nearly 
 
 1.000 
 
 nearly 
 
 000 
 
 
 
140 
 
 NATURAL COSINES 
 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 0° 
 
 1 
 
 2 
 3 
 
 I.OOO 
 
 1.000 
 
 □early 
 
 1.000 
 
 nearly 
 
 1.000 
 
 nearly 
 
 1.000 
 
 nearly 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 000 
 
 
 
 9998 
 
 9994 
 9986 
 
 9998 
 
 9993 
 9985 
 
 9998 
 
 9993 
 9984 
 
 9997 
 9992 
 
 9983 
 
 9997 
 9991 
 
 9982 
 
 9997 
 9990 
 9981 
 
 9996 
 9990 
 9980 
 
 9996 
 9989 
 9979 
 
 9995 
 9988 
 9978 
 
 9995 
 9987 
 
 9977 
 
 000 
 000 
 001 
 
 
 
 1 1 
 1 1 
 
 4 
 5 
 6 
 
 9976 
 9962 
 9945 
 
 9974 
 9960 
 
 9943 
 
 9973 
 9959 
 9942 
 
 9972 
 
 9957 
 9940 
 
 997i 
 9956 
 9938 
 
 9969 
 9954 
 9936 
 
 9968 
 9952 
 9934 
 
 9966 
 9951 
 9932 
 
 9965 
 9949 
 9930 
 
 9963 
 9947 
 9928 
 
 001 
 
 Oil 
 Oil 
 
 1 1 
 1 2 
 1 2 
 
 7 
 
 8 
 
 9 
 
 10 
 
 99 2 5 
 9903 
 9877 
 
 99 2 3 
 9900 
 
 9874 
 
 9921 
 9898 
 9871 
 
 9919 
 
 9895 
 9869 
 
 9917 
 9893 
 9866 
 
 9914 
 9890 
 9863 
 
 9912 
 9888 
 9860 
 
 9910 
 9885 
 9857 
 
 9907 
 9882 
 9854 
 
 9905 
 9880 
 
 9851 
 
 Oil 
 Oil 
 Oil 
 
 2 2 
 2 2 
 2 2 
 
 9848 
 
 9845 
 
 9842 
 
 9839 
 
 ^836 
 
 9833 
 
 9829 
 
 9826 
 
 9823 
 
 9820 
 
 112 
 
 2 3 
 
 11 
 12 
 13 
 
 9816 
 9781 
 9744 
 
 9813 
 9778 
 9740 
 
 9810 
 9774 
 9736 
 
 9806 
 9770 
 9732 
 
 9803 
 9767 
 9728 
 
 9799 
 9763 
 9724 
 
 9796 
 
 9759 
 9720 
 
 9792 
 9755 
 9715 
 
 9789 
 9751 
 97" 
 
 9785 
 9748 
 9707 
 
 112 
 112 
 112 
 
 2 3 
 
 3 3 
 3 3 
 
 14 
 15 
 16 
 
 9703 
 9659 
 9613 
 
 9699 
 
 9655 
 9608 
 
 9694 
 9650 
 9603 
 
 9690 
 9646 
 9598 
 
 9686 
 9641 
 9593 
 
 9681 
 9636 
 9588 
 
 9677 
 9632 
 9583 
 
 9673 
 9627 
 9578 
 
 9668 
 9622 
 9573 
 
 9664 
 9617 
 9568 
 
 112 
 12 2 
 12 2 
 
 3 4 
 3 4 
 3 4 
 
 17 
 18 
 19 
 
 9563 
 95" 
 9455 
 
 9558 
 9505 
 9449 
 
 9553 
 9500 
 
 9444 
 
 9548 
 9494 
 9438 
 
 9542 
 9489 
 9432 
 
 9537 
 9483 
 9426 
 
 9532 
 9478 
 9421 
 
 9527 
 9472 
 
 9415 
 
 9521 
 9466 
 9409 
 
 9516 
 9461 
 9403 
 
 12 3 
 12 3 
 12 3 
 
 4 4 
 4 5 
 4 5 
 
 20 
 
 9397 
 
 939 1 
 
 9385 
 
 9379 
 
 9373 
 
 9367 
 
 9361 
 
 9354 
 
 9348 
 
 9342 
 
 12 3 
 
 4 5 
 
 21 
 22 
 23 
 
 9336 
 9272 
 9205 
 
 9330 
 9265 
 9198 
 
 9323 
 9259 
 9191 
 
 9317 
 9252 
 9184 
 
 93" 
 9245 
 9178 
 
 9304 
 9239 
 9171 
 
 9298 
 9232 
 9164 
 
 9291 
 9225 
 9157 
 
 9285 
 9219 
 9 J 5o 
 
 9278 
 9212 
 9143 
 
 12 3 
 12 3 
 12 3 
 
 4 5 
 
 4 6 
 
 5 6 
 
 24 
 25 
 26 
 
 9135 
 9063 
 8988 
 
 9128 
 9056 
 8980 
 
 9121 
 9048 
 8973 
 
 91 14 
 9041 
 8965 
 
 9107 
 9033 
 8957 
 
 9100 
 9026 
 8949 
 
 9092 
 9018 
 8942 
 
 9085 
 901 1 
 8934 
 
 9078 
 9003 
 8926 
 
 9070 
 8996 
 8918 
 
 12 4 
 I 3 4 
 
 1 3 4 
 
 5 6 
 5 6 
 5 6. 
 
