i .1 v i i ' \ * i i QA37 cop.l Southern Branch of the University of California Los Angeles Form L 1 QA37 K19 cop.l This book is DUE on the last date stamped below AUG 6 1925 NOV 2 1 1932 Form L-9-5m-7.' oc < THE WILEY TECHNICAL SERIES FOR VOCATIONAL AND INDUSTRIAL SCHOOLS EDITED BY JOSEPH M. JAMESON GIRARD COLLEGE THE WILEY TECHNICAL SERIES EDITED BY JOSEPH M. JAMESON MATHEMATICS TEXTS Mathematics for Machinists. By K. W. Burnham, MA 229 pages. 5 by 7. 175 figures. Cloth, $1.75 net. Arithmetic for Carpenters and Builders. By R. Burdette Dale, M.E. 231 pages. 5 by 7. 109 figures. Cloth, $1.75 net. Practical Shop Mechanics and Mathematics. By James F. Johnson. 130 pages. 5 by 7. 81 figures. Cloth, $1.40 net. CASS TECHNICAL HIGH SCHOOL SERIES Mathematics for Shop and Drawing Students. By H. M. Real and C. J. Leonard, vii +213 pages. 4J by 7. 188 figures Cloth, $1.60 net. Mathematics for Electrical Students. Bv H. M. Keal and C. J. Leonard, vii + 230 pages. 41 by 7. 165 figures. Cloth, $1.60 net. Preparatory Mathematics for Use in Technical Schools. By Harold B. Ray and Arnold V. Doub. viii-)-68 pages. 41 by 7. 70 figures. Cloth, $1.00 net. 6-15-21 MATHEMATICS FOR ELECTRICAL STUDENTS BY H. M. KEAL Head of Department of Mathematics, Cass Technical High School, Detroit C. J. LEONARD Instructor in Mathematics, Cass Technical High School, Detroit 46780 NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Limited 1921 Copyright, 1921 BT H. M. KEAL and C. J. LEONARD PREFACE The purpose of these two brief texts is to give to industrial workers and students who have not com- pleted high school, that part of algebra, geometry, and trigonometry which they will need in their work. The parts of algebra, geometry, and trigonometry necessary only for students going on to college are omitted. The texts are planned to present the material in a manner that will enable the student to master it with a minimum of outside help, and it is believed, therefore, that they are suitable for use with evening school and continuation classes, or wherever the work is largely individual. The contents of the course may briefly be outlined as follows: 1. Equations. 2. Slide Rule. 3. Formulas. 4. Positive and Negative Numbers. 5. Proportion. 6. Quadratic Equations. 7. Simultaneous Equations. 8. Graphs. 9. Commonly used Geometric Facts (without proof). 10. Logarithms. 11. Right Triangles. 12. Oblique Triangles. iv PREFACE The work is published in two volumes; one especially adapted to those interested in shopwork and drafting, the other to those interested in electrical work. The theoretical parts of the two are identical, but the one has applications from the shop and the other from electrical work. The books are not shop or electrical texts and cannot be expected to instruct studento along shop or electrical lines. They are mathematics books, written for shop or electrical students and workers, with applications drawn from the shop and electrical work. The main purpose of each is to give not merely a knowledge of mathematics, but to furnish that understanding of mathematics which will increase the student's practical working knowledge. The chapter on geometry is intended to state definitely the geometrical facts that most students know in a more or less general way. All of the geometry ordinarily needed for trigonometry and its uses is given, and the less evident theorems are illustrated by figures. The student should study the entire chapter and solve the problems at the end as a test of his understanding of geometrical facts and processes. The chapter on the slide rule is not a discussion of the theory of the rule but specific directions for its use. The method of presentation is as direct and definite as possible, so that the student can study the text by himself, and master it with little outside help. In courses where in- struction on the slide rule is given, the student should be urged to use the rule as much as possible, especially in the solution of formulas and in checking logarithmic problems. Many of the self-evident and little used definitions are omitted from the text for the sake of clearness. A dictionary of terms used in elementary mathematics is given in the PREFACE V appendix. The attention of students should be called to this dictionary, and they should be urged to use it. Tables of formulas, square roots, and decimal equiv- alents are given in the appendix for the convenience of the student. Should any students, after completing this text, desire to fulfill the college entrance requirements in mathematics, they will find that the knowledge gained from this course will enable them to cover the work in a regular college entrance course rapidly and efficiently, or the following topics added to the work of this course will meet the require- ments : Removal of Parentheses. Multiplication (more advanced problems). Division (Polynomial by Polynomial). Factoring. Fractional Equations (Polynomial Denominator). Literal Equations. Radicals. Equations Containing Radicals. Fractional, Zero, and Negative Exponents. Formal Geometry. Relation between Trigonometric Functions. Functions of the Sum and Difference of two Angles. Functions of twice an Angle and of half an Angle. Proof of the Law of Sines, Law of Cosines, Law of Tangents. H. M. K. C. J. L. Detroit, 1921. TABLE OF CONTENTS CHAPTER PAGE I. The Equation 1 II. The Slide Rule 13 III. Evaluation 29 IV. Positive and Negative Numbers 49 V. Ratio and Proportion 68 VI. Cutting Speed, Pulleys and Gears 79 VII. Electrical Formulas Involving Squares and Square Roots 86 VIII. Quadratic Equations 95 IX. Simultaneous Equations 104 X. The Graph 109 XI. Geometry 119 XII. Logarithms 131 XIII. Right Triangulation 148 XIV. Trigonometric Functions of Any Angle 164 XV. Oblique Triangles 169 XVI. Electrical Applications 177 Appendix Dictionary of Terms 201 Relation between Trigonometric Functions 211 Formulas 213 Decimal Equivalents 215 Logarithms 216 Logarithms of Functions. . . 218 Natural Functions 222 vii MATHEMATICS FOR ELECTRICAL STUDENTS CHAPTER I THE EQUATION 1. Definition. Scales will balance only when equal weights are placed in the pans of the scales. The fact that the 10-lb. weight in one pan, Fig. 1, is exactly balanced by the 5-, 3-, and 2-lb. weights in the other pan may be expressed,. 5+3+2=10. Fig. 1. Fig. 2. In the same manner, the balance of the scales when the unknown weight (x) and 5 lbs. are placed in one pan and 20 lbs. in the other, Fig. 2, may be expressed by the equation, z+5 = 20. The weight of (x) may be found by removing 5 lbs. from each pan of the scales, leaving (x) in one pan and 2 THE EQUATION 15 lbs. in the other and the scales still in balance. This balance may be expressed by the equation, 2=15. The equation x+5 = 20 is a statement that two expres- sions are equal. The expressions 2+5 and 20 are called members of the equation. To solve an equation is to find the value of the unknown which is expressed in the equation by a letter. 2. Subtracting from Both Members of an Equation. Example . Solve x + 5 = 20 2+5 = 20 2=15 (subtracting 5 from both members of the equation) EXERCISE 1 Solve for the unknown in each equation: 1. 2. 3. 4. 5. 2+7 =12. 2+9 =17. w+4i = 8. 2+2.5 = 5.5. Ans. 5. 6. 2+3.3 =7.8. Ans. 8. 7. a+ 2\ =3|. \\. 8. 2 + 31| = 42f. 3f. 9. 2+1.25 = 2.35. 3. 10. 6+1] =2.35. 4.5 H. m i.i i.i -4^n^ 17- Fig. 3. Equations may be used in solv- ing other problems than those of the balance. Example 1. Let x = the missing dimension, Fig. 3, 2+4=17 2=13 CHECKING AN EQUATION Example 2. What number added to 14 equals 30? Let x = the number 14+:r = 30 a; =16 EXERCISE 2 Find the missing dimensions, using equations: 1 2 3 . * n . ! ^ K J >|< i - L_ r< 21 _J 32 Fig. 5. Ans. 12. 4. A vacant lot and $192 were exchanged for another lot worth $1000. Find the value of the first lot. Ans. 3. Checking an Equation. In the equation 3a + 5 = 32, H Fl F1 Fig. 7, if 5 is subtracted | —\ from both members, the equa- A tion, will then be, Fig. 7. 3a = 27. The value of the unknown (a) may then be found by- dividing both members by 3, giving, a = 9. The problem may be proved by substituting 9 in place of (a) in the original equation. If both members reduce THE EQUATION to the same number, the solution is correct. A complete solution is, 3a+5 = 32 3a = 27 (subtracting 5 from both „ members) a = 9 (dividing both members by 3) Check. 3X9+5 = 32 27+5 = 32 32 = 32 EXERCISE 3 Ans. Solve and check: 1. 2a+12= 26. 7. 2. 9y+15 = 96. 9. 3. 12m + 8 = 98. 7.5. 4. 8+9z=116. 12. Find the missing dimensions: 9 10 5. 28*+ 14 =128. 6. 71m +.55 = 9. 07. 7. 113+ 1 = \ 9 -. 8. 7wJ+5f = 12f. Ans. .12. 2. 1. hJ" i t u -^ H — 25 »i Fig. 8. Ans. 2.5. r\ L2 A 25 5 A Fig. 10. Fig. 9. Ans. 6. 4. Adding to Both Mem- bers of an Equation. The 12-lb. weight, Fig. 10, is an up- ward pull on the pan con- taining the weight (w). If the scales balance and the 12-lb. weight is removed, a weight EQUATIONS WITH FRACTIONS 5 of 12 lbs. must bo placed on the other pan to balance the scales. Removing the upward pulling weight is the same as adding the weight to the other pan. Thus, expressed as an equation, w-12 = 25 w = 37 (adding 12 to both members) Example. 5x- 11 = 24 53 = 35 (adding 11 to both members) x = 7 (dividing both members by 5) Check. 5X7-11 = 24 35-11 = 24 24 = 24 EXERCISE 4 Solve and check: 1. 2. 3. •1. s-7 = 10. Ans. 17. 2x- 11 = 13. 12. 12s -34 = 26. 5. 2.1^-3.2 = 3.1. 3. 5. 3z-9| = 8.5. Ans. 6. 6. 17r-3|=13i 1. 7. 43+11 = 17. 1.5 8. 4x-ll = 17. 7. 9. Find the unknown weight in Fig. 11, if the scales balance as shown. Ans. lOf . 10. Three times a number [ less 26 is equal to 73. Find the number. Ans. 33. 5. Equations with Fractions. Example. £3 =10 r~\ LO BBE a Fig. 11. Check. x = 50 (multiplying both members by the least common de- nominator, 5) 1X50=10 THE EQUATION From this example, it can be seen that both members of an equation may be multiplied by the same number. This method is used in equations where fractions occur. Example. fa+£=3f. 5z+6 = 51 (multiplying both members by the L. C, D. 15) 5x = 45 (subtracting 6 from both mem- bers) x= 9 (dividing both members by 5) Check. 3 X " ~rs — o-g Q2_Q2 < • s — o s EXERCISE 5 Solve and check: Ans. 2. ■!■• 2"£ 3 — 3- 2. *x+34 = 9i. 15. 4. 1^-31 = 7. E , - M/ 1 _ C2 3. ^—12 = 161^ means ^/ Ans. 21. 14. Ans. 84. 6. To Solve an Equation. Observe that: I. The same number may be added to both members. £ — 3 = 5 x = 8 II. The same number may be subtracted from both members. a:+7=12 x— 5 777. Both members of an equation may be multiplied by the same number. re = 14 TRANSPOSITION 7 IV. Both members of an equation may be divided by the same number. 3a: =15 x= 5 Note. In x — 5 = 8 x = 8+5 = 13 and z+7 = 12 x = 12-7 = 5 The effect of adding or subtracting the same number from both mem- bers of the equation may be gained by transferring (transposing) the number to the other member and changing the sign before it. V. An equation may be checked by substituting in the original equation, the value found for the unknown. If both members then reduce to the same number, the solution is correct. VI. In an equation, such as 2x-\-o = x-\-9, 2x, 5, x, and 9 are the terms of the equation, 2x+5 and x+9 are the members of the equation separated by the sign of equality ( = ) . Example 1. A man saved a certain amount in one year, twice as much in the second year and three times as much in the third year as in the first. In all he saved $1200. How much did he save during each year? Let x = amount saved first year then 2x = amount saved second year and 3a; = amount saved third year z+2aH-3z=1200 (a; means la;) 6a; = 1200 (combining like terms) x= 200 2x= 400 3a; = 600 1200 8 THE EQUATION Example 2 . x + 7x - 3x = 55 . 5x = 55 (combining like terms) x=ll Check. 11 + 77-33 = 55 Example 3 . 7x — 3x + x = 55 . 5x — 55 (combining like terms) x=U Check. 77-33 + 11=55 Example 4. x— 3x+7:r = 55. 5x = 55 (combining like terms) x=U Check. 11-33+77 = 55 Note. Observe that in combining x—Sx~t7x, x and 7x were first added and the 3x then subtracted. In combining 11—33 + 77, 11 and 77 were added and the 33 then subtracted. The sign before each term belongs to that term. ( + ) is understood before the first term if no sign is expressed. EXERCISE 6 Solve and check: Ans. Ans. 1. x+ll:c = 24. 2. 4. 3z - 7x + lOz =36. 6. 2. 7m+3m=110. 11. 5. x-ll.r+12.r = 32. 16. 3. 17*-3W=45. 3. 6. 2x- 7+3z = 28. 7. Solution. 2x - 7 + 3x = 28 5x-7 = 28 5x = 35 x= 7 Check. 14-7 + 21=28 Note. Observe that the 7 cannot be combined with the 3.r and 2x as it is not a "like term." UNKNOWN IN BOTH MEMBERS 9 7. 13m -2m- 12 = 21. Ans. 3. 8. 4z + 7 + 3x = 91. 12. 9. &c-14z+9 + 7:c = 37. 28. 10. $y-19-17y+12y=U. 10. 11. The sum of three numbers is 32. The second is twice the first and the third is four times the first. Find the numbers. Ans. 4f, 9|, 18f. 12. A man saved $800 in three years. The second year he saved three times as much as the first, and the third year he saved $200. How much did he save the first year? Ans. $150. 13. A man spent $175 in January. He paid a certain sum for room rent, twice as much for board, as much for clothing as for board, and $50 for incidentals. How much did he spend for each? Ans. $25, $50, $50. 14. The sum of \, \, and | of a number is equal to 52. Find the number. Ans. 48. 7. Equations with the Unknown Term in Both Members. Example. 4x = 2x+14. 2x = 14 (subtracting 2x from both members) x = 7 Check. 28=14+14 Example 2. 7x = 48 - bx. \2x = 48 (adding 5x to both members) x= 4 Check. 28 = 48-20 Example 3. 4m — 32 = 2m. 4m = 2m +32 (adding 32 to both mem- bers 2m = 32 (subtracting 2ra from both members) m= 16 10 THE EQUATION Check. 64-32 = 32 Note. Observe that terms containing the unknown may be added to or subtracted from both members. EXERCISE 7 Solve and check: 1. 3^-7 = 2/+5. Ans. 12. 2. 13m -42 = 6 + w. 4. 3. 2^+8+37/4-7 = 18+4?/. 3. 4. 3x — 4 = x. 2. 5. 3x-4=17. 7. 6. %x-2h = 7|-2x (clear of fractions). 3f . 7. 4 m + 12 = 6m 12 = 2m (subtracting 4m fr om both members) m = 6 Check. 24+12 = 36 Note. Observe that the unknown may be collected on either side of the sign of equality.) 8. 3?/+ 18=% +6. Ans. 2. 9. 2y-2$=5y-17l. 5. 10. 17 = 2x- 3. 10. 11. ^+40 = fn-7+4n. 14. 12. lx + 62 = 3a; + 2. 30. 13. |+2=y-lf 19. 14. 7x + 20-3x = 60+4a:-50+8x. \\. 15. 3m + 60=15m+3-2m + 7. 5. 8. Equations with the Unknown in the Denominator. Equations sometimes occur in which the unknown appears in the denominator. In such equations, the unknown becomes part of the common denominator. REVIEW 11 EXERCISE 8 Solve and check: 21 = 3 2a 4' Ans. 14. 42 = 3a (multiplying both members by the L. C. D. 4a) 39 + 10 = 7x (multiplying both members by the L. C. b.Qx) 3. 12. 3. 11_ a = 3f. 12 4 2 4. m m 3' 5. 15,25 2s + 4* *' EXERCISE 9 Review Solve and check: 1. 2a; - -9 = 27. 2. 12z -9z = 14- -Ax. 3. 15- -3aH-lLr = = 39. 8 5 * o, , t A — — — — -— *}_L-I-- 5 3 5 ^ 10 ^4- Write equations and solve: 5. 6. Ans. 18. 2. 3. 6. ^-3- 7. kl25> u-3- *-2a>j -8Ji- -3 ?> | <. i ,> l « .3Cfr .1G3 -3.564- Fig. 12. Fig. 13. Fig. 14. Ans. 1|. Ans. 1.1. Ans. 2.908. 