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Full text of "Mathematics for collegiate students of agriculture and general science"

of the 

University of California 

Los Angeles 



Form L 1 



This book is DUE on the last date stamped below 



NOV 7 






NOV 2 
WOV 4 " RECD 






Form L-9-15m-8,'26 






FOR COLLEGIATE STUDENTS OF 

AGRICULTURE AND GENERAL SCIENCE 
REVISED EDITION 



A SERIES OF MATHEMATICAL TEXTS 

EDITED BY 
EARLE RAYMOND HEDRICK 

THE CALCULUS 

By ELLERY WILLIAMS DAVIS and WILLIAM CHAKLES BRENKE. 

ANALYTIC GEOMETRY AND ALGEBRA 

By ALEXANDER ZIWET and Louis ALLEN HOPKINS. 

ELEMENTS OF ANALYTIC GEOMETRY 

By ALEXANDER ZIWET and Louis ALLEN HOPKINS. 

PLANE AND SPHERICAL TRIGONOMETRY WITH 

COMPLETE TABLES 
By ALFRED MONROE KENYON and Louis INGOLD. 

PLANE AND SPHERICAL TRIGONOMETRY WITH 

BRIEF TABLES 
By ALFRED MONROE KENYON and Louis INGOLD. 

ELEMENTARY MATHEMATICAL ANALYSIS 

By JOHN WESLEY YOUNG and FRANK MILLETS MORGAN. 

COLLEGE ALGEBRA 

By ERNEST BROWN SKINNER. 

MATHEMATICS FOR COLLEGIATE STUDENTS OF 

AGRICULTURE AND GENERAL SCIENCE 
By ALFRED MONROE KENYON and WILLIAM VERNON LOVITT. 

PLANE TRIGONOMETRY 

By ALFRED MONROE KENYON and Louis INGOLD. 

THE MACMILLAN TABLES 

Prepared under the direction of EARLE RAYMOND HEDRICK. 

PLANE GEOMETRY 

By WALTER BURTON FORD and CHARLES AMMERMAN. 

PLANE AND SOLID GEOMETRY 

By WALTER BURTON FORD and CHARLES AMMERMAN. 

SOLID GEOMETRY 

By WALTER BURTON FORD and CHARLES AMMERMAN. 

CONSTRUCTIVE GEOMETRY 

Prepared under the direction of EARLE RAYMOND HEDRICK. 

JUNIOR HIGH SCHOOL MATHEMATICS 

By WILLIAM LEDLEY VOSBURGH and FREDERICK WILLIAM 
GENTLEMAN. 



MATHEMATICS 



FOB COLLEGIATE STUDENTS OF 



REVISED EDITION 



BY 

ALFRED MONROE KENYON 

PROFESSOR OF MATHEMATICS IN PURDUE UNIVERSITY 
AND 

WILLIAM VERNON LOVITT 

ASSOCIATE PROFESSOR OF MATHEMATICS IN COLORADO COLLEGE 



Nefa gorfc 
THE MACMILLAN COMPANY 

LONDON: MACMILLAN & CO., LTD. 

1918 



COPYBIQHT, 1917, 

BY THE MACMILLAN COMPANY. 



Set up and electrotyped. Published December, 1917. 



Nortoooli 
Printed by Berwick & Smith Co., Norwood, Mass., U.S.A. 



ft 4- 2 



PREFACE 

This book is designed as a text in freshman mathematics for 
students specializing in agriculture, biology, chemistry, and 
physics, in colleges and in technical schools. 

The selection of topics has been determined by the definite 
needs of these students. An attempt has been made to treat 
these topics and to select material for illustration so as to put 
in evidence their close and practical relations with everyday 
life, both in and out of college. It is certain that the interest 
of the student can be aroused and sustained in this way. We 
believe also that he can be trained to understand and to solve 
those mathematical problems which will confront him in the 
subsequent years of his college work and in after-life, without 
losing anything in orderly arrangement or in clear and accurate 
logical thinking. 

Reference to the table of contents will indicate the scope and 
proportions of the material presented and something of the 
means employed in relating the material to the vital interests 
of the student and of correlating it to his experience and his 
intellectual attainments. Many of the chapter subjects and 
paragraph headings are traditional. Nothing has been intro- 
duced merely for novelty. Since this course is to constitute the 
entire mathematical equipment of some students, some chapters 
have been inserted which have seldom been available to fresh- 
men; for example, the chapters on annuities, averages, and 
correlation, and the exposition of Mendel's law in the chapter 
on the binomial expansion. 

Particular attention has been given to the illustrative examples 
and figures, and to the grading of the problems in the lists. 
The exercises constitute about one fifth of the text and contain 



yi PREFACE 

a wealth of material. They include much data taken from 
agricultural and other experiments, carefully selected to stimu- 
late thinking and to show the application of general principles 
to problems which actually arise in real life, and in the solution 
of which ordinary men and women are vitally interested. 

The book is intended for a course of three hours a week for 
one year, but it can be shortened to a half-year course. The 
chapters on statics, small errors, land surveying, annuities, 
compound interest law, and as many as is desired at the end, 
can be omitted without breaking the continuity of the course. 

The first two chapters are more than a mere review. This 
matter is so presented as to give the student a new point of 
view. The treatment will show the significance and importance 
of certain fundamental relations among the concepts and 
processes of arithmetic and algebra which the student may 
have used somewhat mechanically .in secondary school work. 
Well prepared students can read these chapters rather rapidly, 
however. 

The four place mathematical tables printed at the end of the 
text have been selected and arranged for practical use as the 
result of long experience and actual use in computing, and are 
adapted to the requirements of the examples and exercises in 
the book. 

The first edition of this book contained problems, formulas 
and other matter taken from a large number of sources. Those 
passages that were directly from other books have now been 
entirely rewritten ; but the book remains indebted to a num- 
ber of others, notably SKINNER, Mathematical Theory of Invest- 
ment, and DAVENPORT, Principles of Breeding. Other references 
occur throughout the text. 

A. M. KENYON, 
W. V. LOVITT. 



CONTENTS 

CHAPTER PAGES ARTICLE 

I. Introduction 1-10 1 

II. Review of Equations 11-35 12 

III. Graphic Representation 36-71 34 

IV. Logarithms 72-90 63 

V. Trigo'nometry 91-138 74 

VI. Land Surveying 139-152 108 

VII. Statics 153-176 122 

VIII. Small Errors 177-189 139 

IX. Conic Sections 190-217 149 

X. Variation . .> 218-225 167 

XI. Empirical Equations 226-242 173 

XII. The Progressions 243-253 178 

XIII. Annuities 254-261 186 

XIV. Averages 262-268 194 

XV. Permutations and Combinations 269-274 201 

XVI. The Binomial Expansion Laws of 

Heredity 275-285 207 

XVII. The Compound Interest Law 286-290 217 

XVIII. Probability 291-303 219 

XIX. Correlation 304-313 232 

TABLES 314-333 

INDEX;. . 335-337 



vu 



MATHEMATICS 



CHAPTER I 
INTRODUCTION 

1. Uses of Mathematics. The applications of mathematics 
are chiefly to determine the magnitude of some quantity such 
as length, angle, area, volume, mass, weight, value, speed, etc., 
from its relations to other quantities whose magnitudes are 
known, or to determine what magnitude of some such quantity 
will be required in order to have certain prescribed relations 
to other known quantities. 

2. Measurement. To measure a quantity is to find its ratio 
to a conveniently chosen unit of the same kind. This number 
is called the numerical measure of the quantity measured. 

The expression of every measured quantity consists of two 
components: a number (the numerical measure), and a name 
(that of the unit employed). For example, we write: 10 inches, 
27 acres, 231 cubic inches, 16 ounces, 22 feet per second. 

3. Arithmetic and Algebra. In arithmetic we study the 
rules of reckoning with positive rational numbers. In algebra 
negative, irrational, and imaginary numbers are introduced, 
letters are used to represent classes of numbers, and the rules 
of reckoning are extended and generalized. Algebra differs from 
arithmetic also in making use of equations for the solution of 
problems requiring the discovery of numbers which shall satisfy 
certain prescribed conditions. 

2 1 



2 MATHEMATICS [I, 4 

4. Positive Numbers. The natural numbers 1, 2, 3, 4, etc., 
are the foundation on which the whole structure of mathe- 
matics is built. They are also called whole numbers, or 
positive integers. Together with the fractions, of which 1/2, 
5/3, .9, 2.31, are examples, they form the class of positive 
rational numbers. 

Every positive rational number can be expressed as a fraction 
whose numerator and denominator are whole numbers. 

Two quantities of the same kind are said to be commensur- 
able when there is a unit in terms of which each has for numer- 
ical measure a whole number. Consequently, their ratio is 
a rational number. If two quantities are not commensurable, 
they are said to be incommensurable. 

The ratio of two quantities which are incommensurable, 
such as the side and the diagonal of a square, or the diameter 
and the circumference of a circle, is an irrational number. 

No irrational number can be expressed as a fraction whose 
numerator and denominator are whole numbers. However, it 
is always possible to find two rational numbers, one less and the 
other greater than a given irrational number, whose difference 
is as small as we please. For example, 

3.162277 < VlO < 3.162278 

and the difference between the first and the last of these num- 
bers is only .000001. Two such rational numbers whose dif- 
ference is still less can easily be found. In all practical appli- 
cations, one of these rational numbers is used as an approximation 
for the irrational number. Thus, we may find the length of the 
circumference of a circle approximately by multiplying its di- 
ameter by 3f. If a closer approximation is needed, the value 
3.1416 is often used. 

The (positive) rational and the (positive) irrational numbers 
make up the class of (positive) real numbers. 



I, 7] INTRODUCTION 3 

5. Negative Numbers. Zero. To every positive real 
number r, there corresponds a negative real number r, called 
negative r. The negatives of the natural numbers are called 
negative integers. The real number zero separates the negative 
numbers from the positive numbers. It is neither positive nor 
negative and corresponds to itself. 

The negatives of negative numbers are the corresponding 
positive numbers; thus, ( 2) =2. 

6. The Four Fundamental Operations. The direct opera- 
tions of addition and multiplication of real numbers are so 
defined that they are always possible, and so that the result 
in each case is a unique real number. These operations are 
subject to the rules of signs and to the following fundamental 
laws of algebra. 

I. The commutative law: 

a + 6 = 6 + a, ab = ba. 

II. The associative law: 

(a + &) + c = a + (b + c), (a6)c = a(6c). 

III. The distributive law: 

a(b + c) = ab + ac- 

The indirect operations of subtraction and division of real 
numbers are always possible, division by zero excepted,* and 
the result is a unique real number. 

7. Involution and Evolution. Involution, or raising to pow- 
ers, is always possible, and the result is unique when the base 
is any real number provided the exponent is a positive integer. 

* Division by zero is excluded because, in general, it is impossible, and when possible 
it is trivial. Thus there is no real number which will satisfy the equation -x = o 4= 
0, and every real number satisfies the equation -x = 0. 



4 MATHEMATICS [I, 7 

Evolution, or extraction of roots,* is not always possible. 
Even when possible, it is not always unique. In particular, 
the square of every real number is a positive real number. 
Hence no negative number can have a real square root. On 
the other hand, every positive real number, a, has two real 
square roots: a positive one, which is denoted by the symbol 
^a; and a negative one, which is denoted by Va. In fact, 
every positive real number has exactly two real nth roots of 
every even index n, denoted by Va and "N/o, respectively. 
Every real number, r, has a unique real nth root of every odd 
index n, denoted by Vr; it is positive when r is positive, and 
negative when r is negative. 

8. Rational Exponents. Involution is extended to frac- 
tional exponents as follows. If a is a positive real number, 
and if m and n are natural numbers, we define a mln by the 
equation 

. i 
a mln -\a m . 

For example, 

8 2/3 = ^ = 4> 
ni 

In particular, a 1 '" = Va denotes the unique real positive nth 
root of a. 

If r is a positive rational number, ( a) r is defined only when 
r, expressed as a fraction m/n, in its lowest terms, has an odd 
denominator and in this case, 

(- a) r = (- l) m a r . 

For example, (- 32) = (- 32) 3/5 = (- 1) 3 (32) 3/5 = - 8, and 
(- 32) = (- 32) 4 / 5 = (- 1)(32) 4 / 5 = 16. 

In particular, ( a) 1/n = a 1/n = Va, if n is odd. 

* The index of a root is always a positive integer. 



I, 9] INTRODUCTION 5 

9. Negative and Zero Exponents. By definition, we write 
a~ b = b and a = 1, 

Q 

provided a =|= 0. Thus a~ b is defined for the same real values 
of a and 6 as is a 6 and the two are reciprocals.* For example, 

1 1 /I \~ 5/2 1 

g-2/3 _ _!_ = L * 

8 2/s 4 > 



A consequence of this definition is the rule: A factor may be 
moved from the numerator to the denominator of a fraction, or vice 
versa, on changing the sign of its exponent. For example, 



a 2 6c~ 3 _ 2- 1 d~ 1 e _ 
2de~ l " a-^-'c 3 ~ 2c 3 d ' 

= (a? - z 2 )-i/ 2 , 



Va 2 x 

or 2 + 2x-*y~ l + 7/- 2 = - + + - . 
x z xy y 2 

EXERCISES 

1. Verify the fundamental laws of algebra by making use of the 
three numbers f, 5-J-, |. 

2. How many real square roots has 24.5? 4.5? 

3. How many real cube roots has 6f? 12? 

4. Find the numerical value of each of the following expressions, 
exactly when rational, correct to three decimal places when irrational. 

(a) 9 s / 2 . (e) (32) s / 5 . (i) (-0.027) / 8 . 

(*>) Gfr) 2/s . (/)(-32). (j) (H)" 4 - 

(c) v< 3 . (g) (-2)"'. (*) (t) 1 ". 

(d) (- 2). (A) (- 0.375)^. (1) (- )w. 

* Two numbers are reciprocals when their product is + 1. Every real number has a 
reciprocal except 0, which has none. 



6 MATHEMATICS [I, 9 

5. Write each of the following expressions without radical signs. 
(a) -^32. (b) Vl28. _c) 



6) 2 . (i) ^4?-8 4 . 0') -v-^ 

6. Write each of the following expressions without negative expo- 
nents and simplify when possible. 



(27 x-V2" 12 )~ 1/3 - (c) (a^ + fc- 2 )- 1 . (/) (a 



ar 1 + y- 1 a" 1 - fe- 1 a;- 1 ?/ + xy 

,.. 3 a-W -S-^tfb ffc x x- l y-*z~ 3 + x*y*z 

3 2 a6- 1 +3a6 ' ar 3 ?/-^- 1 + XT/V ' 

10. Laws of Exponents. The following five laws are useful 
for the reduction of exponential and radical expressions to sim- 
pler forms. They are valid, (1) when the bases are any real 
numbers whatever, provided the exponents are integers or zero, 
and (2) when the exponents are any real numbers whatever, 
provided the bases are positive. 

I. a b - a c = a b+c . 

EXAMPLES. 3 2 tr* = 3~ 2 . ( - |) B ( - f )- 3 = ( - f) 2 . 

(2)-l/3(|)l/2 = (!)l/ 6- g-l/3 . g2 = 8 5/3 . 

11. a c b c = (ab) c . 

EXAMPLES. 2 3 5 3 = 10 3 . (- 3)- 2 (- 5)~ 2 = (15)~ 2 . 

(17)1/3(^)1/3 = 51/8, 

III. (a b ) c = a bc . 

EXAMPLES. (2 8 ) 2 = 2 6 . [(- |)- 2 ]- 3 = (- f) 6 . 

[(f) 1/3 l 6 = (I) 2 - 



I, 11] INTRODUCTION 

a b 

IV. r = a b ~ c . 
a c 

Q2 (_ 5N1/2 

EXAMPLES. = 3'. (_ |j-.,. = (- f>"- 

V. . 

4-2/5 (2\-2 

T?YAU*T>ri?a f4\-2/5 v " x _ f2\-2 

EXAMPLES. 3 _ 2/5 - ($) . 3 ^_ 2 - UJ . 

These laws are readily proved when the exponents are positive 
integers. Thus, to prove law II, when the exponent is a positive 
integer n, we write 

(1)(2)(3) (n)(n (2) (3) (n) 

a n -b n = a- a- a a-b-b-b b 

(1) (2) (3) (n) 

= ab-ab-ab ab = (ab) n . 

Similarly, each of the other laws can be proved when the 
exponents are positive integers. When the exponents are 
negative, we make use of the definition of 9. If they are 
positive fractions we make use of the following lemma: // a 
and b are real numbers of like sign, and if a n = b n , where n is a 
positive integer, then a = b. 

11. Binomial Theorem. By multiplying out, we find the 
following equalities: 

(x + 7/) 2 = x 1 + 2#y + ?/ 2 , 

(x + y) 3 = x 3 + 3z 2 ?/ + 3z?/ 2 + y 3 , 

(x + 2/) 4 = a; 4 + 4:X 3 y + 6.r 2 7/ 2 + 4xy 3 + y 4 , 

(x + yY = x 5 + 5x 4 y + lOo; 3 ?/ 2 + Wx 2 y 3 + 5xy 4 + y 5 . 

By observing the coefficients and the exponents of x and of y 
in the various terms, we observe the law by which these results 
can be written down without the work of multiplying them out. 

In the expansion of (x + y) n for n 2, 3, 4, 5, we note the 
following facts: 

(1) The number of terms is n + 1- 



8 MATHEMATICS [I, 11 

(2) The exponent of x in the first term is n and it decreases 
by 1 in each succeeding term; the exponent of y in the second 
term is 1 and it increases by 1 in each succeeding term. 

(3) The first coefficient is 1; the second is n; the coefficient 
of any term after the second may be found from the preceding 
term by multiplying the coefficient by the exponent of x and dividing 
by a number 1 greater than the exponent of y. 

These three statements constitute the binomial theorem, 
which will be proved in 208, Chapter XVI, for all values of 
x and y no matter how large the positive integer n may be. 
The coefficients which appear in these expansions are called 
binomial coefficients. For example, the numbers 

1, 5, 10, 10, 5, 1 

are the binomial coefficients for the fifth power. The binomial 
coefficients for the second, third, fourth, and fifth powers should 
be memorized. 

EXERCISES 

Use the laws of exponents to combine and simplify the following 
expressions. 

1. g-^-S^-S-^-S 2 -j- 8 3/4 -8 1/12 . 2. 3 2/B -4 2 / B -5 2 / B 4- 15 2 / B -8 2 / 5 . 
3. (3 2 -3 1/2 -5 6/2 ) 2 -i- (7 3 - 10 2 ). 4. (11 -3 2 + 7 4 ) 1 / 2 . 

C3/4 1 03/2 

K (K4 _ 92.96^1/2 A _ 7 *"* 

D< 8 8 ' 12 ' 3 3 / 2 ' 

40 2 " V48 V54 Vl2 

O. iro/o a. ;r~ . 1U. A - . 11. . 

52/3 V3 M36 >/6 



12. 



I, 11] INTRODUCTION 9 

Perform the indicated operations and simplify each of the following 
expressions when possible. 



\ 



16. . 7 ax- 18 

\Wy) UW ' 

19. f"- l -*Y l . 20 f^blY 3 ' 21 

\ *V 2 / ' \8aV/ 

22 / x + 2 \-i. / r &V 

\x*+x-2j \a-b) ' 

24. 63a 4 x 5 -^ 9a 3 x 2 H- 3a 2 x. 25. (o + 6) (a + 6). 

' 




3a 2 6 2 



28 



30 2a 2 + 7ax + 3x 2 ^ 3a 2 + 7ax + 2x 2 




2a + x a + 3x 



31 



a 2 - a - 20 a 2 - 2a - 15 a 2 - a a 2 

33. 



x 2 - 5x + 4 x 2 - lOx +21 x 2 - 9x + 20 

4. a + 3 v a 2 +a-2 . a 2 +3a+2 



6 
Multiply : 

34. a 5/6 _ 

35. a 1 / 2 + 2& 1 / 2 - 3c 1/2 by a 1/2 - 26 1/2 + 3c 1/2 . 

36. x 4 / 3 + 2x + 3x 2 / 3 + 2x 1/3 + x by x 2 / 3 - 2X 1 / 3 + x. 

37. Vx 3 - x 2 y - XT/ 2 + y 3 by Vx 3 + 3x 2 y + 3xy 2 + y 3 . 

38. o 1/4 - 6 1/4 by a 3 / 4 + & 3/4 . 39. x 3 / 5 - ?/ 2/8 by x 2 / 5 + y 3 '*. 

40. (a 1 / 2 ^/ 2 + c 1 / 2 ) 2 by (a 1 / 2 *) 1 / 2 - c 1/2 ) 2 . 

41. (a- 1/2 - 3) 2 by (3O 1 / 2 + I) 2 . 
Divide : 

42. x 8/2 + x 2 - 2X 1 / 2 + 1 by x + x 1 / 2 - 1. 

43. x 3 + 27x - 9x 1/2 - 10 by x - 3x 1/2 + 5. 

44. x-y- 6x 2 / 3 + 12X 1 / 3 - 8 by x 1 / 3 - y 1 ' 3 - 2. 

45. a 6 / 2 - a 2 6 + a 3 2 c - oc + a 1/2 & - 1 by o 1/2 - 1. 

46. a 2 + Sa 1 / 4 + 7 by a 1 / 2 + 2o 1/4 + 1. 



10 MATHEMATICS [I, 11 

Reduce each of the following to its simplest form : 
47. Vl2.25zV. 48. v / 15.625a 6 6 9 . 



49. V343a 10 6 25 . 50. V3a 2 6 - 2a 2 c. 

51. V(a 3 + 53) ( 2 _|_ 52 _ a ty f 52 . Vx 4 ?/ 2 - 2xy + x 2 ?/ 4 . 

53. VVl024. 54. 

55. V27av / 27a6 4 . 

57. \/1.35a 2 V6.25a ;! . 

59. V|05 -2V605+V845. 60. *V 192 - 2v / 375 + #648 

61. V72-V8-V50. 62. ^81 - 2\ y l92 + ^375. 

63. (VI53 - VTI7 + A/52 - V68)(V / 5T + V39). 

64. (Vl2 + V3 + Vs})(vls 

65. (2 + V3 + v / 4)(2 - V3 

66. (3V20 -4V5 + 5V2 -3V8)(V5 + VCL5). 




67. V19 + 3V2- V19-3V2. 68. Vl6 + Vl3- 
Rationalize the denominator of each of the following fractions : 

69. -Iz^L 70. 1^. 71. - 

3-2V2 2+V5 2V5+3V2 



72 x/ 3+ v/ 2 + \ / 2 -V3, 73 2+V5 x 5 -V2. 

\/3-V2 V2+V3 2-Vg 5+V2 

74 V2 - \ 7 3 75 Vl89+3V20_ 



V3^_|_V48-V'50-v / 75 V84-V80 

Expand each of the following expressions : 
76. (2p 
79. (1 - 

82. 

85. (1 + x 2 ) 2 . 

88. (fc 2 +3) 2 . 

91. (Va+Va; 

94. (2x - 37/) 3 

97. (1 + x) 3 . 
100. 



77. 


(5c - 9d) 2 . 


78. 


(4m - 3n) 2 . 


80. 


(1 - *) 2 . 


81. 


(fa + f 6) 2 . 


83. 


(l-i) 2 - 


84. 


(^ - tl/) 2 . 


86. 


(1 - X 2 ) 2 . 


87. 


(1 +Vx) 2 . 


89. 


(2t 2 + 5) 2 . 


90. 


(a 2 + a6) 2 . 


92. 


( a -l/2 + xl /t). 


93. 


(ftl/3 _ 2/ l/2)2_ 


95. 


(a + ^b) 3 . 


96. 


(v^+Vm) 


98. 


(1 - x) 3 . 


99. 


(1 + x 2 ) 3 . 


101. 


(x+y- a) 2 . 


102. 


(a 2 + ab + W 



CHAPTER II 
REVIEW OF EQUATIONS* 

12. Use of Equations. As indicated before, the chief ad- 
vantage of algebra over arithmetic in solving problems lies in 
the method of attack. The algebraic method is to translate 
the problem into an equation and then to solve the equation by 
general methods. 

13. Definition of an Equation. An equation is a statement 
of the equality of two expressions. Each of the expressions 
may contain letters and figures called knowns, representing 
numbers supposed to be given or known; letters called unknowns, 
representing numbers to be found; and symbols of operation 
and combination, such as +, , etc. 

As examples of equations in one unknown, we may write 

(1) x + 13 = 2x - 7, 

(2) x(S - x) = 2(x + l)(z 2 - x + 1), 

(3) x(x + 2) = (x - l)(x - 2) + 5x - 2, 

2x + l 2x - 1 _ 
x- 1 x + l 

(5) 7 Vx 6 + 6 >/3x + 4 = 4x + 3. 

As examples of equations in two unknowns, we may write 

(6) x 2 - if + 2y = 1, 

(7) (2x ?/) 2 5x 2 = 5?/ 2 (x + 2y) z . 
Similarly, we may have equations in more than two unknowns. 

*This chapter is intended for review work. Parts of it may bo omitted at the dis- 
cretion of the instructor, if it appears that the students do not need to review some of 
the topics. 

11 



12 MATHEMATICS [II, 13 

The expression on the left of the equality sign is called the 
left member, or the left side, of the equation. The other is called 
the right member, or right side. 

14. Substitution. It is often necessary to substitute for the 
unknowns in an expression such as one of the members of the 
above equations, certain definite numbers, called values of the 
unknowns. The result of such substitution is, in general, to 
reduce the expression to a single number. 

Thus, if we put 10 for x in equation (1), the left side reduces to 23 
and the right side to 13. If we put 20 for x, each member reduces to 
the same number, 33. 

Again, if we put 1 for x and 1 for y in equation (6), the left side 
reduces to 2 and the right side to 1; but if we put 2 for x and 1 for 
y, each member reduces to 1. 

15. Solution of an Equation. Any set of values of the un- 
knowns which reduces each of the two members of an equation 
to the same number is said to satisfy the equation, and to be a 
solution of the equation. A solution of an equation in one 
unknown is also called a root of the equation. 

The final test to determine whether a set of values of the 
unknowns in an equation is a solution or not, is to substitute 
these values for the unknowns and see whether the equation 
is satisfied or not. 

For example, x = 20 is a solution of equation (1), 13. The value 
x = 10 does not satisfy it. Again, z = |, x = 1, x = 2, are three 
solutions of (2). Every real number is a solution of (3). The value 
x = 2 is a solution of (4). The value x = 15 is a solution of (5). The 
values x = 2, y = 1 constitute a solution of (6). Every pair of real 
numbers constitutes a solution of (7). 

16. Identities. An equation which is satisfied by all values 
of the unknowns (excepting those values if there are any for 
which either member is not defined) is called an identity. An 



II, 16] REVIEW OF EQUATIONS 13 

equation which is not an identity is called a conditional equation, 
or when no ambiguity is likely to arise, simply an equation. 

EXAMPLES. Of the equations in 13, (3) and (7) are identities, the 
others are conditional equations. Also, 



is an identity; it is satisfied by all values of x, except x = 1 for which 
neither side is defined. 

The distinction in point of view between identities and con- 
ditional equations is fundamental. To show that an equation 
is not an identity, we need only find a single set of values of 
the unknown quantities for which both sides are defined, and 
for which the equation is not true. 

EXERCISES 

1. Which of the numbers 3.5, 2, 1, 0, \, 2, satisfy the 
equation 

i x+ 2_ 10 ? 
3 ~2x~+~l { 

2. Which of the numbers T V V7, 2 + V3, Vl4, 2 V3, are solutions 
of the equation x 2 + 1 = 4x? 

3. Which of the following pairs of numbers (0, 0), (1, 3), (4, 2), 
(0, 2), (1, - 1), (3, - 1), (4, 0), (3, 3), satisfy the equation 



Is this equation an identity? 

4. Which of the following pairs of numbers (0, 1), (1, 1), ( 1, 0), 
(2, 3), ( 2, 1), (1, 1), (3, 2), are solutions of the equation 

* + 2y = 1+ V( 1 __ V_ \ ? 

x + y x\ x + y) ' 

Is this equation an identity? 



14 MATHEMATICS [II, 16 

5. Which of the following equations are identities? 

(a) x(x 2 y 2 ) = (x + y)(x 2 xy). 
(6) x(x 2 + y 2 ) = (x - y)(x 2 + xy). 

(c) x(x + 7) - (x + 3)(x + 4) + 12 = 0. 

(d) x(7 - x) + (3 - x)(4 - x) = 12. 

(c) 4x 2 + 7x + 2y = 0. (/) 4x 2 + 7x - 2y = 0. 

(g) x* = (x 2 + l)(x + l)(x - 1) + 1. 

(h) tf = (1 + x 2 )(l + x)(l -x) + 1. 

(i) (ax - b) 2 + (6x + a) 2 = (a 2 + 6 2 )(1 + x 2 ). 

0") (ax - 6) 2 + (ax + &) 2 = (a 2 + 6 2 )(1 + x 2 ). 

(fc) (x - t/) 3 + (y - zY + (z- X? = 3(x - y)(y - 2) (a - x). 

(0 (x + 2/ + z) 3 - (x 3 + y 3 + s ) = 3(x + y)(y + 2) (2 + x). 

fm) V* + *^ . + W = 1 

^ ' (x - y)(x - z) ^ (y - z)(y - x) ^ (z - x)(z - y) 

17. Equivalent Equations. Two equations are said to be 
equivalent when every solution of the first is a solution of the 
second and conversely, every solution of the second is a solu- 
tion of the first. 

For example, the equations 

5-5 = o 
3 7 
and 

7x = 15 

are equivalent; each has the unique solution x = 2f. 
On the other hand 

2x - 3 = x - 1 
and 

(2x - 3) 2 = (x - l) z 

are not equivalent; the latter has the solution 1^, which does not satisfy 
the first. 

18. Transformations of Equations. The following changes 
in an equation lead always to an equivalent equation: 

1. Transposition of terms with change of sign. 

2. Multiplication, or division, of all the terms by the same 
constant (not zero). 



II, 18] REVIEW OF EQUATIONS 15 

If all the terms of an equation be transposed to the left side 
(so that the right member is zero), if the left member be factored, 
and if each of the factors be equated to zero, then the solutions 
of the separate equations so formed are all solutions of the 
original equation, and it has no others. 

EXAMPLE. The equations 

r 3 -4- "vr 
'-~--=x*-x + l and (x - l)(x - 2)(x - 3) = 

are equivalent, and the solutions of the latter are seen by inspection to 
be z = 1, x = 2, x = 3. 

The following changes in an equation lead to a new equation 
which is satisfied by every solution of the given equation, but 
which generally has other solutions also. 

3. Multiplying through by an expression containing un- 
knowns (defined for all values of these unknowns). 

4. Squaring both members, or raising both members to the 
same positive integral power. 

Since the new equation is not, in general, equivalent to the 
given equation, it is necessary to test all results by substituting 
them in the given equation in its original form. 

EXAMPLES. Every solution of the equation 

x 2 1 _ 5x 
6~ H ~ 6" 

is a solution of the equation 

x 3 + 6z = 5x*, 

which is formed by multiplying the first through by 6x; but they are 
not equivalent, since x = satisfies the second but does not satisi'y 
the first. 

Every solution of the equation 

3x - 1 = x - 1 
x + 1 ~ x - 2 



16 MATHEMATICS [II, 18 

is a solution of the equation 

3x 2 - 7x + 2 = x 2 - 1, 

which results from clearing the former of fractions. These two equa- 
tions are in fact equivalent. Each is satisfied by x |, and by x 3, 
and by these only. 

Every solution of the equation 

x - 4 = Vz + 2 
is a solution of the equation 

x 2 - 9z + 14 = 0, 

which results from squaring and transposition in the former; but they 
are not equivalent; the latter equation has the two solutions x = 2, 
x = 7, while the former has only one, x = 7. 

19. Simultaneous Equations. When a common solution of 
two or more equations is sought, the equations are said to be 
simultaneous. For example, each of the equations 

(8) 3z - 2y = 4 
and 

(9) 2x - y = 3 

has an infinite number of solutions: (0, 2), (2, 1), (4, 4), 
(G, 7), etc., satisfy (8), and (1, - 1), (2, 1), (3, 3), (4, 5), etc., 
satisfy (9). But (2, 1) is the only common solution. 

By a solution of a set of equations is meant a common solu- 
tion of all the equations of the set, regarded as simultaneous 
equations. Thus, the set of equations (8) and (9) has a unique 
solution, namely, x = 2, y = 1. 

Two sets of simultaneous equations are equivalent when 
each set is satisfied by all of the solutions of the other set. 

If each of two or more equations from a set of simultaneous 
equations be multiplied through by any constant, or by any 
expression containing unknowns,* and if the resulting equations 

* Defined for all values of the unknowns. 



II, 20] REVIEW OF EQUATIONS 17 

be added or multiplied together, the new equation will be 
satisfied by all the (common) solutions of the given set. 
EXAMPLE. If in the set of simultaneous equations, 

2x 2 + 2y 2 - 3x + y = 9, 
3x 2 + 3yi + x _ y = 14> 

we multiply the first by 3, the second by 2, and add, the resulting 
equation 

llx - 5y = 1 

is satisfied by every solution of the given set. One such solution is 

x = 1, y = 2. 

20. Elimination. By a proper choice of multipliers we can 
use the above principle to secure a new equation lacking a 
certain term, or certain terms, which occur in the given set of 
equations. The missing terms are said to have been eliminated 
and this process is called elimination by addition. 

EXAMPLES. We can eliminate the term in x 2 from the equations, 



- 2 
5 



5x 2 - 9x = 2, 



2x 2 - x = 6, 

by multiplying by 2 and + 5, respectively, and adding. The result 
is 13x = 26. We conclude that if the given equations have a common 
solution, it is x = 2, and we verify that this is a solution of each. 
If we eliminate x 2 from the equations, 

- 2j|5x 2 + 9x = 2, 
5||2z 2 + 5x = 5, 
we obtain 

7x = 21. 

Since x = 3 is not a solution of the given equations, they have none. 
When y is eliminated from the equations, 

2||3x 2 - 4x - 15y + 1 =0, 
- 3 || 2x 2 - 3x - 10y + 1 =0, 
the result is 

x - 1 = 

and on substituting x = 1 in either of the given equations, we find 
y = 0. Therefore (1, 0) is the unique common solution. 
3 



18 MATHEMATICS [II, 20 

When it is possible to solve one of a set of simultaneous equa- 
tions for one of the unknowns, we can eliminate this unknown 
by substituting the value thus found in the other equations of 
the set. This is called elimination by substitution. 

For example, to eliminate t from the set of equations, 
x = a(l+ < 2 ), 
y = o(l + 0, 
solve the second for t and substitute this value in the first. The result is 

x = ^ - 2y + 2a, 

which is equivalent to the equation 

y 2 ax lay + 2a 2 = 0. 

If we can solve each of two simultaneous equations for the 
same unknown, this unknown will be eliminated by equating 
these two values to each other. This is called elimination by 
comparison. 

Thus, if we solve each of the equations 

z 2 - xy - 4x + 2y + 1 =0, 
2x* - 2xy + 3x - 2y + 3 =0, 
for y, and equate these values, the result is 

x* - 4x + 1 = 2x* + 3x + 3 
x -2 2x + 2 

which is equivalent to the equation 

(5z + 8)(x - 1) = 0. 

21. Linear Equations. An equation of the first degree in 
the unknown quantities is called a linear equation. A set of 
linear simultaneous equations can be solved, if they have a 
solution, by successively eliminating the unknowns until a single 
equation in one unknown is obtained. 



II, 21] REVIEW OF EQUATIONS 19 

EXAMPLES. 

3 2x + y = 4, 
1 x - 3y = 9. 

Eliminating y by addition, we obtain 

7x + 0-y = 21. 
Eliminating x, we get 

Q-x + 7y = - 14. 

We conclude that if the given equations have a solution it is x = 3, 
y = 2, and we verify that this is a solution. 

To eliminate x by substitution from the equations 

7x - Qy = 15, 
5x - Sy = 17, 

solve the first for x and substitute this value in the second. The 
result is 

/ n,,. i 1 c \ 

- 8y = 17, 



which is equivalent to y = 4. Substituting 4 f or y in either of the 
given equations, we fird x = 3. Finally, we verify that x = 3, 
y = 4, is a solution of the given set. 

To eliminate x by comparison from the equations 

3x - 7y = 19, 
2x - 5y = 13, 
solve each equation for x, and equate the results. This gives 

7y + 19 5y + 13 
3 2 

which is equivalent to y = I. Substituting this value for y in either 
of the given equations leads to x = 4. 



20 MATHEMATICS [II, 21 

EXERCISES 

1. Solve the following equations and determine whether or not the 
two equations in each pair are equivalent. 

x + 5 x + 1 _ x - 3 _ 1 3x - 7 

(a) ~2~ T~ ~2x~~ ~ 3 ~ ~^x~~ 

n) y~ 7 + 2 - y + 8 2(y ~ 7) 4. ^^ - y + 3 

() 5 10 ' jf + 3 i ,_28 + y-4~y + 7' 

3 - 5 _ 9< - 7 = 2_ 5t + 4 _ lit - 2 

4 12 ~ 3t' 2t 6t 

6x - 1 8x + 3 4x - 3 3.7 

(d) ~l- -fflr - 5 ' 4i + x" 15 ^ 
, 5x + 1 , a; 15x 2 5x 8 



w '2x~3 T 2x-3' 3x 2 + 6x + 4 

(?) x - 1 = V3x - 5, x 2 - 5x + 6 = 0. 



= 5. 



(0 x = 2, x(x - 1) = 2(x - 1). 
0') 2x = 1, Sx 3 - 12x 2 + 6x = 1. 

2. Solve the following simultaneous equations and determine whether 
or not the two sets in each pair are equivalent. 

f 3x + 2y = 32, f 7x - y = 1, 

(a) \ 20x - 3y = 1. t 9x + 4y = 70. 

Ans. (2, 13). The two sets are equivalent. 

3x + 7y = 2, f 2x + 3y = 0, 

7x + 8y = - 2. t 4x + y = - 4. 

2t = - 3, f s + 5t = 3, 



f 
I 



1 i 5s - 3t = - 6. (. s + t = 0. 

f f * + y = i, f 5x + 4y = 22, 

1 1* - fjr - H. I 3x + t/ = 9. 

Ans. These two sets are not equivalent. 

3x - 2y = 1, t x - y = 1, 

(e) { 3x + 4z = 5, | x + z = 1, 

3y + 5z = 4. I y + z = 0. 
Ans. (1, 2, 2). Ans. An infinite number of solutions. 



II, 21] REVIEW OF EQUATIONS 21 

3. Eliminate the x 2 term from the equations 

x 2 - 2y* + 13x +2y = l, 3z 2 + 4?/ 2 - x + Qy = 3. 

4. Eliminate y from the equations 

x 2 + 3xy - x + 1 = 0, 2x + y + 1 = 0. 

5. Eliminate t from the equations 



, 
l+t*' 1+1? . 

6. Eliminate t from the equations 

J 2 x = < 4 + < 2 + 1, ty = P - 1. 

7. Eliminate ra from the equations 

w = mx, x = my. 

m 

8. Clear the following equations of fractions and radicals and 
determine in each whether the resulting equation is equivalent to the 
given one : 



2-3x 3x-l x - 



(d) 



4x9x x 2 9 a; + 3 

(e) x + Vx + 6 = 0. Cf) V'6 - 5x = 




(i) 

9. How must 1% ammonia and 28% ammonia be mixed to get 12 
pints of 10% ammonia? 

Ans. 8 and 4 pints. 

10. Two given mixtures contain respectively p% and q% of a cer- 
tain ingredient. Show that if x units of the first be combined with y 
units of the second so that the resulting mixture contains r% of this 
ingredient, then x:y = r p:q r. 



22 MATHEMATICS [II, 21 

10. Assume that gravel has 45% voids and sand 33%, and that 4 
bags of cement make 3.8 cu. ft., how much cement, sand, and gravel 
are necessary to make 1 cu. yd. of concrete? 

(a) in a 1:2:4 mixture. (fe) in a 1 : 3 : 6 mixture. 
(c) in a 1 : 2 : 3 mixture. (d) in a 1 : 3 : 5 mixture. 

11. How many pounds of skimmilk must be extracted from 12000 
Ibs. of 4% milk to raise the test to 4.5%? 

Am. 1333| Ibs. 

12. How many pounds each of 40% cream and skimmilk are required 
to make 125*pounds of 18% cream? 

Ans. 56.25 Ibs. cream, 68.75 Ibs. skimmilk. 

13. How many pounds each of 25% cream and 3.5% milk are 
required to make 130 pounds of 22.5% cream? 

Ans. 114.8 Ibs. of 25%, 15.2 Ibs. of 3.5%. 

14. How much 25% cream must be added to 1000 pounds of 50% 
cream to reduce it to 40% cream? 

Ans. 666f Ibs. 

15. How many pounds each of 50% and 25% cream must be mixed 
together to produce 1000 pounds of 40% cream? 

Ans. 600 Ibs. of 50%, 400 Ibs. of 25%. 

22. Polynomials. Expressions of the form 

1 - x, x z - 3% + 2, x+ A/3.T 3 + 3.4 + lx z , 
x* - 2x* + x - 5 



, 3x, 



are examples of polynomials in x; 



y - 5 + 4y 2 , z 5 + ^ + A/2z 2 - \ , ^ + a 3 - 1 + 2a, 
5 / o 

are polynomials in y, z, and , respectively 

A polynomial, in x for example, is a sum of terms each con- 
taining a positive integral power of x multiplied by a coefficient 
independent of x, and usually also an absolute term. 

If any number (value of x) be substituted for x, the poly- 
nomial reduces to a number called a value of the polynomial. 



II, 23] REVIEW OF EQUATIONS 23 

To each value of x, which is called the variable, there corre- 
sponds a unique value of the polynomial. For example, the 
values of x 2 3x + 2 which correspond to x = 0, x = 1, 
x = \, are 2, 0, f . 

The degree of any term in a polynomial is the exponent of 
the variable in that term. The degree of a polynomial is the 
degree of the term of highest degree in it. Polynomials are 
usually arranged according to the degrees of the terms and it is 
sometimes convenient to supply with zero coefficients missing 
terms of degree lower than the degree of the polynomial; thus 

3x 2 + 0-x + 2, z 5 + 0-z 4 + 0-z 3 + V2> + fz - f . 

A sharp distinction is to be made between the coefficients 
and the exponents in a polynomial. The coefficients are very 
general: they may be any real numbers whatever, natural 
numbers, rational or irrational numbers, positive, negative, or 
zero. On the other hand the exponents are very special: they 
must be positive integers. Thus while the expressions 

z 2 + 1, ?/ + y/2, z 2 - TTZ + V2, 
are polynomials, the expressions 

3.1/2 + 1? y t _j_ 2/y, z 2 - 32 + Vz, 
are not. 

23. Polynomial of the nth Degree. A polynomial of de- 
gree n in x (n being any given natural number 1, 2, 3, ) can 
be reduced by merely rearranging its terms and adding the 
coefficients of like powers of x to the form 



in which the o's (coefficients) are any real numbers (oo =J= 
but any or all the others may be zero), and the exponent n is a 
positive whole number. 



24 MATHEMATICS [II, 24 

24. Linear Equations. An equation of the first degree, or a 
linear equation, in one unknown, x for example, is the result 
of equating to zero a polynomial of the first degree in x, 

(10) ax + b = (a + 0). 

This equation has one and only one solution. The method of 
finding the solution is already known to the student. 

25. Quadratic Equations. An equation of the second de- 
gree, or a quadratic equation, in x for example, is the result of 
equating to zero a polynomial of the second degree in x, 

(11) ax 2 + bx+c = Q (a 4= 0). 

Any equation which can be reduced to this form by merely 
transposing and combining like terms is also called a quadratic. 
Thus, 

(x - l)(x - 2) = Q(x - 3) 
is a quadratic. 

SOLUTION BY FACTORING. If the polynomial ax z + bx + c 
can be factored into two linear factors in x (i. e., polynomials 
of the first degree in x) the roots of the quadratic equation 
ax 2 + bx + c =0 can be found by inspection. 

EXAMPLE 1. Solve 6x 2 + x = 15. Transpose all terms to the 
left side and factor. In order to do this we seek a pair of numbers 
whose product is 6 and another pair whose product is 15 and such 
that the cross product is 1. The work may be put down as follows: 

6z 2 + x - 15 = 
-5 



This gives the cross product 13, but a few trials of other factors and 
other arrangements quickly leads to the combination 



-3 



II, 25] REVIEW OF EQUATIONS 25 

which gives the cross product 1 as desired. Hence the factors are 
3x + 5 and 2x 3, and we have to solve the equation 

(3x + 5)(2x - 3) = 0. 

On equating the first factor to zero (mentally) and solving we get 
Xi = 5/3 and similarly from the second factor xz = +3/2, and 
these are the two solutions of the given quadratic equation. 

EXAMPLE 2. 

Q o 2_ "" "^ 

3x 2 i x = = - . 

Transposing and combining terms this reduces to 

& + & x = 
which factors by inspection into 

Z(- 2 7 Q X + *V) = 

whence xi = and xz = ?V- 

If there are fractional coefficients in a quadratic it is usually best to 
reduce it to an equivalent equation free from fractions by multiplying 
every term by the least common multiple of all the denominators. 
Thus in Example 2, we could multiply every term by 21 and obtain, 

63x 2 - 14x = 3x 2 - 15x. 

EXERCISES 

Solve the following quadratic equations. 

1. 2z 2 - 5x = 3. 2. 10x 2 + x = 2. 

3. 6x 2 + 5x = 6. 4. 15x 2 - x = 6. 

5. 6x 2 - 5 = 7x. 6. 28x 2 - 15 = x. 

7. 135x 2 + 3x = 28. 8. 78x 2 - x = 2. 

9. 3?/ 2 + y = 10. 10. 147/ 2 + y = 168. 

11. Gy 2 + lly = 35. 12. 15?/ 2 + 4 = 16y. 

13. 6a 2 + a = 5. 14. 2a + 3 = 8a 2 . 

15. 9a(2a + 1) = 14. 16. 10(2a 2 - 3) + a = 0. 

17. 3(2s 2 - 7) = 5s. 18. 15(2i 2 - 1) + 7t = 0. 

19. p(12p - 7) = 10. 20. 5(3r 2 - 8) + r = 0. 



26 MATHEMATICS [II, 25 

21. A z 2 + 2x + 1 T 4 7 = 0. 22. v(y ~ 1} = 3(y + 1) - 2/2. 

23. 2(2 - 1) = ;ft(6z - 1). 24. P + 3 - 1 = - 



25. (1 - e 2 )z 2 - 2px + p2 = 0. 

Clear the following equations of fractions, solve the resulting equa- 
tions and test their solutions in the given equations. 



07 



2x - 7 x - 3 ' x 2 - 1 2(x + 1) 4 ' 

3Oi 1 O O A 

M, X i_ zx K OQ z _ d 

z-5z-3 z-1 x-2 x - 3 x-4' 

26. Solution of a Quadratic by Completing the Suqare. 

If the polynomial on the left of the quadratic equation 

ax 2 + bx + c 

cannot readily be factored by inspection, the equation can 
be solved by transposing the absolute term c, completing the 
square of the terms in x and extracting the square roots of 
both sides. 

To complete the square of ax 2 + bx is to find a number d 
such that ax 2 + bx + d is the square of a linear factor in x and 
it can always be done as follows: 1) extract the square root of 
the first term; 2) double this; 3) divide this into the second term; 
4) square the quotient. 

EXAMPLE. Solve 6x 2 4x 1 =0. 

Transpose 1, and find the number to complete the square of 
6x 2 - 4x by the above four steps: 1) xV6, 2) 2x>/6, 3) 2/V6, 4) 2/3; 
add this to both sides: 

6x 2 - 4x + f = f . 

Extracting the square roots, we have 

-v-v/fi A/1 = 4- A/ 
* ' u 3 3I "? 

whence solving for x, we find 

r, i -L. IA/TO r = l i \/T7) 

^l 3 i^ 6 '-*-", ^2 3 ^^ i(XV/. 



II, 27] REVIEW OF EQUATIONS 27 

The computations are more easily made, if we multiply the given 
equation through by a number which will make the coefficient of x 2 a 
perfect square. In the above example we should have to solve the 
equivalent equation, 

36z 2 - 24x + ( ) = 6. 

The number required to complete the square is 4, 

36z 2 - 24z + 4 = 10. 
Whence 

6z - 2 = VlO 
and 

Si -i- |ViO. 



EXERCISES 

Solve these equations by completing the square. 
1. 4z 2 + 3z = 9. 2. 25z 2 - 14z + 1 = 0. 

3. 50z 2 + 12x = x 2 - i 4. (x 2 + 1) V3 = 4z. 

5. 12z 2 + 5x = 1. 6. 6^ 2 + 1 = 6y. 

7. 32 2 = 13(2 - 1). 8. x + 2 = llar(l - x). 

9. 12< 2 - 4(o + b)t + a6 = 0. 10. 2(?/ 2 + c 2 ) = 5ey. 

27. Solution of a Quadratic by a Formula. By the process 
of completing the square, a formula for the roots of the general 
quadratic equation can be found as follows. Given the equation 

(12) ax* + bx + c = 0, 

multiply through by 4a, transpose 4ac, and complete the square, 

4a 2 z 2 + 4a6x + 6 2 = - 4ac + 6 2 , 
extracting the square roots, we have 



2ax + b = V6 2 - 4oc, 
whence we find 

- & =fc Vfe 2 - 4ac 
~^a^ - 
which gives the two roots 



- 6 + & 2 - 4ac - 6 - 



28 MATHEMATICS [II, 27 

This result may be used as a formula for the solution of any 
quadratic equation by substituting for a, 6, c, of this formula 
their values from the given equation. 

EXAMPLE. Solve 3x 2 + 4x 15 = 0. 
Here a = 3, 6 = 4, c = 15, and by the formula 



_ - 4 V16 + 180 
6 

whence 

_ -2+7_5 

3/1 "~ * -. cLIlCi 3/9 " 

33 a 

EXERCISES 

Solve the following equations by the formula. 

1. 2x 2 + 3x = 4. 2. x 2 = 220 + 9x. 

3. 5x 2 + 3x = 3. 4. 5x 2 + 5x + 1 = 0. 

5. 15?/ 2 = 86y + 64. 6. 5z 2 = 80 + 21. 

7. a 2 + a = 3. 8. p 2 + 3p = 40. 

9. I 2 + 3a 2 = 4o< - 1. 10. 5m 2 + 21m + 4=0. 

Solve the following equations by any method and test all results in 
the given equation. 



11. (2x - 3) 2 = 8x. 12. x 2 - 2 A 

10 " x , X ~r " o 14* C 'l 


!3x + 2 = 0. 


J ' x + 2 ' 2x x 

I K X 1 , 1 * f) 


x - 1* 


x(x - 2) 2x - 2 ' 2x 
ifi 4 ! 3 2 



'x 1 4 x x 2 3 x' 

17. 3x 2 + (9o - l)x - 3a = 0. 18. x 2 - 2ax + a 2 - fc 2 = 0. 
19. c 2 x 2 + c(a - 6)x - afc = 0. 20. x 2 - 4ax + 4a 2 - 6 2 = 0. 

21. x 2 - 6acx + a 2 (9c 2 - 46 2 ) = 0. 

22. (a 2 - 6 2 )x 2 - 2 (a 2 + 6 2 )x + a 2 - ft 2 = 0. 



II, 28] REVIEW OF EQUATIONS 29 

Solve for y in terms of x. 

23. x 2 + 12xy + 9y 2 + 3 = 0. 24. x 2 - 4xy - 4y* + x = 0. 

25. llx 2 + 3Qxy + 25y* = 3. 26. 8x 2 - 12x?/ + 4y 2 = x + 1. 

27. Gx 2 - XT/ - 2y 2 = 0. 28. 21x 2 = xy + lOy 2 . 

29. 30x 2 + 150* = 43xy. 30. 12x 2 + 41xy + 35y 2 = 0. 

31. 2x 2 + 3xy - 2y 2 + x + 7y - 3 = 0. 

32. 3x 2 + lOxy + 8?/ 2 + 4x + 2y - 15 = 0. 

33. 10x 2 + 7xy + r/ 2 - x - 2?/ - 3 = 0. 

34. 12x 2 = 4xy + 2ly 2 + 2x + 29y + 10. 

35. A farmer mows around a meadow 18 X 80 rods. If the swath 
averages 5 ft. 6 in., how many circuits will cut half the grass? 

Ans. 12. 

38. What are eggs worth when 2 more for a quarter lowers the price 
5 cents a dozen? 

37. If the radius of a circle be divided in extreme and mean ratio 
the greater part is the side of the regular inscribed decagon. What 
is the perimeter of the regular decagon inscribed in a circle 2 feet in 
diameter? Ans. 6.180 

38. When a heavy body is thrown upward with an initial velocity 
v ft. per second, its distance from the earth's surface at the end of t 
seconds is given by the equation d vt 16< 2 . If a projectile is shot 
upward with a muzzle velocity of 1000 ft. per second, when will it be 
15,600 ft. high? Ans. 30 and 32^ sec. 

28. Equations in Quadratic Form. The terms of an equa- 
tion which is not a quadratic in the unknown can sometimes 
be grouped so as to make it a quadratic in an expression con- 
taining the unknown. Thus, x 4 13z 2 + 36 = is not a 
quadratic in x but it is a quadratic in x 2 ] again if the terms of 
x* 6x 3 + 7x 2 + Qx = 8 be grouped in the form 



- 2(x 2 - 3z) = 8 
it is seen to be a quadratic in (x 2 3x). 



30 MATHEMATICS [II, 28 

EXAMPLE 1. Solve 6x - 7Vx = 20. 

Transpose 20 and this can be solved by the formula as a quadratic 
in Vx; whence, 

r 7 A/49 + 480 

Vx= "IT" 

and, since the positive square root cannot be negative, 

>lx = 2.5 and a; = 6.25. 
We verify that this satisfies the given equation. 

EXERCISES 

1.x 4 - 13x 2 + 36 = 0. 

2. x + Vx + 6 = 14. Ans. 10. 

3. 2x 2 + 3 Vx 2 - 2x + 6 = 4x + 15. Ans. - 1 and 3. 

= 2. Ans. and 1. 



x + Vl -x 2 

5. s Vx + 18 = 5 3 Vx~ 2 . Ans. 8 and - 729/125. 

6. x 4 - 6X 3 + 7x 2 + 6x = 8. Ans. - 1, 1, 2, 4. 

29. Imaginary Roots. . There are quadratics which are not 
satisfied by any real number. For example, z 2 = 4, x 2 + 2x 
+ 2 = 0. This is because the square of every real number 
(except 0) is positive. If we attempt to solve the equation 
x 2 + 2x + 2 = either by completing the square or by the 
formula we are led to the indicated square root of a negative 
number, and this is not a real number; thus 



These, and other considerations have led to the invention 
of numbers whose squares are negative real numbers; they are 
called imaginary numbers. The imaginary unit is usually 
denoted by i. Hy definition, we have 



II, 29] REVIEW OF EQUATIONS 31 

The number r-i, where r is any real number is called a pure 
imaginary number; e. g., 2i, 5i, 3i, -- %i, i V3, etc. The 
squares of pure imaginary numbers are negative real numbers; 
e. g., (2i) z = 2 2 i 2 = - 4; (- 3i) 2 = (- 3) 2 i 2 = - 9; (i VJF) 2 
= - 3. 

Conversely, the square roots of negative real numbers are 
imaginary numbers; the square roots of 4 are 2i and 2i; 
i. e., V^l =_i VI" = 2i, - - V- 4 = - i VI" = - 2i; V^3 

= i V3, -V 3 = i V3 ; in general, V p = i ^p, where 
p is a real positive number. 

Expressions of the form 2 + 5i, 1 i, 3 2i, I + i, 
etc., indicating the sum of a real and an imaginary number are 
called complex numbers. They may be added, subtracted, 
multiplied, and divided by the laws of algebra as though i were 
a real number and the results simplified by putting 1 for i-, 

- i for i 3 , + 1 for i 4 , etc. 

We can now say that every quadratic equation can be solved. 
The solutions of the equation 

x 2 - 2x + 2 = 
may be found by the formula, 



2 A/4 - 8 
,-- --^ 

whence 

x\ = 1 + i and xz = 1 i; 

and we verify both these answers as follows: 

(1 + i') 2 - 2(1 + i) + 2 = 0, (1 - *)* - 2(1 -0+2 = 

EXERCISES 

i. x- - 4x + 5 = 0. 2. x z + <ox + 13 = 0. 

-4ns. 2 i. Ans. 3 2i. 

3. 36x 2 - 36x + 13 = 0. 4. 2x 2 + 2x + 1 = 0. 

Ans. 1/2 t'/3. -4ns. 1/2 t/2. 



32 MATHEMATICS [II, 29 

5. x 2 + 4 = 0. 6. x 2 + x + 1 = 0. 

Ans. 2i. Ans. - 1/2 i V3/2. 

7. z 2 - 2z + 3 = 0. 8. x 2 - |x + 1 = 0. 

9. x 2 - 2x VJj + 7 = 0. 10. 2z 2 - 2x + 5 = 0. 

11. x 2 + 3x + 2.5 = 0. 12. 49z 2 - 56x + 19 = 0. 

30. The Sum and Product of the Roots. The two roots 
of the quadratic equation 

ax 2 + bx + c = 
are by (14), 27, 



- 6 + V& 2 - 4oc - 6 - Vb 2 - 4oc 

zi = and x 2 = - . 

2a 2a 

The sum of these roots is b/a, and their product is + c/a, as 
may be seen by adding and multiplying them together. 

We can thus find the sum and the product of the roots of a 
given quadratic equation without solving it. Thus in the 
equation, 

36z 2 - 36z + 13 = 

the sum of the roots is 1, and their product is 13/36. 
Again in the equation, 

my z 4ay + 4a6 = 

the sum of the roots is 4a/m, and their product is 4ab/m. 

31. Equation having Given Roots. We have seen that if 
the left member of the quadratic equation 

ax 2 + bx + c = 

can be separated into linear factors, its roots can be found by 
inspection. Therefore if we wish to make up a quadratic 
equation whose roots shall be two given numbers, r and s for 
example, we hare only to write 

a(x r)(x s) =0 



II, 33] REVIEW OF EQUATIONS. 33 

and multiply out. The factor a is arbitrary and may be chosen 
so as to clear the equation of fractions if desired; thus, to make 
an equation whose roots shall be f and f , we write, 

a(z-f)(z+f) =0, 

and if we take a = 10, the resulting equation is 
10z 2 - llz - 6 = 0. 

32. Number of Roots. Conversely, it is readily shown that 
if r and s are roots of the quadratic equation, 

ox 2 + bx + c = 

then the left member can be factored in the form, 
(15) a(x - r)(x - s) = 

and this shows that no quadratic can have more than two 
roots. Some quadratics have only one root; for example 
4z 2 + 9 = 12x is satisfied only by x = 3/2. 

If 6 2 4ct = 0, then the polynomial ax 2 + bx + c is a 
perfect square and the equation ox 2 + bx + c = has only 
one root, and conversely. 

For, if 6 2 4oc = 0, then c = 6 2 /4a and this is precisely 
the number necessary to complete the square of ox 2 + bx. 
Also if 6 2 4oc = 0, the formula (13), 27, gives not two but 
one root. 

33. Kind of Roots. If a, b, and c, are real numbers and if 
6 2 4ac > 0, then the quadratic equation ax* + bx + c = 
has two real roots; but if 6 2 4oc < 0, the equation has two 
imaginary roots. 

This is seen at once on noting the formula (13), 27, which 
gives the roots. 
4 



34 MATHEMATICS [II, 33 

EXAMPLE 1. 4z 2 - 12x + 9 = 0. 

Here b 2 4ac = 144 144 = 0, the left member is a perfect square 
and the equation has only one root. 

EXAMPLE 2. 3x 2 - 5x + 2 = 0. 

Compute b 2 4ac = 25 24 = + 1, which shows that the equa- 
tion has two real roots. 

EXAMPLE 3. x 2 + x + I = 0. 

Here b 2 4oc = 3, which shows that the equation has imaginary 
roots. 

If a, b, and c, are rational numbers, then the roots of the equation 
ax 2 + bx + c = are rational if b 2 4ac is a perfect square, i. e., the 
square of a rational number: in particular if a, b, and c, are integers 
and if b 2 4ac is a perfect square the left member of the equation can 
be factored by inspection. 

EXAMPLE 4. 2x 2 - x - 6 = 0. 

Here b 2 4ac = 1 + 48 = 49; the left member factors into 

(2x + 3)(x - 2) = 
whence the roots are 3/2 and 2. 

EXERCISES 

1. Form the equations whose roots are 

(a) 1, 3, - 5. (6) - 2, 3, - 4, 6. 

(c) 1/3, - 7/2, 3/5. (d) 1, 4. 

(e) V2, V5. (/) 0, - 2, V^"2. 

(?) 3, 5 >/5. (ft) 4 A/3, - 1 VS. 

(i) - a, - 0, - 7. 0') ka > k-P, ky. 

(k) <x+k,0^k y+k. (/) I/a, 1//S, 1/7. 

(TO) a, ft 7. () a 2 , /S 2 , 7 2 - 

(0) a - 0, - 7, 7 - (P) "A 07, 7- 

2. Determine the nature of the roots of the following equations. 
(a) 3x 2 - 4.r - 1 =0. (6) 5x 2 + 6z + 1 = 0. 

(c) 2x 2 + x - 6 = 0. (d) x* - 2x - 1 = 0. 

(e) 5x 2 - 6x + 5 = 0. (/ ) x 2 - 6 V3x - 5 = 0. 

(g) x 2 + x + 1 = 0. (h) 13x 2 - 6 >/3x + 7 = C. 

(1) 3x 2 + 2x + 1 = 0. 0') 2 * 2 - 16x + 9 = 0. 
(*) 5x 2 - 12x - 8 = 0. (0 6x 2 + 4x - 5 = 0. 



II, 33] REVIEW OF EQUATIONS 35 

(m) 5x + 7 = (3x + 2)(x - 1). (n) 5(x 2 + x + 1) = 1 - 16x. 
(o) 2x(x - 3) = 7(3x + 2). (p) 7(x 2 + 5x + 3) = x(l - x). 

(9) 3x(x + 1) = (3 - x)(3 + x). (r) 3x 2 = 13(x - 1). 
t (s) 0, + 2)(V - 2) = 2y - 7. (0 3fo + l)fo - 1) = 4y. 
(M) 60(3 - %) = 19(0 - I) 2 - () 2/ 2 - 2yV3 + 7 = 0. 

3. Without solving find the sum and the product of the roots of each 
of the equations in Ex. 2. 

4. Determine the nature of the roots of the following equations in 
which a, b, c are known real numbers. 

(a) (x - a) 2 = 6 2 + c 2 . (6) (x + a) 2 = 86 2 . 

(c) a (ax 2 + 26x - a) = b(bx 2 - lax - b). 

(d) 7/ 2 = 2a(y -b) + 2b(y - a). 

(e) (a + 6 - c)?/ 2 - 2cy = (a + b + c). 

(/) (a + b - c)x 2 + 4(o + b)x + (a + b + c) = 0. 

(0) (b + c - 2a)x 2 + (c + a - 26)x + (a + 6 - 2c) =0. 

5. Determine values of a for which each of the following quadratic 
equations will have equal roots. 

(a) x(x + 4) + 2a(2x - 1) = 0. (6) (x - I) 2 = 2a(3x - 7) - 20. 
(c) (x - a) 2 = a 2 - 8a + 15. (d) x 2 - 15 = 2a(x - 4). 

(e) 3(x 2 + Sax - a) = x. (/) 9x 2 + 6(0 - 4)x + a 2 = 0. 

(g) 3ax(x - 1) = ax - 2. (h) (a + l)x 2 -(a + 2)x +fa = 0. 

(t) (4a 2 + 3)x 2 + 8a(3 - 2a)x + 4(4a 2 - 12a - 3) = 0. 

6. Find a value of k such that the sum of the roots of the equation 
x 2 3(k + l)x + Qk = shall be one half their product. 

7. Construct equations whose roots shall be greater by 2 (also less 
by 2) than the roots of the equations in Ex. 2. 

8. Construct equations whose roots shall be twice (also half) the 
roots of the equations in Ex. 2. 



CHAPTER III 
GRAPHIC REPRESENTATION 

34. Graphic Methods. The methods studied in plane geome- 
try for constructing various figures when certain of their dimen- 
sions and angles are known are used extensively in making 
designs for machines, plans for buildings and various other 
structures, and also for solving problems that require the deter- 
mination of unknown dimensions, angles, areas, etc. 

These methods often give the desired results with sufficient 
accuracy for practical purposes, and they are more direct and 
rapid than numerical computation. Of even greater importance 
however is their use in checking the results of calculations, since 
there are always possibilities of error even when great care is 
exercised. It should be emphasized that every practical calcu- 
lation (i.e. one which is to be used in construction, or other ac- 
tual work where time, material, and money will be wasted if 
the calculation is incorrect) should always be checked by some 
independent means. 

Two rectilinear figures are similar if their corresponding angles 
are equal and their corresponding dimensions are proportional. 

35. Drawing to Scale. When two plane figures are similar, 
each is said to be a scale drawing of the other. The smaller 
is said to be reduced or drawn to a smaller scale. For example, 
if a drawing be made of a floor plan of a house so that the angles 
in the drawing are equal to those in the house itself, and the 
dimensions of the drawing are -^ of those of the house, it is said 
to be drawn to a scale of ^ inch to one foot. From such a draw- 
ing the builder can read off on a scale divided into quarter inches 
the dimensions of the parts he is about to construct. 

36 




Ill, 36] GRAPHIC REPRESENTATION 37 

This method of drawing figures to scale, reading off their un- 
known angles on a protractor, and their unknown dimensions 
on a conveniently divided scale, furnishes a graphic solution of 
many problems and it has many practical applications. 

EXAMPLE. The distance AB = 98 yards, 
Fig. 1, and the angles PAB = 51, PBA = 63, 
having been measured from one side of a river, 
the triangle can be drawn to scale and the 
width PR of the river can be read off on the 
scale, about 75 yards. 

FIG. 1 
EXERCISES 

1. Find the length of the projection of the altitude of an equilateral 
triangle upon one of its sides. Ans. .75s 

2. Draw two diagonals through the centre of a regular hexagon. 
Find the length of the projection of one of them upon the other. 

Ans. s. 

3. Draw two diagonals through the same vertex of a regular penta- 
gon. Find the length of a projection of one of them upon the other. 

Ans. 1.3s 

4. The pitch of a roof is the ratio of the height of the ridge above 
the plates to the distance between the plates. Find the length of the 
rafters and their inclination for a f pitch roof on a building 28 ft. wide. 

Ans. 21.8, 50. 

5. Find the length of the corner rafters, and also of the middle rafter 
on each side of a square roof on a house 34 X 34 feet, the apex of the 
roof being 12 feet above the top floor. Find also their inclinations. 

Ans. 26.9, 20.8, 26.5, 35. 

6. The roof of a building 36 ft. wide is inclined at an angle of 54 
to the horizon. Find the length of the rafters, allowing 2 ft. overhang. 

Ans. 32.6 ft. 

7. To determine the horizontal distance between two points P and Q 
on the same level but separated by a hill, a point R is selected from which 
P and Q are visible. Then PR = 200 ft., QR = 223 ft., and angle PRQ = 
62 are measured. Draw the figure and scale off PQ. Ans. 210. 



38 MATHEMATICS [III, 36 

8. The steps of a stairway have a tread of 10 in., and a rise of 7 in. 
Find the inclination of the stringers to the floor. Ans. 35. 

9. Plot four points on a sheet of paper. Mark them A, B, C, D. 
Construct a point P one-half the way from A to B, a point Q one- 
third the way from P to C, and a point R one-fourth the way from 
Q to D. Mark the four given points in some other order and repeat 
the construction. What conclusion do you draw ? 

36. Rectangular Coordinates of a Point in a Plane. The 
position of any point in the plane is uniquely determined as 



X' O 

Y' 

FIG. 2 

soon as we know its distance and sense from each of the two per- 
pendicular lines X' X and Y'Y. These lines are taken first, and 
are drawn in any convenient position. 

The distance from X' X (RP = b in the figure) is called the 
ordinate of the point P. The distance from Y'Y (SP = a in 
the figure) is the abscissa of P. 

Abscissas measured to the right of Y'Y are positive, those to 
the left of Y'Y are negative. Ordinates measured above X' X 
are positive, those below negative. 

The abscissa and ordinate taken together are called the 
coordinates of the point, and are denoted by the symbol (a, 6). 
In this symbol it is agreed that the number written first shall 
stand for the abscissa. 

The lines X' X and Y'Y are called the axes of coordinates, 
X' X being the axis of abscissas or the axis of X, and Y'Y the 



HI, 37] 



GRAPHIC REPRESENTATION 



39 



axis of ordinates or the axis of Y; and the point is called the 
origin of coordinates. 

The axes of coordinates divide the plane into four parts 
called quadrants. Figure 3 indicates the proper signs of the 
coordinates in the different quadrants. 



III 



IV 



FIG. 3 



FlG. 4 



37. Plotting Points. To plot a point is to locate it with 
reference to a set of coordinate axes. The most convenient 
way to do this is to first count off from along X'X a number 
of divisions equal to the abscissa, to the right or left according 
as the abscissa is positive or negative. Then from the point 
so determined count off a number of divisions equal to the ordi- 
nate, upward or downward according as the ordinate is positive 
or negative. The work of plotting is much simplified by the 
use of coordinate paper, or squared paper, which is made by ruling 
off the plane into equal squares, the sides being parallel to the 
axes. Thus, to plot the point (4, 3), count off four divisions 
from on the axis of X to the right, and then three divisions 
downward from the point so determined on a line parallel to 
the axis of Y, as in Fig. 4. 

If we let both x and y take on every possible pair of real 
values, we have a point of the plane corresponding to each pair 
of values of (x, ?/). Conversely, to every point of the plane 
corresponds a pair of values of (x, y). 



40 



MATHEMATICS 



[HI, 37 



EXERCISES 

1. Plot the following points (3, 3), (4, 5), (- 2, 3), (- 4, - 2), 
(7, - 2), (0, 4), (0, - 4), (3, 0), (- 3, 0), (0, 0). 

2. What is the y-coordinate of any point on the x-axis? 

3. What is the z-coordinate of any point on the y-axis? 

4. Show that the line joining (5, 4) and (5, 4) is bisected by 
the origin. 

5. Find the distance from the origin to each of the points in Ex. 1. 

6. Find the lengths of the sides of the triangle whose vertices are 
(1, 1), (5, 2), (3, 4). Ans'. Vl7; Vl3; 2V2. 

7. What is the abscissa of any point upon a straight line parallel 
to the y-axis and four units to its right? 

8. What is the ordinate of any point upon a straight line parallel 
to the re-axis and three units above it? 

9. (a) What relation exists between the coordinates of any point 
of a line bisecting the angle between the positive directions of the 
two axes? (6) Between the positive direction of the y-axis and the 
negative direction of the x-axis? 

10. What relation would exist between the coordinates of any 
point of the line in Ex. 9 (a), if it were raised four units parallel to 
itself? If it were lowered five units? 

38. Statistical Data. The following table shows the rainfall 
in inches, as observed at the Agricultural Experiment Station at 
LaFayette, Indiana, by months for 1916, 1917, and the average 
for the past 30 years. 





Jan. 


Feb. 


Mar. 

i~08 


Apr. 


May 


Jun. 


Jul. 


Aug. 


Sep. 


Oct. 

2~25 


Nov. 


Dec. 


1916 . . 


740 


1 16 


1 57 


582 


527 


3 56 


1 81 


222 


225 


4.79 


1917 . . 


1 54 


1 25 


409 


432 


475 


541 


1 47 


409 


1 03 


522 


13 


0.68 


Average . 


3.11 


2.88 


3.78 


3.38 


4.05 


3.75 


3.54 


3.32 


3.03 


2.46 


3.23 


2.71 



While it is possible by a study of this table to compare the 
rainfall month by month in the same year, or for the same month 
in the two years, or any month with the normal for that month, 



Ill, 38] 



GRAPHIC REPRESENTATION 



41 



these comparisons are more easily made and the facts are pre- 
sented much more emphatically by the diagram shown in Fig. 5. 
This is made from the data of the table as follows. The 24 
vertical lines represent the months of the two-year period. The 
altitudes of the horizontal lines represent inches of rainfall. The 
height (ordinate) of the point marked on any vertical line shows 
the rainfall for that month. The points are connected by lines 
to aid the eye in following the march of the rainfall. The full 
line represents the rainfall for 1916 and 1917, the dotted line 
the normal rainfall as shown by the experience of 30 years. 




J F M A M J JASONDJFMAMJJA S O N D 

MONTHS 1910 .MONTHS 1817 

FIG. 5 

Rainfall is a discontinuous phenomenon. Moisture is not pre- 
cipitated continuously, but intermittently. However, if we 
make a similar diagram showing the temperature at each hour 
of the day we might have inserted many other points. We 



42 



MATHEMATICS 



[HI, 38 



think of the change in temperature as a continuous phenome- 
non ; e.g. when the temperature rises from 42 at 8 A.M. to 51 
at 9 A.M., we think of it as having passed through every inter- 
vening degree in that hour. Thus we can think of the points 
which represent the temperature on the diagram from instant 
to instant as lying thick on a continuous curved line. This 
curve is called the temperature curve. 

In making a graph of a discontinuous function like rainfall, 
we connect the points with straight lines as in Fig. 5, but in case 
of a continuous function like temperature, a smooth curve which 
passes through all the plotted points is the best graphic repre- 
sentation of the function. 

EXERCISES 

1. Make a temperature graph from the following data, 



Hour, A. M. 
Temperature 


12 
45 


1 
45 


2 
45 


3 
45 


4 
43 


5 

42 


6 
41 


7 
40 


8 

42 


9 
51 


10 

57 


11 
59 


12 
62 


Hour, p. M. 
Temperature 


1 

66 


2 

70 


3 

74 


4 
76 


5 
76 


6 
75 


7 
74 


8 
73 


9 

72 


10 
70 


11 
69 


12 

68 





2. Determine from Fig. 5 which were the dry months in 1916. In 
1917. To what extent do they agree with each other and with the 
normal ? 

3. Do as directed in Ex. 2 for the wet months. 

4. What straight line in Fig. 5 would represent the average monthly 
rainfall for 1916? For 1917? For the past 30 years? 

5. To what extent does the dotted line in Fig. 5 enable you to pre- 
dict the probable rainfall in any given month subsequent to 1917? 

6. Procure the census data and plot the population graph of the 
United States by decades for a century. 

7. Plot a graph of the attendance of students at your college or Uni- 
versity for as many years back as you can secure the data. 

8. The following data give the Chicago price per bu. of No. 2 corn 
by months from Jan., 1903, to May, 1908. Plot the data using years 
as abscissas and price as ordinates. 



Ill, 38] 



GRAPHIC REPRESENTATION 



43 





Jan. 


Feb. 


Men. 


Apr. 


May. 


June. 


July. 


Aug. 


Sept. 


Oct. 


Nov. 


Dec. 


1903 
1904 
1905 
1906 
1907 
1908 


43 
42 
41 
42 
39 
59 


42 
46 
42 
41 
43 
56 


41 

49 
45 
39 

43 

58 


41 

46 
46 
43 
44 
65 


44 
47 
48 
47 
49 
70 


47 
53 
51 
50 
51 


49 
47 
53 
49 
52 


50 
51 
53 

48 
54 


45 
51 
51 
47 
60 


43 
50 
50 
44 
55 


41 
50 
45 

44 
55 


41 

43 
42 
40 

57 





























9. Find from the graph that month in each year in which the highest 
price occurred. The lowest price. Find the difference for each year 
between the highest and lowest price for that year. Does there appear 
to be any relation between these prices and the period of harvest? 

10. The following data gives the Chicago price of No. 2 oats by 
months from Jan., 1903 to May, 1908. Plot the data using years as 
abscissas and price as ordinates. 





Jan. 


Feb. 


Mch. 


Apr. 


May. 


June. 


July. 


Aug. 


Sept. 


Oct. 


Nov. 


Dec. 


1903 
1904 
1905 
1906 
1907 
1908 


31 
36 
29 
31 
33| 
48 


33 
39 
29 
29 
37 
48 


31 
38 
29 
28 
39 
52 


32 
36 
28 
30 
41 
51 


33 
39 
28 
32 
44 
53 


35 
39 
30 
33 
41 


33 

38 
27 
30 
41 


33 
31 
25 
29 
44 


35 
29 
25 
30 
51 


34 

28 
27 
32 
45 


33 
29 
29 
33 

44 


34 

28 
29 
33 
46 





























11. Handle the data in Ex. 10 as directed in Ex. 9. 

12. A restaurant keeper finds that if he has G guests a day his total 
daily expenditure is E dollars and his total daily receipts are R dollars. 
The following numbers are averages obtained from the books: 



G.. 


210 


270 


320 


360 


E 


16.70 


19.40 


21.60 


23.40 


R 


15.80 


21.20 


26.40 


29.80 













Plot two curves on the same set of axes in each case using G as abscissas. 
For one curve use E as ordinates, for the other use R as ordinates. 

Below what value of G does the business cease to be profitable? 
Connect the points (G, E) by a smooth curve. Continue this curve 
until it cuts the line (7 = 0. What is the meaning of the ordinate E 
for G = 0? Through what point ought the curve connecting the 
points (G, R) to pass? Ans. (0, 0). 



44 



MATHEMATICS 



[III, 38 



13. The population of the United States by decades was as follows. 
Plot, and estimate the population for 1920. 



Year. 


Population. 


Year. 


Population. 