 27 
 28 
 29 
 
 8910 
 8829 
 8746 
 
 8902 
 8821 
 8738 
 
 8894 
 8813 
 8729 
 
 8886 
 8805 
 8721 
 
 8878 
 8796 
 8712 
 
 8870 
 8788 
 8704 
 
 8862 
 8780 
 8695 
 
 8854 
 8771 
 8686 
 
 8846 
 8763 
 8678 
 
 8838 
 
 8755 
 8669 
 
 134 
 134 
 
 1 3 4 
 
 5 7 
 
 6 7 
 6 7 
 
 30 
 
 8660 
 
 8652 
 
 8643 
 
 8634 
 
 8625 
 
 8616 
 
 8607 
 
 8599 
 
 8590 
 
 8581 
 
 13 4 
 
 6 7 
 
 31 
 32 
 33 
 
 8572 
 8480 
 8387 
 
 8563 
 8471 
 
 8377 
 
 8554 
 8462 
 8368 
 
 8545 
 8453 
 8358 
 
 8536 
 8443 
 8348 
 
 8526 
 8434 
 8339 
 
 8517 
 8425 
 8329 
 
 8508 
 
 8415 
 8320 
 
 8499 
 8406 
 8310 
 
 8490 
 8396 
 8300 
 
 2 3 5 
 2 3 5 
 2 3 5 
 
 6 8 
 6 8 
 6 8 
 
 34 
 35 
 36 
 
 8290 
 8192 
 8090 
 
 8281 
 8181 
 8080 
 
 8271 
 8171 
 8070 
 
 8261 
 8161 
 8059 
 
 8251 
 8151 
 8049 
 
 8241 
 8141 
 8039 
 
 8231 
 8131 
 8028 
 
 8221 
 8121 
 8018 
 
 8211 
 8111 
 8007 
 
 8202 
 8100 
 7997 
 
 2 3 5 
 2 3 5 
 2 3 5 
 
 7 8 
 7 8 
 7 9 
 
 37 
 38 
 39 
 
 7986 
 7880 
 7771 
 
 7976 
 7869 
 7760 
 
 7965 
 7859 
 7749 
 
 7955 
 7848 
 7738 
 
 7944 
 7837 
 7727 
 
 7934 
 7826 
 7716 
 
 7923 
 7815 
 7705 
 
 7912 
 7804 
 7694 
 
 7902 
 
 7793 
 7683 
 
 7891 
 7782 
 7672 
 
 2 4 5 
 2 4 5 
 246 
 
 7 9 
 7 9 
 7 9 
 
 40 
 
 7660 
 
 7649 
 
 7638 
 
 7627 
 
 76i5 
 
 7604 
 
 7593 
 
 758i 
 
 757o 
 
 7559 
 
 246 
 
 8 9 
 
 41 
 
 42 
 43 
 
 7547 
 7431 
 7314 
 
 7536 
 7420 
 7302 
 
 7524 
 7408 
 7290 
 
 7513 
 7396 
 
 7278 
 
 75oi 
 
 7385 
 7266 
 
 7490 
 7373 
 7254 
 
 7478 
 7361 
 7242 
 
 7466 
 
 7349 
 7230 
 
 7455 
 7337 
 7218 
 
 7443 
 7325 
 7206 
 
 246 
 246 
 246 
 
 8 10 
 8 10 
 8 10 
 
 44 
 
 7193 
 
 7181 
 
 7169 
 
 7157 
 
 7145 
 
 7133 
 
 7120 
 
 7108 
 
 7096 
 
 7083 
 
 246 
 
 8 10 
 
 N.B. — Numbers in difference columns to be subtracted, not added. 
 
NATUBAL COSINES 
 
 141 
 
 45° 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36' 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 S 
 
 7071 
 
 7059 
 
 7046 
 
 7034 
 
 7022 
 
 7009 
 
 6997 
 
 6984 
 
 6972 
 
 6959 
 
 246 
 
 8 10 
 
 46 
 47 
 48 
 
 6947 
 6820 
 6691 
 
 6934 
 6807 
 6678 
 
 6921 
 6794 
 6665 
 
 6909 
 6782 
 6652 
 
 6896 
 6769 
 6639 
 
 6884 
 6756 
 6626 
 
 6871 
 6743 
 6613 
 
 6858 
 6730 
 6600 
 
 6845 
 6717 
 6587 
 
 6833 
 6704 
 6574 
 
 246 
 246 
 247 
 
 8 11 
 
 9 " 
 9 " 
 
 49 
 50 
 51 
 
 6561 
 6428 
 6293 
 
 6547 
 6414 
 6280 
 
 6534 
 6401 
 6266 
 
 6521 
 6388 
 6252 
 
 6508 
 
 6374 
 6239 
 
 6494 
 6361 
 6225 
 
 6481 
 
 6347 
 621 1 
 
 6468 
 
 6334 
 6198 
 
 6455 
 6320 
 6184 
 
 6441 
 6307 
 6170 
 
 247 
 247 
 
 2 5 7 
 
 9 11 
 9 11 
 9 " 
 
 52 
 53 
 54 
 
 6157 
 6018 
 5878 
 
 6143 
 6004 
 5864 
 
 6129 
 
 599° 
 5850 
 
 6115 
 5976 
 5835 
 
 6101 
 5962 
 5821 
 
 6088 
 5948 
 5807 
 
 6074 
 5934 
 5793 
 
 6060 
 5920 
 
 5779 
 
 6046 
 5906 
 5764 
 
 6032 
 5892 
 5750 
 
 2 5 7 
 2 5 7 
 2 5 7 
 
 g 12 
 9 12 
 9 12 
 
 55 
 
 5736 
 
 572i 
 
 5707 
 
 5693 
 
 5678 
 
 5664 
 
 5650 
 
 5635 
 
 5621 
 
 5606 
 
 2 5 7 
 
 10 12 
 
 56 
 57 
 58 
 
 5592 
 5446 
 5299 
 
 5577 
 5432 
 5284 
 
 5563 
 5417 
 5270 
 
 5548 
 5402 
 5255 
 
 5534 
 5388 
 5240 
 
 5519 
 5373 
 5225 
 
 5505 
 5358 
 5210 
 
 549° 
 5344 
 5195 
 
 5476 
 5329 
 5180 
 
 546i 
 53i4 
 5165 
 
 2 5 7 
 2 5 7 
 2 5 7 
 
 10 12 
 10 12 
 10 12 
 
 59 
 60 
 61 
 
 5150 
 5000 
 
 4848 
 
 5135 
 4985 
 
 4833 
 
 5120 
 4970 
 4818 
 
 5105 
 4955 
 4802 
 
 5090 
 4939 
 4787 
 
 5075 
 4924 
 4772 
 
 5060 
 4909 
 4756 
 
 5045 
 4894 
 4741 
 
 5030 
 4879 
 4726 
 
 5015 
 4863 
 4710 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 10 13 
 10 13 
 