12 THE EQUATION 8. Find the diameters of the three circles, Fig. 15, if the diameter of the second is twice that of the first and of the third H that of the first. Ans. 3|, Of, 5. Fig. 15. 9. Three men together received $335. The first received $11 more than the second and $16 less than the third. Find what each received. Ans. $99, $110, $120. 10. One man has three times as many acres of land as another. After selling; GO acres to the second man, he has still 40 acres more than the other. How many acres had each at first? Ans - 80 > 24() - CHAPTER II SLIDE RULE The following rules and directions for the use of the slide rule are not intended for a complete manual of instruc- tions but are intended to present in a simple readable manner instruction's for solving the problems ordinarily solved on the slide rule. Practice problems are furnished so that the student will have had some practice before he begins to use the slide rule in a practical way. Theory of the rule will not be discussed, but considerable attention will be paid to the principles of operation which make for efficient handling of the rule. The work on the location of the decimal point may be omitted if desired. 9. Common Types of Slide Rules: BBB t lftLi i MJ^ i ffi i!.uu!JuJ...,.,.f..,.,.Li. UUu iA| i ^^ j 44 ' ^ ^d^^^^^^i i tf Fig. 16. — Mannheim Slide Rule. Recommended for general use. Fig. 17. — Polyphase Slide Rule. Recommended for general use and in some problems is more efficient than the Mannheim rule. 13 14 SLIDE RULE -r llwfr w ^l ' l 'l'l' l ' l ' l ' l l l l i mim < t» — 1 [ — * -Uol — , ■ t ■ , iT'i .1 / ri.ri Z3Z Fig. 18. — Polyphase Duplex Slide Rule. Useful in problems involv- ing ir (=3.1416) but not recommended for general use. 10. Scales. The face of most slide rules has four scales, the A, B, C and D scales. The A scale is on the upper part of the rule, the B scale on the upper part of the shde, the C scale on the lower part of the slide and the D scale on the lower part of the rule. The polyphase duplex has the C and D scales but not the A and B, the other two types have the four scales. The C and D scales are the ones in most common use and so will be considered first. Notice on your rule that the C and D scales are alike. Study the following relations on the D scale by reference to your rule. The scale as a whole is divided into 10 parts marked 1, 2, 3, 4, 5, 6, 7, 8 , 9, 1, as in Fig. 19. 3 4 Fig. 19. 7 8 1 These divisions correspond to the numbers 1, 2, 3, 4, etc., but the numbers on the slide rule have no decimal points, so the 3 stands for 3, 30, 300, 3000, etc., and the 4 for 4, 40, 400, etc. Numbers between 1 and 2, 2 and 3, etc., must be read on the unnumbered divisions between the numbered ones or sometimes approximated between the SCALES 15 small division lines. Each interval 1-2, 2-3, 3-4, etc., is divided into 10 main subdivisions indicated by the longest lines between the numbers. The main subdivisions between 1 and 2 are numbered 1, 2, 3, —9; but they are not so numbered on the rest of the rule. Great care must be taken not to confuse these numbers with the numbers on the main part of the scale. Each main division being divided into 10 main subdivisions, these main subdivisions will stand for 11, 12, 13, . . . , 21, 22, 23, 24, . . . ,31, 32, 33, ... , 41, 42, etc. EXERCISE 1 Locate the following numbers on the D scale and have the instructor verify three or four of your answers: 1. 5. 7. 55. 2. 7. 8. 20. 3. 65. 9. 19. 4. 68. 10. 15. 5. 23. 11. 18. 6. 50. 12. 92. The main subdivisions of a slide rule are again sub- divided into ultimate subdivisions. Due to the different lengths of the divisions some of the main subdivisions are divided into 10 spaces, some into 5 spaces, and others into 2 spaces. These ultimate subdivisions are the divisions between 23 and 24, . . . 77 and 78, etc., hence they correspond to the third figure of a number. For example, find 15 and 16 on the D scale. The divisions between 15 and 16 stand for 151, 152, 153, 154, 155, 156, 157, 158, 159. 16 SLIDE RULE EXERCISE 2 Find the following numbers of the I) scale and have the instructor verify three or four of the answers: 1. 125. 6. 115. 2. 146. 7. 105. 3. 181. 8. 103. 4. 195. 9. 108. 5. 175. 10. 199. Between 2 and 4 the main subdivisions are all divided into only five ultimate subdivisions, hence each ultimate subdivision stands for 2 in the third figure of a number. For example find 22 and 23 on the D scale. The divisions between 22 and 23 stand for 222, 224, 226, 228. EXERCISE 3 Find the following numbers on the D scale and have the instructor verify three or four of the answers: 1. 232. 6- 332. 2. 254. 7. 368. 3. 212. 8. 298. 4. 210. 9. 108. 5. 208. 10. 198. Between 4 and the end of the rule, the main subdivisions are divided into only two ultimate subdivisions, hence each ultimate subdivision stands for 5 in the third figure of a number. READING BETWEEN LINES 17 EXERCISE 4 Find the following numbers on the D scale: 1. 435. 6. 863. 2. 565. 7. 995. 3. 670. 8. 715. 4. 675. 9. 710. 5. 885. 10. 705, 11. Numbers Not Represented by a Line. Find 214 and 216 on the D scale; then 215 is midway between these two marks. To find 2155, set the cross mark of the runner f of the way between 214 and 216. Find 420 and 425 on the D scale; then 422 would be f of the way between these two marks. Reading the numbers between the lines on a slide rule is not exact and becomes a matter of judgment and experience. EXERCISE 5 Find the following numbers on the D scale: 1. 225. 9. 25. 2. 365. 10. 125. 3. 485. 11. 175. 4. 477. 12. 1625. 5. 585. 13. 115. 6. 518. 14. 1125. 7. 755. 15. 106. 8. 757. 16. 1065. 18 SLIDE RULE EXERCISE 6 Read the number represented by the positions of Fig. 20. 10 9 \ I'l'l'l 1 ! 11 ! 1 !'!'! 1 1 fl \ 4 5 7 k 8 9 1 111 T 12 6 Fig. 20. 1. Ans. 23. 2. 27. 3. 17. 4. 145. 5. 216. 6. 77. 9. 10. 11. 12. Ans. 495. 284. 1165. 103. 422. 715. 12. Division. To divide 6 by 3, find 6 on the D scale and move the slide to the right until the 3 on the C scale is directly above the 6. Find the answer on the D scale under the end of the C scale. Try this setting on your rule. To divide 20 by 5, find the 20 on the D scale and move the slide until the 5 on the C scale is directly above the 20 on the D scale. The answer is on the D scale under the end of the C scale. Rule. To divide two numbers find the dividend on the D scale and set the divisor (on the C scale) directly above the dividend. The quotient will be found on the D scale under the end of the C scale. MULTIPLICATION 19 EXERCISE 7 Divide: 1. 6-3. Ans. 2. 5. 168-5-2. Ans. 84. 2. 20 -=-4. 5. 6. 256X7. 36.6. 3. 24 -=-8. 3. 7. 653-21. 31.1. 4. 65-5. 13. 8. 768-125. 6.14. 13. Multiplication. Multiplication, in arithmetic, is the opposite of division, hence the operation of multiplication on the slide rule is the opposite of division on the slide rule. In arithmetic the check for division is the quotient times the divisor should equal the dividend. Thus: 6-4-3 = 2. Proof. 6 = 3X2. Therefore to multiply 2X3 on the slide rule the 2 will be in the same position as the 2 in 6—3; that is, on the D scale under the end of the C scale. 3 is found on the C scale and the answer on the D scale under the 3. Try this on your rule. Try also 25X3. Rule. To multiply two numbers, place the end of the C scale on one of the numbers {on the D scale) and find the answer on the D scale under the other number {on the C scale). In division the answer was under one end of the C scale in some problems and under the other end in other problems, and so in multiplication one end of the C scale is used in some problems and the other end in other problems. Use the end which will bring the other number above the D scale. For example, in 4X5 place the left end of the C scale on 4 and look for 5 on the C scale, it comes beyond the end of the D scale, therefore the right end of the C scale must be set on the 4 and under 5 will be found the answer 20. SLIDE RULE EXERCISE 8 1. 2. 3. 4. 5. 2X3. 4X12. 18X5. 5X6. 7X4. Ans. 6. 6. 75X22. 48. 7. 125X55. 90. 8. 223X64. 30. 9. 175X235. 28. 10. 1575X45. 20 Ans. 1650. 6775. 14280. 41100. 70800. 14. Location of the Decimal Point. Heretofore the loca- tion of the decimal point has been left to the student to determine by estimating how large the answer should be. In practice the decimal point is usually located in this manner, but it is well to know the rules. Rule. In multi-plication, if the slide projects to the left, add the number of places in both factors to determine the number of places in the product. If the slide projects to the right, add the number of places in both factors and subtract 1 to determine the number of places in the product. In division if the slide projects to the left, subtract the number of places in the divisor from the number of places in the dividend to deter- mine the number of places in the quotient. If the slide projects to the right, subtract the number of places in the divisor from the number of places in the dividend and add 1 to determine the number of places in the quotient. The following diagram should prove helpful in memo- rizing the above rules: Slide projects to the Left Righl Multiplication Add Add, subtract 1 Division Subtracl Subtract, add 1 In the above rules do not count the figures in the decimal part of a mixed number. In the case of decimals onlv Hi.' number of places is considered negative and is CONTINUED MULTIPLICATION 21 equal to the number of zeros between the decimal point and the first significant figure. EXERCISE 9 Solve the following problems paying attention to the location of the decimal point. 1. 18-M2. Ans. 1.5. 2. 16X8. 128. 3. 18-J-3. 6. 4. 16X2.4. 38.4. 5. 537X38. 20400. 6. 1250X6.25. 7820. 7. 2500X890. 2225000. 8. 234X0.012. 2.808. 9. 5620X3.45. 19400. 10. 25-5-0.0062. 0.155. 15. Continued Multiplication. In the use of the slide rule great care must be taken to develop efficiency or the rule will never become a labor-saving device. In finding the product of 3 factors, as 12X14X24, speed and accuracy can be gained by noting that, after 12 is multiplied by 14 in the usual way, the answer appears on the D scale. It will not be necessary to read this number as it is in place for multiplying it by 24; therefore, mark this product with the crossmark of the runner and then multiply the number represented by this position by 24 by drawing the end of the C scale under the crossmark and find the product on the D scale below the 24 on the C scale. Try this on your rule. Marking the first product in this way saves time and eliminates the error of reading this product and again setting to the number read. 22 SLIDE RULE EXERCISE 10 1. 2. 3. 4. 5. 2X3X4. 12X7X13. 250X3X78. 172X305X450. 751X0.046X231 Ans. 24. 1092. 58500. 23600000. 7980. Kl 16. Formulas of the Type R=~. In problems of this type it is most efficient to first divide K by A and then 25X48 multiply by /, For example, R= — —r — Divide 25 by 32 in the usual way; the result appears on the D scale under the end of the C scale. Then to multiply this number by 48, find 48 on the C scale and the answer, 375, appears on the D scale under the 48 on the C scale. The student should keep in mind the fact that, in problems of this type, it is best to do the division first, since this leaves the rule set for multiplying without moving the slide. All problems of this type can be done with one setting of the rule; but in some cases the second factor will appear off the scale. It is then necessary to change ends with the slide before doing the multiplication. This is accomplished by marking the end of the C scale with the crossmark of the runner and bringing the other end of the C scale under the runner. Changing ends of the slide is not called a slide rule opera- tion. EXERCISE 11 Ans. 15. 17.5X32 15 37.3. PROPORTION 23 225X13.2 3 - " 790 * 376 - 25X3.3X21 8X17 17. Proportion. Set on your rule 2 above 3, and notice that any other pair of corresponding numbers are in the ratio of 2 to 3. For example, 4 is above 6 and 12 is above 18, etc. For any setting of the slide any two pairs of corresponding numbers are in the same ratio. This makes a simple method for solving proportion. For example, 16 = _z 27~64* Place 16 above 27 and find x above 64. In some problems it will be necessary to change ends with the slide after first setting for the one ratio. Observe that: When in any problem a number appears off scale, it is necessary to change ends with the slide. EXERCISE 12 Ans. 6. 30. 85.8. 80.8. 11690. 1. 2 x 3~9' 2. 3_18 5 x ' 125 371 3. x ~255' 4. x 350 289 1250' R 24.5 7500 24 SLIDE RULE 18. The A and B Scales. Most slide rules have an A scale on the upper part of the rule and a B scale on the upper part of the slide. Make sure from the instructor or other- wise, that your rule has these two scales before proceeding with the following directions: The A and B scales are the same and are composed each of two slide rule scales placed end on end. The two scales will be distinguished as the right scale and left scale. Numbers are read as follows: On the left scale 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and on the right scale 10, 20, 30, 40, 50, 60, 70, 80, 90, 100; 100, 200, 300, etc., appear again on the left scale and 1000, 2000, 3000, appear on the right scale. Or, in other words, the 4 on the left scale stands for 4, 400, 40000, etc., while the 4 on the right scale stands for 40, 4000, etc., and not for 4 or 400, etc. Rule. Numbers having a?i odd number of places are found on the left scale, and numbers having an even number of places are found on the right scale. In counting the number of places in a number do not count the figures in the decimal part of a mixed number. In the case of decimals only, the number of places is deter- mined by the number of zeros between the decimal point and the first significant figure. Thus .04 is found on the left scale and .00423 is found on the right scale. A decimal with no zeros before the significant figure is found on the right scale. The A and B scales being shorter than the C and D scales there are not as many ultimate subdivisions. The student should study the scales immediately to determine the number of and value of the ultimate subdivisions. 19. Use of he A and B Scales. The A and B scales are used in connection with the C and D scales to find squares and square roots and to solve problems involving squares and square root. USE OF A AND B SCALES 25 Find several numbers of the D scale as 2, 5, 12, 8, 34, and notice that the squares of these numbers 4, 25, 144, 64, 1156, appear on the A scale directly above the numbers on the D scale. The same relation exists between the B scale and C scale. To find the square root of a number, find the number on the A or B scale, and find the square root on the D or C scale directly below the number. EXERCISE 13 Find the square root of the following numbers: 1. 4. Ans. 2. 4. 563. Ans. 23.7. 2. 36. 6. 5. 1582. 39.8. 3. 144. 12. 6. 75000. 