Year. 


Population. 


1790. . 
1800.... 
1810.... 
1820 


3,929,214 
5,308,433 
7,229,881 
9,663,822 


1840... . 
1850. . . . 

I860.... 
1870 


17,069,453 
23,191,876 
31,443,321 
38,558,371 


1890.. 
1900. . . . 
1910.. . . 


62,669,756 
76,295,200 
91,972,266 


1830 


12,806,020 


1880 


50,155,783 



















14. The football accidents for the years given are as follows: 



Year. 


Deaths. 


Injuries. 


Year. 


Deaths. 


Injuries. 


1901.... 
1902.... 

1903.... 
1904.... 
1905. . . . 
1906 


7 
15 
14 
14 
24 
14 


74 
106 
63 
276 
200 
160 


1907... 

1908.... 
1909.... 
1910.... 
1911.... 


15 
11 
30 
22 
11 


166 
304 
216 
499 
178 















Plot two curves, using the years as abscissas and the deaths and injuries 
respectively as ordinates. 

15. The monthly wages in dollars of a man for each of his first 13 
years of work was as follows: 28, 30, 37.50, 45, 60, 65, 90, 95, 95, 137, 
162, 190, 210. Plot the curve showing the change. Estimate his 
salary for the fourteenth and fifteenth years. Can you be certain of 
his salaries for these years? 

16. Of 100,000 persons born alive at the same time the following 
table shows the number dying in the respective age intervals : 



Months. 


Deaths. 


Months. 


Deaths. 


0-1 


4,377 


6- 7 


579 


1-2 


1,131 


7- 8 


533 


2-3 


943 


8- 9 


492 


3-4 


801 


9-10 


456 


4-5 


705 


10-11 


421 


5-6 


635 


11-12 


389 



Ill, 38] 



GRAPHIC REPRESENTATION 



45 



Years. 


Deaths. 


Years. 


Deaths. 


0- 1 


11,462 


19- 20 


344 


1- 2 


2,446 


29- 30 


479 


2- 3 


1,062 


39- 40 


644 


3- 4 


666 


49- 50 


873 


4- 5 


477 


59- 60 


1,404 


5- 6 


390 


69- 70 


, 1,974 


6- 7 


327 


79- 80 


1,854 


7- 8 


274 


89- 90 


571 


8- 9 


234 


99-100 


25 


9-10 


204 


106-107 


1 



Plot the above data. Make two graphs. In each graph use deaths 
as ordinates; in one use months as abscissas, in the other use years. 
When is the ordinate smallest? largest? Does a small ordinate for the 
years 99-100 and 106-107 indicate a low death rate? Explain. Note 
the continuous decrease in the ordinate of the first curve. 

17. Using the data below and on p. 46, plot a curve using years 
as abscissas and price of corn as ordinates. Do you notice any reg- 
ularity in the number of years elapsing between successive high prices? 
successive low prices? Draw like graphs for the other crops listed? 

18. Plot the prices for the yrs. 74,. 81, 87, 90, 94, 01, 08, 11, 1916. 
What do you observe from this curve as to the tendency in the high 
price of corn? Do you observe any tendency in the lowest prices of 
corn that is in the prices for the yrs. 72, 78, 84, 89, 96, 02, 06, 1910? 

AVERAGE FARM PRICE DECEMBER FIRST 
Data from the year book of the Department of Agriculture 1916 



Year. 


Corn. 


Wheat. 


Oats. 


Barley. 


Rye. 


Potatoes. 


Hay, $ per 
Ton. 


1870. 


49.4 


94.4 


39.0 


79.1 


73.2 


65.0 


12.47 


1871. 


43.4 


114.5 


36.2 


75.8 


71.1 


53.9 


14.30 


1872. 


35.3 


111.4 


29.9 


68.6 


67.6 


53.5 


12.94 


1873. 


44.2 


106.9 


34.6 


86.7 


70.3 


65.2 


12.53 


1874. 


58.4 


86.3 


47.1 


86.0 


77.4 


61.5 


11.94 


1875. 


36.7 


89.5 


32.0 


74.1 


67.1 


34.4 


10.78 


1876. 


34.0 


97.0 


32.4 


63.0 


61.4 


61.9 


8.97 


1877. 


34.8 


105.7 


28.4 


62.5 


57.6 


43.7 


8.37. 


1878. 


31.7 


77.6 


24.6 


57.9 


52.5 


58.7 


7.20 


1879. 


37.5 


110.8 


33.1 


58.9 


65.6 


43.6 


9.32 



Continued on p. 46. 



46 



MATHEMATICS 



[HI, 38 



AVERAGE FARM PRICE, DECEMBER FIRST 

Continued. 



Year. 


Corn. 


Wheat. 


Oats. 


Barley. 


Rye. 


Potatoes. 


Hay, S per 
Ton. 


1880. 


39.6 


95.1 


36.0 


66.6 


75.6 


48.3 


11.65 


1881. 


63.6 


119.2 


46.4 


82.3 


93.3 


91.0 


11.82 


1882. 


48.5 


88.4 


37.5 


62.9 


61.5 


55.7 


9.73 


1883. 


42.4 


91.1 


32.7 


58.7 


58.1 


42.2 


8.19 


1884. 


35.7 


64.5 


27.7 


48.7 


51.9 


39.6 


8.17 


1885. 


32.8 


77.1 


28.5 


56.3 


57.9 


44.7 


8.71 


1886. 


36.6 


68.7 


29.8 


53.6 


53.8 


46.7 


8.46 


1887. 


44.4 


68.1 


30.4 


51 9 


54.5 


68.2 


9.97 


1S88. 


34.1 


92.6 


27.8 


59.0 


58.8 


40.2 


8.76 


1889. 


28.3 


69.8 


22.9 


41.6 


42.3 


35.4 


7.04 


1890. 


50.6 


83.8 


42.4 


62.7 


62.9 


75.8 


7.87 


1891. 


40.6 


83.9 


31.5 


52.4 


77.4 


35.8 


8.12 


1892. 


394 


62.4 


31.7 


47.5 


54.2 


66.1 


8.20 


1893. 


36.5 


53.8 


29.4 


41.1 


51.3 


59.4 


8.68 


1894. 


45.7 


49.1 


32.4 


44.2 


50.1 


53.6 


8.54 


1895. 


25.3 


50.9 


19.9 


33.7 


44.0 


26.6 


8.35 


1896. 


21.5 


72.6 


18.7 


32.3 


40.9 


28.6 


6.55 


1897. 


26.3 


80.8 


21.2 


37.7 


44.7 


54.7 


6.62 


1898. 


28.7 


58.2 


25.5 


41.3 


46.3 


41.4 


6.00 


1899. 


30.3 


58.4 


24.9 


40.3 


51.0 


39.0 


7.27 


1900. 


35.7 


61.9 


25.8 


40.9 


51.2 


43.1 


8.89 


1901. 


60.5 


62.4 


39.9 


45.2 


55.7 


76.7 


10.01 


1902. 


40.3 


63.0 


30.7 


45.9 


50.8 


47.1 


9.06 


1903. 


42.5 


69.5 


34.1 


45.6 


54.5 


61.4 


9.07 


1904. 


44.1 


92.4 


31.3 


42.0 


68.8 


45.3 


8.72 


1905. 


41.2 


74.8 


29.1 


40.5 


61.1 


61.7 


8.52 


1906. 


39.9 


66.7 


31.7 


41.5 


58.9 


51.1 


10.37 


1907. 


51.6 


87.4 


44.3 


66.6 


73.1 


61.8 


11.68 


1908. 


60.6 


92.8 


47.2 


55.4 


73.6 


70.6 


8.98 


1909. 


57.9 


98.6 


40.2 


54.0 


71.8 


54.1 


10.50 


1910. 


48.0 


88.3 


34.4 


57.8 


71.5 


55.7 


12.14 


1911. 


61.8 


87.4 


45.0 


86.9 


83.2 


79.9 


14.29 


1912. 


48.7 


76.0 


31.9 


50.5 


66.3 


50.5 


11.79 


1913. 


69.1 


79.9 


39.2 


53.7 


63.4 


68.7 


12.43 


1914. 


64.4 


98.6 


43.8 


54.3 


86.5 


48.9 


11.12 


1915. 


57.5 


92 


36.1 


51.7 


839 


61.6 


10.70 


1916. 


88.9 


160.3 


52.4 


88.2 


122.1 


146.1 


10.59 



Ill, 40] 



GRAPHIC REPRESENTATION 



47 



39. Other Graphic Methods. The statistical data given in 
the preceding articles has been studied by means of curves or 
graphs drawn on rectangular cross-section paper. There arc 
other important methods of representing statistical data. Of 
these methods we will give names to three: 

(1) Bar diagrams or columnar charts. 

(2) Dot diagrams. 

(3) Circular diagrams. 

These methods are best explained by means of examples. 

40. Bar Diagrams. Below is given a bar diagram or chart 
comparing the average size of farms for the years 1900 and 
1910 for the states indicated. 

Sizes of Farmsdn Hundreds of Ac-res 



3 6 9 12 15 


W (joining 
California 
Arizona 
Nebraska 
Missouri 
Michigan 
Georgia 
Alabama 
New York 
Delaware 










Legend: 
mm&lO 

c=l 1900 

. 



FIG. 6 

EXERCISES 

Make a bar diagram from the following data: 

1. The number of cattle in millions on farms for 1900 and 1910 in 
the following states were as follows. 



Date. 


Texas. 


Iowa. 


Nebraska. 


New York. 


Oklahoma. 


Indiana. 


1910... 


7.0 


4.5 


2.9 


2.4 


2.0 


1.3 


1900. .. 


10.0 


5.5 


3.2 


2.6 


3.3 


1.7 



48 MATHEMATICS [III, 40 

2. The sheep on farms in millions in 1910 and 1900 were as follows. 



Date. 


Texas. 


Iowa. 


Nebraska. 


New York. 


Oklahoma. 


Indiana. 


1910... 


1.6 


1.1 


0.3 


0.9 


0.1 


1.3 


1900. . . 


1.8 


1.0 


0.5 


1.7 


0.15 


1.7 



3. Make a bar diagram comparing the number of hours work re- 
quired by hand and machine labor in producing selected units (U. S. 
labor bulletin 54). 



Description of Unit. 


Number of Hours Worked. 


Hand. 


Machiue. 


Corn 50 bu. husked. Stalk left 


48.44 
223.78 
284.00 
160.63 
247.54 
125.00 
137.50 
115.28 
171.05 


18.91 
78.70 
92.63 
7.43 
86.36 
12.50 
28.33 
80.67 
94.30 


Seed 1000 Ibs. cotton 


Harvesting and baling 8 tons timothy. . . . 
Wheat 50 bu 


Potatoes 500 bu 


Butter 500 Ibs. in tubs 


5000 cotton flour sacks 


Quarry 100 tons limestone 


Mine 50 tons bituminous coal 





4. Make a bar diagram comparing the value of farm property for 
the two years 1900 and 1910. 



Year 


1910. 


1900. 








Land 


28,475,674,169 


13,058,007,995 


Buildings 


6,325,451,528 


3,556,639,496 


Implements and machinery 


1,265,149,783 


749,775,970 


Domestic animals, poultry, and bees 


4,925,173,610 


3,075,477,703 



5. Make a bar diagram of the population of the following states 
for the years 1900 and 1910. 



Date. 


Colorado. 


Nevada. 


Idaho. 


Washington. 


Oregon. 


California. 


1910 
1900 


799,024 
539,700 


81,875 
42,335 


325,594 
161,772 


1.141,990 
518,103 


672,765 
415,536 


2,377,549 
1,485,053 



HI, 41] 



GRAPHIC REPRESENTATION 



41. Double Bar Diagrams. In certain diagrams it is ad- 
vantageous to have the bars extend in both directions from the 
base line as in the following figure which gives the distribution 
by age and sex of the total population for 1910. 



Males 



Females 



'12 10 8 4 2 2 4 6 8 10 12 
Hundreds of Thousands 

FlG. 7 

EXERCISES 

Make corresponding figures for the distribution by age periods and 
sex for 1910, in per cents, of 



1. Native Whites of Native Parentage. 


2. Negroes. 


Age. 


Male. 


Female. 


Male. 


Female. 


Under 5 


6.7 

6.0 
5.5 
5.2 
4.7 
4.1 
3.5 
3.2 
2.6 
2.2 
2.1 
1.6 
1.3 
1.0 
0.6 


6.5 
5.8 
5.3 
5.1 
4.7 
4.0 
3.4 
3.0 
2.4 
2.0 
1.8 
1.4 
1.2 
0.9 
0.6 


6.4 
6.3 
5.9 

5.2 
4.9 
4.3 
3.4 
3.3 
2.3 
2.0 
1.8 
1.2 
1.0 
0.7 
0.4 


6.5 
6.4 
5.9 
5.6 
5.6 
4.7 
3.4 
3.2 
2.3 
1.9 
1.5 
1.0 
0.9 
0.6 
0.4 


5-9 


10-14 


15-19 


20-24 


25-29 


30-34 


35-39 


40-44 


45-49 


50-54 


55-59 


60-64 


65-69 


70-74 . ... 





50 



MATHEMATICS 



[HI, 41 



3. Make a diagram displaying the following data on the average 
yields and values per acre of Iowa farm crops for 1909. 



Crop. 


Yield. 


Value. 


Corn 


37.1 bu. 


$18.60 


Oats 


27.5 


10.54 


Wheat 


15.3 


14.62 


Barley 


19.2 


9.31 


Rve 


13.6 


8.50 


Flaxseed 


9.1 


11.74 


Timothy seed 


4.2 


5.79 


Hay and forage 


32.0 cwt. 


11.76 


Potatoes . . 


86.8 bu. 


39.10 









4. Make a diagram showing the weight in pounds and value of the 
dairy products shipped from Humboldt County, California, in 1913. 



Article. 


Weight in Lbs. 


Value. 


Butter . 


5,793,620 


$1,796,190 


Cheese 


304,570 


54,820 


Condensed milk 


1,302,560 


112,720 


Dry milk 


1,692,100 


157,430 


Fresh cream and buttermilk 


277,800 


6,920 


Casein .... 


1,484,910 


89,100 









42. Dot Diagrams. The following diagram taken from the 
U. S. census reports gives the number of all sheep on farms 
April 15, 1910. 



JV. Dak. ^ 
Q ' 



S. Dak. 



A r cb. 




LEGEND 
200,000 

9 150,000 to 200,000 
3 100,000 to 150,000 
Q 50,000 to 100,000 
O less than 50,000 



FIG. 8 



HI, 43] 



GRAPHIC REPRESENTATION 



51 



EXERCISES 

1. Make a corresponding chart showing all sheep on farms April 15, 
1910 for 



Wyoming 


5,397,000 


Utah 


1,827,000 


Montana 


5,380,000 


Colorado 


1,400,000 


Idaho 


3,010,000 


Nevada 


1,150,000 











2. Make a dot diagram showing all fowls on farms in the states 
given on April 15, 1910. [Here it is convenient to let stand for 
1,000,000.] 



North Dakota 


3,268,000 


Iowa 


23,482,000 


South Dakota 


5,251,000 


Minnesota 


10,697,000 


Nebraska 


9,351,000 


Montana 


966,000 











43. Circular Diagrams. The following diagram shows the 
relative percentage of improved and unimproved land area in 
farms for the total land area of the U. S. 1850-1880-1910. 
(U. S. census report 1910.) 



1850 





1910 




FIG. 9 



FIG. 10 



FIG. 11 



The circles indicate by the size of their sectors the relative 
ratio of lands improved and unimproved in farms to the 
total land area of the U. S. Note the rapid decrease in the 
area not in farms, also the increase in the proportion of improved 
to the unimproved. In 1910 less than 50% of the total area is 
in farms. 



52 



MATHEMATICS 



[HI, 43 



EXERCISES 

1. Make a circular diagram showing in percents the relative im- 
portance of the several countries in the production and consumption of 
cotton. 



United States 60.9 

India 17.1 

Egypt 6.6 

China . . .5.4 



Russia 4.5 

Brazil 1.9 

All others.. ..3.6 



2. Make circular diagrams showing per cent, distribution of foreign 
born population by principal countries of birth for the years indicated. 





1850. 


1870. 


1890. 


1910. 


Germany 


26.0 


30.4 


30.1 


18.5 


Ireland 


42.8 


33.3 


20.2 


10.0 


Canada and New Foundland 


6.6 


8.9 


10.6 


9.0 


Great Britain 


16.9 


13.8 


13.5 


9.0 


Norway, Sweden and Denmark 


0.8 


4.3 


10.1 


9.3 


Austria-Hungary 




1.3 


3.3 


12.4 


Russia and Finland 


0.1 


0.1 


2.0 


12.8 


Italy 


0.2 


0.3 


2.0 


9.9 


All others 


6.6 


7.6 


8.2 


9.1 













3. Make circular diagrams for the years 1900 and 1910 showing per 
cent, of total value of farm property represented by the items men- 
tioned. 





1910. 


1900. 


Land 


69.5 


63.9 


Buildings 


15.4 


17.4 


Implements and machinery T . 


3.1 


3.7 


Domestic animals, poultry, and bees 


12.0 


15.0 









(Compare with Ex. 4, p. 48.) 

44. Different Shadings or Colors are sometimes used in 
maps to represent different statistical facts. The annexed chart 
gives the average value of farm land per acre in Delaware. 
The average for the state is $33.63. 



Ill, 45] 



GRAPHIC REPRESENTATION 



53 




Legend. 

H $10 to $25 per acre 

'3 $50 to $75 per acre 

[^ HJ $75 to $100 per acre 

FIG. 12 



EXERCISES 

1. Draw a map of Connecticut showing the counties and mark to 
show the average value of farm land per acre. Average value for state 
is $33.03. Average value by counties is: Fairfield $75 to $100 per 
acre. New Haven and Hartford $25 to $50 per acre. Litchfield, 
Tolland, Windharn, Middlesex, and New London $10 to $25 per acre. 

2. Draw a map, give legend, and mark to show per cent, of im- 
proved land in farms operated by tenants by states in 1910. Utah, 
less than 10 per cent. Wyoming, 10 to 20 per cent. Colorado and 
Missouri, 20 to 30 per cent. Kansas, Nebraska, and Iowa, 30 to 40 
per cent. Illinois, 40 to 50 per cent. 

45 . Distance between two Points. Let PI and P 2 be the end 
points of a given segment in the plane. PI and P 2 are given 
points, i.e., their coordinates (Xi, YI) and (^2, ^2) are given 
or known numbers. 

We wish to find the length 
of the segment PiP 2 in terms 
of xi, i/i, xz, yz ; or, in other 
words to find the distance 
between two given points. 

Through PI draw a line 
parallel to the a>axis, and 




M, 



FIG. 13 
through PZ a line parallel to the i/-axis intersecting the first 



54 MATHEMATICS [III, 46 

in S. Then whatever the relative positions of PI and P% in the 
plane, the measure of PiS is x* x\, and the measure of SP 2 is 
7/2 - y\ ; also 

pj% = !\s* + isp?. 

Therefore if we let d represent the required distance PiP2, 



(1) d = (x 2 - xtf + (y z - 7/0 2 . 

It is clearly immaterial which of the two points is called PI 
and which P2, so the formula may also be written in the equiva- 
lent form 



(1) d = V( Xl - x 2 Y + (y, - ytf, 

and may be expressed in words thus : The distance between two 
points given by their rectangular coordinates is equal to the square 
root of the sum of the square of the difference of the abscissas and 
. the square of the difference of the ordinates. 

EXAMPLE. The distance from the point (2, 7) to the point (7, 5) is 



d = 5 2 + 12 2 = 13. 

46. Ratio of Division. Let PI and P 2 be two fixed points 
on a line and P any third point. Then the point P is said to 
divide the segment PiP2 in the ratio 



This ratio X is called the ratio of division or the division ratio. 



PI P PI 

FIG. 14 



If we choose an origin on the given line then the abscissas 
Xi of PI and x^ of P2 are known. Let us denote the abscissa of 
P by x. Then we have 

PiP = x - xi, PiP 2 = x 2 - Xi ; 



HI, 47] 



GRAPHIC REPRESENTATION 



55 



hence the abscissa x of P must satisfy the condition 



(3) 



X 



X 2 - 



whence solving for x, 

(4) x = Xi + \(x 2 - Xi). 

If the segments PiP and PiP 2 have the same sense, the 
division ratio is positive and P and P 2 lie on the same side of P\. 
If the segments PiP and PiP 2 are oppositely directed, then the 
division ratio is negative and P and P 2 are on opposite sides 
of PI. Thus, if the abscissas of PI and P* are 2 and 14, the 
abscissas of the points that divide PiP 2 in the ratios 3, ^, f , 
- 1, - 1, - 2 are 6, 8, 10, - 4, - 10, - 22. 

47. Point of Division. To find the coordinates of the point 
which divides the line joining two given points in a given ratio \. 

Let P\(XI, yi) and P 2 (o; 2 , y 2 ) be the two given points, X the 
given ratio, and P(x, y) the required point. 

Draw PiQi, PQ, P 2 Q 2 parallel to the y-axis, and PiRi, PR, 
P 2 R 2 parallel to the z-axis. Then Q and R divide Q\Q 2 and 
R\Ri, respectively, in the ratio X. Now as OQ\ = Xi, OQ 2 = x 2 , 
OQ = x, it follows from (4) 46 that 

(5) x = Xi + \(x 2 - Xi). 
In the same way we find 

(6) y = yi + K(y 2 - yi). 

Thus, the coordinates x, y of P are expressed in terms of the 
coordinates of PI and P 2 and the division ratio X. 




56 MATHEMATICS [III, 48 

48. Middle Point. If P be the middle point of PiP 2 , X = |, 
and 

* = |0i + a*), y = K*/i + 2/s). 

That is, the abscissa of the mid-point of a segment is one half 
the sum of the abscissas of its end points, and the ordinate is one 
half the sum of the ordinates. 

EXERCISES 

1. Find the lengths of the sides of the following triangles : 

(a) (4, 8), (- 4, - 8), (1, 4). (6) (4, 5), (4, - 5), (- 4, 5). 

(c) (2, 1), (- 1, 2), (- 3, 0). (d) (- 2, 1), (- 3, - 4), (2, 0). 

(e) (2, 3), (1, - 2), (3, 8). (/) (5, 2), (- 3, 2), (7, 3). 

What inference can be drawn from the answers to (e) and (/) ? 

2. Find the lengths of the sides and of the diagonals of the quadri- 
lateral (2, 1), (5, 4), (4, 7), (1, 4). 

3. A (0,2), B (3, 0), and C (4, 8) are the vertices of a triangle. 
Show that the distance from A to the mid-point of BC is one-half the 
length of BC. 

4. Show that two medians of the triangle (1, 2), (5, 5), ( 2, 6) are 
equal. What inference can you draw? 

5. The ends of one diagonal of a parallelogram are (4, 2) and 
(4, 4). One end of the other diagonal is (1, 2). Find the other 
end. 

6. The end points of a segment PQ are (1, 3) and (5, 0). Find 
the length of the segment, and the lengths of its projections on the 
x and y axes. 

7. Show that (0, 10), (1, 1), (5, 6) are the vertices of an isosceles 
right triangle. 

8. Find the coordinates of the point 

(a) Two-thirds of the way from ( 1, 7) to (8, 1) ; 
(6) Two-thirds of the way from (8, 1) to (- 1, 7) ; 

(c) Four-sevenths of the way from (1, 7) to (8, 0) ; 

(d) Three-sevenths of the way from (8, 0) to (1, 7). 

9. The segment from (4, 5) to (2, 3) is produced half its length. 
Find the end point. 



Ill, 50] GRAPHIC REPRESENTATION 57 

49. Locus of a Point in a Fixed Plane. If a point is forced 
to move so as to be always equidistant from two fixed points, 
we know that it must lie on the perpendicular bisector of the 
segment joining these points. If a point must be at a constant 
distance from a fixed point, it will lie on a circle. If a point 
must be always equidistant from a fixed point and a fixed line, 
it will lie on a certain curve, called a parabola, which we have not 
yet studied. 

If x and y are the coordinates of a point P, the values of x 
and y change as P moves in the plane. For this reason they are 
called variables. If P is subject to a condition which forces it to 
lie on a certain curve, then x and y must satisfy a certain condi- 
tion which can be expressed as an equation in x and y. 

For example, if P is always equidistant from (1, 2) and (2, 1), 
then, for all positions of P, x y = 0. If P is always equi- 
distant from (0, 2) and the x-axis then x 2 4y + 4 = 0. If P 
is always 3 units from the origin, then x 2 + y z = 9. 

Whenever a plane curve and an equation in x and y are so 
related that every point on the curve has coordinates which 
satisfy the equation, and conversely, every real solution of the 
equation furnishes coordinates of a point on the curve, then the 
equation is called the equation of the curve, and the curve is 
called the locus of the equation. This dual relation between 
equation and curve is the subject of study in Analytic Geometry. 

50. Equation of a Locus. To find the equation of the locus 
of a point which moves in a plane according to some stated law, 
we proceed as follows: First, draw a pair of coordinate axes; 
and locate and denote by appropriate numbers or letters all 
fixed distances, including the coordinates of fixed points. Second, 
mark a point P with coordinates x and y, to represent the mov- 
ing point ; express the conditions of the problem in terms of x, y, 
and the given constants ; and simplify the resulting equation. 



58 MATHEMATICS [III, 50 

Third, show that every real solution of the equation so obtained 
gives a point which satisfies the conditions governing the motion 
of P. 

EXAMPLE. Find the equation of the locus of a point which is always 
equidistant from a fixed line and a fixed point. 

First. We are free to choose the axes where we please. It is conven- 
ient to take the fixed line for the z-axis, and to take the y-axis through 
the fixed point. Then the coordinates of the fixed point may be called 
(0, a). 

Second. The distance from P(x, y) to the fixed line is y, and its dis- 
tance to the fixed point (0, a) is ^x 2 + (y a) 2 . Hence the condition 
expressed in the problem gives y = Vz 2 -|- (y a) 2 . This simplifies to 

x 2 + 2ay = a 2 . 

Third. It is easy to show, by reversing the above prqcess, that if 
x = h, y = k, is any solution of this equation, then the point Q (h, k) is 
equidistant from the re-axis and the point (0, a). 

Therefore x 2 + 2ay = a 2 is the required equation. 

EXERCISES 

1. Find the equation of the locus of a point which moves so that : 
(a) itx is equidistant from the coordinate axes ; 

(6) it is four times as far from the z-axis as from the y-axis ; 

(c) the sum of its distances from the axes is 6 ; 

(d) the square of its distance from the rr-axis is four times its distance 
from the y-axis. 

2. Find the equation of the locus of a point that is always equi- 
distant from (4, - 2) and (7, 3). Ans. 3x + 5y = 19. 

3. Find the equation of the perpendicular bisector of the segment 
joining the two points (a, 6) and (c, d). 

Ans. (a - c)x + (b - d)y = \(a? + b 2 - c 2 - d 2 }. 

4. Find the equation of the locus of a point whose distance from the 
point ( 3, 4) is always equal to 5. .4ns. x 2 + y 2 + 6x 8y = 0. 

5. Find the equation of the circle whose center is (a, b) and whose 
radius is c. 



Ill, 51] 



GRAPHIC REPRESENTATION 



59 



51. Locus of an Equation. In general a single equation in 
x and y has an infinite number of real solutions. Each of these 
solutions furnishes the coordinates of a point on the locus. 

To find solutions and plot points on the curve, solve the equa- 
tion, if possible, for y in terms of x, or vice versa. Determine and 
tabulate a convenient number of solutions by assigning values 
to x and computing the corresponding values of y. Using these 
for coordinates, plot the points which they represent and draw 
a smooth curve through the plotted points. 

EXAMPLE 1. Construct the locus of the equation 

x 2 = 4(x + y). 
Solving the given equation for y we have 



Assigning to x the values 0, 1, 2, 3, etc., 1, 2, 3, etc., and com- 
puting the corresponding values of y, we have the following solutions. 



X. . . 





1 


2 


3 


4 


5 


6 . 


7 


- 1 


- 2 


- 3 


y. . 





- .75 


- 1 


- .75 





1.25 


3 


5.25 


1.25 


3 


5.25 



We choose the axes, as in Fig. 16, so that all these points will go on 
the sheet. 




FIG. 16 



On plotting the points and drawing a smooth curve through them, 
we have a sketch of the locus as shown. 



60 



MATHEMATICS 



[III, 52 



EXAMPLE 2. Plot the curve whose equation is 

x 2 + y 2 = Qx + 2y. 
Solving the given equation for y, we have 



y = 1 Vl + 6z - x 2 , 
and we tabulate solutions as follows. 



X. ... 





1 


2 


3 


4 


5 


6 


7 


- 1 


- 2 


y. . . . 





- 1.45 


2 


- 2.16 


- 2 


- 1.45 





imag. 


imag. 


imag. 




2 


3.45 


4 


4.16 


4 


3.45 


2 









7 



FIG. 16a 



We note that each value of x gives two values of y, i.e. there are two 

points on the curve having the same 
abscissa. We find also that values of 
x ^ 7 do not give real values of y and 
that the same is true for values of 
x ^ 1. 

When these points have been plotted 
and a curve drawn through them we 
have the locus as shown in Fig. 16a. 

52. Study of the Equation. 
Important facts about the shape 
and extent of the locus can be 
learned by a study of its equation. In the first example above, 
the equation is of the first degree in y. From this we infer that 
every value of x, without exception, gives exactly one value of y. 
Therefore every vertical line cuts the curve in one and only 
one point. As x increases beyond 2, y always increases, and 
the curve goes off beyond all limit in the first quadrant. The 
same is true in the second quadrant. On the other hand, the 
equation is of the second degree in x. When solved, it gives 

X = 2 2\/l +y; 

hence every value of y greater than 1 gives two real values 
of x but every value of y less than 1 gives an imaginary value 
of x. Hence every horizontal line above y = 1 cuts the curve 
in two points, but there are no points on the curve below y = 1. 



Ill, 53] GRAPHIC REPRESENTATION 61 

The equation of the second example, when solved for y as 
above, shows that values of x which make 1 + 6x x 2 < 
give imaginary values for y. Hence there are no points on the 
curve to the left of the line x = 3 V 10 = 0.16, nor to the 
right of the line x = 3 + V 10 = 6.16, but every vertical line 
between these limits cuts the curve in two points. 

If we solve the same equation for x, we find 



x = 3 Q +2y -y 2 -, 

hence there are no points below_the line y = 1 VlO = 2.16 
nor above the line y = 1 + VlO = 4.16, but every horizontal 
line between these lines cuts the curve in two points. 

If the equation is a polynomial in x and y equated to zero, 
a glance will show whether or not it passes through the origin. 

The intercepts * can be found by the rule : To find the x-inter- 
cepts let y =0 and solve for x. Similarly find the 7/-intercepts. 

53. Symmetry. Two points A and B are said to be sym- 
metric with respect to a point P when the line AB is bisected by P. 

Two points A and B are said to be symmetric with respect to 
an axis when the line AB is bisected at right angles by the axis. 

If the points of a curve can be arranged in pairs which are 
symmetric with respect to an axis or a point, then the curve 
itself is said to be symmetric with respect to thai axis or point. 

RULE I. // the equation of a locus remains unchanged in form 
when in it y is replaced by y, then the locus is symmetric with 
respect to the axis of x. 

For, if (x, y} can be replaced by (x, y} throughout the 
equation without affecting the locus, then if (a, 6) is on the 

* The intercepts of a curve on the axis of x are the abscissas of the points of inter- 
section of the curve and the z-axis. The intercepts on the j/-axis are the ordinates of 
the points of intersection of the curve and the j/-axis. 



62 MATHEMATICS [III, 53 

locus, (a, 6) is also on the locus, and the points of the locus 
occur in pairs symmetric with respect to the axis of x. 

We can also prove the following rules. 

RULE II. // the equation of a locus remains unchanged in 
form when in it x is replaced by x, then the locus is symmetric 
with respect to the y-axis. 

RULE III. // the equation of a locus remains unchanged in 
form when in it x and y are replaced by x and y, then the 
locus is symmetric with respect to the origin. 

54. Points of Intersection. If two curves whose equations 
are given intersect, the coordinates of each point of intersection 
must satisfy both equations when substituted in them for 
x and y. In algebra it is shown that all values satisfying two 
equations in two unknowns may be found by regarding these 
equations as simultaneous in the unknowns and solving. Hence 
the rule to find the points of intersection of two curves whose 
equations are given. 

Consider the -equations as simultaneous in the coordinates, and 
solve for x and y. 

Arrange the real solutions in corresponding pairs. These will 
be the coordinates of all of the points of intersection. 

EXERCISES 
Plot the loci of the f ollowing equations : 

1. 2z - 3y - 6 = 0. 12. 4z 2 - ?/ 2 = 0. 

2. 4z - Qy - 6 = 0. 13. 6z 2 + 5xy - 6y* = 0. 

3. 6z - 9y + 36 = 0. 14. x 2 + y z = 4. 

4. 2x + 3y + 5 = 0. 15. x 2 - y 2 = 4. 

5. 3x - 2y - 12 = 0. 16. x 2 + y 2 = 25. 

6. 5z + 2y - 4 = 0. 17. (x - 8) 2 + (y - 4) 2 = 25. 

7. y = 7x - 3. 18. (x - 4) 2 + (y - 2) 2 = 5. 

8. 2y - x = 2. 19. 4(x + !) = (?/- 2) 2 . 

9. 2x + 9y + 13 = 0. 20. 10y = (x + I) 2 . 

10. (x - 4)(y + 3) = 0. 21. y = x 3 - 4z 2 - 4x + 16. 

11. (x 2 - 4)(y - 2) = 0. 



Ill, 55] GRAPHIC REPRESENTATION 63 

22. y = x, x z , x 3 , x 4 , , x n . What points are common to these curves? 

23. if = x, x 2 , x 3 , r. 24. y = (x - 1), (x - I) 2 , (x - I) 3 . 
25. y = (x - l)(x - 2)(x - 3). 26. y = (x - l)(x - 2) 2 . 

27. y = (x - 2)'. 28. rf = (x - l)(x - 2)(x - 3). 

29. y"- = (x - l)(x - 2) 2 . 30. y 2 = (x - 2) 3 . 

31. y - -. 32. y = 



x - 1 x + 1 

33- !f---- 34. y- 



x 2 + 1 x 2 + 1 

^ ~ 3 > 



35. y = ~ - . 36. y = 

(x - 2)(x - 4) (x - 2)(x + 4) 

Find the points of intersection of the following curves : 

I2x + y = 5, [x-y = 2, 

' \x + 2y = l. 38 ' t 2* -3y = l. 

+ y' = 18, f x 2 + y 2 = 18, 

40 ' -3*. 



Ans. (3, 3), (- f, - V). Ans. (3, 3), (3, - 3). 

f 3x 2 + 4y 2 = 48, f 3x 2 - 4?/ 2 = 11, 

' ' 



x - y + 1 = 0. ' 1 4x = 3y 2 . 

Ans. (2, 3), (- V, - -V)- ^s. (3, 2), (3, - 2). 

43. I X1J = 2 > 44. 

1 y 2 = 4x. x 2 + r/ 2 - 5. 

45 f xy = x + y + 1, 46 f xy = x + ?/ + 1, 



(xy = 
\4x - 



?/ = x - 1. I 4x - 3j/ + 1 = 0. 

Ans. (3, 2). Ans. (2, 3), (- J, - J). 

47. Find the length of the common chord of the two circles x 2 -f- y z 
= 4x and x 2 + y 2 = 4(x + y- 1). Ans. 2^3. 

48. In what respects are the loci of the following equations sym- 
metric? 

(o) y = x 2 . (e) ?/ 2 = x 2 . (i) x 3 y 3 x y = 0. 

(6) y 2 =x. (/)r/ 2 =x*. (T) XT/ = a. 

(c) y = x 3 . (g) y = X s - x. (fc) ax 2 + by 2 = 1. 

(d) y 2 = x 3 . (A) y = x 4 - x 2 . (I) ax 2 + 26xr/ + ct/ 2 = 1. 

55. Straight Line Parallel to an Axis. Suppose a point 
moves about on a piece of coordinate paper in such a way that 
it is always two units to the right of the axis of y. It would 



64 MATHEMATICS [III, 55 

evidently be on the line A B that is parallel to the y-axis and 
at a distance of two units to the right of 
OF. Every point of the line AB has an 
abscissa of two (x = 2), and every point 
whose abscissa is two lies on the line AB. 
For this reason we say that the equation 

x = 2 



I 2 A 

FIG. 17 



represents the line AB or is the equation of the line AB. 
More generally, the equation 

x = a, 

where a is any real number, represents a straight line parallel 
to the y-axis and at a distance a from it. Similarly, the equation 
y = b represents a line parallel to the z-axis. 

56. Straight Line through the Origin. Suppose a point 
moves about on a piece of coordinate paper in such a way that 
its distance from the x-axis, represented by y, is always equal 
to m times its distance from the ?/-axis, represented by x. The 
equation of the locus of the point is 

y = mx. 

This is the equation of a straight line through the origin. The 
points of this line have the property that the ratio y/x of their 
coordinates is the same number m, wherever on this line the 
point is taken. Moreover for any point Q, not on this line, 
the ratio y/x must evidently be different from m. The number 
m is catted the slope of the line. 

57. Proportional Quantities. Whenever two quantities y 
and x vary in such a manner that their ratio y/x is always 
constant, say m, they are said to be proportional. The constant 
m is called the factor of proportionality. Many instances occur 



HI, 57] 



GRAPHIC REPRESENTATION 



65 



in the applied sciences of two quantities related in this manner. 
It is often said that one quantity varies as the other. Thus 
Hooke's law states that the elon- 
gation E of a stretched wire 
or spring varies as the tension t; 
that is, E = kt, where A; is a con- 
stant. For a given wire, when E 
was expressed in thousandths of 
an inch and t in pounds, the fol- 
lowing relation was found: 

E = .8 
Thus when t = 10, E = 8 and when t = 5, E = 4. 



5 

FIG. 18 



10 T 



EXERCISES 



Draw the lines 



1. x = 1, - 1, 0, 2, 3, - 2, - 3, - 4, 4. ^ 

2. y = 1, - 1, 0, 2, 3, - 2, - 3, - 4, 4. 

3. What is the locus of a point if x > 3? x = 3? x < 3? 

4. What is the locus of a point if2<x<3? 2 < x < 3? 2<x 
< 3? 2 < x < 3? 

5. What is the locus of a point if 2 < x < 3 and 1 < y < 2? 

6. What is the locus of a point if x 2 + y* < 16 and x > 2? 

7. What is the locus of a point if 9 < x 2 + y 2 < 16? 

8. A stand-pipe is filled at the rate of 150 gallons per hour. What is 
the amount A of water in the stand-pipe h hours after filling begins? 

9. A man saves $50 each month and deposits it in a bank. What 
is the amount A which he has in the bank after t months? 

10. A railroad track has a rise of 1 ft. in 20. Give its equation 
and plot. 

11. The extension E in feet of a spiral spring due to a tension I of 
1 lb., was 1 inch. What is the relation connecting E and <? (Use 
Hooke's law.) 



6 



66 



MATHEMATICS 



[III, 58 



58. Slope of a Straight Line. The slope, m, of the line 
passing through two points PI(XI, yi), P 2 (z 2 , 2/2), Fig. 19, is 
given by the formula 



(8) 



m = 



PiR 




59. Equation of a Line through 
two Points. Let the two given 
points be PI(XI, yi), Pz(x 2 , y 2 ). 
Let P(x, y) be any other point on 
the line joining PiP 2 . Draw PiRS 
parallel to the z-axis. Draw P\M\, 
P 2 M 2 , PM, parallel to the y-axis. 



FIG. 19 
Then since the triangles PiSP and PiRP 2 are similar, we have 



SP 
PiS 



RPj 
PiR' 



y - 



?/2 - 



X Xi X- 

which may be written in the form 
(9) V * ~ 



The equation of a straight line with a given slope m and 
passing through a given point (x\, y\) is seen from the last 
equation to be 
(10) y - yi = m(x - jci). 

In particular if the y-intercept is given as b, the equation of 
the straight line having the given intercept and with slope m is 



which reduces to 
(11) 



y b = m(x 0) 
y = mx + b. 



HI, 61] 



GRAPHIC REPRESENTATION 



67 



This last equation is called the slope form of the equation of 
the straight line. 

If both intercepts are given, say Z -intercept = a, ^/-intercept 
= b, we can find the equation of the line by means of the 
equation for a line through two given points. We have 



which reduces to 
(12) 



This is called the intercept form of the equation of the straight 
line. 

60. Parallel Lines. Con- 
sider two parallel lines PiRi 
and P z Rz- Draw R 2 Ri and PzPi 
parallel to the ?/-axis, and RiSi, 
RzSz parallel to the x-axis. 
Then since the triangles RiSiPi 
and RiSiPz are equal, 




and 



i = S Z P Z . 



FIG. 20 



Hence, 



That is the slopes of any two non-vertical parallel lines are equal. 
61. Perpendicular Lines. Consider two perpendicular lines 
LI and Z/2 intersecting at P\(x\, t/i). Let PI(XI + a, y\ + &) be 
a second point on L\\ then since the given lines are perpen- 
dicular, the point Qi(x\ b, yi + a) lies on L 2 as shown by 
construction in the figure. Then the slope of L\ is mi = b/a, 
by the definition of slope, 58; and the slope of L 2 is m 2 =* 



68 



MATHEMATICS 



[HI, 61 



(a/6), for the same reason. It follows that we have 
(13) mim-j = 1. 



This proves the theorem: 
// two non-vertical lines are 
perpendicular, then the prod- 
uct of their slopes is 1. 

The converse is also true: 
// the product of the slopes 
of two lines is 1, then 
they are perpendicular. The 
proof, which is suggested 
by Fig. 21, is left to the 
student. 

62. General Equation of the First Degree. The equation 




V 

FIG. 21 



(14) 



Ax + By + C = 0, 



where A, B, C are constants, is called the general equation of 
the first degree in x and y because every equation of the first 
degree may be reduced to that form. For any values what- 
soever of A, B, and C, provided A and B are not both zero, 
the general equation of the first degree represents a straight 
line. 

EXERCISES 

1. Find the slope of the line joining the points 

(a) (1, 3) and (2, 7). (6) (2, 7) and (- 4, - 4). 

(c) ( A/3, V2) and (- >/2, >/3). (d) (a + b, c + a) and (c + a, 6+c). 

2. Prove by means of slopes that (- 4, - 2), (2, 0), (8, 6), (2, 4) 
are the vertices of a parallelogram. 

3. Prove by means of slopes that (0, - 2), (4, 2), (0, 6), (- 4, 2) 
are the vertices of a rectangle. 

4. What are the equations of the sides of the figures in Exs. 2 
and 3. 



Ill, 62] GRAPHIC REPRESENTATION 69 

5. Find the intercepts and the slope of each of the following 
lines: 

(a) 2x + 3y = 6. (6) x - 2y + 5 = 0. 

(c) 3z - y + 3 = 0. (d) 5x + 2y - 6 = 0. 

(e) 7x - 4y - 28 = 0. (/) 3y - 2x = 8. 

6. Find the equations of the lines satisfying the following conditions: 
(a) passing through (3, 1) and slope = 2. 

(6) having the ^-intercept = 3, y-intercept = 2. 

(c) slope = 3, x intercept = 4. 

(d) x intercept = 3, y intercept = 4. 

(e) passing through the point (2, 3) and with slope = 2. 

7. Find the points of intersection of 

(a) x - 7y + 25 = 0, z 2 + y 2 = 25. 

(b) 2x 2 + 3y 2 = 35, 3z 2 - 4y = 0. 

(c) x 2 + y = 7, y* - x = 7. 

(d) y = x + 5, 9z 2 + 16y 2 = 144. 

8. Find the equations, and reduce them to the general form, of the 
lines for which 

(a) m = 2, b = - 3. (6) m = - 1/2, b = 3/2. 

(c) m = 2/5, b = - 5/2. (d) m = 1, b = - 2. 

(e) a = 3, 6 = 3. (/) a = 4, 6 = 2. 
(0) a = - 3, 6 = - 3. (h) a = 4, & = - 2. 
(t) a = - 3, b = 3. 0') a = 2, 6=4. 

9. Write the equations of the lines passing through the points: 
(a) (- 2, 3), (- 3, - 1). (b) (5, 2), (- 2, 4). 

(c) (1, 4), (0, 0). (rf) (2, 0), (- 3, 0). 

(e) (0, 2), (3, - 1). (/) (2, 3), (- 6, - 5). 

10. Write the equations of the lines passing through the given 
points and with the given slopes: 

(a) (-2,3), m = 2. (6) (5,2), m = 1. 

(c) (1, 4), m = i (d) (2, 0), m = - f. 

(e) (0, 2), w = 0. (/) (3, - 2), m = - 2. 

11. Write the equation of the line which shall pass through the 
intersection of 2y + 2x + 2 = and 3y x 8 = 0, and having a 
slope = 4. Am. IQx 4y + 51 =0. 



70 MATHEMATICS [III, 62 

12. What are the equations of the diagonals of the quadrilateral 
the equations of whose sides are y x + 1 = 0, y = x + 2, y = 3x 
+ 2, and y + 2x + 2 =0? 

13. Required the equation of the line which passes through (2, 1) 
and is 

(a) parallel to 2y - 3x - 5 = 0. Am. 2y - 3x + 8 = 0. 

(6) perpendicular to 2y 3x 5 = 0. Ans. 2x + 3y 1 =0. 

14. Find the equations of the two straight lines passing through 
the point (2, 3), the one parallel, the other perpendicular to the line 
4x - 3y = 6. Ans. 4x - 3y + 1 = 0, 3x + 4y - 18 = 0. 

15. Passing through (4, 2), the one parallel, the other per- 
pendicular to the line y = 2x + 4. Ans. y = 2x 10, x + 2y = 0. 

16. Passing through the point of intersection of 4x + y + 5 = 
and 2x 3y + 13 = 0, one parallel, the other perpendicular to the 
line through the two points (3, 1) and ( 1, 2). 

Ans. 3x - 4y + 18 = 0, 4z + 3y - 1 =0. 

17. Find the equation of the line joining the origin to the point of 
intersection of 2z + 5r/ 4 = and 3x 2y + 2 = 0. 

Ans. y = 8x. 

18. Find the equation of the straight line passing through the 
point of intersection of 2x + 5y 4 = and 2x y + 1 =0 and 
perpendicular to the line 5x Wy = 17. Ans. 6x + 3y = 2. 

19. Find the equations of the lines satisfying the following condi- 
tions : 

(a) through (2, 3), parallel to y = 7x + 3. 

(6) through (4, 1), perpendicular to 2x + 3y = 6. 

(c) through (2, - 1), parallel to 3y 2x = 1. 

(d) through (3, - 6), parallel to 2y + 4x = 7. 

(e) through ( 1, 1), perpendicular to x/2 + y/3 = 1. 
(/) through (2, 2), perpendicular to y = 3x -f 2. 

20. Prove that the diagonals of a parallelogram bisect each other. 

21. Prove that the diagonals of a rhombus bisect each other at right 
angles. 

22. Prove that the diagonals of a square are equal and bisect each 
other at right angles. 

23. A straight line makes an angle of 45 with the x-axis and its y 
intercept = 2; what is its equation? Ans. y = x + 2. 



Ill, 62] 



GRAPHIC REPRESENTATION 



71 



24. The following data gives the height of a plant in inches on 
different days. 



Height 
Day 






28 
40 


33 
60 


36 

80 


40 
100 


52 
120 


62 
140 


66 
160 





















Find the rate of growth after 60 days. 

Find the rate of growth after 110 days. 

[The rate of growth is the slope of the curve. The slope of a curve 
at a given point is defined to be the slope of the tangent line drawn to 
the curve at the given point. Draw the tangent with a ruler and 
with the aid of the eye.] Ans. 7/10 in. per day; 0.55 in. per day. 



CHAPTER IV 

LOGARITHMS 

63. Definitions and Preliminary Notions. In the equa- 
tion 

10 2 = 100, 

three numbers are involved. By omitting each number in turn 
there arise three different problems. If we omit the 100, we 
have the familiar question in involution: 

10 2 = ?. 
If we omit the 10 we have the familiar question in evolution: 

? 2 = 100, 
or, as it is usually written, 

VlOO = ?. 

If we omit the 2 we have the following question 

10 ? = 100, 

which we agree to write in the form, 

\ 

logic 100 = ? 

and we say that 2 = the logarithm of 100 to the base 10. 
In general, if 

(1) IP = N, 

then x = the logarithm of N to the base b, and we write, 

(2) x = log b N. 

(1) and (2) are then simply two different ways of expressing 
the same relation between b, x, and N. (1) is called the ex- 

72 



IV, 63] LOGARITHMS 73 

ponential form. (2) is called the logarithmic form. Either of 
the statements (1) or (2), implies the other. The exponent 
in (1) is the logarithm in (2), a fact which may be emphasized 
by writing 
(3) (base) 10 * arithm = number. 

For example, the following relations in exponential form: 
32 = 9, 2 4 = 16, (1/2) 3 = 1/8, a" = x, 
are written respectively in the logarithmic form: 

2 = logs 9, 4 = Iog 2 16, 3 = logi/z 1/8, y = logo x. 

We shall now give the following 

DEFINITION OF A LOGARITHM. The power to which a given 
number called the base must be raised to equal a second number is 
called the logarithm of the second number. 

EXERCISES 

1. Write the following equations in logarithmic form: 
(a) 9 = 3 2 . (g) 7 = 7 1 . _ 

(6) 64 = 4 3 . (h) 25 = (Vs) 4 . 

(c) 16 = 2 4 . (i) 8 = (V2). 

(d) 243 = 3 5 . 0') 3 = (V3) 2 . 
(c) 64 = 2". (fc) 3 - V9. 
(/) 2401 = 7 4 . (I) 4 = v/64. 

2. Write the following equations in exponential form : 

(a) log, 16 = 4. (g) logioO.l - - 1. 

(b) Iog 4 16 = 2. (A) logz 1/4 : 2. 

(c) logio 1000 = 3. (t) Iog64 2 = 1/6. 

(d) logs 729 = 5. (j) Iog2 1/8 = - 3. 
(c) logs 625 = 4. (fc) logn 1 = 0. 
(/) log 1728 = 3. (0 loga o = l. 

3. Find the numerical value of each of th following : 
(a) Iog 2 64. (e) Iog 26 5. 

(6) logio 0.001. (/) 3 Iog 6 625 + logj 16. 

(c) Iog27 3. (g) logi/2 4. 

(d) logio 100 - | logo.i 100. (h) 5 logz 16-2 log 625. 



74 MATHEMATICS [IV, 64 

64. Properties of Logarithms. Any positive number, ex- 
cept 1, may be the base of a system of logarithms of all the real 
positive numbers. In any such system, 

1) The logarithm of 1 is zero. 
For, 6 = 1, therefore logb 1=0. 

2) The logarithm of the base itself is 1. 
For, 6 1 = 6, therefore logb 6 = 1. 

3) The logarithm of a product is the sum of the logarithms of 
the factors. 

For if logb M = k and logb -Y = I, then M b k and N = b l , 
MN = b k -b l = b k+l , whence 

logb MN = k + I = logb M + logb N. 

This can readily be extended to three or more factors. 

4) The logarithm of a quotient is equal to the logarithm of the 
dividend minus the logarithm of the divisor. 

For, 

' M = y 
N ~ b l '' 
therefore 

logb jf = k - I = logb M - logb N. 

5) The logarithm of the reciprocal of a number is the negative 
of the logarithm of the number. 

For on putting M 1 under (4) above, we have 

logb-r^ = logb 1 - logb N = - logb N, 

since logb 1=0. 

6) The logarithm of the pth power of a number is found by 
multiplying the logarithm of the number by p. 

For, N = b k and N p = (b k ) p = b pk , whence 

logb N p = pk = p logb N. 



IV; 64] LOGARITHMS 75 

7) The logarithm of the rth root of a number is found by dividing 
the logarithm of the number by r. 

For, N = b k and A/A 7 = N llr = (6*) 1/r = b klr , whence 



logs VAT = - = 



N 



r 

EXERCISES 

Express the logarithms of the following numbers in terms of the 
logarithms of integers. In this book, when the base is omitted, 10 is 
to be understood as the base. 

352/3 17 i/4 12 -2 

! ] g iQ2/3.Ai/2 2. log ,,- 7 , . 3. log 



13 2/ 3 . 6 i/ 2 



4. Prove that logs V81V729-9- 2 ' 3 = 31/18. 

Express the logarithms of the following in terms of the logarithms 
of prime numbers. 

(63) 1/4 88~ 1/2 

(25) 2 (72) 1/4 ' (75) 3/4 (12) 2 * 

7. log ^- 3. 8. log (V2 ! V7 2 V6). 

9. Given log 2 = 0.3010, log 3 = 0.4771, log 7 = 0.8451, find the 
logarithms of the following numbers. 

(a) 6. (e) 32. (i) 420. (m) Vf/2. 

(6) 14. (/)10.5. 0') 900. (n) A/504. 

(c) 24. (g) 14?. (k) 35/48. (o) B Vl3.5. 

(d) 28. (h) 2.52. (I) 1/36. (p) >/294. 

10. Express the logarithms of each of the following expressions to 
the base a in terms of logo b, log a c, Iog d. 

11. Prove that 



= 2 loga (X + V-l). 

12. If log 3 = 0.4771, what is the (a) log 30? (b) log 300? (c) log 
3000? (d) log 30,000? What part of these logarithms is the same? 
Why? 



76 MATHEMATICS [IV, 65 

65. Computation of Common Logarithms. While any 
positive number except unity could be used as the base of a 
system of logarithms, only two systems are in general use. 
One, called the natural, or Napierian system is used in analytical 
work and has the number e = 2.71828 + for its base. The 
other, known as the common, or Briggs system is used for all 
purposes involving merely numerical computations and has for 
its base the number 10. Unless specifically stated to the 
contrary the common system will be the one used throughout 
this book. 

In the following discussion of common logarithms, log x is 
written as an abbreviation of logio x. 

Every positive number has a common logarithm, and the 
value of this logarithm may be obtained correct to as many 
places of decimals as may be desired. Negative numbers and 
zero have no real logarithms. 

If we extract the square root of 10, the square root of the 
result thus obtained, and so on, continuing the reckoning in 
each case to the fifth decimal figure, ,we obtain the following 
table : 



1Q1/2 
l l/4 
1Q1/8 
1Q1/16 

10 1/32 
1Q1/64 


= 3.16228, 

= 1.77828, 
= 1.33352, 
= 1.15478, 
= 1.07461, 
= 1.03663, 


1Q1/128 _ 
1Q1/256 _ 
101/512 = 
1Q1/1024 _ 
1Q1/2048 _ 
1Q1/4096 = 


1.01815, 
1.00904, 
1.00451, 
1.00225, 
1.00112, 
1.00056, 



and so on. The exponents 5, |, on the left are the logarithms 
of the corresponding numbers on the right. 

By the aid of this table we may compute the common logar- 
ithm of any number between 1 and 10, and hence of any positive 
number. 



IV, 66] LOGARITHMS 77 

EXAMPLE. Find the common logarithm of 4.26. 

Divide 4.26 by the next smaller number in the table, 3.16228. The 
quotient is 1.34719. Hence 4.26 = 3.16228 X 1.34719. Divide 
1.34719 by the next smaller number in the table, 1.33352. The quo- 
tient is 1 0102. Hence 4.26 = 3.16228 X 1.33352 X 1.0102. Con- 
tinue thus, always dividing the quotient last obtained by the next 
smaller number in the table. We shall obtain by this method an ex- 
pression for 4.26 in the form of a product: 

4.26 = 3.16228 X 1.33352 X 1.00904 X 
= 10 1 / 2 X 10 1/8 X 10 1 / 256 X 

Therefore, 



= .5000 
+ .1250 
+ .0039 

= .6289 

By referring to the table of logarithms at the end of the book we find 
that, correct to four decimal places, 

log 4.26 = .6294 

Hence, by using only three terms in the above approximation we ob- 
tain a result which is in error but 5 units in the fourth decimal place. 

66. Characteristic and Mantissa. If two numbers are un- 
equal, their logarithms are unequal in the same sense; that is if 

a < b < c, 
then 

log a < log b < log c. 
For example 

log 100 < log 426 < log 1000, 
that is, 

2 < log 426 < 3. 



78 MATHEMATICS [IV, 66 

When the logarithm of a number is not an integer it may 
be represented approximately by a decimal fraction correct to 
any desired number of places; thus log 426 = 2.6294 to four 
decimal places. 

The integral part of the logarithm is called the characteristic 
and the decimal part is called the mantissa. In log 426, the 
characteristic is 2 and the mantissa is .6294. For convenience 
in computing it is desirable to have the mantissa positive even 
when the logarithm is a negative number. For example, 
log \ = - 0.3010, but - 0.3010 = 9.6990 - 10, and we write 

log \ = 9.6990 - 10, 

in which the characteristic is 9 10 = 1, but the mantissa 
.6990 is positive. 

It is convenient to write the logarithm of any number N in 
the form 

log # = M - A;- 10, 

in which M is a positive number or zero and A; is a positive 
integer or zero. 

For example, log 426 = 2.6294, log 42.6 = 1.6294, log 4.26 
= 0.6294, log 0.426 = 9.6294 - 10, log 0.0426 = 8.6294 - 10, 
log 0.00426 = 7.6294 - 10. 

Moving the decimal point n places to the right (left) in a number 
increases (decreases) the characteristic of its common logarithm 
by n, but does not affect its mantissa. 

For this has the effect of multiplying (dividing) the number 
by 10", and 

log (N -10") = log N + log 10" = log N + n 
and 

log (N * 10") = log N - n. 

Therefore, the mantissa of the common .logarithm of a number 
is independent of the position of the decimal point. In other 



IV, 66] LOGARITHMS 79 

words, the common logarithms of two numbers which contain 
the same sequence of figures differ only in their characteristics. 
Hence, tables of logarithms of numbers contain only the man- 
tissas and the computer must determine the characteristics 
mentally. This can be done by the following simple rules. 

RULE I. The characteristic of the common logarithm of any 
number greater than 1, is one less than the number of digits before 
the decimal point. 

For if N is a number having n digits in the integral part (i. e. 
before the decimal point), then 

10 n-i < N < 10 n 

and 

n 1 ^ log N < n; 

therefore log N = (n 1) -f (a decimal fraction) and its 
characteristic is n 1. 

On the other hand if N is a decimal fraction (i. e., a positive 
number less than 1), we may move the decimal point 10 places 
to the right and apply Rule I., provided we subtract 10 from 
the resulting logarithm. For example, 

log 0.0006958 = log 6958000 - 10 

and by Rule I. the characteristic is 6 10. 

This process is easily seen to be equivalent to that specified in 

RULE II. To find the characteristic of the common logarithm 

of a number less than 1, subtract from 9 the number of ciphers 

between the decimal point and the first significant figure. From 

the number so obtained subtract 10. 

. A very large number such as the distance in feet from the 
earth to the sun, 490,000,000,000 (correct to two significant 
figures), is conveniently written (on moving the decimal point 
11 places to the left) in the form 

4.9 X 10 11 



80 MATHEMATICS [IV, 66 

and the characteristic of its common logarithm is 11. Similarly 
a very small number such as 0.000,000,453,8 can be written 
(on moving the decimal point 7 places to the right), 

4.538 X 10- 7 

and the characteristic of its logarithm is 7 = 3 10. 

This form of expression is frequently used where only a few 
significant figures are known to be correct, and if the decimal 
point is placed after the first significant figure, the exponent of 
10 is the characteristic of the logarithm of the number. 

EXERCISES 
Find the characteristics of the logarithms of the following numbers: 

(1) 276.35 (5) 0.00072 (9) 73.187 

(2) 0.0495 (6) 4589.5 (10) 8.421 X lO" 26 . 

(3) 1.837 (7) 0.9372 (11) 7.268 X 10 1 *. 

(4) 6.3 X 10 s . (8) 7.32 X 10~ 5 . (12) 0.00008 

67. Use of Tables. 1) The characteristic is not given in the 
table of logarithms. It is to be found by the above two rules. 
It should be written down first, and always expressed even 
though it be zero, in order to avoid error due to forgetting it. 

2) The mantissa of the common logarithms of numbers, 
correct to four decimal places, are printed in Table I., at the 
end of the book. For convenience in printing the decimal points 
are omitted. 

To find the mantissa of a number consisting of one, two, or 
three digits (exclusive of ciphers at the beginning or end, and 
the decimal point), look in the column marked N for the first 
two digits and select the column headed by the third digit; 
the mantissa will be found at the intersection of this row and 
this column. For example, to find the mantissa of 456, we run 
down the column headed N to 45 and then run across the page 



IV, 68] LOGARITHMS 81 

to the column headed 6 where we find the mantissa .6590; again, 
the mantissa of 720 is found opposite 72 in the column headed 0, 
and is 8573. 

EXERCISES 
Look up the following logarithms in Table I. 

(1) log 276 = 2.4409 (11) log .00782 

(2) log 8.64 = 0.9365 (12) log .0856 

(3) log .829 = 9.9186 - 10. (13) log 20. 

(4) log 7.34 X 10 5 = 5.8657 (14) log 8.5 

(5) log 2.30 X 10- 3 = 7.3617 - 10. (15) log 1870. 

(6) log 24700 = 4.3927 (16) log 3.20 X 10~ 12 . 

(7) log 3.7 X 10 12 . .(17) log 5.47 X 10 23 . 

(8) log 9. (18) log 7.58 X 10*. 

(9) log 846000. (19) log 98.3 
(10) log .000172 (20) log 3140000. 

68. Interpolation. If there are more than three significant 
figures in the given number, its mantissa is not printed in the 
table; but it can be found approximately by the principle of 
proportional parts: when a number is changed by an amount 
which is very small in comparison with the number itself, the change 
in the logarithm of the number is nearly proportional to the change 
in the number itself. 

For example, to find the logarithm of 37.68, we find from the 
table, 

Mantissa of 3760 = 5752, 
Mantissa of 3770 = 5763. 

The difference between these mantissas, called the tabular 
difference, is 11. We note that an increase of 10 in 3760 pro- 
duces an increase of 11 in its mantissa and we conclude that an 
increase of 8 in 3760 (to bring it up to 3768, the given digits) 
would produce an increase of .8 X 11 = 8.8 in the mantissa. 
This number 8.8, called the correction, is to be added to the 
7 



82 MATHEMATICS [IV, 68 

mantissa of 3760, but in using a four place table we retain only 
four places in corrected mantissas, so here we add 9 (the integer 
nearest to 8.8) ; thus, 

log 37.60 = 1.5752 
correction = 9 

log 37.68 = 1.5761 

Near the beginning of Table I. the tabular differences are so 
large as to make this process of interpolation inconvenient and 
in some instances unreliable. On this account there are printed 
on the third and fourth pages of Table I., the mantissas of all 
four figure numbers whose first digit is 1. By using these we 
can avoid interpolation at the beginning of the table. Thus, 
on the third page of the table we find, 

log 103.2 = 2.0137, 

but if we find it by interpolation on the first page, 
log 103.2 = 2.0136 

EXAMPLE 1. Find the logarithm of .003467. Opposite 34 in 
column 6 find 5391; the tabular difference is 12; .7 X 12 = 8.4; the 
mantissa is then 5391 + 8 = 5399; hence log .003467 = 7.5399 - 10 

EXAMPLE 2. Find log 2.6582. Opposite 26 in column 5 find 4232; 
the tabular difference is 17; .82 X 17 = 13.9; the mantissa is 4232 
+ 14 = 4246; hence log 2.6582 = 0.4246. 

69. Accuracy of Results. The accuracy of results obtained 
by means of logarithms depends upon the number of decimal 
places given in the tables that are used, and this accuracy has 
reference to the significant figures counted from the left. In 
general, a table will give trustworthy results to as many sig- 
nificant figures, counted from the left, as there are decimal 
places given in the logarithms. For example, four-place 
logarithms would show no difference between 35492367 and 
35490000. 



IV, 70] LOGARITHMS 83 

Neither a four-place nor a five-place table would be of any 
use in financial computations where large sums are involved. 
It would take a nine-place table to yield exact results if the 
sums involved should reach a million dollars. 

70. Reverse Reading of the Table. To find the number 
when its logarithm is known. This is sometimes called finding 
the antilogarithm. For this process we have the following rule. 

RULE III. // the mantissa is found exactly in the table, the 
first two figures of the corresponding number are found in the 
column N of the same row, while the third figure of the number is 
found at the top of the column in which the mantissa is found. 

Place the decimal point so that the rules in 66 are fulfilled. 

EXAMPLE. Given log N = 1.7427; to find N. 

We find the mantissa 7427 in the row which has 55 in column N. 
The column in which 7427 is found has 3 at the top. Thus the sig- 
nificant figures in the number are 553. Since the characteristic is 1 we 
must have 2 figures to the left of the decimal point. Thus N = 55.3. 

If the mantissa of the given logarithm is between two man- 
tissas in the table, we may find the number whose logarithm 
is given by the following 

RULE IV. When the given mantissa is not found in the table, 
write down three digits of the number corresponding to the mantissa 
in the table next less than the given mantissa, determine a fourth 
digit by dividing the actual difference by the tabular difference, 
and locate the decimal point so that the rules for characteristics are 
fulfilled. 

EXAMPLE. Given log N = 0.4675; to find N. 

The mantissa 4675 is not recorded in the table, but it lies between 
the two adjacent mantissas 4669 and 4683. The mantissa 4669 corre- 
sponds to the number 293. The tabular difference is 14. The actual 
difference between 4669 and 4675 is 6. The number 4675 is 6/14 of 
the interval from 4669 to 4683, and the corresponding number N is 



84 MATHEMATICS [IV, 70 

about 6/14 of the way from 293 to 294, or, reducing 6/14 to a decimal, 
about .4 of a unit beyond 293. Hence the corresponding digits are 
2934; hence TV = 2.934. 

The work may be written down as follows: 
log N = 0.4675 
4669 



14)60(4 
N = 2.934 

EXERCISES 

Obtain the logarithm of each of the following numbers. 

1. 3.1416 2. 1.732 3. 2.718 

4. 1.414 5. 39.37 6. 0.4343 

7. 3437 8. 0.0254 9. 0.9144 

10. 0.003954 11. 0.016018 12. 0.0283 

13. 7918. 14. 866500. 15. 92897000. 

Find the antilogarithm of each of the following numbers. 



16. 


0.4563 




17. 


96378 


- 10. 


18. 


5.3144 




19. 


1.7581 




20. 


8.2046 


- 10. 


21. 


6.1126 




22. 


0.4971 




23. 


7.5971 


- 10. 


24. 


4.9365 




25. 


4.6856 - 


10. 


26. 


8.1530 


- 10. 


27. 


8.6123 


- 20. 


28. 


8.4048 - 


10. 


29. 


8.4520 


- 10. 


30. 


0.7318 


- 20. 



71. Cologarithms. The cologarithm of a number is the 
logarithm of the reciprocal of the number. (Compare (5) 64.) 

Thus colog 425 = log - = log 1 - log 425 
4^o 

= - 2.6284 

But since we always wish to have the mantissa of a logarithm 
positive, we write = 10 10, and subtract 2.6284 from this, 

as follows: 

log 1 = 10.0000 - 10 

log 425 = 2.6284 

colog 425 = 7.3716 - 10. 



IV, 72] LOGARITHMS 85 

In practice this is done mentally by beginning at the left not 
omitting the characteristic, and subtracting each digit from 9, 
except the last significant digit, which is subtracted from 10. 

In the process of division subtracting the logarithm of a 
number and adding its cologarithm are equivalent operations 
since dividing by N is equivalent to multiplying by its reciprocal. 

72. Computation by Logarithms. It should be kept in 
mind that a logarithm is unchanged if at the same time any 
given number is added to and subtracted from it. This is useful 
in two cases: 

First. When we wish to subtract a larger logarithm from a 
smaller; 

Second. When we wish to divide a logarithm by an integer. 

EXAMPLE 1. Find the value of 27.4 -f- 652. 

log 27.4= 1.4378 

= 11.4378 - 10 
log 652 = 2.8142 



log x = 8.6236 - 10 
x = 0.04304 

EXAMPLE 2. Find the value of (0.0773) 1 / 3 . 

log 0.0773 = 8.8882 - 10. 

It is convenient to have, after division by 3, 10 after the mantissa; 
hence, before dividing we add 20.0000 - 20. 

log 0.0773 = 28.8882 - 30 (divide by 3), 
log * = 9.6294 - 10 
x = 0.4250 

EXAMPLE 3. Find the value of ' (42>6) 



[" (42.6) (- 3.14) I' 

02.4 



We have no logarithms of negative numbers, but an inspection of 
this problem shows that the result will be negative and numerically 



86 MATHEMATICS [IV, 72 

the same as though all the factors were positive; hence we proceed as 
follows: 

log 42.6 = 1.6294 
log 3.14 = 0.4969 
colog 62.4 = 8.2048 - 10 (add) 

3)0.3311 (divide by 3) 

log (- x) = 0.1104 

- x = 1.290, whence x = - 1.290. 

EXERCISES 

Find approximate values of the following by aid of logarithms. 
1. 231.6 X .0036. 2. 79 X 470 X 0.982. 

3. 13750 X 8799000. 4. (- 9503) X (- 0.008657). 

5. 0.0356 X (- 0.00049). 6. 9.238 X 0.9152 



8075 . 


0.00542 


n 


24617 


364.9' 


0.04708 ' 


' -0.00054' 


10. 


67 X 9 X 0.462 


11. 


9097 X 5.408 


. 12. (2.38S) 5 . 


0.643 X 7095 


- 225 X 593 


X 0.8665 


13. 


(0.57)~ 4 . 


14. 


(19/11) 8 . 


15. 


(1.014) 25 . 


16. 


A/67.54. 


17. 


A/- 0.3089. 


18. 


G V( - 9.718) 3 . 


19. 


8 5/4 . 


20. 


(0.001 ) 2 ' 3 . 


21. 


(29^ 9 r ) 3 / 2 . 


22. 


(6f) 3 - 4 . 


23. 


(- 9306) 3 / 7 . 


24. 


(0.0067) 2 - 5 . 


25. 


vixVif. 


26. 


Vol 


27. 


(0. 00068) ~ 6 / 4 . 


(0.009) 3 / 5 ' 


00 


/ 854 X A/0.042 


! on 3 |7" 4 X 92" X (0.01 )i/ 



A/OOOl ' * (0.00026) s X 5968 1 / 3 

30. V 6 A/0.5804 A/0.2405. 31. (6.89 X lO" 22 ) 16 / 17 . 

Ans. 1.21 X 10- 20 . 

32. (5.67 X 10- 18 ) 9 / 11 . 33. 8. 4 

M[(4.5 X lO-^lO 6 - 58 ] 1 

Ans. 7.76 X lO" 6 . Ans. 5.51 X 10 7 . 

34. The amount a of a principal p at compound interest of rate r 
for n years is given by the formula: a = p(l + r) n . Find the amount 
of $486 in 5 years at five per cent, (r = .05) if the interest is com- 
pounded annually. Ans. $620.27 



IV, 72] LOGARITHMS 87 

35. Find the amount of $384 in 40 years at four per cent., interest 
compounded annually. , Ans. $1,843.59. 

36. Find the simple interest on $6,237.43 for 7 years at six per cent. 
Would the computation made with four-place logarithms, be sufficiently 
accurate for commercial purposes? Explain. Ans. $2619.72. 

37. The weight P in pounds which will crush a solid cylindrical cast- 
iron column is given by the formula 

,73.56 

P = 98920^,, 

where d is the diameter in inches and I the length in feet. What weight 
will crush a cast-iron column 6 feet long and 4.3 inches in diameter? 

[RiBTZ AND CRATHORNE] Ans. 834,200 Ibs. 

The area A in acres, of a triangular piece of ground, whose sides are 
a, b, c, rods, is given by the formula 

_ Vs(s a)(s b)(s c) 
160 

where s = %(a + 6 + c). Compute the areas, in acres, of the follow- 
ing triangles: 



38. a = 127.6, 
39. a = 0.9, 
40. a = 408, 
41. a = 63.89, 


b = 183.7, 
b = 1.2, 
b = 41, 
6 = 138.24, 


c = 201.3. 
c = 1.5. 
c = 401. 
c = 121.15. 



42. The percentage earning power, E, of an individual, in so far as 
it depends upon the eyes is given by Magnus by the formula 



E = c 



where x takes one of the values 5, 7, or 10, C being the maximal central 
visual acuity, VPi the visual field, A/Af the action of the extrinsic 
muscles, Cj and C 2 the central visual acuity of each eye, and A/P 2 the 
peripheric vision. Compute the value of # if C = 1, PI = 1, M = 1, 
Ci = 1, C s = 0.58, x = 10, P 2 = 1. Ans. 97.2% 

43. Compute E if d = 0.41, C 2 = 0.25, x = 5, P 2 = M = Pi = 1, 
C = 0.41. Ans. 33.06%. 



MATHEMATICS 



[IV, 72 



44. The percentage earning ability E, as dependent upon the eyes 
is given by Magnus as 

E = 



where F functional ability, V = necessary knowledge, K the 
ability to compete (demand for him), x has one of the values 5, 7, or 10. 
Compute E for F = 0.78792, V = 1, x = 10, K = 0.39396. 

Am. 71.78%. 

45. Compute E f or F = 0.8254, x = 10, K = 0.4127, 7 = 1. 

Ana. 75.52%. 

46. When w grams of a substance is dissolved in v liters of water at t 
centigrade, the osmotic pressure, p, of the solution and the molecular 
weight, M, of the solute are connected by the equation 

pv = 0.082 (273 + t)w/M. 

Compute the molecular weight of cane sugar from the data 
(a) p = 12.06, v = 1, t = 22.62, w = 171.0 Ans. 343.7 

(6) p = 24.42, = 3,J = 23.56, w = 102.6 Ana. 340.5 

Compute the osmotic pressure for glucose solution, given 

(c) v = 1, t = 26.90, w = 72, M = 180.21 4ns. 9.824 

(d) v = 2, t = 22,20, to = 360, M = 178.46 Ans. 24.36 

73. The Slide Rule. The slide-rule is an instrument for 
carrying out mechanically the operations of multiplication and 
division. It is composed of two pieces, usually about the shape 
of an ordinary ruler; one of the pieces (called the slide,) fits in 
a groove in the other piece. Each piece is marked in divisions 




FIG. 22 



(Fig. 22), such that the distance from one end (e. g., A) is equal 
to the logarithm of the number marked on it. 

To multiply one number (e. g., 2.5) by another (e. g., 2) we 



II, 73] 



LOGARITHMS 



89 



set the point marked 1 on scale B opposite the point marked 2.5 
on scale A (see Fig. 23). Then the product appears on scale A 




FIG. 23 

opposite the point 2 on scale B. Thus 5 on scale A lies oppo- 
site 2 on scale B in Fig. 23. This follows from the fact that 
log 2.5 + log 2 = log 5. 

Likewise, if 1 on scale B is set opposite any number a on 
scale A, then we find opposite any number 6 on scale B the 
number ab on scale A. 

Divisions can be performed by reversing this process. Thus 
if 6 on scale B be set opposite c on scale A, the 1 on scale B 
will be opposite c/b on scale A. 

A little practice with such a slide-rule will make clear the 
actual procedure in any case. 

Scales C and D are made just twice the size of scales A and B. 
It follows that any number on scale C, for example, is exactly 
opposite the square of that number on scale A. This facilitates 
the finding of squares and square roots, approximately. 

Scales C and D may be used in place of scales A and B for 
multiplication and division. Indeed, after some practice, 
scales C and D will be preferred for this purpose. 

More elaborate slide-rules, marked with several other scales, 
are for sale by all supply stores. Descriptions of these and 
full directions for their use will be found in special catalogs 
issued bv instrument makers. 



90 MATHEMATICS [IV, 73 

A simple slide-rule can be bought at a moderate price. One 
sufficient for temporary practice may be made by the student 
by cutting out the large figure printed on one of the fly-leaves 
of this book, and following the directions printed there. 

The student should secure some form of slide-rule and he 
should use it principally in checking answers found by other 
processes. 

As exercises the teacher may assign first very simple products 
and quotients. When the operation of the slide-rule has been 
mastered, the student may check the answers to the exercises 
on p. 86. 



CHAPTER V 



TRIGONOMETRY 

74. Introduction. The sides and angles of a plane triangle 
are so related that any three given parts, provided at least one 
of them is a side, determine the shape and the size of the triangle. 

Geometry shows how, from three such parts, to construct the 
triangle. 

Trigonometry shows how to compute the unknown parts of a 
triangle from the numerical values of the given parts. 

Geometry shows in a general way that the sides and angles 
of a triangle are mutually dependent. Trigonometry begins 
by showing the exact nature of this dependence in the right 
triangle, and for this purpose employs the ratios of the sides. 

75. Definitions of Trigonometric Functions. Let A be any 
acute angle. Place it on a pair of axes as in Fig. 24, with the 
vertex at the origin, one side along 

the ar-axis to the right, and the 
other side in the first quadrant. 
On this side choose any point M (ex- 
cept 0) and drop M N perpendic- 
ular to the or-axis. Let OM = r; 
then by plane geometry, 



x- 



FIG. 24 



r = v ar* 

where x and y are the coordinates of the point M. The differ- 
ent ratios of pairs of the three numbers x, y, and r, are designated 
as follows 

y _ ordinate _ 

r radius 

x abscissa 



(1) 
(2) 
(3) 



r radius 
y _ ordinate _ 
x abscissa 



of angle A, written sin A, 
= the cosine of angle A, written cos A, 



tangent of angle 
91 



> wr i tte n tan A. 