 62 
 
 63 
 64 
 
 4695 
 4540 
 4384 
 
 4679 
 
 4524 
 4368 
 
 4664 
 4509 
 4352 
 
 4648 
 4493 
 4337 
 
 4633 
 4478 
 4321 
 
 4617 
 4462 
 4305 
 
 4602 
 4446 
 4289 
 
 4586 
 4431 
 4274 
 
 4571 
 4415 
 4258 
 
 4555 
 4399 
 4242 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 
 10 13 
 
 11 13 
 
 65 
 
 4226 
 
 4210 
 
 4195 
 
 4179 
 
 4163 
 
 4147 
 
 4131 
 
 4"5 
 
 4099 
 
 4083 
 
 3 5 8 
 
 11 13 
 
 66 
 67 
 68 
 
 4067 
 3907 
 
 3746 
 
 4051 
 3891 
 3730 
 
 4035 
 3875 
 3714 
 
 4019 
 3859 
 3697 
 
 4003 
 
 3843 
 3681 
 
 3987 
 3827 
 3665 
 
 3971 
 3811 
 3649 
 
 3955 
 3795 
 3633 
 
 3939 
 3778 
 3616 
 
 3923 
 3762 
 3600 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 11 14 
 11 14 
 11 14 
 
 69 
 70 
 71 
 
 3584 
 3420 
 3256 
 
 3567 
 3404 
 3239 
 
 3551 
 3387 
 3223 
 
 3535 
 3371 
 3206 
 
 3518 
 3355 
 3190 
 
 3502 
 3338 
 3173 
 
 3486 
 3322 
 3156 
 
 3469 
 3305 
 3140 
 
 3453 
 3289 
 
 3123 
 
 3437 
 3272 
 3107 
 
 3 5 8 
 3 5 8 
 368 
 
 11 14 
 11 14 
 11 14 
 
 72 
 73 
 74 
 
 3090 
 2924 
 2756 
 
 3074 
 2907 
 2740 
 
 3057 
 2890 
 2723 
 
 3040 
 
 2874 
 2706 
 
 3024 
 
 2857 
 2689 
 
 3007 
 2840 
 2672 
 
 2990 
 2823 
 2656 
 
 2974 
 2807 
 2639 
 
 2957 
 2790 
 2622 
 
 2940 
 
 2773 
 2605 
 
 368 
 368 
 368 
 
 11 14 
 11 14 
 11 14 
 
 75 
 
 2588 
 
 2571 
 
 2554 
 
 2538 
 
 2521 
 
 2504 
 
 2487 
 
 2470 
 
 2453 
 
 2436 
 
 368 
 
 11 14 
 
 76 
 77 
 78 
 
 2419 
 2250 
 
 2079 
 
 2402 
 2233 
 2062 
 
 2385 
 2215 
 2045 
 
 2368 
 2198 
 2028 
 
 2351 
 2181 
 201 1 
 
 2334 
 2164 
 1994 
 
 2317 
 2147 
 1977 
 
 2300 
 2130 
 1959 
 
 2284 
 2113 
 1942 
 
 2267 
 2096 
 1925 
 
 368 
 369 
 369 
 
 11 14 
 11 14 
 11 14 
 
 79 
 80 
 81 
 
 1908 
 1736 
 1564 
 
 1891 
 1719 
 1547 
 
 1874 
 1702 
 1530 
 
 1857 
 1685 
 
 1513 
 
 1840 
 1668 
 1495 
 
 1822 
 1650 
 1478 
 
 1805 
 1633 
 1461 
 
 1788 
 1616 
 1444 
 
 1771 
 
 1599 
 1426 
 
 1754 
 1582 
 1409 
 
 369 
 369 
 369 
 
 12 14 
 12 14 
 12 14 
 
 82 
 83 
 84 
 
 1392 
 1219 
 1045 
 
 1374 
 1201 
 1028 
 
 1357 
 1184 
 
 IOII 
 
 1340 
 1167 
 0993 
 
 1323 
 1 149 
 0976 
 
 1305 
 1132 
 
 0958 
 
 1288 
 
 i"5 
 0941 
 
 1271 
 1097 
 0924 
 
 1253 
 1080 
 0906 
 
 1236 
 1063 
 
 0889 
 
 369 
 369 
 
 369 
 
 12 14 
 12 14 
 12 14 
 
 85 
 
 0872 
 
 0854 
 
 0837 
 
 0819 
 
 0802 
 
 0785 
 
 0767 
 
 0750 
 
 0732 
 
 0715 
 
 369 
 
 12 15 
 
 86 
 87 
 88 
 
 0698 
 0523 
 0349 
 
 0680 
 0506 
 0332 
 
 0663 
 0488 
 0314 
 
 0645 
 0471 
 0297 
 
 0628 
 0454 
 0279 
 
 0610 
 0436 
 0262 
 
 0593 
 0419 
 0244 
 
 0576 
 0401 
 0227 
 
 0558 
 0384 
 0209 
 
 0541 
 0366 
 0192 
 
 369 
 369 
 3 6 9 
 
 12 15 
 12 15 
 12 15 
 
 89 
 
 0175 
 
 0157 
 
 0140 
 
 0122 
 
 0105 
 
 0087 
 
 0070 
 
 0052 
 
 0035 
 
 0017 
 
 369 
 
 12 15 
 
 N.B. — Numbers in difference columns to be subtracted, not added. 
 
142 
 
 NATURAL TANGENTS 
 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 36 
 
 42' 
 
 48' 
 
 54' 
 