274. In handling formulas involving squares and square roots do the multiplications and divisions on the A and B scales, using the C and D scales only to locate such numbers as 17 2 , 352 2 , etc., or to find the square root of a product or quotient. The rules for multiplication and division on the A and B scales are the same as for multi- plication and division on the C and D scales. Example 1. 7X5 2 =? Find 7 on the A scale and set the right end of the B scale under it the answer will appear on the A scale above 5 2 on the B scale, that is, above the 5 on the C scale ; use the crossmark on the runner. It is not necessary to read the value of 5 2 . Example 2. -7^=? Find 36 on the A scale place 2 2 on the B scale under it, using the 2 on the C scale to locate the 2 2 on the B scale. The answer is on the A scale above the end of the B scale. 12 2 Example 3. -75- = ? Find 12 2 on the A scale (above 18 26 SLIDE RULE 12 on the D scale); set 18 on the B scale below 12 2 . The answer appears on the A scale. Example 4. Vff=? Or V23X7 = ? Do the multi- plication or division on the A and B scales and the answer will be found on the C or D scale below the result of multiplication or division on the B or A scale. It is not necessary to read the result on the A or B scale. Example 5. — -^ — = ? This is similar to the -j- o A formula, and so 12 should be divided by 5 2 first, as in Example 2. This leaves the rule set for multiplying by 18 without resetting the slide. 20. General Suggestions. The following suggestions should prove helpful in the use of the slide rule. 1. The A and B scales are used in square and square root formulas, the C and D scales being brought into use to locate such numbers as 27 2 , etc., or to find a square root. 2. The principles of operation of the A and B scales are the same as for the C and D scales. 3. In division the dividend may be located on either scale and the divisor on the opposite scale, but the quotient will always appear on the scale with the dividend. 4. In formulas involving more than one operation keep in mind where the result would appear after each operation. It is riot necessary to read these intermediate results. 5. Division leaves the rule set for multiplication without moving the slide. 21. Areas of Circles. Finding the area of circles is similar to Example 1 above, but it will be noticed that -k ( = 3. 1416) is marked with a special mark on the A and B scales. To find the areas of a circle place the end of the B scale on t on the A scale and find the areas on the A scale above the radius square of the B scale (that is, above the radius on READING FROM ONE SETTING 27 the C scale). With this one setting the areas of any number of circles can be read off on the A scale above the radius on the C scale. 22. Ratios of the Same Kind. Different values can be read from one setting of the rule in many other problems. For example, one setting makes it possible to read off the number of centimeters corresponding to any number of inches, or to read the number of pounds corresponding to any number of kilograms, or many other ratios based on the same constant. A table of these various settings and proportions will be found on the back of most single face rules and in the slide rule manual. 1. 27X83. 2. 6X17X35. 3. 167-5-14. . 27X35 EXERCISE 14 Review Ans. 2241. 3570. 11. 93 13 * 5. 16X83X41 125X7 6. 21 2 . 7. Vl34. 8. 32 2 X5. 9. 18X17 2 . 10. 11. (9X64) 2 . V35X72. 12. /l39 \ 22"' 13 27X22 17 2 72.7. 62.2. 441. 11.58. 5120. 5200. 332000. 50.2. 2.514. 2.06. 28 SLIDE RULE 14. ??. 71.3. 16. V 24X169. 63.7. 17. Find the areas of the circles having for radius (1 setting): (a) 3 in. Ans. 28.3. (6) 14 in. 616. (c) 37 in. 4300. (d) 2.68 in. 22.6. 18. Change to centimeters the following inches (1 setting) : (a) 8. Ans. 20.3. (b) 67. 170. (c) 12.3. 31.2. (d) 2.26. 5.74. 19. Change to inches the following centimeters: (a) 27. Ans. 10.63. (6) 19. 7.48. (c) 184. 72.4. (d) 643. 253.1. The work of this chapter is not a complete slide rule manual, but the student should be able to do the problems ordinarily done on the slide rule rapidly and efficiently. He needs still more practice to become accurate, rapid, and confident. There is still much to be learned on the slide rule, and the student should begin to study the manual furnished with slide rules, and to watch his own work to see where he can increase his speed and efficiency. CHAPTER III EVALUATION 23. General Numbers. The area of a rectangle is found by multiplying the base by the altitude. This may be expressed by (b)X(a), in which the value of (6) may be 12 ft., 7 in., 25 rods, or any number of any unit used to measure length, and (a) may be any number of a like unit. Letters which may represent different values in different problems are called general numbers. The unknown letters used in Chapter I are general numbers. EXERCISE 1 1. Find b+a when 6 = 3, a = 7. Ans. 10. 2. Find b+a when 6 = 5, a =12. 17. 3. Find — -\ — when m = 3, n = 4, n y x = 5, y = 8. If. n 4. Find R — -j when # = 5, n = 4, d=8. 4§. 24. Signs. When the multiplication of two or more factors is to be indicated, the sign of multiplication is often omitted or expressed by the sign (•); thus 7XaXbXm is written 7-a-b-m, or more often 7a6m. Care must be taken in the use of the sign ( • ) to distinguish it from the decimal point, 7-9 means 7X9, while 7.9 means 7^. 29 30 EVALUATION EXERCISE 2 1. Find ax when a = 3, x = 5. Ans. 15. 2. Find 3mn when m = 2, n = 7. 42. Note. When addition, subtraction, multiplication and division occur in the same problem, do the multiplication and division first in the order in which they occur, then do the addition and subtraction. 3. Find ax+by when a = 4, x = 7, 6 = 3, ^ = 4. Ans. 40. 4. Find 2nd— 3cd when a=l2, d = 4, c=5. 36. 25. Coefficient. When the multiplication of two factors is expressed as ax, either factor is called the coefficient of the other. EXERCISE 3 What is the coefficient of x in the following: 1. bx. 2. 5x. 3. 3ax. 4. (a+b)x. 5. (2d-S)x. 6. xy. 7. x(m — n). 26. Power. If all the factors of a product are the same, the product is called a 'power of that factor. Thus xxxx is x fourth power and is written x 4 . Similarly, xx =x 2 , and is read x 2d power or x square. xxx =z 3 , and is read x 3d power or x cube. xxxx =x 4 , and is read x 4th power. xxxxx = x 5 , and is read x 5th power. NOTATION 31 27. Exponent. The number which indicates the power is written as a small number above and to the right of the factor and is called its exponent. For example, in x 2 , 2 is the exponent, in y 3 , 3 is the exponent. 28. Base. The number wh ch is used as a factor a number of times is called the base. For example, in r 3 , x is the base and 3 is the exponent; in 5 3 , 5 is the base, 3 is the exponent and the power is equal to 125. EXERCISE 4 1. Find x 2 when x = 5. Ans. 25. 2. Find m 3 when m=2. 8. 3. Find x i — y 2 when x = 3, y = 5. 56. 4. Find orb 3 when a = 4, 6 = 2. 128. 5. Find ax 3 when a = 6, x = 4. 384. 6. Find 5x 3 when x = 4. 320. 29. Signs of Grouping. The sign of grouping most commonly used is the parenthesis ( ). It means that the parts enclosed are to be taken as a single quantity. For example: 3(x+y) means 3 times the sum of x and y; (x+y) 3 means (x+y)(x+y)(x+y). EXERCISE 5 1. Find 3(15-7). Ans. 24. 2. Find 3 (x +y) when x = 2, y = 4. 18. 3. Find 2 (x 2 - a 3 ) when x = 7, o = 3. 44. 4. Find 5(2x+3ay) when x = 4, a = 2, y = 6. 220. 5. Find bx(3m 2 - 2n 2 ) when x = 2, m = 8, n= 5. 1420. 30. Evaluation. Evaluation of an expression is the process of finding its value by substituting definite numbers (figures) for general numbers and performing the operations indicated. 32 EVALUATION EXERCISE 6 Evaluate: 1. 'Sa 2 x :i when a =3, x = 2. Solution. 3 a 2 z 3 3-3 2 -2 3 3-9-8 216. Ans. 2. 2z(a 3 — ?/) when z = 5, a = 4, ?/ = 30. Solution. 2x(a 3 — y) 2-5(4 3 -30) 10(64-30) 10-34 340. Ans. 3. 2:r when a; = 5. Ans. 10. 4. x 2 when rr = 5. 25. 5. 30+?/) when s = 2, y = 4. 18. 6. 4(a+6) 2 when a = 2, 6=4. 144. 7. x 2 y 3 when a- = 3, ?/ = 2. 72. 8. 7x 2 when x = 4. 112. 9. x 3 — a 2 when z = 5, a = 3. 116. 10. 2a% 3 (a-6) when a=12, x=2, 6 = 8. 9216. 11. 1 +-+| when 2 = 2, n = 8, 31. Formulas. A formula is the statement of a rule or principle in terms of general numbers. For example, the area of a rectangle is equal to the product of the base by the altitude. Stated as a formula this becomes A=ba. 32. Evaluation of Formulas. Evaluation of a formula is the process of substituting definite numbers for general FORMULAS 33 numbers, and solving the resulting equation for the one remaining general number. For example, evaluate: A=%bh when 6 = 5, h = 12. Substituting values, A = |-5-12 Whence, A=30 Evaluate, x-\-3r = 5a when r = 2, o = 7 Substituting values, .r+3-2 = 5-7 Multiplying, .r+6 = 35 Whence, .r = 29 33. Electrical Formulas — Resistances in Series. The total, or combined resistance, R, of the three resistances R\, Rj R 2 Rs Fig. 21. R% and Rs in series, Fig. 21, is expressed by the equation R = R l +R2 + R 3 . All are measured in ohms. 34 EVALUATION 1 EXERCISE 7 £va Urate 1 the formula , to find the missing values R Ri R<2 R?. 1. 12 5 8 Ans. 25. 2. G.3 2.9 4.43 13.63 3. 6.8 4.6 2.01 13.41 4. 10 2 1 7. 5. 120 6 70 44. 6. 16 3 4 9. 7. 25 12.5 2.6 9.9. 8. 19.1 14 4.41 0.69. 9. 20 6.4 9.98 3.62. 10. The combined resistance of four lamps in series is 1025 ohms. The first lamp has a resistance of 250 ohms, the second, 260 ohms and the third 290 ohms. What is the resistance of the fourth lamp? Ans. 225. 11. Find the total resistance of three coils in series if the first has a resistance of 2 ohms, the second 5 ohms, and the third 6 ohms. Ans. 13. 12. Four storage batteries are connected in series for charging, 3 have internal resistances of 4 ohms each and the fourth has an internal resistance of 6 ohms. How many ohms must be put in series to make a total resistance of 25 ohms? Ans. 7. 34. Ohm's Law. Ohm's law for the relation of voltage to current and resistance in a circuit is expressed by the formula: E = IR where, E = voltage, 7 = current measured in amperes, R = resistance measured in ohms. ELECTRIC CURRENT 35 EXERCISE 8 Evaluate the formula to find the missing factors: E I R 1. .55 11 Ans. 6.05. 2. .75 9 6.75. 3. 110 220 .5. 4. 110 .75 146.7. 5. 220 450 .488 6. 6 1.5 4. 7. If 7=§ 3 find 7 when E =110 volts and /? = 9ohms. K Ans. 12.2. 8. If R-j, find R when £=110 volts and 7 = 0.5 amperes. Ans. 220. 9. How much current will flow through the windings of an electromagnet of 140 ohms resistance, when placed across a 110 volt circuit? Ans. .185. 10. What is the resistance of an incandescent lamp on a 110 volt circuit if the current is 1.5 amperes? Ans. 73.3. 11. Four lamps in series have resistances of 2 ohms, 3 ohms, 2.5 ohms and 3.25 ohms. If 119 volts are applied find the number of amperes current flowing. Ans. 1 1 . 07. 12. An electric car heater is supplied with 500 volts from the trolley. The current is 2.5 amperes, what is the resistance? Ans. 200. 13. Find the current sent through a circuit of 200 ohms resistance by a Daniell cell cell which has a voltage of 1 . 09 volts. Ans. 0.00545. 35. Work Done by an Electric Current. The work done by an electric current equals the power times the time. Expressed as an equation this is W = PT. The work is measured in watt-hours or kilowatt-hours, the power in watts or kilowatts, and the time in hours. 36 EVALUATION EXERCISE 9 Evaluate the formula to find the missing factors: W P T 1. 110 55 Ans. 2. 2. 75 7 525. 3. 485 3.25 149.23. 4. What will be the cost at 4 cents per kilowatt-hour to run a 4400 watt heater for (i hours? (1 kilowatt = 1000 watts.) Ans. $1.00. 5. How much work is done when a 40 watt lamp is lighted for 4 hours? Ans. 160 W.H. 6. The power used in doing a piece of work is 100 kilo- watts and the time taken is 3 hours. How much work is done? Ans. 300 K.W.H. 7. The work done by a heater is equal to 50 kilowatt- hours, and the time is 200 min. What is the power? Ans. 20 K.W. 8. The power used by an arc lamp is 500 kilowatts and the work done is 2500 kilowatt-hours. Find the time. Ans. 5 Hrs. 36. Heat in an Electric Circuit. The heat developed by an electric current is expressed by the formula: H = 0. 24 Pt, where, 11 = Heat in calories,* P = power in watts, t = time in seconds. The constant 0.24 is called the heat equivalent of elec- tricity. * A calorie is the amount of heat required to raise the temperature of one gram of water one degree Centigrade. TRANSFORMATION OF FORMULAS 37 EXERCISE 10 Evaluate the formula to find the missing factors: H P t 1. 89 1.5 hrs. Ans. 115344. 2 230000 3 hrs. 20 min. 79 . 8. 3. 189000 67 3.26. 4. How much heat is generated per hour in an electric iron using 660 watts? Ans. 570240. 37. Electricity Stored in a Condenser. The electricity stored in a condenser is expressed by the equation : Q = CE, where, Q = quantity of electricity measured in coulumbs, C = capacity of the condenser in farads, E = applied voltage. EXERCISE 11 1. 45 volts are applied to a condenser of 2 microfarads capacity. How much electricity is stored? (1 microfarad = .000001 farad.) Ans. 0.00009. 2. How much voltage is required to store . 005 coulumb in a condenser of 250 microfarads capacity? Ans. 20. 38. Transformation of Formulas. Ohm's law may be expressed in any one of the following forms: 1. E = IR. 2. R=f. 1 is the most convenient for finding E, 2 for finding 46780 38 EVALUATION R, and 3 for finding I. It is not necessary to memorize more than one of these formulas. The other two may be derived from any one as follows: 1. From 1 derive 2: E = IR V J D d = -d- (Dividing both members by R) R K E n = I (Cancellation) 2. From 2 derive 1 : EI IR = -j- (Multiplying both members by /) IR = E (Cancellation) EXERCISE 12 1. From 1 derive 3. 2. From Q = CE derive formulas for C and E. 3. From Q = IT derive formulas for / and T. General Directions. To change the form of a formula first clear the formula of fractions. Then get the letter to be solved for on one side of the equation, and all other letters on the other side. 39. Force of Attraction or Repulsion on a Magnet. A magnet brought into a magnetic field is acted upon by a force expressed by the formula: F=MH, where, F = force of attraction or repulsion in dynes, M = strength of the magnet in unit poles, H = intensity of magnetic field in gausses. RESISTANCES IN PARALLEL 39 E XERCISE 13 Evaluate the formula to find the missing terms. F M H 1. 600 3000 Ans. 1800000. 2. 4000000 2500 1600. 3. 65000 200 325. 4. A magnet of 600 unit poles is placed in a magnetic field of 5000 gausses intensity. What force is exerted upon the magnet? Ans. 300000 dynes. 5. Solve the formula F = MH for M and for H. 40. Resistances in Parallel. The formula p A p A j-> .ttl /V2 113 is used for finding the combined or equivalent resistance of resistances connected in parallel as in Fig. 22. Ri, R 2 , and R3 are the separate resistances, R is the combined resistance. All resistances are measured in ohms. 40 EVALUATION EXERCISE 14 Evaluate the formula to find the missing terms. R /?i R 2 Rs 1. 15 8 12 Ans. 3.(34. 2. 220 3 12 2.37. 3. 210 210 210 70. 4. Three lamps of 195 ohms each are connected in paral- lel. Find the resistance. Ans. 05. 