92 



MATHEMATICS 



[V, 75 



The reciprocals of these rations are also used, 

,., x abscissa 

(4) ~ = : 

y ordinate 



= the cotangent of angle A, written ctn A, 



(5) - = - = secant of angle A, written sec A, 
x abscissa 

(6) - = - = cosecant of angle A, written esc A. 
y ordinate 

These six ratios are called the trigonometric functions of the 
angle A. They do not at all depend upon the choice of the 
point M on the side of the angle but only upon the magnitude 
of the angle itself. 

For if we choose any two points M' and M " on the side of the 





































. 


_ 






















' 
































t 


*.! 






































































/ 


-/' 
































/ 























/ 
































, 


/ 


M 




















/ 






s 


























/ 


























/ 








> 


\j 


i 


















y 


M 
























/ 






y 


' 




^" 


^ 














/ 




























/ 










^ 


x 
















<" 


> 




y 






j 
















/ 


K 


B 


. 


^ 


** 






V 










^ 


/ 


3 





























/ 


\ 


^ 


s 


\ 














o 


,/ 










X 




A 






A 













^ 


^ 




~x 


V 




A' 






\ 




















X 


' 




























X 









































































FIG. 25 

same angle A, and denote their coordinates by (x f , y'} and (x", 
y") respectively, then by similar triangles, 



~. = ^-- = sin A, 



-- = % = tan A, etc. 



But if we take two points M' and M" at the same distance 
r from on the sides of two different angles A and B, then 

y' v" 

sin A = ' ^ '-* = sin B, 
r r 

tan A =^-^^- =tanB, 
x x 

and similarly the other functions of A and B are unequal. 



V, 76] 



TRIGONOMETRY 



93 



From these definitions we deduce the 
following relations which are of fundamen- 
tal importance in computing the unknown 
parts of right triangles. 

In any right triangle, having fixed atten- 
tion on one of the acute angles, 




side adjacent 
FIG. 26 



(7) 
(8) 

(9) 



The side opposite = hypotenuse X sine. 

also = side adjacent X tangent. 
The side adjacent = hypotenuse X cosine. 

also = side opposite X cotangent. 

side opposite 
The hypotenuse = 



also = 



sine 

side adjacent 

cosine 



EXERCISES 

Find the six functions of each of the acute angles in the right tri- 
angle whose sides are : 

1. 3, 4, 5. 2. 9, 40, 41. 3. 60, 91, 109. 

4. 7, 24, 25. 5. 16, 63, 65. 6. 20, 99, 101. 

7. 20, 21, 29. 8. 36, 77, 85. 9. 12, 35, 37. 

10. 2n + 1, 2n(n + 1), 2n 2 + 2n + 1. 11. 2n, n 2 - 1, n 2 + 1. 
12. 2(n + 1), n(n + 2), n 2 + 2n + 2. 13. a(6 2 - c 2 ), 2abc, a(6 2 + c 2 ). 

76. Functions of Complementary Angles. Let A and B be 
the acute angles in any right triangle. Then, 

?/ *T 

B sin A = cos B = - , cos A = sin B = , 
r r 

tan A = ctn # = -, ctn A = tan B = - 

x y 



x 
FIG. 27 



sec A = esc B = , esc A = sec B = - 
x y 



94 



MATHEMATICS 



[V, 76 



Since A -\- C = 90 (i. e., A and C are complementary), the 
above results may be stated in compact form as follows: 

A function of an acute angle is equal to the co-function of its 
complementary acute angle. 

77. Functions of 30, 45, 60. On the sides of a right 
angle lay off unit distances A B and AC and draw BC, forming 
an isosceles right triangle, Fig. 28. The angles at B and C are 
each 45, and the hypotenuse BC is equal to V2~ (why?). 




FIG. 28 



\ 




AID B 

FIG. 29 



From the definitions, 

sin 45 = cos 45 = I/ A/2 = A/2/2. 
tan 45 = ctn 45 = 1. 
sec 45 = esc 45 = A/2/1 = A/2. 

Construct an equilateral triangle whose sides are 2 units long, 
Fig. 29. Bisect one of its angles forming a right triangle ACD, 
in which A = 60, C = 30, and the altitude CD is equal to A/3 
(why?). Then from the definitions, 



sin 60 = cos 30 = A/3/2. 
cos 60 = sin 30 = 1/2. 
tan 60 = ctn 30 = A/3. 



ctn 60 = tan 30 = l/A/3. 
sec 60 = esc 30 = 2. 
esc 60 = sec 30 = 2/ A/3. 



V, 78] 



TRIGONOMETRY 



78. Eight Fundamental Relations. The following relations 
hold for the trigonometric functions of any acute angle A, 

(10) sin A esc A = 1, sine and cosecant are reciprocals ; 

(11) cos A sec A = 1, cosine and secant are reciprocals ; 

(12) tan A ctn A = 1, tangent and cotangent are reciprorocals ; 



(13) tan A = 



sin A 



cos A 

(15) sin 2 A + cos 2 A = 1 ; 

(16) tan 2 A + 1 = sec 2 A 



(14) ctn A = 



cos A 
sin A ' 



(17) ctn 2 A + I = esc 2 A. 



These eight identities are fundamental relations and should 
be thoroughly learned by the student. 

They may be proved as follows: (10), (11), (12) are direct 
consequences of the definitions in 75. To prove (13), we have 

tan A = - , sin A = - , cos A = - , 
x r r 

whence 



sin A 11 x 11 

-=--r-- = -= tan A. 
cos A r r x 



FIG. 30 



Similarly, 

cos A x . y x 

- = - -:- - = - = ctn A. 
sin A r r y 

From Fig. 30, 

(18) x 2 + y 2 = r 2 . 

Dividing through by r 2 , we have 

^+^-1, 



whence cos 2 A + sin 2 A = 1. 

Similarly, dividing (18) through by x 2 , and then by y z we prove 

(16) and (17). 

If the value of one function of an angle is known, the values 
of all the others can be found by means of these relations. 




96 MATHEMATICS [V, 78 

EXAMPLE. Given sin A = 1/2. Then, 

cos A = Vl sin 2 A = A/I = f V3, 
and, by (13), 

tan A = 1/V3 = iV. 

Since the other three functions are reciprocals of these three, we have 
ctn A = V3, sec A = f V, esc A = 2. 

The values of these functions can also be found graphically by con- 
structing a right triangle the ratio of whose 
sides are such as to make the sine of one angle 
equal to 1/2. This can evidently be done 
by making the side opposite equal to 1 and 
the hypotenuse equal to 2; then the side ad- 
jacent is equal to >/3. (Why?) The other 
functions can now be read directly from the 

figure, using the definitions. Thus, tan A = side opposite -f- side ad- 
jacent = l/A/3 = f A/3. 

EXERCISES 

1. Given sin 40 = cos 50; express the other functions of 40 in 
terms of functions of 50. 

2. The angles 45 + A and 45 A are complementary; express 
the functions of 45 + A in terms of the functions of 45 A. 

3. A and 90 A are complementary; express the functions of 
90 A in terms of the functions of A. 

4. Construct a right triangle, having given 

(a) hypotenuse = 6, tangent of one angle = 3/2. 

(b) cosine of one angle = 1/2, side opposite = 3.5. 

(c) sine of one angle = 0.6, side adjacent = 2. 

(d) cosecant of one angle = 4, side adjacent = 4. 

(e) one angle = 45, side adjacent = 20. 
(/) one angle = 30, side opposite = 25. 

5. In Ex. 4, compute the remaining parts of each triangle. 

6. Express each of the following as a function of the complementary 
angle: 

(a) sin 30. (6) tan 89. (c) esc 18 10'. (d) ctn 82 19'. 

(e) cos 45. (/)ctn!5. (g) cos 37 24'. (A) esc 54 46'. 



V, 80] TRIGONOMETRY 97 

7. Express each of the following as a function of an angle less 
than 45: 

(a) sin 60. (6) tan 57. (c) esc 69 2'. (d) ctn 89 59'. 

(e) cos 75. (/)ctn84. (g) cos 85 39'. (ft) esc 45 13'. 

8. Prove that if A is any acute angle, 

(a) sin A -sec A = tan A. (6) sin A- ctn A = cos A. 

(c) cos A -esc A = ctn A. (d) tan A- cos A sin A. 

(e) sin A-sec A-ctn A = 1. (/) cos A-csc A-tan A = 1. 

(g) (sin A + cos A) 2 1 + 2 sin A cos A, 

(h) (sec A + tan A) (sec A tan A) = 1. 

(i) (1 + tan 2 A) sin 2 A = tan 2 A. 

0') (1 - sin 2 A) esc 2 A = ctn 2 A. 

(k) sin 4 A cos 4 A = sin 2 A cos 2 A. 

(0 tan 2 A cos 2 A + cos 2 A = 1. 

(m) (sin A + cos A) 2 + (sin A cos A) 2 = 2. 

(n) sec A cos A = sin A tan A. 

(o) (sin 2 A cos 2 A) 2 = 1 4 sin 2 A cos 2 A. 

(p) (1 - tan 2 A) 2 = sec 4 A - 4 tan 2 A. 

9. Express the values of all the other functions of A hi terms of 
(a) sin A, (6) cos A, (c) tan A, (d) ctn A, (e) sec A, (/) esc A. 

79. Solution of Right Triangles. The values of the six 
trigonometric ratios have been computed for all acute angles, 
and recorded in convenient tables. They are given to four 
decimal places in Table II, at the end of the book. These 
tables, together with the definitions of the functions, enable 
us to solve all cases of right triangles. 

80. General Directions for Solving Right Triangles. 

(1) Draw a diagram approximately to scale, indicating the 
given parts. Mark the unknown parts by suitable letters, and 
estimate their values. 

(2) // one of the given parts is an acute angle, consider the 
relation of the known parts to the one which it is desired to find, 
and apply the appropriate one of formulas (7), (8), (9), p. 93. 

(3) // two sides are given, and one of the angles is desired, 



98 



MATHEMATICS 



[V, 80 



think of the definition of that function of the angle which em- 
ploys the two given sides. 

(4) Check the results. The larger side must be opposite the 
larger angle, and the square of the hypotenuse must be equal to 
the sum of the squares of the other two sides. 

The following examples illustrate the process of solution. 

EXAMPLE 1. Given the hypotenuse = 26, and 
one angle = 43 17'; find the two sides and the other 
acute angle. Do not use logarithms. 

Draw a figure ABC in which AC = 26, A = 43 
17' and denote the unknown parts by suitable let- 
ters, x, y, and C. Find C as the complement of 

A: 

90 00' 
A = 43 17' 
C = 56 43' 




26 



x B 

FIG. 32 



To find x note that it is adjacent to the given angle and that the hypo- 
tenuse is given, 
Then by (8) 75 

x = 26 cos 43 17' 
cos 43 17' = 0.7280 
26 



Similarly by (7) 75 

y = 26 sin 43 17' 



sin 43 17' = 0.6856 
26 



4368 
1456 
x = 18.928 

CHECK: tan A = y/x = 0.9418. 

EXAMPLE 2. An acute angle 
10' and the opposite side is 78. 
Solve by means of logarithms. 
By (8) 75 

x = 78 ctn 62 10' 

log 78 = 11.8921 - 10 
log ctn 62 10' = 9.7226 - 10 
log x = 1.6147 
x = 41.18 



41136 
13712 
y = 17.8256 

tan 43 17' = 0.9418. 

of a right triangle is 62 
Find the other parts. 

By (9) 75 

r = 78/sin 62 10' 

log 78 = 11.8921 - 10 
log sin 62 10' = 9.9466 - 10 
log r = 1.9455 
r = 88.20 




V, 81] 



TRIGONOMETRY 



99 



CHECK: 



r = 88.20 

x = 41.18 

r + x = 129.38 

r - x = 47.02 



log (r + x) = 2.1119 

log (r - x) = 1.6723 

3.7842 

log 78 2 = 3.7842 



EXAMPLE 3. The hypotenuse of a right triangle is 42.7 and one 
side is 18.5. Find the other parts. To find one of the angles, as C, 
note that the hypotenuse and side adjacent are known. Then 




FIG. 34 



ICC 

cos C = ~ = 0.4332 



C = 64 19'.6 



42?7 = 1823.29 
18.5 2 = 342.25 



x 2 = 1481.04 
x = 38.48 



SOLUTION BY LOGARITHMS. 

18.5 
cosC = . 

log 18.5 = 1.2672 
log 42.7 = 1.6304_ 
log cos C = 9.6368 
C = 64 19' 



A = 90 - C = 25 40'.4 

CHECK: x = 18.5 ctn 25 40'.4 

= 18.5 X 2.0803 = 38.48 



= 42.7 2 - 18.5 2 

= 61.2 X 24.2 

log 24.2 = 1.3838 

log 61. 2 = 1.7868 
log x 2 = 3.1706 
log x = 1.5853 
x = 38.48 



81. Graphical Solution. As shown in 35, if the triangle be 
drawn to scale, the unknown sides can be read off on the scale, 
and the unknown angles on a protractor. The results so ob- 
tained will be accurate enough to detect any large errors in the 
computations. 



100 



MATHEMATICS 



[V, 81 



EXERCISES 

Let A, B, C represent the three angles of any triangle and a, b, c 
the sides opposite these angles. 

1. Solve graphically the following triangles: 

(a) a = 5, b = 4, c = 7. Ans. A = 44 30', B = 34, C = 101 30'. 
Ans. A = 22, B = 60, C = 98. 
Ans. A = 38, B = 60, C = 82. 



(6) a = 3, b = 7, c = 

(c) a = 5, b = 7, c = 

(d) a = 8, b = 7, B = 60. 

Ans. Ai = 82, A 2 = 98 

(e) a = 3, b = 5, c = 7. 
(/) a = 7, A = 120, b = 5. 
(0) o = 42, b = 51, A = 55. 

Ans. Bi = 84, B 2 = 96, 



Ci = 38, C 2 = 22, ci = 5, c 2 = 3. 
Ans. 4 = 22, B = 38, C = 120. 
Ans. = 38, C = 22, c = 3. 



= 41, C 2 = 29 
2. Solve the following right triangles (C = 90). 



= 34, c 2 = 25. 



Required parts (Answers). 
B = 60, 
B = 45, 
B = 30, 
A = 65, 
A = 50, 
A = 20, 
A = 45, 
A = 36 52' 
A = 4 46', 
B = 67 
5 = 53, 
A = 48, 

3. The width of the gable of a building is 32 ft. 9 in. The height of 
the ridge of the roof above the plates is 14 ft. 6 in. Find the inclina- 
tion of the roof, and the length of the rafters. 

Ans. 41 32', 21 ft. 10 in. 

4. The steps of a stairway have a tread of 10 in. and a rise of 7 in. ; 
at what angle is the stairway inclined to the floor? Ans. 35. 

5. The shadow of a tower 200 ft. high is 252.5 ft. long. What is the 
angle of elevation of the sun? 



Given parts. 


(a) 


A 


= 30, 


a 


= 12, 


(6) 


A 


= 45, 


b 


= 8, 


(c) 


A 


= 60, 


c 


= 20, 


(d) 


B 


= 25, 


a 


-72, 


(^ 


B 


= 40, 


b 


= 33, 


(!) 


B 


= 70, 


c 


= 81, 


(?) 


a 


= 6, 


b 


= 6, 


(A) 


a 


= 3, 


c 


= 5, 


(*) 


b 


= 12, 


c 


= 13, 


0') 


A 


= 23, 


a 


= 3.246, 


(fc) 


A 


= 37, 


b 


= 7.28, 


(0 


B 


= 42, 


c 


= 1021, 



b 


= 20.78, 


c 


= 24 


a 


= 8, 


c 


= 11.31 


a 


= 17.32, 


6 


= 10 


b 


= 33.57, 


c 


= 79.44 


a 


= 39.33, 


c 


= 51.34 


a 


= 27.70, 


b 


= 76.12 


B 


= 45, 


c 


= 8.484 


B 


= 53 8', 


b 


= 4 


B 


= 85 14', 


a 


= i 


b 


= 7.647, 


c 


= 8.307 


a 


= 5.486, 


c 


= 9.116 


a 


= 758.7, 


b 


= 713.8 



V, 82] TRIGONOMETRY 101 

6. A cord is stretched around two wheels with radii of 7 feet and 1 
foot respectively, and with their centers 12 feet apart. Find the 
length of the cord. Ans. 12 V3 + 10w = 52.2 ft. 

7. Two objects A, B in a rectangular field are separated by a thicket. 
To determine the distance between them, the lines AC = 45 rods, 
BC = 36 rods, are measured parallel to the sides of the field. Find 
the distance AB. Ans. 57.63 

8. One bank of a river is a bluff rising 75 ft. vertically above the 
water. The angle of depression of the water's edge on the opposite 
bank is 20 27'. Find the width of the river. Ans. 201.1 

9. A smokestack is secured by wires running from points on the 
ground 35 ft. from its base to points 3 ft. from its top. These wires 
are inclined at an angle of 40 to the ground, (a) What is the height 
of the smokestack? (6) The length of the wires? (c) What is the 
least number of wires necessary to secure the stack? If they are sym- 
metrically placed, how far apart are their ground ends? (d) How 
far are the lines joining their ground ends from the foot of the stack? 
(e) From the top of the stack? (/) What angle do the wires make 
with these lines? (<?) With each other? (h) What angle does the 
plane of two wires make with the ground? (i) What angle does the 
perpendicular from the foot of the stack on this plane make with the 
ground? (j) What is its length? [DURFEE] 

10. A tree stands on a horizontal plane. At one point in this plane 
the angle of elevation of the top of the tree is 30, at another point 
100 feet nearer the base of the tree the angle of elevation of the top is 45. 
Find the height of the tree. 

11. Find the length of a ladder required to reach the top of a building 
50 ft. high from a point 20 ft. in front of the building. What angle 
would the ladder in this position make with the ground? 

82. General Angles. Rotation. Up to this point we have 
defined and used the trigonometric functions of acute angles 
only. Many problems require the consideration of obtuse angles 
and others, particularly those concerned with the rotating parts 
of machinery, involve angles greater than 180 or 360 even, and 
it is necessary to distinguish between parts in the same or par- 
allel planes which rotate in the same or in opposite directions. 



102 



MATHEMATICS 



[V, 82 



An angle may be thought of as being generated by the rota- 
tion of one of its sides about the vertex; its first position is 

called the initial side, its final 
position the terminal side of 
the angle. An angle gener- 
ated by rotation opposite to 
the motion of the hands of 
a clock (counterclockwise) is 
FIG. 35 said to be positive; an angle 

generated by clockwise rotation is said to be negative. In draw- 
ings a curved arrow may be used to show the direction of rota- 
tion, the arrow head indicating the terminal side. 

83. Trigonometric Functions of any Angle. Let $ = XOP 

be any angle placed with its vertex at the origin and its initial 

side along the positive z-axis. Let P be any point (except 0) 

kY 




J" a> 



FIG. 36 

on the terminal side and let x, y be its coordinates (positive, 
negative, or zero depending upon the position of P in the plane) ; 
let r be the distance from to P (always positive). Then the 
trigonometric functions of $ are defined as follows: 



(19) 



sin (f> = - , 



cos = - . 



The definitions (19) apply to all angles without exception. 



(20) 



tan </> = 



sec d> = - . 
x 



V, 83] 



TRIGONOMETRY 



103 



The definitions (20) apply to all angles except odd multiples of a 
right angle; this exception is necessary because for all such 
angles x is zero. 



(21) 



ctn <A = - 

v 



esc <6 = . 

y 



The definitions (21) apply to all angles except even multiples 
of a right angle; for all such angles y is zero. 

These definitions apply of course to all acute angles and 
give the same values as the definitions in 75. These new 
definitions are more general because they apply to angles to 
which the former do not apply. 

These ratios are independent of the choice of P on the terminal 
side of the given angle. They depend upon the magnitude and 
sign of the angle. For, if we choose a different point P' on the 
terminal side of <, we shall have 



in magnitude and sign and this implies that 



"-. = 2-i etc. 



The signs of the trigonometric functions of an angle depend 
upon the quadrant of the plane in which 
the terminal side of <f> falls when it is placed 
on the axes. An angle < is said to be an 
angle in the first quadrant when its ter- 
minal side falls in that quadrant, and simi- 
larly for the second, third, and fourth 
quadrants. The signs of the sine and the 
cosine of an angle in each of the quadrants 
should be thoroughly learned. The accompanying diagram in- 
dicates these signs. 




FIG. 37 



104 



MATHEMATICS 



[V, 83 



The signs of the other functions are determined by noting that 
tan < is positive when sin < and cos <j> have like signs and 
negative when they have unlike signs; and that reciprocals 
have like signs. 

84. The Fundamental Rela- 
tions. The fundamental identi- 
ties (10) to (18) which were proved 
for acute angles in 78 are valid 
for any angle whatever. The 
proofs which are similar to those 
already given are left to the student. 

85. Quadrantal Angles. Let P be 
a point on the terminal side of an 
angle at a distance r from the origin. 

When (j> = 0, P coincides with PI and its coordinates are 
x = r and y = 0; then by 83 




sin = - = 0, 

r 

tan = - = 0. 
x 



cos = - 
r 



1, 



sec = - = 1. 
x 



The angle has no cotangent nor cosecant. 

When = 90, P coincides with P 2 , x = 0, y = r; then 



sin 90 = = 1, 
r 

ctn 90 = - = 0, 

y 



cos 90 = - = 0, 
r 

esc 90 = - = 1. 

y 



The angle 90 has no tangent nor secant. 

When <j> = 180, P coincides with P 3 , x = - r, y = 0; then 



sin 180 = - = 0, 
r 

tan 180 = - = 0, 



cos 180 = - = - 1, 
r 

sec 180 = 



The angle 180 has no cotangent nor cosecant. 



V, 86] TRIGONOMETRY 105 

When = 270, P coincides with P 4 , x = 0, y = r; then 

sin 270 = - = - 1, cos 270 = - = 0, 

r r 

ctn 270 = - = 0, esc 270 = -=_!. 

v y 

The angle 270 has no tangent nor secant. 

Often it is said that tan 90 = o } but this does not mean 
that 90 has a tangent; it means that as an angle increases 
from to 90, tan increases without limit, and that before $ 
reaches 90. Similar remarks apply to the statements ctn 
= oo , tan 270 = oo , etc. 

86. Line Representations of the Trigonometric Func- 
tions. The trigonometric functions denned in 83 are abstract 
numbers; each is the ratio of two lengths. They are not lengths 
nor lines. They can however very conveniently be represented 
by line segments in the sense that the number of length units in 
the segment is equal to the magnitude of the function, and the sign 
of the segment is the same as the sign of the function. 

Let an angle of any magnitude and sign be placed on the 
axes, Fig. 39. With the origin as center and a radius one unit 
length draw a circle cutting the positive z-axis at A, the positive 
y-axis at B, and the terminal side of at P. Draw tangents 
to this circle at ^4. and at B and produce the terminal side in 
one or both directions from to cut these tangents in T and S 
respectively. Draw PQ perpendicular to the z-axis. Then, if 
we agree that QP shall be positive upward, OQ shall be positive 
to the right, and that OT, or OS, shall be positive when it has 
the same sense as OP and negative when it has the opposite 
sense, 

QP represents sin 0, OQ represents cos 0, 
A T represents tan 0, AS represents ctn 0, 
OT represents sec 0, OS represents esc 0. 



106 



MATHEMATICS 



[V, 86 



For, sin </ = QP/OP = the number of units of length in QP 
since OP = unit length and sin $ and QP agree in sign from 
quadrant to quadrant. Similarly the others may be proved. 



\ 





FIG. 39 



The student is warned against thinking or saying that " QP 
is the sine of "; say " The number of units in QP is sin (/> " 
or, " QP represents sin 0." 

87. Congruent Angles. Any angle formed by adding to or 
subtracting from a given angle <f>, any multiple of 360 is said 
to be congruent to 0; thus 217 and 143 are congruent. 
It is obvious from the definitions and from the line representa- 
tions of the functions of an angle that two congruent angles 
have equal functions. The functions of any angle formed by 
adding to or subtracting from a given angle a multiple of 360 are 
the same as the corresponding functions of the given angle. 



v, 



TRIGONOMETRY 



107 



88. Trigonometric Equations. To solve the equation sin x 
= 1/2 is to find all angles which satisfy it. We know that 
x = 30 is a solution for sin 30 = 1/2; x = 150, x = - 210, 
x = 750, are also solutions. We can find all its solutions by 
the following graphical method. 

1) To solve the equation 
sin x = s. 



where s is a given number between 
- 1 and + 1, draw a unit circle 
center at the origin and on the 
y-Sixis lay off OB = s (above if 
s > 0, below if s < 0) and through 
B draw a parallel to the z-axis cut- 
ting the circle in C and D, Then 
the positive angles 




FIG. 40 



a = AOC and j8 = AOD 

are solutions (and the only solutions between and 360) of 
the given equation. Any angle congruent to a or to /3 is also a 
solution, and there are no others. These results follow directly 
from the line representations of the functions in 86. 
2) To solve the equation 

cos x = c, 

where c is a given number between 1 and + 1, draw a unit 
circle center at the origin, Fig. 41, and lay off on the z-axis 
OB = c (to the right if c > 0, to the left if c < 0) and draw 
through B a parallel to the 7/-axis cutting the circle in C and D. 
Then the positive angles 

a = AOC and = AOD 

are solutions (and the only solutions between and 360) of 
the given equation. Any angle congruent to a or to /3 is also a 
solution and there are no others. 



108 



MATHEMATICS 



[V, 88 





FIG. 41 
3) To solve the equation 



FIG. 42 



tan x = t 

where t is any given number whatever, draw a unit circle center 
at the origin, and lay off on the tangent at A, 

AB = t 

and draw a line through and B cutting the circle in C and D. 
Then the positive angles 

a = AOC, j8 - AOD 

are solutions (and the only solutions between and 360) of 
the given equation. Any angle congruent to a or to /3 is also 
a solution, and there are no others. 

Many other trigonometric equations can be reduced to one 
of these three forms by the transformations given in 78 and 
hence can be solved by the above methods. 

For example, the equation 



is equivalent to 
Again, 



tan x = 3. 

2 sin 2 x cos x = 1 
can be reduced to the form 

(cos x + l)(cos x |) = 



V, 89] 



TRIGONOMETRY 



109 



, 
\ 




0- \ 



f,l3 



by replacing sin 2 x by 1 cos 2 x, transposing all the terms to 
the left side, and factoring. 

8Q. Graphs of the Trigonometric Functions. The varia- 
tion in the sine of a given angle as the angle increases from 
to 360 may be exhibited graphically as follows. 

Divide the circumference of a unit 1X & 

circle into a convenient number of equal 
arcs. In Fig. 43, the points of division 
are marked 0, 1, 2,3, 12. The 
length of the circumference is approxi- 
mately 6.3; lay this off on the z-axis 
(Fig. 44) and divide it into the same 
number of equal parts and number them 
to correspond with the points of division on the circumference. 

At each point of division on the a>axis lay off vertically the 
line representation QP, of the sine of the angle whose terminal 
side goes through the corresponding point of division on the 
circle. Connect the ends of these perpendiculars by a smooth 
curve. This is called the sine curve or the graph of sin x. 



FIG. 43 



* 




A 



23 4 5 6\ 



g\ io\ n\ Si2 is u is 16 x 




FIG. 44 

As the angle increases from to 360, P moves along the 
circle successively through the points 0, 1, 2, 3, , 12, Q' 
moves along the z-axis successively through the corresponding 
points 0, 1, 2, 3, , 12, and P' traces the sine curve. 

The graphs of the other trigonometric functions, cos x, tan x, 



110 



MATHEMATICS 



[V, 89 



etc., are constructed in a similar manner by making use of their 
line representations given in 86. 

If the angle increases beyond 360, P makes a second revolu- 
tion around the circle, and the values of all the trigonometric 
functions repeat themselves in the same order and the graphs 
from x = 6.3 to x = 12.6 will in all cases be a repetition of those 
from x = to x = 6.3. If P goes on indefinitely the graph 
will be repeated as many times as P makes revolutions. 

Functions which repeat themselves as the variable or argu- 
ment increases are called periodic functions. The period is the 
smallest amount of increase in the variable which produces the 
repetition of the value of the function. Thus, sin a; is a peri- 
odic function with a period of 360, while the period of tan x 
is 180. 

90. Functions of Negative Angles. Let AOC = </> be any 
angle placed on the axes; and let AOC' be its negative, <j>; 





FIG. 45 

lay off OP' = OP and draw PP f . Let x, y be the coordinates 
of P and x', y' those of P'; let OP = r and OP' = r'. Then 
no matter what the magnitude or sign of 4>, 



y = - y , 



r = r' 



V, 91] 



TRIGONOMETRY 



111 



and by the definitions, 83 



sin ( 0) = = -- = sin 



> 

cos ( $) =-7 = - = cos </, 
r r 



tan (- <j>) = 7 = - - = - tan 



ctn ($)= = -- = ctn </>, 

y y 



sec ( <f>) =,=- = sec 
a; x 



r r 

esc ( 0) = = -- = esc </>. 



91. The Trigonometric Functions of 90 -f <t>. Let any 

angle </> be placed on the axes; draw a circle, center at the origin, 
with any convenient radius r, cutting the terminal side of in P 
and the terminal side of <f> + 90 in Q. Let the coordinates of P 
be (a, &); then no matter in what quadrant P is, Q is in the 
next quadrant and its coordinates are (6, a), for the right 
triangles OMP and QNO have the hypotenuse and an acute 
angle of the one equal to the hypotenuse and an acute angle of 
the other. Then by the definitions, 83 



C-6,a)\0 





FIG. 46 



MATHEMATICS 



[V, 91 



sin (90 + 0) = - = cos 
r 



cos (90 + 0) = 



- b 



= sin 0, 



tan (90 + 0) = 



ctn (90 + 0) = 



- 6 

- 6 



= ctn 0, 



= tan 0, 



sec (90 + 0) = 



= CSC 0, 



csc (90 + 0) = - = sec 0. 
a 

These formulas hold for all angles.* 

92. Functions of 9, 90 6, 180 0, 270 6. If we 
put for in succession, - 6, 6, 90 - 6, 90 + 6, 180 - 0, 
180 + 6, 270 - 9, 270 + 6, we obtain the values in the 
following table, 6 being any angle.* By drawing diagrams the 
results tabulated can be verified. The student is advised to 
do this. 





90 e. 


90+0. 


180 e. 


180 +0. 


270" 8. 


270 +8. 


360 9. 


-e. 


sin 


cos 


cos 9 


sin 


sin 


cos 6 


cos 


sin 


sin 


cos 


sin 


sin 6 


cos 


cos 6 


sin 


sin 


cos 


cos 


tan 


ctn 6 


ctn 


tan0 


tan 


ctn e 


ctn 


tan 


tan 


ctn 


tan 


tan 


ctn 6 


ctn 6 


tan 


tan0 


ctn 


ctn 


sec 


csc 


csc 


sec 


sec 


csc 


csc 


sec 


sec 


csc 


sec B 


sec 


csc 


csc 6 


sec 5 


sec 


csc 


csc 



If we inspect the table carefully, we find that it can be summed 
up in the two rules that follow. 



* Except that no angle whose terminal side falls on the y-axis has a tangent or 
secant and no angle whose terminal side falls on the z-axis haa a cotangent or cosecant. 



V, 93] 



TRIGONOMETRY 



113 



1. Determine the sign by the quadrant in which the angle would 
lie if 8 were acute; the result holds whether 6 is acute or not. 

2. // 90 or 270 is involved, the function changes name to the 
corresponding cof unction, while if 180 or 360 is involved the 
function does not change name. 

EXAMPLE 1. sin 177 = sin (180 - 3) = + (rule 1) sin (rule 2) 3. 
EXAMPLE 2. cos 177 = cos (90 + 87) = - (rule 1) sin (rule 2) 37. 
EXAMPLE 3. tan300 = tan (180 + 120) = + (rule 1) tan (rule2) 120. 

93. Plotting Graphs from Tables. For many purposes, 
such as the measurement of arcs and the speed of rotations, and 
generally in the calculus and higher mathematics, angles are 
measured in terms of a unit called the radian. 

A radian is a positive angle such that when its vertex is placed 
at the center of a circle the intercepted arc is equal in length 
to the radius. This unit is thus a little less than one of the 
angles of an equilateral triangle, 57. 3 approximately. It is 
easy to change from radians to degrees and vice versa, by 
remembering that 
(22) TT radians = 180 degrees. 

Unless some other unit is expressly stated, it is always under- 
stood that in graphs of the trigonometric functions the radian 
is the unit angle and that 1 unit on the x-axis represents 1 radian. 
These graphs can be constructed from a table of their values 
such as Table III at the end of the book. Thus to plot the 
graph of sin x, draw a pair of rectangular axes on squared paper 



FIG. 47 



114 MATHEMATICS [V, 93 

and mark the points 1, 2, 3, on the x-axis. These unit 
lengths are divided by the rulings of the cross-section paper 
into tenths. At each of these points of division on the x-axis 
lay off parallel to the y-axis the sine of the angle from the table, 
e. g., at 1 we plot AP = .84 = sin 1 (radian). The curve may 
be extended beyond the first quadrant by the principles of 92. 
Similarly the graphs of cos x and tan x can be plotted from 
their tabulated values. 

EXERCISES 

1. Express each of the following functions as functions of angles 
less than 90. 

(a) sin 172, (6) cos 100, (c) tan 125, (d) ctn 91, (e) sec 110, 
(/) esc 260, (g) sin 204, (h) cos 359, (i) tan 300, (j) ctn 620. 

2. Express each of the preceding functions as functions of an angle 
less than 45. 

3. Express each of the following functions in terms of the functions 
of positive angles less than 45. 

(a) sin (-160), (6) cos (- 30), (c) esc 92 25', 

(d) sec 299 45', (e) sin (- 52 37'), (/) cos (- 196 54'), 

(g) tan 269 15', (h) ctn 139 17', (i) sec (- 140), 

(j) ctn (- 240), (ft) esc (- 100), (Z) sin (- 300), 

(m) cos 117 17', (n) sin 143 21' 16", (o) tan 317 29' 31", 

(p) ctn 90 46' 12", (q) sec (- 135 14' 11"), (r) cos (- 428). 

4. Simplify each of the following expressions. 

(a) sin (90 + x) sin (180 + x) + cos (90 + x) cos (180 - x). 
(6) cos (180 + x) cos (270 - y) - sin (180 + x) sin (270 - y). 
(c) sin 420 cos 390 + cos (- 300) sin (- 330). 

5. Prove each of the following relations, 
(a) cos \(x - 270) = + sin x/3. 

(6) sec ( x 540) = sec x. 

6. Verify each of the following equations, 
(a) cos 570 sin 510 - sin 330 cos 390 = 0. 

(6) cos (90 + a) cos (270 - a) - sin (180 - a) sin (360 - a) 

= 2 sin 2 a. 



V, 93] TRIGONOMETRY 115 

(c) 3 tan 210 + 2 tan 120 = - >/3. 

(d) 5 sec 2 135 - 6 ctn 2 300 = 8. 

(e) sin (90 + ) sin (180 + x) + cos (90 + x) cos (180 - x) = 0. 

tan (90 + tt) + C8c2 (270 -) = L+ ^c 2 . 



7. Construct a table containing the functions of the eighths and 
twelfths of 360. 

8. In each of the following equations find graphically the two solu- 
tions which are between and 360 and compute the values of the 
other five functions of each of these angles. 

(a) sin x = 3/5. (6) sin x = 1/3. (c) cos x = 1/3. 

(d) ctnx = - 3. (e) sec x = - 5/3. (/) esc x = 13/5. 

(0) esc x = -^3. (h) tan x = V?. (i) tan x = 2.5. 

9. Verify each of the following equations. 

(a) sin 90 + cos 180 = 0. (g) sec 270 + esc = 0. 

(6) sin 270 + cos = 0. (h) sin 120 + sin 300 = 0. 

(c) esc 90 + sec 180 = 0. (i) cos 150 + cos 330 = 0. 

(d) esc 270 + sec = 0. (j) tan 135 + tan 225 = 0. 

(e) sin + cos 270 = 0. (fc) ctn 315 + ctn 45 = 0. 
(/) sin 180 + cos 90 = 0. (1) sin 120 + cos 210 = 0. 

10. Find graphically another angle between and 360 which has 
the same 

(a) sine as 140, (6) sine as 220, (c) cosine as 330, 

(d) tangent as 230, (e) cotangent as 110, (/) secant as 160. 

11. Find the values of 6 between and 360 which satisfy the 
following equations. 

(a) sin = sin 320. (d) cos = - cos 50. 

(6) tan 9 = tan 125. (c) ctn = - ctn 220. 

(c) sec = sec 80. (/) esc = - esc 340. 

12. In what quadrant does an angle lie if sine and cosine are both 
negative? if cosine and tangent are both negative? if cotangent is 
positive and sine negative? 

13. In finding cos x from the equation cos x = =*= Vl sin 2 x, 
when must we choose the positive and when the negative sign ? 

14. Plot the graphs of each of the following functions and determine 
its period. 



116 



MATHEMATICS 



[V, 93 



(a) cos x. 
(d) sec x. 
(g) cos (- x). 



(6) tan x. 
(e) esc x. 
(K) sin (90 + x). 



(c) ctn x. 
(/) sin (- x). 
(i) sin x cos x. 



15. Plot the graph of each of the following functions. 

(a) x + sin x. (6) x 2 + sin x. (c) sin x + cos x. 

(d) x + cos x. (e) x cos x. (/) x 1 + sin x. 

94. Sine and Cosine of the Sum of two Angles. Let 

AOB = x, BOC = y, then AOC = x + y. With as center 
and a convenient radius r > 0, strike an arc cutting OC in P. 
Drop PQ perpendicular to OB, also PR and QS perpendicular to 




o R S 




it o s 



FIG. 48 



CM. Through Q draw a parallel to OA cutting Pfl in T. Then 
by (7), 75, 

r sin (x + y) = RP = SQ + TP. 
Now by (7) and (8), 75, we have 

OQ = r cos y and <SQ = OQ sin x = r cos y sin x, 
PQ - r sin i/ and TP = PQ cos x = r sin y cos 2. 

Hence we may write 

r sin (a: + y) = r cos y sin x + r sin y cos a:, 
and 
(23) sin (x + y) = sin x cos y + cos x sin y. 



V, 95] TRIGONOMETRY 117 

Similarly, we may write 

r cos (x + y) = OR = OS - TQ. 
Then as before, 

OS = OQ cos x = r cos y cos x, 
TQ = PQ sin x = r sin y sin x. 

Hence we may write 

r cos (x + T/) = r cos ?/ cos x r sin y sin .r, 
and 

(24) cos (x + y) = cos x cos y sin x sin y. 

The above formulas, therefore, hold true for all acute angles 
x and y. They are called the addition formulas. 

It is readily proved that if x = a and y = /3 are any two 
acute angles for which these formulas hold good they will hold 
good for any two of the angles a, ft, a + 90, a - 90, /3 + 90, 
/3 90. Therefore, since we have found that they hold good 
for all acute angles, they hold good for all positive or negative 
angles of any magnitude whatever. 

The addition formulas may be translated into words as follows: 

I. The sine of the sum of two angles is equal to the sine of the 
first times the cosine of the second, plus the cosine of the first times 
the sine of the second. ' 

II. The cosine of the sum of two angles is equal to the cosine of 
the first times the cosine of the second minus the sine of the first 
times the sine of the second. 

95. Tangent of the Sum of two Angles. This can be de- 
rived from the addition formulas as follows 

sin (x + y) sin x cos y + cos x sin y 

tan (x + y) = - : : . 

cos (x + y) cos x cos y sin x sin y 

If we divide each term of the numerator and denominator of 



118 MATHEMATICS [V, 95 

the last fraction by cos x cos y, we have 

sin x sin y 

cosx cos y 
tan (x + y) = 



sin x sm 



cos z cos y 
that is 

(25) (. + )- *" + "' . 

1 tan x tan y 

This formula holds good for all angles such that z, y, and z + y 
have tangents. 

96. Functions of Twice an Angle. If we put z for y in 
(23), (24), 94, and (25), 95, these formulas give 

(26) sin 2z = 2 sin z cos z. 

(27) cos 2z = cos 2 z sin 2 z. 

(28) = 2 cos 2 z - 1. 

(29) =1-2 sin 2 z. 

2 tan z 



(30) tan2x = 



1 tan 2 x 



97. Functions of Half an Angle. The preceding formulas 
are true for all values of x for which they have a meaning. Hence 
we may replace x by any other quantity. If we write x/2 in 
place of x in (28) and (29), 96, and solve the resulting equa- 
tions for sin (z/2) and cos (z/2), we find 



, ._ cos z 

(31) sm \x = db 



. /I + cos z 
(32) cos \x = db ^ g ' 

Whence on dividing (31) by (32) 

fl cos z 1 cos z sin z 



(33) tan |z = x/ , 

1 cos z sin z 1 + cos z 



V, 97] TRIGONOMETRY 119 

The positive or 'negative sign is to be chosen according to the 
quadrant in which z/2 lies. 

EXERCISES 

1. Putting 75 = 45 + 30, find cos 75 and tan 75. 

2. ^Putting 15 = 45 + (- 30), find sin 15, cos 15, and tan 15. 

3. 'Putting 15 = 60 + (- 45), find sin 15, cos 15, and tan 15. 

4. Putting 90 = 60 + 30, find sin 90 and cos 90. 

5. Show that sin (x y) = sin x cos y cos x sin y. 

6. Show that cos (x y) = cos x cos y + sin x sin y. 

7. Putting 15 = 60 - 45, find sin 15. 

8. Show that sin 3x = sin x(3 4 sin 2 x) = sin x(4 cos 2 x 1). 

9. Show that cos 3x = cos x(4 cos 2 x 3) = cos z(l 4 sin 2 x). 

10. Find sin 4x; cos 4x; tan 4x. 

11. Show that tan (45 + A) = ? + ^ A A . 

1 tan A 

12. Show that 

, . tan x tan y 

(a) tan (x y) = , 

1 + tan x tan y ' 

, . ctn x ctn y 1 
Ctn(x + y) = ^nT + inT- 

13. From the trigonometric ratios of 30, find sin 60, cos 60, tan 60. 

14. Express sin 6,4, cos 6 A, tan GA in terms of functions of 3A. 

15. Find sin 22|, cos 22-J- , and tan 22, from cos 45. 

16. Find sin 15, cos 15, and tan 15, from cos 30. 

17. Find cos (x + y), having given sin x = 3/5 and sin y = 5/13, 
x being positive acute, y being positive obtuse. Ans. 63/65. 

18. Verify the following: 

(a) sin (60 + x) - sin (60 - x) = sin x. 

(6) cos (30 + y) - cos (30 - y) = - sin y. 

(c) cos (45 + x) + cos (45 x) = V2 cos x. 

(d) cos (Q + 45) + sin (Q - 45) = 0. 

(e) sin (x + y) sin (x y) = sin 2 x sin 2 y. 

. , , sin (x + y) tan x + tan y 2 tan x 

(f ) ~ 7 (0) sin 2x = 

^* ' otr\ tfm 1*1 -for* />- *.,!,! ' 



sin (x y) tan x tan y ' 1 + tan 2 x ' 

,, , esc 2 x sin $x 

(h) sec 2x = ^. (i) tanjx = = r-. 

esc 2 x 2 1 + cos \x 

(fi ctn ix - sin ^ (H tn ' A - l ~ C08 A 
n * X 1-cos^x* sin A ' 



120 MATHEMATICS [V, 97 

(I) 2 esc 2s = sec s esc s. 

(m) tan (x + 45) + ctn (x - 45) = 0. 

19. Prove each of the following identities, 
(a) cos (A + B) cos (A - B) = cos 2 A - sin 2 B. 
(6) sin (A + B) cos B cos (A + B) sin B = sin A. 

(c) sin (A + B) + cos (A - B) = (sin A + cos A) (sin B + cos B). 

(d) cos 4 A = f + 5 cos 2A + | cos 4A. 

(e) sin 4 A = f 5 cos 2A + | cos 4A. 
(/) sin 2 A cos 2 A = | - | cos 4A. 

(g) sin 2 .A cos 4 A. = ^ + jj cos 2A ^ cos 4A ^ c s 6A. 
(/i) cos (x y -\- z) = cos a; cos y cos 2 + cos x sin ?/ sin z 

sin x cos y sin z + sin x sin y cos z. 

(i) cos sin (y z) + cos y sin (z x) + cos z sin (x y) = 0. 
0') sin A + sin 5 = 2 sin f(A + 5) cos f (A - B). 
(fc) sin A - sin B = 2 cos i(^ + B} sin |(A - B). 
(1) cos A + cosB =2 cos \(A + B) cos f(A - B). 
(m) cos A cos 5 = 2 sin |(A + .B) sin %(A B). 
(n) sin A cos (B C) sin cos (A. C) = sin (A. JB) cos C. 
(o) cos 2 %<f>(l + tan |0) 2 = 1 + sin 0. 
(p) sin 2 |x(ctn |x I) 2 = 1 sin x. 

, . 2 sec A . x . 2 sec A 

(g) sec 2 JA = - : -. . (r) esc 2 A = s . 

1 + sec A sec A 1 

98. Solution of Oblique Triangles. One of the chief uses 
of trigonometry is to solve triangles. That is, having given 
three parts of a triangle (sides and angles) at least one of which 
must be a side, to find the others. In plane geometry it has 
been shown how to construct a triangle, having given 

CASE I. Two angles and one side. 

CASE II. Two sides and the angle opposite one of them. 

CASE III. Two sides and the included angle. 

CASE IV. Three sides. 

When the required triangle has been constructed by scale and 
protractor the parts not given may be found by actual measure- 
ment. The results obtained by such graphic methods are not, 
however, sufficiently accurate for many practical purposes. 



V, 99] TRIGONOMETRY 121 

Nevertheless, they are very useful as a check upon the com- 
puted values of the unknown parts. Other checks are fur- 
nished by the theorems of plane geometry that the sum of the 
angles of any triangle is 180, and that if two sides (angles) are 
unequal the greater side (angle) lies opposite the greater angle 
(side). The properties of isosceles triangles can also be used in 
certain special cases. 

The direction solve a triangle tacitly assumes that a sufficient 
number of parts of an actual triangle are given. A proposed 
problem may violate this assumption and there will be no 
solution. Thus, there is no triangle whose sides are 14, 24, 
and 40 ; likewise, there is no triangle of which two sides are 9 
and 10 and the angle opposite the former is 64 10'. Any tri- 
angle which can be constructed can be solved. 

Any oblique triangle can be divided into right triangles by a 
perpendicular from a vertex upon the opposite side, and this 
method when applied to the various cases leads to three laws, 
called the law of sines, the law of cosines, and the law of tan- 
gents, by means of which the unknown parts of any oblique 
triangle can be computed. We proceed to prove these 
laws. 

99. Law of Sines. Any two sides of a triangle are to each 
other as the sines of the opposite angles. 

In any oblique triangle let a, b and c be the measures of the 
lengths of the sides and A, B, and C the measures of the angles 
opposite. Drop the perpendicular CD = p from the vertex 
of angle C to the opposite side. 

Two possible cases are shown in Figs. 49, 50. In either of 
these figures, 

p = b sin A. 

In Fig. 49, 

p = a sin B. 



122 



MATHEMATICS 



[V, 99 



In Fig. 50, 

p = a sin (180 - B) = a sin B. 

Therefore, whether the angles are all acute, or one is obtuse 
a sin B = b sin A, 




D B 



FIG. 49 




whence dividing first by sin A sin B, and second by 6 sin B, 



(34) 



sin A sin B ' 



or 



sin A 
sin B ' 



Similarly, by drawing perpendiculars from A and B to the 
opposite sides, we obtain 

be a c 



Hence, 

(35) 



sin B sin C" sin A sin C ' 

a b c 

sin A sin B sin C * 



It is evident that a triangle may be solved by the aid of the 
law of sines if two of the three known parts are a side and its 
opposite angle. The case of two angles and the included side 
being given, may also be brought under this head, since we 
may find the third angle which lies opposite the given side. 

100. Law of Cosines. In any triangle, the square of any 
side is equal to the sum of the squares of the other two sides minus 
twice the product of these two sides into the cosine of their included 
angle. 



V, 100] 



TRIGONOMETRY 



123 



Let ABC be any triangle. Drop a perpendicular BD from B 
on AC or AC produced. Two possible cases are shown in 




FIG. 51 
Figs. 51, 52. Then we have either 




or else 



and 



CD = b - AD (Fig. 51), 
= 6 c cos A, 

CD = b + AD (Fig. 52) 
= 6 + c cos (180 - A) 
= 6 c cos A, 

p = c sin A (Fig. 51), 

p = c sin (180 - A) = c sin A (Fig. 52). 



Hence, in either figure, we may write 

CD = b c cos A and p = c sin A. 
Again, in either figure, 
a 2 = CD 2 + P 2 

= (b c cos A) 2 -f- (c sin A)- 

= b 2 - 2bc cos A + c 2 (sin 2 A + cos 2 A) 

= b- 2bc cos A + c 2 



that is 
(36) 



COS 



In like manner it may be proved that the law of cosines applies 
to the side b or to the side c. 



124 MATHEMATICS [V, 100 

These formulas may be used to find the angles of a triangle 
when the three sides are given and also to find the third side 
when two sides and the included angle are given. 

101. Law of Tangents. The sum of any two sides of a tri- 
angle is to their difference as the tangent of half the sum of their 
opposite angles is to the tangent of half their differ&nce. 

From the law of sines, we have 

a sin A 
b = sin B' 

whence, by division and composition in proportion, we find 

o + b _ sin A + sin B 
a b sin A sin B ' 

Let x + y = A and x y = B. Then we have 

2x = A + B, and x = f (A + B), 
2y = A - B, and y = %(A - B). 

Hence, substituting in (37), we find 

sin A + sin B sin (x + y) + sin (x y) 
sin A sin B sin (x + y) sin (x y) 

_ 2 sin x cos y _ tan x 
2 cos x sin y tan y 

_ tan |(A + B) 

~ tan |(A - B} ' 

From (37) and the preceding result, we have 

(38) + b = tan %(A + B) 
a- b tan %(A - B) ' 

Since 

tan \(A + B) = tan |(180 - (7) = tan (90 - C) = ctn |C, 
we may write the law of tangents in the form 

(39) tan %(A - B) = ?^-ctn \C. 

a + o 



V, 103] TRIGONOMETRY 125 

As a check, (38) is the more convenient form, while for solving 
triangles, (39) is preferred by some computers. If 6 > a, then 
B > A. The formula is still true, but to avoid negative num- 
bers the formula in this case should be written in the form 

. b + a = tan %(B + A) 

b - a ~ tan \(B - A) ' 

When two sides and the included angle are given, as a, 6, C, 
the law of tangents may be employed in finding the two unknown 
angles A and B. 

102. Methods of Computation. The method to be used in 
computing the unknown parts of a triangle depends on what 
parts are given. In what follows triangles are classified ac- 
cording to the given parts and the methods of computation are 
stated and illustrated by examples. 

103. Case I. Given two Angles and one Side. There is 
always one and only one solution, provided the sum of the 
given angles is less than 180. 

The third angle is found by subtracting the sum of the two 
given angles from 180. The unknown sides are found, suc- 
cessively, by the law of sines. 

EXAMPLE. In a triangle given two angles 38 
and 75 43', and the side opposite the former 
180; find the other parts. 

Construct the triangle approximately to scale 
and denote the unknown parts by suitable let- 
ters as in Fig. 53. A 

First compute the third angle C = 66 17'. FIG. 53 

To compute b use the law of sines, 

_6_ sin 75 43' 
180 sin 38 * 

In any proportion imagine the means and the extremes to be paired 
by lines crossing at the equal sign, 




126 MATHEMATICS [V, 103 



then the rule: Multiply the pair of knowns and divide by the known in 
the other pair; or, Add the logarithms of the pair of knowns and the co- 
logarithm of the known in the other pair. 

FIRST METHOD: without logarithms. 

sin 75 43' = 0.9691 
180 



775280 
9691 

sin 38 = 0.6157)174.4380(283.3 
12314 



51298, etc. 
whence 6 = 283.3. 

SECOND METHOD: with logarithms. 

log 180 = 2.2553 

log sin 75 43' = 9.9864 - 10 

colog sin 38 = 0.2107 

log b = 2.4524 

18 



15)60(4 b = 283.4. 

Similarly we may compute c. Using logarithms, we find c = 267.7. 
Not using logarithms, we find 267.6. The difference in the two answers 
is due to the slight inaccuracy caused by our using only four decimal 
places. 

EXERCISES 

1. Given two angles 43 and 67 and the included side 51; find the 
other parts. Ans. 70, 49.96, 37.02. 

2. Given two angles 24 14' and 43 13' and the side opposite the 
latter 240; find the other parts. Ans. 112 33', 143.9, 323.8. 

3. Solve the triangle ABC being given A = 17 17', B = 102 25', 
and a = 36.84. Ans. C = 60 18', c = 107.7, 6 = 121.1. 

4. Solve the triangle LMN being given L = 28, M = 51, I = 6.3. 

Ans. N = 101, n = 13.17, m = 10.43. 



V, 104] 



TRIGONOMETRY 



127 



104. Case II. Given two Sides and the Angle opposite 
one of Them. This case sometimes admits two solutions and 
on this account is called the ambiguous case. The number of 
solutions can be determined by constructing the triangle to 
scale as follows. 

To fix our ideas, let the given angle be A, the given opposite 
side a, and the given adjacent side 6. Construct the given 
angle A, and on one of its sides lay off AC = b, the given 
adjacent side, and drop a perpendicular CP, of length p, from 
C to the other side of the given angle A. With C as center 
and with radius o, the given opposite side, strike an arc to 
determine the vertex of the third angle B. Several possible 
cases are shown in Fig. 54. 




t. One Solution 



S, One .Solution 



FIG. 54 



A study of these diagrams shows that there will be two 
solutions when, and only when, the given angle is acute and the 
length of the given opposite side is intermediate between the 
lengths of the perpendicular and the given adjacent side; that is 

A < 90 and p < a < b. 

The two triangles to be solved are AB\C and AB 2 C. Since 



128 



MATHEMATICS 



[V, 104 



the triangle BiCB 2 is isosceles, the obtuse angle BI (i. e., angle 
ABiC) is the supplement of the acute angle B- 2 . 

The following examples illustrate the method of computing 

the unknown parts in Case II. 

EXAMPLE 1. One angle 
of a triangle is 34 23', the 
side opposite is 44.24 and 
another side is 60.35; find 
the other parts. 

On constructing the tri- 

j. B^~ p <B angle to scale as in Fig. 55, 

p IG 55 it appears that there are two 

solutions. This is verified by 

computing p = 60.35 sin 34 23'. Noting from the tables that sin 35 < .6, 
it is evident that p < 40. 

Let us solve first the triangle AB 2 C, the angle B 2 being acute. By 
the law of sines, 




60.35 sin 



44.24 sin 34 23' 
B 2 = 50 23' 



log 60.35 = 1.7807 
s sin 34 23' = 9.7518 - 10 
colog 44.24 = 8.3542 - 10 
log sin 5 2 = 9.8867 - 10 
64 



10)30(3 



Then find C 2 (i. e., angle ACBJ = 95 14'. To find c 2 (i. e., side 
use the law of sines again, 

c 2 sin 95 14' 



44.24 sin 34 23' 



c 2 =78.02 



log 44.24 = 1.6458 
log sin 95 14' = 9.9982 - 10 
colog sin 34 23' = 0.2482 
log c 2 = 1.8922 
21 



6)10(2 



To solve the triangle AB&, we first find BI = 129 37' being the 
supplement of 5 2 , and then the third angle Ci = 16 00'. To find Ci 
(i. e., the side ABi) use the law of sines, 



V, 104] 



TRIGONOMETRY 



129 



C] 



sin 16 



44.24 ein 34 23' 



d = 21.59 

CHECK. 

c 2 = 78.02 
ci = 21.59 



log 44.24 = 1.6458 
log sin 16 = 9.4403 - 10 
colog sin 34 23' = 0.2482 
log ci = 1.3343 



= 2(44.24 cos 50 23') 



c 2 - 



= 56.43 



log 2 = 0.3010 
log 44.24 = 1.6458 
log cos 50 23' = 9.8046 - 10 
log i 2 = 1.7514 



= 56.41 



EXAMPLE 2. One angle of a triangle is 34 23', the side opposite is 
60.35 and another side is 44.24. Solve. 

There is only one solution 
as shown by constructing. 

44.24 sing 

60.35 ~ sin 34 23' ' 

whence B = 24 27' and the 
third angle C = 121 10'. 

c 




FIG. 56 



sin 121 10' 



60.35 sin 34 23' ' 
whence c = 91.46 

EXERCISES 

1. Two sides of a triangle are 17.16 and 14.15 and the angle opposite 
the latter is 42. Find the other parts. 

Ans. 125 46', 12 14', 4.483, or 54 14', 83 46', 21.02 

2. In the triangle AGK, A = 31 14', a = 54, g = 48.6. Find the 
other parts. Ans. 27 49', 120 57', 89.3 

3. A 50 ft. chord of a circle subtends an angle of 100 at the center. 
A triangle is to be inscribed in the larger segment having one side 
40 ft. long. How long is the third side? How many solutions? 

Ans. 65.22 

4. If the triangle of Ex. 3 is to have one side 60 ft. long, how many 
solutions? How long is the third side. Ans. 18.88 or 58.25 

10 



130 



MATHEMATICS 



[V, 104 



105. Case III. Given two Sides and the included Angle. 

There is always one and only one solution. The third side 
can be found by the law of cosines and if the angles are not 
required, this is a convenient method of solution, especially if 
the given sides are not large. 

EXAMPLE 1. Two sides of a triangle are 2.1 and 3.5 and the in- 
cluded angle is 53 8'. Find the third side. 

x 2 = 27P + iTB 2 - 2 (2.1) (3.5) cos 53 8' 

= 4.41 + 12.25 - 14.7 X 0.6000 = 7.84, 
whence x = 2.8. 

If the other two angles as well as the third side are required, 
the two angles should be found by the 
law of tangents and then the third side 
can be found by the law of sines. 
Both these computations can be made 
by logarithms. 




EXAMPLE 2. In the triangle ARK, 
a = 23.45, r = 18.44, and K = 81 50'. 



Find the other parts. 
By the law of tangents, 



a+r tan 



+ R) 



_ 
a r tan 5 (A R) ' 

The actual computation may be arranged as follows. 



a = 23.45 

r = 18.44 

a + r = 41.89 

a - r = 5.01 



180 00' 

K = 81 50' 

A + R = 98 10' 

1(A + R) = 49 5' 



41.89 



tan 49 5' 



5.01 tan \(A-R) 

log 5.01 = 0.6998 
log tan 49 5' = 0.0621 

colog 41.89 = 8.3779 - 10 
log tan $(A - R) = 9.1398 - 10 

i(A - R) = 7 51' 
HA + B) = 49 5' 



A = 56 56' 



R = 41 14' 



V, 106] TRIGONOMETRY 131 

CHECK. 

23.45 = sin 56 56' 
18.44 sin 41 14' 

log 23.45 = 1.3701 log 18.44 = 1.2658 

log sin 41 14' = 9.8190 - 10 log sin 56 56' = 9.9233 - 10 
1.1891 1.1891 

To compute k use the law of sines, 

k sin 81 50' 
23.45 ~ sin 56 56' ' 
whence k =27.70 

EXERCISES 

1. In the triangle ABC given a = 52.8, b = 25.2, C = 124 34'; 
find the other parts. Ans. 38 15', 17 11', 70.2 

2. Given I = 131, m = 72, N = 39 46', find n, L, M. 

Ans. 88.57, 108 54', 31 20'. 

3. Given u = 604, v = 291, W = 106 19', find U, V, w. 

Ans. 51 32', 22 9', 740.4 

4. To find the distance between two objects A and B, separated by 
a swamp, a station C is selected so that CA = 300 ft., CB = 277 ft., 
and angle ACB = 65 47', can be measured. Compute AB. 

Ans. 313.9 

5. Two sides of a parallelogram are 23.47 and 62.38 and one angle 
is 71 30'. Find its diagonals. Ans. 59.27 and 73.29 

106. Case IV. Given the three Sides. There is one and 
only one solution, provided no side is grea- 
ter than the sum of the other two. 

The angles can be computed, in succes- 
sion, by the law of cosines. 

EXAMPLE 1. The sides of a triangle are 5, 7, 
8. Find the angles. 

49 = 25 + 64 - 2 X 5 X 8 cos A, 




132 MATHEMATICS [V, 106 

whence cos A = |, A = 60. 

25 = 49 + 64 - 2 X 7 X 8 cos B, 
cos B = -B = 0.7857, B = 38 13'. 
64 = 25 + 49 - 2 X 5 X 7 cos C, 
cos C = | = 0.1429, C = 81 47'. 
CHECK. 60 + 38 13' + 81 47' = 180 00'. 

The law of cosines is not adapted to logarithms but can be 
transformed as follows. The three sides of a triangle ABC, 
being given, then 

a 2 = & 2 + c 2 - 2bc cos A, 
whence 

6 2 + c 2 - a 2 
(41) cosA =--2bT~' 

To adapt this to logarithmic computation, subtract each 
member from unity 

& 2 + c 2 - a 2 2bc - b 2 - c 2 + a 2 



1 cos A = 1 



2bc 

a 2 - (b - c) 2 



2bc 
Hence we have 

(42) 2sin4A = l-cosvl= (a + fe - 



If we now set a + & + c = 2s, we have 

a + & c = 2(s c), 
a - 6 + c = 2( - 6). 

Substituting these values in (42) we find 



(43) sm 

Similarly, 



- a)(s - c) -rfi r (s - a)(g - 6) 

, sin' ^o = - - -. 
ac ab 



V, 106] TRIGONOMETRY 133 

Again, adding each member of (41) to unity, 

52 + C 2 _ Q 2 (6 + c)2 - a 2 
1 + cos A = 1 + - -^- -2- - 

_ (b + c + a)(& + c - a) 

26c 
Therefore, 

2 cos* |A = 1 + cos A = 2 ' ( V" a) , 

oc 

whence 

(44) cos* 4A = '-^^ . 

Similarly, 

, ID (* - 6) , ir< s(s - c) 

cos 2 5 - , cos 2 \C = -- - . 
ac ab 

Dividing sin 2 %A by cos 2 \A, we have, by (43) and (44). 

tan 2 %A = (s b)(s - c)/s(s - a) 

= (s - a)(s - b)(s - c)/s(s - a) 2 . 
It follows that 

1 KS -a)(s -b)(s-c) 

(45) tan \A = - \/ . 

s a \ s 

If we now set 

(46) r = V(* - a)(s - b)(s - c)/s, 

the equation (45) becomes 

(47) tan \A = ^ . 

s a 

Similarly, 

7" * 

tan %B = - tan 



s b s c 



It will be shown in 107 that r is the radius of the circle in- 
scribed in the given triangle. 



134 



MATHEMATICS 



[V, 106 



EXAMPLE. The sides of a triangle are 77, 123, 130. Find the 
angles. 

Us - a)(s - b)(s -c) log (s - a) = 1.9445 
= V s log (s - 6) = 1.6232 

log (s - c) = 1.5441 

colog s = 7.7825 - 10 



tan \A = 



s a 
a = 77 
b = 123 
c = 130 



2)2.8943 



2s = 330 

s = 165 

s - a = 88 

s - b = 42 

s - c = 35 

CHECK 165 



logr = 1.4472 
log tan \A = 9.5027 - 10 
log tan \E = 9.8240 - 10 
log tan \C = 9.9031 - 10 
\A = 17 39' 
\B = 33 42' 
\C = 38 40' 
CHECK 90 01' 



Therefore A = 35 18', B = 67 24', C = 77 20'. 
The sum of the half angles should check within 3'. 

107. Area of a Triangle. It is shown in plane geometry 
that the area of a triangle is equal to 
one half the product of any side and 
the perpendicular from the opposite ver- 
tex upon that side. 

If two sides and their included angle 
are given, say b, c, and A, then 

p = 6 sin A 




FIG. 59 



and 

(48) Area = \bc sin A, 

whence, the area of a triangle is equal to one half the product of 
any two sides and the sine of their included angle. 

If the three sides are given, a formula for the area can be 
deduced from (48) as follows. From (26), 96, we have 

sin A = 2 sin |A cos %A 

9 Vs(.9 a)(s b)(s c) 
~bc~ 



V, 107] TRIGONOMETRY 

by (43) and (44), 106. It follows that 
(49) 



135 



Area = Vs(s a)(s V)(s c), 

in which s denotes one half the perimeter. 

Let r be the radius of the inscribed circle of the triangle 
whose sides are a, b, c. Then since the area of the triangle 




FIG. 60 

ABC is equal to the sum of the areas of the triangles AOB, 

BOG, CO A, we have, 

(50) Area = \cr + \ar + \br = rs. 



Equating (49) and (50), and dividing through by s, 
(51) r 



c) 



which proves that the r of 106 is in fact the radius of the in- 
scribed circle. 

EXERCISES 
1. Solve each of the following triangles. 

(a) a = 50, A = 65, B = 40. 

Ans. C = 75, 6 = 35.46, c = 53.29 

(b) a = 30, b = 54, C = 46. 

Ans. A = 33 6', B = 100 54', c = 39.56 

(c) a = 872.5, b = 632.7, C = 80. 

Ans. A = 60 36', B = 39 24', c = 986.2 



136 



MATHEMATICS 



[V, 107 



(d) a = 120, b = 80, B = 35 18'. 



(a) A 


= 21 30', 


(b) A 


= 62 15', 


(c) A 


= 53 25', 


(d) a 


= 30, 


(e) a 


= 25.8, 


(/) a 


= 37, 


(fiO a 


= 25.3, 


(h) a 


= 42, 


() a 


= 3, 


(j) a 


= 640, 


(fc) a 


= .0428, 


(0 a 


= 12, 


(m) a 


= 6.02, 


(a) C 


= 83 30', 


(&) c 


= 69, 


(c) C 


= 56, 


(d) Ci 


= 125 14', 


C 2 


= 14 46', 


(e) c 


= 30.57 


(/) No solution 


(gr) No solution 


(A) B 


= 56, 


(0 A 


= 111 44', 


(?) A 


= 51 58', 


(/c) A 


= 30 58', 


(0 A 


= 32 10', 


(m) A 


= 47 24', 



Ans. A = 60, C = 


84 42', 


c = 137.9 


39, C = 72 15'. 






Ans. A = 51 15', B = 


56 30', 


c = 95.24 


following triangles. 






Given parts. 






B = 75, 


a = 


31.24 


B = 48 45' 


6 = 


402.3 


B = 70 35', 


c = 


6.031 


6 = 50, 


A = 


20. 


b = 40, 


A = 


40 10'. 


b = 25, 


A = 


37. 


6 = 54, 


A = 


28. 


b = 42, 


A - 


56. 


b =2, 


C = 


30. 


6 - 800, 


C = 


48 10'. 


c = .0832, 


B = 


58 30'. 


6 = 16, 


c = 


22. 


& = 4.82, 


c = 


8.12 


Answers : Required parts 






b = 82.32, 


c = 


84.68 ' 


a = 473.4, 


c 


499.4 


a = 5.841, 


6 = 


6.861 


B l = 34 46', 


Ci = 


71.63 


B 2 = 145 14', 


C 2 = 


1.577 


C = 50, 


B = 


90. 


C = 68, 


c = 


46.97 


B = 38 16', 


c 


2.403 


B = 79 52', 


c = 


605.4 


C = 90 32', 


b = 


.0709 


B = 45 12', 


/-Y 


102 38'. 


B = 36 8', 


C = 


96 24'. 



V, 107] 



TRIGONOMETRY 



137 



3. Find the areas of each of the following triangles. 



(a) Given a = 40, 
(6) Given a = 502, 

(c) Given a = 27.2, 

(d) Given a = 38, 




FIG. 61 



b = 13, c = 37. Ans. Area = 240. 

b = 62, c = 484. Ans. Area = 14,590. 

b = 32.8, C = 65 30'. Ans. Area = 406. 
c = 61.2, 5 = 6 56'. Ans. Area = 1,078. 

4. Venus is nearer to the Sun than the Earth. Assume that the 
orbit of Venus is a circle with the Sun at its center. The distance from 
the Earth to the Sun is 92.9 millions of miles. What is the distance 
from Venus to the Sun if the greatest angular distance of Venus from 
the sun as seen from the Earth is 46 20'? Ans. 67,200,000 mi. 

5. On a clear day, twilight ceases when the sun 
has reached a position 18 below the horizon 
(HAS = 18) . Find the height AE of the atmos- 
phere which is sufficiently dense to reflect the 
sun's rays. Take OC = 4,000 miles. The result 
must be diminished by 20% to allow for re- 
fraction. [MORITZ] . Ans. 40 miles. 

6. The mean distances of the Earth and Mars from the sun are 92.9 
and 141.5 millions of miles respectively. How far is Mars from the 
Earth when its angular distance from the sun is 28 10' ? 

Ans. 21,280,000 mi. 

7. From two points on the same meridian, the 
zenith distances of the moon are 35 25' and 
40 11'. The difference in latitude between the 
points of observation is 74 26'. Find the dis- 
tance of the moon from the earth, assuming the 
FIG. 62 radius of the earth as 3,959 miles. [MORITZ] 

Ans. 239,000 miles, approximately. 

8. A search light 20 feet above the edge of a tank is directed to a 
point on the surface of the water 40 feet from the edge. If the tank 
is 15 feet deep how far will be the illuminated spot on the floor of the 
tank from the edge, the index of refraction being 4/3? Ans. 62.5 ft. 

9. A man whose eye is 6 feet above the edge of a tank 10 feet deep sees 
a coin in a direction making an angle of 34 with the surface of the 
water. If the index of refraction is 4/3, how far is the coin from the 
side of the tank? Ans. 16.83 ft. 




138 MATHEMATICS [V, 107 

10. Three forces of 12, 16, and 22 pounds in equilibrium can be 
represented by the 3 sides of a triangle taken in order. Find the 
angles which they make with each other. 

Am. 77 22', 134 48', 147 50'. 

11. A sharpshooter and an enemy are 220 feet apart and on the 
same side of a street 100 feet wide. Both are concealed by buildings. 
A bullet striking a building on the opposite side of the street at an angle 
x is deflected from the building at an angle y so that 3 sin a; = 4 sin y. 
Find x so that the sharpshooter may be able to hit the enemy. 

Ans. 40 6'. 

12. A ship is going 15 miles per hour. How far to the side of a target 
1 mile distant must the gunner aim if the shot travels 2000 ft. per 
second and the shot is fired when directly opposite? 

Ans. 38' or 58 ft. 

13. An aeroplane is observed from the base and from the top of a 
tower 40 feet high. The angles of elevation are found to be 10 40' 
and 9 50'. Find the distance from the base to the plane and the 
height of the plane. Ans. 2713 ft., 502.4 ft. 

14. To determine the distance of a hostile fort A from a place B, a 
line BC and the angles ABC and BCA were measured and found to be 
1006.6 yd., 44, and 70, respectively. Find the distance AB. 

Ans. 1,036 yd. 

15. In order to find the distance between two objects, A and B, 
separated by a pond, a station C was chosen, and the distance CA 
= 426 yd., CB = 322.4 yd., together with the angle ACB = 68 42', 
were measured. Find the distance from A to B. Ans. 430.9 yd. 

16. A surveyor wished to find the distance of an inaccessible point 
from each of two points A and B, but had no instrument with which 
to measure angles. He measured A A' = 150 ft. in a straight line with 
OA, and BE' 250 ft. in a straight line with OB. He then measured 
AB = 279.5 ft., BA' = 315.8 ft,, A'B' = 498.7 ft. From these 
measurements find each of the distances AO and BO. 

Ans. 152.3 ft., 319.7 ft. 

17. Two stations, A and B, on opposite sides of a mountain, are both 
visible from a third station C. The distance AC = 11.5 mi., BC = 9.4 
mi., and angle ACB = 59 30'. Find the distance between A and B. 

Ans. 10.5 mi. 



CHAPTER VI 



LAND SURVEYING 

108. The Surveyor's Function. Land surveying consists 
in measuring distances and angles and marking corners and lines 
upon the ground, and in recording these measurements in field 
notes from which a map can be drawn and the area computed. 

The original survey of a tract of land having been made and 
recorded, a surveyor may subsequently be called upon to find 
the corners, to relocate them if lost, to retrace the old boundaries, 
and to renew the corner posts and monuments if decayed or 
destroyed. This is called a resurvey. 

A surveyor may make a resurvey of a tract of land in order 
to divide it by new lines and to 
map and compute the areas of 
the subdivisions. 

109. Instruments. Distances 
on the ground are measured with 
the chain or tape. The land sur- 
veyor's chain is 66 feet (4 rods) 
long and is divided into 100 links 
each 7.92 inches long. The steel 
tape is usually 100 feet long, sub- 
divided to hundredths of a foot. 

Angles, horizontal or vertical, 
are usually measured with the 
transit. This is an instrument 

mounted on a tripod, and composed of the following parts: (a) 
the telescope provided with cross hairs to determine the line of 
sight, a sensitive spirit level, and a graduated circle on which the 

139 




FIG. 63 



' 140 MATHEMATICS [VI, 109 

angular turn of the telescope in the vertical plane is read; (6) 
the alidade, carrying the telescope, provided with spirit levels 
to bring its base into the horizontal plane and a large gradu- 
ated circle on which is read the angular turn of the telescope in 
measuring horizontal angles; and (c) the magnetic compass. 

110. Bearing of Lines. The direction of a line on the ground 
may be given by its bearing; this is the angle between the line 
and the meridian through one end of it. For example, a line 
bearing N 26 E is one which makes an angle of 26 on the east 
side of north; one bearing S 85 W makes an angle of 85 on the 
west side of south. The bearing of a line which is run by the 
transit is read off on the compass circle but is subject to a cor- 
rection depending upon the time and place since the magnetic 
needle does not point due north at all times and places. 

111. Government Surveys. In government surveys of the 
public lands, a north and south line called a principal meridian 
is first accurately laid out and marked by permanent monuments. 
From a convenient point on the principal meridian a base line 
is run east and west and carefully marked. North and south 
lines, called range lines, are then run from points six miles apart 
on the base line. Then township lines six miles apart are run 
east and west from the principal meridian. 

The land is thus divided into townships six miles square. A 
tier of townships running north and south is called a range. 
Ranges are numbered consecutively east and west from the 
principal meridian. Townships are numbered north and south 
from the base line. 

In deeds and records a township is located, not by the county, 
but as " Township No. north (or south) of a certain base line 
and in range No. east (or west) of a certain principal me- 
ridian. Townships are divided into thirty-six sections each 
one mile square containing 640 acres, and are numbered from 



VI, 111] 



LAND SURVEYING 



141 



1 to 36 as shown in Fig. 64. The sections are often subdivided 
into halves, quarters, eighths, etc., as illustrated in Fig. 65. 



6 


S 


4 


3 


2 


1 


7 


8 


9 


10 


11 


12 


18 


17 


16 


15 


14 


13 


19 


20 


21 


22 


23 


24 


30 


29 


28 


27 


26 


25 


31 


32 


33 


34 


35 


36 





Iff A. 


NE 1 


s.j-x.w.j 


160 A. 


80 A. 






t 


I 




2 , 

s.w.4 


S~E. i 




4 


160 A. 




80 A. 





FIG. 64 



FIG. 65 



The first principal meridian runs north from the junction of 
the Ohio and Big Miami rivers on the boundary between Ohio 
and Indiana. The second coincides with 86 28' of longitude 
west of Greenwich running north from the Ohio river near the 
towns of English, Bedford, Lebanon, Culver, Walkerton, and 
Warwick, Indiana. The surveys in Indiana (with the exception 
of certain lands in the southeast corner) are governed by this 
second principal meridian and a base line in latitude 38 28' 20" 
crossing this meridian about 5 miles south of Paoli, in Orange 
County.* Thus a certain parcel of land is described in the 
Indiana records as " E \ of NW \ of Section nineteen (19), 
Township twenty-three (23) N, Range four (4) W." 

The surveys extending east from one meridian will not gener- 
ally close with those extending west from the preceding meridian; 
the same is true of the ranges of townships extending north 

* The first six principal meridians are designated by number ; some twenty -odd others 
by name. E. g., the Mount Diablo meiidian, 120 54' 48" W, which governs sur- 
veys in California and Nevada. The first six base lines are neither numbered nor 
named but all subsequent ones are named. The locations of all the principal merid- 
ians and base lines is given in the Manual of Instructions for the Survey of the Public Lands 
issued from timo to time by the GENERAL LAND OFFICE, Washington. D. C. For de- 
tails and a historical sketch see also, PENCE AND KETCHUM, Surveying Manual. 



142 MATHEMATICS [VI, 111 

and south from the base lines. These circumstances and the 
presence of rivers and lakes give rise to fractional townships and 
sections. 

112. Corners. In an original survey one of the most im- 
portant of the surveyor's duties is the marking of corners in 
such a manner as to perpetuate their location as long as possible. 
The Manual of Instructions (see 1894 edition, p. 44) says, " If 
the corners be not perpetuated in a permanent and workman- 
like manner, the principal object of the surveying operations 
will not have been attained." 

The Instructions prescribe in detail the kind of monument and 
the mark to be put upon it to establish each of the various kinds 
of corners that are located in the government surveys. Wooden 
posts and stakes, stones, trees, and mounds of earth are used. 
Witness trees or witness points are trees or other objects located 
near the corner, suitably marked and described in the field notes 
to make easy a subsequent relocation of the corner. 

If called upon to make a resurvey of land that was originally 
laid out under the direction of the General Land Office, the 
surveyor will do well to make a careful study of the instructions 
concerning corners that were in force when the original survey 
was made, as the practice has varied somewhat from time to 
time. 