 12 3 
 
 4 5 
 
 0° 
 
 .0000 
 
 0017 
 
 Q035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 0157 
 
 369 
 
 12 14 
 
 1 
 
 2 
 3 
 
 •0175 
 •0349 
 .0524 
 
 0192 
 0367 
 0542 
 
 0209 
 0384 
 0559 
 
 0227 
 0402 
 0577 
 
 0244 
 0419 
 
 0594 
 
 0262 
 
 0437 
 0612 
 
 0279 
 
 0454 
 0629 
 
 0297 
 0472 
 0647 
 
 0314 
 0489 
 0664 
 
 0332 
 0507 
 0682 
 
 369 
 369 
 369 
 
 12 15 
 
 13 15 
 12 15 
 
 4 
 S 
 6 
 
 .0699 
 .0875 
 .1051 
 
 0717 
 0892 
 1069 
 
 0734 
 0910 
 1086 
 
 0752 
 0928 
 1 104 
 
 0769 
 
 0945 
 1122 
 
 0787 
 0963 
 1139 
 
 0805 
 0981 
 "57 
 
 0822 
 0998 
 "75 
 
 0840 
 1016 
 1 192 
 
 0857 
 1033 
 1210 
 
 369 
 369 
 369 
 
 12 15 
 12 15 
 12 15 
 
 7 
 8 
 9 
 
 .1228 
 .1405 
 .1584 
 
 1246 
 1423 
 
 1602 
 
 1263 
 1441 
 1620 
 
 1281 
 
 1459 
 1638 
 
 1299 
 1477 
 1655 
 
 1317 
 1495 
 1673 
 
 1334 
 1512 
 1691 
 
 1352 
 1530 
 1709 
 
 1370 
 1548 
 1727 
 
 1388 
 1566 
 1745 
 
 369 
 
 369 
 369 
 
 12 15 
 
 12 15 
 
 12 15 
 
 10 
 
 •1763 
 
 1781 
 
 1799 
 
 1817 
 
 1835 
 
 1853 
 
 1871 
 
 1890 
 
 1908 
 
 1926 
 
 369 
 
 12 is 
 
 11 
 12 
 13 
 
 .1944 
 .2126 
 .2309 
 
 1962 
 2144 
 2327 
 
 1980 
 2162 
 2345 
 
 1998 
 2180 
 2364 
 
 2016 
 2199 
 
 2382 
 
 2035 
 2217 
 2401 
 
 2053 
 2235 
 2419 
 
 2071 
 2254 
 2438 
 
 2089 
 2272 
 2456 
 
 2107 
 2290 
 
 2475 
 
 369 
 369 
 369 
 
 12 15 
 
 12 15 
 
 12 IS 
 
 14 
 15 
 16 
 
 .2493 
 .2679 
 .2867 
 
 2512 
 2698 
 2886 
 
 2530 
 2717 
 2905 
 
 2549 
 2736 
 2924 
 
 2568 
 2754 
 2943 
 
 2586 
 
 2773 
 2962 
 
 2605 
 2792 
 2981 
 
 2623 
 2811 
 3000 
 
 2642 
 2830 
 3019 
 
 2661 
 2849 
 3038 
 
 369 
 369 
 369 
 
 12 l6 
 
 13 16 
 13 16 
 
 17 
 18 
 19 
 20 
 
 •3057 
 •3249 
 ■3443 
 
 3076 
 3269 
 3463 
 
 3096 
 3288 
 3482 
 
 3"5 
 
 3307 
 3502 
 
 3134 
 3327 
 3522 
 
 3153 
 3346 
 354i 
 
 3172 
 3365 
 356i 
 
 3191 
 
 3385 
 358i 
 
 32" 
 3404 
 3600 
 
 3230 
 3424 
 3620 
 
 3 6 jo 
 3 6 10 
 3 6 10 
 
 13 16 
 13 16 
 
 13 17 
 
 •3640 
 
 3659 
 
 3679 
 
 3699 
 
 3719 
 
 3739 
 
 3759 
 
 3779 
 
 3799 
 
 3819 
 
 3 7 10 
 
 13 17 
 
 21 
 22 
 23 
 
 •3839 
 .4040 
 
 •4245 
 
 3859 
 4061 
 4265 
 
 3879 
 4081 
 4286 
 
 3899 
 4101 
 
 4307 
 
 39 x 9 
 4122 
 4327 
 
 3939 
 4142 
 
 4348 
 
 3959 
 4163 
 4369 
 
 3978 
 4183 
 4390 
 
 4000 
 4204 
 44" 
 