5. Five lamps are connected in series, each has a resist- ance of 80 ohms. Another series of 4 similar lamps is con- nected in parallel with the first series. Find the total resistance. Ans. 177.7. 6. If 220 volts are applied to the above circuit, find the current. Ans. 1.237. 41. Computing Resistance of a Conductor from Its Size and Length. The resistance in ohms of a conductor of known material may be computed from the formula, Kl where, R = resistance of the conductor in ohms, Z = length of the conductor in feet, A = area of the conductor in circular mils, K = a constant depending on the material in the conductor, and is equal to the resistance per mil foot in ohms. One circular mil is the area of a circle 1 mil in diameter, A mil is one thousandth of an inch. A circle 2 mils in diameter has an area of 4 circular mils, and a circle 3 mils in diameter has an area of 9 circular CIRCULAR MILS 41 mils. The area of any circle in circular mils is equal to the square of the diameter in mils. The circular mil is a con- venient unit for measuring the areas of wires since it elimi- nates the use of tt (3. 1410). 1 circular mil 4- circul ar mils Fig. 23. EXERCISE 15 9 circulaj - mils Evaluate the formula to find the missing factors: R A' I A Ans. 1. 10.4 1000' 16510 .630. 2. 10.4 275' 810 3.525. 3. 10.4 1 mile 6530 8.41. 4. 0.528 25' 509 10.8. 5. 7.79 25 409 127.4. 6. 13.27 10.4 404 515'. 7. o.o 10.4 850' 1607. 8. 10.4 1 mile 10381 5.29. 9. Find the resistance of 680 feet of No. 25 B. S. gauge German silver wire (K = 125, A = 320 . 4) . Ans. 265 . 5. 10. Find the resistance of 20 miles of trolley wire made of No. 00 B. S. gauge copper wire (iv = 10.4, A = 133080). Ans. 8.26. 42 EVALUATION 11. Find the area of cross-section of copper wire having a resistance of 140 ohms per mile. Ans. 392.2. 12. It is desired to transmit 200 amperes to a point 2500 feet from the generator with not more than 4 volts line drop. What diameter copper wire must be used? Ans. 1140 mils. 13. It is desireu to transmit 50 amperes 5 miles with a line drop of not more than 10 volts. What size copper wire must be used? Ans. Area = 1372800 circular mils. K1 14. Solve the formula -R = -r for K and for A. v RA , Kl Ans. K—-J-, A= — . 42. Resistance of a Conductor at Different Temper- atures. When the temperature of most conductors changes, their resistances change also. The formula used to find resistance change with change of temperature is R^Ri+Rtat, where, R t = initial resistance, R f = resistance at any other temperature higher than the original temperature, t = change of temperature in degrees centigrade, a = temperature coefficient of resistance. EXERCISE 16 Find the missing terms* Rf Ri a t 1. 200 .00406 25 Ans. 220.3. 2. 5000 .00406 85 0725.5. 3. 50.8 10 .0034 1200. 4. 60 55 .00406 22.39 5. 220 .00406 25 199.7. 6. 25 .0034 650 7.78. FIELD INTENSITY INSIDE OF A COIL 43 7. The resistance of a coil of copper wire at 39° is 300 ohms. What will be the resistance of the coil at 60° (a =.003613). Ans. 332.76. 8. The resistance of a coil of copper wire is 200 ohms at 40°. What will it be at 90° (a = . 003605) . Ans. 236.05. 9. What will be the resistance of a copper wire at 25° if the resistance at 48° is 2 . 08 ohms? (a = . 00383 1) . Ans. 1.91. 10. Solve the formula R / =Ri J rR i at for a and for t. . Rf—Ri , Rf—Rt Ans - a = -7^' ^^aT- 43. Field Intensity Inside a Coil. The formula used for finding the field intensity inside a coil of wire is where, H — the field intensity inside the coil measured in gausses, N = the number of turns of wire, I = current through the coil in amperes, L = length of the coil in centimeters. EXERCISE 17 Find the missing terms: H N I L Ans. 1. 80 2 12 16.8 2. 40 115 1.5 5.434 3. 33.35 1.75 1575 238.2. 4. 22.31 76 16.34 3.806. 5. 85 .85 20.1 4.53 6. 38.5 65.5 .741 1.59 EVALUATION Field Assembly of a Six-Pole Direct Current Generator with Slotted Poles. TOTAL FLUX IN MAGNETIC CIRCUIT 45 7. Find the field intensity inside a coil of wire 12 cm. long having 100 turns and carrying a current of 0.5 ampere. Ans. 5.25. 8. A coil of wire is to have a field intensity of 40 gausses and is to be 15 cm. long, and carry a current of 1.2 amperes. Find the number of turns necessary. Ans. 396.8. 9. A coil 18 cm. long has 75 turns. What current will be necessary to give a field intensity of (350 gausses? Ans. 123.8. 44. Total Flux in a Magnetic Circuit. The following formulas are given for finding the total flux through a mag- netic circuit : AwNI * = ■ R where, $ = total flux through the magnetic circuit, N — number of turns of wire in the magnetizing coil, / = current in amperes in the magnetizing coil, R = reluctance of magnetic circuit, computed as ex- plained in the following equation. In the above equation, the value of R, the reluctance, is found as follows: where, I = average length of magnetic circuit in centimeters. U. — permeability of circuit . A = average cross-sectional area of magnetic circuit. When the circuit is made up of different parts, as the frame of the motor, air gaps, and core of the armature, Joint reluctance, or R = R\-\-Ro-\-Rs~\- . . . h j h fX\A\ /X2-4-2 46 EVALUATION where R\, R2 and Rz are reluctances of the different parts of the circuit and similarly U, l>, etc., refer to the lengths in centimeters of different parts of the circuit. EXERCISE 18 1. Find the total flux through a circuit composed of iron and an air gap, magnetized by 50 turns of wire carrying a current of 4 amperes, if the length of the iron is 36 cm., cross-section area 12 sq. cm., and /x = 2000; and the length of the air gap is 2 cm., area 15 sq. cm., and /*=1. Ans. 1863. 2. Find the total flux in a motor if the path consists of the frame of the machine, the core of the coils, two air gaps between the coils and armature, and the armature. The magnetizing current is 24 amperes and has 200 turns. The cores of the machine have a cross section of 150 sq. cm., a length (including both cores) of 45 cm., and /x = 200(). The frame of the machine has a length of 75 cm., an average area of 175 sq. cm., and ai=1500. The two air gaps have an area of 180 sq. cm., each has a length of 1.5 cm., and p=l. The armature has an area of 160 sq. cm., a length of 40 cm., and M = 2000. Ans. 350,079. 45. Armature Winding. The following formula is some- times used in armature winding. y= -^ — ±2w. When the positive sign is used in the fraction, use the positive sign before 2m, etc. ARMATURE WINDING 47 Partly Wound Armature showing Method of Assembling Coils. Details of Commutator Construction. 48 EVALUATION EXERCISE 19 1. Evaluate the formula of Section 45, when N= 120, b = 18, p=3, ra = 3. Aris. 29 or 11. 2. Evaluate the formula of Section 45, when A r = 80, 6 = 24, p = 4, wz=2. Ans. 17 or 3. CHAPTER IV POSITIVE AND NEGATIVE NUMBERS 46. Negative Numbers. The numbers used in arith- metic consist of a complete system of numbers and fractions, ranging in value from zero up. The order and value of these numbers ma}' be represented along a straight line, as in Fig. 24. Fig. 24. This system of numbers fails to express completely some numbers encountered in algebra. For example: 1. In the case of the thermometer there is a 5 above zero and a 5 below zero, represented on the scale as in Fig. 25. The 5 above zero and the 5 below zero can be distinguished conveniently by +5 (called posi- tive 5) and —5 (called negative 5). Then +12 means 12 above and —21 means 21 below zero. Fig. 25. W E \+S H5 -6 +6 Fig. 26. 2. Two distances in opposite directions, as shown in Fig. 26, can be represented conveniently by positive and negative numbers, as +6 and —6. 49 50 POSITIVE AND NEGATIVE NUMBERS Observe that 1. The ( + ) (post l ire) and ( — ) (negative) signs have a new meaning, being used to distinguish things of opposite nature. The ( + ) and ( — ) will continue to be used as signs of addition and subtraction as well as signs of quality. —5 -4 -3 —2 — 1 6 +'l +2 + 3 +4 +5» +3 +2 +1 -1 Fig. 27. 2. 77ie positive numbers are the same as the numbers used in arithmetic, and when no sign is expressed the ( + ) sign is understood. 3. The relative order and value of positive and negative numbers can be represented by the scale of Fig. 27. 47. Addition. Algebraic addition is the combination of positive and negative numbers. For example: 1. A man travels 7 miles east and then 3 miles east from that point. This may be represented as in Fig. 28: ALGEBRAIC ADDITION 51 -1 3 4 5 6 7 Fig. 28. 9 10 or (+7) + (+3) = +10. 2. A man travels 7 miles west and then 3 miles west from that point. This may be represented as in Fig. 29: K 1 1 K — 3P — 2 -1 1 —10 -9 -8 -7 -6 —5 —4 —3 -2 -1 +1 Fig. 29. or (-7) + (-3)=-10. 3. A man travels 7 miles east and then 3 miles west from that point. This may be represented as in Fig. 30: —3 -2 -1 -1 3 4 Fig. 30. or (+7) + (-3) = +4. 4. A man travels 3 miles east and then 7 miles west from that point. This may be represented as in Fig. 31: -7 -6 -5 -4 -,3 -2 -1 or -5 -4 -3 -2-10 1 Fig. 31. (+3) + (-7) = -4. 52 POSITIVE AND NEGATIVE NUMBERS The problems above illustrate: 1. (+7) + (+3) = +10. 2. (-7) + (-3) = -10. 3. (+7) + (-3) = + 4. 4. (+3) + (-7) = - 4. where the signs within the parentheses are signs of quality and the signs between the parentheses indicate algebraic addition. From these examples the following rule can be stated : Rule. 1 . To add two numbers of like signs add the numbers as in arithmetic and give the residt the common sign. 2. To add two numbers of opposite signs subtract the smaller from the larger and give to the result the sign of the larger. EXERCISE 1 Add 1. +5, +12. Ans. +17. 2. -5, -12. -17. 3. +5, -12. -7. 4. -5, +12. +7. 5. -29x, Ux. -I5x. 6. -15d, S2d. lid. 7. —3.712/, -5.23y. -8.94?/. 8. +9.21, -4.356. 4.854. 9. +22, -13, +24, -8. +25. Note to Prob. 9. This problem can be solved most efficiently by adding all the positives and all the negatives and combining results. 10. +34, -45, -17, +12. Ans. -16. 11. 5x, -7x, Ux. \2x. 12. -13.T 2 , -5x 2 , 29.T 2 . Ux 2 . 13. (a 3 +3a 2 b+3a& 2 +6 3 ) + (2a 3 +4a 2 6-7«6 2 ) + (-5a 3 -a6 2 +46 3 ). SUBTRACTION 53 Solution. Arrange with similar terms in a column and add columns thus: cP+3a 2 b+ZdP+ 6 3 2a 3 +4a 2 6-7a& 2 -5a 3 - ao 2 +46 3 -2a 3 +7a 2 6-5a6 2 + 56 3 14. (5x 2 - 3xy + 2y 2 ) 4- (2x 2 + 3xy - y 2 ) + (x 2 - 4xy) . Ans. 8x 2 — Axy-\-y 2 . 15. (3a 4- 56 - 4c) 4- (2a + 3c) + (a -126). Ans. 6a- 76- c. 16. (oax 2 - 3aij 2 +2bz 2 ) + (7a?/ - 6a.x 2 ) 4-(8bz 2 +2ax 2 -ay 2 ). Ans. az 2 +3a?/ 2 -f-106z 2 . 48. Subtraction. Subtraction means to find the differ- ence between two numbers, that is, the distance between the two numbers on the number scale. By reference to the number scale for positive and negative numbers find: 1. The distance to +5 from +3. Fig. 32. -101234567 Fig. 32. This is seen to be 2 in the positive direction. It may be computed by subtracting the second number from the first. Therefore (4-5) -(+3) = 4-2. 2. The distance to +3 from +5. Fig. 33. -10+1+2 3 4 5 Fig. 33. 54 POSITIVE AND NEGATIVE NUMBERS This is seen to be 2 in the negative direction. It is equivalent to subtracting the second number from the first. Therefore ( + 3)-( + 5)=-2. 3. The distance to -3 from +5. Fig. 34. -3-2-10 123 45 6 Fig. 34. This is seen to be 8 in the negative direction. It is equivalent to subtracting the second number from the first. Therefore (_3)-(+5) = -8. 4. The distance to +5 from -3. Fig. 35. -3 -2 -1 1 5 5 I 5 6 Fig. 35. This is seen to be 8 in the positive direction. It is equivalent to subtracting the second number from the first. Therefore (+5) -(-3) = +8. EXERCISE 2 1. Find the following distances: To -5 + 5 -10 +3 -8 +7 From +5 -5 - 3 -8 +3 +4 2. Add -5 +5 -10 +3 -8 +7 -5 + 5 + 3 +8 -3 -4 SUBTRACTION 55 Compare the results in the two problems above and note the following rule : Rule. To subtract one number from another change the sign of the subtrahend and apply the rules for addition. EXERCISE 3 Ans. 12. Subti 1. ■act: +32 + 20 2. -24 -18 3. -42 + 16 4. + 18 -22 5. + 17:r 2 + bx 2 6. -29ax + 16ax 7. -\-hay 2 -lay 2 8. +2a-36 + a+ b -58. +40. 12x 2 . —45ax. Ylay 2 . a- 46. 9. (-5)-(-7). 2. 10. (+8)-(-3). 11. 11. (-12)-(+16). -28. 12. (3x+7y)-(2x+Sy). x+4y. 56 POSITIVE AND NEGATIVE NUMBERS 13. (Zr i -4xy+7y') - (2a?+3xy- I2y 2 ). Ans. x 2 — 7xy-\-l9y 2 . 14. (5a 2 + 6a6 + 2b 2 ) - (3a 2 - lab + b 2 ) - (2a 2 - Sab + 36 2 ) . Ans. (k/ 2 + 12ao. 15. (\2x 3 -7x 2 +4x) - (2x 3 +3x 2 -8x) + (4.T 3 -rvr- , + 6a;). Ans. 14a: 3 -15.r 2 + 18a;. 1G. (a+b)-(c-d). Ans. a+6-c+d. Note. Subtraction will remove the parentheses but there a re no similar terms that can be combined. 17. x 2 +3x-(-5+y). Ans. x 2 + 3x4-5 — #. 18. 4+3x-(2s-5). 9+x. Solve and check: 19. 10.r-(3:c-4) = (4x+4)+7. 2i Note. First remove the parentheses by changing the signs of the subtrahend thus: 10x-3x+4 = 4.r+4+7. 20. 6-(3w-4) = (2n-3)-(n-l). 21. (2x+3)-(-3x-2) = 25. 22. (5x-3)-7=(2x+5). Ans. 3. 4. 5. 7T Ft Fig. 36. 49. Multiplication. When a force is applied to a lever, it will cause the lever to turn or to have a tendency to turn. If the force is doubled, the tendency to turn will be doubled; or if the length of the lever is doubled, the tendency to turn will be doubled. The tendency of a lever to turn is called the turning moment or leverage. The length of the lever from the fulcrum (the turning point) to the force is called the lever arm. See Fig. 36. LAW OF SIGNS 57 The leverage caused by a force is equal to the force times the arm. An arm to the right of the fulcrum is considered positive and an arm to the left of the fulcrum is considered negative. An upward pulling force is considered positive and a down- ward pulling force is considered negative. These assump- tions are the same as the numbers on the number scale. Fig. 37. — Positive Leverage. A positive force on a positive arm must give a positive leverage since (+4)(4-7) = +21 (Arithmetic). See Fig. 37. Note. A leverage in the direction opposite to the motion of the hands of a clock (counter clockwise direction) is a positive leverage. Then a leverage in the clockwise direction must be considered negative. A negative force on a negative arm will cause the lever to turn in the same direction (counter clockwise) giving likewise a positive leverage, Fig. 38. Therefore (-3)(-7) = +21. 71 Fig. 38. — Positive Leverage. A negative force on a positive arm will cause the lever 58 POSITIVE AND NEGATIVE NUMBERS to turn in the opposite direction (clockwise) giving a nega- tive leverage, Fig. 39. Therefore (+3)(-7)--21. 7X If Fig. 39. — Negative Leverage. A positive force on a negative arm will cause the lever to turn in the clockwise direction, Fig. 40. Therefore (-3)(+7)=-21. Fig. 40. — Negative Leverage. The four preceding examples show that 1. (+7)(+3) = 4-21. 2. (-7)(-3) = +21. 3. (-7)(+3) = -21. 4. (+7)(-3)=-21. From these problems a law of signs for multiplication can be stated. 50. Law of Signs for Multiplication. 1. If two factors have like signs their product is positive. 2. If two factors have unlike signs their product is negative. LAW OF EXPONENTS 59 EXERCISE 4 Multiply: 1. (3)( + 12). Ans. +36. 