113. Judicial Functions of Surveyors.* Many years have 
elapsed since the greater part of the government surveys were 
made and in many cases the original corner marks have entirely 
disappeared. The first settlers and original owners often failed 
to fix their lines accurately while the monuments remained, and 
the subsequent owners have no first hand knowledge of their 
location. When in such cases a surveyor is called upon to 

* This topic is based upon a paper of the same title by Justice Cooley of the Michi- 
gan Supreme Court, published in the Michigan Engineer's Annual for 1880-81. 
pp. 18-25. 



VI, 114] LAND SURVEYING 



1* 



make a resurvey, it is his duty to find if possible where the 
original corners and boundary lines were, and not at all where 
they ought to have been. However erroneous the original sur- 
vey may have been, the monuments that were set must never-^ 
theless govern, for the parties concerned have bought with refer- 
ence to these monuments and are entitled only to what is 
contained within the original lines. 

If the original monument and all the witness trees and other 
identification marks mentioned in the field notes of the original 
survey have disappeared, the corner is lost and it is the duty of 
the surveyor to relocate it in the light of all the evidence in the 
case, including the testimony of persons familiar with the 
premises, existing fences, ditches, etc., at the point where this 
evidence shows it most probably was. 

In making a resurvey the surveyor has no authority to 
settle disputed points ; if the disputing parties do not agree 
to accept his decision, the question must be settled in the 
courts. In a controversy between adjacent owners over the 
location of corners and division lines, it is well established in law 
that a supposed boundary line long accepted and acquiesced 
in by both parties is better evidence of where the real line should 
be than any survey made after the original monuments have 
disappeared. It is common belief that boundary lines do not 
become fixed by acquiescence in less than 21 years, but there is 
no particular time that must elapse to establish boundary lines 
between private owners where it appears that they have ac- 
cepted a particular line as their boundary and all concerned 
have claimed and occupied up to it. 

114. Measuring on Level Ground. The line to be meas- 
ured is first ranged out and marked with range poles or its 
direction is determined by the line of sight of the transit 
set on the line. The leader sticks a pin at the starting point, 
takes ten in his hand and steps forward on the line dragging the 




44 MATHEMATICS [VI, 114 

chain behind him. At a signal from the follower, given just 
before the full length has been drawn out, he turns, aligns, and 
levels the chain, stretches it to the proper tension, and, while 
the follower holds the rear end at the starting point, sticks a pin 
at the forward end on the line determined by the follower and 
a range pole or by the transitman. At a signal from the 
leader the follower pulls his pin and both move forward on the 
line another chain's length and set the next pin. This process 
is repeated until the leader has set his tenth pin, when the 
follower goes forward, counting his pins as he goes and, if there 
are ten, hands them to the leader who also counts them. The 
count of tallies is kept by both. When the end of the line is 
reached the follower walks forward and reads the fraction of 
the chain at the pin and notes the number of pins in his hand to 
determine the distance from the last tally point which is re- 
corded in the field notes. 

115. Measuring on Slopes. The horizontal distance which 
is required can be found on slopes by leveling the chain and 

plumbing down from the end 
off ground. On steep slopes 
only a part of the chain can 
be used as at A and B in Fig. 
66. The part used should be 
a multiple of ten links and 
great caution must be used 

by both leader and follower to avoid mistakes and confusion in 

the count of pins. 

116. Offsets. In case measurements cannot be made on the 
desired line on account of a fence, hedge, pond or other obstacle, 
a perpendicular to the line, called an offset, is measured, suf- 
ficiently long to avoid the obstruction and the measurements are 




VI, 117] 



LAND SURVEYING 



145 



made on an auxiliary line parallel to the required line. Stakes 
may then be set on the required line by offsets from the auxiliary 
line. See Fig. 67. 

c D 





FIG. 68 



FIG. 67 

From any point on a line a right angle (or any other required 
angle) can be laid off with the tran- 
sit. An angle of 90 or 60 can be 
laid off in a clear space with chain or 
tape and pins as shown in Fig. 68, 
from the facts that (1) a triangle 
whose sides are to each other as 3 : 
4 : 5 has a right angle opposite the 
longest side; and (2) an equilateral triangle has three 60 angles. 

117. Passing Obstacles. An obstacle in the line can be 
passed and the line prolonged beyond it by means of perpen- 
dicular offsets as shown in Fig. 67, if the nature of the locality 
makes it convenient. 

The same result can be accomplished by a triangle as shown 
in Fig. 69. The angle HAB, the distance AB, the angle KBC, 

are measured; then the distance BC 
and the angle MOD are computed; 
the distance BC is measured off and 
the point C is located and the angle 
at C is turned off and the direction 
CD established; AC is computed 
and the point D is taken a whole 
The angles at A and B and the 




FIG. 69 



number of chains from A 
11 



146 MATHEMATICS [VI, 117 

distance AB are arbitrary and may be taken so as to avoid 
difficulties of the surroundings. If the circumstances permit 
the angle HAB may be made 60, and angle KEG = 120; 
then the triangle ABC will be equilateral and computations will 
be avoided. 

1 18. Random Lines. When it is desired to mark out a long 
line, such as AB, Fig. 70, whose end points are established but 

A^ k T & T s T t T 5 B 

FIG. 70 

are invisible each from the other, a line AC, called a random line, 
is run as nearly in the direction of A B as can be determined 
and stakes Si, S 2 , 83, etc., are set at regular measured distances. 
On coming out near B a perpendicular is let fall from B to AC 
precisely locating the point C. The lengths of the offsets 
SiTi, $2^2, $3^3, etc., all perpendicular to AC, can be computed 
and on retracing CA, stakes can be set at T$, T t , T 3 , etc., on 
the desired line AB. For example, if the stakes on AC are 12 
chains apart, if S$C = 6.46 chains, and if BC = 54 links, then 
the offset, in links, at any stake S, is found by multiplying its 
distance AS, in chains, from A, by the ratio 54/66.46 = 0.8125. 
Thus the offset S 4 T 4 = 48 X 0.8125 = 39. It is left to the 
student to show that A B is longer than AC by less than 1/4 a 
link and that the stakes on AB are practically 12 chains apart. 

119. Computing Areas. If the boundaries of a tract are all 
straight lines, i. e., if its perimeter is a polygon, the area can be 
computed by dividing it into triangles, or into rectangles and 
triangles, provided enough measurements are made so that the 
required dimensions of each part are known or can be com- 



VI, 121] 



LAND SURVEYING 



147 




puted. It is customary to measure more lines on the ground 
than is theoretically necessary in order to check the computa- 
tions. These extra measurements are called proof lines in the 
field notes. 

120. Irregular Areas by Offsets. When one side of a field 
is not straight as occurs if the boundary is a stream or curved 
road, a line may be run cutting off the 

irregular part and leaving the remain- 

der of the field in a shape whose area 

is easily computed; as AD in Fig. 71. 

Stakes are set at regular measured 

intervals on AD and the offsets AB, 

SiTi, SzTz, SzTs, etc., are measured. 

The area can be approximated by considering each of the strips 

to be a trapezoid. On computing and adding we are led to the 

following rule. 

RULE: From the sum of all the offsets subtract half the sum of 
the extreme ones and multiply the remainder by the common 
distance between them. 

121. Areas by Rectangular Coordinates. If the irregular 

side of the field is a broken 
line or if the nature of the 
place makes it inconvenient 
to measure the offsets at regu- 
lar intervals the area can be 
found by measuring the rec- 
tangular coordinates of the 

points A, B, C, D, E, Fig. 72, referred to the axes OX and OY. 
Let the coordinates of A, B, C, be (0, y ), (zi, T/I), (x 2 , 7/2), 

respectively. Then the sum of the areas of the trapezoids is 




FIG. 72 



(1) 



2/2) 



148 



MATHEMATICS 



[VI, 121 



where n is the number of trapezoids. On combining terms 
this reduces to 

(2) |[{zi(ffe - 2/2) + 2(2/1 - 2/3) + 

+ n _l(2/n-2 - 2/n)} + n (2/n-l + 2/n)]- 

Hence we have the following rule. 

RULE: From each ordinate subtract the second succeeding 

ordinate and multiply the remainder by the abscissa of the inter- 

mediate point; also multiply the sum of the last two ordinates by 

the last abscissa; and divide the 
algebraic sum of the products by 
two. 

If the coordinates of the ver- 
tices of a closed polygon are 
known its area can be computed 
as follows. Consider the con- 
vex pentagon shown in Fig. 73. 
The area included may be found 
by adding the trapezoids under 

the sides ED and DC and subtracting those under the other 

three sides; this gives 

(3) |[(z 4 - 5X2/4 + 2/5) + (3 - 4X2/3 + 2/4) 

(3 2X2/3 + 2/2) (2 1X2/2 + 2/0 

- (xi - x 5 )(yi + 2/5)]., 

Combining like terms, we find that this reduces to either 

(4) $[xi(yt - 2/5) + 2(2/3 - 2/i) + 3(2/4 - 2/2) 

+ 4(2/5 - 2/s) + 5(2/1 - 2/4)], 




TJI ~o 



or 



(5) ?[yi(x 6 2) + yz(x\ 3 ) + 2/3(^2 4) 

5) 



1)]. 



VI, 121] 



LAND SURVEYING 



149 



These formulas are easily extended to convex polygons of 
any number of sides and prove the following rule. 

Multiply each abscissa by the difference of its adjacent ordinatcs, 
always making the subtractions in the same sense around the perim- 
eter, and take one-half the algebraic sum of the products. 

The result will be the same (except as to sign) if in this rule 
the words abscissa and ordinate be interchanged. 

EXERCISES 

1. Find the area of a field in the form of a right triangle. 

(a) Base = 31.28 ch., Altitude = 16.25 ch. Ans. 25.42 A. 

(6) Base = 28.46 ch., Altitude = 38.65 ch. Ans. 55.00 A. 

2. Find the area of a triangular field, 

() whose three sides are 24.50, 10.40, and 21.70 ch. 
(6) having two sides 35.60, 23.70 ch., and their included angle 42 30'. 

Ans. (a) 11.27 A. (6) 28.50 A. 

3. How many acres in a rectangular field whose dimensions are 
17.44 and 32.65 ch. Ans. 56.94 A. 

4. One side of a 200 acre rectangular field is 33.60 chains. Find the 
other side. Ans. 62.50 ch. 

5. What is the length of one side of a square field which contains 
36 acres? Ans. 18.97 ch. 

6. The diagonals of a four-sided field measure 21.40 and 24.50 ch., 
and they cross at an angle of 74 40'. Find the area. Ans. 25.28 A. 

7. One diagonal of a quadrangle runs N. 36 20' E. 22.40 ch., and the 
other S. 69 30' E. Find its area. Ans. 25.22 A. 

8. Find the areas of the fields whose boundaries are given. 



Sta- 
tion. 


Bearing. 


Distance. 


A 


North 


9.14 ch. 


B 


S. 73 25' E. 


8.27 


C 


S. 28 15' E. 


10.04 


D 


N. 80 45' W. 


12.84J 



Sta- 
tion. 


Bearing. 


Distance. 


P 


West 


19.66 ch. 


Q 


North 


13.77 


R 


N. 64 15' E. 


16.66 


S 


S. 12 30' E. 


21.51 



Ans. (a) 8.74 A. (6) 30.97 A. 



150 



MATHEMATICS 



[VI, 121 



9. Find the areas of the fields whose boundaries are given, 
(a) (6) 



Sta- 
tion. 


Bearing. 


Distance. 


A 


N. 25 30' E. 


10.50 ch. 


B 


N. 76 50' E. 


7.00 


C 


S. 19 30' E. 


7.92 


D 


S. 53 34' W. 


11.90 


E 


N. 64 30' W. 


4.20 



Sta- 
tion. 


Bearing. 


Distance. 


1. . 


N. 12 46' W. 


6.80 ch. 


2. . 


N. 49 10' E. 


2.40 


3. . 


S. 40 50' E. 


6.00 


4. . 


S. 10 30' W. 


4.00 


5. . 


N. 85 50' W 


4.52| 



Ana. (a) 10.09 A. (6) 3.30 A. 
10. The coordinates, in chains, of the vertices of a broken line are: 



Vertex. 


A. 


B. 


c. 


D. 


E. 


F. 


X 


0.00 


2.95 


1.10 


0.60 


2.20 


1.80 


y 


10.00 


8.12 


7.25 


5.00 


4.50 


0.00 

















Find the area included by the broken line and the axes. Ans. 2.36 A. 
11. The coordinates, in chains, of the corners of a field are: 



Vertex. 


1. 


2. 


3. 


4. 


5. 


6. 


X 


0.00 


7.00 


12.50 


18.00 


15.00 


10.00 


y 


6.00 


12.00 


20.00 


15.00 


8.25 


0.00 

















Make a plot and find the area. 



Ans. 16.175 A. 



12. Starting on the bank of a river a line is run across a bend 20.00 
ch., to the bank again. Offsets are measured every two chains as fol- 
lows: 1.61, 2.27, 1.96, 4.23, 3.70, 2.92, 3.26, 2.50, 1.25 ch. Make a 
plot of the land between the line and the river and find the area. 

Ans. 4.74 A. 

13. Find the measurements so as to run a line from the vertex A 
of a triangle ABC to a point D on the side BC = 8.75 ch., so as to cut 
off 2/5 of the area next to B. Ans. BD = 3.50 ch. 

14. Find the measurements so as to run a line through a point E 
on BC of the triangle of Ex. 13, parallel to AB so as to cut off 2/5 of 
the area in the trapezoid. Ans. CE = 6.78 ch. 



VI, 121] 



LAND SURVEYING 



151 



15. Two lines meet at P. PA bears S. 65 30' E., PB bears N. 78 
15' E. Determine measurements to run a line perpendicular to PA 
so as to cut off five acres. Ans. Base = lOVctn 36 15' = 11.68 ch. 

16. A triangular field contains 6 A. Show how to find on the plot 
a point inside the triangle from which lines drawn to the vertices will 
divide it into three triangular fields of 1, 2, and 3 A., and so that the 
smallest and largest shall be adja- 
cent respectively to the smallest and 

largest sides of the field. 

17. If the bases of a trapezoid 
are a and b, a < 6, and the slant 

sides are c and d, as in Fig. 74, de- FIG. 74 

termine measurements to run a line 

parallel to the bases to cut off, adjacent to the shorter base a, a frac- 
tion /, of the whole area. 




Ans. x = Va 2 + /(b 2 - a 2 ), y = c 



b -a' 



b -a' 



18. Given a = 20, b 30, c = 54.40 ch., determine x and y to cut 
off \ the area, Fig. 74. Ans. x = 23.80, y = 20.69 ch. 

19. In a four sided field ABCD, AB runs S. 8.40 ch., BC, E. 9.24 ch., 
and CD, N. 5.68 ch. 

(a) Run a north and south line so as to divide it into two parts whose 
areas shall be to each other as 2 : 3 with the smaller on the east. 

(b) Run a north and south line so as to cut off 3 A. on the west. 

Ans. (a) 4.14 ch. from the east; (6) 5.40 ch. from the east. 

20. A tract of land A BCD, lies 
between two converging streets as 
shown in Fig. 75. AB = 1980 ft., 
AC = 590 ft., BD = 1380 ft. De- 
termine the measurements for run- 
ning lines PQ, RS, etc., perpen- 
dicular to AB, so as to divide the 
tract into ten lots of equal area. 

[HINT. Use the method of Ex. 17 
to find PQ and AP. Or otherwise, 




1 



^^ 


S 




Q 




c 






A 


P 


R 



FIG. 75 
find the tangent of the angle between the streets AB and CD ; find the 



MATHEMATICS 



[VI, 121 



Corner. 


Bearing. 


Distance. 


1. . 


N. 75 30' W. 
N. 5 15' E. 
S. 68 10' E. 
S. 23 E. 


30.08 ch. 


2 ... 


3 


4 




Area = 139.84 acres 



area of CAPQ in terms of x( = AP) ; this leads to a quadratic equation 

in x. Find the positive root.] 

Ans. AP = 300.11 ft. PQ = 709.74 ft. Area CAPQ = 4.477 A. 

21. From the notes in Ex. 8 (6), make a plot and (a) run a line from S 
to a point M on PQ so as to divide the field into two parts of equal areas, 
(6) run a line from R to a point N on SP so as to cut off 10 acres in the 
triangle. 

22. From the accompany- 
ing notes from a farm survey 
compute the lengths of the 
first, second, and fourth sides. 

[HiNT. Produce the second 
and fourth sides to form a 
triangle.] Ans. 51.38, 36.56, 40.16. 

23. Suppose the lengths of the first and fourth sides of the field 
in Ex. 9 (a) are unknown. Compute them from the other data if the 
area is 10.094 acres. 

24. It is desired to mark out and measure a line PQ. A random 
line PR is run and stakes are set on it every 100 ft. The perpendicular 
from Q upon PR is 48.82 ft. long and meets it at R, 22.18 ft. beyond the 
42nd stake. Compute the offsets for setting the stakes over on PQ, 
their distance apart, and the length of PQ. 

25. To prolong a line AB past an obstacle 0, a right turn 40 is 
made at B, 400 ft. is measured to C, and a left turn of 116 is made. 
Compute the distance to D on AB produced through O and the right 
turn which must be made at D. How far from D should hundred foot 
stakes be resumed? 



CHAPTER VII 
STATICS 

122. Statics. Statics treats of bodies at rest and of bodies 
whose motion is not changing in direction or in speed. A body 
whose motion is not changing is said to be in equilibrium. The 
chief problem of statics is to find the conditions of equilibrium. 

123. Mass. The weight, W, of a body is not constant. For 
instance a body weighs less on a mountain top than at sea level. 
Also the acceleration, g, due to gravity is not constant. It 
likewise is less on a mountain top than at sea level. An increase 
in acceleration is accompanied by a proportional increase in 
weight. But the ratio W/g is constant. The constant number 
represented by this ratio is called the mass of the body. A unit 
of mass is the gram, and is 1/1000 of the mass of a certain piece 
of platinum which is preserved at Paris. Another unit of mass 
is the avoirdupois pound. One thousand grams is equal to 
2.20462125 Ibs. The mass of any body is then the number ex- 
pressing the ratio of its weight to the weight of a unit of mass. 
The weight is to be determined by means of a spring balance. 

124. Momentum. When a given mass is in motion, we 
require to know not only the magnitude of the mass, but also 
its velocity. The product of the mass of a body and its velocity 
is called its momentum. 

125. Force. If a body possesses a certain amount of momen- 
tum, it is impossible for it to alter its motion in any manner 
unless acted upon by some other body which pushes or pulls it. 

Force is that which tends to produce a change of motion in a 
body on which it acts. This change of motion is proportional to 
the force and takes place in the direction of the straight line in 

153 



154 MATHEMATICS [VII, 125 

which the force acts. Thus, to increase the speed of an auto- 
mobile, the driving force must be increased. The greater the 
force, the greater the rate of increase in the speed. 

This illustrates the fact that forces are of different magnitudes. 

If a motionless croquet ball is struck, its subsequent motion 
depends upon the direction of the stroke. This illustrates the 
fact that forces have different lines of action. If a billiard ball 
is struck, the motion of the ball depends upon the point at which 
the cue struck the ball. This illustrates the fact that we must 
take into account the point of application of the force. 

A force is said to be completely determined if we know (a) its 
magnitude; (6) its line of action; (c) its direction along the 
line of action; (d} its point of application. 

In practice forces are never applied at a point. The force is 
applied over an area such as the pressure of a thumb on the 
head of a tack or the pressure of a book on a table. A force 
may act throughout an entire volume as is the case with at- 
traction. These forces are called distributed forces. In prac- 
tice we often consider the forces which applied at a point would 
produce the same effect as the given distributed forces. Such 
forces are termed concentrated forces. 

126. Unit of Force. The unit of force is sometimes taken 
as the weight of a unit mass. This unit of force is not constant. 
It changes both with altitude and with latitude. These changes 
are small but for scientific purposes cannot be neglected. To 
obtain a constant unit it is sufficient to make the following 
definition : 

The unit of force is the weight of a unit of mass at a fixed place, 
say at London, Paris, or Washington. 

127. Graphic Representation of Forces. A force P is com- 
pletely determined if we know its magnitude, its line of action, 
its direction along this line, and its point of application. It 



VII, 128J STATICS 155 

follows that a force can be completely represented by anything 
which possesses these attributes. It can, for example, be repre- 
sented by a directed segment of a 
straight line. For we may let any 
point 0, Fig. 76, represent the point 
of application. From draw any 
line segment OA the number of units 

in whose length is the same as the number of units in the given 
force. The length of the segment represents then the magnitude 
of the force. The line of which OA is a part represents the line 
of action of the force. We can represent the direction along the 
line by an arrowhead placed on OA. 

128. Composition of Forces. Parallelogram of Forces. 
If two or more forces act in the same straight line and in the 
same direction, their resultant, or sum, is obtained by adding 
the numbers representing the magnitudes of the forces. 

If the forces act in the same straight line but in opposite 
directions, the resultant is equal to their difference, that is to 
their algebraic sum. 

When the forces do not act in the same straight line the 
total or resultant force is found by means of a rule called the 
parallelogram of forces: If two forces not in the same straight 
line are represented in direction and in magnitude by two adjacent 
sides of a parallelogram, the single force which would produce the 
same effect as the two given forces is represented in direction and 
in magnitude by that diagonal of the parallelogram which passes 
through the same vertex as the two given forces. 

In Fig. 77, the forces FI and F 2 are represented by the lines 
AB and AC, respectively. Their resultant R is represented by 
AD. The magnitude of the resultant is given by the equation 



(1) R = VFS + F 2 2 + 2FiF 2 cos 0, 

where 6 stands for the angle BAC. 



156 MATHEMATICS [VII, 128 

It will be noted that BD, being parallel and equal to AC, 
represents the magnitude and the direction (but not line of 




FIG. 77 

action) of the force F 2 . If we let a equal the angle BAD, we 
have, from the triangle BAD 



sine sin a' 
whence 

F 2 sin 9 
(3) sin a = ^, 



The direction a of the resultant force may be found from this 
equation. Thus R is completely determined. 
When 6 = 90, equation (1) reduces to 



(4) R = fi* + Ft. 
We also have 

F 2 F l 

(5) sin a = , cos a = . 

ti T 

Two forces which have a given force for their resultant are 
called the components of this force. Thus FI and F 2 are com- 
ponents of R. The process of finding the resultant of any 
number of forces is known as the composition of forces. The 
process of finding the components of a given force is called the 
resolution of forces. Two systems of forces acting on a particle 
and having the same resultant are said to be equivalent. 



VII, 129] 



STATICS 



157 




FIG. 7& 



129. Rectangular Components of a Force. Frequently, 
it is desired to resolve a force 
into components which are, re- 
spectively, parallel and perpen- 
dicular to a given line. Such 
components are called rectan- 
gular components. In this case 
the magnitudes of the compon- 
ents may be found by the solu- 
tion of equations (5), or directly from a figure. See Fig. 78. 
Thus we find 
(6) Fi = R cos a, F 2 = R sin a. 

These formulas give FI and F 2 as the rectangular components 
of R. 

Similarly the component of any given force along any given 
line is equal to the magnitude of the force multiplied by the 
cosine of the angle between the line and the force. 

EXERCISES 

1. Given F! = 48.7, F 2 = 69.8, 6 = 65 20 , find R and a. 

2. Given FI = 20.3, F 2 = 60.2, = 135 10'; find R and a. 

3. Given FI = 60.3, F 2 = 30.2, =90, find R and . 

4. Given F! = 26.7, F 2 = 45 7, = 60; find R and a. 

5. R = 140, a = 15; find Fi and Fi. 

Ans. Fi = 135.2, F, = 36.2 

6. R = 125, a = 25; find Fi and FI. 

Ans. Fi = 113.3, F, = 52.8 

7. R = 325, a = 35; find Fi and F,. 

Ans. F! = 266.2, F, = 186 4 

8. R = 600, a = 55; find Fi and F 2 . 

Ans. F, = 344.1, F, = 491.5 

9. A particle is acted upon by two forces, of 8 and 10 pounds re- 
spectively, making an angle of 30 with each other. Find the mag- 
nitude of the resultant. Ana. 17.39 



158 MATHEMATICS [VII, 129 

10. A boat is being towed by two ropes making an angle of 60 with 
each other. The pull on one rope is SCO pounds, the pull on the other 
is 300 pounds. In what direction will the boat tend to move? What 
single force would produce the same result? [MILLER-LILLY] 

Ans. 21 47' with force of 500 Ibs.; 700 Ibs. 

11. Let a raft move in a straight line down stream with a uniform 
speed of 2 feet per second; suppose a man upon the raft walks at a 
uniform speed of 4 feet per second in a direction making an angle of 60 
with the direction of movement of the raft. Find the speed and 
direction qf_the man relative to the earth. 

Ans. V28 ft. per sec. at an angle of 40 54' with direction of raft. 

12. A river one mile wide flows at a rate of 2.3 miles per hour. A 
man, who in still water can row 4.2 miles per hour, desires to cross to a 
point directly opposite. Find in what direction he must row and how 
long he will be in crossing. 

Ans. Upstream at an angle of 56 48' with direction of stream; 17 
minutes approximately. 

13. A man in a house observes rain drops falling with a speed of 
32 feet per second. The direction of descent makes an angle of 30 
with the vertical. Find the velocity of the wind. 

Ans. 18.5 ft. per sec. 

14. A motor boat points directly across a river which flows at the 
rate of 3.5 miles per hour; the boat has a speed in still water of 10 miles 
per hour. Find the speed of the boat and the direction of its motion. 

Ans. 10.59 mi. per hr., 70 43' with direction of stream. 

15. From a railway train going 40 mi. per hour a bullet is fired 
2,000 ft. per second at an angle of 65 with the track ahead. Find its 
speed and direction. 

130. Triangle of Forces. It will be seen at once on re- 
ferring to Fig. 77 that the sum or 
resultant of the two forces FI and F 2 
could be obtained more easily by 
7g drawing a triangle ABD, as in Fig. 79; 

when applied to find the resultant of 

two forces the triangle ABD is called the triangle of forces. 
Referring again to Fig. 79, it is evident that if a force equal 




VII, 130] STATICS 159 

and opposite to the resultant R were applied at A, this force 
and the forces FI and F z would balance, and the point A would 
be in equilibrium. Another way of stating the proposition 
would be as follows. 

// three concurrent forces are in equilibrium, their magnitudes 
arc proportional to the three sides of a triangle whose sides, taken 
in order, are parallel to the directions of the given forces. Con- 
rcrwly, if the magnitudes of three concurrent forces are propor- 
tional to the three sides of a triangle and their directions are paral- 
lel to the sides taken in order, these forces will be in equilibrium. 

EXERCISES 

1. Draw a triangle ABC whose sides BC, CA, AB are 7, 9, 11 units 
long. If ABC is a triangle for three forces in equilibrium at a point P, 
and if the force corresponding to the side BC is a force of 21 Ibs., show 
in a diagram how the forces act, and find the magnitude of the other 
two forces. Ans. 27, 33. 

2. Draw two lines AB and AC containing an angle of 120, and sup- 
pose a force of 7 units to act from A to B and a force of 10 units from 
A to C. Find by construction the resultant of the forces, and the 
number of degrees in the angle its direction makes with AB. 

Aris. V79; 77, approximately. 

3. Draw an equilateral triangle ABC, and produce BC to D, making 
CD equal to BC. Suppose that BD is a rod (without weight) kept at 
rest by forces acting along the lines AB, AC, AD. Given that the 
force acting at B is one of 10 units acting from A to B, find by con- 
struction (or otherwise) the other two forces, and specify them com- 
pletely. 

4. Find the resultant of two velocities of 9 and 7 ft. per second 
acting at a point at an angle of 120. Ans. ^G7. 

5. Find the magnitude and direction of the resultant of two velocities 
of 5 and 4 ft. per second acting at a point at an angle of 45. 

Ans. 8.32; 19 52'. 

6. A certain clothes line which is capable of withstanding a pull of 
300 pounds, is attached to the ends A and B of two posts 40 feet apart, 



160 



MATHEMATICS 



[VII, 130 



A and B being in the same horizontal line. When the rope is held 
taut by a weight W, attached to the middle point, C, of the line, C is 
four feet below the horizontal line AB. Find the weight of the heaviest 
boy it will support without breaking. [MILLER-LILLY] 

Ans. 117.7, Ibs. 

7. A street lamp weighing 100 pounds is supported by means of a 
pulley which runs smoothly on a cable supported at A and B, on oppo- 
site sides of the street. If A is 10 feet above B, and the street 60 feet 
wide, and the cable 75 feet long, find the point on the cable where the 
pulley rests, and the tension in the cable. [MILLER-LILLY] 

8. A particle of weight W lies on a smooth plane which makes an 
angle a with the horizon. Show that P = W sin a, R W cos a, 
where P is the force acting along the plane to keep the particle from 
slipping and R is the reaction of the plane. 

131. The Simple Crane. One of the most useful applica- 
tions of the triangle of forces is the case of an ordinary crane. 
It has a fixed upright member AB called the crane post, a member 
AC called the jib, and a tie-rod BC, A weight W suspended 




FIG. 80 

rigidly at C is kept in position by three forces in equilibrium. 
These forces are (a) the weight W, (b) the pull in the tie-rod, 
and (c) the thrust in the jib. To determine their magnitudes 
construct to scale a force triangle EFG. Draw EF parallel to 
the line of action of the weight W and equal to W in magni- 
tude. From F draw F G parallel to the jib and from E draw 



VII, 131] STATICS 161 

EG parallel to the tie-rod. The lengths of EG and FG to the 
same scale on which EF was drawn represent the thrust in the 
jib and the pull in the tie-rod. The directions of the forces 
acting along the tie-rod and jib are given by following around 
the triangle in order from E to F to G to E. 

When a crane is used to raise or lower a weight, the weight 
is held by a rope passing over a pulley at C. The tension of the 
rope must now be taken into account. 

Suppose a chain or rope supporting the weight is made to 
pass over a pulley at C, and is then led on to a drum at A round 
which the rope or chain is coiled. The pull in the rope and 
tie-rod together is the same as before and is represented by EG. 
The tension in the rope is the same on each side of the pulley. 
Therefore if we mark off on EG a distance HE equal to EF, 
this distance will represent the pull in the rope, thus leaving 
GH to represent the pull in the tie-rod. 

EXERCISES 

Find the pull in the tie-rod and the thrust in the jib of a crane when 
the dimensions and weight are as given below. (Weight suspended 
rigidly at C.) 

1. AB = 10, BC = 24, AC = 31, W = 12 tons. 

2. AB =6, BC = 12, AC = 16, W = 6 tons. 

3. AB = 15, BC = 50, AC = 45, W = 5 tons. 

4. AB = 9, BC = 16, AC = 21, W = 4 tons. 

5. The jib of a crane is subjected to a compressive force equal to 
the weight of 24 tons, the suspended load being 10 tons. If the in- 
clination of the jib to the horizontal is 60, find the tension in the tie- 
rod. Ans. 16.1 tons. 

6. In a crane the pull in the tie-rod inclined at an angle of 60 to 
the vertical is 18 tons. If the weight lifted be 8 tons, find the thrust 
in the jib. Ans. 23.06 tons. 

7. In exercises 1-4, find the forces acting in each member of the 
crane when the load is suspended, but not rigidly, at the jib head, for 

12 



162 



MATHEMATICS 



[VII, 131 



the two cases when the rope passes from the pulleys to the drum (a) par- 
allel to the tie-rod, (6) parallel to the jib. 

8. The jib of a crane is subjected to a compressive force equal to the 
weight of 4000 Ibs., the suspended load being 2000 Ibs. If the inclina- 
tion of the jib to the horizontal is 45, find the tension in the tie-rod. 

9. In a crane the pull in the tie-rod inclined 45 to the vertical is 
1000 Ibs. Find the thrust in the jib if the weight is 2000 Ibs. 

10. In Ex. 9 find the thrust in the jib if the weight is 1000 Ibs. 

11. The thrust in the jib inclined 60 to the vertical is 1800 Ibs. 
The load is 900 Ibs. Find the tension in the tie-rod. 

132. Polygon of Forces. The resultant of three or more 
concurrent forces lying in the same plane may be found by 
repeated applications of the triangle of forces. 

Let a particle at be acted upon by any number of forces, 
Fi, F 2 , ; to be definite, say Fi, F 2 , F 3 , F 4 . To find their 
resultant proceed as follows. For the forces FI and F 2 con- 
struct the triangle of forces OAB (Fig. 81). Then OB is the 




FIG. 81 

resultant of FI and F 2 . For the forces OB and F 3 construct 
the triangle of forces OBC. The sum is given by OC. In a 
similar manner combine OC and F 4 . The resultant is R = OD. 
The construction of the lines OB and OC is unnecessary and 
should be omitted. The figure OABCDO is called the polygon 
of forces. OD, the closing side, is called the resultant. It will 
be noticed that the arrows on the vectors representing the 



VII, 133] 



STATICS 



163 



given forces all run in the same sense around the polygon, 
while the arrow of the resultant runs in the opposite sense. 

If any number of forces acting at a point can be represented 
by the sides of a closed polygon taken in order, the point is in 
equilibrium and the resultant is zero. 

From the above discussion we obtain the following rule for 
finding the resultant of any number of forces. 

From any point draw a line OA to represent in magnitude 
and direction the force F\. From the extremity A draw a linc~ AB 
to represent in magnitude and direction the force F%. Continue 
thix process for each of the given system of forces. Then the line 
which it is necessary to draw from to close the polygon represent* 
the resultant in magnitude and direction. 

133. Resultant of Several Concurrent Forces. Analytic 
Formula. Let any number of forces Fi, F 2 , , lying in the 
same plane, act on a particle at 0. To fix the ideas, suppose 
there are three forces. With as origin refer the forces to a 
pair of coordinate axes, OX and OY (Fig. 82). Resolve each 




force into two components, one along OX and one along OY. 
The components of F! will be OA and OB; of F z , OC and OE; 



164 MATHEMATICS [VII, 133 

of FS, OD and OF. If a\, a^, a 3 represents the angles which 
FI, FZ, FS make respectively with the axis X, we have : 

Xt = OA = FI cos ai, F! = OB = F l sin a h 

Xz = OE = F z cos 2 , F 2 = OC = Fz sin a 2 , 

Z 3 = 0Z> = F 3 cos a s , Y 3 = OF = F 3 sin a 3 . 

If a component acts upward or toward the right we will assume 
it to be positive ; if downward or toward the left, negative. 

The given system of forces is equivalent to another set con- 
sisting of the rectangular components of the forces of the given 
system. Let us use the letters X and Y to represent the sum 
of these components along the x-axis and the 7/-axis, respectively. 
Then 



(7) 



X = FI COS i + Fz COS 2 + FZ COS 3 

= the sum of all the horizontal components. 

Y = FI sin i + FZ sin 2 + F 3 sin a 3 

= the sum of all the vertical components 



The two forces X and Y acting at right angles to each other 
are equivalent to the given system of forces. The single force 
R which is the resultant of X and Y is also the resultant of the 
given system of forces. We have 



(8) R = X 2 + P. 

The resultant R is always thought of as being positive. We 
now have the magnitude of the resultant force. To find the 
line of action we have 

Y 

(9) tana = , 

Ji. 

where a is the angle between the positive direction of the x-axis 
and the positive direction of the resultant R. 



VII, 134] STATICS 165 

To find the direction along the line of action the two following 
equations are used : 

Y IT 

(10) sin a = , cos a = . 

It is obvious that equations (10) determine both the line of 
action and the direction along that line. 

EXERCISES 

1. If four forces of 5, 6, 8, and 11 units make angles of 30, 120, 
225, and 300 respectively, with a fixed horizontal line, find the mag- 
nitude and the direction of the resultant. Ans. 7.39 ; 81 6'. 

2. Forces P, 2P, 3P, and 4P act along the sides of a square taken 
in order. Find the magnitude, the direction, and the line of action of 
the resultant. 

Ana. 2V2P, - 45 with line of force of 4P, through (- 2a, - 4a) 
where side of square is 4o and origin of coordinates is intersection of 
3P, 4P. 

3. A particle is acted on by five coplanar forces ; a force of 5 Ibs. 
acting horizontally to the right, and forces of 1, 2, 3, 4 Ibs. making 
angles of 45, 60, 225, and 300 respectively with the 5-lb. force. 
Find the magnitude and the direction of the resultant. 

Ans. R = 7.31, = 334 28'. 

4. Find the resultant of the following concurrent, coplanar forces : 
(a) (14, 45), (6, 120), (5, 240). 

(6) (2, 0), (3, 50), (4, 150), (5, 240). 

(c) (2, - 30), (3, 90), (4, 135), (5, 225). 

(d) (5, - 30), (6, 270), (4, 120), (3, 135). 

134. Resultant of Parallel Forces. Let FI and F 2 be two 
parallel forces acting in the same direction and with their 
points of application at the points A and B, Fig. 83. At A 
and B apply two equal and opposite forces, AS and BT, whose 
line of action coincides with AB. These will balance and will 
not change the effect of the other forces. Find the resultant 
AD of AS and FI, and the resultant BE of B T and F 2 , by con- 
structing the parallelograms of forces. Then by constructing a 



166 



MATHEMATICS 



[VII, 134 



parallelogram of forces at 0, the intersection of AD and BE 
produced, we may find their resultant OR, which is evidently 
the resultant of FI and F 2 . Draw MK parallel to AB. Then 

E 




since OM is equal to AD in magnitude and in direction and MR 
is equal to BE in magnitude and direction, it follows that the 
triangles OMK and ADFi are equal, and the triangles MKR 
and BTE are equal. Hence the resultant OR is equal to 
FI + F 2 , and its line of action is a line through the point 
parallel to the lines of action of FI and F 2 . 

Let C be the intersection of AB and OR. Then from the 
pairs of similar triangles OCA and AFiD, and OCB and BF 2 E, 
we have 

AC AS BC _BT _ AS 

oc~J\ oc " FT ~ 77 ' 

Hence 

Fi BC 

(ii) F = IF" 

r 2 ^10 

A similar proof can be given for the case of unequal parallel 
forces acting in opposite directions. Both results may be 
combined into the following theorem. 

The resultant of any two parallel forces, acting in the same 
direction, or of two unequal forces acting in opposite directions, 



VII, 136] STATICS 167 

is parallel to the forces and equal to their algebraic sum and cuts 
a line joining their points of application into segments, the lengths 
of which are inversely proportional to the magnitudes of the forces. 

135. Moment of a Force. The moment of a force with re- 
spect to a point, called the center of moments, is the product of 
the magnitude of the force and the perpendicular distance, called 
the arm, from the point to the line of action of the force. 

Geometrically the moment of a force is represented by twice 
the area of a triangle whose base is the line representing the given 
force and whose vertex is the center of moments. 

The moment of a force in a given plane with respect to a line 
perpendicular to that plane is the moment of the force with 
respect to the foot of that perpendicular. The line is called the 
axis of moments. 

Moments are positive or negative according as they tend to 
produce counter clockwise or clockwise rotation about the axis 
of moments. 

136. Composition of Moments. The algebraic sum of the 
moments of any two forces with respect to any point of their plane 
is equal to the moment of their resultant with respect to the same 
point. 

There are two cases. 

CASE 1. When the lines of action of 
the forces are not parallel. 

PROOF. Let OP, OQ be two forces 
acting at 0, and OR their resultant; and 
let A be any point in the plane about 

which moments are to be taken. Join AO, AP, AQ, and AR. 
Then 

Area &AOQ = Area &APR + Area ARPO,* 

*By convention areas are positive or negative according as their boundaries are 
travel sed in counterclockwise or clockwise direction. 




168 MATHEMATICS [VII, 136 

since they have equal bases OQ and PR, and the altitude of 
AAOQ is equal to the sum of the altitudes of APR and RPO. 

Area AAOR = Area AAOP + Area &APR + Area ARPO, 
for obvious reasons it follows that 

Area AAOR = Area LAOP + Area AAOQ. 

Therefore the moment of OR about A is equal to the sum 
of the moments of OP and OQ about A. 

Frequently it is easier to determine the moment of a force 
by computing the sum of the moments of its components than 
to determine it directly. 

CASE II. When the lines of action of the forces are parallel. 
We exclude the case in which the forces are equal and opposite. 

Suppose that two forces P and Q act on the body at the 
points A and B, Fig. 85. From any point 0, draw OACB per- 
pendicular to the lines of action of the forces. Let OA = p, 
AC = x. Then by 134, CB = Px/Q. Taking moments about 
we find 

moment of P = P p, moment of Q = Q(p + x + 
moment of P + moment of Q = Pp -\-Qp-\-Qx-\-Px 



= moment of R. 

If P and Q are in opposite directions the proof is similar to the 
above and is left to the student. The proof in case P and Q 
are equal but opposite in direction is given in the following 
section. 





f 


It 


r 


- 






FIG. 86 




6 


A 

FIG. 85 


B 



137. Couples. A system of two parallel forces, equal irt 



VII, 138] STATICS 169 

magnitude and opposite in direction, is called a couple. The 
perpendicular distance between the lines of action of the forces 
is called the arm of the couple; and the plane containing the 
forces is called the plane of the couple. 

The moment of a couple is the algebraic sum of the moments 
of its forces about any axis perpendicular to its plane and is 
equal to the product of either force and the length of the arm. For, 
let be any axis, perpendicular to the plane of the couple, and 
OA and OB, the moment arms of the forces with respect to 0. 
Taking moments about 0, we have 

F-OB - F-'OA = F-AB. 

The sign of the couple is plus if it tends to turn with clock- 
wise rotation, and minus if it tends to turn with counter-clock- 
wise rotation. 

138. Conditions of Equilibrium. 

(a) Concurrent coplanar forces. In order that the forces of 
a system may balance each other, the resultant must be equal 
to zero, that is 



(12) R = V(SZ) 2 + (S7) 2 = 0. 
Hence we have also 

(13) SX = 0, and 2Y = 0. 

The algebraic sum of the moments of the forces (written SAf) 
about any point is equal to the moment of the resultant. If 
the forces are in equilibrium, R = 0; therefore 

(14) SM = 0. 

These conditions are used in the second method of Ex. 1, below. 

(6) System of parallel forces. If the algebraic sum of a sys- 

tem of parallel forces is not zero, the resultant is a single force 

and the system is not in equilibrium. Hence a necessary con- 



170 



MATHEMATICS 



[VII, 138 



dition for equilibrium is that 



= 0, 



where F represents the magnitude of a force. If the algebraic 
sum of the moments of the forces about any point is not zero, 
while the algebraic sum of the forces is zero, the resultant is a 
couple, and the body is not in equilibrium. Hence a necessary 
condition for equilibrium is that 
(15) 2F x = 0, 

where x is the moment arm of the force F. 

EXERCISES 

BALANCED SYSTEMS OF FORCES ACTING THROUGH THE SAME POINT 

1. A triangular frame ABC (Fig. 87) carries a load of 1000 Ibs. at A. 
Find the stresses in the members AB and AC. 




1000 




FIG. 87 



FIG. 88 



SOLUTION. We have in this problem a balanced system of forces 
acting through the point A, namely, the load of 1000 Ibs. and the forces 
FI and F 2 in the members AC and AB. Both AC and AB are subjected 
to a compression. Hence both members exert a thrust in the direction 
indictated by the arrows. The problem is to determine the magnitude 
of two unknown forces in a balanced system of three forces, the direc- 
tions of the forces being known. This problem may be solved in any 
one of the three following ways. 

FIRST METHOD. (Triangle of Forces.) The forces may be repre- 



VII, 138] 



STATICS 



171 



sented by the sides of a triangle taken in order, Fig. 88. If the figure 
is drawn to scale the magnitudes of the unknown forces F\ and Ft may 
be obtained directly from the figure by measurement. 




woo 



FIG. 89 



1000 
FIG. 90 



If the lengths of all of the members of the frame ABC are known or 
can be computed, we can obtain the magnitudes of FI and Ft by pro- 
portion, since the triangle ABC and the force triangle are similar. 

In this particular problem we observe that the force triangle is right- 
angled and one acute angle is 60. Hence 

F! = 1000 sin 60 = 866 Ibs., F 2 = 1000 cos 60 = 500 Ibs. 

SECOND METHOD. (Resolution of Forces.) Refer the forces to a 
system of coordinate axes, Fig. 88, and use the conditions (13) of equi- 
librium. We have 

SX = F 2 cos 30 - F! cos 60 = 0, 

2F = Ft sin 30 + F l sin 60 - 1000= 0. 

The solution of these equations gives, 

F l = 866 Ibs., Ft = 500 Ibs. 

THIRD METHOD. (Moments.) The sum of the moments of all the 
forces about any arbitrarily chosen point leads to one equation contain- 
ing the unknowns. If we take the sum of the moments of all the forces 
about as many arbitrary points as there are unknowns then we will have 
as many equations as unknowns. The solution of these equations gives 
the magnitudes of the unknown forces. It is often advantageous to 
choose for the points about which moments are taken, points on the lines 
of action of the unknown forces, one on each line. 

Taking moments about B we find 



172 



MATHEMATICS 



[VII, 138 



whence 



SAf = 8V3F, - 1000 X 12 = 0, 
Fi = 866 Ibs. 



Taking moments about C we find 

SM = 1000 X 4 - 8F 2 = 0, 



whence 



F 2 = 500 Ibs. 




2. Find the stresses in the members AB and AC, of the triangular 

frame ABC, Fig. 91, the load at A being 
1000 Ibs. 

[HiNT. Use the triangle of forces.] 
Ans. AB, 739.1 Ibs.; AC, 922.2 

3. S Ive Ex. 2 (a) by using the 
method of resoluti n of forces ; (6) by 
the method of moments. 

4. Assuming that the frame in Ex. 
2 is supported by a vertical force at B, find the magnitude of the force 
and the stress in BC. 

5. A crane is loaded with 3000 Ibs. at C. Determine the stresses 
in the boom CD, the tie BC, the mast BD 

and the stay AB, Fig. 92. 

[HiNT. Use the triangle of forces.] 

Ans. CD, 6250 Ibs. (compression) ; BC, 
4250 Ibs. (tension) ; AB, 5858 Ibs. (tension) ; 
BD, 2500 Ibs. (compression). 

6. Solve Ex. 5, using the method of reso- 
lution of forces. 

7. Find the horizontal and vertical components of the supporting 

forces at A and D, Ex. 5. 

8. Find the stresses in the members of the 
crane in Ex. 5, when the boom makes an an- 
gle of 15 with the horizontal. 

9. What is the smallest force F which will 
prevent a weight of 150 Ibs. from slipping 

down the incline represented in Fig. 93 if friction is neglected? 

Ans. 212.2 Ibs. 





FIG. 



VII, 138] STATICS 173 

10. Let F = 150 Ibs. (Fig. 93) and let the weight also be 150 Ibs 
What will be the largest angle between the inclined plane and the hori 
zontal at which the weight will not slip ? Ans. 30. 

11. Experiments indicate that a horse exerts a pull on his traces 
equal to about one-tenth of his weight, when the working day does not 
exceed 10 hours. The draft of a certain wagon is due to (a) axle 
friction = 5 Ibs. per 2000 Ib. load ; (6) gradient or hills ; (c) rolling draft 
depending on height of wheel, width of tire, condition of road-bed, etc. 

How large a load can a team of horses each weighing 1000 Ibs. pull 
up a 10% grade if the rolling draft is zero. (A 10% grade is a rise of 
10 feet for each 100 feet measured horizontally along the roadway.) 

Ans. 1961 Ibs. 



FIG. 94 

12. What extra pull must a horse exert on his traces (assumed 
horizontal) if on a level road the wheel, 4 feet in diameter, strikes a 
stone 2 inches high, the load being 1000 Ibs. Ans. 436 Ibs. 

13. A carriage wheel whose weight is W and whose radius is r rests 
on a level road. Show that any horizontal force acting through the 
center of the wheel greater than 



r h 

will pull it over an obstacle whose height is h. 

14. In Ex. 13, let P = 100 Ibs., W = 1000 Ibs., r = 2 feet. Find h. 

Ans. 0.126 in. 

15. A 50 Ib. boy swings on the middle of a clothes line which is 50 feet 
long. The lowest point is 2 feet below either end. Find the tension 
in the rope. Ana. 625 Ibs. 

16. A wire 90 feet long carries a weight of 25 Ibs. at each of its trisec- 
tion points. When the wire is taut each weight is 5 feet below the hori- 
zontal line connecting the points of support. Find the tension in each 
segment of the wire. Ans. 150 ; 147.9 ; 150 Ibs. 



174 



MATHEMATICS 



[VII, 138 



17. Steam in the cylinder of an engine exerts a pressure of 20,000 
pounds on the piston-head. The guides N, Fig. 95, are smooth. What 







N 

1 




; i 




N 
FIG. 95 




is the thrust in the connecting rod when it makes an angle of 20 with 
the horizontal? What is the pressure on the guides N ? [MILLER-LILLY] 

PARALLEL FORCES ACTING IN THE SAME PLANE 
18. Determine the resultant R of each of the following systems of 
parallel forces. 

50 20 30 



Y~4 -!< - 



6' ->f -5-' 






20 



40 



80 



(a) FIG. 96 

10 



60 

(6) FIG. 97 



70 



500 
U-3./4- +__ ... . 

800 

(c) FIG. 98 



19. Let AB (Fig. 99) represent a beam carrying the weights indi- 
cated and supported by the vertical forces FI and F 2 . Find FI and F 2 . 



1000 2000 
>*-* 4+ 4* 



F = 2500 



FIG. 99 



20. The system of parallel forces in Fig. 100 is in equilibrium. Find 
the magnitudes and directions of the unknown forces FI and F t . 
4P F, 




FIG. 100 



VII, 138] 



STATICS 



175 



21. If a horse exerts a pull on his traces equal to one-tenth of his 
weight, where should the single-tree for each of two horses weighing 
1200 and 1600 Ibs., respectively, be fastened to a double-tree in order 
that each horse shall do his proper share of the work ? 

22. The center clevis pin A, of a double-tree is a inches in front of the 
mid-point B, of the line connecting the end clevis pins C and D, which 
are b inches apart. If one horse is pulling c inches ahead of the other 
what fraction of the load L is each horse pulling, Fig. 101 ? 

1 _ ac 
2 



Ans. 

2 



- c 2 



O 




FIG. 101 

23. Find what fractional part of the load each horse is pulling if 
a = 2, when 

(a) b = 41, c = 9. (b) b = 39, c = 15. 

(c) b = 34, c = 16. (d) 6 = 52, c = 20. 

(e) b = 37, c = 12. (/) b = 50, c = 14. 

(jr) b = 61, c = 11. (h) b = 36, c = 4. 

24. In Ex. 22, if the evener makes an angle with the tongue, what 
fractional part of the load is pulled by each horse ? 



Ans. + 



\ - \ tan 9. 



176 MATHEMATICS [VII, 133 

25. In Ex. 24 put a 2, b 40. Plot a curve using values of 9 as 
abscissas and values of the load pulled by one horse as ordinates. What 
can you say about the part of the load pulled by this horse as increases ? 

26. In each of the cases of Ex. 23 find the pounds of pull exerted by 
each horse if the total pull on the load is 362.88 Ibs. 

27. The middle clevis pin A of a three-horse evener is a inches in front 
of the point B of the line connecting the end clevis pins C and D. The 
end clevis pins are b and 26 inches from the point B. Find what frac- 
tional part of the load is borne by the horse on the longer end when it is 
c inches behind the other horses. 



28. Find what fractional part of the load the horse on the long end is 
pulling if a = 2, when 

(a) b = 24, c = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 

(b) b =25, c = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 

(c) b = 26, c = 2, 4, 6, 8, 10, 12, 14. 

29. In Ex. 27, if the evener makes an angle with the tongue, what 
fractional part of the load is pulled by the horse on the long end. 

Ans. - + --tau6. 
3 3b 

30. In Ex. 29 put a = 2, b = 25. Plot a curve using values of 
as abscissas and fractional parts of the load pulled by the horse on the 
long end as ordinates. Discuss the problem. 

31. A steel rail 60 ft. long weighs 1595 Ibs. Where must a fulcrum 
be placed so that a 180 Ib. man at one end can raise 4 tons at the other? 

Ans. 6 ft. 



CHAPTER VIII 
SMALL ERRORS 

139. Errors of Observation. Suppose that we measure the 
length of a building and record the result. Such a record is 
called a reading or an observation. Suppose that we measure 
the same length and record the reading on each of several suc- 
cessive days. On comparison it is likely we shall find that 
they do not exactly agree. What then is the true length? 
Whatever the actual length may be the difference between it 
and any observation of it is called an error of observation. 

Suppose that we measure the length of a building with a tape 
whose smallest division is one foot. If the length is not a whole 
number of feet, we estimate by the eye the fraction of a foot left 
over. This estimate will almost certainly be in error. If we 
measure the same length with a tape divided to eighths of an inch, 
the end of the building may coincide with a division of the tape 
or we may have to estimate the fraction of an eighth. Subse- 
quent readings are not likely to agree exactly with the first, and 
even if they do all agree we cannot be sure that we have the true 
length. Inattention and lack of precision of the observer, in- 
experience in using the measuring instrument, or the use of an 
instrument which is defective or out of adjustment, all tend 
to introduce errors. It is important to keep in mind that such 
errors are always present, in greater or less degree, in every set 
of observations. 

If a is the recorded reading of a measurement of an unknown 
quantity u, a measure of the error in this reading is a positive 
number m, such that u lies between a m and a + m. The 
actual error may be very much less than its measure m. For 
example if a rod of (unknown) length / be measured with a scale 

177 



178 MATHEMATICS [VIII, 139 

divided to tenths of an inch and the reading is 47.8, it is fairly 
certain that 47.7 < / < 47.9, and we write I = 47.8 0.1. 

It is evident that any number will be in error if it is derived by 
computation from other numbers which are inexact. Approxi- 
mations are used in computations not only for recorded meas- 
urements but also in the case of irrational numbers, such as 
surds, most logarithms, trigonometric functions, TT, etc. We 
have 3-place, 5-place, 7-place, 10-place tables in order to secure 
the degree of accuracy desired in the computed result. In what 
follows it is shown how to find a measure of the error in a number 
computed by some of the simpler processes of arithmetic from 
given numbers the measures of whose errors are known. 

140. Error in a Sum. Suppose that in measuring two quan- 
tities whose actual (and unknown) values are u and v, we make 
errors Aw and Av respectively, and record the readings a and b. 
Then u = a Aw, v = b A and their sum lies between 
a + b (Aw + A0) and a + b + (Aw + Au). 

Whence, w + - = a + 6 (Aw + Aa). 

That is, the error in the sum of two readings is measured by the 

sum of their errors. 

This result is readily extended to the sum of more than two 
readings. The error in the difference of two readings is never 
greater than the sum of their errors, though it may be greater 
than their difference. 

EXAMPLE. Find the sum and difference of 46.8 0.65 and 12.4 
0.15. Here the readings are 4.68, 12.4 and the measures of their errors 
are 0.65, 0.15 respectively. The measure of the error of their sum is 

065 + 0.00 = 0.80 ; 

whence (46.8 0.65) + (12.4 0.15) = 59.2 0.8 
and (46.8 0.65) - (12.4 0.15) = 34.4 0.8 

141. Error in a Product. With the same notation as above, 
the product uv lies between 

ah (aA0 + feAw + Aw At)) and ab + (aAw + 6Aw -f Aw Ac), 



VIII, 142] SMALL ERRORS 179 

whence, neglecting the small term Aw A, we have approxi- 
mately, 

uv = ab (aAv + 6 Aw). 

That is, a measure of the error in the product of two readings is the 
first times the error of the second plus the second times the error of 
the first. 

142. Error in a Fraction. The quotient of u divided by v 
evidently lies between 

a Au i a + AM 

6 + A b At)' 

that is between 



a aAfl + bAu j n . aAy + 



b b(b + At)) 6 b(b - At>) 

whence a measure of the error in the fraction is 

aAt) -f feAw aAa -f &Aw 

- -- , approximately. 

b(b At)) fr 2 

That is, a measure of the error in the .quotient of two readings is a 
measure of their product divided by the square of the divisor. 

EXAMPLE. Find the product and quotient of 12.4 0.15 and 
46.8 0.65. By 141, a measure of the error in the product is (12.4) 
(0.65) + (46. 8) (0.15) = 15.08 and the error in their quotient is meas- 
ured by 15.08/(46.8) 2 = 0.0069. 

Whence, (12.4 0.15) (46.8 0.65) = 580.32 15.08 

and (12.4 0.15)/(46.8 0.65) = 0.265 0.0069 

EXERCISES 

Make each of the following computations and state the result so as to 
show a measure of the error in it. 

1. (123 0.2) (241 0.1). 2. (222 0.5) (111 0.4). 
3. (217 0.2J(117 0.3). 4. (1267 0.5)(1342 0.4). 

5. (163 0.2)/(25 0.5). 6. (732 0.3)/(21 0.4). 

7. In Ex. 3 and 4 compute the term Au Ay neglected. 

8. In Ex. 5 and 6 compute " ~ A>1 and a + A ' ; \ Find the differ- 

+ Ay b Aj 



180 



MATHEMATICS 



[VIII, 142 



ence between the error thus computed and those computed in exercises 
5 and 6 and consider the influence of this difference upon the quo- 
tient. 

9. A line is measured with a chain (100 links each 1 ft. long). After- 
wards, it is found that the chain is one foot too long. If the measured 
length was 10.36 chains, what is its true length if the error is assumed 
to be distributed through the chain? Ans. 10.4636 chains. 

10. A line is measured with a 100-ft. tape and found to be 723.36 
feet long. The tape is afterwards found to be 0.02 of a foot short. 
What is the true length of the line? Ans. 723.22 ft. 

11. A certain steel tape is of standard length at 62 F. A tape will 
expand or contract sixty-five ten millionths of its length for each 
Fahrenheit degree change of temperature. A line is measured when 
the temperature of the tape is approximately 80 and found to be 
323.56 feet long. What is its true length? Is it necessary to know 
the nominal or standard length of the tape to solve this problem? 

Ans. 323.52 ft. 

12. What change in temperature is necessary to change a 100-foot 
tape by 0.01 of a foot, or 1 in 10,000? Ans. 15.38 

13. A certain 100-foot steel tape, standard length at 62 F., is used 
to measure from the monuments (Fig. 102) to the point A, in a line 



X Monument 



so'-; 





7th 

St. 


1 


2 


3 


4 


5 


6 


8th 
St. 



Monument X 



FIG. 102 

between lots 2 and 3 extended, when the temperature is 40 F. As- 
suming that the map distances are correct, what lengths must be 
measured from 7th street and 8th street monuments respectively to 
locate the point A, the monuments being in the center lines of the 
streets? Ans. 160.02; 280.04 

Show that if x, y, and z are small that 

14. (1 + z)(l + y} is nearly equal to 1 + x + y. 

15. (1 + x)l(\ + y) is nearly equal to 1 + x y. 



VIII, 143] SMALL ERRORS 181 

16. (1 + x)(l + ?/)(l + z) is nearly equal to 1 + x + y + z. 

17. Show that (1 + 0.03) (1 - 0.05) = 0.98 nearly. 

18. Compute (1.04)(1.06)(0.95). Ans. 1.05 

19. Compute (a) 1.03/1.02; (6) (1.03)(1.02). 

Ans. (a) 1.01; (6) 1.05 

20. Compute (a) (1.03) (0.98); (6) 1.03/0.98. 

Ans. (a) 1.01; (b) 1.05 

21. Draw a figure (rectangle) to represent (4.03) (9.02) and indi- 
cate 4 X 0.02; 9 X 0.03; 0.03 X 0.02; 4X9. 

22. Show that the error in abc due to errors Aa, Ab, Ac in a, b, and c 
respectively, is be Aa + ac Ab + ab Ac. 

23. Compute 2.01 X 4.02 X 3.02 Draw a figure (parallelepiped) 
to represent this product and indicate 3 X 4 X .01; 2 X 3 X .02; 
2 X 4 X .02; .01 X .02 X .02; 2X4X3. Ans. 24.4 

143. Data derived from Measurements. The preceding 
results apply immediately to the case in which numbers ob- 
tained by measurement are stated without any accompanying 
indication of the probable error. 

In such cases it is understood that the given figures are all 
reliable, i. e., that we stop writing decimal places as soon as 
they are doubtful. The last figure written down should be as 
accurate as is possible. Then the error will surely not be 
more than 5 in the next place past the last one actually written. 

Thus, if a certain length is reported to be 2.54 ft., we would 
understand that the true length is not more than 2.545 ft., and 
not less than 2.535 ft. For if the true length is more than 2.545 
ft., it should be given as 2.55 ft.; and so on. 

It may happen that the last figure written down is 0. This 
means that that place is reliable. Thus, to say that a given 
length is 2.4 ft. means that the true length is between 2.35 ft. 
and 2.45 ft. But to say that a given length is 2.40 ft. means 
that the true length is between 2.395 ft. and 2.405 ft. 

In computations based upon numbers obtained by measure- 
ment, these facts must be kept in mind, and the result of any 



182 MATHEMATICS [VIII, 143 

calculation should not be stated to more decimal places than 
are known to be reliable. 

EXAMPLE 1. Find the area of a rectangle whose sides are found, 
by actual measurement, to be 2.54 ft. and 6.24 ft., respectively. 

Since the error in writing 2.54 ft. may be as great as .005, we must 
write for the length of this side (2.54 .005) ft. Likewise, we must 
write for the other side (6.24 .005) ft. Hence, by the rule of 141, 
the error in the product may be as large as 

2.54 X .005 + 6.24 X .005, 

that is .043. Hence we are not justified in expressing the answer to 
more than one decimal place; although 

2.54 X 6.24 = 15.8496, 

we must sacrifice all the figures past 15.8, and write 
2.54 X 6.24 = 15.8 .1 

since the true answer may be as large as 15.894 Even the figure 8 
in the first decimal place is not reliable, since the true area may be 
nearer 15.9 than 15.8 sq. ft. 

EXERCISES 

1. Assuming that the numbers stated below are the results of 
measurements, and that each of them is stated to the nearest figure 
in the last place, find the required answer and state it so that it also is 
correct to the nearest figure in the last place you give, or else to within 
a stated limit of possible error. 

(a) 2.74 -f 3.48 + 11.25 + 7.34 Ans. 24.8 

(6) 3.25 - 7.348 + 4.26 - 6.1 Ans. 20.9 .1 

(c) 6.27 X 3.14 (g) 61.54 X 45.2 + 14.81 

(d) 26.5 X 11.4 (A) 8.26 -=- 2.14 

(e) 7.32 X 5.4 (i) 43.7 + 5.4 

(/) 36.4 X 2.78 0') (6-42 X 2.35) -?- 4.5 

2. The sides of a rectangle are measured, and are found to be 4 ft. 
6.3 in. by 3 ft. 5.4 in. Express correctly the area of the rectangle. 

3. The three sides of a rectangular block are measured and are 
found to be 7.4 in. by 3.6 in. by 4.7 in. Express the volume. 



VIII, 146] SMALL ERRORS 183 

4. Suppose that the dimensions of a bin are measured roughly to 
the nearest foot, and that they are 8 ft. by 4 ft. by 3 ft. How large 
may the volume actually be? How small may it be? 

Ans. 118.1 cu. ft., 65.6 cu. ft. 

5. The floor of a room is found by measurement to be 22 ft. X 15 ft., 
each dimension being to the nearest foot. How should the area be 
stated? Ans. 330 18 sq. ft., or 300 sq. ft. 

6. If, in Ex. 5, the height of the room is 9 ft. to within the nearest 
foot, express the volume of the room. 

144. Error in a Square. If a is an observed value of an un- 
known quantity u, then it follows directly from 141 that a 
measure of the error in w 2 is approximately 

aAw + aAw 2aAw, and we write 

w 2 = a 2 2aAw. 

That is, a measure of the error in the square of a reading is twice 
the reading times its error. 

145. Error in a Square Root. With the same notation as 

above, u = a Aw is nearly equal to 

2 

"la"' 

since the last term is small. This is a perfect square and hence 
the positive square root of w is approximately 

Va-^=. 
2V a' 

That is, a measure of the error in the positive square root of a read- 
ing is equal to its error divided by twice its square root. 

EXAMPLE. Find Vl25 0.5 A measure of the error is 

0.5/2(11.18) = .022 and Vl25 0.5 = 11.18 0.022 
Again V2400 = V2401 - 1 = 49 - & = 49 - 0.0102 

146. Errors in Trigonometric Functions. Suppose a ex- 
pressed in radians is an observed value of an unknown angle a. 



184 MATHEMATICS [VIII, 146 

Then a = a Aa and by 94, 

sin a = sin (a Aa) = sin a cos Aa cos a sin Aa. 
Now if Aa is small, cos Aa is nearly equal to 1, and sin Aa is nearly 
equal to Aa. Whence we have, approximately, 
sin a = sin a cos a Aa, 

and the smaller Aa is, the better the approximation. Hence, 
a measure of the error in the sine of an angle is the error in the 
reading (expressed in radians) multiplied by the cosine of the 
reading. 

Similarly we can show that a measure of the error in the cosine 
of an angle is the error in the reading multiplied by the sine of the 
reading. 

By means of these results and the principles of 142 we can 
readily find a measure of the error in the other trigonometric 
functions. For example 

_ sin a _ sin (a Aa) 

tan a - ~ 

cos a cos (a Aa) 

and by 142, a measure of the error in tan a is 

Aa(sin 2 a + cos 2 a)/cos 2 a = sec 2 a Aa 

EXAMPLE, sin (36 40' 5') = sin 36 40' .00145 cos 36 40' 

= .5972 .0012 

cos (36 40' 5') = cos 36 40' .00145 sin 36 40' = .8021 .0009 
tan (36 40' 5') = tan 36 40' .00145 sec* 36 40' = .7445 .0023 

147. Computation of Error from Tables. This will be illus- 
trated by an example. To find sin (36 40' 10') we look in 
a table of sines and find sin 36 50' = .5995, sin 36 40' = .5972, 
sin 36 30' = .5948 ; the difference between the first and second 
is .0023 and that between the second and third is .0024. Choos- 
ing the larger we write sin (36 40' 10') = .5972 .0024. 

This method applies to tables of logarithms, squares, square 
roots, etc., in fact to any tables giving the values of a function 



VIII, 148] 



MATHEMATICS 



185 



In practical 




4 



FIG. 103 



for regularly spaced values of the argument. For example, a 
measure of the error in log u = log (o Aw) is the greater of 
the differences log (a + Aw) log a and log a log (a Aw). 
Thus to find log (17.4 0.7) we look up in the table log 16.7 = 
1.2227, log 17.4 = 1.2405, log 18.1 = 1.2577. The larger 
difference is 0.0178 and we write 

log (17.4 0.7) = 1.2405 0.0178 
148. Errors in Computed Parts of Triangles, 
applications, e.g. in surveying, 
the given parts of triangles are 
subject to errors of measure- 
ment and consequently the com- 
puted parts are also in error. 
Suppose the base AB of the tri- 
angle ABC in Fig. 103 is 23.4 
0.02, the side AC = 15.6 0.04, and the angle A = 32 30' 
10'. Then the altitude 

CD = (15.6 0.04) sin (32 30' 10') 

= (15.6 0.04) (0.5373 0.0025) 146, 147 

= 8.382 0.060 141 

Again, the area is given by the following computation. 

Area = (23.4 0.02) (8.382 0.06) = 98.069 0.786. 
Similarly a measure of the error in any computed part of a 
triangle may be found by the foregoing principles of this chapter. 

EXERCISES 

Calculate the error and the per cent, error of the square in each of 
the following numbers. Where no estimate of the error is expressed 
the error is supposed to be not greater than 5 in the next place past 
the last one written ( 143). 

1. a = 76 0.1 4. a = 432 0.03 

2. a = 101 0.4 5. a = 2.46 

3. a = 32 0.04 6. a = 13.4 



186 MATHEMATICS [VIII, 148 

Find the error and the per cent, error in the square root of each of the 
following: 

7. 121 0.4 11. 216 0.03 

8. 169 0.5 12. 165 0.2 

9. 144 0.02 13. 43.7 
10. 625 0.01 14. 6.45 

15. Show that the error of the cube of a Aa is 3a 2 -Aa. Hence 
find a correct expression for the volume of a cube of side 2.6 ft. 

16. Show that the error of the fourth power of a Aa is 4o 3 -Aa. 

17. Show that the error of the cube root of a Aa is Aa/3a 2/3 . 

18. Find by the use of the tables and by use of the results of Ex. 17 
the error in the cube root of (a) 1728 2; (6) 15625 1; (c) 343 0.2 

Ans. .005; .0006; .014 

19. By applying twice the formula for the error of the .square root 
of a Aa, show that the error of the fourth root of a Aa is Aa/4a. 
Find the error in the fourth root of 256 1. Ans. 0.001 

20. Find the error by both methods of sin a for each of the following: 

(a) 26 10'. (6) 45 15'. 

(c) 80 30'. (d) 10 db 10'. 

Ans. .0026; .0031; .0015; .0028 

21. Find the error of (a) cos a; (6) tan ; (c) ctn a; (d) sec a; (e) 
esc a due to an error Aa in a. 

22. Find by the use of the tables the error of (a) cos (26 db 25') ; 
(6) tan (20 3'); (c) ctn (70 20'); (d) sec (24 10'); (e) esc (46 
10'). 

Ans. (a) .0032; (b) .0009; (c) .0066; (d) .0014; (e) .0039 

23. Find the error of the area of the triangle for each of the following : 

(a) a = 120 db 0.3 rod, 6 = 144 0.2 rod, y = 47 10'. 

(6) a = 80 0.1 rod, b = 160 0.5 rod, y = 89 30'. 

(c) a = 40 0.5 rod, 6 = 60 0.3 rod, y = 45 d= 10'. 

(d) a = 32 0.4 rod, b = 146 0.8 rod, y = 26 5'. 

24. If A, B, C denote the angles and a, b, c the sides opposite in a 
plane triangle and if a, A, B are known by measurement, then 

b = a sin B/sin A. 



VIII, 148] SMALL ERRORS 187 

Show that the error, called the partial error in b due to a (written A 6), 
in the computed value of b due to an error Aa in measuring a is, approxi- 
mately, 

A & = sin B esc A Aa. 
Likewise show that 

A A b = a-sin B-csc A-ctn A-&A, and Agb = a cos B-csc A-&B, 
and that the total error is, approximately, 

A6 = A a & + A A b + AB&. 

Note that A and B are to be expressed in radian measure. 

25. The measured parts of a triangle and their probable errors are 

a = 100 0.01 ft.; A = 100 1'; B = 40 1'. 
Show that the partial errors in the side b are 

A Q b = 0.007 ft.; A A b = 0.003 ft.; A B b = 0.023 ft. 

If these should all combine with like signs, the maximum total error 
would be A6 = 0.033 ft. 

26. If a = 100 ft., B = 40, A = 80, and each is subject to an error 
of 1 %, find the per cent, of error in b. 

27. Find the partial and total errors in angle B, when 

a = 100 db 0.01 ft., b = 159 0.01 ft., A = 30 10'. 

28. The radius of the base and the altitude of a right circular cone 
being measured to 1%, what is the possible per cent, of error in the 
volume? Ans. 3%. 

29. The formula for index of refraction is m = sin i/sin r, where i 
denotes the angle of incidence, and r the angle of refraction. If i = 50 
and r = 40, each subject to an error of 1%, what is m, and what its 
actual and percentage error? 

30. Water is flowing through a pipe of length L ft., and diameter 
D ft., under a head of // ft. The flow in cubic feet per minute, is 



Q = 2356 J- * 



IL + 30D 

If L = 1000, D = 2, and H = 100, determine the change in Q due to 
an increase of 1% in H; in L; in D. 

31. The formula for the area of a triangle hi terms of its three sides 



188 MATHEMATICS [VIII, 148 



is A = Vs(s - a)(s - b)(s c) where s = \(a + b + c). A tri- 
angular field is measured with a chain that is afterwards found to be 
one link too long. The sides as measured are 6 chains, 4 chains, and 
3 chains respectively. What is the computed area, and what is the 
true area? 

32. Show that the erroneous area of a field, determined from measure- 
ments with an erroneous tape, will be to the true area as the square of 
the nominal length of the tape is to the square of its true length. 

33. An irregular field is measured with a chain three links short. 
The area is found to be 36.472 acres. What is the true area? 

34. The acceleration of gravity as determined by an Atwood's 
machine is given by the formula: g = 2s /I 2 . Find approximately the 
error due to small errors in observing s and t. 

Ans. A- s g = 2As/P; A t g = - 4s/l 3 . 

35. A right circular cylinder has an altitude 12 ft. and the radius 
of its base is 3 ft. Find the change in its volume (a) by increasing 
the altitude by 0.1 ft., and (6) the radius by 0.01 ft. (c) By increasing 
each simultaneously. Ans. (a) 2.83; (6) 3.02; (c) 5.85 

36. The period of a simple pendulum is 



.2, jr. 

\<7 



9 

Show that AT IT = %Al/l %Ag/g and hence a small positive error 
of k per cent, in observing I will increase the computed time by k/2%, 
and a small positive error of k'% m the value of g will decrease the 
computed time by k'/2 per cent. 

37. Let Wi denote the weight of a body in air, and w-i its weight in 
water; then the formula 

o 

Wi Wz 

gives the specific gravity of a body which sinks in water. If 
wi = 16.5 0.01, w> 2 = 12.3 0.02, 

find the error in S due to the error in w\; due to the error in w 2 ; the 
total error in S; the relative error AS/S. 

38. The specific gravity S of a floating body is given by the expression 

S = ^ > 



VIII, 148] SMALL ERRORS 189 

where Wi is the weight of the body in air, w 2 is the weight of a sinker 
in water, and w 3 is the weight in water of the body with sinker attached. 
Determine the specific gravity of a body and the probable error if 

wi = 16.5 0.01 

w z = 182.2 0.03 

w a = 176.5 0.02 [RIETZ AND CRATHORNE] 

39. To determine the contents of a silo I measure the inside diameter 
and height in feet and inches and find D = 8 ft. 2 in., h = 21 ft. 6 in. 
Find the error in the computed contents if there are errors AD = 0.4 
in., A/i = 0.3 in. in the measured dimensions. Ans. 2.22 cu. ft. 

40. My neighbor wants to buy the wheat from one of my bins. 
The measurements are: length = 12 feet; width = 6 feet; depth of 
wheat in bin = 8 ft. I make a mistake however of 1 /4 inch in measur- 
ing each 2 feet of linear measure. Find the error of contents in cubic 
inches. Find the error in bushels if 2150.4 cu. in. make 1 bushel. 
A more accurate value is 2150.42 Find the error due to using 2150.4 
instead of 2150.42 Find the error if 2150 is used. 

41. I decide to sell to a neighbor by measurement my corn in the 
crib. I measure with a yard stick placing my thumb to mark the 
end of the yard and holding my thumb in place proceed to measure 
beyond it thus making an error of 1/2 inch. My measurements are 
length = 30 ft. 3 in.; width = 11 ft. 9 in.; height 13 ft. 6 in. Find the 
error in cubic inches due to my method of measuring. 

42. The quantity of water in cubic feet per second flowing through 
a rectangular weir is given by the formula. 

Q =* 3.33 [L - 2h]hw, 

where h is the depth in feet of water over the sill of the weir, and L 
the length in feet of the sill. Find Q and the error hi Q if L = 26 0.1, 
h = 1.6 0.02 



CHAPTER IX 

CONIC SECTIONS 

149. Derivation. The circle, the ellipse, the parabola, and 
the hyperbola, are curves which can be cut out of a right circular 
conical surface by planes passing through it in various directions. 
For this reason, they are called also conic sections. Being 
plane curves, however, they can be defined and studied as the 
locus of a point moving in a plane under certain conditions. 

150. The Circle. The circle is the locus of a point moving at 
a fixed distance r from a fixed 

point C. 

The fixed distance r is called 
the radius; the fixed point C is 
called the center. 

EQUATION OF THE CIRCLE. 
Given the center, C(x , y ) and 
the radius, r, of a circle, to de- 
duce its equation. 

Let P(x, y) be any point on the locus (Fig. 104). Then by 

(D 45, 

CP = V(x z ) 2 + (y 2/o) 2 , 

and by the definition of the circle CP = r. Hence, squaring 
and equating the two values of CP , we find 

(1) (x - x ) 2 + (y - y Y = r\ 

Conversely, let Q(x\, y\) be any point which satisfies (1); i. e., 




~7 



FIG. 104 



- x ) 2 + (t/i - y ) = 
190 



IX, 152] CONIC SECTIONS 191 

whence 



(*i - so) 2 + (</! - y o y = r, 

but this says that CQ = r, and therefore Q is on the circle. 
Therefore (1) is the equation of the circle. 

If the center is at the origin, x = yo = 0, and the equation 
reduces to 

(2) x 2 + y 2 = r 2 . 

151. Equation of the Second Degree. The most general 
equation of the second degree in x and y is of the form 

(3) a.-c 2 + bxy + cy 2 + dx + cy + f = 0, 

in which the coefficients are real numbers and a, 6, c, are not 
all zero. The equation of the circle which we have obtained 
is of this form and has always 6 = and a = c. Conversely, 
the special equation of the second degree 

(4) ox 2 + ay 2 + dx + ey + f = 0. 

is the equation of a circle or of no locus. To show this we 
have only to complete the square of the terms in x and of the 
terms in y. This process will reduce it to the form of (1) 150, 
as is shown in the next paragraph. 

152. Determination of Center and Radius. When the 
equation of a circle is given, the center and radius can be found 
by transposing the constant term to the right and completing 
the square of the terms in x and also of the terms in y. 