 4020 
 4224 
 4431 
 
 3 7 10 
 3 7 10 
 3 7 10 
 
 13 17 
 
 14 17 
 14 17 
 
 24 
 25 
 26 
 
 •4452 
 .4663 
 .4877 
 
 4473 
 4684 
 4899 
 
 4494 
 4706 
 4921 
 
 4515 
 4727 
 4942 
 
 4536 
 4748 
 4964 
 
 4557 
 4770 
 4986 
 
 4578 
 
 479i 
 5008 
 
 4599 
 4813 
 5029 
 
 4621 
 4834 
 5051 
 
 4642 
 4856 
 5073 
 
 4 7 10 
 4 7 11 
 4 7 n 
 
 14 18 
 
 14 18 
 
 15 18 
 
 27 
 28 
 29 
 
 •5095 
 •5317 
 ■5543 
 
 5117 
 5340 
 5566 
 
 5139 
 5362 
 5589 
 
 5161 
 
 5384 
 5612 
 
 5184 
 5407 
 5635 
 
 5206 
 543° 
 5658 
 
 5228 
 5452 
 5681 
 
 5250 
 
 5475 
 5704 
 
 5272 
 5498 
 
 5727 
 
 5295 
 5520 
 575° 
 
 4 7 11 
 4 8 11 
 
 4 8 12 
 
 IS 18 
 
 15 19 
 
 IS 19 
 
 30 
 
 •5774 
 
 5797 
 
 5820 
 
 5844 
 
 5867 
 
 5890 
 
 5914 
 
 5938 
 
 5961 
 
 5985 
 
 4 8 12 
 
 16 20 
 
 31 
 32 
 33 
 
 .6009 
 .6249 
 .6494 
 
 6032 
 6273 
 6519 
 
 6056 
 6297 
 6544 
 
 6080 
 6322 
 6569 
 
 6104 
 6346 
 6594 
 
 6128 
 6371 
 6619 
 
 6152 
 
 6395 
 6644 
 
 6176 
 6420 
 6669 
 
 6200 
 6445 
 6694 
 
 6224 
 6469 
 6720 
 
 4 8 12 
 4 8 12 
 4 8 13 
 
 16 20 
 
 16 20 
 
 17 21 
 
 34 
 35 
 36 
 
 ■6745 
 .7002 
 .7265 
 
 5771 
 7028 
 7292 
 
 6796 
 7054 
 7319 
 
 6822 
 7080 
 7346 
 
 6847 
 7107 
 7373 
 
 6873 
 7133 
 7400 
 
 6899 
 
 7159 
 7427 
 
 6924 
 7186 
 7454 
 
 6950 
 7212 
 748i 
 
 6976 
 
 7239 
 7508 
 
 4 9 "3 
 
 t 9"3 
 
 5 9 14 
 
 17 21 
 
 18 22 
 18 23 
 
 37 
 38 
 39 
 
 •7536 
 ■7813 
 .8098 
 
 7563 
 7841 
 8127 
 
 759° 
 7869 
 8156 
 
 7618 
 7898 
 8185 
 
 7646 
 7926 
 8214 
 
 7673 
 7954 
 8243 
 
 7701 
 7983 
 8273 
 
 7729 
 8012 
 8302 
 
 7757 
 8040 
 8332 
 
 7785 
 8069 
 8361 
 
 5 9 14 
 S 10 14 
 5 10 IS 
 
 18 23 
 
 19 24 
 
 20 24 
 
 40 
 
 .8391 
 
 8421 
 
 8451 
 
 8481 
 
 8511 
 
 8541 
 
 8571 
 
 8601 
 
 8632 
 
 8662 
 
 5 'o IS 
 
 20 25 
 
 41 
 42 
 43 
 
 .8693 
 .9004 
 •9325 
 
 8724 
 9036 
 9358 
 
 8754 
 9067 
 
 9391 
 
 8785 
 9099 
 9424 
 
 8816 
 9131 
 9457 
 
 8847 
 9163 
 9490 
 
 8878 
 9195 
 9523 
 
 8910 
 9228 
 9556 
 
 8941 
 9260 
 9590 
 
 8972 
 
 9293 
 9623 
 
 5 10 16 
 
 5 " 16 
 
 6 11 17 
 
 21 26 
 
 21 27 
 
 22 28 
 
 44 
 
 .9657 
 
 9691 
 
 9725 
 
 9759 
 
 9793 
 
 9827 
 
 9861 
 
 9896 
 
 9930 
 
 9965 
 
 6 11 17 
 
 23 29 
 
NATUBAL TANGENTS 
 
 143 
 
 1 
 
 36' 42' 
 
 0' 
 
 6' 
 
 12' 
 
 18' 
 
 24' 
 
 30' 
 
 48' 
 
 54' 
 
 12 3 
 
 46' 
 