2. (-3X-12). +36. 3. (-!)(+!). i 2* 4. (2.5)(-4). -10. 5. (-5.6)(-2, 3). 12.88. 6. (3f)(-5). m 51. Law of Exponents. By definition of an exponent x 3 means xxx and r 1 means xxxx. Therefore (x 3 ) (x 4 ) = (xxx) (xxxx) = X 7 . From this and similar problems a law of exponents can be stated: Rule. To multiply powers of the same base add their exponents. EXERCISE 5 Multiply: 1. x 7 x 4 . Ans. x n . 3. (a 3 )(-o 2 ). -a 5 . 3. (-v 2 )(-y). +y 3 . 4. x 3 x 5 x. x 9 . 52. Multiplication of Monomials. When different bases occur in the factors, their product can be indicated only. Thus (x 3 )(if)=x 3 y 2 . 60 POSITIVE AND NEGATIVE NUMBERS When several different bases occur in each factor, as (3a 2 & 3 r)(4a 3 t 4 a; 6 ), only the powers of the same base can be combined. For example, (3a 2 6 3 r) (4a 3 6% 6 ) = 3 X4a 2 a 3 6 3 6 4 r:r 6 = I2a 5 b 7 rx*. Note. Expressions whose parts are not separated by the (+) or ( — ) signs are called monomials. From the above example a rule for the multiplication of monomials can be stated. Rule. To multiply two monomials multiply their numerical coefficients and annex all the different bases, giving to each base an exponent equal to the sums of the exponents of that base in the two factors. EXERCISE C Multiply: 1. (4x 3 )(5a: 4 ). Ans. 20x 7 . 2. (I2x 2 y 3 )(2x 3 y 5 ) 24x 5 /y 8 . 3. (5a 3 xV)(6a 2 a;2 2 ). 30a 5 zV2 2 4. (-2ab 2 )(lZb 2 c). - 26a6 4 c. 5. (5xy)( — 4xz). - 20x 2 yz. 53. Law of Leverages. If two forces act upon the same lever at the same time, the lever will be in balance when the BALANCED LEVERS 61 positive leverages equal the negative leverages; that is, when the sum of all the leverages equals zero. Thus in Fig. 41: + 8 Fig. 41 12 The leverage caused by the force 8 = ( — 3) ( — 8) = +24. The leverage caused by the force 12=(— 12)(-(-2)= —24. (+24) + (-24)=0, = 0. Therefore the lever will balance. EXERCISE 7 1. Find the sum of all the leverages in Fig. 42. « h L \ } •3 6*' Fig. 42. Will the lever balance? Why? 2. Find the value of x that will make the levers balance in Figs. 43 (a)-43 (e). 62 POSITIVE AND NEGATIVE NUMBERS A" 17 t«-i-*i (a) -5.5- -3.3- 125 k- ^1T"^ (c) (d) {20 (•) Fig. 43. >U — 9 rlO Solution. 4z 4-2-50 = 4z = 48 x=12. Ans. 75. T^^ Ans. 10. Ans. 6f. Ans. 35. 54. Multiplication of Polynomials by Monomials. The perimeter of a rectangle is 2 (a -\-b) and also 2a 4- 26. Fig. 44. Therefore 2(a+b) = 2a+2b. b Fig. 44. Expressions like (a 4- 6) and (x 2 — 3x 4-4) which consist of two or more parts added or subtracted, are called polynomials. MULTIPLICATION OF POLYNOMIALS 63 From the above problem a rule for multiplication of a polynomial by a monomial can be stated: Rule. To multiply a polynomial by a monomial multiply each term of the polynomial by the monomial. EXERCISE 8 Multiply: 1. a 2 +3a&+46 2 by 5. 2. x 2 — 3xy+4y 2 by 5x. 3. a 3 -a 2 b-2ab 2 by -2ab. 4. x 2 — 7x 2 y 2 +4y 2 by 3x 2 y 3 . Ans. 5a 2 +15a6+206 2 . 5r 3 — 1 bx 2 y -\-2Qxy 2 . -2a 4 6+2a 3 6 2 +4a 2 6 3 . Zx 4 y 3 -21x 4 y 5 +12x 2 y 5 . xy 6x 27/ 12 55. Multiplication of a Polynomial by a Polynomial. The product (y+6)(a; + 2) can be represented by the rect- V angle, Fig. 45, whose length is y-\-Q and whose width is x-\-2. The area of the rectangle is the product (s/+6)(x+2). The area is also the sum of the areas of Fig. 45. the four small rectangles, that is, xy-{-Qx-\-2y+l2. Therefore (2/+6)(z+2)=:cy4-6z+2?/+12 From the above problem the following rule can be stated : Rule. To multiply a polynomial by a polynomial mul- tiply each term of the one by every term of the other and combine the results. 64 POSITIVE AND NEGATIVE NUMBERS The work can be arranged conveniently thus: x -2 x + 3 x 2 — 2x (multiplying x— 2 by x\ 3x— 6 (multiplying x — 2 by 3) ar + x — 6 (combining similar terms) EXERCISE 9 Multiply: 1. (x+3)(s+4). Ans. z 2 +7z+12. 2. (*-5)(s-2). x 2 -7z+10. 3. (x-y)(x-y). x 2 — 2.T4/+?/ 2 . 4. (2a +36) (3a- 46). 6a 2 + a6-12&2 5. (a -6) (a -6). a 2 -2a6 + 6 2 . 6. (z + 2)(x + 2). z 2 +4z+4. 7. (y-S(y-S). ?/ 2 — 6?/+9. 8. (n-5)(n-5). w 2 -10n+25. 56. Division. The division of positive and negative numbers requires a law of signs and a law of exponents which are developed by a study of the laws of signs and exponents for multiplication. Division is the opposite of multiplication, hence, Since ( + 5)(+6) = +30 it follows (+30) Since (+5) (-6)= -30 it follows (-30) Also (-30) Since ( - 5) ( - 6) = (+30) it follows (+30) (+6) = +5. (-6) = +5. ( + 5)= -6. (-0) = -5. From these problems a law of signs for division can be stated : DIVISION 65 57. Law of Signs for Division. 1. // two numbers have like signs their quotient is positive. 2. // two numbers have unlike signs their quotient is negative. EXERCISE 10 Divide : 1. ( + 25) 4- (+5). Ans. +5. 2. (-63) -K-3). +21. 3. (-24)h-(+5). -4.8. 4. (+38)-*-(-4). -9.5. (-4). +9. 5. (-36) 6. (-24) (-3). 58. Exponents in Division. By the law of exponents for multiplication, x 7 x* = x 11 Therefore X 11 +x 7 = x 4 and X U -trX 4: =X 7 From this problem the following law of exponents for division can be stated: 1 . To divide powers of the same base subtract the exponent of the divisor from the exponent of the dividend. 2. The quotient of powers of different bases can be indicated only. 59. Division of Monomials. Example. Divide 24a 3 rVz 4 by Sa 2 bx 2 y 5 xz 6 . 66 POSITIVE AND NEGATIVE NUMBERS Solution. 3 a y* M

'/// 4. -75x 7 b 4 c 3 by -10x& 4 . 7 . 5x 6 c?. 5. 38r 2 s 3 c by - 19/-Vc. — 2. 6. 27x 3 by 9 if. 3-". 60. Division of a Polynomial by a Monomial. By the rules for multiplication 5 (x -2y) = 5x-10y Therefore : (5x-10y)+5=x-2y Rule. To divide a polynomial by a monomial divide each term of the polynomial by the monomial. The work can be arranged conveniently thus: lOsV- 15sY + 25sV = 2 3 _ 3^+5^ 5xy/- DIVISION 67 EXERCISE 12 Divide : L r 3 -2z 2 +a ^ a a_ 2x+lm x 2. °y 2 -^+ 4q . x = amount of water 2x+3z=14 gallons 5x=U x= 2| 2x = 5f gallons of acid 2>x = 8f gallons of water EXERCISE 4 1. Divide 35 into two parts in the ratio of 2 to 3. Ans. 14, 21. 2. Divide 180 in the ratio 4 to 5. Ans. 80, 100. 3. Bronze is composed of 11 parts of tin to 39 parts of copper. Find the number of lbs. of tin and copper in 625 lbs. of bronze. Ans. 137 . 5, 487 . 5. 4. Eighteen carat gold is composed of 18 parts of pure gold to 6 parts of other metal. How much pure gold is contained in 4.8 oz. of the alloy? Ans. 3.6. 5. In an electric circuit the fall of potential over any two of the parts is in the ratio of the resistances. 24 ohms and 40 ohms are joined in a circuit. The drop of the poten- tial over both resistances is 90 ohms, find the drop over each resistance. Ans. 33f , 56|. 6. In a divided circuit the current divides in the ratio of the resistances, the larger current taking the path of the smaller resistance. If the two resistances of a divided circuit are 24 ohms and 36 ohms,, and the total current flowing is 8 amperes, find how much current will flow in each branch. Ans. 4.8, 3.2. 65. Proportion. In two samples of the same kind of bronze the ratios of the copper to the tin are equal. When two ratios are equal chey form a -proportion. A proportion PROPORTION 73 Wheatstone Bridge. Leeds and Northrup Potentiometer. 74 RATIO AND PROPORTION is written as f = -rr, and read the ratio of 3 to 7 equals the ratio of 9 to 21; or 3 : 7 :: 9 : 21, and read 3 is to 7 as 9 is to 21. The first and last terms of a proportion are called extremes and the second and third are called means. A proportion is used in finding ratios of quantities when some other ratio between the same two quantities is known. For example: In bronze 11 parts of tin combine with 39 parts of copper. The same ratio holds for any quantity of bronze. How many parts of tin will combine with 260 lbs. of copper? Solution. Let rr = the number of pounds of tin. 11 -. x Then the ratio ttk = the ratio ^^ 39 200 11 = x 39 ~~ 260 2860 = 39s z=73.33+ lbs. of tin Observe that: 1. A proportion is an equation. 2. A proportion can be cleared of fractions by multiplying both members by the product of the denominators. 3. To clear a proportion of fractions write the product of the means equals the product of the extremes. 66. The Wheatstone Bridge the unknown resist- ance is placed in a Wheatstone bridge, as shown in Fig. 46, where any three of the resistances are k n o w n a n d .the fourth resistance is the resistance to be In measuring resistance Fig. 46. — Wheatstone Bridge. FRACTIONS OF A GIVEN DENOMINATOR 75 measured. The three known resistances are adjusted to /? /?• make yt = ~d^- The galvanometer G indicates when the /12 Ra resistances are so adjusted that the ratios are equal. The proportion ~- = ■=- is used as a formula. Ro R± EXERCISE 5 Find the missing numbers in the following: Ri R 2 R 3 R* Ans. 1. 5 4 3 6.67 2. 1.5 3.5 2.5 1.07 3. 9.85 6.8 9.9 6.84 4. 1.5 2.35 5 3.19 5. 8.7 9.5 45 41.5 6. 160 32 43 8.6 7. 650 80 160 19.7 8. 60 235 25 6.38 9. 4.75 6.843 5 3.47 10. 275 6.625 9.375 195.1 11. If ^ = 100 and #4 = 4.26, find tf 3 . Ans. 426. Ri 67. Reduction to Fractions of a Given Denominator. It is often necessary to express a fraction or a decimal in halves, fourths, sixteenths, etc. Example. How many eighths in f of an inch? Solution. If there are x eighths, then x _A 8~5 5z = 32 z = 6.4 76 RATIO AND PROPORTION EXERCISE 6 1. How many 04ths in f of an inch? Ans. 38.4. 2. How many 32ds in two °f an inch? 23. 3. How many 64ths in .365 of an inch? 23.36. 4. How many 16ths in .82 of an inch? 13.12. 5. How many 64ths in T V of an inch? 37.3. 6. How many thousandths in ^ T of an inch? 15.625. 7. How many thousandths in t\ of an inch? 187.5. 68. Writing a Proportion. In writing a proportion two types of problems will be encountered. First, when the two quantities are so related that an increase or a decrease in one will produce the same kind of a change on the other. For example: If 20 men assemble 8 machines in a day, more men could assemble more machines, and less men less machines. This is called a direct proportion. Secondly, when two quantities are so related that an increase or a decrease in the one will produce the opposite change in the other. For example: It takes 30 men 12 days to assemble a machine, more men could do it in less time and for less men it would take more time. This is called an inverse proportion. Example 1. 12 feet of angle iron weighs 44 lbs., how much will 30 ft. weigh? , . 12 ft 44 lbs. solution. 30 ft x lbs. Solve the equation for x. Ans. 110. Every ratio must be « comparison of similar things, and DIRECT AND INVERSE PROPORTION 77 in every proportion both ratios must be written in the same order of value, that is: Small leng th Small we ight Large length Large weight or Large length _ Large we ight Small length Small weight Example 2. If 30 men do a piece of work in 12 days, how long will it take 20 men to do the same work? Solution. Large number of men Large number of days Small number of men ~ Small number of days That is 30 = ^ 20~12 Solve for a. Ans. 18. Observe that: 1. In the direct proportion the 44 #>s. corresponds to 12 ft. and x lbs. to 30 ft. and the corresponding numbers are arranged directly across from each other thus: 12 < _ S "44 30 < >x 2. In the inverse proportion the 30 men correspond to 12 days and 20 men to x days and the corresponding numbers are arranged diagonally thus: >fc^ >< >^r2 78 RATIO AND PROPORTION EXERCISE 7 Solve by proportion: 1. If a steel rail 3 ft. long weighs 112 lbs., how much will a rail 20 ft. long weigh? Ans. 746.7. 2. 20 men do a piece of work in 54 days. How many men would it take to do it in 30 days? Ans. 36. 3. The volume of a quantity of gas is inversely pro- portional to the pressure upon it. If a quantity of gas measures 350 cu. ft. at 15 lbs. pressure, how many cubic feet will it measure at 25 lbs. pressure? Ans. 210. 4. The pressure of 627 cu. ft. of gas is to be reduced from 35 lbs. to 24 lbs., what will be the volume. Ans. 1164.37. 5. The volume of a gas is directly proportional to the absolute temperature when the pressure is constant. If a quantity of gas occupies 125 cu. ft. at 278°, what will be its volume at 316°? Ans. 124.09. 6. If a quantity of gas occupies 28 cu. ft. at a tempera- ture of 250°, what will be the temperature when it occupies 30 cu. ft.? Ans. 267.85,. 7. If the resistance of 25 ft. of wire is 11.2 ohms, what will be the resistance of 83 ft. of the same wire? Ans. 37.1841 8. An investment produces an income of $600 at 3? per cent. What would it produce at 5 per cent? Ans. 857.14. CHAPTER VI CUTTING SPEED, PULLEYS AND GEARS 69. Rim Speed. When work is turned in a lathe, the work must pass by the point of the cutting tool at a speed which will complete the work in the shortest possible time without injury to the work or the tools used. When the work is turned one complete revolution, a point on the surface of the work travels a distance equal to the circumference of the work. In one min- ute the point would travel a distance equal to the circumference Fig. 47. of the work times the number of revolutions per minute (R.P.M.). The same applies to emery wheels, grindstones, and pulleys. The distance traveled in one minute by a point on the circumference of any revolving object is called rim speed, surface speed or cutting speed. Rim speed must be expressed in feet per minute. Rule. To find rim speed multiply the circumference of the revolving object by the number of revolutions per minute. By formula: RS=C- (R.P.M.) 79 SO CUTTING SPEED, PULLEYS AND GEARS where C = circumference in feet, and RS = rim speed. Or C- (R.P.M.) KS = 12 where C = circumference in inches. But since C = ird, where d is the diameter, 7Tf/(R.P.M.) RS= 12 Example 1. The diameter of a bolt being turned in a lathe is 3 ins. If it makes 160 R.P.M., what is the rim speed? Solution. na 7Td(R.P.M.) Hb- I 2— „„ 3. 1416-3- 160 , a . ... ,. v RS = rx (Substitution.) 72/S= 125.664 ft, per min. Example 2. At how many revolutions per minute must a 4-in. bolt be turned to give a cutting speed of 60 ft. per min.? Solution. 7rrf(R.P.M.) RS = 12 _ 3. 1416-4- (R. P.M.) , a . ... .. v 60 = — r^ (Substitution.) 60- 12 = 3. 1416-4- (R.P.M.) (Clearing of fractions.) _60 12 RpM (Dividing by the coefficient of R.P.M.) R.P.M. = 57. 6+in. RIM SPEED 81 Example 3. A pulley on a shaft turning at 140 R.P.M. is to furnish a rim speed of 1500 ft. per min. Find the diameter of the pulley that must be placed on the shaft. Solution. xrf(R.P.M.) RS= 12 ir M 3.1416-cM40 , a , ... ,. x 15.00 = r^ — (Substitution.) 1500 -12 = 3. 1416 • d- 140 (Clearing of fractions.) = d (Dividing by the coefficient of d.) 3.1416-140 rf=40.9 + inches Note. In rough work where the exact answer is not required the following formula is used: „ _ rf(R.P.M.) 3.1416 . ... . . , R.S. = — — -, since — — — is approximately equal to j. ■ 4 i.1 EXERCISE 1 Find the exact results in the following: 1. Find the cutting speed of a 5-in. cylinder being turned at 75 R.P.M. Ans. 98.175 ft. per min. 2. The armature of a dynamo is 16 ins. in diameter and runs at 1350 R.P.M. Find the rim speed. Ans. 5654.9 ft. per min. 3. At how many R.P.M. should a 6-in. cylinder be turned to give a cutting speed of 60 ft. per minute? Ans. 38.2. 