EXAMPLE 1. Find the center and radius of the circle 

x 2 + y* - 3x - 2y - 3 = 0. 
To reduce this equation to the form (1) we complete the squares as 

follows: 

(z 2 - 3x + ) + (y 2 - 2y + ) = 3, 

(z 2 - 3x + |) + & - 2y + 1) = 3 + | + 1, 

(x - f) 2 + (y - I) 2 = (f) 2 



192 MATHEMATICS [IX, 152 

Comparing this with the standard equation (1), we see that the center 
is at (3/2, 1) and r = 5/2. 

EXAMPLE 2. Examine the equation 

9x 2 + 9?/ 2 - 6x + 12y + 6 = 0, 
We complete the squares as follows : 

& + y 2 - lx + f y + f = 0, 
X* - \x + $ + 2/2 + f y + f = - f + i + f, 
(s ~ i) 2 + (2/ + ) 2 = ~ i 

But since the square of a real number is positive (or zero), this shows 
that there are no points in the plane which satisfy the given equation. 
Therefore it has no locus. 

EXAMPLE 3. Examine the equation 

225x 2 + 225?/ 2 - 270x - 300t/ + 181 = 0. 
We complete the squares as follows: 

x 2 + y 2 - fx - fa + iH = 0, 
x 2 - fx + A + 2/ 2 - f !/ + I = - Mi + & + *, 

(x - f ) 2 + (y - I) 2 = o. 

This shows that the given equation is satisfied by the point (3/5, 2/3) 
and by no other point in the plane. This case may be looked upon as 
the limiting case of a circle whose center is at (3/5, 2/3), and whose 
radius is zero. 

EXERCISES 

1. Write the equation of the circle determined by each of the follow- 
ing conditions. 

(0) Center (2, 4), radius = 3. (6) Center ( 1, 3), radius = 5. 

(c) Center (-2, -3), radius = 3. (d) Center (3, - 2), diameter = 7. 
(e) Center (a, a), diameter a. (f) Center (r, 0), radius = r. 

(g) Center (4, 6) passes through the point (0, 3). 
(h) Abscissa of center = 1, passes through the points (0, 1), (0, 7). 

(1) The segment from (1, 3) to (7, 5) is a diameter. 

0) Center is on the line x = y, tangent to x-axis at ( 6, 0). 



IX, 152] CONIC SECTIONS 193 

2. Write the equation of a circle of radius 6 when the origin is (a) at 
the highest point of the circle ; (6) at the lowest point ; (c) at the left- 
most point ; (d) at the rightmost point ; (e) when the origin divides 
the horizontal diameter from left to right in the ratio 1/3. 

3. Determine which of the following equations represent circles; 
find the center and the radius in each case. 

(a) x 2 + y 2 = 4x. (6) x 2 + y 2 = 6y. 

(c) x 2 + 8y = 4x - y 2 . (d) 3z 2 + 3y 2 = 14y. 

(e) x* + y 2 + 4x + 7 = 0. (/) x 2 + y 2 + 3x + 5y = 0. 

(0) x 2 + y 2 = 2(y + 4). (h) x 2 + y 2 = 4(x - 2). 

(1) x 2 + y 2 - 4x - 6y + 9 = 0. 
0') * 2 + y 2 + 101 = 87y - 20x. 
(A;) 2z 2 + 27/ 2 + 15y = 12x + 7. 
(1) 9x 2 + 9y 2 + 6y = 24x + 47. 
(TO) 16.x 2 + 167/ 2 = 24x + 40y - 34. 
(n) 49x 2 + 49y 2 + 28x - 2% + 9 = 0. 

(o) 4a(ax 2 + 6x - by) + b 2 + 4o(ay 2 - ex - cy) + c 2 = 0. 

4. Show that if the coefficients of x 2 and y 2 in the equation of a 
circle are each + 1, the coordinates of the center can be found by 
taking negative one-half the coefficient of x and negative one-half the coef- 
ficient of y. 

For example, the center of the circle 

x 2 + y 2 - 5x + 4y - 3 = 

is (5/2, - 2). 

5. Find the coordinates of the center of each of the following circles, 
by the process of Ex. 4. 

(a) x 2 + y 2 - 4x - 6y + 9 = 0. (d) x 2 + y 2 - 2x + 4y + 1 = 0. 
(6) x 2 + y 2 + 6x + 4y + 9 = 0. (e) x 2 + y 2 - 3x + 5y + 3 = 0. 
(c) x 2 + ?/ - 4y = 0. (/) 2x 2 + 2y 2 + 4x - 6y + 1 =0. 

6. The value of the polynomial P = x 2 + y 2 2x 4y + 3 at any 
point of the xy-plane is found by substituting the coordinates of the 
point for r and y in P. Thus at (3, 2), P = 2. Show that all points 
at which P is positive lie outside a certain circle, and all points at 
which P is negative lie inside the same circle. With respect to this 
circle, where are the points (0, 1), (1, 2), (2, 3), (4, 5), (0, 3), (1, 4), 
(2, 2)? 

14 



194 



MATHEMATICS 



[IX, 153 



153. Translation of Axes. Given a pair of axes OX and 
Y, a curve C, and its equation in terms of the coordinates 
x = OA and y = AP. (Fig. 105.) Move the origin to the 

point 0' whose coordinates 
referred to the old axes are 
(h, k} and draw new axes 
O'X' and O'Yf parallel to the 
x old axes. The curve is not 
moved or changed but the 

Y 

-- coordinates of all its points 
are changed, and its equation 
is changed. 




4-- 



FIG. 105 



From the figure we see that 



and 



x = x' + h 

y = y' + k. 



These equations are true no matter which way nor how far the 
origin is moved if the new axes are parallel to the old ones. 
These values substituted in the old equation of the curve, 
give the new equation. Hence, to find the new equation, 
substitute in the old equation, in the place of x, the new x plus 
the abscissa of the new origin and in the place of y, the new y plus 
the ordinate of the new origin. 

EXAMPLE. Translate the origin to the point (1, 2) on the circle 

3x 2 + 3?/ 2 - 5x + 2y = 6. 
The new equation is 

3(x' + I) 2 + 3(y' - 2)' - 5(z' + 1) + 2(y' - 2) = 6, 
and this reduces to 

3x' 2 + 3y' 2 + x' - 1<V = 0. 



IX, 155] 



CONIC SECTIONS 



195 



154. Parabola. The parabola is the locus of a point which 
moves so as to be always equidistant from a fixed point F and a 
fixed line L. 

The fixed point F is called the focus. The fixed line L is 
called the directrix. 

155. Equation of the Parabola. Let F be the focus and 
RS the directrix of a parabola. (Fig. 106.) Draw FD per- 
pendicular to the directrix. The 

midpoint between D and F 
is on the parabola. Take for 
the origin, OF for the re-axis, 
and take OY parallel to the 
directrix for ?/-axis. Let the 
distance DO = OF = p. Then 
the coordinates of the focus 
are (p, o). Let P(x, y) be any 
point on the parabola. By 
definition, FP = NP; but 




FIG. 106 



and 
whence 



FP = (z - p) 2 + y 2 , 
NP = x + p, 



Squaring this, we find 
(5) 



p) 2 + 2/ 2 = x + p. 



= ipx. 



We have now proved that every point on the parabola satis- 
fies the equation (5). It follows that the parabola has no 
points on the left of the y-axis, for negative values of x cannot 
satisfy the equation (5). 

Conversely, let PI(XI, y\) be a point which satisfies (5); then 



?/i 2 = 4pzi, and (x\ p) 2 = (xi p) 2 , 



196 



MATHEMATICS 



[IX, 155 



whence, adding, we have 

(xi - p) 2 + yS = (xi + p) 2 , 
that is 

FP? = N\P?. 

Therefore PI is on the parabola. This completes the proof 
that (5) is the equation of the parabola. 

The parabola is symmetric with respect to the line through 
its focus perpendicular to its directrix. This line is called the 
axis of the parabola. The point where the parabola crosses 
its axis is called its vertex. The chord through the focus 
perpendicular to the axis of the parabola is called its latus 
rectum. Let the student show that the length of the latus 
rectum is 4p. 

The parabola y 2 = 4px crosses every horizontal line exactly 
once, and every vertical line to the right of the 7/-axis twice, 
once above and once below the z-axis. The farther the vertical 
line is to the right, the farther from the z-axis does the curve 
cut it. 

By analogy to (5) it is evident that the equations of the 
parabolas shown in Figs. 107, 108, 109 are, respectively, 




FIG. 107 




FIG. 109 



(6) t/ 2 = - 4pz, (7) x* = 4py, (8) z 2 = - 4py. 

The position of each of these curves should be related to its 
equation as follows: y z = 4px is a parabola tangent to the y-axis 
at the origin, having its focus on the x-axis to the right. The 
student should make similar statements concerning equations 
(6), (7), and (8). 



IX, 156] CONIC SECTIONS 197 

156. Vertex not at the Origin. Each of the equations 

(9) (y - k) 2 = db 4p(z - h), 

(10) (x - h)* = 4p(y - k) 

represents a parabola whose vertex is at (h, k) and whose axis is 
either horizontal (equation (9)) or vertical (equation (10)). For, 
on translating the axes to this point they reduce to one of 
the types (5), (6), (7), or (8) considered above. 
In particular, the equation 

(11) y = ax*+bx + c (fl+0) 

represents a parabola whose axis is vertical. It is concave up 
or down according as a is positive or negative, and the vertex, 
focus, and directrix can be found by completing the square of 
the terms in x and reducing it to the form (10). 

EXAMPLE 1. Locate the parabola y = 2x 2 8x + 5. Transposing, 

2x 2 - 8x = y - 5; 
dividing by 2, 

x 2 4x = \y \ ; 
adding 4, 

x 2 - 4x + 4 = \y + f ; 



Hence the vertex is the point (2, 3), and p = |. The parabola is 
concave upwards; its focus is | above the vertex, and its directrix is 
below the vertex. 

EXAMPLE 2. Examine the equation y = 2x 2 + 4x. We may 
write successively the equations 



x 2 -2x=-|y, x 2 -2x + l = 

Hence the vertex is at the point (1, 2), and p = |. The parabola is 
concave downwards, its focus is below the vertex, and its directrix 
is j above the vertex. 

Similarly, the equation x = ay 2 -\-by-\-c can be reduced to 
the type (9) by completing the square of the terms in y, and 
from this a sketch of the parabola can be made. 



198 MATHEMATICS [IX, 156 

EXERCISES 

1. Sketch each of the following parabolas, write the equation of its 
directrix, and the coordinates of its focus and vertex: 

(a) y* = Sx. (d) % 2 = 3z. (g) (x + 3) 2 = 5(3 - y). 

(b) x* = Gy. (e) 2y* = 25z. (h) x 2 = I0(y + 1). 

(c) y 2 = - 3z. (/) (y - 2) 2 = 8(s - 5). (i) (y + 4) 2 = - 6z. 

2. Sketch each of the following parabolas, and find the coordinates 
of the vertex and focus and the equations of the directrix and axis. 

(a) y 2 - 2y - 4x + 6 = 0. (b) y* + y - 6x = 0. 

(c) x 2 + 4z + 6y - 8 = 0. (d) a; 2 - x + y = 0. 

(e) 4z 2 - 12x + 3y - 2 = 0. (/) 3y 2 + 6?/ - 7x - 10 = 0. 

3. Sketch the parabolas with the following lines and points as direc- 
trices and foci, respectively; and find their equations. 

(a) x - 3 = 0, (6, - 3). (b) x = 0, (- 2, - 2). 

(c) y + 4 = 0, (- 2, 0). (d) y - 26 = 0, (0, 0). 

4. Find the parabolas with the following points as vertices and foci, 
respectively. 

(a) (0, 0), (2, 0). (6) (1, 1), (3, 1). 

(c) (- 2, - 2), (- 4, - 2). (d) (3, 2), (3, 6). 

5. Find the parabola with vertex at the origin and axis parallel to 
the x-axis, and passing through the point : 

(4,1); (2,3); (1,1); (-1,2); (2, - 4); (- 2, - 5). 

6. The cable of a suspension bridge assumes the shape of a parabola 
if the weight of the suspended roadbed (together with that of the cables) 
is uniformly distributed horizontally. Suppose the towers of a bridge 
240 ft. long are 60 ft. high and the lowest point of the cables is 20 ft. 
above the roadway. Find the vertical distances from the roadway to 
the cables at intervals of 20 ft. 

7. An arch in the form of a parabolic curve is 29 ft. across the 
bottom and the highest point is 8 ft. above the horizontal. What is 
the length of a beam placed horizontally across it, 4 ft. from the top? 

8. A parabolic reflector is 8 inches across and 8 inches deep. How 
far is the focus from the vertex? Ans. 2 in. 



IX, 157] 



CONIC SECTIONS 



199 



157. Ellipse. An ellipse is the locus of a point which moves 
so that the sum of its distances from two fixed points is constant. 

The fixed points F and F' (Fig. 110) are called the foci. Let 
the constant distance be 2a; 
this cannot be less than F'F. 
If it is just equal to F'F the 
locus is evidently the seg- 
ment F'F. Hence we assume 
that 2a > F'F. Take the 

x-axis through the foci, and 

FIG. 110 
the origin midway between 

them. Then for all positions of the moving point P, we have 







(12) 



F'P + FP = 2a. 



One position of P is a certain point A on the z-axis to the 
right of F, and by (12), 



and 



F'A + FA = 2a 
OA = $(F'A + FA) = a. 



Similarly the point A' to the left of F' such that A'O = a, is a 
point on the ellipse. The points A and A' are called the vertices. 
The segment A' A is called the major axis of the ellipse. 

Another position of P is a point B on the ?/-axis above and 
OB is denoted by b. By (12), we have 

F'B + FB = 2a, 

and since B is on the perpendicular bisector of F'F, 
F'B = FB = a. 

Similarly, the point B' below such that B'O = b, is a point on 
the ellipse. The distance B'B is called the minor axis. The 



200 MATHEMATICS [IX, 157 

intersection of the major and minor axes is called the center of 
the ellipse. 

The rectangle formed by drawing lines perpendicular to the 
major and minor axes at their extremities is called the rectangle 
on the axes. 

Let a denote the acute angle OFB. Then cos a is called the 
eccentricity of the ellipse, and is denoted by e. It is evident 
that e = OF JO A. Hence, from the right triangle OFB, we 
have 

6 2 

(13) < e < I and - = sin 2 a = 1 - e 2 . 

a 2 

Since OF = ae the coordinates of the foci F and F' are (ae, o) 
and ( ae, o), respectively. 

Then for all positions of the moving point P, by (12), we have 



(14) V(z + ae} 2 + y 2 + V(z - ae) 2 + y 2 = la. 

Transposing the second radical, squaring, and reducing, we find 



(15) V(z - ae) 2 + y 2 = FP = a - ex, 

which is the right-hand focal radius. 

Similarly, on transposing the first radical in (14), we obtain 
the equation 



(16) (a + ae) 2 + i/ = F'P = a + ex, 

which is the left-hand focal radius. Squaring either (15) or (16) 
and reducing, we find 

(17) (1 - e 2 )x 2 + y 2 = a 2 (l - e 2 ), 
whence, by (13), 

-j.2 n/2 

a* + V = L 
We have now proved that every point on the ellipse satisfies 



IX, 157] 



CONIC SECTIONS 



201 



(18). It can be proved, conversely, that every point which 
satisfies (18) is on the ellipse. Hence we may state the fol- 
lowing theorem. 

The equation of the ellipse whose semi-major axis is a, whose 
semi-minor axis is b, whose center is at the origin, and whose foci 
are on the x-axis, is 



(19) 



W 



The numbers a, b, e, are positive, a > 6, e < 1, & 2 /a 2 = 1 e 2 . 
The coordinates of the foci are (ae, o) and ( ae, 0). The focal 
distances of any point on the ellipse are a ex and a + ex, 
respectively. 

The equation shows that the curve is symmetric with respect 
to the x-axis and also with respect to the ?/-axis. It follows 
that the curve is symmetric with respect to the origin. It is 
only necessary to plot that part of the curve which lies in the 
first quadrant to determine the shape of the whole curve, which 
is as shown in Fig. 111. 





FIG. 112 



The ellipse can be drawn by the continuous motion of a pencil 
point by means of a pair of tacks set at the foci and a loop of 
string around them as shown in Fig. 112. This 1 is the best 
method of tracing an ellipse on a drawing board. It can be 
used to lay out an ellipse of any desired size on the ground. 
Let the student show that the length of the loop of string is 
2o(l + e). 



202 



MATHEMATICS 



[IX, 158 



158. Auxiliary Circle. A comparison of the equation of the 
ellipse (19) with that of the circle 



(20) 



shows that any ordinate of the ellipse is to the corresponding 
ordinate of the circle as b is to a. The 
diameter of this circle (20) is the 
major axis of the ellipse. For this 
reason, the circle (20) is called the 
major auxiliary circle, or simply the 
auxiliary circle. The points where 
any ordinate cuts the ellipse and the 
auxiliary circle are called correspond- 
ing points. 




FIG. 113 



159. Area of an Ellipse. Since the horizontal dimensions 
of the ellipse and its auxiliary circle are the same, and since 
their vertical dimensions are in the ratio b : a, we have 



(21) 



Area of ellipse 6 

Area of auxiliary circle a 



Hence, since the area of the circle is known to be ira z , the area, 
of an ellipse whose semi-axes are a and b is irab. 

160. Projection. If a circle of 
radius a be drawn on a plane making 
an angle a with the horizontal plane, 
then the vertical projection of this 
circle on the horizontal plane is an 
ellipse whose semi-major axis is a and 
whose semi-minor axis is a cos a, 
since its ordinates are to the corre- 
sponding ordinates of the circle as a cos a is to a. 




FIG. 114 



IX, 160] CONIC SECTIONS 203 

EXAMPLE 1. Reduce the equation of the ellipse 3x 2 + 4y 2 = 48 to 
standard form; find a, b, and c, the coordinates of the foci, the focal 
distances to the point (2, 3), and the area of the ellipse. 

Dividing through by 48, we find 



Then, by comparison with (19), we have o 2 = 16 and b 2 = 12, whence 
a = 4 and b = 2V3. From (13) we find e = ; hence ae = 2. It 
follows that the foci are (2, 0) and (2, 0). The right-hand focal 
distance to (2, 3) is a ex = 3 and the left-hand focal distance is 
a + ex = 5. The area is irab = 87rA/3 = 43.53 + 

EXAMPLE 2. Reduce the equation 15x 2 + 28y 2 = 12. 

Dividing by 12, we have 



_ 

4 "3 ' 



or 



Hence, by comparison with (19), we have o = VH and b = f V21. 

EXERCISES 

1. Find the semi-axes, the eccentricity, locate the foci, and find the 
focal distances to any point (x, y) on the curve; construct the rectangle 
on the axes, and sketch the curve: 

(a) 4x 2 + 9i/ 2 = 36. (b) x* + 25y* = 100. 

(c) 9x 2 + 25y 2 = 225. (d) 9x 2 + 16?/ 2 = 144. 

(e) x 2 + 2y 2 = 4. (/) 6x 2 + 9?/ 2 = 20. 

2. In each of the following cases find the values of a, b, e, if they 
are not given. Locate the foci, and write the equation of the ellipse. 
Construct the rectangle on the axes and sketch the curve. 



(a) o = 10, b = 6. (g) b = 2^, e = 1/2. 

(6) a = 10, 6=8. (h) a = 5, e = 2/3. 

(c) o = 5, b = 3. (t) a = 6, e = 0. 

(d) a = 13, e = 12/13. (j) b = 8, e = 3/5. 

(e) a = 7, e = 5/7. (Jfc) 6 = 12, e = 5/13. 
(/) a = 10, e = 3/5. (06=2, e = 1/3. 



204 MATHEMATICS [IX, 160 

3. Find the area of each of the ellipses in Ex. 1. 

4. Show that any oblique plane section of a circular cylinder is an 
ellipse. 

5. Find the semi-axes and the area of the section formed by cutting 
off a log 14 inches in diameter by a plane making an angle of 60 with 
its length. 

6. Design a flashing (sheet metal collar) for a four inch soil pipe 
projecting vertically through a roof whose pitch is 1/3. 

7. A circular window in the south wall of a building is 4 ft. in diam- 
eter. Light from the sun passes through the window and falls on the 
floor. Find the area of the bright spot at noon, when the angle of 
elevation of the sun is (a) 60, (6) 45, (c) 30. 

8. An ellipse whose semi-axes are 10 and 9 is in a horizontal position. 
Through what angle must it be rotated about its minor axis hi order 
that its projection on a horizontal plane shall be a circle. 

Ans. 25 50'. 

161. Hyperbola. A hyperbola is the locus of a point which 
moves so that the difference of its distances from two fixed points is 
constant. 

The fixed points are called the foci. Other terms are defined 
in a manner analogous to those for the ellipse. 

By an analysis similar to that given in 157 for the ellipse, 
it can be shown that the equation of the hyperbola whose semi- 
transverse axis is a, whose semi-conjugate axis is 6, whose 
center is at the origin and whose foci are on the cc-axis, is 



a* V 

The curve consists of two branches and is symmetric with 
respect to both axes and with respect to the origin, as shown in 
Fig. 115. The quantities a, b, and e (= sec a), are positive, 
a = b, e > 1, 6 2 /a 2 = e 2 1; the coordinates of the foci are 
(ae, o) and ( ae, o) ; the focal distances to a point on the 
right branch are ex a and ex + a, and to a point on the 
left branch, the negatives of these. The diagonals OC and OC", 



IX, 162] 



CONIC SECTIONS 



205 



of the rectangle on the axes are called the asymptotes of the 
hyperbola, and the curve approaches nearer and nearer to 




FIG. 115 

them as the moving point recedes from the vertices, 
equations of the asymptotes are 



The 



(23) 



and 



y- --*. 



162. Rectangular or Equilateral Hyperbola. If the semi- 
axes of a hyperbola are equal, b = a, its equation reduces to 
the form 
(24) x 2 - y 2 = a 2 . 

The rectangle on the axes is a square, the eccentricity is sec 45 
= V2, and the asymptotes are the two perpendicular lines 
y = x and y = x. This is called a rectangular or equilateral 
hyperbola. It plays a role among hyperbolas analogous to that 
played by the circle among ellipses. 

The product of the distances of any point on an equilateral 
hyperbola to its asymptotes is constant. For the distance to 
the asymptote y = x is (x y) cos 45, and the distance to 
the asymptote y = . x is (x + y) cos 45; hence the product 
of these distances is a 2 cos 2 45 = a 2 . 



206 



MATHEMATICS 



[IX, 162 



It follows from this property that if the asymptotes of an 
equilateral hyperbola be taken for coordinate axes the equation 
of the curve will be 
(25) xy = a positive constant, 

when the branches are in the first and 
third quadrants, as shown in Fig. 116; 
and the equation will be 

(26) xy = a negative constant, 

when the branches of the curve are in 
the second and fourth quadrants. 




FIG. 116 



EXAMPLE. Reduce the equation of the hyperbola lQx z 9?/ 2 = 144 
to standard form. 

Dividing by 144, we find 

i 2 _ yL = 

9 16 

Hence, by comparison with (22), we have a = 3, b = 4. From b?/a 2 
= e 2 1 we find e = 5/3. 

It follows that the coordinates of the foci are (5, 0) and ( 5, 0). 
The focal distances to a point on the right branch are 

ex a = l(5x 9) and ex + a = |(5a; + 9). 

For example to the point (6, 4V3) they are 7 and 13. The equations 
of the asymptotes are 



\ 



rf 



A' 



y = fx and y = fz. 

To sketch the curve, lay off OA = 3, OB 
= 4, Fig. 117, construct the rectangle on the 
axes, locate the foci by circumscribing a circle 
about this rectangle. Sketch in the curve 
free hand in four parts beginning each time 
at a vertex, using the asymptotes as guides, 
the curve approaching them in distance and direction. 



FIG. 117 



IX, 162] CONIC SECTIONS 207 

EXERCISES 

1. Find the semi-axes, the eccentricity, the coordinates of the foci, 
the focal distances to the point indicated, the equations of the asymp- 
totes; construct the rectangle on the axes and the asymptotes, and 
sketch each of the following hyperbolas. 

(a) 4x 2 - 9?/ 2 = 36, (Vl3, 4/3). 

(6) 4x 2 -7/ 2 -8, (-3/2, 1). 

(c) 3x 2 -i/ 2 = 9, (3, -3>/2). 

(d) 3x 2 - 4y 2 = 1, (- V?; V5). 

(e) 144x 2 - 25y 2 = 3600, (10, - 12 V3). 
(/) 9x 2 - 16y 2 = 576, (12, 31/5). 

(g) 25x 2 - y 2 = 100, (- V29, 25). 

(h) 225x 2 - 647/ 2 = 14400, (17, 28 J). 

(i) x 2 - y 2 = 9, (- 5, 4). 

0') x 2 - ?/ 2 = 400, (101, 99). 

2. Plot on the same axes the curves xy = c, for c = 1, 4, 6, 1, 
- 4, - 6. 

3. Find the equation of the locus of a point which moves so that 
the difference of its distances from the two points (1,1) and ( 1, 1) 
is constant and equal to 2. 

4. Find the locus as in Ex. 3, when the foci are (a, a) and ( a, a) 
and the constant is 2a. 

5. Find the locus of a point where two sounds emitted simultaneously 
at intervals one second apart at two points 2,000 ft. apart are heard at 
the same time, the speed of sound in air being 1,090 ft. per second. 

6. On a level plain the crack of a rifle and the thud of the bullet 
on the target are heard at the same instant. The hearer must be on a 
certain curve; find its equation. (Take the origin midway between the 
marksman and the target.) 

7. By translation of the axes ( 153) find the equation of the ellipse 
.(a) whose foci are ( 4, 2) and (0, 2), and whose eccentricity is 5. 

Ans. 3x 2 + 4y 2 + 12x - IQy = 20. 
(6) whose vertices are (2, 2) and (4, 2), and which passes through 

the point (1,4). Ans. 4x 2 + 9y 2 - 8x - 36?/ + 4 = 0. 

(c) whose semi-axes are 5 and 3, whose right-hand focus is at (4, 4), 
and whose left-hand vertex at ( 5, 4). 

Ans. 9x 2 + 25r/ 2 + 200y + 175 = 0. 



208 MATHEMATICS [IX, 162 

[HINT. Start with the equation of the same curve when its center is 
at the origin.] 

8. By the method of Ex. 7, find the equation of the hyperbola 
(a) whose vertices are (2,2) and (4, 2), and whose eccentricity is 5/3. 

Ans. 16x 2 - 9?/ 2 - 32x + 36y = 164. 

(6) whose semiminor axis is 15, whose left-hand vertex is at ( 15, 3) 
and whose right-hand focus is at (10, 3). 

Ans. 225x 2 - 64y 2 + 3150x + 384y = 3951. 

(c) which passes through the origin and whose asymptotes are the 
lines x = 2 and y = 1. Ans. xy = x + y. 

163. Intersection of Loci. If a point lies on a curve, its 
coordinates must satisfy the equation of that curve. Con- 
versely, any pair of values of x and y which satisfy an equation 
determines a point on the locus of that equation. If the same 
pair of values of x and y satisfies two equations, it locates a 
point which is common to the two curves, i. e., a point of inter- 
section. Hence, to find the points of intersection of two curves, 
solve their equations simultaneously to find all their common 
solutions. 

EXAMPLE 1. Find the intersections of the line 3x y = 5 and the 
ellipse 4x 2 + 9?/ 2 = 25. 

Solving the first equation for y = 3x 5, substituting this in the 
second and reducing, we have 

17x 2 - 54x + 40 = 0. 
We can factor this quadratic by inspection : 

(17x - 20) (x - 2) = 0, 
whence 

xi = 20/17 and x 2 = 2. 

Substituting these values in the equation 3x y 5, gives 7/1 = (25/17) 
j/s =1. Therefore the points of intersection are (20/17, 25/17), and 
(2, 1). 

Let the student plot the curves on the same axes and verify these 
results. 



IX, 163] CONIC SECTIONS 209 

EXAMPLE 2. Where does the parabola 

3y = x 2 - 5x + 12 
intersect the ellipse 

4x 2 + 3y 2 = 48? 

Substituting the value of y from the first equation in the second 
and reducing, we get 

x 4 - lOx 3 + 61x 2 - 120x = 0. 
Factoring this equation, we have 

x(x -3)(x 2 - 7x + 40) = 

and we see by inspection that Xi = and x 2 = 3 are roots. The quad- 
ratic x 2 7x + 40 has imaginary roots. 

Substituting these values of x in the first given equation we find 
?/i=4 and 7/ 2 = 2. Hence the points of intersection are (0, 4) and (3, 2)- 

EXERCISES 

Find the points of intersection of the following pairs of curves. 

1. x 2 + Qxy + 9j/ 2 =4, 4x + 3y = 12. 

Ans. (14/3, - 20/9), (10/3, - 4/9). 

2. x 2 - if =0, 3x - 2y = 4. Ans. (4, 4), (4/5, - 4/5). 

3. y 2 +x=0, 2y + x = 0. Ans. (0, 0), (- 4, 2). 

4. x 2 5y = 0, x y = 1. 

Ans. x = i(5 V5), y = |(3 V5). 

5. 2x + 3y = 5, 4x 2 + 9(/ 2 + 16x - 18^ -11=0. 

Ans. (1,1), (-2,3). 

6. x - y + 1 = 0, (x + 2) 2 - 4y = 0. Ans. (0, 1). 

7. y - 2x = 0, x 2 + y* - x + 3y = 0. 

Am. (0,0), (- 1, - 2). 

8. x - 2y + 4 = 0, 5x 2 - 4y 2 + 20 = 0. Ans. (I, 2). 

9. y = 2x - 3, 4y 2 = (x + 3)(2x - 3). 

Ans. (3/2, 0), (1, - 1). 

10. 4i/ 2 = x 2 (x + 1), y 2 = x(x + I) 2 . Ans. (0, 0), (- 1, 0). 

11. 2x 2 - 3i/ 2 = - 58, 3x 2 + t/ 2 = 111. 

Ans. (5, 6), (- 5, 6), (5, - 6), (-5, - 6) 

12. x 2 = 4ay, y = 8n 3 /(x 2 + 4o J ). Ans. ( 2a, a). 
ITu x 2 + y 2 = 2, x 2 + 7/ 2 - 6x - 6y + 10 = 0. Ans. (1, 1). 

15 



210 



MATHEMATICS 



[IX, 154 



164. Straight Line and Conic. The equations of the circle, 
parabola, ellipse, and hyperbola, are all of the second degree in 
x and y. Conversely, it can be shown that every such equa- 
tion represents a conic section, if it represents any curve at all. 
Given a straight line and a circle we know that one of three 
things will happen, 1) there may be two intersections, 2) there 
may be no intersection, or 3) there may be only one point in 
common and then the line is a tangent. The same three cases 
occur with the intersections of a straight line with any conic 
section.* 

When we solve simultaneously the equation of a straight line 
with the equation of a conic, we may begin by substituting the 
value of y from the first equation in the second. The result is 
a quadratic equation in x. This quadratic equation ( 32, 33), 
(27) Ax 2 + Ex + C = 

will have 1) two real roots when B 2 4 AC > 0, or 2) no real 
roots when B z 4 AC < 0, or 3) one real root when B 2 4AC 

= 0. These algebraic cases cor- 
respond exactly to the geometric 
cases enumerated above. 

EXAMPLE. Of the three parallel 
lines 8z - 9y = 20, 8z - Qy = 30, 
and 8x 9y = 25, the first cuts 
the ellipse 4x 2 + 9?/ 2 = 25 in two 
points (5/2, 0) and (7/10, - 8/5), 
the second does not intersect it at 
all, and the third intersects it at 
(2, 1) only, i. e. it is tangent at 
that point. 
The resulting quadratics are, respectively, 

20x 2 - 64x + 35 = 0, B 2 - 4AC = 1296, 

. 20x 2 - 96x + 135 = 0, B 2 - 4AC = - 1584, 

x 2 - 4x + 4 = 0, B* - 4AC = 0. 

* There is one exception to this rule: any line parallel to the axis of a parabola h&t> 
one and only one point in common with the curve, but no such line is a tangent to 
the parabola. 




IX, 165] CONIC SECTIONS 211 



1. Show that one of the three lines 4x + 25 = Wy, 4x + 27 = lOy, 
4x + 21 = lOy, intersects the parabola y 2 = 4x in two points, another 
is tangent, and the third does not intersect it at all. 

2. Determine whether the following given lines are tangent, secant, 
or do not meet, the corresponding given conic. 

(a) x + y + 1 = 0, x 2 = 4y. 

(6) x - 2y + 20 = 0, x 2 + y* = 16. 

(c) 2x + 3y =8, y 2 = 4x. 

(d) x + 2y = 5, x 2 + y 2 = x + 2y. 

(e) 2x = 3y, 4x 2 - 3y 2 + 8x = 16. 
(/)x + 7/ = 8, 4x 2 +?/ 2 = 16x. 

3. Find the points in which the circle x 2 + y 2 = 45 is cut by the lines 
(a) 2x - y = 15, (6) 2x - y = 0, (c) 2x - y = - 15. 

Ans. (a) (6, - 3), (6) (3, 6), (- 3, - 6), (c) (- 6, 3). 

4. Find the points in which the circle x 2 + y z 6x Qy + 10 = 
is cut by the lines (a) x + y = 2, (6) x + y = 6, (c) x + y = 10, 

(d) x - y = 0. 

Ans. (a) (1, 1), (6) (1, 5), (5, 1), (c) (5, 5), (d) (1, 1), (5, 5). 

5. Find the points in which the parabola 3y = 2x 2 8x + 6 is cut 
by the lines (a) 4x + 3y = 4, (6) 4x + 3y = 6, (c) 4x + 3y = 12. 

Ans. (a) (1, 0), (6) (0, 2), (2, - f), (c) (3, 0), (- 1, 5|). 

6. Find the points in which the ellipse 3x 2 + 4t/ 2 = 48 is cut by the 
lines (a) x + 2y = 0, (6) x + 2y = 4, (c) x + 2y = 8, (d) x + 2y = -4, 

(e) x + 2y = -_8. 

Ans. (a) (2\3, - V3), (-2^3, V3), (b) (4, 0), (- 2, 3), (c) (2, 3), 
(rf) (-4,0), (2, -3), (e) (-2, -3). 

165. Tangent and Normal. Focal Properties. The 

equation of the tangent to a conic can be found by the principles 
of 164 if the slope of the tangent is known, or if the coordinates 
of one point on the tangent are known. This given point may 
be the point of contact or some other point through which the 
tangent is to pass. 

The perpendicular to the tangent at the point of contact is 
called the normal to the conic at that point. When the slope 



212 MATHEMATICS [IX, 165 

of the tangent is known or can be found, the equation of the 
normal can be written by the principles of 59 and 61. 

EXAMPLE 1. Find the equation of the tangent to the parabola 
y z = 24x which is perpendicular to the line x + 3y + 1 =0. 

By (13) 61, the slope of the required tangent is 3, and by (11) 59, 
y = 3x + b is parallel to it no matter what value b has. Proceeding 
to find the points where this line intersects the parabola we are led to 
the quadratic equation, 

9z 2 + 6(6 - 4)z + 6 2 = 0. 

By 32, this quadratic will have only one root and the line will be 
tangent to the parabola, if 

36(6 - 4) 2 - 366 2 = 0. 

This gives 6=2; whence, the equation of the required tangent is 
y = 3x + 2. 

EXAMPLE 2. Find the equation of the tangent and of the normal 
to the ellipse 3z 2 + 4y 2 = 48 at the point (2, 3). 

We' first verify that the given point is in fact on the ellipse. Then 
by (10) 59, y 3 = m (x 2) is the equation of a line through (2, 3) 
no matter what value m has. Solving this simultaneously with the 
equation of the ellipse we get the quadratic equation, 

(4m 2 + 3)z 2 + 8m(3 - 2m)x + 4(4m 2 - 12m - 3) = 0. 

This equation will have only one root and the line will be tangent to 
the ellipse if (32), 

64m 2 (3 - 2m) 2 - 16 (4m 2 + 3) (4m 2 - 12m - 3) =0, 

that is if, 

4m 2 + 4m + 1 =0, 

whence m = \ and the equation of the required tangent is 

y - 3 = - \(x - 2), or x + 2y = 8, 

and the equation of the normal (whose slope by (13) 61 is 2) is 
y - 3 = 2(x - 2), or 2x - y = 1. 

The normal at any point .P on a parabola bisects the angle 
between the focal radius FP, and the line through P parallel 
to the axis of the curve. 



IX, 165] 



CONIC SECTIONS 



213 



We learn in Physics that light is reflected by a mirror in such 
a way that the angle of incidence is equal to the angle of reflection. 
Hence, a ray of light emanating from a source at the focus and 





FIG. 119 



Fia. 120 



striking the parabola at any point, will be reflected parallel to 
the axis. This is the principle of parabolic reflectors which 
are extensively used for head lights. It is easily seen that if 
the light be moved slightly beyond the focus, the reflected rays 
will tend to illuminate the axis. 

The normal at any point of an ellipse bisects the angle between 
the focal radii to that point, Fig. 120. It follows that rays 
of light, or sound, emanating from one focus F, will after re- 
flection by the ellipse, converge at the other focus F". Hence 
the name focus. This is the principle of whispering galleries. 

EXERCISES. 

1. Find the equations of the tangents and normals to the following 
curves at the points indicated: 

(a) y 2 = 8x, (2, 4), (6) x 2 - y 2 = 64, (10, 6), 

(c) x 2 + 3J/ 2 = 21, (3, - 2), (d) 28?/ 2 = 27x, (2J, 1J). 

2. Find the equations of the two tangents which can be drawn to 
the parabola y* + 8x = from the point (2, 1) and verify that they 
are perpendicular. 

3. Find the equations of the tangent and normal to the circle x 2 + y* 
- 6x - Gy + 10 = at the point (1, 1). 

Ans. x + y = 2,x y = 0. 



214 MATHEMATICS [IX, 165 

4. Find the equations of the tangents from the point (9, 3) to the 
circle x 2 + y 2 45. Ans. 2x y = 15, x + 2y = 15. 

5. Find the equations of the tangent and normal to the circle x z + y 2 
= 6z + 2y at the point (2, 4). 

Ans. x - 3?/ + 10 = 0, 3x + y = 10. 

6. Find the equations of the tangent and normal to the hyperbola 
xy = 6 at the' point (2, 3). Ans. 3x + 2y = 12, 2x - 3y + 5 = 0. 

7. Find the tangent to the parabola y 2 = 12x which makes an 
angle of 60 with the z-axis. 

8. Find the tangent to the parabola y 2 = 6x which makes an angle 
of 45 with the re-axis. 

9. Find the equations of the tangents to the circle 

(a) x 2 + y 2 = 4 parallel to 2x + 3y + 1 =0, 

(b) x 2 + y 2 = 16 parallel to 3x - 2y + 2 = 0. 

10. Find the tangents to the ellipse Qx 2 + 16?/ 2 = 144 which make 
an angle of 30 with the x-axis. 

11. Find the equations of the tangents to the following conies which 
satisfy the condition indicated. 

(a) y 2 = 4x, slope = 1/2. (/) x 2 + y 2 = 25, at (4, - 3). 

(6) x 2 + y 2 = 16, slope = - 4/3. (g) x 2 + 4?/ 2 = 8, at (- 2, 1). 

(c) 9z 2 + l&y 2 = 144, slope = - 1/4. (h) x 2 - y 2 = 16, at (- 5, 3). 

(d) x 2 = 4y, passing through (0, - 1). (i) 2y 2 - x 2 = 4, at (2, - 2). 

(e) x 2 = Sy, passing through (0, 2). 0') 2/ 2 = % x > at ( 2 16 )- 

12. Determine the condition for tangency of the following pairs of 
curves. 

(a) x 2 y 2 = a 2 , y kx. Ans. k = 1. 

(6) x 2 + y 2 = r 2 , 4y - 3x = 4Je. Ans. 16/b 2 = 25r 2 . 

(c) 4x 2 + y 2 - 4x - 8 = 0, y = 2x + k. Ans. k 2 + 2k - 17 = 0. 

(d) xy + x - 6 = 0, x = ky + 5. Ans. k 2 + 14fc + 25 = 0. 

13. A parabolic reflector is 12 inches across and 8 inches deep. 
Where is the focus? 

14. The ground plan of an auditorium is elliptic in shape. The 
extreme length is 2,725 ft. and the width is 2,180 ft. By what path 
will a sound made at one focus arrive first at the other focus, i. e., directly 
or by reflection from the walls? How much sooner if sound travels 
1,090 ft. per second? 



IX, 166] 



CONIC SECTIONS 



215 




166. Intersection of Conies. Simultaneous Quadratics. 

Two conies intersect, in general, in four points. Since their 
equations are of the second degree in x and y, this corresponds 
to the fact that two quadratics in x and y have, in general, four 
solutions. In some cases these solutions are not all real, or 
there may be less than four so that 
the conies represented intersect in 
less than four points. 

As shown in Fig. 121a, the hyper- 
bola x 2 y- = 5 intersects the ellipse 
x 2 + 4?/ 2 = 25 in the four points (3, 2), 
(- 3, 2), (-3, - 2), and (3, - 2). 
The parabola 4x 2 = Qy cuts the same 
ellipse only in (3, 2) and ( 3, 2), as 
shown in Fig. 1216. 

FIG. 121 

Certain types of these equations 

can be solved by elementary methods. The most important 
cases will now be explained. 

CASE I. When all the terms (except the constant terms) are 
of the second degree in x and y. 

Eliminate the constant terms and factor the result into two 
linear factors. 

X 2 - 7/ 2 = 5, 

x 2 + 4?/ 2 = 25. 
Multiplying the first equation by 5 and subtracting, we -have 

4x 2 - 9y 2 = 0, 
whence 

(2x - 3y)(2x + 3y) = 0. 

Now solving simultaneously the two pairs of equations 

\ 2x 3y = 0. \ 2x + By = 0. 

We find that the solutions of (a) are (3, 2) and ( 3, 2) ; and those 
of (6) are (3, 2) and ( 3, 2). It is easy to verify that these are all 
solutions of the given equations by actual substitution. 



EXAMPLE 1 



{ 



216 MATHEMATICS [IX, 166 

f x 2 + 3xy = 28, 
EXAMPLE 2. < ' 

t 4?/ 2 + xy = 8. 

Eliminating the absolute terms, we have 

2 X 2 - X y - 282/2 = 0, 
whence 

(2x + 7y)(x - 4y) = 0. 



This gives the two pairs of simultaneous equations 

(\ S 4 2/ 2 + W = 8 ' / 4 ^ 2 + sy = 8, 
W I 2x + 7y = 0, W 1 x - 4y = 0. 

The solutions are therefore (14, - 4), (- 14, 4), (4, 1), (- 4, - 1). 
Verify each of these by actual substitution. 

SPECIAL METHOD, CASE I. If there is no term in xy the 
equations can be solved as linear equations considering z 2 
and i/ 2 as the unknowns. 



r x 2 

EXAMPLE. ! 

t x 2 



- y 1 = 5, 

+ 42/ 2 = 25. 

Eliminate x 2 and solve for y 2 . This gives y 2 = 4, whence y = 2. 
Then eliminate y 2 and solve for x 2 . This gives x 2 = 9, whence x = 3. 
Verify that (3, 2), (- 3, 2), (-3, - 2), (3, - 2), are all solutions of 
the given equations. 

CASE II. When the equations are symmetric in x and y; 
i. e., when the interchange of x and y leaves the equations 
unchanged. 

/ 2 _i_ q.2 "1 O 

EXAMPLE. -\ 

L xy = 6. 

Substituting s + t = x and s t = y in the given equations, we find 
2s 2 + 2> = 13, 

S 2 _ ft - Q f 

Solving these equations, we have 

s = 5/2, t = 1/2. 

Hence the values of x and y are x = 3 or 2, y = 3, or 2. 
Testing these values in the given equations we verify that (3, 2), (2, 3) 
(2, 3), (3, 2), are solutions. 



IX, 166] CONIC SECTIONS 217 



EXERCISES 

Solve the following pairs of simultaneous equations. 

J 4x 2 + 4xy - y 2 = 7x - y, | x 2 - y 2 + 16 = 0, 

i 4x + 3y = 1. 1 (x + I) 2 = (y + I) 2 . 

f 3x 2 + 4y 2 = 48, J 5x 2 + 7y 2 = 225, 

1 y 2 = 3(1 - x). \ 2x + 3y = 9. 

f 4x 2 + 3xy = 10, f a? + y 2 = 153> 

I 3y 2 + 4xy = 20. 1 xy = 36. 

J 2x 2 + 2xt/ + 5y 2 = 40, f x 2 + y 2 - x - y = 204, 

1 x 2 - 

f 2x 2 + xy + 37/ 2 = 12, x 2 - 2y* + 1 = 0, 



y 2 + 2x - 2y = 0. xy + x + y = 129. 



, 
1 2x + y = 0. 



y = 0. 2x 2 - 3y 2 - 23 = 0. 

3x 2 + 4?/ 2 = 48, j 4x 2 + 6xi/ + 4y 2 = 46, 

\ X 2 + ^ = 34. 

x 2 + 2xy = 407, f x 2 + 2?/ 2 = 123, 

7/ 2 + 2xy = 455. 1 y 2 + 2x 2 = 99. 



, , 

= 2x + 5y. 1 2x + y = 3. 

f x(3x + y) = j/(r/ + 3), f 3x(x - 4) = y - 5, 

1 (3x - y)(x - 2y + 3) = 0. 1 2x + y = 30. 

2 + 4j/ 2 = 48, 



10 \ 20 

V + y 2 = 58 U + 3(x + l) =0. 

f x 2 + xy + y 2 = 7, f 2x(2x - 3) = 184, 

1 x 2 - xy + y 2 = 19. 1 9y(2x + y) = - 135. 

f x 2 - 3xy + y 2 = 1, . f x 2 + xy + y 2 = 7, 

1 (x+y+2)(2x- 2/ + l) =0, 1 y 2 - x 2 = 5. 

' x 2 + 3xy + y 2 - 4x - 2y - 1 = 0, 

0O 



15x + 4y - 1 = 0. 
26 ' I x + 2xy + y - 17 = 0. 



CHAPTER X 
VARIATION 

167. Function and Variables. One of the most common 
scientific problems is to investigate the causes or effects of 
certain changes. The change or variation of one quantity in 
the problem is produced or caused by changes in other variable 
quantities and is said to depend upon, or be a function of these 
variables. Thus the growth of a plant depends on the amount 
of certain constituents in the soil, upon the temperature and 
humidity of the soil and of the atmosphere, upon the intensity 
of the light, and doubtless upon several other variables. The 
volume of gas contained in an elastic bag depends on the pressure 
and the temperature. The circumference of a circle depends 
only on the radius. 

To study the effect of any one variable upon a function of 
two or more variables, we try to arrange conditions so that 
all the other variables of the problem shall remain constant, 
while this one varies. Thus we keep the temperature of a gas 
constant to find the effect on the volume of a change of the 
pressure. To study the effect of carbonate of lime on the 
growth of alfalfa, we arrange a series of plats of soil so that 
they shall have all the other constituents the same, and all 
be subject to the same conditions of light, heat, and moisture, 
but differ from plat to plat by known amounts of pulverized 
limestone. 

The precise form of the relation between a function and its 
variables is often very complicated and difficult or impossible 
to obtain. Often, the best that can be done is to record the 
results of experiments and to study these records to deduce 

218 



X, 170] VARIATION 219 

general effects. Such results are called empirical. This is 
especially true of the so-called applications of science to the 
processes of nature. 

168. Direct Variation. One of the simplest relations that 
can exist between two variables is called direct variation. 
When the ratio of two variables is constant, each is said to 
vary directly as the other. 

The statement that y varies directly as x or simply y varies 
as x, is written 



which means that the ratio y/x is constant and implies the 
equation 

y = kx, 

where k is called the constant of variation. 

The circumference of a circle varies as the radius; i. e., C c r, 
or C = kr. The constant of variation is known to be k = 2ir 
6f , approximately. 

169. Inverse Variation. When the product of two variables 
is constant, each is said to vary inversely as the other. If y 
varies inversely as x, then 

xy = k, or y = k I - ) , 

\ * / 

whence y varies directly as 1/x, the reciprocal of x. 

EXAMPLE. The volume v, of a gas kept at constant temperature, 
varies inversely as the pressure p; i. e. 

k k 

pv = k, or v = - , or p = - . 
p v 

170. Joint Variation. When a function z depends upon two 
variables x and y, in such a manner that z varies as the product 
xy, i. e., z = kxy, then z is said to vary jointly as x and y. Thus, 
the area of a rectangle varies jointly as the length and the 



220 



MATHEMATICS 



[IX, 170 



breadth. This definition may be extended to functions of three 
or more variables. A function /, depending upon several vari- 
ables x, y, - , z, is said to vary jointly as x and y, , and z, 
when it varies as their product, i. e., / = kx-y - - z. Thus, 
simple interest varies jointly as the principal, and the rate, 
and the time. 

It is evident that if one variable z depends on two other 
variables x and y, and if z varies as x when y is constant, and z 
varies as y when x is constant, then z varies jointly as x and y 
when x and y vary simultaneously. Thus, the area of a triangle 
varies as the altitude when the base is constant and varies as 
the base when the altitude is constant; therefore the area varies 
jointly as the base and the altitude. 

This principle is readily extended to functions of three or 
more variables. Thus, simple interest varies as the principal 
when rate and time are constant, as the rate when principal 
and time are constant, and as the time when principal and rate 
are constant; therefore simple interest varies jointly as the prin- 
cipal, the rate, and the time. 

171. Graphic Representation. When y varies directly as 
x, the graph of the relation, y = kx, which connects them is 



1 






































































































X 


x* 






























X 


X 






























S 


X 






























^ 


^ 






























X 


x 




























f 


n-"* 1 


^ 






























^ 


? 




























o 


f* 


x 






























.Y 


^ 




1 


2 


? 


J 


5 


























1 


1 



















FIG. 122 

a straight line through the origin whose slope is k. The position 
of this line is fixed and the value of k can be determined if we 



X, 171] 



VARIATION 



221 



know one other point on the line, i. e., one pair of simultaneous 
values of x and y; and values of y corresponding to any given 
values of x, can be read directly from the graph. Then k is the 
difference of two values of y divided by the difference of the 
corresponding values of x ( 58). 

EXAMPLE. Given that y varies as x and that y = 1 when x = 2. 

Plotting the point (2, 1) and drawing the line OP we have the graph 
of the relation between x and y. From this we read off y = | when 
x = 1, y = 3i when x = 6|, etc. Fig. 122. 

When y varies inversely as x, the graph of their relation 
xy = k is a rectangular hyperbola asymptotic to the x and y 
axes. Here again one point is sufficient to determine k and fix 
the curve. 

EXAMPLE. Given that volume v, varies inversely as pressure p, 
and that v = 12 when p = 3. 

Then pv k, 3-12 = k, pv = 36. The graph of this is shown in 



10 



10 



FIG. 123 

Fig. 123 for positive values of p and v. From this we can read off 
t; = 6 when p = 6, v = 4 when p = 9, etc.* 

* When z varies jointly as x and y, the graph of the relation z = kxy, in three 
dimensions, is a surface called a hyperbolic paraboloid with which the student is not 
yet familiar. 



24 




222 MATHEMATICS [X, 172 

172. Determination of the Constant. By substituting in 
an equation of variation a set of simultaneous values of the 
variables, the constant of variation can be determined. 

EXAMPLE 1. Given, y varies as x and y = 8 when x = 10. We 
may write y = kx, as in 168. Substituting x = 10 and y = 8, we 
find 8 = ft- 10. From this equation, we can find k by dividing both 
sides by 10. This gives k = 4/5. Hence we have y = (4/5)x. 

From this equation, the value of y corresponding to any given 
value of x can be found. Thus, y = If when x = 2. 

EXAMPLE 2. A light is 24 inches above the cen- 
ter of a table. The illumination I at any point P of 
the surface of the table varies directly as the cosine 
of the angle of incidence, i, of the ray LP, and also 
p ^24 inversely as the square of the distance LP = x to the 

light. If the illumination at C is 10, what is it at 
any point P of a circle of radius 18 inches about C? 
SOLUTION. The illumination 7 at any point is 

T _ , cos i 
: ^>?~' 

but x = 24 sec i and therefore 

i* 

-r A/ . . 

7 = m co#t. 

Since 7 = 10 when i = 0, k = 5760, and hence 

7 = 10 cos' i. 
Now when CP = 18, cos i = 4/5, and 7 at P is equal to 5.12 

EXERCISES 

1. Write equations equivalent to each of the following statements; 
determine the constant of variation and construct the graph. 

(a) y varies as x; y = 7 when x = 3. 

(6) y is proportional to x; y = 3 when x = 2|. 

(c) y varies inversely as x; y = 1J when x = 1-|. 

(d) v varies inversely as p] v =3 when p 2. 

2. Write equations equivalent to each of the following statements 
and find the value asked for in each case. 



X, 172] VARIATION 223 

(a) y varies as x 2 ; y = 81 when x = 3 ; find y when x = 51. 

(b) y varies as sin x ; y 2 when x = 30 ; find y when x = 150. 

(c) u varies inversely as v ; u = 8 when v = 2 ; find u when v = 6. 

(d) z varies jointly as x and y;z = Q when x = 2, y = 7 ; find 2 when 

x = 4, y = 6. 

(e) y varies directly as r and inversely as s ; y = 16 when r = 10, s = 8 ; 

find y when r = 7, s = 12. 
(/) M varies jointly as x, and y 2 , and z" 1 ; tt = 6 when x = 2, y = 3, 

2 = 4; find w when x = 10, ?/ = 15, 2 = 25. 
(0) 2 varies directly as x and inversely as y 2 ; z = 2 when x = 17, y = 3 ; 

find x when 2 = 6, y = 4. 

3. Express each of the following by means of an equation. 

(a) The volume of a cone varies directly as the height when the radius 

of the base is constant. 

(b) The volume of a cone varies directly as the square of the radius of 

the base when the height is constant. 

(c) The number of calories of heat produced when a moving body is 

stopped varies jointly as the mass and the square of the velocity. 

(d) The squares of the periods of the planets vary directly as the cubes 

of their mean distances from the sun. 

4. With the statement of Ex. 3 (c) find the heat generated by a mass 
of 8 kilograms striking the sun with a velocity of 500 miles per second 
if a body weighing one kilogram and moving with a velocity of 380 
miles per second on striking the sun produces 45,000,000 calories of heat. 

5. The simple interest due on P dollars varies jointly as the amount 
P, the rate, and the time. If $1000 yields $30 interest in six months 
find the interest on $1200 for eight months at 7%. 

6. The amount of heat received by a given planet varies inversely 
as the square of its distance from the sun and directly as the square of its 
radius. 

(a) What is the effect of doubling the distance? 

(b) Mercury has a diameter of 3000 miles and is 36 million miles 
from the Sun. The Earth has a diameter of 8000 miles and is 93 million 
miles from the Sun. Compare the amounts of heat they receive. 

7. With the statement in Ex. 3(d), taking the distance of the Earth 
from the Sun as the unit and the period of the Earth as the unit of time, 



224 MATHEMATICS [X, 172 

find the period of Neptune whose distance from the Sun is known to be 
30 units. Ans. 165 yrs. 

8. The amount of heat received on a surface of given size varies in- 
versely as the distance from the source. One body is twice as far as 
another from the source. Compare the amounts of heat received. 

9. The resistance offered to a rifle bullet varies directly as the square 
of the velocity. Discuss the effect of doubling the velocity. 

10. The maximum load P that a rectangular beam supported at one 
end will hold without breaking varies directly as the breadth, the square 
of the depth and inversely as the length. A beam 4" X 2" X 10' 
supports 300 pounds. What load will the same beam support when 
placed on edge? 

11. The deflection y in a rectangular beam supported at the ends and 
loaded in the middle varies directly as the cube of the length, inversely 
as the breadth, and inversely as the cube of the depth. A beam 6 inches 
wide, 8 inches deep, 15 feet long, supporting 1000 Ibs., has a deflection 
of \ inch at the middle. Find the deflection in a beam 4 inches wide, 
4 inches deep, 10 feet long, supporting 800 Ibs. 

12. With the data of Ex. 10, find the load which a beam 4 inches wide, 
6 inches deep, and 16 feet long will support. 

13. With the data of Ex. 10, find the longest beam 4 inches wide and 
4 inches deep which will support 100 Ibs. 

14. With the data of Ex. 10, find the least depth of a beam 12 feet 
long and 4 inches wide that will support 400 Ibs. 

15. With the data of Ex. 10, find the least breadth of a beam 12 feet 
long and 4 inches deep that will support 500 Ibs. 

16. Evaporation from a surface varies directly as its area. 

(a) Of two square vats the side of one is 10 times that of the other. 

What is the ratio of evaporation? 
(5) Of two circular vats one evaporates 10 times as fast as the other. 

Compare their radii. 

17. The distance traversed by a falling body varies directly as the 
square of the time. If a body falls 144 feet in 3 seconds, how far will 
it fall in 5 seconds? 

18. The area of a triangle varies jointly as the length of the base b 
and the altitude a. Write the law if the area is 12 square inches when 
a = 6 inches and b = 4 inches. 



X, 172] VARIATION 225 

19. Similar figures vary in area as the squares of their like dimensions. 
A new grindstone is 48 inches in diameter. How large is it in diameter 
when one-fourth of it is ground away? 

20. A circular silo has a diameter of a feet. What must be the 
diameter of a circular silo of the same height to hold 4 times as much? 

21. What is the effect on the area of a regular hexagon if the length 
of each side of the hexagon is doubled. 

22. Similar solids vary in volume as the cubes of their like dimen- 
sions. A water pail that is 10 inches across the top holds 12 quarts. 
Find the volume of a similar pail that is 12 inches across the top. 

23. Using the rectangular pack, 432 apples 2 inches in diameter can 
be put in a box 12 X 12 X 24. How many 3 inch apples can be packed 
in the same box? How many 4 inch apples? Ans. 128; 54. 

24. If a lever with a weight at each end is balanced on a fulcrum, 
the distances of the two weights from the fulcrum are inversely propor- 
tional to the weights. If 2 men of weights 160 Ibs. and 190 Ibs. respect- 
ively are balanced on the ends of a 10 foot stick, what is the length 
from the fulcrum to each end? Ans. 4^ ft.; 5f ft. 

25. A wire rope 1 inch in diameter will lift 10,000 Ibs. What will 
one f inches in diameter lift? Ans. 1,406 Ibs. 

26. Two persons of the same build are similar in shape; their weights 
should vary as the cube of their heights. A man 5| ft. tall weighs 
150 Ibs. Find the weight of a man of the same build and 6 feet tall. 

Ans. 194.74 Ibs. 

27. A man 5 ft. 5 in. tall weighs 140 Ibs., and one 6 ft. 2 in. tall weighs 
216 Ibs. Which is of the stouter build? 

28. The size of a stone carried by a swiftly flowing stream varies as 
the 6th power of the speed of the water. If the speed of a stream is 
doubled, what effect does it have on its carrying power? What effect 
if trebled? 



16 



CHAPTER XI 

EMPIRICAL EQUATIONS 

173. Empirical Formulas. In practice, the relations be- 
tween quantities are usually not known in advance, but are to 
be found, if possible, from pairs of numerical values of the 
quantities discovered from experiment. 

In order to determine the relation between these quantities 
it is useful to first plot the corresponding pairs of values upon 
cross-section paper, and draw a smooth curve through the 
plotted points. If the curve so drawn resembles closely one 
of the following types of curves: 

(1) y = ax + b (straight line), 

(2) y = a + bx + ex 2 (parabola), 

(3) x = a + by + c?/ 2 (parabola), 

(4) y = kx n (parabolic in form), 

(5) xy = c (hyperbola), 

(6) y = c!0 kx (exponential curve), 

we assume that the relation connecting the quantities is the 
corresponding equation of the above set and it remains to 
determine the constants of the equation. 

If the plotted data does not fit any of the type curves men- 
tioned above, a general method of procedure is to assume an 
equation of the type 

(7) y = OQ + a\x + a 2 z 2 + + a n x n (nth degree curve) . 

The coefficients OQ, a\, a%, , a n can be found from any n + 1 
pairs of values of x and y. 

Since the measurements made in any experiment are liable 

226 



XI, 174] 



EMPIRICAL EQUATIONS 



227 



to be in error, errors will occur in the computed values of the 
coefficients. The curve represented by the final equation will 
not in general pass through the points representing the ob- 
served data. Some of these points will be on one side and 
some on the other. All will be near the curve. 

174. Computation of the Coefficients in the Assumed 
Formula. In case the plotted points appear to be upon a 
straight line, a parabola, or a curve of the nth degree, the 
corresponding equation is assumed and we proceed to determine 
the coefficients by a method which is illustrated in the following 
example. 

EXAMPLE 1. Let the observed values of x and y be 



X 


43 


85 


127 


169 


V . . 


17 


33 


49 


65 













Plotting this data, the points will be seen to lie roughly on a straight 
line. Hence we assume a relation of the form 

y = ax + b. 



y 



10 



o 



40 



120 



160 



80 

FIG. 125 

In this equation replace x and y by their observed values. In this way 



228 MATHEMATICS [XI, 174 

we obtain four equations connecting a and b : 
43a + b = 17, 
85a + b = 33, 
127a + b = 49, 
169a + b = 65. 

Two equations are necessary and sufficient for the determination of the 
two unknowns a and b. In general if we have more equations than 
unknowns the equations are not consistent. That is, the values of 
a and b as determined from the first two equations are not the same 
as those obtained from the last two, or from the second and third, etc. 
Our problem then is to derive from the given set two equations such 
that the values of a and b obtained therefrom when used as coefficients 
in the assumed equation will give us a straight line which fits closely 
the points plotted from the observed data. There are in common use a 
number of ways of doing this. 

FIRST METHOD. Multiply each equation in turn by the coefficient 
of a in that equation and add. This gives one equation containing a 
and b. Multiply each equation in turn by the coefficient of b in that 
equation and add. This gives a second equation containing a and b. 
Using the data in (8) above we find in this way the following equations : 
53764a + 4246 = 20744, 
424a + 46 = 164. 
The solution of these equations for a and 6 gives 

(10) a = 0.39, 6 = - 0.34 
Substituting these values of a and 6 in the assumed equation, we find 

(11) y = 0.39* - 0.34. 

SECOND METHOD. When on plotting it is clear that a straight line 
is the best fitting curve, draw a straight line among the points so that 
about half are above and half below. The y coordinate of the inter- 
section of this line with the y-axis can then be read directly from the 
graph and gives the value of 6 in the equation y = ax + b. Measure 
the angle a that this line makes with the z-axis and then a = tan a. 

In ca*se different scales are used on the two axes select two points 
(xi, yi) (x 2 , 2/2) on the line, then 

(12) =W2-H-Wl. 

Xt Xi 



XI, -174] 



EMPIRICAL EQUATIONS 



229 



THIRD METHOD. Suppose the best fitting curve is a straight line, 
i.e. that the equation should be of the form 

y = ax + b. 

Use for a and b the values obtained on solving the first and last of equa- 
tions (8). The straight line so found actually passes through the first 
and last points. 

If the points are so distributed that one of the forms (2) or (3) 173 
should be used, proceed to find a, b, and c by using the first, middle, and 
last points only. 

If an equation of degree n [(7), 173], i.e. an equation of the form 

y = QO 4- o,\x + a 2 x 2 + + a n x n 

should be assumed, use n + 1 points evenly distributed along the 
curve. This method gives us always the same number of equations as 
there are unknown coefficients to be determined. 

FOURTH METHOD. If it is known that the curve is a straight line 
through the origin then 

y = kx. 

Substitute the observed pairs of values of x and y in this equation, 
add the resulting equations and solve for k. See 172. 

EXERCISES 

1. In the following example a series of observed values of y and x 
are given. The variables are known to be connected by a relation 

of the form 

y = ax + b. 

Ans. a = 0.498, b = 0.96 



Find a and b. 



11. . 


6 


10.8 


16.1 


20.6 


26 


X 


10 


20 


30 


40 


50 















2. The following table gives the density 8 of liquid ammonia at vari- 
ous degrees centigrade. Find a relation of the form 

5 = at + b. 
i.e. determine the values of a and b. 



t 





5 


10 


15 


s 


.6364 


.6298 


.6230 


.6160 













Ans. d = 0.6364 - 0.0014 t 



230 



MATHEMATICS 



[XI, 174 



3. The following table gives the specific heat s of hot liquid ammonia 
at various degrees Fahrenheit. Find a relation of the form s = at + b. 



t . . 


5 


10 


15 


20 


25 


s 


1 090 


1 084 


1 078 


1.072 


1 066 















Ans. s = 1.096 - 0.0012i 

4. In an experiment to determine the coefficient of friction between 
two surfaces (oak) the following values of F were required to give steady 
motion to a load W. Plot F and W on squared paper, and find M where 
M = F/W. [CASTLE] Ans. M = 3.302 



F 


5 


10 


15 


20 


25 


30 


35 


40 


W 


2 


3 




6} 


74 




10V 


111 








2 






2 







5. In the following examples a series of values of x and y are given. 
In each case the variables are connected by an equation of the form 
y = ax + b. Find a and b. 



(a) 



(b) 



(c) 



(d) 

Ans. a = 0.33, b = 0.7 

In the two following sets of data plot the values of E (Electromotive 
force) and R (Resistance), and determine an equation of the form 

E = aR + b. 



11. . 


5 


7.8 


11.1 


14.2 


17 


X 


9 


18 


27 


36 


45 




















Ans. 


a = 0.337 


, b = 1.9 


y 


2 


3.1 


4 


5.2 


6.2 


x 


4 


8 


12 


16 


20 




















Ans. a 


= 0.2625, 


6 = 0.95 


y . . 


5 


6.1 


8.2 


10 


12.1 


X . . 


1 


2 


3 


4 


5 




















Ans. 


a = 1.81, 


b = 2.85 


y 


4 


7 


11 


14 


17 


X 


10 


20 


30 


40 


50 















XI, 174] 



EMPIRICAL EQUATIONS 



231 



E. . 







5 


1 


1 5 


2 


9 5 




3 


3 


5 


4 




45 


5 


R 




7 


5 


18 


28 


38 


49 


t 


>9 


f 


58 


8(1 




90 


100 


































































E.. 
R.... 


1 


3 
4 


4 

( 


[.5 

28 


6 

42 


7.75 
56 


9.5 
70 




1] 

& 


I 
I 


IS 

g 


.5 

8 




13.5 
112 


15 
126 



6. A wire under tension is found by experiment to stretch an amount 
I, in thousandths of an inch, under a tension T, in pounds, as follows: 



T. . 


10 


15 


20 


25 


30 


1 


8 


12.5 


155 


20 


23 















Find a relation of the form I = kT (Hooke's law) which best represents 
these results. 

7. In an experiment with a Weston differential pulley block, the 
effort E, in pounds, required to raise a load W, in pounds, was found 
to be as follows: 



W. . 


10 


90 


30 


40 


50 


60 


70 


80 


90 


100 


E 


H 


4f 


61 


74 


9 


10| 


m 


13f 


15 


16i 

























Find a relation of the form E = aW + b. 

8. If 6 denotes the melting point (Centigrade) of an alloy of lead 
and zinc containing x per cent, of lead, it is found that 



x 


40 


50 


60 


70 


80 


90 


e 


186 


205 


226 


250 


276 


304 

















Find a relation of the form = a + bx + ex 2 . 

9. The readings of a standard gas-meter S and those of a meter T 
being tested on the same pipe line were found to be 



S . 


3,000 


3 510 


4022 


4 533 


T 





500 


1,000 


1,500 













Find a formula of the type T = aS + b which best represents these 
data. What is the meaning of a? of fe? 



232 



MATHEMATICS 



[XI, 174 



10. An alloy of tin and lead containing x per cent, of lead melts at 
the temperature (Fahrenheit) given by the values 



X 


25 


50 


75 


e 


482 


370 


356 











Determine a formula of the type 6 = a + bx + ex 2 . 

11. A restaurant keeper finds that if he has G guests a day his total 
daily expenditure is E dollars, and his total daily receipts are R dollars. 
The following numbers are averages, obtained from the books 



G. . 


210 


270 


320 


360 


E 


16.7 


19.4 


21.6 


23.4 


R 


15.8 


21.2 


26.4 


29.8 













Find the simple algebraic laws which seem to connect E and R with 
G. [R = mG; E = aG + b.] What are the meanings of m, a, and fe? 
Below what value of G does the business cease to be profitable? 

12. The following statistics (taken from Bulletin 110, part 1 of the 
Bureau of Animal Industry, U. S. Dept. of Agriculture) give the changes 
in average egg production between 1899 and 1907: 



Year. 


Birds 
Competing 
per Year. 


Eggs Laid. 


Actual 
Average 
Production. 


Added to 
Actual 
Average. 


Modified 
Average 
Due to 
Abnormal 
Conditions. 


1899-1900 


70 


9,545 


136.36 





136.36 


1900- 01 


85 


12,192 


143.44 





143.44 


01- 02 


48 


7,468 


155.58 





155.58 


2- 3 


147 


19,906 


135.42 


23.73 


159.15 


3- 4 


254 


29,947 


117.90 


11.24 


129.14 


4- 5 


283 


37,943 


134.07 





134.07 


5- 6 


178 


24,827 


140.14 


13.95 


154.09 


6- 7.. 


187 


21.175 


113.24 


29.53 


142.77 



With the actual and modified averages in hand we may inquire: 
what has been the general trend of the mean annual egg production 
during the period covered by the investigation? The clearest answer 
to this question may be obtained by plotting the figures in the fourth 
and sixth columns of the above table, and then striking through each 



XI, 175] 



EMPIRICAL EQUATIONS 



233 



of the two zigzag lines so obtained the best fitting straight line, as 
determined by the method of least squares. The equations of the 
two straight lines are as follows: 



actual averages: 
modified averages: 



y = 148.48 - 3.10z, 
y = 144.13 + 0.043z. 



In these equations y represents the mean annual egg production and 
x the year. The origin for x is at 1898-99. Verify these two equations. 

13. The following table, taken from the same bulletin, gives the 
percentage of the flocks laying (a) less than 45 eggs, and (6) 195 or more 
eggs in a year. 



Annual Egg Production. 


1899- 
1900. 


1900- 
81. 


01-'02. 


02- 
'03. 


03- 
'04. 


04-'05. 


'05- 
06. 


06- 
07. 


Less than 45 in % . . 


4.29 


1.18 





1.36 


6.70 


7.07 


0.56 


4.81 


195 or more in % ... 


4.29 


10.60 


18.75 


6.12 


0.79 


12.71 


5.06 






Plot this data, using years for abscissa and percentages for ordinates, 
making two curves and find by the method of least squares the best 
fitting lines. 

Poor layers: y = 1.795 + 0.3225x. 

Good layers: y = 11.639 - 0.966x. 

Interpret the sign of the coefficient of x in each equation, and give the 
meaning of the constant term in each equation. 

175. Substitution. If on plotting the given values of x 
and y the plotted points are seen to be approximately on a 
branch of a rectangular hyperbola with vertical and horizontal 
asymptotes we assume a relation of the form 

(14) (x - a)(y -b) =c, 

where (a, 6) are the coordinates of the intersection of the asymp- 
totes, and proceed to determine a, b, and c. 

In many of the cases in which this form appears both a and 6 
are zero and the equation (14) becomes 

y = c/x. 



234 MATHEMATICS [XI, 175 

In some cases a is zero and equation (14) becomes 
y = b + c/x. 

There are many curves which resemble closely the curve 
given by equation (14), but whose equation is somewhat differ- 
ent. In order to determine whether (14) is the best equation 
to represent the plotted data, obtain from the figure an approxi- 
mate value of a. In many cases a = 0. Make the substitution 
l/(x a) = u and plot the new points (u, y). If these are 
approximately upon a straight line then 

y b + cu* 

and equation (14), in one of its forms, is the proper relation to 
assume. 

If on plotting the observed values of x and y the plotted points 
appear to be on a parabola with axis parallel to one of the axes 
and vertex on that axis then call that axis the ?/-axis and assume 
(15) y = a + bx 2 . ' 

The determination of the coefficients a and b can be reduced 
to that of finding the coefficients in the linear form 

y = a + bu, 

where u = x z . As a check that (15) is the correct form to as- 
sume plot pairs of values of u and y. If these points appear to 
be on a straight line then equation (15) is the correct form to 
assume. 

EXAMPLE. The distance s, in feet, passed over by a falling body in t 
seconds is found by experiment to be 



s 






5 
.5 


16 
1 


35 
1.5 


65 

2 


t 




Find a law connecting s and t. 



*This is sometimes called the reciprocal curve. 



XI, 175] 



EMPIRICAL EQUATIONS 



235 



Upon plotting this data, the points are seen to fall on a parabola with 
vertex upward and at the origin. This suggests that we assume the 
relation of the form 

s = aP. 

As a check on this assumption we plot the points (, s) given in the 
following table: 



5 
.5 

.25 



16 
1 
1 



35 
1.5 
2.25 



65 
2 
4 



These points are approximately 
upon a straight line s = au. The de- 
termination of a by the method of least 
squares gives a = 16.9, whence 

s = 16.9< 2 . 
EXERCISES 



10 



In 



1. The following data on the relation 
of temperature to insect life gives the 
number of days at a given temperature 
to complete a given stage of develop- 
ment and is taken from Technical Bul- 
letin, No. 7, Dec. 1913 of the New 
Hampshire College Ag. Exp. Station, 
each case the plotted points are on a curve of the type 



FIG. 126 



y b = c/x (x = days, y = temperature). 

The term developmental zero is used to designate that point at 
which an insect may be kept, theoretically at least, without change for 
an indefinite period. The developmental zero for the insect and stage 
approximates the point where the reciprocal curve (calculated from the 
time factor) intersects the temperature axis. (6 = developmental 
zero.) 

For each set, plot the data, and the reciprocal curve; find the 
developmental zero, and obtain an equation of the form y b = c/x 
connecting the data. 



236 MATHEMATICS [XI, 175 

(a) Malacosoma americana, pupal stage. Developmental zero = 11C. 



y . 


32.4 


32 


26.1 


20 


16 


X 


9.7 


10 


13.2 


22.5 


54 















(b) Tenebrio molitor, incubation of eggs. Developmental zero = 9.5 C. 





31.1 


26.6 


21 


11 6 


X 


6 


7.4 


12.1 


57 













(c) Leptinotarsa decemlineaia, incubation of eggs. Developmental zero 
= 6. 





32.2 


26.7 


18.6 


X 


3 


3.9 


6.3 











(d) Toxoptera graminum, birth to death. Developmental zero = 5 



V 


32.5 


26.5 


21 


15.5 


10 


x 


10 


12 


20 


30 


58 















(e) Incubation period of eggs of codling moth. Developmental zero 
= 6. 



y 


28 


25 


22 


18 


16 


15 


X . ... 


4.5 


6 


7 


9 


12 


16 

















(/) Toxoptera graminum, birth to maturity. Developmental zero = 5. 



11 


26'.5 


21 


15.5 


10 


x 


6.5 


9 


15 


32 













Ans. y 5 = 



150 (nearly) 



2. Determine a relation of the form y = a + fez 2 that best represents 
the values. 



X 


1 


2 


3 


4 


5 


6 


11. 


14.1 


25.2 


44.7 


71.4 


105.6 


147.9 

















3. The pressure p, measured in centimeters of mercury, and the 
volume v, measured in cubic centimeters, of a gas kept at constant 
temperature, were found to be as follows. 



XI, 176] 



EMPIRICAL EQUATIONS 



237 



V 


145 


155 


165 


178 


191 


V 


117.2 


109.4 


102.4 


95 


88.6 















Determine a relation of the form pv = k. 

4. Find a formula of the type u = kv 2 that best represents the 
following values. 



u 


3.9 


15.1 


34.5 


61.2 


95.5 


137.7 


187.4 


V 


1 


2 


3 


4 


5 


6 


7 



















5. If a body slides down an inclined plane, the distance , in feet, 
that it moves is connected with the time t, in seconds, after it starts 
by an equation of the form s = kP. Find the best value of k con- 
sistent with the following data. 



s 


2.6 


10.1 


23 


40.8 


63.7 


t 


1 


2 


3 


4 


5 















Am. k = 2.556. 

6. Find approximately the relation between s and t from the fol- 
lowing data. 



s 


3.1 


13 


30.6 


50.1 


79.5 


116.4 


/ 


.5 


1 


1.5 


2 


2.5 


3 

















176. Logarithmic Plotting. In case the plotted points (x, y} 
appear to lie on one of the parabolic or hyperbolic curves of the 
family 

(16) y = bx m 

there is a distinct advantage in taking the logarithm (base 10) 
of both sides: 

(17) log y = m log x + log 6, 

and then substitute 



(18) 



X for log x, Y for log y, B for log b 



238 



MATHEMATICS 



[XI, 176 



so that the equation (17) becomes, 
(19) F = mX + 5. 

If the values of x and y are tabulated in columns, and their 
logarithms X and Y are looked up and written in parallel 
columns opposite, then the points (X, Y) should lie on a straight 
line to justify the assumption of equation (16). And if they do 
lie fairly on a line, its slope and y-intercept determine the constants 
m and b of equation (16). This can often be done graphically 
from the drawing with sufficient accuracy, but if greater ac- 
curacy is required they can be determined from the data by 
least squares. 



EXAMPLE. 



X. 


y- 


A'= log z. 


r=io g i/. 