 1.0000 
 
 0035 
 
 0070 
 
 0105 
 
 0141 
 
 0176 
 
 0212 
 
 0247 
 
 0283 
 
 0319 
 
 6 12 18 
 
 46 
 47 
 48 
 
 1-0355 
 1.0724 
 1.1 106 
 
 0392 
 0761 
 "45 
 
 042S 
 0799 
 
 1184 
 
 0464 
 0837 
 1224 
 
 0501 
 0875 
 1263 
 
 0538 
 0913 
 1303 
 
 0575 
 0951 
 
 1343 
 
 0612 
 0990 
 
 1383 
 
 0649 
 1028 
 1423 
 
 0686 
 1067 
 1463 
 
 6 12 
 
 6 13 19 
 
 7 13 20 
 
 49 
 50 
 51 
 
 1.1504 
 
 1.1918 
 1.2349 
 
 1544 
 1960 
 
 2393 
 
 1585 
 2002 
 
 2437 
 
 1626 
 2045 
 2482 
 
 1667 
 2088 
 2527 
 
 1708 
 2131 
 2572 
 
 175° 
 2174 
 2617 
 
 1792 
 2218 
 2662 
 
 1833 
 2261 
 2708 
 
 1875 
 2305 
 2753 
 
 7 14 21 
 
 7 14 22 
 
 8 15 23 
 
 52 
 63 
 54 
 
 1.2799 
 1.3270 
 I-3764 
 
 2846 
 3319 
 3814 
 
 2892 
 3367 
 3865 
 
 2938 
 3416 
 3916 
 
 2985 
 
 3465 
 3968 
 
 3032 
 
 3514 
 4019 
 
 3079 
 3564 
 4071 
 
 3127 
 3613 
 4124 
 
 3175 
 3663 
 4176 
 
 3222 
 
 3713 
 4229 
 
 8 16 23 
 
 8 16 25 
 
 9 17 26 
 
 55 
 
 1.428 1 
 
 4335 
 
 4388 
 
 4442 
 
 4496 
 
 4550 
 
 4605 
 
 4659 
 
 4715 
 
 4770 
 
 9 18 27 
 
 66 
 67 
 
 58 
 
 1.4826 
 
 1-5399 
 1.6003 
 
 4882 
 5458 
 6066 
 
 4938 
 5517 
 6128 
 
 4994 
 5577 
 6191 
 
 5051 
 5637 
 6255 
 
 5108 
 5697 
 6319 
 
 5166 
 
 5757 
 6383 
 
 5224 
 5818 
 6447 
 
 5282 
 5880 
 6512 
 
 5340 
 5941 
 6577 
 
 zo 19 29 
 
 10 SO 30 
 
 11 21 32 
 
 59 
 60 
 61 
 
 1.6643 
 1. 7321 
 i.8o4< 
 
 6709 
 
 7391 
 8115 
 
 6775 
 7461 
 8190 
 
 6842 
 7532 
 8265 
 
 6909 
 7603 
 8341 
 
 6977 
 7675 
 8418 
 
 7045 
 7747 
 8495 
 
 7"3 
 7820 
 8572 
 
 7182 
 7893 
 8650 
 
 7251 
 7966 
 8728 
 
 11 23 34 
 
 12 24 36 
 
 13 26 38 
 
 62 
 
 63 
 
 64 
 
 1.8807 
 1.9626 
 2.0503 
 
 8887 
 9711 
 0594 
 
 8967 
 9797 
 0686 
 
 9047 
 9883 
 0778 
 
 9128 
 9970 
 0872 
 
 9210 
 6057 
 0965 
 
 9292 
 
 6145 
 1060 
 
 9375 
 0233 
 "55 
 
 9458 
 0323 
 1251 
 
 9542 
 0413 
 
 1348 
 
 14 27 41 
 
 15 29 44 
 
 16 31 47 
 
 65 
 
 2.1445 
 
 1543 
 
 1642 
 
 1742 
 
 1842 
 
 1943 
 
 2045 
 
 2148 
 
 2251 
 
 2355 
 
 17 34 51 
 
 66 
 67 
 68 
 
 2.2460 2566 
 
 2.3559 3673 
 2.4751 4876 
 
 2673 
 3789 
 5002 
 
 2781 
 3906 
 5129 
 
 2889 
 4023 
 
 5257 
 
 2998 
 4142 
 5386 
 
 3109 
 4262 
 5517 
 
 3220 
 4383 
 5649 
 
 3332 
 4504 
 5782 
 
 3445 
 4627 
 
 59i6 
 
 18 37 55 
 20 40 60 
 
 22 43 65 
 
 69 
 70 
 71 
 
 2.6051 
 
 2-7475 
 2.9042 
 
 6187 
 7625 
 9208 
 
 6325 
 7776 
 9375 
 
 6464 6605 
 7929 8083 
 
 9544 
 
 9714 
 
 6746 
 8239 
 9887 
 
 688g 
 8397 
 6061 
 
 7034 
 8556 
 6237 
 
 7179 
 8716 
 
 0415 
 
 7326 
 8878 
 °595 
 
 24 47 7i 
 26 52 78 
 29 58 87 
 
 72 
 73 
 74 
 
 3-0777 
 3.2709 
 
 3-4874 
 
 0961 
 2914 
 5105 
 
 1 146 
 3122 
 
 5339 
 
 1334 1524 
 
 3332 
 
 3544 
 
 5576 5816 
 
 1716 
 
 3759 
 6059 
 
 191c 2106 
 
 3977 
 6305 
 
 4197 
 6554 
 
 2305 
 4420 
 6806 
 
 2506 
 4646 
 7062 
 
 32 64 96 
 36 72 108 
 
 41 82 Z22 
 
 75 
 
 3-7321 
 
 7583 
 
 7848 
 
 8118 
 
 8391 
 
 8667 
 
 8947 
 
 9232 
 
 9520 
 
 9812 
 
 46 94139 
 
 76 
 77 
 78 
 
 4.0108 
 4-3315 
 4-7046 
 
 0408 
 3662 
 7453 
 
 0713 
 4015 
 7867 
 
 1022 
 
 4374 
 8288 
 
 1335 
 4737 
 8716 
 
 1653 
 5107 
 9152 
 
 1976 
 5483 
 9594 
 
 2303 
 5864 
 6045 
 
 2635 
 6252 
 0504 
 
 2972 
 
 6646 
 0970 
 
 53 107 160 
 62 124 186 
 73 146 219 
 
 79 
 80 
 81 
 82 
 83 
 J54 
 
 _85 
 86 
 87 
 
 J? 8 . 
 
 89 
 
 5.1446 
 5-67I3 
 6.3138 
 
 1929 
 7297 
 3859 
 
 2422 
 7894 
 4596 
 
 2924 
 8502 
 
 5350 
 
 3435 
 9124 
 6122 
 
 2066 
 
 2636 
 
 5144J9.677 
 
 7-«54 
 8.1443 
 9 
 
 3002 
 3863 
 9-845 
 
 3962 
 5126 
 
 4947 
 6427 
 10.20 
 
 H-43 
 
 11.66 
 
 11.91 
 
 12.16 12.43 
 
 14.30 
 19.08 
 28.64 
 
 57.29 
 
 14.67 
 19.74 
 30-14 
 
 63.66 
 
 15.06 
 20.45 
 31-82 
 
 15.46 
 21.: 
 
 33 
 
 69 35 
 
 15.89 
 
 .02 
 80 
 
 71.62 
 
 81.85 
 
 95-49 
 
 3955 
 9758 
 6912 
 
 4486 
 0405 
 7920 
 
 5026 
 1066 
 
 8548 
 
 5578 
 1742 
 9395 
 
 6140 
 2432 
 0264 
 
 87 17s 262 
 
 5958 
 7769 
 iQ-39 
 
 6996 
 9152 
 10.58 
 
 8062 
 
 0579 
 10.78 
 
 9158 
 2052 
 
 io- 99 
 
 0285 
 3572 
 11.20 
 
 12.71 
 
 13.00 13.30 
 
 13-62 
 
 13-95 
 
 16.35 
 22.90 
 38.19 
 
 16.83 
 
 23 
 40.92 
 
 17-34 
 go 
 
 144-07 
 
 8624. 
 
 17.89 
 26.03 
 47-74 
 
 18.46 
 27.27 
 52.08 
 
 Difference - col- 
 umns cease to be 
 useful, owing to 
 the rapidity with 
 which the value 
 of the tangent 
 changes. 
 
 1 14.6 
 
 143-2 
 
 igl.O 
 
 286.5 
 
 573-0 
 
ANSWERS TO PROBLEMS 
 
ANSWERS TO PROBLEMS 
 
 CHAPTER I 
 
 i. 2a+66 + 6c — 3d. 4. — 3X + 6y+ 4-Z + a. 
 
 2. — ga-\- b — 6c. 5. — 86+90 — 2 c. 
 
 3. 30" — z+ 146 — 100. 6. — 8a;— 60 + 46+11 y. 
 
 7. 2X — 2y+ 28 z 
 
 
 
 
 CHAPTER H 
 
 
 
 I. 
 
 2. 
 
 3- 
 
 18 o 2 6 2 . 
 48 oW. 
 90^3^. 
 
 
 5. 06c 2 . 
 oW 
 
 7. 0*6%. 
 
 
 a 2 c 2 z 
 9- -gr- 
 
 40 o r 
 
 10.. ^ • 
 
 ^c 2 
 
 4- 
 
 144 o 8 6 6 c 2 . 
 
 
 8. oW. 
 CHAPTER HI 
 
 
 11. %• 
 
 54 ad 
 
 1. 
 
 go^e 
 4a; 
 
 
 7. 6 a 2 — a 
 
 8. a - 6. 
 
 6+50C- 
 
 -2b 2 +6bc— 4c' 
 
 2. 
 
 6c 
 
 18 d 
 aWc'x 
 
 
 9. a 2 + 2 
 
 a+b 
 
 10. 7- 
 
 0—6 
 
 06 + 6 2 . 
 
 ^ 
 
 3- 
 4- 
 
 ,6? 
 2ox 2 +i5xy+ 
 
 3^-3^ + 
 
 IOZZ. 2 0C+2 0rf- 
 
 Sac 2 — 3 ocrf 
 -2c 2 — 2cd 
 
 5- 
 6. 
 
 4 a + 2 o 2 6 2 — 
 a 2 -J 2 . 
 
 6. 
 