4. An 8-in. emery wheel has a rim speed of 4000 ft. per minute. How many R.P.M. does it make. Ans. 1909. 5. A shaft is running at 175 R.P.M. How large a pulley must be placed on this shaft to give a rim speed of 2250 ft. per minute? Ans. 49.11 ins. 82 CUTTING SPEED, PULLEYS AND GEARS 6. A 30-in. pulley runs at 250 R.P.M. Find the rim speed. Ans. 1963.5. 7. The pulley of Problem 6 is belted to a 15-in. pulley. What is the speed of the belt? What is the rim speed of the 15-in. pulley? Ans. 1963.5; 1963.5. 8. Find the R.P.M. of the 15-in. pulley. Ans. 500. 7 and 8 above illus- is belted to a 15-in. 70. Pulley Speeds. Problems 6, trate the fact that if a 30-in. pulley pulley the R.P.M. of the 15-in. pulley will be twice the R.P.M. of the 30-in. pulley. Or, in general : Rule. When two pul- leys are belted together the R.P.M. vary inversely as the size of the pulleys. Example 1. A 20-in. pulley running at 180 R.P.M. drives an 8-in. pulley. Find the R.P.M. of the 8-in. pulley. Solution. Let z = the R.P.M. of the 8-in. pulley. Then since 180 = R.P.M. of the 20-in pulley, 600 R.P.M. 250 R.P.M. Fig. 48. X 180 : 20 8z = 3600 rr = 450 R.P.M. EXERCISE 2 1. A 32-in. pulley running at 150 R.P.M. drives a 22-in. pulley. Find the R.P.M. of the 22-in. pulley. Ans. 218 + . 2. A 30-in. pulley is to drive a 12-in. pulley at 800 R.P.M. Find the R.P.M. of the 30-in. pulley. Ans. 320. GEARS 83 3. A 22-in. pulley on a shaft running at 264 R.P.M. is to drive a machine at 570 R.P.M. Find the size of the pulley on the machine. Ans. 10. 17. 4. A line shaft running at 150 R.P.M. is to drive a machine having a 14-in. pulley at 375 R.P.M. Find the size pulley that will be required on the shaft. Ans. 35 ins. 5. An electric motor running at 1250 R.P.M. and having a 16-in. pulley is to drive a line shaft at 175 R.P.M. Find the size of the pulley on the line shaft. Ans. 114.3 ins. 6. A 30-in. pulley running at 240 R.P.M. drives a 24-in. pulley. Find the R.P.M. of the 24-in. pulley. Ans. 300. 71. Gears. In machines where driving is done by means of gears it will be seen that, if two gears are meshed together, the smaller gear will have the greater R.P.M. Sizes of gears are measured by the number of teeth. Rule. When two gears run Fig. 49.— Gears in Mesh, together, the R.P.M. varies invers- ely as the number of teeth. EXERCISE 3 1. A 48-tooth gear is driving a 72-tooth gear. Find the R.P.M. of the 72-tooth gear if the 48-tooth gear is running 160 R.P.M. Ans. 106.7. 2. A 72-tooth gear running at 190 R.P.M. is to drive another gear at 360 R.P.M. Find the number of teeth in the second gear. Ans. 38. 3. A 26-tooth gear running at 105 R.P.M. is to drive a 14-tooth gear. Find the R.P.M. of the 14- tooth gear. Ans. 195. 84 CUTTING SPEED, PULLEYS AND GEARS EXERCISE 4 (Miscellaneous Problems) 1. A locomotive has a 6-foot drive wheel. Find the R.P.M. of the wheel when the engine is running at 50 miles per hour. Ans. 700.2. 2. Cone pulleys: 1050 R.P.M. Fig. 50. From Fig. 50 find the three speeds of the lower pulley. Ans. 600, 1260, 2800. 3. Pulleys: Fig. 51. Pulley 1 is a 16-in. pulley and runs at 125 R.P.M. Pulley 2 is a 10-in. pulley. REVIEW PROBLEMS 85 Pulley 3 is an 18-in. pulley. Pulley 4 is an 8-in. pulley. Pulleys 2 and 3 are on the same shaft. Find the R.P.M. of pulley 4. 4. Gears: Ans. 450. Gear 1 has 42 teeth and should be run at 800 R.P.M. Gear 2 has 96 teeth. Gear 3 has 48 teeth. Gear 4 is on a shaft running at 300 R.P.M. How many teeth has gear 4? Ans. 56. 5. Dynamo under a passenger car: Fig. 53. Circle 1 represents a 40-in. car wheel. Circle 2 represents a 14-in. pulley on the axle of the car. Circle 3 represents a 6-in. pulle3 r on a dynamo and is driven by a belt from the pulley on the car axle. Find the R.P.M. of the dynamo when the car is running at 30 miles per hour. Ans. 588. CHAPTER VII ELECTRICAL FORMULAS INVOLVING SQUARES AND SQUARE ROOTS 72. Power in a Direct Current Circuit. In a circuit carrying direct current, P=EI=PR=^, E = IR K where, P = power in watts; E = EM.F. in volts; I = current in amper es; R = resistance in ohms. EXERCISE 1 Find the missing terms: P E I R Ans. 1. 110 .85 93.5, 129.41 2. 220 .425 93.5, 517.64 3. 1.2 220 316.8, 264. 4. 110 225 53 . 77, 488. 5. 220 32 1512.5 ,6.875. 6. .55 200 60.5, 110. 7. 40 .38 105.26 ,276.3. 8. 75 240 134.2, 558. 9. 28. 125 80 47.43, 592. 10. 100 55 1.818, 30.25. 86 DIRECT CURRENT CIRCUIT 87 11. Three resistances of 220, 234 and 431 ohms respect- ively are connected in series and a pressure of 550 volts is applied. How much power is used? Ans. 341.8 watts. 12. If the above resistances are connected in parallel, how many volts must be applied to get the same power? Ans. 175.19. 13. Which uses the more power, a lamp with a resist- ance of 220 ohms on a 110 volt circuit or lamp on a 110 volt circuit that uses 2 . 24 amperes? Ans. Second one. 14. A 110 volt arc light requires 12 amperes to operate. How many watts are used? What is the resistance? Ans. 1320 watts, 9.17 ohms. 15. An arc light requires 20 amperes to operate at 45 volts. How many horse-power does it take? (1 horse power = 746 watts.) Ans. 1.206. 16. A 12 candle power lamp on a 110 volt circuit takes 0.25 ampere. How many watts per candle power are used? Ans. 2.29. 17. A generator is producing 30,000 watts at 225 volts. What is the current flowing? Ans. 133.3. 73. Force Between Two Magnets. The force between two magnet poles is expressed by the formula: MiM 2 *- D 2 > where, F = force in dynes between two magnets; M i and M2 = pole strengths of the two magnets in unit poles; D = distance between the two magnet poles in centi- meters, (1 inch = 2. 54 centimeters). 88 ELECTRICAL FORMULAS EXERCISE 2 r in< 1 the missing terms: F Mi M 2 D 1. 2 8 4 Ans. 1. 2. 4 12.5 8 5. 3. 12 3 2 16. 4. 9 7 2 15.75 5. 10 10 12 144. 6. 20 20 16 4. 7. 4 6 1 24. 8. Change to centimeters: 3", 4", 12". Ans. 7.62, 10.14, 30.45. 9. Change to inches: 10 cm., 50 cm., 75 cm. Ans. 3.94", 19.7", 29.55". 10. If a magnet pob of 10 unit poles is placed 4 cm. from another magnet of 24 units pole strength, what force will they exert upon each other? Ans. 15 dynes. 74. Horse-power of an Electric Motor. The horse- power of a motor is expressed by the formula: 2*nT 33000' where, H.P. = horse power; 7T = 3.1410; n = R.P.M. of the motor; T — torque in lbs. ft. HORSE-POWER OF AN ELECTRIC MOTOR 89 Six-Pole Direct Current Generator. Armature of a Direct Current Generator. 90 ELECTRICAL FORMULAS EXERCISE 3 Find the missing terms: H.P. n T Ans. 1. 1500 18.4 5.255, 2. 5 1800 14.6. 3. 1.5 1500 5.25. 4. 10 39.6 1328. 5. 8.5 1650 27.1. 6. 6.454 1200 28.25. 7. A motor running at 1040 R.P.M. develops a torque of 931 lbs. ft. Find the horse power? Ans. 185. 8. A motor is to run at 1680 R.P.M. If it has a 12" pulley and is to furnish a pull of 200 pounds on the belt, what horse power will be required? Ans. 32. 9. A 10 H.P. motor with an 8" pulley is to drive a 30" pulley at 480 R.P.M. Find the torque. Ans. 29. 18. 10. An 80 horse power motor developing a torque of 1231 lbs. ft. and having a 10" pulley drives a 48" pulley. Find the R.P.M. of the 48" pulley. Ans. 71. 11. An electric car has a gear ratio of 14 teeth on the motor sprocket to 65 teeth on the axle sprocket. The wheels are 33". The car has a 20 H.P. motor which gives a torque of 84.50 lbs. ft. Find the maximum speed of the car if the motor is 75 per cent efficient. Ans. 19 . 7. 75. Voltage of a D. C. Generator. Voltage for direct current generators is expressed by the formula: F= NZp 10V60' MURRAY LOOP FORMULA 91 where, # = E.M.F. in volts; Z = number of active conductors on the armature; <{> = flux in maxwells; p = number of poles; n = number of revolutions per minute; p' = number of brush arms. EXERCISE 4 Fi nd the missing numbers : E Z n p V' Ans. 1. 200 1000000 100 2 2 70. 2. 220 226 1500 2 2 3020000. 3. 116 24 2400 8 2 4840000, 4. 120 200 1671000 2 2 2160. 5. 1040 2400 2600000 1200 2 2. 6. 112 400000 2400 4 4 700. 7. 1040 260 4000000 1500 8 ft 2. 8. A generator has a speed of 1980 R.P.M., an E.M.F. of 50 volts, 4 poles, 200 turns, and 4 brushes. What will be the flux? Ans. 1515000. 9. How many poles will be required in a generator for a voltage of 520, 2000 surface conductors, flux of 1300000 maxwells and a speed of 1200 R.P.M. and 4 brush arms? Ans. 4. 76. Murray Loop Formula. The formula X R2 = Ri L-X' is used to find the distance from a station along a telephone 92 ELECTRICAL FORMULAS line to the place where a wire is grounded; thus in Fig. 54, X Fig. 54. Station Ground Ri and Ro are resistances in a bridge at the station: L = total length of the line and return ; X = distance from the station to the place where the wire is grounded. EXERCISE 5 Find the missing terms: Ri R-2 L X 1. 75 50 25 miles Ans. 10. 2. 84 20 163 miles 31.35. 3. 128 8 10 miles .588 4. Find the formula for X in terms of Ri, Ro, and L. . „ RoL *"■■ X= 5T+B 2 - 77. Designing an Inductance Coil. The formula _ 4iT 2 n 2 r 2 u 1 . 26 n 2 uA wh~ or ~m ' is given for finding the inductance of a coil; L = inductance in henries; n = number of turns; r = radius of the coil in centimeters; , ... . . ( =1 for air u — permeability ot the core \ -, r ™ r i = loOO for iron. A = cross section area of the core. DESIGNING A CONDENSER 93 EXERCISE 6 1. Find the inductance of the primary coil of a trans- former having 400 turns if the iron core has a cross section area of 300 sq. cm. and is 60 cm. long. Ans. 15 . 1 henries. 2. Design the dimensions of an inductance coil wound on an iron core that will have an inductance of 25 henries and be of good shape. 78. Designing a Condenser. The following formula is given for the capacity of a condenser _ 885Xa(ft-l) tmf ~~~W"d ' C mf = capacity of a condenser in microfarads; K — dielectric constant ; a = area of one plate; n = number of metal plates ; d = thickness of dielectric plate in centimeters. EXERCISE 7 1. An aluminum and air condenser has 25 aluminum plates of diameter 12 cm., separated by 2 mm. of air. Find the capacity if K = 1 for air. Ans. .0012. 2. A condenser is to be made of mica and tinfoil to have a capacity of .32 microfarad. The sheets are 20 cm. square and the mica is .05 cm. thick, K for mica = 6. How many sheets are needed? Ans. 76. 3. Design the dimensions of a condenser of tinfoil and glass that will have a capacity of 0.002 microfarad. Assume that the glass available has a thickness of 0.25 cm. K = 6 for glass. 94 ELECTRICAL FORMULAS EXERCISE 8 General Formulas. Evaluate the following formulas: - -b-Vb 2 +4ac . , , . 1. x = — — ^ — — , when a=l,b = 4,c = 5. Ans. 1. -b-Vb 2 +4ac , , , . B 2. x=— — - — — , when a = l, 6=4, c = 5. Ans. —5. 3. x = — — ~— — , when a = 3, b= — 14, c= — 6. Ans. 4.1893 and 0.4774. 4. 6 = Va 2 +c 2 -2a'c, when a = 7, c = 5, a' = 3. Ans. 6.633. 5. A = Vs(s - a ) (s-b) (s-c) , when a = 16, b = 12, c=20 and s=§(a+6+c). Ans. 96. 6. A = Vs(s-a) (s-6) (s-c), when a = 22, 6 = 30, c=26ands = |(a+&+c). Ans. 278.5. CHAPTER VIII QUADRATIC EQUATIONS 79. Definitions. Some formulas and problems require the use of an unknown letter raised to the second power, giving an equation containing x 2 or y 2 , etc. An equation that contains an unknown in the second power as the highest power of the unknown is a quadratic equation. If a quadratic equation contains only the second power, of the unknown, it can be put in the form x 2 = 25. (Axiom : If two expressions are equal their square roots are equal.) Therefore, since x 2 = 25 x=zk 5 (± is read + or — ) Observe that: 1. The sign ± means that both +5 and —5 check the equation. 2. The square root of a number may be positive or negative because ( + 5) 2 = 25 and (-5) 2 = 25. 3. A quadratic equation has two answers. EXERCISE 1 Solve the following equations: 1. z 2 =121. Ans. £=±11. 2. x 2 = 50. £=±7.071. 3. R 2 = l #=±.866. 4. d 2 = U. d=±%. 9G QUADRATIC EQUATIONS y 8- 6. ?/ 2 -12 = 37. 7. 5a; 2 =180. 8. 3a- 2 +13 = 1 GO. Ans. y= ±935. I/=± 7. z=± 0. x = ± 7. 80. Quadratic Equations with Both First and Second Powers. A quadratic equation which contains the first and second powers of the unknown cannot be solved by the above method. For example, in x 2 -{-6x = 40, the square root of x 2 -\-6x cannot be found. The form of the equation can be changed so that the square roots of both members can be found. EXERCISE 2 Expand the following: 1. (x+1) 2 . Ans. x 2 4-2x4-1. 2. + 2) 2 3. (x-1) 2 4. (x-4) 2 5. + 5) 2 6. (x + i) 2 7. Or + f) 2 8. (x-h) 2 Observe that: 1. The aiibivers to all the 'problems in Exercise 2 contain three terms. An x 2 term, an x term, and a numerical term. 2. The x 2 term and the numerical term are perfect squares and the x term is twice the product of the two terms of the problem. (For example, in (x — 4) 2 = x 2 — 8x-\-lQ, x 2 and 16 are the squares of x and 4, while —8x is 2x( — 4).) TRINOMIAL SQUARES 97 81. Trinomial Squares. An expression having two terms which aie perfect squares and the other term twice the product of the square roots of these two terms is a trinomial square. The square root of a trinomial square can be found. The process is the reverse of the method used in Exercise 2. Rule. To find the square root of a trinomial square take the square root of the two -perfect square terms and connect them by the sign of the other term. Example 1. V~i 2 + Qx+ 9=±(z+3). Example 2. Vx 2 - 10a; +25 = ± (x- 5). EXERCISE 3 Find the square root of the following; give positive results only: 1. z 2 +2a;+l. Ans. x+1. 2. :r 2 -4x+4. x-2. 3. x 2 -Gx+9. 4. x 2 +14x+49. 5. ?/ 2 -l 6?/ + 64. 6. x 2 +20x+100. 7. R 2 -R + l 4 4 8. a 2 — ^a-\-q. 82. To Complete a Trinomial Square. A quadratic equation of the form z 2 + 6a: = 40 can be changed to an equation having a trinomial square in the first member by adding 9 to both members, thus: x 2 + Sx =40 z 2 +6z+9 = 49 98 QUADRATIC EQUATIONS The number 9 which must be added to z 2 +6;r to make it a trinomial square is the square of \ of 6 or 9. (This follows from Exercises 2 and 3.) Rule. To change an expression of the form x 2 -\-ax into a trinomial square add to it the square of \ the coefficient of x. Example 1. Find the number which will make x 2 -\-\()x a trinomial square. Solution. The coefficient of x is 10, § of 10 = 5, 5 2 = 25. Therefore 25 added to x 2 +lQx will make it a trinomial square. Example 2. Find the number which will make x 2 -7x a trinomial square. Solution The coefficient of x is 7, \ of 7 = $, (1) 2 = Y- Therefore ^ added to x 2 - Ix will make it a trinomial square. EXERCISE 4 Find the number which will add t o the following to make rinomial squares: 1. x 2 + 2x. Ans. 1. 2. x 2 -(jx. 9. 3. x 2 +U)x. 25. 4. x 2 + 5x. 25 4~- 5. x 2 — 9x. 8 1 6. x 2 -12z. 36. 83. Solution of Quadratic Equations by Completing the Square. Example 1. :r 2 + 6a; = 40 af'-f 6:c+9 = 49 (Adding 9 to both members to make the first member a tri- nomial square) z+3 = ±7 (Extracting the square root of both members) TRINOMIAL SQUARES 99 Then z+3=+7 and x+3=- 7 x = 4 x=-10 Check. 