2 
4 
8 
16 


6.000 
24.60 
70.80 
338.8 


0.3010 
0.6020 
0.9030 
1.2040 


0.7782 
1.3909 
1.8500 
2.5299 



FIG. 127 



and these values in equation (16) give, 

y = 1.574X 1 -"* 



The points (X, F) lie nearly on 
a line BD, Fig. 127. Graphically, 
we scale off from the figure, 

B = the t/-intercept OB = 0.2, 

CD 

m the slope = -^ 

-DO 

- 47 -188 
-25- 1 ' 88 

By least squares, putting the 
data into equation (19), we find 

B = 0.1970 = log 6; 

hence 

b = 1.574, 

m = 1.914, 



XI, 177] 



EMPIRICAL EQUATIONS 



239 



In case the quantities x and y are connected by a relation of 
the form 

(20) y = cW kx , 

it is advantageous to compute Y = log y and plot x and Y. 
If these new values when plotted appear to be on a straight line 
we write 

(21) F = kx + log c 

and determine k and log c by the method of least squares. 

177. Logarithmic Paper. Paper, called logarithmic paper, 
may be bought that is ruled in lines whose distances, horizontally 
and vertically, from a point are proportional to the logarithms 
of the numbers 1, 2, 3, etc. 

Such paper may be used instead of actually looking up the 
logarithms in a table. For if the given values be plotted on this 
new paper, the resulting "figure is identically the same as that 
obtained by plotting the logarithms of the given values on ordi- 
nary squared paper. 

The use of logarithmic paper is however not essential; it is 
merely convenient when one has a large number of problems 
of this type to solve. 

EXERCISES 

1. A strong rubber band stretched under a pull of p kg. shows an 
elongation of E cm. The following values were found in an experiment: 



p 


05 


1 


1.5 


?n 


?5 


30 


3 5 


40 


45 


5.0 


E 


1 


03 


Of> 


0.9 


1 3 


1 7 


?,?, 


2.7 


33 


3.9 

























Find a relation of the form E = kp n . Ans. E = .Sp 1 -' 

2. The amount of water A, in cu. ft., that will flow per minute 
through 100 feet of pipe of diameter d, in inches, with an initial pressure 
of 50 Ibs. per sq. in., is as follows: 



d 


] 


1.5 


2 


3 


4 


6 


A 


4.88 


13.43 


27.50 


75.13 


152.51 


409.54 

















Find a relation of the form A = kd n . 



Ans. A = 4.88eP- 473 



240 



MATHEMATICS 



[XI, 177 



3. In testing a gas engine corresponding values of the pressure p, 
measured in Ibs. per sq. ft., and the volume v, in cubic feet, were obtained 
as follows: 



V 


7.14 


7.73 


859 


J> 


54.6 


50.7 


45.9 











Find a relation of the form p = kv n . Ans. p = 387.6#~- 938 

4. Find a relation between p and v from the following data: 



v 


6.27 


534 


3 15 


V 


20.54 


25.79 


54.25 











Ans. pv lM = 273.5 

5. The intercollegiate track records for foot-races are as follows, 
where d means the distance run, and t the record time: 



d 


100 yds. 


220 yds. 


440 yds. 


860 yds 


1 mi 


2 mi 


t 


0:094 


0:214 


0:48 


l-54f 


4-151 


9-241 

















Find a relation of the form t = kd n . What should be the record time 
for a race of 1,320 yds.? 

6. In each of the following sets of data find a relation of the form 
y = kx n connecting the quantities. 



(a) 



(6) 



V 


1 


2 


3 


4 


5 


v . . 


137.4 


62.6 


39.6 


286 


226 


























u 


12.9 


17.1 


23 1 


285 


30 


v 


63.0 


27 


13 8 


85 


6 9 















(d) 



e 






2 


212 


390 




K <7( 


1 


7 


50 




1100 


c 




f 


5.09 


2.69 


2.90 




9 C 


IS 


s 


09 




3 28 






















































X . . 


1 ^ 




2 5 


3 5 


4 5 


[ 


5 


6 L 




7 5 




8 5 


V. . 


3(1 


5 


3 92 


4 65 


5 30 




5 82 


6 ' 


10 


6 85 




7 25 





























Ans. y = 2.5x 1/2 . 



XI, 177] 



EMPIRICAL EQUATIONS 



241 



7. Draw each of the following curves: 

(a) y = x 1 / 2 . (6) y = 2x 2 . 

(c) y = 2s 1 / 2 . (d) y = 3X 3 / 2 . 

(e) y = 8x~ 3 / 2 . (/) y = 1.5x 2 ' 3 . 

(g) y = 9.2X- 2 / 3 . (h) y = log x 2/3 . 

(i) 2/ = 10. tf) y = 2-10* 2 . 

(fc) y = 10*/ 2 . (Q y = 10* +2 . 

8. Find an empirical equation connecting the x and y values given 
in the following tables. 



(a) 



x 


0.2 


0.4 


0.6 


0.8 


u . . 


3.18 


3.96 


5.00 


6.30 













Ans. y = 2.51(10"*). 



(d) 



x 


0.2 


0.4 


0.6 


0.8 


y . 


5.8 


4.4 


3.4 


2.6 






















X 





14.4 


28.4 


42.2 


y . . 


180 


24 


3 


0.7 






















X 





41.4 


83.6 


126.2 


y 


180 


92 


46 


22 













9. Given age in years and diameter in inches of a tree If feet from 
the ground as follows. 



Age 


19 


58 


114 


140 


181 


229 


Diameter 


3 


7 


13.2 


17.9 


24.5 


33 

















Plot the data and determine a relation of the form y = kx n . 
10. Given age in years and height in feet of a tree as follows : 



Age 


13 


34.4 


50.5 


218 


247 


Height 


13.4 


27.5 


38.4 


72.5 


73 















Plot the data and determine a relation of the form y = kx n . 

11. Following are vapor pressures, in mm. of mercury, of methyl 
alcohol at various temperatures: 
17 



242 



MATHEMATICS 



[XI, 177 



t 


6 


13 


21 


30 


40 




' 42 


64 


100 


160 


260 















Represent these by an empirical formula. 

12. The safe load W in tons of 2000 Ibs. for a beam 4 inches wide when 
the distance between the supports is 12 feet is given by 

W = KD\ 
where D is the depth in inches. Find K from the following table : 



D. . 


10 


12 


14 


16 


18 


W 


1.85 


2.67 


3.63 


4.74 


6.00 















13. Plot a curve from the following data, find its equation, and esti- 
mate the price of 36-inch pipe. 



Diameter of Sewer 
Pipe . . . 


8 


10 


19! 


14 


16 


18 


20 


22 


24 


Price in i per linear 
ft 


?6 


7,1 


30 


36 


50 


68 


93 


125 


150 























14. Plot a curve from the following data, find its equation, and 
estimate the pressure for a velocity of 110 miles per hour. The pressure 
is given in pounds per square foot of cross section of the first car in a 
train of ten, and the velocity in miles per hour. 



V. . . . 

p 


32 
.97 


37 
1.35 


43 
1.80 


48 
2.25 


55 
3.32 


64 
4.18 


68 
4.83 


83 
6.75 


88 
7.72 


91 

8.37 


95 
9.01 



CHAPTER XII 
THE PROGRESSIONS 

178. Arithmetic Progression. A sequence of numbers in 
which each term differs from the preceding one by the same 
number is called an arithmetic progression (denoted by A. P.). 
The common difference is that number which must be added 
to any term to obtain the next one. 

To determine whether or not a given sequence is an arith- 
metic progression we find and compare the successive differences 
of consecutive terms. Thus 

3, 10, 17,24,31, 
is an A. P. in which the common difference is 7. 

5,8, 11, 15, 18, 
is not an A. P. 

179. Notation. The following symbols are commonly used 
to denote five important numbers, called elements, which are 
considered in connection with arithmetic progressions. 

a or ai = the first term 

n = the number of terms 
I or a n = the last or rith term 

d = the common difference 
5 or s n = the sum of the first n terms 

180. Formulas. If the terms of an arithmetic progression 
are written down and numbered as follows, 

Terms : a, a + d, a + 2rf, a + 3d,- 

Number of term : 1, 2 , 3 , 4 , 

243 



244 MATHEMATICS [XII, 180 

we observe that the coefficient of d in each term is one less than 
the number of the term. Hence for the last or nth term we have 
(1) I = a + (n- i) d 

We may write the progression in which I is the last term as 
follows: 

a, a + d, a + 2d, , I Id, I d, I. 

The sum of an arithmetic progression is found by adding the 
n terms together: 

s = a + (a + d) + (a + 2d) + + (I - 2d) + (I - d) + I. 
Inverting the order of the terms 

s = I + (I - d} + (I - 2d) + + (a + 2d) + (a + d) + a. 
By addition of corresponding terms, we have 

2s = (a + 1) + (a + Z) + (a + Z) + + (a + 1) + (a + I) 
= n(a + 0- 




EXAMPLE. Find the sum of an arithmetic progression of six terms 
whose first term is 4 and whose common difference is 2. 

Since n = 6, we have I = 4 + 5-2 = 14. Hence s = |(4 -}- 14) 
= 54. 

Given any three of the elements a, n, I, d, s, either of the other 
two can be found by substituting in (1) or (2) and solving. If 
n is to be found, the given elements must be such that the 
formula will be satisfied by a positive integral value of n. 

EXAMPLE. Given d = 5, Z = f, s= - 1 /; find a and n. Sub- 
stituting in (1) and (2), we have 

3 - i-lfci i\ 15 - 1 

W 2 ~ a + 2 (n ~ l) ' ~ 2" ~ 2 



XII, 181] THE PROGRESSIONS 245 

Eliminating a, 

n* - 7n - 30 = 0. 
Solving for n, 

n = 10 or - 3. 

The value n = 3 is inadmissible. Substituting n = 10 in (3), we 
obtain a = 3. Hence n = 10, a = 3, and the arithmetic pro- 
gression is - 3, - 2\, - 2, - 1|, - 1, - i, 0, i, 1, H. 

181. Arithmetic Means. The terms of an arithmetic progres- 
sion between the first and last terms are called arithmetic means. 
Between any two numbers as many arithmetic means as desired 
can be inserted. To do this we can use equation (1) to compute 
the common difference d, for a and / are known and n is two more 
than the number of terms to be inserted. Then the required 
means are. a + d, a + 2d, etc. 

The problem of inserting one arithmetic mean between two 
numbers is the same as the problem of finding the average of 
two numbers. If m is the average of o and b, then 



and a, m, b form an arithmetic progression. For this reason 
m is called the arithmetic mean of a and 6. 

EXAMPLE. Insert 4 arithmetic means between 7 and 20. Here 
a = 7, I = 20, n = 6. Substituting these values in (1), we have 
20 = 7 + 5-d, whence d = 2|. Hence, the required means are 9$, 
12i, 14f, 17|. 

EXERCISES 

Determine which of the following suites of numbers form arithmetic 
progressions. 

1. 1, 7, 9, 12, . 2. x, x\ 3x, 

3. 5, 8, 11, 14, 4. a - 26, a, a + 26, 

5. 3, 7, 11, 15, 6. 4, 2, 0, - 2, 

7. 2, 4, 6, 9, - 8. 5, 3, 1, - 1, 



246 MATHEMATICS [XII 181 

Find I and s for the following progressions : 
9. - 2, - 6, - 10, to 17 terms. 

10. 3, 10, 17, to 50 terms. 

11. 5, 7.5, 10, to 36 terms. 

12. 2, |, V> 4, to 48 terms. 

13. Solve formula (1) for a, n, and d in turn. 

14. Solve formula (2) for a, n, and I in turn. 

15. Given n = 20, a = 1, d = 7 ; find I and s. 

16. Given n = 1000, I = 500, d = ; find a and s. 

17. Given n = 16, a = 2, Z = 3 ; find d and s. 

18. Given a = 2, I = 3, s = 100 ; find n and d. 

19. Given n = 9, a = 1, s = 37; find d and Z. 

20. Given a = 4, d = 0.1, Z = 8; find n and s. 

21. Given n = 10, d = 0.2, s = 78 ; find a and Z. 

22. Given n = 12, I = - 3, s = 140 ; find a and d. 

23. Given d = 3, I! = 22, s = 87 ; find a and n. 

24. Given a = 8, d = 8, s = 80 ; find I and n. 

25. Insert 3 arithmetic means between 1 and 17. 

26. Insert 4 arithmetic means between 2 and 18. 

27. Insert 5 arithmetic means between 3 and 38. 

28. Insert 6 arithmetic means between 4 and 6. 

29. Eight stakes are to be set at equal distances between the two cor- 
ners of a 60 ft. lot. How far apart must they be? Ans. 6 ft. 8 in. 

30. I desire to close up one side of crib 12 feet 4 inches high, with 6 
inch boards. I have just 21 boards. I desire to leave a 1 inch crack 
at top and bottom. How far apart must I place the boards to have 
them equally spaced? Ans. 1 inch. 

31. At the end of each year for 10 years a man invests $200 on which 
he collects annual interest at 6%. Find the total interest received. 

Ans. $540. 

32. The population of a certain town has made a net gain of the same 
number of people each year for the last 30 years. In 1893 it was 1523 ; 
in 1906 it was 2212. What was it in 1890 ? in 1902 ? in 1916 ? Predict 
the population for 1925. 

33. What will it cost to erect the steel work of a 20 story building at 
$3000 for the first story and $250 more for each succeeding story than 
for the one below? Ans. $107500. 



XII, 183] THE PROGRESSIONS 247 

34. I drop a rock over a cliff 400 ft high. How long before I hear it 
strike bottom if it falls 16 ft. the 1st second, 48 ft. the 2d second, 80 ft. 
the 3d second, etc., and sound travels 1090 ft. per second in air? 

Ans. 5f sec. nearly. 

35. A ball rolling down an incline goes 2 ft. the first second and 6 ft., 
10 ft., 14 ft., respectively in the next three seconds, starting from rest. 
How far will it roll in 15 seconds? Ans. 450 ft. 

36. A clock strikes the hours and also 1, 2, 3, 8, respectively, at the 
quarter hours. How many strokes does it make in a day ? Ans. 422. 

37. A farmer is building a fence along one side of a quarter section. 
The post holes are dug one rod apart and the posts are piled at the first. 
How far will he walk to distribute them one at a time and return to set 
the first one? Ans. 20| miles. 

38. Find the sum of all multiples of 7 less than 1000. Ans. 71071. 

39. Find two numbers whose arithmetic mean is 11 and the arith- 
metic mean of their squares is 157. 

40. Show that if an A. P. has an odd number of terms the middle term 
is the arithmetic mean of the first and last. 

41. If the sum of any number of terms of the A. P. 8, 16, 24, be 
increased by 1, the result is a perfect square. 

182. Geometric Progression. A sequence of numbers in 
which each term may be found by multiplying the preceding 
term by the same number is called a geometric progression 
(denoted by G. P.). The constant multiplier is called the 
common ratio. Thus 

3, 15, 75, 375, 
is a G. P. in which the common ratio is 5. 

The elements of a geometric progression are the first term a 
or oi, the number of terms n, the last or nth term / or a n , and the 
sum s or s n of the first n terms. 

183. Formulas. If the terms of a geometric progression be 
written down and numbered as follows, 

Term : a, ar, ar 2 , ar 3 , 

Number of term: 1, 2, 3, 4 , 



248 MATHEMATICS XII, 183 

we see that the exponent of r in each term is one less than the 
number of the term. Hence for the nth or last term we have 

(4) I = ar"- 1 

The sum of the first n terms of the preceding geometric pro- 
gression is 

s = a + ar + ar 2 + + ar n ~ l 
Multiplying both sides by r, 

sr = ar + ar 2 + ar 3 + + ar n 

By subtraction, we have 

sr s = ar n a. 
Solving the last equation for s, we get 



r - 1 1 - r 

From (4) we obtain rl = ar n . Hence (5) may also be written 

a - rl 



(6) 



1 - r 



The two fundamental formulas (4) and (6) contain the five 
elements a, I, n, r, s, any two of which may be found if the 
other three are given. 

EXAMPLE 1. Find s if a = 1, n = 7, r = 4. 
Substituting these values in (5), we get 

4 7 - 1 16384 - 1 
S = T ^ -3- =5461. 

184. Geometric Means. If three positive numbers are in 
geometric progression the middle one is said to be the geo- 
metric mean of the other two. It is easy to see that the geo- 
metric mean of two numbers is the square root of their product. 
Thus 3 is the geometric mean of 2| and 4. 

If several numbers are in geometric progression all the inter- 



XII, 184] THE PROGRESSIONS 249 

mediate terms are said to be geometric means between the first 
and last terms. We can insert as many geometric means as we 
wish between any two positive numbers. To do this we use 
equation (4), 183, to compute r; a, I, and n being known. 
Then the desired means are ar, ar 2 , ar 3 , etc. 

EXAMPLE. Insert three geometric means between 4 and 16. Since 
16 is to be the 5th term we have a = 4. ar 4 = 16, whence r 4 = 4 and 
r = V2 ; hence the five terms are 4, 4V2, 8, 8 V2, 16. 

EXERCISES 

Which of the following sets of numbers form geometric progressions ? 
1. 3, - 6, 12, - 24, 2. 4, 6, 9, 13.5, 

3. 7, 18, 40, 4. 8, 12, 18, 26, 

5. a, 2a, 3a, 4a, 6. a, a?, a 3 , 

7. V3 - 1, V2. V3 + 1, - 8. 8, 4, 2, 1, 

9. a, - a 2 , a 3 , - a, 10. \/2, 2, 2^2, 4, 

11. \/2, V6, 3^2, 12. 9, 3, 1, i 

13. Solve formula (4) for a, n, and r in turn. 

14. Solve formula (6) for a, I, and r in turn. 

15. Given a = 2, r = 3, n = 12 ; find I and s. 

16. Given a = 3, r = 5, n = 10 ; find I and s. 

17. Given a = 4, n = 6, s = 252 ; find I and r. 

18. I = 486, a = 2, n = 6 ; find r and s. 

19. Given a = 15, r = 3, I = 3645 ; find n and s. 

20. Given n = 5, r = \, I = 512 ; find a and s. 

21. Insert two geometric means between 2 and 128. 

22. Insert 3 geometric means between 2 and 162. 

23. Insert 2 geometric means between "N/2 and 108. 

24. What is the geometric mean between a/6 and 6/a? 

25. Find the 6th term and the sum of the series 2, 4, 8, . 

26. It takes 32 nails to shoe a horse. A blacksmith agrees to drive 
them as follows : 2 cents for the first, 4 cents for the second, 8 cents for 
the third, etc. What is the total cost? Ans. $85,899,345.90 

27. Find the amount of $500 in 5 years at 6% compounded annually j 
compounded semiannually. Ans. $669.10; $672.45 



250 MATHEMATICS [XII, 184 

28. In how many years will $100 amount to $200, interest at 8% 
compounded annually ? In how many years with interest at 6 % com- 
pounded annually? 

Ans. 9 years approximately ; 12 years approximately. 

29. A man promises to pay $10,000 at the end of 5 yr. What amount 
must be invested each year at 6 % compound interest so that at the end 
of the time the debt can be paid? 

30. A premium of $104 is paid to an insurance company each year 
for 10 years. 

What is the value of these amounts at the end of the time if accumu- 
lated at 3% compound interest? 

31. A premium of $91 is paid to an insurance company each year for 
10 years. 

What is the value of these amounts at the end of the time if accu- 
mulated at 3% compound interest? 

What is the value if accumulated at 4% compound interest? 

32. An insurance company agrees to pay me $1000 a year for 10 
years, or an equivalent cash sum to myself or heirs at the end of the 
period. 

Compute the equivalent cash sum if money is worth 6% compound 
interest. 

33. A father invests $100 each year for a newborn son, beginning 
when he is one year old. 

If money is worth 4% compounded annually, what sum is due the 
son on his twenty-first birthday ? 

What does he receive on his twenty-first birthday if the amounts in- 
vested bear 5% compound interest? 

34. A potato cuts into 4 parts for planting, each piece produces 
5 good sized potatoes, 80 of which make a bushel. If I plant each 
year all that I raised the preceding year, how many bushels of potatoes 
will I have at the end of the fifth year? How much are they worth 
at $4.00 per bu. ? Ans. $160,000. 

35. One kernel of corn planted produces a stalk with 2 ears with 
16 rows each, 50 kernels to the row. Suppose 100 ears make a bushel 
and that I plant each year one-half of all that I raised the preceding 
year and that one-half of the kernels grew and produced. How many 



XII, 184] 



THE PROGRESSIONS 



251 



bushels would I have at the end of the fifth year? (Assume two kernels 
planted the first year.) 

36. I have one sow. Let us suppose that the average litter of pigs 
is 6, sexes equally distributed, and that I keep all of the sows each 
year but sell all the others. How many sows in the sixth generation? 
How many pigs will have been sold after I have disposed of 1 /2 of the 
last or 5th litter? Am. 243; 363. 

37. The common housefly matures and incubates a new litter every 
3 weeks. There are approximately 200 to a litter evenly distributed 
as to sex. What will be the number of descendents of one female fly 
in 12 weeks? Ans. 2 X 10 8 . 

38. Grasshoppers hatch yearly a brood of 100 evenly distributed 
as to sex. Assuming that none are destroyed, what will be the number 
of descendants of one female grasshopper at the end of 5 years? 6 years? 

39. The apple aphis matures and incubates in 10 days. The progeny, 
all females, are 5 in number. The female propagates 5 each day for 
30 days. What will be the number of descendants of one female at 
the end of 30 days? 

40. If the population doubled every 40 years, how many descend- 
ants would one person have after 800 years? Ans. 1,048,576. 

41. Find the amount of money that could profitably be expended 
for an overcoat which lasts 5 years provided it saved an annual doctor 
bill of $5, money being worth 6% compound interest. 

42. The effective heritage contributed by each generation and by 
each separate ancestor according to the law of ancestral heredity as 
stated by Galton is shown in the following table from Davenport. 



Generation Back- 
ward. 


Eflective Contribu- 
tion of Each Gen- 
eration. 


Number of Ancestors 
Involved. 


Effective Contribution 
of Each Ancestor. 


1 


1/2 


2 


1/4 


2 


1/4 


4 


1/16 


3 


1/8 


8 


1/64 


4 


1/16 


16 


1/256 


5 


1/32 


32 


1/1024 











Compute the effective contribution of the last 20 generations. The 
number of ancestors involved in the 20th generation backward and 
the total number of ancestors involved. The effective contribution of 
each ancestor in the 20th generation backward. 



252 MATHEMATICS [XII, 185 

185. Infinite Geometric Series. A geometric progression can 
be extended to as many terms as we please, since on multiplying 
any term by the common ratio we obtain the next one. Any 
series which has no last term and can be indefinitely extended is 
called an infinite series. 

Suppose the terms of a geometric series are all positive. If 
we begin at the first and add term after term the sum always 
increases. If r > 1, this sum becomes infinite, i.e., if we choose 
a positive number N no matter how large it is possible to add 
terms enough that the sum will exceed N. If however r < 1, 
the case is quite different. The sum does not become infinite ; 
it converges to a limit, i.e., it is possible to find a number L such 
that the sum will exceed any number whatever less than L, but 
it will never reach L. For example the sum obtained by adding 
terms of the geometric series 

1 +i+i+^ + 

in which r = -|, will never reach 1.5, but terms enough can be 
added to make the sum exceed any number less than 1.5. If, 
e.g., we wish to make the sum greater than 1.49, five terms are 
sufficient. 

A geometric series in which r < 1 is called a decreasing geo- 
metric series. The limit to which the sum of the first n terms of 
a decreasing geometric series converges is a/(\ r), i.e., the first 
term divided by one minus the ratio. 

For by (5) 183, 

8 = a(l - r") = a a ^ ^ 

1 - r 1 - r I - r 

Now as we add more and more terms, the n in this formula gets 
larger and larger, a and r remain fixed. Since r < 1, it follows 
that r 2 < r, r 3 < r 2 , etc., and r n converges to zero when n is 
taken larger and larger. Therefore the second term on the 
right converges to zero, and s n converges to a/(l r). This 



XII, 185] THE PROGRESSIONS 253 

limit is sometimes called the " sum " (although strictly it is not 
a sum) of the infinite decreasing geometric series 

a + ar + or 2 + , 
and we write 

(7) s =^' r<1 - 



EXAMPLE. The repeating decimal .666 can be written thus 
.6 + .06 + .006 + . It is therefore an infinite geometric series 
whose first term is .6 and whose common ratio is .1. Hence 

.6 _2 

fin 3- 

EXERCISES 
Find the sum of the following infinite series : 

1. 1+0.5 +0.25 + -. 6. 1+I + H + -- 

2. 1 -0.5 + 0.25 -0.125 + -. 7. 3 + f + T 3 5 + . 

3. l + i + i + . 8. 100 + 1 +0.01 +. 

4. 1 - i + I - iV + 9. 3 + 0.3 + 0.03 + -. 

5.. 1 + f + | + -. 10. 0.23 + 0.023 + 0.0023 + -. 
Find the value of the following repeating decimals : 

11. .1111 -. 17. .00032525 . 

12. .2222 . 18. .1234512345 . 

13. .252252-. 19. 20.2020. 

14. 1.2424 . 20. 5.312312 -. 

15. 2.53131 . 21. 6.4141 -. 

16. 2.3452345 -. 22. 3.214214 . 



CHAPTER XIII 
ANNUITIES* 

186. Definitions. Suppose you take out a life insurance 
policy on which you agree to pay a premium of $100 at the end 
of each year for 10 years. Such an annual payment of money 
for a stated time is termed an annuity. Instead of paying $100 
a year you may prefer to pay $24 at the end of every three 
months or $206 at the end of every two years. In any case the 
stated amount paid at the end of equal intervals of time is called 
an annuity. 

Suppose the stated sums are not paid when due and that after 
the lapse of say 5 years you desire to pay off your indebtedness 
with interest compounded. The sum due is called the amount 
of the annuity for the five years. 

Suppose you buy a house and agree to pay $1000 at the end 
of each year for 4 years. This is an annuity. An equivalent 
cash price at the time of sale is called the present value of the 
annuity. 

187. Notation. The letter r stands for the rate of interest, 
e.g. 6 ; the letter f ( = r/100) stands for the annual interest on one 
dollar, e.g. .06. 

The symbol S^ stands for the amount of an annuity of one 
dollar paid at the end of each year for n years. 




are indebted for many ideas, methods, and exercises. 

254 



XIII, 189] ANNUITIES 255 

The symbol S^ stands for the amount of an annuity of one 
dollar paid at the end of each pth part of a year for n years. 

The symbol a^\ stands for the present value of one dollar paid 
at the end of each year for n years. 

The symbol a^ stands for the present value of an annuity of 
one dollar paid at the end of each pth part of a year for n years. 

188. Amount of an Annuity. It is sufficient to consider an 
annuity of one dollar since the amount for any other sum will 
be proportional to this. 

The first payment of one dollar made at the end of the first 
year will bear interest for n 1 years, and at the end of the 
period the amount due will be (1 + t)" 1 . The second payment 
will bear interest for n 2 years and will increase to (1 + i) n ~ 2 . 
The next to the last payment will bear interest for one year and 
will increase to 1 + i. The last payment will be one dollar and 
it will bear no interest. The total amount S^, due at the end of 
n years is therefore 

1 + (1 + i} + (1 + t) 2 + . + (1 + t)"- 2 + (1 + i}"- 1 . 

In this geometric progression the first term is 1, the last term 
is (1 + i) n ~S and the ratio is 1 + i. Substituting these values 
in the formula (5) 183 for the sum of a geometric progression, 
we find 



189. Partial Payments. Suppose that the payments instead 
of being made at the end of each year are made at the end of 
each pth part of a year for n years. Consider an annuity of one 
dollar. 

The payment to be made at each payment period is l/p. The 
first payment will bear interest for n \/p years. The second 
payment will bear interest for n 2/p years, and so on. The 
next to the last payment will bear interest for l/p years. The 



256 MATHEMATICS [XIII, 189 

last payment will bear no interest. The total amount due is 
then 

i + 1 (i + o* + - a + o* + - + - a + *r* . 

p P P P 

i 

In this geometric progression the common ratio is (1 + i} v , 
and by (5), 183, the sum of the terms is 



As shown in 145 for the square root, the pth root of 1 + i 
is nearly equal to 1 + i/p. In fact it is customary in comput- 
ing the amount of one dollar at interest compounded p times a 
year, to use 1 + i/p instead of Vl + i- See 217. If this ap- 
proximate value be used in formula (2), the right member 
reduces to 



^ 
which shows that S^ is approximately equal to S^. 

EXERCISES 

1. Find the amount of an annuity of $200 for 10 years at 3% ; 4% ; 
5% ; 6% ; 8%. Ans. For 3% $2292.78 

2. The semiannual premium on an insurance policy is $50. Find 
the amount of this annuity for 10 years at 4%. Ans. $606.37 

3. The quarterly premium on a policy is $62.10. Find the amount 
of this annuity for 10 years at 3%. Ans. $719.11 

4. The annual rent of a house is $480. Find the amount of this 
annuity for 20 years at 6%. Find the amount if the rent is paid 
monthly. Ans. $17657.08 

5. A man saves and at the end of each year for 40 years deposits $100 
in a savings bank which pays 4% compounded annually. Find the 
amount. Ans. $9502.55 

6. A man saves $500 a year and invests savings and interest in bonds 
yielding 6%. What will his accumulations amount to in 10, 15, 20, 
30 years? Ans. $6590.40 



XIII, 190] ANNUITIES 257 

190. Given the Amount of an Annuity to find the Annuity. 
Let the annual payment be x. The first payment made one 
year from the beginning of the term of the annuity will bear in- 
terest for n 1 years and will increase to x(l + i) n ~ l . Like- 
wise, the second will increase to x(l + i} n ~~, the third to 
(1 + z')"~ 3 > and so on, while the last payment x will bear no 
interest. If the sum of the amounts due at the end of n years 
is $1, we have 

x[(\ + t)"- 1 + (1 + i)"- 2 + " + (1 + i) + i] = 1. 
The expression within the square brackets is a geometric pro- 
gression of n terms with ratio (1 + i} ; hence, by (5), 183, we 
have 



or 

'"(TT^" 1 ' 

which gives the annuity whose amount after n years is $1. This 
formula for x may be written symbolically in the form 

(4) x=-f. 

S*i 

EXERCISES 

1. In 10 years a man desires to be worth $30,000. What sum must 
he set aside yearly to realize that amount if money is worth 8% ? 

2. An auto truck costing $2000 lasts 5 years. What sum must be 
set aside annually at 6% to replace the truck when worn out? 

3. An automobile costs $1500 and lasts 5 years. What is the equiva- 
lent annual expenditure, money worth 6%? 

4. A city decides to pave some of its streets. For this purpose bonds, 
bearing 6% interest, to the amount of $50,000 are issued. The bonds 
are due in 10 years. What sum must be collected yearly in taxes and 
invested at 6% to pay off the bonds when due? 



258 MATHEMATICS [XIII, 191 

191. Present Value of an Annuity. The present value of 
one dollar due in one year is (1 + *) -1 , 
one dollar due in two years is (1 + i)~ 2 > 

one dollar due in n years is (1 + i}~ n . 

The present value of one dollar paid at the end of each year 
for n years will then be 

(1 + i)~ l + (1+ i)- 2 + - + (1 + ^ 

The sum of this geometric progression is the present value 
sought. Hence the present value, a$j-,, of an annuity of $1 is 

(5) a . - (1 + ^ - (1 + fl" 1 

(1 + i)- 1 - 1 

Multiplying numerator and denominator by 1 + i we find 



EXERCISES 

1. A man buys a farm, agreeing to pay $1500 cash and $1500 at the 
end of each year for three years. What would be the equivalent cash 
value of the farm if money is worth 6%? 

2. A man buys a farm, agreeing to pay $2000 cash and $2000 at the 
end of each year for ten years. What would be the equivalent cash value 
of the farm if money is worth 6%? 

3. A contractor performs a piece of work for a city and takes bonds 
in payment. The bonds do not bear interest, and are payable in 10 
equal annual installments of $2000, the first payment to be made one 
year from date. Money being worth 6%, payable annually, what is 
the cash value of the bonds on the date of issue ? 

4. Prove that the present value of one dollar paid at the end of each 
pth part of a year for n years is 

1 



1 + 



(1 + i)" 



and show that this is approximately equal to a^. See 189. 



XIII, 192] ANNUITIES 259 

5. A man contracts to buy a house paying $200 every three months 
for 8 years. Find the equivalent cash price, money being worth 6%. 

6. Find the cash value of semiannual payments of $500 for 5 years, 
money being worth 6%. 

192. Cost of an Annuity. A man desires to provide for his 
family, in event of his death, an annuity of $5000 a year for 20 
years. What amount must he set aside in his will to provide 
for this annuity, assuming that money is worth 6%. 

The cost of an annuity of one dollar per year for n years is 
^/fi 187, 191. Whence the cost C, of an annuity of P dollars 
per year for n years is 

(7) 



i 

From this we compute that the man should set aside in his will 
about $57350. 

EXERCISES 

1. What will be the cost of an annuity of $500 a year for 10 years, 
money being worth 4%? Ans. $4055 

2. A man agrees to pay $700 a year for 5 years for a house. What is 
the cash value of the house, money being worth 6%. Ans. $2948.66 

3. A man agrees to pay $700 a year for 20 years for a farm. What 
is the cash value of the farm, money being worth 5%? Ans. $8723.55 

4. A man 70 years old has $3000. His expectation of life being 8 
years, what annuity can an insurance company offer him, money being 
worth 4% ? 

5. A man with $10,000 pays it into a life insurance company which 
agrees to pay him or his heirs a stated sum each year for 20 years. 
What is the yearly payment, money being worth 4%? 

6. A man buys a house for $4000. What annual payment will can- 
cel the debt in 5 years, money being worth 6%? Ans. $949.60 

7. How long will it take a man to accumulate $100,000, by saving 
$1000 a year and investing it at 6%. Ans. 33 yrs. 

8. A man inherits $20,000 which is invested at 4%. If $1000 a 
year is spent, how long will the inheritance last. Ans. 41 yrs. 



260 MATHEMATICS [XIII, 193 

193. Perpetuities. In the previous problems treated in this 
chapter the payments continued over a fixed number of years 
and then stopped. The annual amount expended for repairs 
on a gravel road does not stop at the end of a given period, but 
continues forever. Such payments constitute an endless an- 
nuity, which is called a perpetuity. Other examples are the 
annual repairs on a house, taxes, annual wage for a flag man, 
annual pay of a section gang. The amount of an annuity would 
evidently increase indefinitely as time went on. The present 
value of a perpetuity, however, has a definite meaning. The 
present value of a perpetuity is a sum which put at interest at 
the given rate will produce the specified annual income forever. 
Denote by V the present value of the perpetuity and by P the 
annual payment. Then 

(8) F i = P. 

If the payments are made every n years instead of yearly, 
the present value of the perpetuity is denoted by V n ; its value 
will be 

(9) V n = P[(l + i)- + (1 + i)- 2n + - + (1 + i~)~ pn + ]. 

This is an infinite geometric progression whose first term is 
P(l + i)-" and whose ratio is (1 + t)"". Hence, by (7), 185, 
the present value of the perpetuity is 

(1 + i)- P 



(10) V n = P 



1 - (1 + i)- (1 + i) w - 1 



EXERCISES 



1. What is the present cash value of a perpetual income of $1200 per 
year, money being worth 6% ? Ans. $20,000. 

2. How much money must be invested at 6% to provide for an in- 
definite number of yearly renewals of an article costing $24? 

3. How much money must be invested at 4% to provide for the pur- 
chase every 4 years of a $1000 truck? 



XIII, 193] ANNUITIES 261 

4. What is the cash value of a farm that yields an average annual 
profit of $2400, money being worth 6%? 

5. The life of a certain farming implement costing $100 is 6 yrs. 
Find what sum must be set aside to provide for an indefinite number of 
renewals, money being worth 4%. 

6. The life of a University building costing $100,000 is 100 years. 
A man desires to will the University enough money to erect the building 
and to provide for an indefinite number of renewals. How much must 
he leave the institution? 



CHAPTER XIV 
AVERAGES * 

194. Meaning of an Average. In referring to a group of 
individuals, a detailed statement of the height of each would 
take considerable time, when large numbers are involved. In 
comparing two or more groups, such a mass of detail might fail 
to leave a definite impression as to their relative heights. What 
is needed is a single number, between that of the shortest and 
that of the tallest, which is representative of the group with 
respect to the character measured. Such an intermediate 
number is called an average. 

The idea of an average is in use in everyday affairs. We 
hear mentioned frequently such expressions as the average rain- 
fall, the average weight of a bunch of hogs, the average yield 
of wheat per acre for a county or state, the average wage, the 
average length of ears of corn, the average increase in popula- 
tion, etc. Often these expressions are used with only an indefi- 
nite idea as to what is really meant. 

In this Chapter we shall discuss some of the averages in com- 
mon use, and we shall explain the circumstances under which 
each is to be used. 

195. Arithmetic Average. The arithmetic average is the 

* The authors of this book are indebted for many ideas in this Chapter and for some 
of its methods to an Appendix by H. L. RIETZ to E. DAVENPORT, Principles of Breeding, 
Ginn and Co. Some use has been made also of ZIZEK, Statistical Averages, Henry Holt 
and Co. ; PEARSON, Grammar of Science; BOWLEY, Elements of Statistics; and SECRIST, 
Introduction to Statistical Methods, Macmillan/ 

262 



XIV, 196] AVERAGES 263 

number obtained by dividing the sum of the measurements taken 
by the number of those measurements : 

/1N .,, ,. sum of all measurements 

arithmetic average = . 

number of measurements 

Thus, if we measure seven ears of corn and find their lengths 
to be 6, 7, 8, 9, 10, 11, 12 inches, the arithmetic average of their 
lengths is 9 inches. Again, the arithmetic average of 6, 7, 8, 12, 
12 is 9. This example shows that the arithmetic average gives 
no indication of the distribution of the items and that there 
may be no item whose measurement coincides with the average. 
However, it is influenced by each of the items, and it is easily 
understood and computed. It should seldom be used except in 
conjunction with other forms of averages. When used alone it 
should be for descriptive purposes only. 

196. Weighted Arithmetic Average. In measuring the given 
items it frequently happens that there are 

n\ items with the same measurement /i, 
HZ items with the same measurement 1%, 

n t items with the same measurement l^. 
Then the weighted arithmetic average is given by the formula 

(2) weighted arithmetic average = ni/1 + n ' 2/2 + -+"***. 

ni + w 2 + " + n k 

In the simple case mentioned above, the weighted arithmetic 
average gives the same result as the arithmetic average. Its 
chief advantage is that it facilitates computations. For 
example the average length of the ears of corn whose individual 
lengths are 6, 7, 8, 12, 12 can be found as follows : 

average length = 1X6 + 1X7 + 1X8 + 2X12 = 

1+1 +1+2 
There may be other reasons, however, for counting one item 



264 MATHEMATICS [XIV, 196 

several times. Thus, in measurements, an item that is known 
to be particularly trustworthy may be counted doubly or triply. 
In such cases, the weighted average differs from the arithmetic 
average. 

197. The Median. If we arrange the numbers representing 
the measurements of the items in order of magnitude, the 
middle number is called the median. Thus, the median length 
of the ears of corn whose lengths are 6, 7, 8, 12, 12 inches is 8 
inches. In case there are an even number of items the median 
is midway between the two middle terms. Thus if the lengths 
of four ears of corn are 6, 7, 9, 10 inches, the median length is 
8 inches. There is no ear of this length among those measured. 

The median is often used because it is so easily found. Like 
the arithmetic mean, it gives no indication of the distribution. 
It can be used even when a numerical measure is not attached 
to the various items. For example, ears of corn can be ar- 
ranged in order of length without knowing the numerical length 
of any ear ; clerks can be ranked in order of excellence ; shades 
of gray may be arranged with respect to darkness of color ; etc. 
The median is the central one of a group and is unaffected by the 
relative order of the other members of the group. Thus it is 
used when the primary interest is in the central members. 

198. The Mode. In measuring the items of a given set it 
may happen that some one measurement occurs more frequently 
than any other. This measurement is called the mode. Thus, 
the modal length of six ears of corn whose lengths are 6, 7, 8, 12, 
12, 13 inches is 12 inches. A set of measurements may have 
more than one mode. Thus in a given factory there might be 
few men who received $2 per day, a large number who received 
$3, a small number who received $4, and a large number who 
received $5, while few received more than $5. There would then 
be two modes for wages, namely $3, and $5. 



XIV, 199] AVERAGES 265 

If a curve be plotted using measurements as abscissas and the 
number of items corresponding to each frequency as ordinates, 
the mode corresponds to the maximum ordinate or ordinates. 
(See 225.) 

Unlike the arithmetic average, and the median, the mode is 
always the value of one individual measurement. Extreme 
measurements have no effect upon it. 

In measuring heights of men we might place all those over 
4.5 and under 5.5 feet at 5 feet. For this distribution the mode 
would necessarily fall at one of the integers. If we arrange 
the heights in three-inch intervals the mode might not appear 
as an integer, although it would be near the mode first obtained. 
Thus it is seen that the mode depends upon the grouping of the 
measurements. 

The existence of a mode shows the existence of a type. It is 
the mode that we have in mind when we speak of the average 
height of a three-year-old apple tree, the average price of land, or 
the average interest rate. 

199. The Geometric Average. The geometric mean of two 
positive numbers has been defined in 184. By analogy we 
may define the geometric average of n positive numbers as the 
nth root of their product. 

If a growing tree doubles its diameter in 20 years what is its 
annual percentage rate of increase ? It is not 5%, for an increase 
of 5% a year would give the following diameters at the end of the 
1st, 2d, 3d, . . ., 20th year 

which would give a final diameter greater than 2.6d. Evi- 
dently what is wanted is a rate r such that 

/ f \20 

M -i--L- \ 2 

I J. | , 

v 100; 

whence r = 100(V2 1) = 3.53 + . Hence an annual increase 



266 MATHEMATICS [XIV, 200 

of about 3|% will double anything in 20 years. The geometric 
average is used in many practical affairs. Knowing the average 
rate of growth of a city in the past the geometric average is used 
to predict its future growth. When a new school building is 
being designed, for example, it should be made large enough to 
meet the future growth of the community as shown by this 
geometric average. 

200. Conclusion. Given a set of items numerically measured 
or not, we should first determine whether or not the data is such 
as to warrant any kind of an average. Then the decision 
whether one or another kind of average is to be employed de- 
pends upon the use to which the result is to be put. If the data 
is not complete, the arithmetic average cannot be used. If we 
desire to characterize a type in such a case, we may find the 
mode, for which the data need not be complete. 

Frequently it is best to make use of more than one kind of 
average in describing a distribution. It must be remembered that 
any average at best conveys only a general notion and never con- 
tains as much information as the detailed items which it repre- 
sents. 

EXERCISES 

1. From the heights of the members of your class, find each of the 
following kinds of average height : (a) arithmetic, (6) median, (c) mode. 

2. Determine in the following cases which average is meant : mean 
daily temperature ; average student ; average price of butter ; average 
of a flock with respect to egg production ; average salary for all of the 
teachers of a state ; average number of bushels of corn per acre for a 
state or nation ; normal rainfall ; average number of pigs per litter ; 
average number of hours of sunshine per day ; average speed of train 
between two stops ; average wind velocity ; mean annual rainfall ; aver- 
age sized apple ; average price of oranges when arranged according to 
sizes ; average date of the last killing frost in the spring ; average price 
of land per acre in a given locality ; average gain in weight per day of 
a hog. 



XIV, 200] AVERAGES 267 

3. What kind of an average is meant in each of the following cases : 
one fly lays on an average 120 eggs; 63% of the food of bobolinks is 
insects ; every sparrow on the farm eats j oz. of weed seed every day ; 
the average gas bill is $2 per month ; the average price received for lots 
in a subdivision was $800; repairs, taxes, and insurance on a house 
average $100 per year ; the average amount of material for a dress pat- 
tern is 8 yards, 36 inches wide ; a college graduate earns on an average 
$1125 a year, while the average yearly earnings of a day laborer, who 
has no more than completed the elementary school, is $475. 

4. Suppose that we consider 5 millionaires and 1000 persons who are 
in poverty. Find the arithmetic average, the median, and the mode of 
the wealth of this group. Which best portrays conditions? 

5. In the Christian Herald for March 10, 1915, p. 237, it is stated that : 
"The average salary of ministers of all denominations is $663. The 
few large salaries bring up the average." Which average is used here? 
Is it the best to portray conditions? Is the result too high or too low 
to represent conditions properly? 

6. Compute for the members of your family the mean age, and arith- 
metic average. Is there a mode? 

7. On a given street ascertain the number of houses per block for 5 
blocks. Find the arithmetic average and the median. Is there a mode ? 

8. On a given business street ascertain the number of stories of each 
business house for one block. Find the arithmetic average and the 
median. Is there a mode? 

9. Proceed as in Ex. 8 for a residence street. Is there a mode ? 

10. In 4 years the number of motorists killed at railroad crossings 
doubled. Find the annual rate of increase, using the geometric average. 

Ans. 19%. 

11. If in the last 20 years the number of deaths in the U. S. due to 
consumption has increased 50%, find the annual rate of increase, using 
the geometric average. Ans. 2%. 

12. Land increased in value from $40 to $150 per acre from 1890 to 
1915. What was the average yearly increase? 

13. Find the average (arithmetic) word, sentence, and paragraph 
length, of some one of the writings of Longfellow, Holmes, Whittier, 
Poe ; of some short story ; of some newspaper article. 

14. The total of the future years which will be lived by 100,000 



268 MATHEMATICS [XIV, 200 

persons born on the same day are 5,023,371. If the total number of 
-years to be lived is divided by the number of persons the quotient will 
be the average number of future years to be lived by each person. 
What kind of an average is this ? What average age does it give ? 

15. Out of 100,000 males born alive on the same date about one-half, 
namely 50,435, attain age 59. This is then an average age attained. 
What kind of an average is it? 



CHAPTER XV 
PERMUTATIONS AND COMBINATIONS 

201. Introduction. In how many ways can I make a selec- 
tion of two men to do a day's work if there are 3 men available 
for the forenoon and 4 for the afternoon? Having hired one 
man for the forenoon, I can hire any one of 4 for the afternoon, 
and since this is true for each of the three, there are 3 X 4 = 12 
ways of making the selection. This reasoning is general ; that 
is, it does not depend upon the special properties of the numbers 
3 and 4. Hence we see that if there are p ways of doing a first 
act, and if corresponding to each of these p ways there are q ways 
of doing a second act, then there are pq ways of doing the sequence 
of two acts in that order. 

It is evident also that this principle applies to a sequence of 
more than two acts and we may say, 

If there are p ways of doing a first act; and if after this has 
been done in any one of these p ways there are q ways of doing a 
second act; etc.; and if after all but the last of the sequence have 
been done there are r ways of doing the last act, then all the acts of 
the sequence can be done in the given order in pq r ways. 

EXERCISES 

1. With 4 acids and 6 bases, how many salts can a student make? 

2. A ranchman has 5 teams, 4 drivers, and 3 wagons. In how many 
ways can he make up one outfit? 

3. There are 6 routes from Chicago to Seattle, 4 from Seattle to Port- 
land, 3 from Portland to San Francisco. How many ways are there of 
going from Chicago to San Francisco via Seattle and Portland? 

269 



270 MATHEMATICS [XV, 202 

202. Combinations and Permutations. A group of things 
selected from a larger group is called a combination. The 
things which constitute the group are called elements. Two 
combinations are alike if each contain all the elements of the 
other irrespective of the order in which they appear. Two 
combinations are different if either contains at least one element 
not in the other. 

A permutation of the elements of a group or combination, or 
simply a permutation, is any arrangement of these elements. 
Two permutations are alike if, and only if, they have the same 
elements in the same order. Thus, eat, tea, and ate are the same 
combination of three letters a, e, t ; but they are different per- 
mutations of these three letters. 

203. Number of Permutations. The number of permuta- 
tions of three elements taken all at a time is 6, as may be seen by 
writing them down and counting them : 

abc, acb, bac, bca, cab, cba. 

The number of permutations of 4 elements taken 2 at a time is 
12. Thus, 

ab, ac, ad ; ba, be, bd ; 

ca, cb, cd ; da, db, dc. 

If the number of elements is large the process of counting is 
tedious. It is possible to derive general formulas for the num- 
ber of permutations of any number of elements by which the 
number can be easily computed. 

204. Permutations of n Things. A rule for the number of 
permutations of n things taken all at a time is easily deduced 
by means of the principle of 201 . We have n elements and n 
places to fill. We may think of a row of cells numbered from 
1 to n. 



XV, 205] PERMUTATIONS AND COMBINATIONS 271 



1 


2 


3 


4 




n 















The first cell can be filled in n different ways and after it has 
been filled the second cell can be filled in n 1 ways. There- 
fore the first two can be filled in n(n 1) ways. When they 
have been filled in any one of these possible ways the third cell 
can be filled in (n 2) ways. Therefore the first three cells 
2an be filled in n(n l}(n 2) ways. Continuing thus we 
see that the first k cells (k < n) can be filled in n(n l}(n 2) 
(n k + 1) ways, and that all the n cells can be filled 
in n(n l}(n 2) -"2 1 ways. This product of all the 
natural numbers from 1 to n is called factorial n, and is denoted 
by n ! or \n. Thus, 2 ! = 2, 3 ! = 6, 4 ! = 24, 10 ! = 3,628,800. 
Therefore, 

The number of permutations of n things taken all at a time is 
factorial n. 

For example, 4 horses can be hitched up in 24 ways ; 10 cows 
can be put into 10 stanchions in 3,628,800 ways. 

By the same reasoning the number of permutations of n 
things k at a time (k ^ n) is the number of ways that k cells 
can be filled from n things. The symbol n P t is used to denote 
this number. Then, as shown above, 

(1) n P k = n(n - l)(n - 2) (n - k + 1) 

To remember this formula, note that the first factor is n and the 
number of factors is k. Thus B ^3 = 5 4 3 = 60. The 
number of ways in which 4 stanchions can be filled out of a herd 
of 10 cows is 10 P 4 = 10 9 8 7 = 5040. In this notation 
we should write for the number of permutations of n things 
all at a time 

(2) n P n = n\ 

205. Repeated Elements. The above reasoning assumes that 
the elements are all distinct. If some of the n elements are alike, 



272 MATHEMATICS [XV, 205 

the number of distinguishable permutations is less than n \ 
For example, the number of distinct permutations that can be 
made out of the 7 letters of the word reserve is not 7 ! The 
number of permutations of the 7 characters ri, ei, s, 62, r 2 , v, 63 
is indeed 7 ! ; but when the subscripts are dropped the permuta- 
tions TI e\ s e^rzv 63 and r^ 2 s e 3 r\ v e\ become identical. 

Let x be the number of different permutations of the letters 
of the word reserve. For each of these x there will be 2 ! per- 
mutations of the characters r\ e s e r<z v e and for each of these 
x 2 ! there will be 3 ! permutations of the characters r\ c\- 
s cz r z v 63, making x 2 ! 3 ! in all. It follows that 

x 2 I 3 ! = 7 ! and x = -^~ 
2!3! 

This reasoning can be extended to show that the number of 
distinguishable permutations of n elements of which p are alike, 
q others are alike, etc., , r others are alike, is equal to 



(3) 



n ! 



p ! q I r I 



EXERCISES 



1. How many 3-letter words can be formed from the letters a, p, <? 
How many 2-letter words ? How many of each are used in the English 
language ? 

2. How many different 2-digit numbers can be made from the ten 
digits 0, 1, 2, , 9 ? How many if repetitions are allowed ? How many 
of these are used? 

3. Find the number of permutations of the letters in each of the fol- 
lowing words : (a) degree, (6) natural, (c) Indiana, (d) Mississippi, 
(e) Connecticut, (/) Kansas, (g) Pennsylvania, (h) Philadelphia, 
(i) Onondaga, (j) Cincinnati. 

4. In how many ways can a pack of 52 cards be dealt into four piles 
of 13 each? 

5. With 15 players available, in how many ways can the coach fill 
the various positions on a baseball team? 



XV, 206] PERMUTATIONS AND COMBINATIONS 273 

6. How many different signals of two flags, each one above the other, 
can be made with five different colored flags ? 

7. How many different sounds can be made by plucking the five 
strings of a banjo one or more at a time? 

8. How many football signals can be given with four numbers, no 
repetitions being allowed? 

9. In how many ways can four fields be cropped with corn, oats, wheat, 
and clover, one field to each? 

10. A seed store offers 12 varieties of garden seeds. My garden 
has 8 rows. In how many ways can I plant one row of each variety 
selected? 

11. In how many ways can a gardener plant 2 rows of lettuce, 3 of 
onions, 3 of beans, 4 of potatoes, if his garden has 12 rows? 

12. How large a vocabulary could be formed with 9 letters, no repe- 
titions being allowed? How many with ten? How many with 
twenty-six? (There are about 100,000 words in Webster's dictionary. 
The average man has a vocabulary of less than 5000 words.) 

206. Combination of n Things k at a Time. The symbol 
n Ck or (2) is used to denote the number of different combinations 
( 202) that can be made from n elements taken k at a time. 

A combination of k elements can be arranged into k ! permuta- 
tions of these elements. That is, there are k I times as many 
permutations as there are combinations of k elements taken all 
at a time. Whence 

n P k = k\ n C k . 

Making use of the value of n P t , (1), 203, and solving for n Ck 
we have, 

(M r _"(" ~l)(n -2)--(n -fc + 1) 

1.2.3-* 

To remember this formula note that the first factor of the nu- 
merator is n, and that there are k factors in the numerator and 
k in the denominator. 

Another useful form of this result is obtained by multiplying 



274 MATHEMATICS [XV, 206 

both numerator and denominator of (4) by (n k}(n k 1) 
(n -k - 2) 2 1. This gives 

(5) n C k = ' 1 -TW 

k i(n k) I 

We note that the interchange of Jc and n k leaves (5) un- 
altered and hence conclude that 

(6) n^n-k = n Ck- 

This is what we should expect when we think that the numbers 
of ways that k things can be selected from a group of n must be 
the same as the number of ways that n k can be rejected. 

EXERCISES 

1. From a pack of 52 cards how many different hands can be dealt? 

2. How many combinations of 5 can be drawn from 42 dominoes? 

3. How many different tennis teams can be made up from 6 players 
(a) singles ; (6) doubles ? 

4. How many straight lines can be drawn through 8 points, no three 
of which lie on a straight line ? How many circles ? 

5. How many diagonals has a convex polygon of n vertices? 

Ans. n Ci. n. 

6. Two varieties of corn are planted near each other. How many 
varieties will be harvested? Ans. zCz + 2. 

7. If four varieties of oats are sown near each other, how many varie- 
ties will be harvested? Ans. 4^2 + 4. 

8. A starts with two kinds of pure-bred chickens. How many kinds 
will he have at the end of the third hatching if all stock is sold when 
one year old? Ans. Cz + 6. 

9. In how many ways can 15 gifts be made to 3 persons, 5 to each? 

Ans. isCe i C 5 . 

10. In how many ways can 15 gifts be made to 3 persons, 4 to A, 
5 to B, 6 to C? Ans. 630,630. 

11. Given (a) n C 2 = 45; (6) B C 2 = 190; (c) n C 2 = 105; find n. 

12. In how many different ways can 500 ears of corn be selected from 
505 ears? 

13. Compute: (a) loooCW; (&) mC m ; (c) 10002^10000. 



CHAPTER XVI 

THE BINOMIAL EXPANSION LAWS OF 
HEREDITY 

207. Product of n Binomial Factors. If the indicated mul- 
tiplications are performed and terms containing like powers of x 
are collected, 



(1) (x + Oi)(a; + a 2 )(.r + a,)(x + o 4 ) (x + a n ) 

= X n + CiX 1 + C Z X n ~* + C 3 X n ~ 3 + + Cn-iX + C n 

in which the coefficients have the following values: 

Ci = 01 + a 2 + 3 + + a n . 
The number of these terms is n. 
C 2 = Oi0 2 + aia n + a z a 3 +'+ a 3 a 4 + + a n -in. 

The number of these terms is the number of combinations 
that can be made from n a's, 2 at a time, i. e., n C 2 . 

Cs = aia 2 a 3 + aia 2 a 4 + + a 2 a 3 a 4 + * + a n _ 2 a n _iO n . 

The number of these terms is the number of combinations 
that can be made from n a's, 3 at a time, i. e., n C 3 . 

d = Oia 2 a 3 a 4 + aia 2 a 3 a 6 + + a n _ 3 o n _ 2 a n _ia n . 
The number of these terms is n Ct. 

C r = aiO 2 o 3 a r + 
The number of these terms is C r . 

C n = aidzds ' a n , and consists of one term. 

275 



276 MATHEMATICS [XVI, 208 

If now each of the a's be replaced by y, it is evident that, 

Ci = ny, C z = n C 2 y*, C 3 = n C 3 y 3 , 
r - r if r - n n 

\sr n^ry , > ^n y j 

and therefore 

(2) (x + y) n = x n + nx n ~ l y + n C 2 x"- 2 z/ 2 + n C 3 x n ~ 3 y 3 + 

+ n C r x n ~ r y r + + nxy n ~ l + y n . 

This is known as the binomial expansion, or binomial formula. 

208. Binomial Theorem. If x and y are any real (or imagin- 
ary) numbers and if n is a positive integer, then the binomial 
formula (2) is valid. The following observations will be of 
value. 

(1) The exponent of x in the first term is 1 and decreases by 
1 in each succeeding term. 

(2) The exponent of y in the second term is 1 and increases 
by 1 in each succeeding term. 

(3) The coefficient of the first term is 1, that of the second 
term is n. The coefficient of any term can be found from the 
next preceding term by multiplying the coefficient by the exponent 
of x and dividing by one more than the exponent of y. 

(4) The (r + l)th term is n C r x n - r y r , i. e., 

n(n - l)(w - 2) (n - r + 1) 

__i 1J ' : ' ' ~n r,,r 

r! 

The coefficient of this (r + l)th term is the product of the 
first r factors of factorial n, divided by factorial r. 

(5) The sum of the exponents of x and y in any term is n. 

(6) The number of terms is n -\- 1. 

To prove the rule in statement (3) apply it to the (r + l)th 
term, 

n . ~nr*,r 
n^r X y . 



XVI, 210] THE BINOMIAL EXPANSION 



277 



It gives 



n r n(n 



r + 1 r! 

n(n - l)(n - 2) (n - r) 



2) (n r + 1) n r 
' r +1 



(r + l)l 

but this is precisely n CV+i, which was to be proved. 

209. Binomial Coefficients. The coefficients in the bi- 
nomial expansion are called binomial coefficients. Their values 
are given in the following table for a few values of n. This 
table is called Pascal's triangle. 

TABLE OP BINOMIAL COEFFICIENTS, n C r . PASCAL'S TRIANGLE 





r=0 


r = l 


r=2 


r=3 


r=4 


r=5 


r=6 


r=7 


r -8 


r=9 


r=lO 


r=ll 


n= 1 


1 


1 






















n= 2 


1 


2 


1 




















n= 3 


1 


3 


3 


1 


















n= 4 


1 


4 


6 


4 


1 
















n= 5 


1 


5 


10 


10 


5 


1 














n= 6 


1 


6 


15 


20 


15 


6 


1 












n= 7 


1 


7 


21 


35 


35 


21 


7 


1 










n= 8 


1 


8 


28 


56 


70 


56 


28 


8 


1 








n= 9 


1 


9 


36 


84 


126 


126 


84 


36 


9 


1 






n = 10 


1 


10 


45 


120 


210 


252 


210 


120 


45 


10 


1 




n = ll 


1 


11 


55 


165 


330 


462 


462 


330 


165 


55 


11 


1 


etc. 


etc. 


etc. 

















































NOTE. If any number in the table be added to the one on 
its right, the sum is the number under the latter. 

210. Sum of Binomial Coefficients. A great many uses for 
binomial coefficients and a great many relations among them 
have been discovered. Two of these are as follows. 

(1) The sum of the binomial coefficients of order n is 2 n . We 
verify from the above table that 

1 + 1- 2 1 ; 1+2 + 1= -2 2 ; 1 + 3 + 3 + 1 = 2 3 ; etc. 
To prove it for any value of n, put x = 1 and y = 1, in the 



278 MATHEMATICS [XVI, 211 

binomial formula: 

(1 + l) ra = 1 + n Cl + n C 2 + + Cn-l + n C n 

which proves the statement. 
Transposing 1, we have 

Ci + n C 2 + n C 3 + ' + n C n = 2 n -I 

i. e., the total number of combinations of n things taken 1,2, 3, 
" ' , n, at a time is 2 n 1. 

(2) The sum of the odd numbered coefficients is equal to the 
sum of the even numbered ones and each is 2"" 1 . 

We verify from the table, that 

1 = 1, 1 + 1=2, 1+3 = 3 + 1, 
1+6 + 1=4 + 4, etc. 

To prove it for any value of n, put x 1, y = 1, in the bi- 
nomial formula: 

(1 - 1)" = 1 - n d + n C 2 - n C 3 + n C 4 - n C n 

whence 

1 + nC'2 + nC* + = n C\ + n Cs + nCg + ' . 

211. Use of the Binomial Theorem. In expanding a bi- 
nomial with a given numerical exponent, the student is urged 
to find thq successive coefficients by using the statement (3) 208, 
and not by substitution in a formula. This is illustrated in the 
following examples. 

EXAMPLE 1. Expand (2z 3?/) 5 . 

(2x - Si/) 5 = (2x) + 5(2a;)(- 3?/) 1 + 10(2z) 3 (- Sy) 2 

+ 10(2x) 2 (- 3?/) 3 + 5(2z)(- ZyY + (- 3y) fi . 



XVI, 211] THE BINOMIAL EXPANSION 279 

The coefficients are computed mentally as follows, 

the 3d coefficient from the 2d term : 5X4/2 = 10, 
the 4th " " " 3d term : 10 X 3/3 = 10, 

the 5th " " " 4th term : 10 X 2/4 = 5, 

the 6th " " " 5th term : 5X1/5 = 1. 

Simplifying the terms, we have 

(2x - 3?/) 5 = 32z 8 - 240x 4 t/ + 720z 3 2/ 2 - 2160z 2 ?/ 3 + 810zy 4 - 243t/ 5 . 
EXAMPLE 2. Expand (3 |) 6 . 

(3 - |) 6 = 3 + 6(3) 6 (- i)i + 15(3)<(- W + 20(3)(- i) 3 

+ 15(3) 2 (- |)< + 6(3)'(- I) 5 + (- ) 6 . 
The coefficients are computed as follows: 

6X5/2 = 15, 15 X 4/3 = 20, 20 X 3/4 = 15, etc. 
Simplifying, we have 

+ 

729. 729. 

303.75 67.5 

8.4375 0.5625 



0-015625 797.0625 

1041.203125 
797.0625 
(2|) 6 = 244.140625 

EXAMPLE 3. Expand (a + b + c) 3 . 
[(a + 6) + c] 3 = (a + 6) 3 + 3 (a + 6) 2 c + 3 (a + 6)c 2 + c 3 

= a 3 + 3a 2 6 + 3afe 2 + b 3 + 3 (a 2 + 2ab + 6 2 )c 

+ 3(o + b)c 2 + c 3 
= a 3 + 6 3 + c 3 + 3a 2 6 + 3a 2 c + 36 2 c + 36 2 a + 3c 2 a 

+ 3c 2 6 + 6o6c. 
EXERCISES 

Expand the following expressions by the binomial theorem. 

1. (x + 3) 5 . 2. (y - 4). 3. (2 - a:) 4 . 

4. (2z + 3y) 3 . 5. (3x - 4y) 3 . 6. (3o + x 2 ) 6 . 

7. (x*+ y*)*. 8. (or 1 + 2ay~ 1 )*. 9. (a" 1 - x~ 2 )*. 



280 MATHEMATICS [XVI, 212 



10. (a 2 - b 2 ) 8 . 


11. (3a 2 b + 2C 3 ) 8 . 


12. (1 + x) 10 . 


(<! 9 V 
l+f) . 


14. (2x - ) 9 . 


15. (5 + i) 8 - 


16. (4.9) 3 . 


17. (1.01) B . 


18. (0.99). 


19. (1.9) 5 . 


20. (1.02) 4 . 


21. (15/8) 7 . 



22. Expand (1 + i) 5 and (2 - f) 5 and check results. 

23. Prove that any binomial coefficient, counted from the first, is 

equal to the same numbered one, counted from the last. 

212. Selected Terms. To select a particular term in the 
expansion of a binomial without computing the preceding terms, 
we can use the formula for the (r + l)th term, namely, 

the first r terms of n ! 

n C r x n ^ r y r = - x n ^y r . 

r\ 

(x \ 20 
9 ~~ 2y ) . 

Here r + 1 = 10, r = 9, n = 20, and the required term is 

20- 19- 18- 17- 16- 15- 14- 13- 12 /x\" _ _ 41990xlly9 

9-8-7-6.5-4. 3-2-1 \2 ) ( 

(r- 1 V 3 
\x + - j which contains x 2 . 

The (r + l)th term is i S C f r (x 1 / 2 ) 13 - r (x- 1 ) r = n C r -x 3 - 3r ^ 2 , whence r 
must be 3 and the 4th term is required. It is 



EXERCISES 

1. Find the 4th term of (4a - 6) 12 . 

2. Find the llth term of (2x - y) 17 . 

3. Find the 6th term of (xVy + yVx) 9 . 

4. Find the middle term of (x + 3?/) 8 . 

(x 2 V 
- + - j which does not contain x. 

f x y2\12 

6. Find the term of I *s. I which contains neither x nor 



XVI, 213] THE BINOMIAL EXPANSION 281 

213. The Binomial Series. The binomial theorem and the 
symbols n C r for the number of combinations of n things taken r 
at a time, have no meaning except when n and r are positive 
integers. On the other hand we know that such expressions as 

(1 + i)*/ 2 , (2 + 5)- 2 , (32 + 3) 1 /*, (1 - O.I)- 1 / 2 , 

have perfectly definite meanings; e. g., (2 + 5)~ 2 = 1/49. 

If we should expand a binomial whose exponent is not a 
positive integer by the binomial theorem (that is form the 
coefficients and exponents by the same rules as though the 
exponent were a positive integer), we should get a non-termi- 
nating series of terms. For example, 



(32 + 3) 1/5 = 32 1/5 

+ Tfy(32)-""(3) ---- . 

Now it is shown in advanced courses in mathematics, that 
this binomial series is actually valid, provided the numerical value 
of the first term of the binomial is greater than the numerical value 
of the second term. It is then valid, in the sense that if we 
begin at the first and add term after term, the more terms we 
take the nearer the sum approaches to the true value sought 
and that, by taking terms enough, the sum which we are com- 
puting will approximate the true value as nearly as we please. 

EXAMPLE. Find VlO by the binomial series. 
VlO = (8 + 2) 1 / 3 = 2(1 + i) 1 / 3 



Whence computing, we have 



+ 

1.0000- 
.0833- 
.0010- 
.0000- 


0.0069- 
.0002- 
.0000- 


1.0843--- 
.0071 


1.0772--- 
2 


1.0843--- 


0.0071 


2.1544- 



= VIo. 



282 MATHEMATICS [XVI, 214 

The student should note carefully that while the binomial 
series for (1 + ) 1/3 is valid, that for (- + 1) 1/3 is not. 

EXERCISES 

Expand the following in binomial series and simplify five terms. 

1 (1 4- x) 1 ^ 2 2 (1 -r- x)" 1 / 2 3 (1 x)" 1 / 3 

4. (0.98) 1/3 . 5. (1.02) 1/2 . 6. (0.99) 1/2 . 

7. V96. 8. v/30. 9. v'tKJ. 10. -\/33~. 
11. \/15. 12. v/65. 13. \ / 732. 



14. 


V1025. 


15. ^2400. 






16 


. ^125. 


17. 


flVimv thnt , 


11 1 ,2 1 


1 . 3^ 


,4 , 1 


3- 


5 


** + 


... 




T 2* ~T 


2- 4 


' 2 


4- 


6 


18. 


01, 1 


_1_ 1 _L 


1 4 . 


' 1 x 


4- 


7 


C 3 + 


... 


VI - x 


1 + 3 X ~T 


3- 6 


1 3 


6- 


9 


19. 


Show that (1 a;)" 2 


= 1 + 2x + 


3x 2 + 


4x 3 -i 


..... 








20. 


O how that 1 


-1 *x+- 


3 r2 


1- 


3- 


i 


' + ' 


". 



Vl +x 

214. Mendel's Law.* An Austrian monk by the name of 
Mendel planted some sweet peas of different colors in the garden 
of the monastery. These blossomed and produced seed. This 
seed was gathered and planted the following year. The flowers 
produced the second summer contained all of the colors of the 
first summer, but other colors were present. By observing and 
counting the number of flowers of each color Mendel discovered 
the law which bears his name. In its simplest form it may be 
explained as follows. 

Suppose a bed of sweet peas with blossoms half of which are 
red and half of which are white. Fertilization of the flowers 
by wind and insects will take place without selection. That is, 
pollen from a white flower is equally likely to fertilize a red or a 
white flower. If pollen from a white flower fertilizes a white 

* The following articles ( 214-216) are based largely upon Chapter XIV of E. 
DAVENPORT, Principles of Breeding, Ginn and Co. Much additional information may 
be found there. 



XVI, 215] 



LAWS OF HEREDITY 



283 



flower the seed produced is of pure stock and will produce pure 
white flowers the following year. Such flowers let us denote by 
W 2 . If pollen from a white flower fertilizes a red flower, or vice 
versa, the seed produced will be mixed stock and the following 
year will show its mixed character by producing flowers which are 
neither red nor white but some intermediate shade. Such 
flowers let us denote by RW. The symbol R 2 is now self-explan- 
atory. On counting the flowers which are pure white, mixed, 
and red, we would discover their numbers to be approximately 
in the ratio 1:2: 1. These are the coefficients in the expansion 
of (R + W) 2 . This is what one might have expected beforehand, 
as is seen from the adjoined table. Observe that there are twice 
as many flowers of mixed color as of either of the pure colors. 



Color of fertilizing flower. 


Color of flower fertilized. 


R 


W 


R... 


R? 
RW 


RW 

W 2 


w 





Result of mixing : R 2 + 2RW + W 2 . 

215. Successive Generations. Let R* denote the result of 
fertilizing R 2 with R 2 ; R S W denote the result of fertilizing R 2 
with RW, and so on. Then the results of indiscriminate fertili- 
zation of the flowers will be shown in the second generation, but 
in the third year, as given in the following table. 



Color of fertilizing 
flower and its relative 
numbers. 


Color of flower fertilized and their relative numbers. 


fi 


2RW 


W* 


&.. 


R 4 
2R 3 W 
R*W* 


2R 3 W 
4R?W 2 
2RW 3 


/PTP 
2RW 3 
W* 


2RW 


JP 





Result of mixing : R* + 4RW + 6RW 2 + 4RW 3 + W* 
Observe that the result in the second generation of mixing is 
the binomial expansion of (R + W) 4 . 



284 MATHEMATICS [XVI, 216 

Similarly we can show that the result in the third generation 
of mixing is given by (R + W) 8 , and so on. 

216. Mixing of Three Colors. Make a table, as above, but 
for three colors. Suppose the third color to be blue (B). Then 
a complete expression for the effect, in the first generation after 
mixing, is the following : 

(R + W + BY = R 2 + W 2 + B 2 + 2RW + 2RB + 2WB. 

In case the ratio of the number of white flowers to red flowers 
is as 2 to 3 then the result in the first generation after mixing 
is as follows : 

(2W + 3fl) 2 = 4PF 2 + 12WR + 9fl 2 . 

Mendel's law of heredity, as illustrated above by the dis- 
tribution of color in the successive generations of plants, applies 
to other transmissible characters in both plants and animals. 
That this distribution follows the mathematical laws of the 
binomial formula is due to the fact that each individual plant 
or animal inherits the characteristics of two parents, and hence 
the number two and its mathematical properties have their 
analogies in the laws of biology. 

EXERCISES 

1. Plot a few graphs, using binomial coefficients as ordinates and 
the number of the corresponding term as abscissas. 

2. How many varieties of sweet peas are produced by sowing in the 
same bed three different strains (a) first year ; (6) second year. 

Ans. (a) 6; (6) 14. 

3. A farmer buys two different kinds of thoroughbred chickens but 
allows them to mix freely. How many different kinds of chickens will 
he have at the end of (a) the first, (6) the second, (c) the third year of 
hatching? Ans. (a) 3, (6) 5, (c) 9. 

4. Four different varieties of wheat are planted side by side. How 
many different varieties will be harvested? Ans. 10. 

5. Plot graphs as indicated in Ex. 1 for the results of Ex. 3. 



XVI, 216] LAWS OF HEREDITY 285 

6. What varieties and in what proportion are obtained by freely 
mixing the first and second generations? 

7. I plant 8 sweet pea seeds 4 red, 4 white. Each seed produces 
16 flowers each flower matures 2 seeds which germinate and grow 
the following season. Find the total number of flowers, the proportion 
and number of the different kinds of flowers, in the (a) first, (6) second, 
and (c) third generations. 



CHAPTER XVII 
THE COMPOUND INTEREST LAW 

217. Compound Interest. Suppose one dollar to be loaned 
at compound interest at r% per annum payable annually. The 
interest i, due at the end of the first year, is r/100. The amount 
due is 1 + i- If interest is payable semiannually the amount 
due at the end of the first half year is 1 + i/2* If the interest 
is payable quarterly the amount due at the end of the first quar- 
ter is 1 + i/4. 

In general terms if the interest is payable p times a year at 
r% per annum compound, the amounts due on a principal of 
one dollar at the end of the 1st, 2d, , pth period are respec- 
tively, 



and the amounts due at the end of the 1st, 2d, , nth years are 
respectively, 

(i\ p 
1+*) 
pJ \ p \ p 

The amount A at the end of n years at r% per annum payable 
p times a year on a principal of P dollars is given by the formula 

* The amount of one dollar for n years compound interest at r% payable annually 
is (1 + i) n . If a settlement is made between two interest dates there is some divergence 
of practice in computing the interest for the fractional part of a year. The amount of 
one dollar for the pth part of a year by analogy to (1 + t) w would be (1 +t) l / p = Vl + i, 
but 1 + - is often used instead. When, however, by the terms of the note the interest 
is payable p times a year, and is to be compounded, it is clear that the amounts due at 
the end of 1, 2, , n periods are 



+LY .., 

p/ 



286 



XVII, 218] THE COMPOUND INTEREST LAW 287 



(,) A - P( 



218. Continuous Compounding. The larger p is the shorter 
the interval between the successive interest paying dates. As 
p increases without bound this interval approaches zero ; i.e. 
we can take p large enough to make this interval as small as we 
please. In the limit interest is said to be compounded contin- 
uously. While this state is never realized in financial affairs 
it is closely approximated. For example, large retail stores 
sell goods over the counter very nearly continuously and con- 
tinuously replenish their stock. 

Let us see what form equation (1) takes when p becomes 
infinite. Put x for i/p which approaches zero when p becomes 
infinite. Then (1) becomes 

in 1 

(2) A = P(l + x}* = P[(l + x) x ] in . 

Now it is shown in books on the Calculus that as x approaches 

i 

zero, the quantity (1 + x) x converges to a certain number be- 
tween 2 and 3. This number is the base of the natural or 
Napierian system of logarithms and is usually denoted by e. 
To five decimal places c = 2.71828. It can be shown that the 
following steps are justifiable, although the proof will not be 
given here. By the Binomial Formula, 



x 






As x approaches zero the terms on the right converge respectively 
to the terms of the series 



288 MATHEMATICS [XVII, J 218 

If we begin at the first and add the terms of this series, the 
more terms we add the nearer the sum comes to e. 00000 

The sum of the first ten terms is 2.71828, as is shown 

u.ouuuu u 

7 
7 



in the adjoining computation. ~ 

Then we conclude that as x approaches zero 



i 



0.04166 



(1 + x) x converges to e. 

Returning now to equation (2) we see that as p 0.00138 9 

becomes infinite and x approaches zero, A converges 0.00019 9 

to Pe in . Hence we say that when interest is com- 0.00002 

pounded continuously, the amount of P dollars at 0-00000 2 



r% per annum for n years is given by the equation 2.71828 
(3) A = Pe in , 

in which i = r/100 is the simple interest on one dollar for one 
year. This equation is said to represent the compound interest 
law. 

Scientific investigations reveal many examples of quantities 
whose rate of increase (or decrease) varies as the magnitude of 
the quantity itself. For example, the number of bacteria in a 
favorable medium, or the growth of an organic body by cell 
multiplication ; again the rate of decrease in atmospheric pres- 
sure in ascending a mountain is proportional to the pressure, 
and the rate of change in the volume of a gas expanding against 
resistance varies as the volume. The proverbial phrases, the 
rich grow richer, the poor poorer; nothing succeeds like success; 
a stitch in time saves nine; are expressions in popular language 
which show a recognition of this law in crude form.* 

In general terms if y and x are two varying quantities such 
that the rate of change in y (as regards a change in x) is known to 
vary directly as y itself, then they are connected by an equation 
of the form 

* See DAVIS, The Calculus, 81. 



XVII, 218] THE COMPOUND INTEREST LAW 289 



(4) y = 

in which c and k are constants. 

EXAMPLE. Suppose that atmospheric pressure at the earth's surface 
is 15 Ibs. per square inch and that it is 10 Ibs. per square inch at a height 
of 12,000 ft. If now it be assumed that the rate of decrease in the pres- 
sure is proportional to the pressure, we have from equation (4) 

p = ce trt . 

Substituting p = 15 when & = 0, we find c = 15; then substituting 
p = 10, h = 12,000, c = 15, we find 



12000 ' 



and these values of c and k give 



12000 



p = 

by means of which the pressure at any height h can be computed. 

This example illustrates the method of solving similar problems which 
fall under the compound interest law. We assume an equation of the 
form of (4) and determine the constants c and k by substituting in known 
pairs of values of x and y. Having determined the constants we insert 
them in the assumed formula which is then in form to give the value 
of y corresponding to any value whatever of x. 