 <?ba 
 
 12. 
 
 12 
 
 
 
 147 
 
148 ANSWERS TO PROBLEMS 
 
 8 a + 6 2 + 4 c 4 — 12 a + d?c 
 ,I3 - 4! I4 ' 6* 
 
 IS- 
 
 120 a 2 c + 3 6c — 6 6a; + 2 6^ 
 126c 
 
 3a6 — ac+ 26" 5 a 2 — 2 a— 26 
 
 4 aft X sa 2 + sa6 
 
 CHAPTER IV 
 
 1. 3, 2, 5, a, a, 6. 11. (2 a; - 3 y) (2 z - 3 y). 
 
 2. 3, 2, 2, 2, 2, a, a, a, a, c. 12. (9a + 56) (9a + 56). 
 
 3. 3, 2, 5, *, *, y, y, y, y, z 3- (4c - 6a) (4c- 6a). 
 
 z,z,z. 14. #, y, (4^ + 523;— 10 z). 
 
 4. 3, 3, 2, 2, 2, 2, a;, a;, a, a. *5- S*(6a + 3ac — c). 
 
 11 11, l6 - (9*y-s«)(9»y + s«)- 
 
 5. 3, 2, a, a ,-, a,-, -1 6, i7 ( 2 +46 2)( a+26) ( a _ 26)< 
 
 1 x x . . , 18. (i2» 2 y+80)(i2 3c 2 y— 8z). 
 
 * 19. (a 2 — 2 ac + c) 2, 2. 
 
 6. 2,5,i,a;,i, i >y,y,-- 2 °" ^y + z) (4? + *)- 
 
 ' °'2 'a; a; -"-"-y c 1 \ /• \ 
 
 y 21. (3y + 2a;) (2y — 3a;). 
 
 7. (a-c)( 2 a + 6). 22 ( 4a+s j)( a _ 2 j). 
 
 8. (3 * + y) (x + c). 23- ( 33 ,_ 2a .)( 2y _ 3 »). 
 
 9. (2a; + sy)(a; + z). 24. (2 a + 6) (a - 3 6). 
 10. (a — 6) (a — 6). 25. (2a+ 56) (a+ 26). 
 
 
 Square roots. 
 
 CHAPTER V 
 
 Cube roots. 
 
 I. 
 
 2. 
 
 3- 
 4- 
 5- 
 
 4 * + 3 y- 
 
 2a + 6. 
 6 a; + 2 y. 
 
 5 a — 2 6. 
 a + 6 + c. 
 
 i. 2X+ sy. 
 2. x+2y. 
 3- 3<* + 36- 
 
ANSWERS TO PROBLEMS 149 
 
 CHAPTER VI 
 
 1. * = 4§- ioy 
 
 14. 2 = - 
 
 2. x = 2$. 
 
 y + 2 
 
 3. * = 4. *5- 5<* + 3 = * + <* + 3- 
 
 4. z = -A. l6, 6«*-5:y=5-">*- 
 
 5. jc=^. , J 7- iSZ 2 + 4*= 12-ioy. 
 
 6. * = 30. 18. 6 a + 2 <f = 4. 
 
 7. x=6tfi- I0 - 3*-* = 3^-y. 
 s 9g+9&-«y-*y 20. 8s -10^=203,. 
 
 1 X? X? 
 
 21: 
 
 3 (a-6)+ 2g 2 ' (e-<9 (3 0+6) 3 (c-d) 
 9 ' a5-_ 2a(a-J)(a+i)" =2fl+6. 
 
 _ io(fl !! -6 2 ) , 22 - * = ~i 
 
 IO ' * 2 a 23. Coat costs $28.57. 
 
 11. 2d*x+2db-a X *-bx= Gun costs $57.14. 
 
 <*»-fc+io«-io&. Hat costs * I 4.»9- 
 
 ^ cy 24. Horse costs $671.66. 
 
 12. "T + "* = "J "t"^~ " Carriage costs $328.33. 
 
 25. Anne's age is 18 years. 
 
 x 3' ° — — c + 3 26. 24 chairs and 14 tables. 
 
 CHAPTER VH 
 
 1. y = 4, x=2. 6. a; = -15, y = 15. 
 
 2. a; =5, y=2. 7- a; = --084, y= -10.034. 
 
 3. a: = 1, y = 2. 8. a; = 5^, y = — rfV- 
 4- *=5> y= 2 . z = 3- 9- a; = -i.i, y = 6.1. 
 5. x = 3, y = 2, z = 4. 10. a; = i&, y = 2^. 
 
iS° 
 
 ANSWERS TO PROBLEMS 
 CHAPTER Vm 
 
 i. * = 2 or x — i. 
 
 — 2 ± 2V19 
 
 2. X = 
 
 3 
 
 3. x = 2. 
 
 4. a; = 4 or —2. 
 
 5. a; = 3 or 1 
 
 6. a; = ± 2 or ± V^6. 
 
 7 . ,__i±^S. 
 
 X = • 
 
 14 a 
 
 /i-l 
 
 8. * = - 
 1 — 3 a=fc V51 a 2 — 6 a + 1 
 
 azfcV^aff + a 2 
 26 
 
 2 a 
 
 10. x 
 
 _ , 3 (a + 6)±V8(a + 6) + o(a+&) 2 , 
 
 11. x = — * 
 
 205 
 
 17. X = — 
 
 3±' 
 
 12. a; = — 3. 
 
 13. a; = 4(2±V / 3 ~). 
 
 14. a; = — — • 
 
 4a 
 
 18. * = - 
 
 :±VT 
 
 299 
 
 19. a; = 63. 
 
 20. a; = 100 a 2 — 301 a+ 225. 
 
 IS- * = 
 
 16. x = — 
 
 21. a; = 
 
 a 2 ± a Va 2 + 4 
 
 a + b 
 
 27 ± V2425 
 16 
 
 CHAPTER IX 
 
 -5 ±"^5 
 22. x = — 3— 2. 
 
 
 
 1. k = 50. 
 2.6= V, 
 
 3. & = 60. 
 
 4. a = 192. 
 
 5- c = 5- 
 
 CHAPTER X 
 
 1. 96 sq. ft. 6. V301 ft. long. 
 
 2. 180 sq.ft. 7 27.6 ft. long. 
 