4 2 4-6.4 = 40 (-10) 2 +6(-10)=40 16+24 = 40 100-60 = 40 40 = 40 40 = 40 Example 2. 2x 2 - Qx = 12 4- 4x 2x 2 -10.r=12 x 2_ 5£ = g (Th e coefficient of x 2 must be made equal to 1) x 2 -hx-\- 2 i=^i- (Adding ?-£- to both mem- bers to make the first member a trinomial square) Then »—'!=+$ and x—%= -r — 1-2 r = _ 2 £ = 6 z= — 1 Observe that: 1. A quadratic equation has two answers. 2. Both answers must check. 3. Before completing the square the first term must have the coefficient 1. EXERCISE 5 Solve the following equations: 1. z 2 +4:c = 45. Ans. x = 5 or -9. 2. z 2 + 6:r = 27. a; = 3 or -9. 3. x 2 -5.r = 24. x = 8 or -3. 4. 2x 2 -7.r = 30. x = 6 or -2\. 5. 2x 2 -7x = 34. z = 6.229 or 2.729. 6. Zx 2 -\ ".r = 8. x=l or -2f. 100 7. x 2 +x 2 +10x = 22 + 3x. ^_7z = 51 8 - 2 8 4 - QUADRATIC EQUATIONS Ans. x = 2 or or -4 84. Solution of Quadratic Equations by Formula. Any quadratic equation can be expressed in the form ax 2 -\-bx = c, where a, b and c are general numbers. If the equation ax 2 + bx = c is solved for x by completing the square, it will be found thai -6±\/6 2 +4ac x = 2a The above result should be memorized and used as a formula for solving other quadratic equations of the form ax 2 + bx = c. Observe that a is the coefficient of x 2 , b is the coefficient of x and c is the term that does not contain x. Problems. = c. Solutions. 1. ax 2 -\-bx = -b±Vb 2 +4ac X ~ 2a = 18 a - '2 b= 5 c=18 2. 2x 2 + 5x = -5±V5 2 +4-2-18 X ~ 2-2 Simplifying: -5±V25 + 144 4 -5±Vl69 X = 4 -5±13 *~ 4 8 -18 £ = t or — t— 4 4 x = 2 or -4| SOLUTION BY FORMULA 101 Substitute in the formula. a = 3 3. 3z 2 -4:r=15 b=- 4 c= 15 4. a; 2 — 2a; — 5 = 45 — 3a\ a; 2 — 5a: = 50. (Arranging in the standard form.) a= 1 b=- 5 c= 50 Substitute in the formula. EXERCISE 6 Solve bv the formula method: 1. 2x 2 -\-5x = S. 2. 5a; 2 +3x = 2. 3. a; 2 +4a; = 5. 4. a- 2 -2x =15. 5. a; 2 = 4a;+12. 6. 2.r 2 -3a;=18. 7. 7x 2 -4a;-8=10. 8. 5t 2 -6z = 41. 9 ^_ = _^. 3x-7 a;+84 10. a; 2 + 3a; = 9. 11. 3a; 2 + 5x = 2. Ans. -3, +.5. I -1. 1, -5. 5, -3. 0, -2. 3.84, -2.34. 1.889, -1.318. 3.526, -2.326. 14, -10. -4.854, 1.854. — 2 i 85. Application of the Quadratic Equation to the Right Triangle. If the hypotenuse of a right triangle (Fig. 55) 102 QUADRATIC EQUATIONS is c and the sides arc a and b the relation between the sides, and the hypotenuse is expressed by the formula, (? = a?+b 2 . EXERCISE 7 1. v+L Find the sides. Ans. 12, 16. 2x + a Find the sides. Ans. 5, 13. 3. One side of a right triangle is 2 more than the other side and the hypotenuse is 10. Find the sides. Ans. 6, 8. Hint. Let x = ovc side then £+2 = the other side 4. One side of a rectangular lot is 31 rods more than the other side and the area is 360 square rods. Find the dimen- sions. Ans. 9, 40. RIGHT TRIANGLES 103 Fig. 58. 5. Find the side of the square in Fig 58. Ans. 18. 6. Find the inside and outside radius of a ring that has an area of 34.5576 square inches and a thickness of 1 inch. Ans. 5, 6. Fig. 59. Find the size plug that will fit as shown in Fig. 59. Ans. Radius = 1.0858. CHAPTER IX SIMULTANEOUS EQUATIONS 86. Definition. In some problems which involve two or more unknown quantities it is often difficult to express both unknowns in terms of one letter. Such problems can be solved conveniently by using two or more unknown letters. For example, The sum of two numbers is 29 and the difference is 11. Find the numbers. Let z = one of the numbers y = the other number Then x+y = 29 x — y — 11 Neither one of these two equations can be solved inde- pendently; they must be solved together. Two or more equations involving two or more unknowns which must be solved together are called simultaneous equations. 87. Solution of Simultaneous Equations. Two simul- taneous equations can be solved by eliminating one of the unknowns. (a) Elimination by Addition or Subtraction: Example 1. The two simultaneous equations above can t)o solved by adding the first member of the one to the first member of the other, and the second member to second 104 ELIMINATION BY ADDITION OR SUBTRACTION 105 member. The sums will be equal. (Axiom: If equals are added to equals the sums are equal.) .T + 7/ = 29 ....... (1) x-y^ll (2) 2.t = 40 Adding (1) and (2) (3) a; = 20 20 + */= 29 Substituting value of x V= 9 in (1), Check. 20+9 = 29 29 = 29 20-9=11 11 = 11 Note. When one unknown is found the other may be found by substituting the value found in either of the original equations. To check simultaneous equations the values found must check both equations. Example 2. x+Zy= 8 2x+y=l\ 2z+6y=16 Multiplying both mem- bers of (1) by 2. (1) (2) (3) -5y = -5. y= 1. x+3= 8. Subtracting (3) from (2). Substituting in (1). Check. 5+3 = 10+1 = 11. 11=11. 106 SIMULTANEOUS EQUATIONS Example 3. 7z+3?/=16 (1) Zx+2y= 9 (2) Ux+Gij = 32. (1)X2 (3) 9z+6?/ = 27. (2)X3 (4) 5x = 5. (3)- (4). x= 1. 7 + 3?/= 16. Substituting in (1), 3?/= 9. 2/=3. CTiecfc. 7+9=16. 16=16. 3 + 6= 9. 9= 9. Observe that: 1. Either unknown may be eliminated. 2. To eliminate an unknown it may be necessary to mul- tiply both members of one equation by some number, and in some problems it may be necessary to multiply both members of both equations by some number. Both equations do not have to be multiplied by the same number. 3. In checking simultaneous equations the values found must check both equations. EXERCISE 1 Solve the following simultaneous equations: 1. 2x+3i/=16. 2x- y = 8. Ans. x = 5, y = 2. 2. 2x+ y= 4. 3s- ?/ = 21. x=5, y=-Q. 3. &r+ y= 7. llx+2?/ = 28. x=-2$, y = 29l ELIMINATION BY SUBSTITUTION 10, 4. x-2y=-12. Ax— y = 1. 5. 3z + 5?/ = 33. 4x-2y=18. Ans. x = 2, y-7. x = Q, ?/ = 3. (6) Elimination by Substitution. Some simultaneous equations can be solved more efficiently by another method called substitution. In this method, solve either equation for either unknown in terms of the other unknown (choose the one giving the simplest solution), and substitute the value found in the other equation. Example 1. x+y = 5 2x+3i/=13 x = 5 — y Solving (1) for a;. 2(5-y)+Sy= 13 Substituting in (2), 10-2y+3y=13 y= 3 z+3 = 5 x= 2 Check. 2+3= 5 5= 5 2-2+3-3 = 13 4+9 = 13 13=13 (1) (2) (3) Example 2. 3x+2y = 25 . 2x — 5y= 4 2x = 4 + 5?/ 4+5?/ (1) (2) From (2), x = 3(4+57/) +2?/ = 25 Substituting in (1), 3(4 + 57/)+4?7 = 50 108 SIMULTANEOUS EQUATIONS 12+15*/+4y = 50 19y = 38 y= 2 2x-10= 4 2x=14 x= 7 EXERCISE 2 Solve by the substitution method: 1. 2z+3?/ = 26. x — 5y = 0. Ans. z=10, y=2. 2. 3i/= 5s- 18. 3y a: = 6, y = 4. 3. as=2y+3. z=7y-12. • z = 9, ?/ = 3. 4. ?/-3z=2. £ 2 +y 2 = 4. x = rr=-l|. O 01 ' 13 «/=2 2/=-l|. 5. 4^+2/2 = 5. y=2° T y=-2. CHAPTER X THE GRAPH 88. Plotting a Graph. The number of revolutions per minute, that will give a cutting speed of 45 ft. per minute to the following size stock, may be found and the results expressed in tabular form thus: R.P.M. Diameter of Stock. 1. \" 343 2. |" 229 3. 1" 172 4. \\" 137 5. \\" 114 6. 2" 86 7. 2§" 69 8. 3" 57 These results may also be represented graphically by plotting the diameters along a horizontal line (called an axis) and the R.P.M. along a line perpendicular to this line (also an axis), as in Fig. 64. In Fig. 64 point 1 corresponds to a diameter of \ in. and 343 R.P.M. Similarly point 2 may be located, etc. Observe that: 1. All of the points are connected by a smooth regular curve. 109 110 THE GRAPH 2. All points on the curve represent some corresponding diameter and R.P.M. that will give a cutting speed of 45 ft. per minute. 3. When either diameter of stock or R.P.M. is given, the other can be found from the figure. The horizontal axis is called the X-axis and the vertical l 2 Diameter of Stock Fig. 60. axis is called the Y-axis. The curve connecting the points located is a graph. Example. -From Fig. 60, find the R.P.M. necessary to give a 2;j-in. stock a cutting speed of 45 ft. per minute. Solution. Find 2 \ in. on the X-axis. Find the point on the graph directly above this point and read the R.P.M. on the Y-axis (75) corresponding to this point. CUTTING SPEED 111 EXERCISE 1 From Fig. 60 find the R.P.M. necessary to give a cutting speed of 45 ft. per minute to the following stock : Diameter R p M of Stock. 1. 2f" 2. 2*" 3. 4. ■8 1" 8 EXERCISE 2 1. Make a table of diameters and R.P.M. as on page 109, for a cutting speed of 85 ft. per minute. (The slide rule can be used to advantage here.) 2. Plot on squared paper a graph of the data computed in Problem 1. Note. The value of the units on the squared paper must be so chosen that the curve will all lie on the sheet of paper, and the units must be sufficiently large to permit reading nearly accurate values from the curve. (The larger the curve the more accurate the results.) 3. From the curve of Problem 2, find the R.P.M. neces- sary to give the following stock a cutting speed of 85 ft. per minute: Diam. R.P.M. (a) I" (b) If" (c) 2f " (d) 2} " 4. Students having use in the shop for a set of these curves should plot a curve for several other cutting speeds in common use. These curves can all be plotted on the same scale and axes. 112 THE GRAPH EXERCISE 3 1. A storage battery is discharged in 100 minutes and the voltage furnished by the battery measured at different times during the discharge. Following is a table of the data read: Voltage. Time (in minutes). 6 2 5.9 4 5.8 10 5.7 12 5.7 17 5.6 22 5.5 27 5.5 32 5.5 37 5.4 42 5.3 52 5 62 4.7 72 4.3 82 3.7 92 2.7 94 2.5 97 1.9 100 1.3 (a) Plot a graph of the discharge, laying off the time on the X-axis and the voltage on the Y-axis. Connect the points plotted by a smooth curve. (6) Read from the curve the voltage at the end of 30 minutes. (c) What does the shape of the curve indicate? COORDINATES 113 EXERCISE 4 1. Compute the weight per foot of round steel, and fill in the following table (1 cis. in. of steel weighs .28 lb.): Diam. (Inches). We 1. l 8 2. 1 4 3. 3 8 4. 1 2 5. 3 4 6. 1 7. u 8. n 9. 2 10. 2\ 11. 3 2. Plot a curve from the above data. Read from the curve the missing values in the following table: Diameter. Weight (a) 7 8 (b) 13.4 (c) 2| 89. Coordinates. Many problems arise in which it is necessary to plot or locate points with reference to one • another. A point is generally located, as in the previous exercises, by giving its distance from each of two perpen- dicular lines. Such a system is a rectangular coordinate system. The two distances are the coordinates of the point. The two perpendicular lines are axes: the horizontal line is the X-axis, and the vertical line is the Y-axis. The Ill THE GRAPH point where the axes meet is the origin. The axes are drawn and lettered as in Fig. 61. The horizontal distance of a point from the origin is the abscissa of the point. The vertical X 1 -; X distance of a point from the origin is the ordinate of the point. The abscissa is usually represented by x and the ordinate by ?/. Y' Fig. 61. 90. Location of Points. Any point is represented by the symbol (x, y) and a particular point is represented by the symbol (2, 7), (—3, 5), etc., the abscissa always being given first. Negative abscissas are plotted to the left of the origin and negative ordinates below the origin. To locate a point, lay off the axes (squared paper is the most convenient to use) ; then lay off the abscissa along the X-:ixis, and lay off the ordi- nate perpendicular to the . X-axis at this point. Example 1. Locate the point (3, 5). Solution. Measure 3 from the origin along the X-axis to the right, then 5 above this point. This is the point (3, 5), as shown in Fig. 62. Y' Fig. 62. GRAPH OF AN EQUATION 115 Example 2. Locate the point (4, -5). Solution. Measure 4 from the origin along the X-axis to the right then 5 below this point, as in Fig. 63. EXERCISE 5 Locate the following points: 1. (5, 3). 2. (-4,2). 3. (2, 7). 4. (-5, -2). 5. (4, -3). 6. (-4, 0). 7. (0, 4). x- 8. (3i, 4f). Y' Fig. 63. 91. Graph of an Equation. An equation containing two unknowns, as x-\-y = 7, cannot be solved for a value of x and y, since many different pairs of values would check, as x = 3, y = 4, or .r = 2, y = 5, etc. By plotting several points representing corresponding values of x and y and connecting these points, a line is obtained which expresses the relation between x and y in the equation. The line is the graph of the equation. Example 1. Construct the graph of the equation v- 2x -3. Solution. Find several pairs of corresponding values of x and y. This can be done most efficiently by assuming values of x and computing by evaluation the corresponding 116 THE GRAPH values of y. For example, when x = 2, then ?/ = 2-2 — 3 = 1. Record the values found in a table thus: Y / / Y' X y -3 1 -l 2 i 3 3 4 5 Fig. 64. Plot the points on squared paper, as in Fig. 64. The line connecting the points is a graph of the equation y = 2x — 3, and represents the relation between x and y. The coordinates of every point on the line satisfy the equation, and every point whose coordinates satisfy the equation lies on the line. Any number of corresponding values of x and y can be read from the figure. EXERCISE 6 Draw the graph of the following equations: 1. y = 3x. 2. y = 2x-l. 3. x-\-y = 5. (Solve for y first.) 4. 2x+3y =13. 92. Graphs which Are Not Straight Lines. All the above graphs are straight lines, and when a few points are located the whole graph can be drawn. If the graph of an equation is not a straight line, more points will have to be plotted to determine the shape of the graph. Connect the points by a smooth curve. (A French curve will assist in this.) GRAPH OF AN EQUATION 117 EXERCISE 7 Draw the graphs of the following curves: 1. y = 2x 2 +l. Note. x = +3 or —3 will give the same value of y, therefore x has two values giving two points one above the X-axis and the other below the X-axis. 2. x 2 + if = 25. 3. x 2 -y 2 =W. 4. 4.r 2 +V = 36. 93. Graphs of Simultaneous Equations. The graph of the equation 2x = Sy — 5 is the line A B, Fig. (65). The graph of the equation '$x-\-2y = 12 is the line CD, Fig. (65). Observe that: 1. The coordinates of all points on the line AB check the equation Q . Fig. 65. 2x — 6y — o. 2. The coordinates of all points on the line CD check the equation Y "R .F s Y' 3x+2?/=12. 3. The coordinates of the point P {2, 8) check both equations. 4. The coordinates of the point P are the same as the values of x and y when the two equations are solved simultaneously. 94. Solution of Simultaneous Equations by Graphs. If. the two equations are graphed, using the same axes, the coordinates of the points of intersection of the two graphs check both equations and are the values of the unknowns found .when the equations are solved simultaneously. 118 THE GRAPH Example 1. Solve the equations x 2 +y 2 = 2b and 4x = 3y by plotting the graphs and reading the coordinates of the points of intersection of the graphs, and check by solving as simultaneous equations. From Fig. 66 it will be seen that there are two points of intersection P (3, 4) and P' (-3, -4). Check. :r 2 + ?/ = 25 4x = 3y x == ~ti (f 2 /)2+y2 == 25 V 16 V + 16?/ 2 = 400 25?/ 2 = 400 if= 16 Z/=±4 , to 3(±£) Fig. 66. 4 4 Y (T \j -'/ Y' +?