EXERCISES 

1. Do you see any relation between the growth of plants, or the 
increase in population, and the compound interest law? Is the relation 
exact? What circumstances tend to limit its application? 

2. Is there any relation between your ability to acquire knowledge 
and to think clearly and the compound interest law? 

3. The population of the state of Washington was 349,400 in 1890 and 
in 1900 it was 518,100. Assume the relation P = ce T , where P = 
population, T = time in years after 1890, and predict the population 
for 1910. 

4. Using the data of Ex. 3, find the average annual rate of increase 
from 1890 to 1900. Assuming the same average rate to be maintained 
for the next 10 years, predict the population for 19^0. 



290 MATHEMATICS [XVII, 218 

5. When heated, a metal rod increases in length according to the 
compound interest law. If a rod is 40 ft. long at C., and 40.8 ft. long 
at 100 C., find (a) its length at 300 C; (&) at what. temperature its 
length will be 41 ft. 6 in. Ans. (a) 42.448 ; (&) 185.8 

6. The rate of increase in the tension of a belt is proportional to the 
tension as the distance changes from the point where the belt leaves the 
driven pulley. If the tension = 24 Ibs. at the driven pulley, and 32 Ibs. 
ten feet away, what is it six feet away? fAns. 28.52 

7. Assuming that the rate of increase in the number of bacteria in a 
given quantity of milk varies as the number present, if there are 10,000 
at 6 A.M., 60,000 at 9 A.M., how many will there be at 2 P.M.? At 
3 P.M.? At 6 P.M.? Ans. 2 P.M., 1,188,700. 

8. In the process of inversion of raw sugar, the rate of change is pro- 
portional to the amount of raw sugar remaining. If after 10 hours 1000 
Ibs. of raw sugar has been reduced to 800 Ibs., how much raw sugar will 
remain at the end of 24 hours? Ans. 586 Ibs. 



CHAPTER XVIII 
PROBABILITY 

219. Definition of Probability. // an event can happen in h 
ways, and fail in f ways, the total number of ways in which the 
event can happen and fail is h +/. Then h/(h +/) is said to 
be the probability that the event will happen, andf/(h-\-f) is said 
to be the probability that the event will fail. 

For example, suppose we have a box containing 4 red marbles 
and 5 white ones. Let us determine the chance of drawing a 
red marble the first time. This event can happen in 4 ways, and 
fail in 5 ways, while the total number of ways in which the 
event can happen and fail is nine. Then by the preceding defi- 
nition the probability of drawing a red marble is 4/9, and the 
probability of not drawing a red marble is 5/9. Observe that 
one of these things is certain to happen. The measure of this 
certainty is the sum of the probabilities of the separate events. This 
sum is 1 . Hence, if p is the probability that an event will happen, 
the probability q that it will not happen is 1 p. 

220. Statistical Probability. In a throw of a penny, before 
the event takes place, there is no reason to suspect that heads 
are more likely to turn up than tails. In a throw of a die any one 
of the six faces is equally likely to turn up and this probability 
does not depend upon the particular die used. The probability 
of a man's making a safe hit in a game of baseball, and that of 
not making a safe hit are not equal. Here the individuality 
of the batter enters and before the event takes place, if the batter 

291 



292 MATHEMATICS [XVIII, 220 

is unknown, we have nothing on which to make an estimate. 
If the batter is known, our estimate is based on his past perform- 
ance and this, unlike a throw of dice, depends upon the particu- 
lar individual at bat. If out of the last 60 times at bat, he has 
made a safe hit 20 times, then we say that the probability of his 
making a safe hit this time at bat is 1/3. 

Again what is the probability that a man aged 70 will die 
within the next year? Clearly this depends upon the individual, 
his present state of health, his habits, etc. In this case, how- 
ever, we can construct a measure of his probability of dying 
which is independent of these personal elements. From the 
American Experience Mortality Table (see Tables, p. 329), we 
find that out of 38,569 persons living at age 70, within the year 
2,391 die. Hence the probability that a man aged 70 will die 
within the year is 2,391 -4- 38,569. 

To derive the probability of an event from statistical data divide 
the number of cases h in which the event happened by the total num- 
ber n of cases observed. 

221. Expectation. If p is the probability that a man will 
win a certain sum s of money, then the product sp is called the 
value of his expectation. 

Thus the value of a lottery ticket in which the prize is $25 
and in which there are 500 tickets is $25 X 1/500, or 30 cents. 

EXERCISES 

1. According to the mortality table (p. 329) it appears that of 
100,000 persons at the age of 10, only 5,485 reach the age of 85. What is 
the probability that a child aged 10 will reach the age of 85? 

2. On 200 of 240 school days a student has had a grade of 90. What 
is the probability that his grade will be 90 on the 241st day? 

3. The weather bureau predicts rain for to-day. What is the prob- 
ability that it will rain, if on the average 90 out of every 100 predictions 
are correct? 



XVIII, 223] PROBABILITY 293 

/ 4. Compute the probability of throwing with 2 dice a sum of (a) 
seven, (6) eight, (c) nine, (d) ten, (e) eleven. 

Ans. (a) i; (6) &; (c) *; (d) & ; (e) &. 

5. Find the probability in drawing a card from a pack that it be (a) an 
ace, (6) a spade, (c) a face card, (d) not a face card. 

Ans. (a) A; (6) i; (c) &; (d) . 

6. Find the expectation of a man who is to win $300 if he holds one 
ticket out of a total of 1000 tickets. Ans. 30 cents. 

222. Mutually Exclusive Events . Two events are said to be 
mutually exclusive if the occurrence of one of them precludes the 
occurrence of the other. For example, in a race between A, B, 
and C, if A wins, B and C do not win. 

// the probabilities of the mutually exclusive events E\, E%, , 
E n are p\, pz, , p n , then the probability that some one will occur 
is the sum of the probabilities of the separate events. 

The meaning will be made clear by means of the following 
illustration. A bag contains 3 red, 4 white, and 5 blue balls. 
What is the probability that in a first draw we obtain a red or 
a white ball? There are 12 balls in all and 7 cases are favorable, 
namely 3 red and 4 white balls. Then from the definition of 
probability the chance of drawing a red ball or a white ball is 
7/12. But the probability of drawing a red ball is 3/12 and that 
of drawing a white ball is 4/12 and (3/12) + (4/12) = 7/12. 

223. Dependent Events. Events are said to be dependent 
if the occurrence of one influences the occurrence of the other. 
// the probability of a first event is p\ ; and if after this has happened 
the probability of a second event is p^; etc., ; and if after all 
those have happened the probability of an nth event is p n ; then 'the 
probability that all of the events will happen in the given order is 
Pi, P2 " p n . 

For, if the first event can happen in hi ways and can fail in /i 
ways ; and if after this has happened the second can happen in h% 
ways and can fail in fa ways ; etc., ; and if after these have hap- 



294 MATHEMATICS [XVIII, 223 

pened the nth event can happen in h n ways and can fail in / 
ways ; then they can all happen and fail in (hi + fi)(hz + f%) - 
(h n + /) ways. Now all the events can happen together in the 
given order in hi A 2 h n ways. Then by the definition of prob- 
ability the chance that all of the dependent events will take 
place in the given order is 

hi hz "- h n _ h\ hi h n 

' " 



(hi +fi)(h 2 +/ 2 ) -. (h n +/) hi +/! A 2 +/ 2 +/ 

= PlP* '" Pn- 

Thus the problem of drawing 2 red balls in succession from a 
bag containing 3 red and 2 black balls is (3/5) X (2/4) = 3/10. 
For after drawing one red ball and not replacing it the probability 
of drawing a red ball the second time is 2/4. 

224. Independent Events. Events are said to be independent 
when the occurrence of any one of them has nothing to do with 
the occurrence of the others. 

The probability that all of a set of independent events will take place 
is the product of the probabilities of the independent simple events. 

This follows as a corollary from the theorem of 223. 

Thus the probability of throwing a deuce twice in succession 
is (1/6) X (1/6) = 1/36. 

EXERCISES 

1. If the batting average of Tyrus Cobb is 0,400 what is the chance 
that in any single time at bat he will make a safe hit ? 

2. What is the probability of holding 4 aces in a game of whist? 

Ans. 1/270,725. 

3. Suppose I enter 2 horses for a race and that the probabilities of 
their winning are respectively | and j. What is the probability that 
one or the other will win the race? Ans. 3/4. 

4. Does Ex. 3 teach us anything with respect to diversified farming? 
Discuss the probability of crop failure of a single crop as compared with 
that of two or more different crops. 

5. Three men A, B, C go duck hunting. A has a record of one bird 



XVIII, 223] PROBABILITY 295 

out of two, B gets two out of three, C gets three out of four. What is 
the probability that they kill a duck at which all shoot at once ? 

Ans. 23/24. 

6. What is the chance of drawing a white and red ball in the order 
named from a bag containing 5 white and 6 red balls? Ans. 3/11. 

7. In a certain zone in times of war 23 out of 5000 ships are sunk by 
submarine in one week. What is the chance that a single vessel will 
cross the zone safely? What is the chance that all of 4 vessels which 
enter the zone at the same time will cross in safety ? What is the chance 
that of these 4 exactly 3 will cross in safety ? That at least 3 will cross 
in safety? 

8. In certain branches of the army service 2% of the men are killed 
each year. Three brothers enlist in this branch of the service for a 
period of two years. Compute the probability that (a) all will survive, 
(b) exactly 2 will survive, (c) at least 2 will survive, (d) exactly one will 
survive, (e) at least one will survive, (/) none will survive. 

9. At the time of marriage the probabilities that a husband and wife 
will each live 50 years are \ and j respectively. Compute the probabil- 
ity that (a) both will be alive, (b) both dead, (c) husband alive and wife 
dead, (d) wife alive husband dead. 

10. From the American Experience Table of Mortality (Tables, 
p. 329) compute your chances of living 1, 10, 20, 30, 40, 50 years. 

11. From the American Experience Table of Mortality (Tables, 
p. 329) find that age to which you now have an even chance of living. 

12. Find from the same table that age to which a person aged 20 has 
an even chance of living. Ans. 66 + . 

13. Three horses are entered for a race. The published odds are 5 : 4 
for A ; 3:2 against B ; 4:3 against C. Is it possible to place bets in 
such a way that I win some money no matter which horse wins ? 

Ans. Yes. 

14. Suppose n horses entered for a race, and let the published odds 
be (a 1) to 1 against the first ; (6 1) to 1 against the second, (c 1) 
to 1 against the third and so on. A man bets (a l)/a to I/a against 
the first; (b l)/6 to 1/6 against the second, etc. Show that whatever 
horse wins his gains are represented algebraically by the formula 

f 5+; + 



296 



MATHEMATICS 



[XVIII, 225 



225. Frequency Distribution Curves.* A sample of 400 oats 
plants were taken from an experimental plot and measured as to 
height in centimeters with the following results : f 



Height, H 


45 
50 


50 
55 


55 
60 


60 
65 


65 
70 


70 
75 


75 

80 


80 
85 


85 
90 


90 
95 
























Frequencies, F 


2 


9 


21 


34 


97 


m 


89 


?4 





1 

























Let us plot this data with heights as abscissas and frequencies 
as ordinates. Construct rectangles, with bases on the horizontal 



rr 1 














































^, 


































































/ 


\ 










































/ 






v 














































\ 






















/ 




80 
















/ 
/ 










V 




































/ 




- 






\ 
\ 




















-GO 


























\ 
































/ 














j 


















10 














/ 
/ 














\ 
\ 














































i 
j 




























/ 


' 








































/ 
























^ 


















.- 


"" 




























\ 
5 










s i 


'i 


45 


5 





5 


5 


6 





f 


5 


7 





' 


5 


E 





f 





<J 





9 


5 


/* 

















































FIG. 128 

axis. Let the width of the base in each case be 5 units, which 
agrees with the grouping of the measurements as to height. 

* In the remainder of this Chapter ( 225-231), the authors are indebted for many 
ideas to E. DAVENPORT, Principles of Breeding [Chapter XII and Appendix (H. L. 
HIETZ)]. Other books containing similar matter are JOHNSON, Theory of Errors and 
Method of Least Squares: WRIGHT AND HAYFORD, Adjustments of Observations; MERRI- 
MAN, Textbook of Least Squares; WELD, Theory of Errors and Least Sqiiares; etc. 

t MEMOIR No. 3, CORNELL UNIVERSITY AGRICULTURAL EXPERIMENT STATION, 
Variation and Correlation of Oats, by H. H. LOVE and C. E. LEIGHTY, Aug., 1914. 



XVIII, 226] PROBABILITY 297 

Let the height of the individual rectangles be representative of 
the frequency for the corresponding heights of plants, as shown 
in Fig. 128. 

The upper parts of these rectangles form an irregular curve 
made up of segments of straight lines. A smoother curve is 
obtained by connecting the middle points of the upper bases of 
these rectangles by segments of straight lines as shown by the 
dotted line in Fig. 128. Instead of the dotted line we may draw 
a smooth curve as near as possible to the middle points of the 
upper bases. Any curve drawn as nearly as possible through 
a series of plotted points representing a distribution with respect 
to a given character is called a frequency distribution curve. 

Such curves are useful in presenting to the eye some of the 
features of a distribution. The type of character most fre- 
quent is represented by the mode ( 198), which is the value 
of the abscissa corresponding to the highest point of the curve. 
The median measurement of the group ( 197) is represented by 
the abscissa of that ordinate on either side of which there are 
equal areas under the curve. The arithmetic average ( 195) 
is the abscissa of the center of gravity of the area under the 
curve. 

Frequency distribution curves are plotted for a great variety 
of things, such as frequency distribution of people with respect 
to height, weight, or age ; grains of wheat with respect to weight ; 
alfalfa with respect to duration of bloom in days ; cherry trees 
with respect to earliness of bloom ; pigs with respect to size of 
litter; diphtheria with respect to time of year; women with 
respect to age of marriage ; etc. 

226. Probability Curve. If a large number of measurements 
are made upon the same item, they will not in general agree. 
Let us plot as abscissas the measurements observed and as ordi- 
nates their relative frequencies. In most cases, the positive 



298 



MATHEMATICS 



[XVIII, 226 



and negative errors are equally likely to occur, and small errors 
are more numerous than large ones. The frequency curve for 
the observed data would then have its highest point at the true 
value of the measured magnitude, would be symmetric about an 
ordinate through this highest point, and would rapidly approach 
the axis of abscissas both to the right and left of this maximum 
ordinate. If we take the vertical through the highest point as an 
axis of y, then abscissas will represent errors of observation and 
ordinates will represent frequency of error. 

The curve so drawn is well represented by the equation 



(1) 



y = 



in which cr is what we shall call the standard deviation, e = 
2.71828 the base of Napierian logarithms, n the number of 
observations, x the error of a reading, y the probability of an 




error x. This curve is called the probability curve or curve 
of error. 

While the theoretical curve (1) is symmetric, the curves ob- 
tained by plotting the results of statistical study are often 
not symmetric. However the formulas developed in this chapter 
for the symmetric case can be used for approximate results in the 
non-symmetric cases. 

227. Standard D e viation . It is not enough to know the value 



XVIII, 227] PROBABILITY 299 

of the arithmetic average or the mode. It is important to have 
a measure of the tendency to deviate from the average or from 
the mode. 

The general theory will be explained by means of the data of 
225, which represents the measurements of the heights of 400 
oat plants. From this data the average height of oat plant is 
70.8 centimeters. Compute the deviation, D, of these plants 
from their average height. Multiply the square of each devia- 
tion by its corresponding frequency and add the results. We 
get 19,320. Divide by the sum, 400, of the frequencies. The 
quotient is 48.3. We next extract the square root since the de- 
viations have all been squared in the above calculations. We 
get 6.95~, and this is called the standard deviation. 

In general, to find the standard deviation, 

Compute the deviation of each frequency from the arithmetic 
average. Multiply the square of each deviation by its corresponding 
frequency and add the results. Divide by the sum of thefrequenci.es. 
Extract the square root. 

This rule is symbolized in the following formula : 



(2) 

The curve A in Fig. 129 represents the distribution when a is 
small, and the curve B represents the distribution when a is large. 

For example, the two sets of numbers 7, 7, 8, 8, 8, 8, 9, 9 and 
5, 6, 7, 8, 8, 9, 10, 11 have the same arithmetic mean. The 
second set, however, shows a greater tendency to vary from 
the arithmetic average (type) than does the first. This greater 
tendency to vary is shown by the larger value for cr for the 
second set. The values of a- are 0.706 and 1.87 respectively. 

Again, suppose two men are shooting at a mark, and that we 
compute the standard deviation for each. The man for whom <r 
is smallest is said to be the more consistent shot. 



300 MATHEMATICS [XVIII, 228 

228. Coefficient of Variability. A comparison of the standard 
deviations of two different groups conveys little information 
as to their respective tendencies to deviate from the arithmetic 
average. This is due to two causes : (1) the measurements may 
be in different units, as centimeters and grams, (2) one average 
may be much larger than the other, for example the average 
height of a group of men would be larger than the average length 
of ears of corn. We need then a measure of variability which 
is independent of the units used and takes into account the 
relative magnitudes of the means. Such a measure is the 
coefficient of variability, which is denoted by C and is determined 
by the formula, 

/ O x r _ Standard deviation _ <r 

\y) v "7 r~j : 

Arithmetic average ra 

For example, the coefficient of variability in height of the 400 
oat plants considered in 225 is 6.95/70.8, or approximately 
10%. 

229. Probable Error of a Single Measurement. Any indi- 
vidual measurement is likely to be in error. This error is ap- 
proximately the difference between this measurement and the 
arithmetic average of all the measurements. Compute these 
errors for all the measurements, some positive, some negative. 
Give them all positive signs and arrange them in order of magni- 
tude. The median of this list is called the probable error of a 
single measurement of the set and is denoted by E s . It is 
shown in the theory of probability that 

(4) E s = 0.67450-. 

230. Probable Error in the Arithmetic Average. Take a 
sample of 500 ears of corn from a crib. Compute the arithmetic 
average of their lengths. We use this to represent the mean 
length of all the ears in the crib. Quite likely it differs from their 
true arithmetic average. We now find by means of equation 



XVIII, 231] PROBABILITY 301 

(5) below, a number E m , called the probable error in the arith- 
metic average. This is a number such that it is equally likely 
whether or not the computed arithmetic average of the 500 ears 
selected lies between ra E m and m + E m , where m denotes the 
(unknown) true arithmetic average for all the ears in the crib. 
In other words if a very large number of persons take a sample 
of ears and each computes an average length, then, in a sufficiently 
large number of cases, one half of these averages will be within 
the limits set and one half will be without. 
In treatises on probability it is shown that 

P E, 0.6745o- 

\P) &m - ~j= = T= 

Vn Vn 

This formula shows that in order to double the precision of the 
computed arithmetic average it is necessary to take four times as 
many observations. 

231. Probable Error in the Standard Deviation. Compute 
the standard deviation, 227, of the lengths of 500 ears of corn 
from a crib. This will differ slightly from the true standard 
deviation <r, of the lengths of all the ears in the crib. Next find, 
by means of equation (6) below, the probable error E ay of the 
standard deviation. Then for a sufficiently large number of 
samples from the crib, the computed standard deviations will 
fall one half within the limits a- E* and a- + E y , and one half 
without. The formula for the probable error in the standard 
deviation is 
ttrt F .- E m _ 0.6745Q- 

\P) &v j= . 

V2 V2n 

EXERCISES 

1. Compute E,, E m , E for the data in 225. 

2. Compute <r, C, E,, E m , E<r for the following sets of measurements, 
(a) 5, 6, 7, 8, 8, 9, 10, 11 ; (6) 5, 5, 5, 7, 9, 10, 11, 12. 

(c) 1, 6, 8, 8, 8, 8, 10, 15 ; (d) 51, 56, 58, 58, 58, 58, 60, 65. 



302 



MATHEMATICS 



[XVIII, 231 



3. Compute <r, C, E,, E m , E* for the following distribution of oat 
plants with respect to height in centimeters [LOVE-LEIGHTY]. 



Height 


60 


65 


70 


75 


80 


85 


90 


















Frequency .... 


2 


11 


45 


140 


122 


73 


7 



(a)- 



(6) 



Height 


60 


65 


70 


75 


80 


85 


90 


95 




















Frequency. . . . 


11 


36 


60 


94 


99 


102 


68 


18 



4. Compute from the following data the mode, the mean, the coef- 
ficient of variability, the standard deviation, the probable error in the 
mean, and the probable error in the standard deviation. 



Lbs. of butter fat . . 
No. of cows 


400 
1 


375 

?, 


350 
4 


325 
5 


300 

7 


275 
6 


250 
5 


225 

2 


200 
1 























Draw the distribution curve. 

5. The following table is taken from BULLETIN 110, PART 1, Bureau 
of Animal Husbandry, U. S. Dept. of Agriculture on "A BIOMETRICAL 
STUDY OF EGG PRODUCTION IN THE DOMESTIC FOWL" and shows the 
frequency distribution for hens in first-year egg production. 



Annual Egg 
Production^ 


A 


* 


IS 


M 


60 
7? 


If 


90 
ITJ? 


m 


120 
134 


m 


ill 


i-n 


180 

y? 


195 
25'J 


fl 
23 


IIS 


1902-03 
1903-04 


7 


2 
5 


5 


1 

10 


5 
10 


8 
?0 


17 

?4 


18 
W 


17 

5? 


26 
37 


17 
W 


18 
16 


9 

8 


2 
2 


6 


1 


1905-06 






r 


?, 


4 


q 


13 


?5 


?4 


?,?, 


3?! 


17 


90 


q 






1906-07 
(a) 


2 


2 


5 


5 


q 


16 


30 


39 


?6 


?1 


1Q 


1? 


1 








00 


10 


8 


8 


15 


29 


32 


48 


39 


36 


25 


18 


6 


5 




2 





From this data compute for each year the mean, the median, and the 
mode for egg production. Compute ff, C, E<r, E m , E,. Draw the dis- 
tribution curve. 

6. From Table I at the end of Chapter XIX compute for each weight 
(length) the mean, the median, and the mode for length (weight). 
Compute <r, C, E<r, E m , E, of weight (length) for each length (weight). 

7. For Table II (p. 312) follow the directions as given in Ex. 6 for 
Table I, reading however number of kernels instead of weight. 



XVIII, 231] PROBABILITY 303 

8. For Table III (p. 312) follow the directions as given in Ex. 6 
for Table I, reading yield and number of culms in place of weight 
and length. 

9. For Table IV (p. 313) follow the directions as given in Ex. 6. 
Read height of mid-parent and height of adult children in place of 
weight and length. 



CHAPTER XIX 
CORRELATION* 

232. Meaning of Correlation. Whenever two quantities 
are so related that an increase in one of them produces or is ac- 
companied by an increase in the other and the greater the in- 
crease in the one the greater the increase in the other, these 
quantities are said to be correlated positively. If an increase 
in one produces, or is accompanied by, a decrease in the other, 
they are said to be correlated negatively. If a change in one is 
not accompanied by any change in the other, there is no corre- 
lation, and the quantities are said to be unrelated. Perfect 
positive correlation is represented by the number + 1, perfect 
negative correlation by 1, no correlation by zero. There 
is perfect positive correlation between the area of a rectangular 
field and its length, the extension of a spiral spring and the sus- 
pended load. There is perfect negative correlation between 
the pressure and volume of a perfect gas. No relation exists 
between the price of coal and the length of ears of corn. 

There are quantities, common in everyday life, such that a 
change in one is not accompanied by a proportionate change 
in the other, but a given change in one is always accompanied by 
some change in the other. Such quantities are still said to be 
correlated. The degree of relationship may be anywhere be- 
tween complete independence and complete dependence, that is 

* Throughout this Chapter, the authors have consulted the following books, and are 
indebted to them for ideas: E. DAVENPORT, Principles of Breeding (Chap. XIII); 
ZIZEK, Statistical Averages; SECRIST, Introduction to Statistical Methods; PEAKSON, 
Grammar of Science; BOWLEV, Elements of Statistics. 

304 



XIX, 232] 



CORRELATION 



305 



between zero and + 1 or between zero and 1. For example 
we may mention the effect of potato prices on acreage, and vice 
versa. 

^Ve desire a numerical measure for this correlation. Any 
adequate expression must be such that it becomes zero when 
there is no correlation, 1 when there is perfect negative corre- 
lation, + 1 for perfect positive correlation, and which is always 
between 1 and + 1. Yule has proposed a formula which 
satisfies these conditions. Arrange the observed data with refer- 
ence to the two quantities in question as in the following dia- 
gram : 





x present. 


x absent. 


y present 


U 


V 


y absent 


T 


S 









Then a measure m of the correlation existing is given by the 
equation 

(1) 



If either r or v is zero 
If either u or s is zero 
If us = rv 



us + rv 

m = + 1. 
m = 1. 
m = 0. 

EXERCISES 



1. Compute from the following table the degree of effectiveness of 
vaccination against diphtheria : 





Recoveries. 


Deaths. 


Vaccinated . 


2843 


106 


Not vaccinated 


254 


225 









2. Compute from the following table the correlation between prohi- 
bition and the arrests per day in a given city for one year : 



306 



MATHEMATICS 



[XIX, 232 





Days with more than 20 arrests. 


Less than 20. 


Wet 


281 


84 


Dry 


142 


223 









3. Compute the correlation between use of fertilizer and yield of 
potatoes in bushels per acre when the results from fifty plats are as 
follows : 





Yield over 100 bushels.) 


Under 100 bushels. 


Fertilizer. 


47 


3 


No fertilizer 


14 


36 









Ans. 0.95 

This high value of correlation is considered evidence of some connec- 
tion between use of fertilizer and yield. 

233. Correlation Table. Let it be proposed to find the de- 
gree of correlation, if any, between the lengths of ears of corn and 
their weight, between their lengths and number of rows of 
kernels, between length and circumference, between length and 
yield per acre, between length of head of wheat and yield per 
acre, between height of wheat and yield per acre. The problem 
is now more complex. Let us take for example a given number 
of ears of corn and examine them as to weight in ounces and 
length in inches. The measurements may be tabulated as shown 
in the accompanying table. Each column is a frequency dis- 
tribution of lengths for a constant weight. Each row is a fre- 
quency distribution of weights for a constant length. The 
distribution of the ears of length 8 inches with respect to weight 
is 3, 7, 19, 25, 17, 22, 17, 3, 1. 

It is to be noticed that the table extends across the enclosing 
rectangle from the upper left-hand corner to the lower right- 
hand corner. Whenever data tabulated with respect to two 
measurable characters show this skew arrangement, correlation 
exists. In the accompanying table weights increase from left 



XIX, 234] 



CORRELATION 



307 



to right and lengths increase as we move downward. We 
have in this case positive correlation. An extension of the array 
from the upper right-hand corner to the lower left would have 
indicated negative correlation. 

234. Coefficient of Correlation. The method of obtain- 
ing the correlation coefficient may be explained in connection 

CORRELATION BETWEEN WEIGHT AND LENGTH OF EAR * 







Weight of Ear in Ounces. 






2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


is 


19 


20 


21 




3 


1 


2 




1 




































3.5 




4 




1 




































4 


3 


5 


5 


1 


































o5 


4.5 




6 


5 


4 






1 




























QJ 




5 




2 


4 


7 


2 


4 






























o 
a 


5.5 




2 


9 


15 


14 


8 


4 


1 


























s 


6 




1 


2 


12 


16 


13 


13 


6 


1 
























a 

iH 


6.5 






1 


6 


11 


26 


11 


8 


6 


1 






















03 


7 






1 


2 


2 


12 


18 


12 


12 


11 


4 


1 


















H 


7.5 








1 


2 


4 


20 


12 


13 


21 


11 


6 


6 


1 


1 












s 


8 












3 


7 


19 


25 


17 


22 


17 


3 


1 














,D 


8.5 












1 


1 


12 


9 


23 


30 


26 


26 


5 


1 












1 


9 
















1 


7 


10 


23 


35 


26 


24 


12 


1 


2 


1 






a 

CJ 


9.5 


















1 


4 


14 


19 


29 


17 


10 


1 


3 




1 


1 


^ 


10 


















1 


1 


3 


8 


18 


10 


(I 


4 


2 










10.5 
























2 


3 


6 


7 


2 


5 


1 








11 


























1 


1 








2 


1 






11.5 


































] 









with the above table. Find the arithmetic mean of each char- 
acter involved in this case mean length of ear, MI, and mean 
weight of ears, M w . Find the deviation DI of ear length from 
mean length, and the deviation D w of weight from mean weight, 
for each ear tabulated. For each ear tabulated find the product 
of DI and D w and then add all of these products. This sum we 
will indicate by 3DiD w . Find in the usual way the standard 
deviation of length of ears, <TI, and the standard deviation of 
weight of ears, <r w . Then the coefficient of correlation, r, is 

* E. DAVENPORT, Principles of Breeding, p. 458. 



308 



MATHEMATICS 



[XIX, 234 



given by the formula 

(2) 



where n is the number of things observed, in this case the total 
number of ears. 

A convenient arrangement for computing DI for each ear length and 
D w for each ear weight is shown in the table below. 

The row labeled 6.5 inches (table 233), gives the frequency distri- 
bution of ears with respect to weight. There is one ear of weight 
4 oz., 6 ears of weight 5 oz., 11 ears of weight 6 oz., 26 ears of weight 
7 oz., etc.; a total of 70 ears, fi, of length 6.5 inches. 
fiVi = 1X4 + 6X5 + 11X6 + 26 X7 + 11X8 + 8X9 

+ 6 X 10 + 1 X 11 = 455.0 

The mean length of ear is obtained by adding the numbers in the 
column headed fiVi and dividing this sum by the total number, 
n = 993, of ears. 

CORRELATION OF WEIGHT TO LENGTH OF EARS OF CORN 



Length, 


ji 


SiVi 


DI 


Weight, 


t 


fvVw 


D w 


DiD v 


Inches. 








Ounces. 










3 


4 


12.0 


- 4.8 


2 


4 


8 


-8.7 


143.0 


3.5 


5 


17.5 


- 4.3 


3 


22 


66 


-7.7 


156.9 


4 


14 


56.0 


- 3.8 


4 


27 


108 


-6.7 


394.4 


4.5 


16 


72.0 


- 3.3 


5 


50 


250 


-5.7 


347.2 


5 


19 


95.0 


- 2.8 


6 


47 


282 


-4.7 


297.6 


5.5 


53 


291.5 


- 2.3 


7 


71 


497 


-3.7 


618.9 


6 


64 


384.0 


- 1.8 


8 


75 


600 


-2.7 


465.8 


6.5 


70 


455.0 


- 1.3 


9 


71 


639 


-1.7 


306.8 


7 


75 


525.0 


- 0.8 


10 


75 


750 


-0.7 


110.8 


7.5 


98 


735.0 


- 0.3 


11 


88 


968 


0.3 


14.9 


8 


114 


912.0 


0.2 


12 


107 


1,284 


1.3 


1.4 


8.5 


134 


1,139.0 


0.7 


13 


114 


1,482 


2.3 


129.6 


9 


142 


1,278.0 


1.2 


14 


112 


1,568 


3.3 


466.3. 


9.5 


100 


950.0 


1.7 


15 


65 


975 


4.3 


564.4 


10 


53 


530.0 


2.2 


16 


37 


592 


5.3 


431.0 


10.5 


26 


273.0 


2.7 


17 


8 


136 


6.3 


364.0 


11 


5 


55.0 


3.2 


18 


13 


234 


7.3 


107.2 


11.5 


1 


11.5 


3.7 


19 


4 


76 


8.3 


27.0 




993 


7,791.5 




20 


2 


40 


9.3 










21 


1 


21 


10.3 




M 


7791.5 


5 




993 


10,576 






1 993 




M w 


10,576 


= 10.65 


993 



* E. DAVENPORT, Principles of Breeding, p. 461. 



XIX, 235] CORRELATION 309 

All of the symbols used have been defined with the exception of the 
following : <r/ is the standard deviation of length ; /, is the number (fre- 
quency) of ears of same weight w ; Vi stands for the value of length of 
ears with given frequency ; V w represents the value of weight of ears 
with given frequency. This gives MI = 7.85. In the row labeled 6.5 
and in the column headed DI we write the difference between this mean 
length 7.85 and the length 6.5. This gives the number 1.3 of the col- 
umn headed D/. The number 306.8 in the last column is obtained as 
follows : 

(- 1.3)[1(- 6.7) + 6(- 5.7) + 11(- 4.7) + 26(- 3.7) 

+ 11(- 2.7) + 8(- 1.7) + 6(- 0.7) + 1(0.3)] = 306.8 

That is, the ear of weight 4 oz. deviates from the mean weight by 6.7 oz., 
the 6 ears of weight 5 oz. deviate from the mean weight by 5.7 oz., 
the 11 ears of weight 6 oz. deviate from the mean weight by 4.7 oz., 
etc. 

The number 306.8 represents the sum of the products of the cor- 
responding length and weight deviations for every individual in the 
horizontal row to which the number belongs. To find the correlation 
coefficient add the numbers in the column headed DiD w , obtaining in 
this case 4947.2. 

Divide this number 4947.2 by n X <TI X . In this case n = 993, 
and <TI, ff w have been computed to be 1.57 and 3.63 respectively. 
This gives the correlation coefficient 

- - 4947 ' 2 = 0.87 



993(1.57) (3.63) 

235. The Regression Curve. For each recorded height (see 
table, 233) compute the arithmetic average of length of ears. 
Thus the ears of weight 4 oz. have an average length of 5.1 inches. 
The ears of weight 5 oz. have an average length of 5.46 inches, 
etc. Plot a curve using for abscissas the weights, and for 
ordinates the computed average lengths. The curve so plotted 
is called a regression curve. In many cases this curve is a 
straight line. It can be shown that the straight line which best 
represents the plotted data is given by the equation 



310 MATHEMATICS [XIX, 235 

(2) M t = r^lw. 

a w 

Another regression curve can be plotted for the same data, 
using lengths as ordinates and mean weights for abscissas. This 
curve does not in general coincide with the first. Its equation is 

M w = r^l. 
&1 

By means of these curves the mean value of one character can be 
read off when a fixed value is given to the other character. 

EXERCISES 

1 . Find, for the correlation table in 233 : 

(a) the regression of weight relative to length ; 

(b) regression of length relative to weight. 

Ans. (a) 2.03 (6) 0.38 

2. Find the equation of the line of regression in both cases of Ex. 1. 

3. Plot the line of regression in Ex. 2 from the equation found there 
and then again plot the line from the data as suggested in 235. 

4. From Table II, p. 312, which gives the correlation of height of oat 
plants with the average number per plant of kernels per culm, compute 
the mean height, the mean number of kernels per culm, the standard 
deviation with respect to height, the standard deviation with respect to 
number of kernels per culm, the correlation coefficient, and the regres- 
sion coefficients. 

5. Examine Table IV, p. 313, which gives the number of children of 
various statures born of 205 mid-parents of various statures. From 
this table compute : 

M p = mean height of mid-parents, 

M e mean height of adult children, 
ff p = standard deviation of height of mid-parents, 
ff e = standard deviation of height of adult children, 
r = the correlation coefficient, and both regression coefficients. 

6. For Ex. 4 plot the lines of regression (a) from their equations, 
(6) from the data directly. 

7. For Ex. 5 plot the lines of regression (a) from their equations, 
(6) from the data directly. 



XIX, 235] 



CORRELATION 



311 



8. From the following table find a measure of the effectiveness of 
vaccination against smallpox. 





Recoveries. 


Deaths. 


Total. 


Vaccinated 


3,951 


200 


4,151 


Not vaccinated 


278 


274 


552 


Total 


4,229 


474 


4,703 











9. Construct a correlation table from your own observations on 
length and breadth of leaves, (a) Use 30 classes for length, (fe) Use 
15 classes for length, thus making the class interval twice as large. 
Compute in each case the correlation coefficient. 

10. From Table I, below, which gives the conslation of lengths and 
weights of ears of corn, compute the mean length, the mean weight, 
the standard deviation with respect to length, the standard deviation 
with respect to weight, the correlation coefficient, and both regression 
coefficients. 

11. The same as Ex. 10 after writing number of kernels in place of 
weight, using Table II, p. 312, in place of Table I. 

I. CORRELATION OP LENGTH AND WEIGHT OF EARS OF CORN 



Length 
In Inches. 


Weight In Ounces. 


2 


3 


4 


5 


c 


7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


is 


3.0 




1 


1 


1 




























3.5 




1 


2 


2 


1 


























4.0 


2 


3 


5 


4 


1 


























4.5 




4 


5 


6 


2 


1 
























5.0 




4 


7 


8 


6 


4 


1 






















5.5 




3 


9 


12 


13 


8 


3 


1 




















6.0 




1 


5 


10 


15 


12 


9 


5 


2 


















6.5 






2 


6 


12 


26 


14 


10 


5 


3 


1 














7.0 






1 


3 


4 


14 


18 


15 


10 


7 


2 


1 












7.5 








1 


2 


6 


13 


17 


19 


13 


9 


6 


4 


2 








8.0 












2 


7 


10 


13 


19 


7 


6 


2 


1 








8.5 














1 


3 


9 


14 


25 


17 


8 


5 


1 






9.0 
















1 


4 


7 


19 


25 


16 


11 


3 






9.5 


















2 


3 


8 


18 


20 


15 


6 


1 




10.0 




















1 


3 


9 


18 


13 


7 


5 


2 


10.5 
























2 


3 


7 


5 


4 


1 


11.0 


























1 


2 


3 


2 





312 



MATHEMATICS 



[XIX, 235 



II. CORRELATION OP AVERAGE HEIGHT OF OAT PLANTS IN CENTI- 
METERS AND AVERAGE NUMBER OF KERNELS PER CULM PER 
PLANT. [LOVE-LEIGHTY.] r = 0.73. 





Number of Kernels. 


Height. 
























30 


40; 


50 


GO 


70 


80 


90 


100 


110 


120 




40 


50 


60 


70 


80 


90 


100 


110 


120 


130 


55-60 


1 




1 
















60-65 




4 


7 
















65-70 






7 


22 


9 


6 


1 








70-75 






1 


13 


30 


59 


32 


5 






75-80 








2 


16 


40 


38 


23 


3 




80-85 










1 


12 


26 


23 


9 


2 


85-90 
















3 


2 


2 



III. CORRELATION OF NUMBER OF CULMS PER OAT PLANT AND 
TOTAL YIELD OF PLANT IN GRAMS. [LOVE-LEIGHTY.] 
r = 0.712 



Yield 


Number of Culms per Plant. 


2 


3 


4 


5 


6 


7 


0-1 


3 

28 
18 
1 


19 
66 

42 

7 


3 
20 
58 
59 
26 

1 


1 

7 
11 
14 
4 

1 


1 

3 
2 
3 

1 


1 


1-2 


2-3 


3-4 


4-5 . 


5-6 


6-7 


7-8 . 


8-9 





XIX, 235] 



CORRELATION 



313 



IV. CORRELATION OF HEIGHTS OF ADULT CHILDREN AND PARENTS 
DATA FOR CHILDREN OF 205 MID-PARENTS* OF VARIOUS STATURES 



Heights of 
Mid-parents. 


Heights of Adult Children In Inches. 


Above. 


73.2 


72.2 


71.2 


70.2 


69.2 


68.2 


67.2 


66.2 


65.2 


64.2 


63.2 


62.2 


Below. 


Above 




3 


1 
























72.5 


4 


2 


7 


2 


1 


2 


1 
















71.5 


2 


2 


9 


4 


10 


5 


3 


4 


3 


1 










70.5 


3 


3 


4 


7 


14 


18 


12 


3 


1 


1 




1 




1 


69.5 


5 


4 


11 


20 


25 


33 


20 


27 


17 


4 


16 


1 






68.5 




3 


4 


18 


21 


48 


34 


31 


25 


16 


11 


7 




1 


67.5 






4 


11 


19 


38 


28 


38 


36 


15 


14 


5 


3 




66.5 










4 


13 


14 


17 


17 


2 


5 


3 


3 




65.5 






1 


2 


5 


7 


7 


11 


11 


7 


5 


9 




1 


64.5 
















5 


5 


1 


4 


4 


1 


1 


Below 












1 


1 


2 


2 


1 


4 


2 




1 



* Height of mid-parent is the mean height of the two parents. 
[GALTON-DAVENPORT] 



GREEK ALPHABET 



LETTERS NAMES 



A a Alpha 


H, 


B Beta 


90 


T 7 Gamma 


It 


A 6 Delta 


KK 


E Epsilon 


AX 


Z f Zeta 


MM 




LETTERS NAMES 


LETTERS NAMES 


N v 


Nu 


TT Tau 


H 


Xi 


T v Upsilon 


Oo 


Omicron 


> Phi 


UTT 


Pi 


X X Chi 


Pp 


Rho 


\ff Psi 


So-s 


Sigma 


Q w Omega 



314 



FOUR PLACE TABLES 

PAGES 

I. LOGARITHMS OF NUMBERS 316-319 

II. VALUES AND LOGARITHMS OF TRIGONOMETRIC 

FUNCTIONS 320-324 

III. RADIAN MEASURE TRIGONOMETRIC FUNCTIONS 325 

IV. SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS 326 
V. IMPORTANT CONSTANTS 327 

VI. DEGREES TO RADIANS 327 

VII. COMPOUND INTEREST 328 

VIII. AMERICAN EXPERIENCE MORTALITY TABLE. . . 32!) 

IX. HEIGHTS AND WEIGHTS OF MEN 330 

EXPLANATION OF TABLE II . . 331-333 



315 



Table I. Logarithms of Numbers 



N. 





1 


2 


3 


4 


5 


6 


7 


8 9 


Prop. Parts 







0000 


3010 


4771 


6021 


6990 


7782 


8451 


9031 


9542 




22 


21 




1 


0000 


0414 


0792 


1139 


1461 


1761 


2041 


2304 


2553 


2788 


l 


2.2 


2.1 


2 


3010 


3222 


3424 


3617 


3802 


3979 


4150 


4314 


4472 


4624 


2 


4.4 


4.2 


3 


4771" 


4914 


5051 


5185 


5315 


5441 


5563 


5682 


5798 


5911 


3 

4 


6.6 
8.8 


6.3 
8.4 
























5 


11.0 


10.5 


4 


6021 


6128 


6232 


6335 


6435 


6532 


6628 


6721 


6812 


6902 


6 


13.2 


12.6 


5 


6990 


7076 


7160 


7243 


7324 


7404 


7482 


7559 


7634 


7709 


7 


15.4 


14.7 


6 


7782 


7853 


7924 


7993 


8062 


8129 


8195 


8261 


8325 


8388 


8 
9 


17.6 
19.8 


16.8 
18.9 


7 


8451 


8513 


8573 


8633 


8692 


8751 


8808 


8865 


8921 


8976 




20 


19 


8 


9031 


9085 


9138 


9191 


9243 


9294 


9345 


9395 


9445 


9494 


1 


2.0 


1.9 


9 


9542 


9590 


9638 


9685 


9731 


9777 


9823 


9868 


9912 


9956 


2 
3 


4.0 
6 


3.8 

5 7 


10 


0000 


0043 


"0086 


0128 


0170" 


0212 


0253" 


0294 


0334 


0374 


4 


8.0 


7^6 


11 


0414 


0453 


0492 


0531 


0569 


0607 


0645 


0682 


0719 


0755 


5 


10.0 


9.5 


12 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038 


1072 


1106 


6 

7 


12.0 
14 


11.4 
13 3 


13 


1139 


1173 


1206 


1239 


1271 


1303 


1335 


1367 


1399 


1430 


8 


ie!o 


15.2 
























9 118.0 


17.1 


14 


1461 


1492 


1523 


1553 


1584 


1614 


1644 


1673 


1703 


1732 




IB 


17 


15 


1761 


1790 


1818 


1847 


1875 


1903 


1931 


1959 


1987 


2014 


1 


lo 

1.8 


i / 

1 7 


16 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


2279 


2 


3.6 


3.4 
























3 


5.4 


5.1 


17 


2304 


2330 


2355 


2380 


2405 


2430 


2455 


2480 


2504 


2529 


4 


7.2 


6.8 


18 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 


2742 


2765 


5 


9.0 


8.5 


19 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 


2967 


2989 


6 

7 


10.8 
12.6 


10.2 
11.9 


20 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


8 


14^4 


13.6 


21 


3222 


3243 


3263 


3284 


3304 


3324 


3345 


3365 


3385 


3404 


9 


16.2 


15.3 


22 


3424 


3444 


3464 


3483 


3502 


3522 


3541 


3560 


3579 


3598 




16 


15 


23 


3617 


3636 


3655 


3674 


3692 


3711 


3729 


3747 


3766 


3784 


1 


1.6 


1.5 
























2 


3.2 


3.0 


24 


3802 


3820 


3838 


3856 


3874 


3892 


3909 


3927 


3945 


3962 


3 


4.8 


4.5 


25 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4116 


4133 


4 
5 


6.4 
8.0 


6.0 
7.5 


26 


4150 


4166 


4183 


4200 


4216 


4232 


4249 


4265 


4281 


4298 


6 


9^6 


9^0 
























7 


11.2 


10.5 


27 


4314 


4330 


4346 


4362 


4378 


4393 


4409 


4425 


4440 


4456 


8 


12.8 12.0 


28 


4472 


4487 


4502 


4518 


4533 


4548 


4564 


4579 


4594 


4609 


9 


14.4 13.5 


29 


4624 


4639 


4654 


4669 


4683 


4698 


4713 


4728 


4742 


4757 




14 


13 


30 


4771 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


1 


1.4 

O o 


1.3 


31 


4914 


4928 


4942 


4955 


4969 


4983 


4997 


5011 


5024 


5038 


3 


.O 

4.2 


2.6 
3.9 


32 


5051 


5065 


5079 


5092 


5105 


5119 


5132 


5145 


5159 


5172 


4 


5.6 


5.2 


33 


5185 


5198 


5211 


5224 


5237 


5250 


5263 


5276 


5289 


5302 


5 


7.0 


6.5 
























6 


8.4 


78 


34 


5315 


5328 


5340 


5353 


5366 


5378 


5391 


5403 


5416 


5428 


7 


9.8 


9.1 


35 


5441 


5453 


5465 


5478 


5490 


5502 


5514 


5527 


5539 


5551 


8 
9 


11.2 
12.6 


10.4 
11.7 


36 


5563 


5575 


5587 


5599 


5611 


5623 


5635 


5647 


5658 


5670 
































12 


11 


37 


5682 


5694 


5705 


5717 


5729 


5740 


5752 


5763 


5775 


5786 


1 


1 2 


1.1 


38 


5798 


5809 


5821 


5832 


5843 


5855 


5866 


5877 


5888 


5899 


2 
3 


2.4 
3.6 


2.2 
3.3 


39 


5911 


5922 


5933 


5944 


5955 


5966 


5977 


5988 


5999 


6010 


4 


4^8 


4.4 


40 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 


5 


6.0 

7 A 


5.5 

ft ft 


41 


6128 


6138 


6149 


6160 ; 6170 


6180 


6191 


6201 


6212 


6222 


7 


. 

8.4 


D.n 

7.7 


42 


6232 


6243 


6253 


6263 6274 


6284 


6294 


6304 


6314 


6325 


8 


9.6 


8.8 


43 


6335 


6345 


6355 


6365 


6375 


6385 


6395 


6405 


6415 


6425 


9 


10.8 


9.9 


























9 


8 


44 


6435 


6444 


6454 


6464 


6474 


6484 


6493 


6503 


6513 


6522 


1 


0.9 


0.8 


45 


6532 


6542 


6551 


6561 6571 


6580 


6590 


6599 


6609 


6618 


2 


1.8 


1.6 


46 


6628 


6637 


6646 


6656 6665 


6675 


6684 


6693 


6702 


6712 


3 


2.7 


2.4 
























4 


3.6 


3.2 


47 


6721 


6730 


6739 


6749 


6758 


6767 


6776 6785 


6794 


6803 


5 


4.5 


4.0 


48 


6812 


6821 


6830 


6839 


6848 


6857 


6866 6875 


6884 


6893 


6 

7 


5.4 
6.3 


4.8 
5.6 


49 


6902 


6911 


6920 


6928 


6937 


6946 


6955 6964 


6972 


6981 


8 


7^2 


6^4 


50 


6990 


6998 


7007 


"7016" 


7024 


7033 


7042 7050 


7059 


7067 


9 8.1 


7.2 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 





316 



Table I. Logarithms of Numbers 



N. 





1 


2 


3 


4 


5 


6 | 7 


8 


9 


Prop. Parts 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 


7067 






51 


7076 


7084 


7093 


7101 


7110 


7118 


7126 


7135 


7143 


7152 




9 


52 


7160 


7168 


7177 


7185 


7193 


7202 


7210 


7218 


7226 


7235 


2 


1 8 


53 


7243 


7251 


7259 


7267 


7275 


7284 


7292 


7300 


7308 


7316 


3 


2.7 
























4 


3.6 


54 


7324 


7332 


7340 


7348 


7356 


7364 


7372 


7380 


7388 


7396 


5 


4.5 


55 


7404 


7412 


7419 


7427 


7435 


7443 


7451 


7459 


7466 


7474 


6 


5.4 


56 


7482 


7490 


7497 


7505 


7513 


7520 


7528 


7536 


7543 


7551 


7 
8 


6.3 

7.2 
























9 


8.1 


57 


7559 


7566 


7574 


7582 


7589 


7597 


7604 


7612 


7619 


7627 




58 


7634 


7642 


7649 


7657 


7664 


7672 


7679 


7686 


7694 


7701 




59 


7709 


7716 


7723 


7731 


7738 


7745 


7752 


7760 7767 


7774 




8 


60 


7782 


7789 


7796 


7803 


7810 


7818 


7825 


7832 


7839 


7846 


1 


0.8 


61 


7853 


7860 


7868 


7875 


7882 


7889 


7896 


7903 


7910 


7917 


3 


2.4 


62 


7924 


7931 


7938 


7945 


7952 


7959 


7966 


7973 


7980 


7987 


4 


3.2 


63 


7993 


8000 


8007 


8014 


8021 


8028 


8035 


8041 


8048 


8055 


5 


4.0 
























6 


4.8 


64 


8062 


8069 


8075 


8082 


8089 


8096 


8102 


8109 


8116 


8122 


7 


5.0 


65 


8129 


8136 


8142 


8149 


8156 


8162 


8169 


8176 


8182 


8189 


8 
g 


7 9 


66 


8195 


8202 


8209 


8215 


8222 


8228 


8235 


8241 


8248 


8254 




67 


8261 


8267 


8274 


8280 


8287 


8293 


8299 


8306 


8312 


8319 




7 


68 


8325 


8331 


8338 


8344 


8351 


8357 


8363 8370 8376 


8382 


1 


0.7 


69 


8388 


8395 


8401 


8407 


8414 


8420 


8426 8432 8439 


8445 


2 


1.4 


70 


8451 


8457 


8463 


8470 


8476 


84 S2 


8488 8494 


8500 


8506 


3 
4 


2.1 
2.8 


71 


8513 


8519 


8525 


8531 


8537 


85-13 


8549 


8555 


8561 


8567 


5 


3.5 


72 


8573 


8579 


8585 


8591 


8597 


8603 


8609 


8615 


8621 


8627 


6 


4.2 


73 


8633 


8639 


8645 


8651 


8657 


8663 


8669 


8675 


8681 


8686 


7 


4.9 
























8 


5.6 


74 


8692 


8698 


8704 


8710 


8716 


8722 


8727 


8733 


8739 


8745 


9 


6.3 


75 


8751 


8756 


8762 


8768 


8774 


8779 


8785 


8791 


8797 


8802 




76 


8808 


8814 


8820 


8825 


8831 


8837 


8842 


8848 


8854 


8859 




6 


77 


8865 


8871 


8876 


8882 


8887 


8893 


8899 


8904 


8910 


8915 


1 
2 


0.6 
1.2 


78 


8921 


8927 


8932 


8938 


8943 


8949 


8954 


8960 


8965 


8971 


3 


1.8 


79 


8976 


8982 


8987 


8993 


8998 


9004 


9009 


9015 


9020 


9025 


4 


2.4 


80 


!K.)31 


9036 9042 


9047 


9053 


905X 


9063 


9069 


9074 


9079 


6 


3.6 


81 


9085 


9090 


9096 


9101 


9106 


9112 


9117 


9122 


9128 


9133 


7 


4.2 


82 


9138 


9143 


9149 


9154 


9159 


9165 


9170 


9175 


9180 


9186 


8 


4.8 


83 


9191 


9196 


9201 


9206 


9212 


9217 


9222 


9227 


9232 


9238 


9 


5.4 


84 


9243 


9248 


9253 


9258 


9263 


9269 


9274 


9279 


9284 


9289 






85 


9294 


9299 


9304 


9309 


9315 


9320 


9325 


9330 


9335 


9340 






86 


9345 


9350 


9355 


9360 


9365 


9370 


9375 


9380 


9385 


9390 


2 


1.0 
























3 


1 5 


87 


9395 


9400 


9405 


9410 


9415 


9420 


9425 


9430 


9435 


9440 


4 


2.0 


88 


9445 


9450 


9455 


9460 


9465 


9469 


9474 


9479 , 9484 


9489 


5 


2.5 


89 


9494 


9499 


'.).-,( )4 


9509 


9513 


9518 


9523 


9528 


9533 


053S 


e 


3.0 


90 


9542 


9547 


9552 


9557 


9562 


9566 


9571 


9576 


95S1 


9586 


8 


4.0 


91 


9590 


9595 


9600 


9605 


9609 


9614 


9619 


!)<>_> i 


9<>L>S 


9633 


9 


4.5 


92 


9638 


9643 


9647 


9652 


9657 


9661 


9666 


9671 


9675 


9680 




93 


9685 


9689 


9694 


9699 


9703 


9708 


9713 


9717 


9722 


9727 




























4 


94 


9731 


9736 


9741 


9745 


9750 


9754 


9759 


9763 


9768 


9773 


1 


04 


95 


9777 


9782 


9786 


9791 | 9795 


9800 


9805 


9809 9814 9818 


3 


1 2 


96 


9823 


9827 


9832 


9836 


9841 


9845 


9850 


9854 


9859 


9863 


4 


1.6 
























5 


2.0 


97 


9868 


9872 


9877 


9881 


9886 


9890 


9894 


9899 


9903 


9908 


G 


2.4 


98 


9912 


9917 


9921 


9926 


9930 


9934 


9939 


9943 


9948 


9952 


7 


2.8 


99 


9956 


9961 


<)'.)< if, 


9969 


9974 


0!>7S 


9983 ; 9987 ; 9991 


9996 


8 
9 


3.2 
3.6 


100 


0000 


0004 


0009 


0013 


0017 


0022 


0026 ; 0030 


0035 


0039 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 





317 



Table I. Logarithms of Numbers 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Parts 


100 


0000 


0004 


0009 


0013 


0017 


0022 


0026 


0030 


0035 


0039 




101 


0043 


0048 


0052 


0056 


0060 


0065 


0069 


0073 


0077 


0082 




102 


0086 


0090 


0095 


0099 


0103 


0107 


0111 


0116 ,0120 


0124 




103 


0128 


0133 


0137 


0141 


0145 


0149 


0154 


0158 


0162 


0166 




104 


0170 


0175 


0179 


0183 


0187 


0191 


0195 


0199 


0204 


0208 




105 


0212 


0216 


0220 


0224 


0228 


0233 


0237 


0241 


0245 


0249 




106 


0253 


0257 


0261 


0265 


0269 


0273 


0278 


0282 


0286 


0290 




5 
























1 


0.5 


107 


0294 


0298 


0302 


0306 


0310 


0314 


0318 


0322 


0326 


0330 


2 


1.0 


108 


0334 


0338 


0342 


0346 


0350 


0354 


0358 


0362 


0366 


0370 


3 


1.5 


109 


0374 


0378 


0382 


0386 


0390 


0394 


0398 


0402 


0406 


0410 


5 


2.5 


110 


0414 


0418 


0422 


0426 


0430 


0434 


0438 


0441 


0445 


0449 


6 


3.0 


111 


0453 


0457 


"0461 


04~65" 


0469" 


0473" 


0477 


0481 


0484 


0488 


7 
g 


3.5 
1 


112 


0492 


0496 


0500 


0504 


0508 


0512 


0515 


0519 


0523 


0527 


9 


4.5 


113 


0531 


0535 


0538 


0542 


0546 


0550 


0554 


0558 


0561 


0565 




114 


0569 


0573 


0577 


0580 


0584 


0588 


0592 


0596 


0599 


0603 




115 


0607 


0611 


0615 


0618 


0622 


0626 


0630 


0633 


0637 


0641 




116 


0645 


0648 


0652 


0656 


0660 


0663 


0667 


0671 


0674 


0678 




























4 


117 


0682 


0686 


0689 


0693 


0697 


0700 


0704 


0708 


0711 


0715 


1 


0.4 


118 


0719 


0722 


0726 


0730 


0734 


0737 


0741 


0745 


0748 


0752 


2 


0.8 


119 


0755 


0759 


0763 


0766 


0770 


0774 


0777 


0781 


0785 


0788 


3 


1.2 


120 


0792 


0795 


0799 


0803 


0806 


0810 


0813 


0817 


0821 


0824 


5 


2.0 


121 


0828 


0831 


0835 


0839 


0842 


0846 


0849 


0853 


0856 


0860 


6 


2.4 


122 


0864 


0867 


0871 


0874 


0878 


0881 


0885 


0888 


0892 


0896 


7 


2.8 


123 


0899 


0903 


0906 


0910 


0913 


0917 


0920 


0924 


0927 


0931 


8 
9 


3.2 
3.6 


124 


0934 


0938 


0941 


0945 


0948 


0952 


0955 


0959 


0962 


0966 




125 


0969 


0973 


0976 


0980 


0983 


0986 


0990 


0993 


0997 


1000 




126 


1004 


1007 


1011 


1014 


1017 


1021 


1024 


1028 


1031 


1035 




127 


1038 


1041 


1045 


1048 


1052 


1055 


1059 


1062 


1065 


1069 






128 


1072 


1075 


1079 


1082 


1086 


1089 


1092 


1096 


1099 


1103 




3 


129 


1106 


1109 


1113 


1116 


1119 


1123 


1126 


1129 


1133 


1136 


2 


0.6 


130 


1139 


1143 


1146 


1149 


1153 


1156 


1159 


1163 


1166 


1169 


3 


0.9 


131 


1173 


1176 


1179 


1183 


1186 


1189 


1193 


1196 


1199 


1202 


5 


1 5 


132 


1206 


1209 


1212 


1216 


1219 


1222 


1225 


1229 


1232 


1235 


6 


1.8 


133 


1239 


1242 


1245 


1248 


1252 


1255 


1258 


1261 


1265 


1268 


7 


2.1 
























8 


2.4 


134 


1271 


1274 


1278 


1281 


1284 


1287 


1290 


1294 


1297 


1300 


9 


2.7 


135 


1303 


1307 


1310 


1313 


1316 


1319 


1323 


1326 


1329 


1332 




136 


1335 


1339 


1342 


1345 


1348 


1351 


1355 


1358 


1361 


1364 




137 


1367 


1370 


1374 


1377 


1380 


1383 


1386 


1389 


1392 


1396 




138 


1399 


1402 


1405 


1408 


1411 


1414 


1418 


1421 


1424 


1427 






139 


1430 


1433 


1436 


1440 


1443 


1446 


1449 


1452 


1455 


1458 




2 


140 


1*461 


1464 


1467 


1471 


1474 


1477 


1480 


1483 


1486 


1489 


o 


0.4 


141 


1492 


1495 


1498 


1501 


1504 


1508 


1511 


1514 


1517 


1520 


3 


0.6 


142 


1523 


1526 


1529 


1532 


1535 


1538 


1541 


1544 


1547 


1550 


4 


0.8 


143 


1553 


1556 


1559 


1562 


1565 


1569 


1572 


1575 


1578 


1581 


5 
6 


1.0 
1.2 
























7 


1 4 


144 


1584 


1587 


1590 


1593 


1596 


1599 


1602 


1605 


1608 


1611 


8 


1.6 


145 


1614 


1617 


1620 


1623 


1626 


1629 


1632 


1635 


1638 


1641 


9 


l.S 


146 


1644 


1647 


1649 


1652 


1655 


1658 


1661 


1664 


1667 


1670 




147 


1673 


1676 


1679 


1682 


1685 


1688 


1691 


1694 


1697 


1700 




148 


1703 


1706 


1708 


1711 


1714 


1717 


1720 


1723 


1726 


1729 




149 


1732 


1735 


1738 


1741 


1744 


1746 


1749 


1752 


1755 


1758 




150 


1761 


1764 


1767 


1770 


1772 


1775 


1778 


1781 


1784 


1787 




No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 





318 



Table I. Logarithms of Numbers 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Parts 


150 


1761 


1764 


1767 


1770 


1772 


1775 


1778 


1781 


1784 


1787 




151 


1790 


1793 


1796 


1798 


1801 


1804 


1807 


1810 


1813 


1816 




152 


1818 


1821 


1824 


1827 


1830 


1833 


1836 


1838 


1841 


1844 




153 


1847 


1850 


1853 


1855 


1858 


1861 


1864 


1867 


1870 


1872 




154 


1875 


1878 


1881 


1884 


1886 


1889 


1892 


1895 


1898 


1901 




155 


1903 


1906 


1909 


1912 


1915 


1917 


1920 


1923 


1926 


1928 




156 


1931 


1934 


1937 


1940 


1942 


1945 


1948 


1951 


1953 


1956 


3 
























1 0.3 


157 


1959 


1962 


1965 


1967 


1970 


1973 


1976 


1978 


1981 


1984 


2 0.6 


158 


1987 


1989 


1992 


1995 


1998 


2000 


2003 


2006 


2009 


2011 


3 0.9 


159 


2014 


2017 


2019 


2022 


2025 


2028 


2030 


2033 


2036 


2038 


4 1.2 
5 15 


160 


2041 


2044 


2047 


2049 


2052 


2055 


2057 


2060 


2063 


20(10 


6 1.8 


161 


2068 


2071 


2074 


2076 


2079 


2082 


2084 


2087 


2090 


2092 


7 2.1 


162 


2095 


2098 


2101 


2103 


2106 


2109 


2111 


2114 


2117 


2119 


9 2.7 


163 


2122 


2125 


2127 


2130 


2133 


2135 


2138 


2140 


2143 


2146 




164 


2148 


2151 


2154 


2156 


2159 


2162 


2164 


2167 


2170 


2172 




165 


2175 


2177 


2180 


2183 


2185 


2188 


2191 


2193 


2196 


2198 




166 


2201 


2204 


2206 


2209 


2212 


2214 


2217 


2219 


2222 


2225 




167 


2227 


2230 


2232 


2235 


2238 


2240 


2243 


2245 


2248 


2251 




168 


2253 


2256 


2258 


2261 


2263 


2266 


2269 


2271 


2274 


2276 




169 


2279 


2281 


2284 


2287 


2289 


2292 


2294 


2297 


2299 


2302 




170 


2304 


2307 


2310 2312 


2315 


2317 


2320 


2322 


2:525 


2327 




171 


2330 


2333 


2335 


2338 


2340 


2343 


2345 


2348 


2350 


2353 




172 


2355 


2358 


2360 


2363 


2365 


2368 


2370 


2373 


2375 


2378 




173 


2380 


2383 


2385 


2388 


2390 


2393 


2395 


2398 


2400 


2403 




174 


2405 


2408 


2410 


2413 


2415 


2418 


2420 


2423 


2425 


2428 




175 


2430 


2433 


2435 


2438 


2440 


2443 


2445 


2448 


2450 


2453 




176 


2455 


2458 


2460 


2463 


2465 


2467 


2470 


2472 


2475 


2477 


2 
























1 0.2 


177 


2480 


2482 


2485 


2487 


2490 


2492 


2494 


2497 


2499 


2502 


2 0.4 


178 


2504 


2507 


2509 


2512 


2514 


2516 


2519 


2521 


2524 


2526 


3 0.6 


179 


2529 


2531 


2533 


2.-,:;(> 


2538 


2541 


2543 


2545 


2548 


2550 


5 1.0 


180 


2553 


2555 


2558 


2560 


2562 


2565 


2567 


2570 


2572 


2574 


6 1.2 


181 


2577 


2579 


2582 


2584 


2586 


2589 


2591 


2594 


2596 


2598 


7 1.4 
8 16 


182 


2601 


2603 


2605 


2608 


2610 


2613 


2615 


2617 


2620 


2622 


9 1.8 


183 


2625 


2627 


2629 


2632 


2634 


2636 


2639 


2641 


2643 


2646 




184 


2648 


2651 


2653 


2655 


2658 


2660 


2662 


2665 


2667 


2669 




185 


2672 


2674 


2676 


2679 


2681 


2683 


2686 


2688 


2690 


2693 




186 


2695 


2697 


2700 


2702 


2704 


2707 


2709 


2711 


2714 


2716 




187 


2718 


2721 


2723 


2725 


2728 


2730 


2732 


2735 


2737 


2739 




188 


2742 


2744 


2746 


2749 


2751 


2753 


2755 


2758 


2760 


2762 




189 


2765 


2767 


2769 


2772 


2774 


2776 


2778 


2781 


2783 


2785 




190 


27SH 


2790 


2792 


2794 


2797 


2799 


2801 


2804 


2806 


2808 




191 


2810 


2813 


2815 


2817 


2819 


2822 


2824 


2S2(> 


2S2S 


2831 




192 


2833 


2835 


2838 


2840 


2842 


2844 


2847 


2849 


2851 


2853 




193 


2856 


2858 


2860 


2862 


2865 


2867 


2869 


2871 


2874 


2876 




194 


2878 


2880 


2882 


2885 


2887 


2889 


2891 


2894 


2896 


2898 




195 


2900 


2903 


2905 


2907 


2909 


2911 


2914 


2916 


2918 


2920 




196 


2923 


2925 


2927 


2929 


2931 


2934 


2936 


2938 


2940 


2942 




197 


2945 


2947 


2949 


2951 


2953 


2956 


2958 


2960 


2962 


2964 




198 


2967 


2969 


2971 


2973 


2975 


2978 


2980 


2982 


2984 


2986 




199 


2989 


2991 


2993 


2995 


2997 


21MHI 


3002 


3004 


3006 


3008 




200 


3010 


3012 


3015 


3017 


3019 


3021 


3023 3025 


3028 


3030 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 





,519 



Table II. Values and Logarithms of Trigonometric Functions 

[Characteristics of Logarithms omitted determine by the usual rule from the value] 



RADIANS 


DEGREES 


SINE 


TANGENT 


COTANGENT 


COSINE 










Value Log 10 


Value Lo<r 10 


Value Logio 


Value Log 10 






.0000 


000' 


0000 


0000 




1.0000 .0000 


90 00' 


1 .5708 


!(X)29 


10 


.0029 .4637 


.0029 .4637 


343.77 .5363 


1.0000 .0000 


50 


1^5679 


.0058 


20 


.0058 .7(548 


.0058 .7648 


171.89 .2352 


1.0000 .0000 


40 


1.5650 


.0087 


30 


.0087 .9408 


.0087 .9409 


114.59 .0591 


1.0000 .0000 


30 


1.5621 


.0116 


40 


.0116 .0658 


.0116 .0658 


85.940 .9342 


.9999 .0000 


20 


1.5592 


.0145 


50 


.0145 .1627 


.0145 .1627 


68.750 .8373 


.9999 .0000 


10 


1.5563 


.0175 


100' 


.0175 .2419 


.0175 .2419 


57.290 .7581 


.9998 .9999 


89 00' 


1.5533 


.0204 


10 


.0204 .3088 


.0204 .3089 


49.104 .6911 


.9998 .9999 


50 


1.5504 


.0233 


20 


.0233 .3668 


.0233 .3(569 


42.964 .6331 


.9997 .9999 


40 


1.5475 


.02(52 


30 


.0262 .4179 


.0262 .4181 


38.188 .5819 


.9997 .9999 


30 


1.5446 


.021)1 


40 


.0291 .4637 


.0291 .4638 


34.368 .5362 


.9996 .9998 


20 


1.5417 


.0320 


50 


.0320 .5050 


.0320 .5053 


31.242 .4947 


.9995 .9998 


10 


1.5388 


.0349 


2 00' 


.0349 .5428 


.0349 .5431 


28.636 .4569 


.9994 .9997 


8 8 00' 


1.5359 


.0378 


10 


.0378 .5776 


.0378 .5779 


2(5.432 .4221 


.9993 .9997 


50 


1.5330 


.0407 


20 


.0407 .6097 


.0407 .6101 


24.542 .3899 


.9992 .9996 


40 


1.5301 


.0436 


30 


.0436 .6397 


.0437 .6401 


22.904 .3595) 


.9990 .9996 


30 


1.5272 


.0465 


40 


.0465 .6677 


.0466 .6682 


21.470 .3318 


.9989 .9995 


20 


1.5243 


.0495 


50 


.0494 .6940 


.0495 .6945 


20.206 .3055 


.9988 .9995 


10 


1.5213 


.0524 


3 00' 


.0523 .7188 


.0524 .7194 


19.081 .2806 


.9986 .9994 


87 00' 


1.5184 


.0553 


10 


.0552 .7423 


.0553 .7429 


18.075 .2571 


.9985 .C993 


50 


1.5155 


.0582 


20 


.0581 .7645 


.0582 .7652 


17.169 .2348 


.9983 .9993 


40 


1.5126 


.0611 


30 


.0610 .7857 


.0612 .7865 


16.350 .2135 


.9981 .9992 


30 


1.5097 


.0640 


40 


.0640 .8059 


.0(541 .8067 


15.605 .1933 


.9980 .9991 


20 


1.5068 


.0669 


50 


.0669 .8251 


.0670 .8261 


14.924 .1739 


.9978 .9990 


10 


1.5039 


.0698 


4 00' 


.0698 .8436 


.0699 .8446 


14.301 .1554 


.9976 .9989 


86 00' 


1.5010 


.0727 


10 


.0727 .8613 


.0729 .8624 


13.727 .1376 


.9974 .9989 


50 


1.4981 


.0756 


20 


.0756 .8783 


.0758 .8795 


13.197 .1205 


.9971 .9988 


40 


1.4952 


.0785 


30 


.0785 .8946 


.0787 .8960 


12.706 .1040 


.9969 .9987 


30 


1.4923 


.0814 


40 


.0814 .9104 


.0816 .9118 


12.251 .0882 


.9967 .9986 


20 


1.4893 


.0844 


50 


.0843 .9256 


.0846 .9272 


11.826 .0728 


.9964 .9985 


10 


1.4864 


.0873 


5 00' 


.0872 .9403 


.0875 .9420 


11.430 .0580 


.9962 .9983 


85 00' 


1.4835 


.0902 


10 


.0901 .9545 


.0904 .9563 


11.059 .0437 


.9959 .9982 


50 


1.4806 


.0931 


20 


.0929 .9682 


.0934 .9701 


10.712 .0299 


.9957 .9981 


40 


1.4777 


.09(50 


30 


.0958 .9816 


.0963 .9836 


10.385 .0164 


.9954 .9980 


30 


1.4748 


.0989 


40 


.0987 .9945 


.0992 .95X56 


10.078 .0034 


.9951 .9979 


20 


1.4719 


.1018 


50 


.1016 .0070 


.1022 .0093 


9.7882 .9907 


.9948 .9977 


10 


1.4690 


.1047 


6 00' 


.1045 .0192 


.1051 .0216 


9.5144 .9784 


'.9945 .9976 


84 00' 


1.4661 


.1076 


10 


.1074 .0311 


.1080 .0336 


9.'_'5. r )3 .9664 


.9942 .9975 


50 


1.4632 


.1105 


20 


.1103 .0426 


.1110 .0453 


9.0098 .9547 


.9939 .9973 


40 


1.4(503 


.1134 


30 


.1132 .0539 


.1139 .05(57 


8.7769 .94a3 


.9936 .9972 


30 


1.4573 


.1164 


40 


.1161 .0648 


.1169 .0678 


8.5555 .9322 


.9932 .9971 


20 


1.4544 


.1193 


50 


.1190 .0755 


.1198 .0786 


8.3450 .9214 


.9929 .9969 


10 


1.4515 


.1222 


7 00' 


.1219 .0859 


.1228 .0891 


8.1443 .9109 


.9925 .9968 


83 00' 


1.4486 


.1251 


10 


.1248 .0961 


.1257 .0995 


7.9530 .9005 


.9922 .9966 


50 


1.4457 


.1280 


20 


.1276 .10(50 


.1287 .109(5 


7.7704 .8904 


.9918 .9964 


40 


1.4428 


.1309 


30 


.1305 .1157 


.1317 .1194 


7.5958 .8806 


.9914 .9963 


30 


1.4399 


.1338 


40 


.1334 .1252 


.1346 .1291 


7.4287 .8709 


.9911 .9961 


20 


1.4370 


.1367 


50 


.13(53 .1345 


.1376 .1385 


7.2(587 .8615 


.9907 .9959 


10 


1.4341 


.1396 


8 00' 


.1392 .1436 


.1405 .1478 


7.1154 .8522 


.9903 .9958 


82 00' 


1.4312 


.1425 


10 


.1421 .1525 


.1435 .15(59 


6.9682 .8431 


.9899 .9956 


50 


1.4283 


.1454 


20 


.1449 .1612 


.1465 .1(558 


6.8269 .8342 


.9894 .9954 


40 


1.4254 


.1484 


30 


.1478 .1697 


.1495 .1745 


6.6912 .8255 


.9890 .9952 


30 


1.42-24 


.1513 


40 


.1507 .1781 


.1524 .1831 


6.5606 .8169 


.9886 .9950 


20 


1.4106 


.1542 


50 


.1536 .1863 


.1554 .1915 


6.4348 .8085 


.9881 .9948 


10 


1.41(56 


.1571 


9 00' 


.1564 .1943 


.1584 .1997 


6.3138 .8003 


.9877 .9946 


81 00' 


1.41 ."7 






Value Log 10 


Value Lojr ln 


Value Log 10 


Value Log 10 


DEGREES 


RADIANS 






COSINE 


COTANGENT 


TANGENT 


SINE 







320 



Table II. Values and Logarithms of Trigonometric Functions 

[Characteristics of Logarithms omitted determine by the usual rule from the value] 



UADIANS 


DEGREES 


SINE 


TANGEN-T 


COTANGENT 


COSINE 










Value Log 10 


Value Loi^m 


Value L<> 10 


Value Log la 






.1571 


9 00' 


.1504 .1943 


.1584 .1997 


6.3138 .8003 


.9877 .9946 


81 00' 


1.4137 


.1600 


10 


.1593 .2022 


.1014 .2078 


0.1970 .7922 


.9872 .9944 


50 


1.4108 


.1629 


20 


.1622 .2100 


.1044 .21.78 


6.0844 .7842 


.98(58 .9942 


40 


1.4079 


.1658 


30 


.1050 .2170 


.1673 .22:50 


5.9758 .77(54 


.9863 .9940 


30 


1.4050 


.1687 


40 


.1679 .22.-. 1 


.1703 .2313 


5.8708 .7(587 


.9858 .9938 


20 


1 .4021 


.1710 


50 


.1708 .2324 


.1733 .2389 


5.7094 .7611 


.9853 .99:1(5 


10 


1.3992 


.1745 


10 00' 


.1736 .2397 


.1763 .2463 


5.6713 .7537 


.9848 .9934 


80 00' 


1.39(53 


.1774 


10 


.1765 .2468 


.1793 .25: JO 


5.5704 .7464 


.9843 .99: :i 


50 


1.3934 


.1804 


20 


.1794 .25:38 


.1823 .2(509 


5.4845 .7391 


.9838 .9! 129 


40 


1.3904 


.1833 


30 


.1822 .2606 


.1853 .2(580 


f).:;9.-)5 .7320 


.9833 .9927 


30 


1.3875 


.1862 


40 


.1851 .2674 


.1883 .2750 


5.3093 .7250 


.9827 .9924 


20 


1.384fi 


.1891 


50 


.1880 .2740 


.1914 .2819 


5.2257 .7181 


.9822 .9922 


10 


1.3817 


.1920 


1100' 


.1908 .2806 


.1944 .2887 


5.1446 .7113 


.9816 .9919 


79 00' 


1.3788 


.1949 


10 


.1937 .2870 


.1974 .2!r,:; 


5.0658 .7047 


.9811 .9917 


50 


1.3759 


.1978 


20 


.1965 .2934 


.2004 .3020 


4.9894 .6980 


.9S05 .9914 


40 


1.3730 


5007 


30 


.1994 .2997 


.2035 .3085 


4.9152 .(5915 


.9799 .9912 


30 


1.3701 


.2036 


40 


.2022 .3058 


.20(55 .3149 


4.8430 .6851 


.9793 .9909 


20 


1.3672 


.2005 


50 


.2051 .3119 


.2095 .3212 


4.7729 .6788 


.9787 .9907 


10 


1.3643 


.2094 


12 00' 


.2079 .3179 


.2126 .3275 


4.7046 .6725 


.9781 .9904 


78 00' 


1.3614 


.'21--':$ 


10 


.2108 .:52:;s 


.2150 .3336 


4.6382 .6664 


.9775 .9901 


50 


1.3584 


5153 


20 


.2136 .3290 


.2186 .3397 


4.5736 .6(503 


.9769 .9899 


40 


1.3555 


.2182 


30 


.2164 .3353 


.2217 .3458 


4.5107 .6542 


.9763 .9896 


30 


1.3526 


.2211 


40 


.2193 .3410 


.2247 .3517 


4.4494 .0483 


.9757 .9893 


20 


1.3497 


.21-40 


50 


.2221 .3466 


.2278 .357(5 


4.3897 .6424 


.9750 .9890 


10 


1.3468 


5269 


13 00' 


.2250 .3521 


.2309 .3634 


4.3315 .6366 


.9744 .9887 


77 00' 


1.3439 


5298 


10 


.2278 .3575 


.2339 .3091 


4.2747 .0:509 


.97:57 .9884 


50 


1.3410 


.2327 


20 


.2306 .3629 


.2370 .3748 


4.2193 .02.72 


.9730 .'.issi 


40 


1.3381 


.2356 


30 


.2334 .30H2 


.2401 .3804 


4.1053 .6196 


.9724 .9878 


30 


1.3352 


5380 


40 


.2:503 .3734 


.2432 .3859 


4.1120 .6141 


.9717 .9S7.- 


20 


i .:;:;23 


.2414 


50 


.2391 .3786 


.2462 .3914 


4.0(511 .6086 


.9710 .9872 


10 


1.3294 


.2443 


14 00' 


.2419 .3837 


.2493 .3908 


4.0108 .6032 


.9703 .98(59 


76 00' 


1.32(55 


5473 


10 


.2447 .3887 


.2524 .4021 


3.9(517 .5979 


.9090 .'.'sro 


00 


L.323B 


5502 


20 


.2470 .3937 


.2555 .4074 


3.9130 .5926 


.<iOs9 .9863 


40 


1.3200 


.2331 


30 


.2504 .3986 


.2586 .4127 


3.81 5G7 .5873 


.9681 .9S59 


30 


1.3177 


5560 


40 


.2532 .4035 


.2617 .4178 


3.8208 .5822 


.9674 .98.-.0 


20 


1.31 48 


.2589 


50 


.2560 .4083 


.2648 .4230 


3.7700 .5770 


.9007 .9853 


10 


1.3119 


.2618 


1500' 


.2588 .4130 


.2679 .4281 


3.7321 .5719 


.9(559 .9849 


75 00' 


1.3090 


.2647 


10 


.2616 .4177 


.2711 .4331 


3.6891 .5(569 


.9052 .'.MI; 


no 


1.3061 


5676 


20 


.2641 .4223 


.2712 .4381 


3.0470 .5019 


.9(544 .984:5 


40 


1.3032 


5708 


30 


.2672 .420!) 