 3. 254.469 sq. ft. _ ., 
 
 o 3* t y -1 — g 7 g S7 . 7 miles. 
 
 4. Hypotenuse = V117 /vo ' ' 
 
 ft. long. 9- Altitude = 7.5 ft. 
 
 5. 62.832 ft. long. 10. Altitude = 4 ft. 
 
ANSWERS TO PROBLEMS 
 CHAPTER XI 
 
 I5i 
 
 1. sine = .5349; cosine = .8456; tangent = .6330. 
 
 2. sine = .9888; cosine = .1495; tangent = 6.6122. 
 
 3. 25° 36'. 7. c = 600ft.; 6 = 519.57 ft. 
 4- 79° 25'- 8. j.a = 57 47'; c = 591.01 ft. 
 
 5. 36 59'. 9. a = 1231 ft.; b = 217 ft. 
 
 6. 28 54'. 10. £a = 6i° 51'; a = 467.3 ft. 
 
 
 
 CHAPTER XII 
 
 
 
 I. 
 
 3-5879- 
 
 10. 
 
 9,802,000. 
 
 19. 
 
 1,198,000. 
 
 2. 
 
 1.8667. 
 
 11. 
 
 24,860,000. 
 
 20. 
 
 18,410,000 
 
 3- 
 
 - 3-9948. 
 
 12. 
 
 778,500,000. 
 
 21. 
 
 275>5°°- 
 
 4- 
 
 4-6155- 
 
 13- 
 
 .000286. 
 
 22. 
 
 .00001314. 
 
 5- 
 
 666.2. 
 
 14. 
 
 .0001199. 
 
 23- 
 
 549-7- 
 
 6. 
 
 7443°- 
 
 15- 
 
 3 2 -34- 
 
 24. 
 
 4.27. 
 
 7- 
 
 •2745- 
 
 16. 
 
 in. 6. 
 
 25- 
 
 .296. 
 
 8. 
 
 .00024105. 
 
 i7- 
 
 •0323. 
 
 26. 
 
 46.86. 
 
 9- 
 
 2302.5. 
 
 18. 
 
 .03767. 
 
 
 
 
 
 CHAPTER XIH 
 
 
 
 Get cross-section paper and plot the following correspond- 
 ing values of x and y and the result will be the line or curve 
 as the case may be. 
 
 This is a straight line and only 
 two pairs of corresponding val- 
 ues of x and y are necessary to 
 draw it. 
 
 This is also a straight line. 
 
 1. x = 0; 
 
 y = -3h 
 
 y = o; 
 
 X = 10. 
 
 x = 22; 
 
 y = 4. 
 
 X = — 2 
 
 y = - 4- . 
 
 2. X = O; 
 
 y = 3- 
 
 y = 0; 
 
 * = 7i 
 
152 
 
 ANSWERS TO PROBLEMS 
 
 3- x = o; 
 y = o; 
 
 4. x = o; 
 -y = o; 
 
 5. a; = o; 
 y = o; 
 x = 1; 
 x = 2; 
 
 * = 4; 
 
 x= 5; 
 
 y = — 2. 
 x = 4. 
 
 y = - tV 
 
 X = - 2§ . 
 
 y =±6. 
 x=±6. 
 
 y =± v 32. 
 y =± V27. 
 y = ± V 20. 
 y =± V11. 
 
 32. 
 
 6. y = o; x = o. 
 y — 2 ; x =± 
 y = 4; x =± 8. 
 
 y = 6; x =±V96. 
 
 7. y = o; x=±4. 
 y==fci; x=±vi7. 
 y=±3; x=±S. 
 
 y =±5; x=± V4T. 
 
 x=db V7. 
 y =+7 or— 3 
 y = 2 ± V22. 
 y = 2 ± V 13. 
 
 y = o; 
 x = o; 
 x = 1; 
 x = 2; 
 
 A straight line. 
 
 A straight line. 
 
 This is a circle with its 
 center at the intersection 
 of the x and y axes and 
 with a radius of 6. 
 
 This is a parabola and 
 to plot it correctly a great 
 many corresponding values 
 of x and y are necessary. 
 
 This is an hyperbola and 
 a great many corresponding 
 values of x and y are neces- 
 sary in order to plot the 
 curve correctly. 
 
 This is an ellipse with its 
 center at +2 on the y axis. 
 A great many corresponding 
 values of x and y are neces- 
 sary to plot it correctly. 
 
ANSWERS TO PROBLEMS 153 
 
 Intersections of Curves 
 
 *= 2?; y = &■ 
 
 This is the intersection of 2 
 straight lines. 
 
 2 - y = ~ 5 ± v ^; I This is the intersection of a 
 x = 5 ± V^. J straight line and a circle. 
 
 3. The roots are here imaginary showing that the two 
 curves do not touch at all, which can be easily shown by 
 plotting them. 
 
 CHAPTER XIV 
 
 I. 
 
 6 x 2 dx. 
 
 14. 
 
 When x = at which 
 
 2. 
 
 24 x dx. 
 
 
 time y also = 0. 
 
 3- 
 
 40 x* dx. 
 
 IS- 
 
 t. 
 
 4- 
 
 6af9a; + 4 dx = isr'da;. 
 
 
 2 
 
 5- 
 
 8 y dy — 3 x v dy. 
 
 16. 
 
 5*. 
 
 6. 
 7- 
 
 42 y*x? dx + 56 ay dy. 
 2 yx dx — # 2 dy 
 f 
 
 i7- 
 18. 
 
 3 
 
 5 a:* 2 + 5 x 3 + 3 X. 
 —3 cos X. 
 
 8. 
 
 4ydy-4qx v dy. 
 
 X 
 
 19. 
 20. 
 
 2 sine a;. * 
 117. 
 
 9- 
 
 y x = 
 
 2y 
 
 21. 
 
 8-7795- 
 
 10. 
 
 y x = - 3 * 2 - 
 
 22. 
 
 10 cosine x dx. 
 
 11. 
 12. 
 
 a; 
 
 y» = — 
 y 
 
 y 
 
 23- 
 24. 
 
 cos 2 xdx — sin 2 a; dx. 
 -dx. 
 
 X 
 
 2 x?y dy — 2 y 2 x dx 
 
 13- 
 
 41 48' 10". 
 
 25. 
 
 X* 
 
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