.2773 .4430 


30059 .5570 


.9(5:50 


30 


1.3003 


.2734 


40 


.2700 .4314 


.2805 .417!) 


3.5050 .5521 


.902S .'.is:,-; 


20 


1.2974 


.2763 


50 


.2728 .4:9 


.2836 .4527 


3.5261 .5473 


.9621 .9S32 


10 


15948 


5793 


16 00' 


.2756 .4403 


.2S07 .4575 


3.4874 .5125 


.9613 .9-S2S 


74 00' 


1.2918 


5822 


10 


.2784 .4447 


.2*99 .4(522 


3.4495 ..T.J7S 


.9605 .9*2". 


60 




.2851 


20 


.2812 .4491 


.29:51 .4(509 


3.1124 .5331 


.959(5 .9821 


40 


1.2857 


.2880 


30 


.2840 .4533 


.2902 .471(5 


3.3759 .52S4 


.!i:,ss .9817 


30 


1.2828 


.2909 


40 


.2868 .4571! 


.2994 .4762 


3.3402 ..72: IS 


.95SO .9814 


20 


1.2799 


.2938 


50 


.2896 .4618 


.3026 .4808 


3.3052 .5192 


.9572 .9810 


10 


1.2770 


.2967 


17 00' 


.2924 .4659 


.3057 .48.-,:? 


3.2709 .r,117 


.9503 .9SO<; 


73 C 00' 


1.2741 


5996 


10 


,2!i:,2 .4700 


.3089 .4S9S 


3.2371 .5102 


.'.).-,.-,.-, .9M>2 


50 


1.2712 


.3028 


20 


.297!) .4741 


..-5121 .4943 


3.2041 .r,0.-,7 


.9.140 .9798 


40 


1.2683 


.3054 


30 


.3007 .4781 


.3153 .4987 


3.1 7 10 ..Vi 115 


.9537 .97d! 


30 


1.2664 


.3083 


40 


.:;(:,;, .4821 


.3185 .5(): ;i 


3.1:197 .4909 


.9528 .9790 


20 


1.2025 


.3113 


50 


.3062 .4861 


.3217 .5075 


3.1084 .4925 


.9520 .978(5 


10 


1.2595 


.3142 


18 00' 


.3090 .4900 


.:-.2i9 ..-11* 


3.0777 .4882 


.9511 .97S2 


72 00' 


1.2500 






Value Logu 


V III Ui- I.l'lTi,, 


Value Loffio 


Value I."L', n 


DEGREKB 


UADIANS 






Oomra 


COTANGENT 


|'AXI;ENT 


SINK 







321 



Table II. Values and Logarithms of Trigonometric Functions 

[Characteristics of Logarithms oniitii-d ilctcnnim' !>y the usual rule IVom tlic valucj 



RADIANS 


DEGREES 


SINK 


TANGENT 


COTANGENT 


COSINE 










Value Log 10 


Value Log 10 


Value Log- 10 


Value Lopr 10 






.3142 


18 00' 


.3090 .4900 


.3249 .5118 


3.0777 .4882 


.9511 .9782 


72 00' 


1.2500 


.3171 


10 


.3118 .4939 


.3281 .5101 


3.0475 .4839 


.9502 .9778 


50 


1.2537 


.3200 


20 


.3145 .4977 


.3314 .5203 


3.0178 .4797 


.9492 .9774 


40 


1.2508 


.3221 i 


30 


.3173 .5015 


.3340 .5245 


2.9887 .4755 


.9483 .9770 


30 


1.2479 


.3258 


40 


.3201 .5052 


.3378 .5287 


2.9(500 .4713 


.9474 .9705 


20 


1.2450 


.3287 


50 


.3228 .5090 


.3411 .5329 


2.9319 .4671 


.9465 .97(51 


10 


1.2421 


.3316 


19 00' 


.3250 .5120 


.3443 .5370 


2.9042 .4030 


.9455 .9757 


71 00' 


1.2392 


.3345 


10 


.3283 .5103 


.3476 .5411 


2.8770 .4589 


.944(5 .9752 


50 


1.2363 


.3374 


20 


.3311 .5199 


.3508 .5451 


2.8502 .4549 


.943(5 .9748 


40 


1.2334 


.3403 


30 


.3338 .5235 


.3541 .5491 


2.8239 .4509 


.9420 .9743 


30 


1.23! '5 


.3432 


40 


.33<55 .5270 


.3574 .5531 


2.7980 .4409 


.9417 .9739 


20 


1.2275 


.341 W 


50 


.3393 .5300 


.3007 .5571 


2.7725 .4429 


.9407 .9734 


10 


1.22-tfi 


.3401 


20 00' 


.3420 .5341 


.3(540 .5011 


2.7475 .4389 


.9397 .9730 


70 00' 


1 .2217 


5520 


10 


.3448 .5375 


.3073 .5050 


2.7228 .4350 


.9387 .9725 


50 


1.2188 


.3649 


20 


.3475 .5409 


.370(5 .5089 


2.6985 .4311 


.9377 .9721 


40 


1.2150 


.3578 


30 


.3502 .5443 


.3739 .5727 


2.0740 .4273 


.9367 .9716 


30 


1.2130 


.3007 


40 


.3529 .5477 


.3772 .5766 


2.0511 .4234 


.9350 .9711 


20 


1.2101 


.3030 


50 


.3557 .5510 


.3805 .5804 


2.6279 .419(5 


.9340 .9700 


10 


1.2072 


.3005 


21 00' 


..3584 .5543 


.3839 .5842 


2.6051 .4158 


.9336 ,9702 


69 00' 


1.2043 


.3694 


10 


.3011 .5570 


.3872 .5879 


2.5826 .4121 


.9325 .9097 


50 


1.2014 


.3723 


20 


.3(538 .5009 


.3900 .5917 


2.5(505 .4083 


.9315 .9092 


40 


1.1985 


.3752 


30 


.3I505 .5041 


.3939 .5954 


2.5386 .4046 


.9304 .9087 


30 


1.1950 


.3782 


40 


.3092 .5(573 


.3973 .5991 


2.5172 .4009 


.9293 .9(582 


20 


1.1920 


.3811 


50 


.3719 .5704 


.4000 .0028 


2.4960 .3972 


.9283 .1)077 


10 


1.1897 


.3840 


22 00' 


.3746 .5730 


.4040 .6064 


2.4751 .3936 


.9272 .9(572 


68 00' 


1.1808 


.3860 


10 


.3773 .5707 


.4074 .6100 


2.4545 .3900 


.9201 .1X507 


50 


1.1839 


.3808 


20 


.3800 .5798 


.4108 .0130 


2.4342 .38(54 


.9250 .9001 


40 


1.1810 


.3027 


30 


.3827 .5828 


.4142 .6172 


2.4142 .3828 


.9239 .1)050 


30 


1.1781 


.35)66 


40 


.3854 .5859 


.4176 .6208 


2.3945 .3792 


.9228 .9651 


20 


1.1752 


.3983 


50 


.3881 .5889 


.4210 .6243 


2.3750 .3757 


.9216 .9046 


10 


1.1723 


.4014 


23 00' 


.3907 .5919 


.4245 .6279 


2.3559 .3721 


.9205 .9640 


67 00' 


1.1094 


.4043 


10 


.3934 .5948 


.4279 .6314 


2.3309 .3086 


.9194 .9(535 


50 


I.IK;.-, 


.4072 


20 


.3901 .5978 


.4314 .6348 


2.3183 .3652 


.9182 .9629 


40 


1.10.30 


.4102 


30 


.3987 .0007 


.4348 .6383 


2.2998 .3(517 


.9171 .9024 


30 


1.1000 


.4131 


40 


.4014 .0036 


.4383 .6417 


2.2817 .3583 


.9159 .9018 


20 


1.1577 


.41(50 


50 


.4041 .0005 


.4417 .6452 


2.2637 .3548 


.W147 .9013 


10 


1.1548 


.4189 


24 00' 


.4067 .0093 


.4452 .6486 


2.24(50 .3514 


.9135 .9007 


66 00' 


1.1519 


.4218 


10 


.4094 .6121 


.4487 .0520 


2.2286 .3480 


.9124 .9602 


50 


1.1400 


.4247 


20 


.4120 .6149 


.4522 .(5553 


2.2113 .3447 


.9112 .9596 


40 


1.1401 


.4270 


30 


.4147 .0177 


.4557 .6587 


2.1943 .3413 


.9100 .9590 


30 


1.1432 


.4300 


40 


.4173 .6205 


.4592 .6620 


2.1775 .3380 


.9088 .9584 


20 


1.1403 


.4334 


50 


.4200 .6232 


.4628 .0654 


2.1609 .3346 


.9075 .9579 


10 


1.1374 


.4303 


25 00' 


.4220 .0259 


.4063 .6687 


2.1445 .3313 


.9063 .9573 


65 00' 


1.1346 


.4392 


10 


.4253 .6280 


.4699 .0720 


2.1283 .3280 


.9051 .95(57 


50 


1.1310 


.4422 


20 


.4279 .6313 


.4734 .0752 


2.1123 .3248 


.9038 .95(51 


40 


1.128H 


.4451 


30 


.4305 .6340 


.4770 .0785 


2.09(55 .3215 


.9020 .9555 


30 


1.1257 


.4480 


40 


.4331 .6:366 


.4800 .0817 


2.0809 .3183 


.9013 .9549 


20 


1.1228 


.4501) 


50 


.4358 .0392 


.4841 .0850 


2.0(555 .3150 


.9001 .9543 


10 


1.1190 


.4538 


26 00' 


.4384 .6418 


.4877 .0882 


2.0503 .3118 


.8988 .9537 


64 00' 


1.1170 


.4507 


10 


.4410 .6444 


.4913 .0914 


2.0353 .3080 


.8975 .95:50 


50 


1.1141 


.4590 


20 


.4436 .6470 


.4950 .0940 


2.0204 .3054 


.89(52 .9524 


40 


1.1112 


.4(525 


30 


.4402 .6495 


.4986 .15977 


2.0057 .3023 


.8949 .9518 


30 


1.1083 


.4054 


40 


.4488 .6521 


.5022 .7009 


1.9912 .21)91 


.8930 .9512 


20 


1.1054 


.4(583 


50 


.4514 .6546 


.5059 .7040 


1.9768 .29(50 


.8923 .9505 


10 


1.1025 


.4712 


27 00' 


.4540 .6570 


.5095 .7072 


1.9026 .2928 


.8910 .9499 


63 00' 


1.0996 






Value Logjo 


Value Login 


Value Log 10 


Value Logjo 


DEGREES 


RADIANS 






COSINE 


COTANGENT 


TANGENT 


SINE 







322 



Table II. Values and Logarithms of Trigonometric Functions 

[Characteristics of Logarithms omitted determine by the usual rule from the value] 



RADIANS 


DEGREEK 


SlXE 


TANGENT 


COTANGENT 


COSINE 










Value Log 10 


Value Log 10 


Value Log 10 


Value Log 10 






.4712 


27 00' 


.4540 .6570 


.5095 .7072 


1.9(526 .2928 


.8910 .9499 


63 00' 


1.0996 


.4741 


10 


.45(56 .6595 


.5132 .7103 


1.9486 .2897 


.8897 .9492 


50 


1.09(5(5 


.4771 


20 


.4592 .6620 


.5169 .7134 


1.9347 .28(56 


.8884 .948(5 


40 


1.0937 


.4800 


30 


.4617 .6644 


.5206 .7165 


1.9210 .2835 


.8870 .9479 


30 


1.0908 


.4829 


40 


.4643 .66(58 


.5243 .7196 


1.9074 .2804 


.8857 .9473 


20 


1.0879 


.4858 


50 


.4669 .6692 


.5280 .7226 


1.8940 .2774 


.8843 .9466 


10 


1.0850 


.4887 


28 00' 


.4695 .6716 


.5317 .7257 


1.8807 .2743 


.8829 .9459 


62 00' 


1.0821 


.4916 


10 


.4720 .(5740 


.5354 .7287 


1.8676 .2713 


.8816 .9453 


50 


1.0792 


.4945 


20 


.4746 .6763 


.5392 .7317 


1.8546 .2683 


.8802 .944(5 


40 


1.07(53 


.4974 


30 


.4772 .6787 


.5430 .7348 


1.8418 .2652 


.8788 .9439 


30 


1.0734 


.5003 


40 


.4797 .6810 


.5467 .7378 


1.8291 .2(522 


.8774 .9432 


20 


1.0705 


.5032 


50 


.4823 .6833 


.5505 .7408 


1.8165 .2592 


.8760 .9425 


10 


1.0676 


.5061 


29 00' 


.4848 .6856 


.5543 .7438 


1.8040 .2562 


.8746 .9418 


61 00' 


1.0647 


.5091 


10 


.4874 .6878 


.5581 .7467 


1.7917 .2533 


.8752 .9411 


50 


1. 0(517 


.5120 


20 


.4899 .6901 


.5619 .7497 


1.7796 .2503 


.8718 .9404 


40 


1.05S8 


.5149 


30 


.4924 .6923 


.5658 .7526 


1.7675 .2474 


.8704 .9397 


30 


1.0559 


.5178 


40 


.4950 .6946 


.6696 .7556 


1.7556 .2444 


.8689 .9390 


20 


1.0530 


.5207 


50 


.4975 .6968 


.5735 .7585 


1.7437 .2415 


.8675 .9383 


10 


1.0501 


.5236 


30 00' 


.5000 .6990 


.5774 .7614 


1.7321 .2386 


.8660 .9375 


60 00' 


1.0472 


.5265 


10 


.5025 .7012 


.5812 .7644 


1.7205 .2356 


.8646 .9368 


50 


1.0443 


.521)4 


20 


.5050 .7033 


.5851 .7673 


1.7090 .2327 


.8631 .9361 


40 


1.0414 


.5323 


30 


.5075 .7055 


.5890 .7701 


1.6977 i2299 


.8(516 .9353 


30 


1.0385 


.6352 


40 


.5100 .7076 


.5930 .7730 


1.6864 .2270 


.8601 .9346 


20 


1.0356 


.5381 


50 


.5125 .7097 


.5969 .7759 


1.6753 .2241 


.8587 .9338 


10 


1.0327 


.5411 


31 00' 


.5150 .7118 


.6009 .7788 


1.6643 .2212 


.8572 .9331 


59 00' 


1.0297 


.5440 


10 


.5175 .7139 


.6048 .781(5 


1.6534 .2184 


-S557 .9323 


50 


1.0268 


.5469 


20 


.5200 .7160 


.6088 .7845 


1.6426 .2155 


.8542 .9:;i5 


40 


1.0239 


.5498 


30 


.5225 .7181 


.6128 .7873 


1.6319 .2127 


.8526 .9308 


30 


1.0210 


.5527 


40 


.5250 .7201 


.6168 .7902 


1.6212 .2098 


.8511 .9:500 


20 


1.0181 


.5556 


50 


.5275 .7222 


.6208 .7930 


1.6107 .2070 


.8496 .9292 


10 


1.0152 


.5585 


32 00' 


.5299 .7242 


.6249 .7958 


1.6003 .2042 


.8480 .9284 


58 00' 


1.0123 


.5(514 


10 


.5324 .7262 


.6289 .798(5 


1.5900 .2014 


.84(55 .927(5 


50 


1.0094 


.5643 


20 


.5348 .7282 


.6330 .8014 


1.5798 .19S6 


.8450 .9268 


40 


1.0065 


.5(572 


30 


.5373 .7302 


.6371 .8042 


1.5697 .1958 


.8434 .9260 


30 


1.003(5 


.5701 


40 


.5398 .7322 


.6412 .8070 


1.55! )7 .1930 


.8418 .9252 


20 


1.0007 


.5730 


50 


.5422 .7342 


.6453 .8097 


1.5497 .1903 


.8403 .9244 


10 


.9977 


.5760 


33 00' 


.5446 .7361 


.6494 .8125 


1.5399 .1875 


.8387 .92:!6 


57 00' 


.9948 


0789 


10 


.5471 .7380 


.(55.'56 .8153 


1.5301 .1847 


.8371 .9228 


50 


.9919 


.5818 


20 


.5495 .7400 


.6577 .8180 


1.5204 .1820 


.8.'555 .9219 


40 


.9890 


.5847 


30 


.5519 .7419 


.(',619 .8208 


1.5108 .1792 


.8339 .9211 


30 


.98(51 


.5876 


40 


.5544 .743S 


.6(561 .8235 


1.5013 .17C.5 


..s: !23 .9203 


20 


.9632 


.5905 


50 


.5568 .7457 


.(5703 .8263 


1.4919 .1737 


.8307 .9194 


10 


.9803 


.5934 


34 00' 


.5592 .7476 


.6745 .8290 


1.4826 .1710 


.8290 .918(5 


56 00' 


.9774 


.5963 


10 


.5(516 .7494 


.6787 .8317 


1.4733 .1683 


.8274 .9177 


50 


.9745 


.5992 


20 


.5640 .7513 


.68:50 .8344 


1.4641 .1656 


.8258:. 91(19 


40 


.9716 


.6021 


30 


.5(5(54 .7531 


.6873 .8371 


1.4650 .1629 


.8241 .91(50 


30 


.'.H1S7 


.6050 


40 


.5<;ss .7550 


.6916 .8398 


1.44(50 .1602 


.-S-J25 .9151 


20 


.9657 


.6080 


60 


.5712 .7568 


.6959 .8425 


1.4370 .1575 


.8208 .9142 


10 


.9628 


.6109 


35 00' 


.5736 .7586 


.7002 .8452 


1.4281 .1548 


.8192 .9134 


55 00' 


.9599 


.6138 


10 


.57(50 .7(504 


.Tdlil .8479 


1.4193 .1521 


.8175 .9125 


50 


.9570 


.6167 


20 


.5783 .7622 


.7089 .8506 


1.4106 .1494 


.8158 .9116 


40 


.9541 


.6196 


30 


.5S07 .7640 


.7133 .S5. ,:; 


1.4019 .14(57 


.8141 .9107 


30 


.9512 


.6225 


40 


.5831 .7(157 


.7177 .8559 


1.3934 .1441 


.8124 .9098 


20 


.9483 


.6254 


5i > 


.5854 .7675 


.7221 .8586 


1.3848 .1414 


.8107 .9089 


10 


.9454 


.6283 


36 00' 


.5878 .7692 


.7265 .8613 


1.37(54 .1387 


.8090 .9080 


54 00' 


.9425 






Value Log 10 


Value Log 10 


Value Lop 10 


Value Log 10 


D EG BEES 


RADIANS 






COSINB 


COTANGENT 


TANGENT 


SINE 







323 



Table n. Values and Logarithms of Trigonometric Functions 

[Characteristics of Logarithms omitted determine by the usual rule from the value] 



KAIHA N 


DEGREE 


SINK 


TANGENT 


COTANGENT 


COSINE 










Value Logj 


Value LOJBTJ 


Value Log le 


Value Log t 






.6283 


36 00 


.5878 .7692 


.7265 .8613 


1.3764 .1387 


.8090 .9080 


54 00 


.9425 


.6312 


10 


.5901 .7710 


.7310 .8639 


1.3680 .1361 


.8073 .9070 


50 


.9396 


.6341 


20 


.5925 .7727 


.7355 .8666 


1.3597 .1334 


.8056 .9061 


40 


.9367 


.6370 


30 


.5948 .7744 


.7400 .8692 


1.3514 .1:308 


.8039 .91152 


30 


.9338 


.6400 


40 


.5972 .7761 


.7445 .8718 


1.3432 .1282 


.8021 .9042 


20 


.9308 


.6429 


50 


.5995 .7778 


.7490 .8745 


1.3351 .1255 


.8004 .9033 


10 


.9279 


.6458 


37 00 


.6018 .7795 


.7536 .8771 


1.3270 .1229 


.7986 .9023 


53 00 


.9250 


.6487 


10 


.6041 .7811 


.7581 .8797 


1.3190 .1203 


.7969 .9014 


50 


.9221 


.6516 


20 


.6065 .7828 


.7627 .8824 


1.3111 .1176 


.7951 .9004 


40 


.9192 


.6545 


30 


.6088 .7844 


.7673 .8850 


1.3032 .1150 


.7934 .8995 


30 


.9163 


.6574 


40 


.6111 .7861 


.7720 .8876 


1.2954 .1124 


.7916 .8985 


20 


.9i;34 


.6603 


50 


.6134 .7877 


.7766 .8902 


1.2876 .1098 


.7898 .8975 


10 


.9105 


.6632 


38 00' 


.6157 .7893 


.7813 .8928 


1.2799 .1072 


.7880 .8965 


52 00 


.8076 


.6661 


10 


.6180 .7910 


.7860 .8954 


1.2723 .1046 


.7862 .8955 


50 


.9047 


.6690 


20 


.6202 .7926 


.7907 .8980 


1.2647 .1020 


.7844 .8945 


40 


.9018 


.6720 


30 


.6225 .7941 


.7954 .9006 


1.2572 .0994 


.7826 .8935 


30 


.8988 


.6749 


40 


.6248 .7957 


.8002 .9032 


1.2497 .09(58 


.7808 .8925 


20 


.8959 


.6778 


50 


.6271 .7973 


.8050 .9058 


1.2423 .0942 


.7790 .8915 


10 


.8930 


.6807 


39 00' 


.6293 .7989 


.8098 .9084 


1.2349 .0916 


.7771 .8905 


51 00 


.8901 


.6836 


10 


.6316 .8004 


.8146 .9110 


1.2276 .0890 


.7753 .8895 


50 


.8872 


.6865 


20 


.6338 .8020 


.8195 .9135 


1.2203 .0865 


.7735 .8884 


40 


.8843 


.6894 


30 


.6361 .80:35 


.8243 .9161 


1.2131 .0839 


.7716 .8874 


30 


.8814 


.6923 


40 


.6383 .8050 


.8292 .9187 


1.205!) .0813 


.7698 .8864 


20 


.8785 


.6952 


50 


.6406 .8066 


.8342 .9212 


1.1988 .0788 


.7679 .8853 


10 


.8756 


.6981 


40 00' 


.6428 .8081 


.8391 .9238 


1.1918 .0762 


.7660 .8843 


50 00 


.8727 


.7010 


10 


.6450 .8096 


.8441 .9264 


1.1847 .0736 


.7642 .8832 


60 


.8698 


.7039 


20 


.6472 .8111 


.8491 .9289 


1.1778 .0711 


.7623 .8821 


40 


.8668 


.7069 


30 


.6494 .8125 


.8541 .9315 


1.1708 .0685 


.7604 .8810 


30 


.8639 


.7098 


40 


.6517 .8140 


.8591 .9341 


1.1640 .0659 


.7585 .8800 


20 


.8610 


.7127 


50 


.6539 .8155 


.8642 .9366 


1.1571 .0634 


.7566 .8789 


10 


.8581 


.7156 


41 00' 


.6561 .8169 


.8693 .9392 


1.1504 .0608 


.7547 .8778 


49 00 


.8552 


.7185 


10 


.6583 .8184 


.8744 .9417 


1.1436 .0583 


.7528 .8767 


50 


.8523 


.7214 


20 


.6604 .8198 


.8796 .9443 


1.1369 .0557 


.7509 .8756 


40 


.841)4 


.7243 


30 


.6626 .8213 


.8847 .9468 


1.1303 .0532 


.7490 .8745 


30 


.8465 


.7272 


40 


.6648 .8227 


.8899 .9494 


1.1237 .0506 


.7470 .8733 


20 


.8436 


.7301 


50 


.6670 .8241 


.8952 .9519 


1.1171 .0481 


.7451 .8722 


10 


.8407 


.7330 


42 00' 


.6691 .8255 


.9004 .9544 


1.1106 .0456 


.7431 .8711 


48 00' 


.8378 


.7359 


10 


.6713 .8269 


.9057 .9570 


1.1041 .0430 


.7412 .86<)9 


50 


.8348 


.7389 


20 


.6734 .8283 


.9110 .9595 


1.0977 .0405 


.7392 .8688 


40 


.8319 


.7418 


30 


.6756 .8297 


.9163 .9621 


1.0913 .0379 


.7373 .8676 


30 


.8290 


.7447 


40 


.6777 .8311 


.9217 .9646 


1.0850 .0354 


.7353 .8665 


20 


.8261 


.7476 


50 


.6799 .8324 


.9271 .9671 


1.0786 .0329 


.7333 .8653 


10 


.8232 


.7505 


43 00' 


.6820 .8338 


.9325 .9697 


1.0724 .0303 


.7314 .8641 


47 00' 


.8203 


.7534 


10 


.6841 .8351 


.9380 .9722 


1.0661 .0278 


.7294 .8629 


50 


.8174 


.7563 


20 


.6862 .8365 


.9435 .9747 


1.059!) .0253 


.7274 .8618 


40 


.8145 


.7592 


30 


.6884 .8378 


.9490 .9772 


1.0538 .0228 


.7254 .8606 


30 


.8116 


.7621 


40 


.6905 .8391 


.9545 .9798 


1.0477 .0202 


.7234 .8594 


20 


.8087 


.7650 


50 


.6926 .8405 


.9601 .9823 


1.0416 .0177 


.7214 .8582 


10 


.8058 


.7679 


44 00' 


.6947 .8418 


.9657 .9848 


1.0355 .0152 


.7193 .8569 


46 00' 


.8029 


.7709 


10 


.6967 .8431 


.9713 .9874 


1.0295 .0126 


.7173 .8557 


50 


.7999 


.7738 


20 


.6988 .8444 


.9770 .9899 


1.0235 .0101 


.7153 .8545 


40 


.7970 


.7767 


30 


.7009 .8457 


.9827 .9924 


1.0176 .0076 


.7133 .8532 


30 


.7941 


.7796 


40 


.7030 .8469 


.9884 .9949 


1.0117 .0051 


.7112 .8520 


20 


.7912 


.7825 


50 


.7050 .8482 


.9942 .9975 


1.0058 .0025 


.7092 .8507 


10 


.7883 


.7854 


45 00' 


.7071 .8495 


1.0000 .0000 


1.0000 .0000 


.7071 .8495 


45 00 


.7854 






Value Log 10 


Value Log in 


Value Log ]0 


Value Log 10 


)EGREES 


vADIANS 






COSINE 


COTANGENT 


TANGENT 


SINE 







324 



Table III. Radian Measure Trigonometric Functions 



Had. 


Deg. Mln. 


sin. 


cos. 


tan. 


Rad. 


Deg. Mln. 


sin. 


cos. 


tan 


0.0 








1 





3.2 


183 20.8 


-.058 


-.998 


.058 


0.1 


5 43.8 


.100 


.995 


.100 


3.3 


189 4.6 


-.158 


-.987 


.161 


0.2 


11 27.5 


.199 


.980 


.203 


3.4 


194 48.3 


-.255 


-.967 


.264 


0.3 


17 11.3 


.296 


.955 


.309 


3.5 


200 32.1 


-.351 


-.936 


.375 


0.4 


22 55.1 


.389 


.921 


.423 


3.6 


206 15.9 


-.443 


-.897 


.493 


0.5 


28 38.9 


.479 


.878 


.546 


3.7 


211 59.7 


-.530 


-.848 


.625 


0.6 


34 22.6 


.565 


.825 


.684 


3.8 


217 43.4 


-.612 


-.791 


.774 


0.7 


40 6.4 


.644 


.765 


.842 


3.9 


223 27.2 


-.688 


-.726 


.947 


0.8 


45 50.2 


.717 


.697 


1.030 


4.0 


229 11.0 


-.757 


-.654 


1.158 


0.9 


51 34.0 


.783 


.622 


1.260 


4.1 


234 54.8 


-.818 


-.575 


1.424 


1.0 


57 17.7 


.841 


.540 


1.557 


4.2 


240 38.5 


-.872 


-.490 


1.778 


1.1 


63 1.5 


.891 


.454 


1.965 


4.3 


246 22.3 


-.916 


-.401 


2.286 


1.2 


68 45.3 


.932 


.362 


2.572 


4.4 


252 6.1 


-.952 


-.307 


3.096 


1.3 


74 29.1 


.964 


.267 


3.602 


4.5 


257 49.9 


-.978 


-.211 


4.638 


1.4 


80 12.8 


.985 


.170 


5.798 


4.6 


263 33.6 


-.994 


-.112 


8.859 


1.5 


85 56.6 


.997 


.071 


14.101 


4.7 


269 17.4 


-1.00 


-.012 


80.713 


1.6 


91 40.4 


1.000 


-.029 


-34.233 


4.8 


275 1.2 


-.996 


.088 


-11.385 


1.7 


97 24.2 


.992 


-.129 


- 7.700 


4.9 


280 45.0 


-.982 


.187 


- 5.267 


1.8 


103 7.9 


.974 


-.227 


- 4.286 


5.0 


286 28.6 


-.959 


.284 


- 3.381 


1.9 


108 51.7 


.946 


-.323 


- 2.927 


5.1 


292 12.5 


-.926 


.378 


- 2.449 


2.0 


114 35.5 


.909 


-.416 


- 2.185 


5.2 


297 56.3 


-.883 


.469 


- 1.885 


2.1 


120 19.3 


.863 


-.505 


- 1.710 


5.3 


303 40.1 


-.832 


.554 


- 1.501 


2.2 


126 3.0 


.808 


-.588 


- 1.374 


5.4 


309 23.8 


-.773 


.635 


- 1.217 


2.3 


131 46.8 


.746 


-.666 


- 1.119 


5.5 


315 7.6 


-.706 


.709 


- .996 


2.4 


137 30.6 


.675 


-.737 


.917 


5.6 


320 51.4 


-.631 


.776 


- .814 


2.5 


143 14.4 


.598 


-.801 


.747 


5.7 


326 35.2 


-.551 


.835 


- .600 


2.6 


148 58.1 


.516 


-.857 


- .602 


5.8 


332 18.9 


-.465 


.886 


- .525 


2.7 


154 41.9 


.427 


-.904 


- .473 


5.9 


338 2.7 


-.374 


.927 


- .403 


2.8 


160 25.7 


.335 


-.942 


- .356 


6.0 


343 46.5 


-.279 


.960 


- .291 


2.9 


166 9.5 


.239 


-.971 


- .246 


6.1 


349 30.3 


-.182 


.983 


- .185 


3.0 


171 53.2 


.141 


-.990 


.143 


6.2 


355 14.0 


-.083 


.997 


- .083 


3.1 


177 37.0 


.042 


-.999 


.042 


6.3 


360 57.8 


+.017 


1.000 


+ .017 



325 



Table IV. Squares and Cubes Square Roots and Cube Roots 



No. 


SQUARE 


C'UUE 


SQUARE 
UOOT 


CUBE 

ItOOT 


No. 


SQUARE 


CUBE 


SQUARE 
KOOT 


ClT.E 

HOOT 


1 


1 


1 


1.000 


1.000 


51 


2,601 


132,651 


7.141 


3.708 


2 


4 


8 


1.414 


1.260 


52 


2,704 


140,608 


7.211 


3.733 


3 


9 


27 


1.732 


1.442 


53 


2,809 


148,877 


7.280 


3.75(5 


4 


16 


64 


2.000 


1.587 


54 


2,916 


157,464 


7.348 


3.780 


5 


25 


125 


2.236 


1.710 


55 


3,025 


166,375 


7.416 


3.803 


6 


36 


216 


2.449 


1.817 


56 


3,136 


175,616 


7.483 


3.826 


7 


49 


343 


2.646 


1.913 


57 


3,249 


185,193 


7.550 


3.84!) 


8 


64 


512 


2.828 


2.000 


58 


3,364 


195,112 


7.616 


3.871 


9 


81 


729 


3.000 


2.080 


59 


3,481 


205,379 


7.681 


3.893 


10 


100 


1,000 


3.162 


2.154 


60 


3,600 


216,000 


7.746 


3.915 


11 


121 


1,331 


3.317 


2.224 


61 


3,721 


226,981 


7.810 


3.936 


12 


144 


1,728 


3.464 


2.289 


62 


3,844 


238,328 


7.874 


3.968 


13 


109 


2,197 


3.606 


2.351 


63 


3,969 


250,047 


7.937 


3.979 


14 


196 


2,744 


3.742 


2.410 


64 


4,096 


262,144 


8.000 


4.000 


15 


225 


3,375 


3.873 


2.466 


65 


4,225 


274,625 


8.062 


4.021 


16 


256 


4,096 


4.000 


2.520 


66 


4,356 


287,496 


8.124 


4.041 


17 


.289 


4,913 


4.123 


2.571 


67 


4,489 


300,763 


8.185 


4.062 


18 


324 


5,832 


4.243 


2.621 


68 


4,624 


314,432 


8.246 


4.082 


19 


361 


6,859 


4.359 


2.668 


69 


4,761 


328,509 


8.307 


4.102 


20 


400 


8,000 


4.472 


2.714 


70 


4,900 


343,000 


8.367 


4.121 


21 


441 


9,261 


4.583 


2.759 


71 


5,041 


357,911 


8.426 


4.141 


22 


484 


10,648 


4.690 


2.802 


72 


5,184 


373,248 


8.485 


4.160 


23 


529 


12,167 


4.796 


2.844 


73 


5,329 


389,017 


8.544 


4.17H 


24 


576 


13,824 


4.899 


2.884 


74 


5,476 


405,224 


8.602 


4.198 


25 


625 


15,625 


5.000 


2.924 


75 


5,625 


421,875 


8.660 


4.217 


26 


676 


17,576 


5.099 


2.962 


76 


5,776 


438,!>76 


8.718 


4.236 


27 


729 


19,683 


5.196 


3.000 


77 


5,929 


456,533 


8.775 


4.254 


28 


784 


21,952 


5.292 


3.037 


78 


6,084 


474,552 


8.832 


4.273 


29 


841 


24,389 


5.385 


3.072 


79 


6,241 


493,039 


8.888 


4.291 


30 


900 


27,000 


5.477 


3.107 


80 


6,400 


512,000 


8.944 


4.:50!> 


31 


961 


29,791 


5.568 


3.141 


81 


6,561 


531,441 


9.000 


4. :','21 


32 


1,024 


32,768 


5.657 


3.175 


82 


6,724 


551,368 


9.055 


4.:U4 


33 


1,089 


35,937 


5.745 


3.208 


83 


6,889 


571,787 


9.110 


4.3(52 


34 


1,156 


39,304 


5.831 


3.240 


84 


7,056 


592,704 


9.165 


4.380 


35 


1,225 


42,875 


5.916 


3.271 


85 


7,225 


614,125 


9.220 


4.397 


36 


1,2'W 


46,656 


6.000 


3.302 


86 


7,396 


636,056 


9.274 


4.414 


37 


1,3(59 


50,653 


6.083 


3.332 


87 


7,569 


658,503 


9.327 


4.431 


38 


1,444 


54,872 


6.164 


3.362 


88 


7,744 


681,472 


9.381 


4.448 


39 


1,521 


59,319 


6.245 


3.391 


89 


7,921 


704,969 


9.434 


4.4(;r, 


40 


1,600 


64,000 


6.325 


3.420 


90 


8,100 


729,000 


9.487 


4.481 


41 


1,681 


68,921 


6.403 


3.448 


91 


8,281 


753,671 


9.588 


4.49S 


42 


1,764 


74,088 


6.481 


3.476 


92 


8,464 


778,688 


9.592 


4.614 


43 


1,849 


79,507 


6.557 


3.503 


93 


8,649 


804,a57 


9.644 


4.631 


44 


1,<>:!6 


85,184 


6.633 


3.530 


94 


8,836 


HlJu,.-^ 


9.698 


4.547 


45 


2,025 


91,125 


6.708 


3.557 


95 


9,025 


857,370 


!>.747 


4.56.", 


46 


2,116 


97,336 


6.782 


3.583 


96 


9,216 


884,736 


9.798 


4.579 


47 


2,20! 


103,823 


6.856 


3.609 


97 


9,409 


912,673 


9.849 


4.595 


48 


2,304 


110,592 


6. 928 


3.634 


98 


9,604 


941,192 


9.899 


4.610 


49 


2.401 


117,649 


7.000 


3.<>59 


99 


9,801 


970,299 


9.950 


4.626 


50 


2,500 


125,000 


7.071 


3.684 


100 


10,000 


1,000,000 


10.000 


4.1142 



For a more complete table, see THE MACMILLAN TABLES, pp. 94-111. 



326 



Table V. Logarithms of Important Constants 



A r = NUMBER 


VALUE OF JV 


LOGjo ff 


f 


3.14159265 


0.49714987 


1 + * 


0.31830989 


9.50285013 


X* 


9.86960440 


0.99429975 


V^r 


1.77245385 


0.24857494 


e = Napierian Base 


2.71828183 


0.43429448 


M = logio e 


0.43429448 


9.63778431 


1 -H M = log e 10 


2.30258509 


0.36221569 


! 180 -i- r = degrees in 1 radian 


57.2957795 


1.75812262 


IT -=- 180 = radians in 1 


0.01745329 


8.24187738 


IT -T- 10800 = radians in 1' 


0.0002908882 


6.4637261 


x H- 648000 = radians in 1" 


0.000004848136811095 


4.68557487 


sin 1" 


0.000004848136811076 


4.68557487 


tan 1" 


0.000004848136811152 


4.68557487 


centimeters in 1 ft. 


30.480 


1.4840158 


feet in 1 cm. 


0.032808 


8.5159842 


inches in 1 m. 


39.37 


1.5951654 


pounds in 1 kg. 


2.20462 


0.3433340 


kilograms in 1 Ib. 


0.453593 


9.6566660 


g 


32.16 ft. /sec. /sec. 


1.5073160 




= 981 cm. /sec. /sec. 


2.9916690 


weight of 1 cu. ft. of water 


62.425 Ib. (max. density) 


1.7953586 


weight of 1 cu. ft. of air 


0.0807 Ib. (at 32 F.) 


8.9068735 


cu. in. in 1 (U. S.) gallon 


231. 


2.3636120 


ft. Ib. per sec. in 1 H. P. 


550. 


2.7403627 


kg. m. per sec. in 1 H. P. 


76.0404 


1.8810445 


watts in 1 H. P. 


745.957 


2.8727135 



Table VI. Degrees to Radians 



1 


.01745 


10 


.17453 


100 


1.74533 


6' 


.00175 


6' 


.00003 


2 


.03491 


20 


.34907 


110 


1.91986 


7' 


.00204 


7' 


.00003 


3 


.05236 


30 


.52360 


120 


2.09440 


8' 


.00233 


8' 


.00004 


4 


.06981 


40 


.69813 


130 


2.26893 


9' 


.00262 


9' 


.00004 


5 


.08727 


50 


.87266 


140 


2.44346 


10' 


.00291 


10' 


.00005 


6 


.10472 


60 


1.04720 


150 


2.61799 


20' 


.005S2 


20' 


.00010 


7 


.12217 


70 


1.22173 


160 


2.79253 


30' 


.00873 


30' 


.00015 


8 


.13963 


80 


1.39626 


170 


2.96706 


40' 


.01164 


40' 


.00019 


9 


.15708 


90 


1.57080 


180 


3.14159 


50' 


.01454 


50' 


.00024 



327 



Table VII. Compound Interest Table 

Amount of One Dollar Principal with Compound Interest at Various Rates. 



a 
< 
H 
P 


2} Per 
Cent. 


3 Per 

Cent. 


3J Per 
Cent. 


4 Per 
Cent. 


4.^ Per 
Cent. 


5 Per 

Cent. 


5J Per 
Cent. 


6 Per 
Cent. 


6 Per 
Cent. 


7 Per 
Cent. 


8 Per 
Cent. 


1 


$1.025 


$1.030 


$1.035 


$1.040 


$1.045 


$1.050 


$1.055 


$1.060 


$1.065 


$1.070 


$1.800 


2 


1.051 


1.061 


1.071 


1.082 


1.092 


1.103 


1.113 


1.124 


1.134 


1.145 


1.166 


3 


1.077 


1.093 


1.109 


1.125 


1.141 


1.158 


1.174 


1.191 


1.208 


1.225 


1.260 


4 


1.104 


1.126 


1.148 


1.170 


1.193 


1.216 


1.239 


1.262 


1.286 


1.311 


1.360 


5 


1.131 


1.159 


1.188 


'1.217 


1.246 


1.276 


1.307 


1.338 


1.370 


1.403 


1.469 


6 


1.160 


1.194 


1.229 


1.265 


1.302 


1.340 


1.379 


1.419 


1.459 


1.501 


1.587 


7 


1.189 


1.230 


1.272 


1.316 


1.361 


1.407 


1.455 


1.504 


1.554 


1.606 


1.714 


8 


1.218 


1.267 1.317 


1.369 


1.422 


1.477 


1.535 


1.594 


1.655 


1.718 


1.851 


9 


1.249 


1.305 1 1.363 


1.423 


1.486 


1.551 


1.619 


1.689 


1.763 


1.838 


1.999 


10 


1.280 


1.344 


1.411 


1.480 


1.553 


1.629 


1.708 


1.791 


1.877 


1.967 


2.159 


11 


1.312 


1.384 


1.460 


1.539 


1.623 


1.710 


1.802 


1.898 


1.999 


2.105 


2.332 


12 


1.345 


1.426 


1.511 


1.601 


1.696 


1.796 


1.901 


2.012 


2.129 


2.252 


2.518 


13 


1.379 


1.469 


1.564 


1.6G5 


1.772 


1.886 


2.006 


2.133 


2.267 


2.410 


2.720 


14 


1.413 


1.513 


1.619 


1.732 


1.852 


1.980 


2.116 


2.261 


2.415 


2.579 


2.937 


15 


1.448 


1.558 


1.675 


1.801 


1.935 


2.079 


2.232 


2.397 


2.572 


2.759 


3.172 


16 


1.485 


1.605 


1.734 


1.873 


2.022 


2.183 


2.355 


2.540 


2.739 


2.952 


3.426 


17 


1.522 


1.653 


1.795 


1.948 


2.113 


2.292 


2.485 


2.693 


2.917 


3.159 


3.700 


18 


1.560 


1.702 


1.857 


2.026 


2.208 


2.407 


2.621 


2.854 


3.107 


3.380 


3.996 


19 


1.599 


1.754 


1.923 


2.107 


2.308 


2.527 


2.766 


3.026 


3.309 


3.617 


4.316 


20 


1.639 


1.806 


1.990 


2.191 


2.412 


2.653 


2.918 


3.207 


3.524 


3.870 


4.661 


21 


1.680 


1.860 


2.059 


2.279 


2.520 


2.786 


3.078 


3.400 


3.753 


4.141 


5.034 


22 


1.722 


1.916 


2.132 


2.370 


2.634 


2.925 


3.248 


.3.604 


3.997 


4.430 


5.437 


23 


1.765 


1.974 


2.206 


2.465 


2.752 


3.072 


3.426 


3.820 


4.256 


4.741 


5.871 


24 


1.809 


2.033 


2.283 


2.563 


2.876 


3.225 


3.615 


4.049 


4.533 


5.072 


6.341 


25 


1.854 


2.094 


2.363 


2.666 


3.005 


3.386 


3.813 


4.292 


4.828 


5.427 


6.848 


26 


1.900 


2.157 


2.446 


2.772 


3.141 


3.556 


4.023 


4.549 


5.142 


5.807 


7.396 


27 


1.948 


2.221 


2.532 


2.883 


3.282 


3.733 


4.244 


4.822 


5.476 


6.214 


7.988 


28 


1.996 


2.288 


2.620 


2.999 


3.430 


3.920 


4.478 


5.112 


5.832 


6.649 


8.627 


29 


2.046 


2.357 


2.712 


3.119 


3.584 


4.116 


4.724 


5.418 


6.211 


7.114 


9.317 


30 


2.098 


2.427 


2.807 


3.243 


3.745 


4.322 


4.984 


5.743 


6.614 


7.612 


10.063 


31 


2.150 


2.500 


2.905 


3.373 


3.914 


4.538 


5.258 


6.088 


7.044 


8.145 


10.868 


32 


2.204 


2.575 


3.007 


3.508 


4.090 


4.765 


5.547 


6.453 


7.502 


8.715 


11.737 


33 


2.259 


2.652 


3.112 


3.648 


4.274 


5.003 


5.852 


6.841 


7.990 


9.325 


12.676 


34 


2.315 


2.732 


3.221 


3.794 


4.466 


5.253 


6.174 


7.251 


8.509 


9.978 


13.690 


35 


2.373 


2.814 


3.334 


3.946 


4.667 


5.516 


6.514 


7.686 


9.062 


10.677 


14.785 


36 


2.433 


2.898 


3.450 


4.104 


4.877 


5.792 


6.872 


8.147 


9.651 


11.424 


15.968 


37 


2.493 


2.985 


3.571 


4.268 


5.097 


6.081 


7.250 


8.636 


10.279 


12.224 


17.246 


38 


2.556 


3.075 


3.696 


4.439 


5.326 


6.385 


7.649 


9.154 


10.947 


13.079 


18.625 


39 


2.620 


3.167 


3.825 


4.616 


5.566 


6.705 


8.069 


9.704 


11.658 


13.995 


20.115 


40 


2.685 


3.262 


3.959 


4.801 


5.816 


7.040 


8.513 


10.286 


12.416 


14.974 


21.725 


41 


2.752 


3.360 


4.098 


4.993 


6.078 


7.392 


8.982 


10.903 


13.223 


16.023 


23.462 


42 


2.821 


3.461 


4.241 


5.193 


6.352 


7.762 


9.476 


11.557 


14.083 


17.144 


25.339 


43 


2.892 


3.565 


4.390 


5.400 


6.637 


8.150 


9.997 


12.250 


14.998 


18.344 


27.367 


44 


2.964 


3.671 


4.543 


5.617 


6.936 


8.557 


10.547 


12.985 


15.973 


19.628 


29.556 


45 


3.038 


3.782 


4.702 


5.841 


7.248 


8.985 


11.127 


13.765 


17.011 


21.002 


31.920 


46 


3.114 


3.895 


4.867 


6.075 


7.574 


9.434 


11.739 


14.590 


18.117 


22.473 


34.474 


47 


3.192 


4.012 


5.037 


6.318 


7.915 


9.906 12.384 


15.466 


19.294 


24.046 


37.232 


48 


3.271 


4.132 


5.214 


6.571 


8.271 


10.401 13.065 


16.394 


20.549 


25.729 


40.211 


49 


3.353 


4.256 


5.396 


6.833 


8.644 


10.921 


13.784 


17.378 


21.884 


27.530 


43.427 


50 


3.437 


4.384 


5.585 


7.107 


9.033 


11.467 


14.542 


18.420 


23.307 


29.457 


46.902 



328 



Table VIII. American Experience Mortality Table 



Based on 100,000 living at age 10. 



At 


Number 




At 


Number 




At 


Number 




At 


Number 




Age. 


Surviving. 


Deaths. 


Age. 


Surviving. 


Deaths. 


Age. 


Surviving. 


Deaths. 


Age. 


Surviving. 


Deaths. 


10 


100,000 


749 


35 


81,822 


732 


60 


57,917 


1,546 


85 


5,485 


1,292 


11 


99,251 


746 


36 


81,090 


737 


61 


56,371 


1,628 


86 


4,193 


1,114 


12 


98,505 


743 


37 


80,353 


742 


62 


54,743 


1,713 


87 


3,079 


933 


13 


97,762 


740 


38 


79,611 


749 


63 


53,030 


1,800 


88 


2,146 


744 


14 


97,022 


737 


39 


78,862 


756 


64 


51,230 


1,889 


89 


1,402 


555 


IS 


96,285 


735 


40 


78,106 


765 


65 


49,341 


1,980 


90 


847 


385 


16 


95,550 


732 


41 


77,341 


774 


66 


47,361 


2,070 


91 


462 


246 


17 


94,818 


729 


42 


76,567 


785 


67 


45,291 


2,158 


92 


216 


137 


18 


94,089 


727 


43 


75,782 


797 


68 


43,133 


2,243 


93 


79 


58 


19 


93,362 


725 


44 


74,985 


812 


69 


40,890 


2,321 


94 


21 


18 


20 


92,637 


723 


45 


74,173 


828 


70 


38,569 


2,391 


95 


3 


3 


21 


91,914 


722 


46 


73,345 


848 


71 


36,178 


2,448 








22 


91,192 


721 


47 


72,497 


870 


72 


33,730 


2,487 








23 


90,471 


720 


48 


71,627 


896 


73 


31,243 


2,505 








24 


89,751 


719 


49 


70,731 


927 


74 


28,738 


2,501 








25 


89,032 


718 


50 


69,804 


962 


75 


26,237 


2,476 








26 


88,314 


718 


51 


68,842 


1,001 


76 


23,761 


2,431 








27 


87,596 


718 


52 


67,841 


1,044 


77 


21,330 


2,369 








28 


86,878 


718 


53 


66,797 


1,091 


78 


18,961 


2,291 








29 


86,160 


719 


54 


65,706 


1,143 


79 


16,670 


2,196 








30 


85,441 


720 


55 


64,563 


1,199 


80 


14,474 


2,091 








31 


84,721 


721 


56 


63,364 


1,260 


81 


12,383 


1,964 








32 


84,000 


723 


57 


62,104 


1,325 


82 


10,419 


1,816 








33 


83,277 


726 


58 


60,779 


1,394 


83 


8,603 


1,648 








34 


82,551 


729 


59 


59,385 


1,468 


84 


6,955 


1,470 









329 



Table IX. Heights and Weights of Men 

Light-face figures are 20 per cent, under and over the average. 



AGES. 


S 




01 

I 


* 

n 

i 


O 

n 
i 

IO 

w 

105 
131 

157 


1 


OJ 

I 

# 

107 
134 

161 


s 
s 

107 
134 

161 



I 

10 




i< 
| 

cq 


OJ 
<N 

I 

C* 


* 


A 

123 
154 
185 


a 
3 

00 


5 

i 


OJ 

2 

* 

129 
161 
193 


50-54 


o 

B 
1 

10 

130 
163 

196 


Ft. 
5 


In. 



96 
120 

144 


100 
125 

150 


102 
128 
154 


106 
133 

160 


107 
134 
161 


Ft. 
5 


In. 
8 


117 
146 
175 


121 
151 
181 


126 
157 

188 


128 
160 

192 


130 
163 

196 




1 


98 
122 
146 


101 
126 
151 


103 
129 
155 


105 
131 

157 


107 109 
134 136 
161 163 


109 109 
136 136 
163 163 




9 


120 124 127 
150155 159 

180186191 


130 132 
162 165 
194 198 


133 
166 

199 


134 
167 

200 


134 
168 

202 





2 
3 
4 
5 


99 
124 
149 
102 
127 
152 


102 
128 
154 
105 
131 
157 


105 
131 
157 


106 
133 
160 
109 
136 
163 


109 
136 
163 


110 
138 
166 

113 
141 
169 


110 
138 
166 
113 
141 
169 


110 
138 
166 
113 
141 
169 




10 

11 


123 
154 
185 
127 
159 
191 


127 131 
159 164 

191 197 


134 
167 

200 


136 137 
170 171 
204205 


138 
172 
206 
142 
177 
212 


138 
173 
208 
142 
178 
214 


107 
134 
161 


111 
139 

167 




131 
164 

197 


135 
169 

203 


138 
173 
208 


140 142 
175 177 
210212 


105 108 110 
131 135 138 

157:162 166 


112 
140 

168 


114 
143 
172 


115 116 
144 145 
173 174 


116 
145 
174 


6 





132 136 140 143 
165 170 175 179 
198 204 210 215 


144 146' 146 146 
180 183 182 183 

216220218220 




107 
134 
161 


110 
138 
166 


113 
141 
169 


114 
143 
172 
118 
147 
176 


117 
146 
175 


118 
147 
176 


119 
149 
179 


119 
149 
179 




1 


136142 
170 177 
204 ; 212 


145 148 
181 185 
217 222 


149 151 
186 189 

223 227 


150 151 
188 189 

226 227 




6 


110114 
138 142 
166170 


iir, 
145 
174 


120 
150 
180 
124 
155 
186 


121 
151 
181 


122 
153 

184 


122 
153 
184 




2 


141 147 
176 184 
211 221 


150 
188 
226 


154 155! 157 
192 194 196 
230:233 235 


166 
194 

233 


155 
194 
233 




7 


114118 
142 147 

170 176 

i 


120 
150 

180 


122 
152 

182 


125 
156 

187 


126 
158 

190 


126 
158 
190 




3 


145 
181 

217 


152 
190 

228 


156 
195 
234 


160 
200 
240 


1621163 
203204 
244,245i 


161 

201 
241 


158 
198 
238 



330 



FOUR PLACE TABLES 33 J 

EXPLANATION OF TABLE II* 
VALUES AND LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 

1. DIRECT READING OF THE VALUES. This table gives the sines, 
cosines, tangents and cotangents of the angles from to 45; and by 
a simple device, indicated by the printing, the values of these functions 
for angles from 45 to 90 may be read directly from the same table. 
For angles less than 45 read down the page, the degrees and minutes 
being found on the left; for angles greater than 45 read up the page 
the degrees and minutes being found on the right. 

To find a function of an angle (such as 15 27', for example) we 
employ the process of interpolation. To illustrate, let us find tan 15 
27'. In the table we find tan 15 20' = .2742 and tan 15 30' = .2773; 
we know that tan 15 27' lies between these two numbers. The process 
of interpolation depends on the assumption that between 15 20' and 
15 30' the tangent of the angle varies directly as the angle; while this 
assumption is not strictly true, it gives an approximation sufficiently 
accurate for a four-place table. Thus we should assume that tan 
15 25' is halfway between .2742 and .2773. We may state the problem 
as follows: An increase of 10' in the angle increases the tangent .0031; 
assuming that the tangent varies as the angle, an increase of 7' in the 
angle will increase the tangent by .7 X .0031 = .00217. Retaining only 
four places we write this .0022. Hence 

tan 15 27' = .2742 + .0022 = .2764. 

The difference between two successive values in the table is called 
the tabular difference (.0031 above). The proportional part of the 
tabular difference which is used is called the correction (.0022 above), 
and is found by multiplying the tabular difference by the appropriate 
fraction (.7 above). 

Example 1. Find sin 63 52'. 

\Vefind 

sin 63 50' = .8975. 
tabular difference = .0013 (subtracted mentally from the table). 

correction = .2 X .0013 = .0003 (to be added). 
Hence, 

sin 63 52' = .8978. 

* The use of Table I. is explained on pages 80-86 of the text. 



332 MATHEMATICS 

Example 2. Find tan 37 44'. 

tan 37 40' = .7720 
tabular difference = .0046 

correction = .4 X .0046 = .0018. 
Hence, 

tan 37 44' = .7738. 

Example 3. Find cos 65 24'. 

cos 65 20' = .4173 
tabular difference = 26; .4 X 26 = 10 

(to be subtracted because the cosine decreases as the angle increases). 
Hence 

cos 65 24' = .4163. 
Example 4. Find ctn 32 18'. 

ctn 32 10' = 1.5900 

tabular difference = 102; .8 X 102 = 82 (to be subtracted). 
Hence, 

ctn 32 18' = 1.5818. 

Rule. To find a trigonometric function of an angle by interpolation: 
select the angle in the table which is next smaller than the given angle, and 
read its sine (cosine, tangent, or cotangent as the case may be) and the 
tabular difference. Compute the correction as the proper proportional 
part of the tabular difference. In case of sines or tangents ADD the cor- 
rection: in case of cosines or cotangents, SUBTRACT it. 

2. REVERSE READINGS. Interpolation is also used in finding the 
angle when one of its functions is given. 

Example 1. Given sin x = .3294, to find x. 

Looking in the table we find the sine which is next less than the given 
sine to be .3283, and this belongs to 19 10'. Subtract the value of the 
sine selected from the given sine to obtain the actual difference = .0011; 
note that the tabular difference = .0028. We may state the problem 
as follows: an increase of .0028 in the function increases the angle 10'; 
then aa increase of .0011 in the function will increase the angle 11/28 
of 10 = 4 (to be added). Hence x = 19 14'. 

Example 2. Given cos x = .2900, to find x. 

The cosine in the table next less than this is .2896 and belongs to 
73 10'; the tabular difference is 28; the actual difference is 4; correction 
= 4/28 of 10 = 1 (to be subtracted). Hence x = 73 9'. 



FOUR PLACE TABLES 333 

Rule. To find an angle when one of its trigonometric functions is 
given: select from the table the same named function which is next less than 
the given function, noting the corresponding angle and the tabular differ- 
ence: compute the actual difference (between the selected value of the func- 
tion and the given value), divide it by the tabular difference, and multiply 
the result by 10; this gives the correction which is to be added if the given 
function is sine or tangent, and to be subtracted if the given function is 
cosine or cotangent. 

3. THE LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS. If it is 

required to find log sin 63 52', the most obvious way is to find sin 63 52' 
= .8978, and then to find in Table I, log .8978 = 9.9532 - 10, but this 
involves consulting two tables. To avoid the necessity of doing this, 
Table II gives the logarithms of the sines, cosines, tangents, and co- 
tangents. The student should note that the sines and cosines of all 
acute angles, the tangents of all acute angles less than 45 and the 
cotangents of all acute angles greater than 45 are proper fractions, 
and their logarithms end with 10, which is not printed in the table, 
but which should be written down whenever such a logarithm is used. 

Example 1. Find log sin 58 24'. 

In the row having 58 20' on the right and in the column having 
sine at the bottom find log sin 58 20' = 9.9300 - 10; the tabular differ- 
ence is 8; correction = .4 X 8 = 3 (to be added). Hence 

log sin 58 24' = 9.9303 - 10. 

(In case of sine and tangent add the correction.) 
Example 2. Find log cos 48 38'. 

log cos 48 30' = 9.8213 - 10, tabular difference 15; 
.8 X 15 = 12 (subtract) therefore log cos 48 38' = 9.8201 - 10. 

(In case of cosine and cotangent, subtract the correction.) 

Example 3. Given log tan x = 0.0263, to find x. 

The log tan in Table II next less than the given one is 0.0253 and 
belongs to 46 40'; actual difference is 10; tabular difference is 25; 
correction = 10/25 of 10 = 4. Hence x = 46 44'. 

Example 4. Given log cos x = 9.9726 10, to find x. 

The logarithmic cosine next less than the given one is 9.9725 10 
and belongs to 20 10'; actual difference = 1; tabular difference = 5; 
correction = 1/5 X 10 = 2 (subtract). Hence x = 20 8'. 



INDEX 



Abscissa, 38 

Addition formulas, 116 

Angles, 91, 94, 101, 104, 106 

trigonometric functions of, 

91, 102 
Annuity, amount of, 254, 255 1 

cost of, 259 

present value of, 254, 258 
Area, by offsets, 147 

by rectangular coordinates, 
147 

of ellipse, 202 

of triangle, 134 
Associative law, 3 
Asymptote, 205, 206 
Auxiliary circle, 202 
Average, 262 

arithmetic, 262, 263 

weighted arithmetic, 263 

geometric, 265 
Axis, 38, 61, 196, 199 

Bearing, 140 

Binomial coefficients, 8, 277 

series, 281 

theorem, 7, 276, 278 

Characteristic, 77 
Circle, 190 

auxiliary, 202 
Coefficients, 23 

binomial, 8, 277 

variability, 300 
Cologarithm, 84 
Combinations, 270, 273 
Commutative law, 3 
Components, of a force, 156 

rectangular, 157 
Compound interest, 255, 256, 

286 

Conic sections, 190 
Coordinates, 38 
Corners, 142 
Correlation, 304 

coefficient of, 307 



Correlation, measure of, 305 

table, 306 
Cosecant, 92, 103 
Cosines, 91, 102 

law of, 122 
Cotangent, 92, 103 
Couple, 168 
Crane, 160 

Degree, 23 

Deviation, standard, 298 
Diagrams, 41, 47, 49, 50, 51 
Directrix, 195 

Distance between two points, 53 
Distributive law, 3 
Division, point of, 55 
ratio of, 54 

Eccentricity, of ellipse, 200 

of hyperbola, 204 
Elimination, 17 
Ellipse, 190, 199 

area of, 202 
Equation, of a circle, 190 

of a curve, 58 

of an ellipse, 201 

of a hyperbola, 204, 205, 
206 

of a parabola, 195, 197 

of a straight line, 63, 64, 66, 

67 
Equations, definition of, 11 

conditional, 13 

empirical, 226 

equivalent, 14 

general, 68, 191 

having given roots, 32 

in quadratic form, 29 

linear, 18, 24, 68 

quadratic, 24 

simultaneous, 16 

trigonometric, 107 

transformation of, 14 
Equilibrium, conditions of, 169 
Error, 177, 181 



335 



336 



INDEX 



Error, curve of, 298 

in a fraction, 179 

in parts of a triangle, 185 

in a product, 178 

in a square, 183 

in a square root, 183 

in a sum, 178 

in trigonometric functions, 
183 

probable, 300 
Evolution, 4 
Expectation, 292 
Exponents, 4, 5, 6, 23 

Focal properties, 211, 213 
Focus, 195, 199, 201, 204 
Forces, components of, 156 
concentrated, 154 
distributed, 154 
graphical representation of, 

154 

moments of, 167 
parallelogram of, 155 
resolution of, 156 
Frequency distribution curves, 

296 
Functions, 218 

of complementary angles, 93 
of half an angle, 118 
of negative angles, 110 
of twice an angle, 118 
periodic, 110 
trigonometric, 91, 102, 105, 

109 
Fundamental relations, 95, 104 

Graphical solution, 36, 99, 154, 

158, 220, 228, 238 
Graphs, 41, 113, 154, 220 

Hyperbola, 190, 204 

equilateral or rectangular, 
205 

Identities, 12 
Imaginary numbers, 30 
Intercepts, 61 
Interpolation, 81, 331 
Intersection, points of, 62 

of conies, 215 

of loci, 62, 208 



Involution, 3 
Irrational numbers, 2 

Latus rectum, 196 
Law, associative, 3 

commutative, 3 

distributive, 3 

of cosines, 122 

of exponents, 6 

of sines, 121 

of tangents, 124 
Lines, base, 140 

bearing of, 140 

parallel, 63, 67 

perpendicular, 67 

random, 146 

range, 140 

slope of, 66 

through the origin, 64 

through two points, 66 

township, 140 
Locus, of a point, 57 

of an equation, 58, 59 
Logarithmic paper, 239 

plotting, 237 
Logarithms, Briggs, 76 

computation by, 87 

computation of, 76 

definition of, 72 

Napierian, 76 

properties of, 74 

Mantissa, 77 

Mass, 153 

Mean, arithmetic, 245, 262, 263 

geometric, 249, 265 
Measurement, 1, 181 

on level ground, 143 

on slopes, 144 

of force, 154 
Median, 264 
Mendel's law, 282 
Middle point, 56 
Mode, 264 
Moments, of force, 167 

center of, 167 

composition of, 167 
Momentum, 153 

Normal, 211 
Number, 2, 3, 30, 31 



INDEX 



337 



Oblique triangles, solution of, 

120, 125 
Offsets, 144 
Ordinate, 38 
Origin, 38 

Parabola, 190, 195 
Parallelogram of forces, 155 
Periodic functions, 110 
Permutations, 270 
Perpetuity, 260 
Point, of division, 55 

of intersection, 62 
Polygon of forces, 162 
Polynomial, 22 
Principal meridian, 140 
Probability, 291 

curve, 297, 298 

Probable error, in a single 
measurement, 300 

of arithmetic average, 300 

of standard deviation, 301 
Progression, arithmetic, 243 

geometric, 247, 252 
Proportional quantities, 64, 219, 
220 

Quadrantal angles, 104 
Quadratic equation, 24 

kind of roots, 30, 33 

number of roots, 33 

solution of, 26, 27 

sum and product of roots, 
32 

Radian, 113 
Ratio of division, 54 
Rational number, 2 
Rectangular components, 157 
coordinates, 38 



Rectangular hyperbola, 205, 206 

Regression curve, 309 

Resolution of forces, 156 

Resultant, 155 

of concurrent forces, 163 
of parallel forces, 165 

Right triangles, solution of, 97 

Root, 12 

Scales, 36 
Secant, 92, 102 
Series, binomial, 281 

infinite geometric, 252 
Sines, 91, 102 

law of, 121 
Slide rule, 88 
Slope, 66 

Statistical data, 40 
Substitution, 12, 233 
Symmetry, 61 

Tabular difference, 81, 331 
Tangents, 91, 102, 211 

law of, 124 

Translation of axes, 194 
Triangle, oblique, 120, 125 

of forces, 158 

right, 97 

Trigonometric functions, of an 
acute angle, 91 

of any angle, 102 

graphs of, 109 

line representation of, 105 

Variable, 218 
Variation, 219 

direct, 219 

inverse, 219 

joint, 219 

constant of, 219, 222 



Printed in the United States of America. 



TRIGONOMETRY 

BY 

ALFRED MONROE KENYON 

PROFESSOR OF MATHEMATICS, PURDUE UNIVERSITY 

AND LOUIS INGOLD 

ASSISTANT PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF 
MISSOURI 

Edited by EARLE RAYMOND HEDRICK 

Trigonometry, flexible cloth, pocket size, long I2mo (xi- s rij2 pp.) with Complete 

Tables (xviii + 124 pp.), $t-5O 

Trigonometry (xi-\- 132 pp.) with Brief Tables (xviii -{- 12 pp.), $1.20 
Macmillan Logarithmic and Trigonometric Tables, flexible cloth, pocket size, long 

izmo (xviii + 124 pp.) , $60 

FROM THE PREFACE 

The book contains a minimum of purely theoretical matter. Its entire 
organization is intended to give a clear view of the meaning and the imme- 
diate usefulness of Trigonometry. The proofs, however, are in a form that 
will not require essential revision in the courses that follow. . . . 

The number of exercises is very large, and the traditional monotony is 
broken by illustrations from a variety of topics. Here, as well as in the text, 
the attempt is often made to lead the student to think for himself by giving 
suggestions rather than completed solutions or demonstrations. 

The text proper is short; what is there gained in space is used to make the 
tables very complete and usable. Attention is called particularly to the com- 
plete and handily arranged table of squares, square roots, cubes, etc. ; by its 
use the Pythagorean theorem and the Cosine Law become practicable for 
actual computation. The use of the slide rule and of four-place tables is 
encouraged for problems that do not demand extreme accuracy. 

Only a few fundamental definitions and relations in Trigonometry need be 
memorized; these are here emphasized. The great body of principles and 
processes depends upon these fundamentals; these are presented in this book, 
as they should be retained, rather by emphasizing and dwelling upon that 
dependence. Otherwise, the subject can have no real educational value, nor 
indeed any permanent practical value. 



THE MACMILLAN COMPANY 

Publishers 64-66 Fifth Avenue New Tork 



ELEMENTARY MATHEMATICAL 
ANALYSIS 

BY 

JOHN WESLEY YOUNG 

PROFESSOR OF MATHEMATICS IN DARTMOUTH COLLEGE 

AND FRANK MILLETT MORGAN 

ASSISTANT PROFESSOR OF MATHEMATICS IN DARTMOUTH COLLEGE 



Edited by EARLE RAYMOND HEDRICK, Professor of Mathematics 
in the University of Missouri 

Cloth, i2tno, 542 pp., $2.60 

A textbook for the freshman year in colleges, universities, and 
technical schools, giving a unified treatment of the essentials of 
trigonometry, college algebra, and analytic geometry, and intro- 
ducing the student to the fundamental conceptions of calculus. 

The various subjects are unified by the great centralizing 
theme of functionality so that each subject, without losing its 
fundamental character, is shown clearly in its relationship to the 
others, and to mathematics as a whole. 

More emphasis is placed on insight and understanding of 
fundamental conceptions and modes of thought ; less emphasis 
on algebraic technique and facility of manipulation. Due recog- 
nition is given to the cultural motive for the study of mathe- 
matics and to the disciplinary value. 

The text presupposes only the usual entrance requirements in 
elementary algebra and plane geometry. 



THE MACMILLAN COMPANY 

Publishers 64-66 Fifth Avenue New Tcrk 



ANALYTIC GEOMETRY 

BY 

ALEXANDER ZIWET 

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN 

AND LOUIS ALLEN HOPKINS 

INSTRUCTOR IN MATHEMATICS IN THE UNIVERSITY OF MICHIGAN 

Edited by EARLE RAYMOND HEDRICK 

Flexible cloth. 111., izmo, viii + j6g pp., $fj6o 

Combines with analytic geometry a number of topics, tradi- 
tionally treated in college algebra, that depend upon or are 
closely associated with geometric representation. If the stu- 
dent's preparation in elementary algebra has been good, this 
book contains sufficient algebraic material to enable him to 
omit the usual course in College Algebra without essential 
harm. On the other hand, the book is so arranged that, for 
those students who have a college course in algebra, the alge- 
braic sections may either be omitted entirely or used only for 
review. The book contains a great number of fundamental 
applications and problems. Statistics and elementary laws of 
Physics are introduced early, even before the usual formulas 
for straight lines. Polynomials and other simple explicit func- 
tions are dealt with before the more complicated implicit equa- 
tions, with the exception of the circle, which is treated early. 
The representation of functions is made more prominent than 
the study of the geometric properties of special curves. Purely 
geometric topics are not neglected. 



THE MACMILLAN COMPANY 

Publishers 64-66 Fifth Avenue New Tork 



Analytic Geometry and Principles of Algebra 

BY 

ALEXANDER ZIWET 

PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF MICHIGAN 

AND LOUIS ALLEN HOPKINS 

INSTRUCTOR IN MATHEMATICS, THE UNIVERSITY OF MICHIGAN 

Edited by EARLE RAYMOND HEDRICK 

Cloth, viii + 369 pp., appendix, answers, index, I2mo, $i-7j 

This work combines with analytic geometry a number of topics traditionally 
treated in college algebra that depend upon or are closely associated with 
geometric sensation. Through this combination it becomes possible to show 
the student more directly the meaning and the usefulness of these subjects. 

The idea of coordinates is so simple that it might (and perhaps should) be 
explained at the very beginning of the study of algebra and geometry. Real 
analytic geometry, however, begins only when the equation in two variables 
is interpreted as defining a locus. This idea must be introduced very gradu- 
ally, as it is difficult for the beginner to grasp. The familiar loci, straight 
line and circle, are therefore treated at great length. 

In the chapters on the conic sections only the most essential properties of 
these curves are given in the text ; thus, poles and polars are discussed only 
in connection with the circle. 

The treatment of solid analytic geometry follows the more usual lines. But, 
in view of the application to mechanics, the idea of the vector is given some 
prominence; and the representation of a function of two variables by contour 
lines as well as by a surface in space is explained and illustrated by practical 
examples. 

The exercises have been selected with great care in order not only to fur- 
nish sufficient material for practice in algebraic work but also to stimulate 
independent thinking and to point out the applications of the theory to con- 
crete problems. The number of exercises is sufficient to allow the instructor 
to make a choice. 

To reduce the course presented in this book to about half its extent, the 
parts of the text in small type, the chapters on solid analytic geometry, and 
the more difficult problems throughout may be omitted. 



THE MACMILLAN COMPANY 

Publishers 64-66 Fifth Avenue New York 



THE CALCULUS 

BY 

ELLERY WILLIAMS DAVIS 

PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF NEBRASKA 

Assisted by WILLIAM CHARLES BRENKE, Associate Professoi ol 
Mathematics, the University of Nebraska 

Edited by EARLE RAYMOND HEDRICK 

Cloth, semi-flexible, xxi + 3&3 PP- + Tables (63), szmo, $2.10 
Edition De Luxe, flexible leather binding, India paper, $2.30 

This book presents as many and as varied applications of the Calculus 
as it is possible to do without venturing into technical fields whose subject 
matter is itself unknown and incomprehensible to the student, and without 
abandoning an orderly presentation of fundamental principles. 

The same general tendency has led to the treatment of topics with a view 
toward bringing out their essential usefulness. Rigorous forms of demonstra- 
tion are not insisted upon, especially where the precisely rigorous proofs 
would be beyond the present grasp of the student. Rather the stress is laid 
upon the student's certain comprehension of that which is done, and his con- 
yiction that the results obtained are both reasonable and useful. At the 
same time, an effort has been made to avoid those grosser errors and actual 
misstatements of fact which have often offended the teacher in texts otherwise 
attractive and teachable. 

Purely destructive criticism and abandonment of coherent arrangement 
are just as dangerous as ultra-conservatism. This book attempts to preserve 
the essential features of the Calculus, to give the student a thorough training 
in mathematical reasoning, to create in him a sure mathematical imagination, 
and to meet fairly the reasonable demand for enlivening and enriching the 
subject through applications at the expense of purely formal work that con- 
tains no essential principle. 



THE MACMILLAN COMPANY 

Publisher! 64-66 Fifth Avenue Hew Tork 



GEOMETRY 

BY 
WALTER BURTON FORD 

JUNIOR PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF 
MICHIGAN 

AND CHARLES AMMERMAN 

THE WILLIAM McKiNLEY HIGH SCHOOL, ST. Louis 

Edited by EARLE RAYMOND HEDRICK, Professor of Mathematics 

in the University of Missouri 

Plane and Solid Geometry, doth, izmo, 319 pp., $125 
Plane Geometry, cloth, I2mo, 213 pp., $ .80 
Solid Geometry, doth, I2mo, 106 pp., $ .80 

STRONG POINTS 

I. The authors and the editor are well qualified by training and experi- 
ence to prepare a textbook on Geometry. 

II. As treated in this book, geometry functions in the thought of the 
pupil. It means something because its practical applications are shown. 

III. The logical as well as the practical side of the subject is emphasized. 

IV. The arrangement of material is pedagogical. 

V. Basal theorems are printed in black-face type. 

VI. The book conforms to the recommendations of the National Com- 
mittee on the Teaching of Geometry. 

VII. Typography and binding are excellent. The latter is the reenforced 
tape binding that is characteristic of Macmillan textbooks. 

"Geometry is likely to remain primarily a cultural, rather than an informa- 
tion subject,"" say the authors in the preface. " But the intimate connection 
of geometry with human activities is evident upon every hand, and constitutes 
fully as much an integral part of the subject as does its older logical and 
scholastic aspect." This connection with human activities, this application 
of geometry to real human needs, is emphasized in a great variety of problems 
and constructions, so that theory and application are inseparably connected 
throughout the book. 

These illustrations and the many others contained in the book will be seen 
to cover a wider range than is usual, even in books that emphasize practical 
applications to a questionable extent. This results in a better appreciation 
of the significance of the subject on the part of the student, in that he gains a 
truer conception of the wide scope of its application. 

The logical as well as the practical side of the subject is emphasized. 

Definitions, arrangement, and method of treatment are logical. The defi- 
nitions are particularly simple, clear, and accurate. The traditional manner 
of presentation in a logical system is preserved, with due regard for practical 
applications. Proofs, both foimal and informal, are strictly logical. 



THE MACMTLLAN COMPANY 

Publishers 64-66 Fifth Avenue New York 



SLIDE-RULE 






r 



(1) (*) (3) 




I la' 



LLL.I 



DIRECTIONS 

A reasonably accurate slide-rule 
may be made by the student, for 
temporary practice, as follows. 
Take three strips of heavy stiff 
cardboard 1".3 wide by 6" long; 
these are shown in cross-section in 
(1), (2), (3) above. On (3) 
paste or glue the adjoining cut 
of the slide rule. Then cut strips 
(2) and (3) accurately along the 
lines marked. Paste or glue the 
pieces together as shown in (4) 
and (5). Then (5) forms the 
slide of the slide-rule, and it will 
fit in the groove in (4) if the work 
has been carefully done. Trim 
off the ends as shown in the large 
cut. 



m o 



UNIVERSITY OF CALIFORNIA LIBRARY 

Los Angeles 
This book is DUE on the last date stamped below. 



RHTD LD-UR0 

FEB161971 

16 1971 



Form L9-Series 444 



UC SOUTHERN REGIONAL LIBRARY FACILITY 



A 000933189 3