/ 2 = 25 EXERCISE 8 Solve the following equations graphically and check by solving simultaneously : 1. 2x+ y=Q. x + Zy=l3. 2. 3x+2y=12. y = 4x—5. 3. x 2 + y 2 = 3G. x+ y= 4. 4. 4z 2 +97/ 2 = 72. 3y = 2z. Ans. x=l, y = 4. z = 2, ?/ = 3. s=;5.742,y= — 1.742 or -x= 1.742, ?/ = 5. 742. z = 3, ?/ = 2, or x= —3, y= —2. CHAPTER XI GEOMETRY 95. Angles. If the line OA (Fig. 67) revolves about as center, to the position OB, the two lines form an angle. The point is the vertex of the angle. ^ The lines OA and OB are the sides of the angle. An angle is read by reading the letter at the vertex between the letters at the ends of the sides. Thus, the angle in Fig. 67 is read angle AOB. The size of the angle is independent of the length of the sides, and is measured by the fractional part of a revolution made in turning from OA to OB. 96. Right Angle. If the line OB makes one-fourth of a complete revolution, as Fig. 68, the angle AOB is a right angle. Fig. 67. B Fig. 68. 97. Straight Angle. If the line OB makes one-half of a complete revolution, as in Fig. 69, the angle AOB is a straight angle. 119 120 GEOMETRY 98. Perigon. If the line OB makes one complete revolution, the angle AOB is a perigon. 99. Degree. A right angle is divided into 90 equal parts called degrees. A degree is divided into 60 equal parts called minutes. A minute is divided into 60 equal parts called seconds. Degrees, minutes, and seconds are indicated thus: 50 degrees, 32 minutes, 24 seconds, or 50° 32' 24". 100. Protractor. A protractor (Fig. 70) is an instrument for measuring and constructing angles. 101. Supplementary Angles. If the sum of two angles is a straight angle or 180°, the angles are supplementary angles. Each is the supple- ment of the other, i.e., repre- sents what must be added to make 180°. Fig. 70. 102. Complementary Angles. If the sum of two angles is 90°, the angles are complementary angles. Each angle is the complement of the other, i.e., represents what must be added to make 90°. 103. Angle Theorems. 1. All right angles are equal. 2. All straight angles are equal. 3. Complements of equal angles are equal. 4. Supplements of equal angles are equal. 5. The sum of the angles about a point on one side of a straight line is equal to 180°. ANGLE THEOREMS 121 6. The sum of the angles of a triangle is equal to 180°. 7. If two straight lines intersect, as in Fig. 71, the oppo- site or vertical angles are equal. That is, Z1=Z2 and Z3= Z4. 8. If two angles have their sides parallel right side to right side and left side to left side, as in Fig. 72, they are equal. Fig. 71. Fig. 72. Fig. 73. 9. If two angles have their sides perpendicular, right side to right side and left side to left side, as in Fig. 73, they are equal. 104. Congruent Triangles. Triangles equal in all re- spects are congruent. 1. If two triangles have two sides and the included angle of the one equal to two sides and the included angle Fig. 74. of the other, the triangles are congruent. That is, if AC = DF, AB = DE and Zi=ZD, Fig. 74, triangles I and II are congruent. 122 GEOMETRY 2. If two triangles have two angles and the included side of the one equal to two angles and the included side of the other, the triangles are congruent. That is, in Fig. 75, if \C aF Fig. 75. ZA-= ZD, ZB= ZE and AB = DE, triangles I and II are congruent. 3. If two triangles have three sides of the one equal to three sides of the other, the triangles are congruent. That is, in Fig. 76, if AB = DE, AC = DF, BC = EF, triangles I and II are congruent. B D Fig. 76. 4. Corresponding parts of congruent triangles are equal. 105. Isosceles Triangle Theorems. 1. In an isosceles triangle (Fig. 77) the angles opposite the equal sides are equal. 2. In an isosceles triangle the bisector of the vertex angle is the perpendicular bisector of the base. Fig. 77. 3. If two angles of a triangle are equal PARALLEL LINES 123 the sides opposite the equal angles are equal and the triangle is isosceles. 106. Parallel Lines. Straight lines in the same plane that will not meet, however far they are extended, are parallel lines. 1. A perpendicular to one of two parallel lines (Fig. 78) is perpendicular to the other also. Fig. 78. 2. If two parallel lines are cut by a third straight line, (Fig. 79): Fig. 79. (a) The alternate interior angles are equal (i.e., Z2= Z7 and Z4 = Z5). (6) The alternate exterior angles are equal (i.e., Zl= Z8 and Z3= Z6). (c) The corresponding angles are equal (i.e., Z 1= Z5, etc.). 124 GEOMETRY (d) The interior angles on the same side of the third line are supplementary (i.e., Z2+Z5 = 180°, etc.). 3. If two lines are cut by a third line making (a) The alternate interior angles equal, (b) The alternate exterior angles equal, (c) The corresponding angles equal, or (d) The interior angles on the same side of the third line supplementary, the lines are parallel. Note. No. 3 is the converse of No. 2. 4. Two lines perpendicular to the same straight line (Fig. 80) are parallel. Fig. 80. 107. Quadrilaterals. A plane figure having four straight sides is a quadrilateral. A trapezium is a quadrilateral having no two sides parallel. Trapezium Trapezoid Fig. 81. A trapezoid s a quadrilateral having one pair of parallel sides. QUADRILATERALS 125 Rhombus Fig. 82. A quadrilateral having its oppos'te sides parallel s a parallelogram. A rectangle is a parallelogram having four right angles. A square is a rectangle having four equal sides. A rhombus is a parallelogram having four equal sides and four oblique angles. 108. Parallelogram Theorems. 1. The opposite sides of a par- allelogram are equal. 2. The opposite angles of a parallelogram are equal. 3. Two adjacent angles of a parallelogram are supple- mentary. 4. If a pair of opposite sides of a quadrilateral are equal and parallel, the figure is a parallelogram. 109. Circles. 1. The diameter of a circle bisects the circle. 2. In the same circle or in equal circles, radii that form equal angles at the center intercept equal arcs on the circumference, and conversely. FlG - 83 - That is, if circle I is equal to circle II (Fig. 84), and if Zl= Z2, then arc l = arc 2 or if arc 1 = arc 2, then Zl= Z2. Fig. 84. 126 GEOMETRY 3. In the same circle or in equal circles equal chords subtend equal arcs, and conversely. If chord A£ = chord CD (Fig. 85), then arc AB = arc CD, and conversely. 4. In the same circle or in equal circles equal chords are equal distances from the center (Fig. 86), and conversely, if AB = CD, then OX=MN, and conversely. Fig. 85. 5. A line fulfilling any two of the following conditions fulfills the other two (Fig. 87): Fig. SO. Fig. 87. (a) Passes through the center of the circle. (6) Bisects the chord. (c) Is perpendicular to the chord. (d) Bisects the arc. 6. A tangent to a circle is perpen- dicular to the radius drawn to the point of tangency (Fig. 88), and con- versely. 7. Arcs included between parallel lines are equal. CIRCLE THEOREMS 127 8. An inscribed angle is measured by one-half the inter- cepted arc (Fig. 89). Fig. 89. Fig. 90. 9. An angle formed by a tangent and a chord Fig. (90) is measured by one-half the intercepted arc. 10. An angle formed by two secants meeting outside a circle (Fig. 91) is measured by one-half the difference of the intercepted arcs. Fig. 91. Fig. 92. 11. Two tangents drawn to a circle from an external point (Fig. 92) are equal. 110 Ratio and Proportion. 1. If two quantities are in proportion the product of the extremes equals the product of the means: i.e., if a :b :: c : d, then ad = be. 128 GEOMETRY 2. A line parallel to one side of a triangle divides the other two sides proportionally. 3. The medians of a triangle meet in a point two-thirds of the distance from the vertex to the mid-point of the opposite side. 4. The bisector of the angle of a triangle (Fig. 93) divides the opposite side into segments proportional to the Fig. 93. adjacent sides. 111. Similar Triangles. Triangles having their sides proportional and angles equal are similar. 1. Two triangles are similar if two angles of the one are equal to two angles of the other. 2. Two triangles are similar if the three sides of the one are proportional to the three sides of the other. 3. Triangles having their corresponding sides parallel or perpendicular are similar. 112. Right Triangles. 1. Two right triangles having an acute angle of the one equal to an acute angle of the other are similar. 2. The square on the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. 3. The perpendicular from the vertex of the right angle of a right triangle to the hypotenuse divides the triangle into two triangles similar to each other and similar to the original triangle. 4. The perpendicular to the hypotenuse of a right triangle is the mean proportional between the segments of the hypotenuse. 5. Either side is the mean proportional between the REVIEW PROBLEMS 129 whole hypotenuse and the segment of the hypotenuse adja- cent to that side. EXERCISE 1 1. Find the other angles of the parallelogram, Fig. 94. Ans. 5=126°. C = 54°, D=126°. Fig. 94. 2. Find the value of the following angles in the regular inscribed hexagon Fig. 95: (a) BOC. Ans. 60° (6) BEC. 30° (c) AFE. 120° (d) ECD. 30° F^ " v x E Fig. 95. 3. If a stick 10 ft. long casts a shadow 8 ft., how high, is a tree that casts a shadow 55 ft.? Ans. 68.7 ft. 130 GEOMETRY 4. In the right triangles, Fig. 96, find x. Ans. x = 4. 2896. 10.724 Fig. 96. 5. In the triangle, Fig. 97, find the hypotenuse. Ans. 20 ft. Fig. 97. 6. Find the segments of the hypotenuse. Ans. 7|, 12i. 7. Find the perpendicular to the hypotenuse. Ans. 9|. 8. Find the area. Ans. 96. CHAPTER XII LOGARITHMS 113. Aids to Computation. Various means have been devised for simplifying the work of long multiplication and division, such as necessarily occurs in the use of trigo- nometric functions and in other places. Three of these means are in common use: I. Calculating Machines. They are always expensive and not easily carried about. II. Slide Rules. Inexpensive instruments and easily carried, but not sufficiently accurate for some work. III. Logarithms. Tables of logarithms are very nex- pensive and easily carried about and give more accurate results than the slide rule. Logarithms were first put into practical use by John Napier in 1614, later they were improved by Henry Briggs. Logarithms are used extensively in computations because they reduce multiplication, division and finding roots to easy, rapid calculations. Logarithms are also necessary in solving certain equations and are used in some formulas. 114. Common Logarithms. The common logarithm of a number is the exponent of 10 which makes the power of 10 equal to that number. For example, 10 2 =100, there- fore 2 is the logarithm of 100. 10 is the base of the logarithm in this definition; other bases are used in higher work. The student must keep in mind the fundamental conception of a logarithm, namely, that " a logarithm is an exponent." 131 132 LOGARITHMS 115. Finding the Logarithm of a Number. By the definition of a logarithm, 10 2 =100 therefore log 100 = 2, 103 = 1000 therefore log 1000 = 3. Hence the logarithm of 254 must lie between 2 and 3. That is, it must be 2+; carried to five places it is 2.40483. A logarithm is composed of two parts, a whole number and a decimal. The decimal part of the logarithm is the mantissa and can be found in the table of logarithms. The mantissa is always positive, and is independent of the position of the decimal point in the number. That is, the mantissa is determined by the figures only. For example, the mantissa of 3472 is the same as the mantissa of 34.72. The whole number part of a logarithm is the charac- teristic. The characteristic cannot be found in the table, but must be determined by rule. The characteristic is sometimes positive and sometimes negative, as in the case of decimals. If the characteristic is —3 and the mantissa is .44632, the whole logarithm cannot be written —3.44632, as this would indicate that the mantissa also is negative. The mantissa is never negative and so the following nota- tion is used, 3.44632, which means that the characteristic alone is negative.* The student should learn to distin- guish between characteristic and mantissa. 116. Finding the Characteristic. The characteristic can be determined by definition thus: 10 2 =100 therefore log 100 =2, 103=1000 therefore log 1000 =3, 10 4 = 10000 therefore log 10000 = 4. Hence the characteristic of the logarithm of all numbers * Note. The following notation is sometimes used for negative characteristics: 7.44G32-10. CHARACTERISTIC 133 between 100 and 1000 is 2 and of all numbers between 1000 and 10000 is 3, etc. If the number of figures in the whole number is increased or decreased by 1, the characteristic of the logarithm of the number is increased or decreased by 1. Hence: For numbers between The characteristic is 3 2 1 -1 -2 -3 This rule is too cumbersome for efficient use so the following rule will be used. Observe the following table of numbers and their logarithms: Number. Logarithm. 1000 and 10000 100 and 1000 10 and 100 1 and 10 .1 and 1 .01 and . 1 .001 and .01 0.00254 3.40483 0.0254 2.40483 0.254 1.40483 2.54 0.40483 2 5.4 1.40483 25 4 . 2.40483 254 0. 3.40483 2540 0. 4.40483 Observe that the characteristics in the above table could be found by counting from the units figure (the figure in the column between the lines) to the 2. Rule. To find the characteristic count from {not includ- ing) the units figure to the first significant figure. 134 LOGARITHMS Observe that: 1. If the count is to the left, the characteristic is positive, and if to the right, the characteristic is negative. 2. A significant figure means a figure other than zero, and the first significant figure means the first figure (other than zero) on the left of a number. Example 1. What is the characteristic of 7934? Count 3 2 1 from the 4 to the 7, thus 7934. The characteristic is 3. Example 2. What is the characteristic of 0.000853? Count from the first to the 8, thus 0.000853. The characteristic is —4. EXERCISE 1 What is the characteristic in the following numbers? 1. 25 6. 0.0623 2. 354 7. 0.00043 3. 3540 8. 0.0004 4. 75.84 9. 0.421 5. 66319 10. 6.3 117. Finding the Mantissa. The mantissa must be found in the table of logarithms. The first two figures of a number will be found in the column marked N at the left of the page and the third figure will be found at the top of the page. The mantissa will be found in the row with the first two figures and in the column of the third figure. Example 1. Find the mant ssa o 567. Look for 56 in the column at the left of the page and find the mantissa (.75358) in that row and in the column headed 7. The complete logarithm is 2.75358. In finding a logarithm the following steps should be followed: INTERPOLATION 135 1. Write the number thus, log 567 = 2. Determine the characteristic, log 567 = 2. 3. Find the mantissa, log 567 = 2.75358. For efficiency in finding the mantissa, fix the entire mantissa in mind before writing any of it. EXERCISE 2 Find the logarithms of the following: 1. 567. 7. 9. 2. 983. 8. 0.234. 3. 75.4. 9. 0.0056. 4. 6330. 10. 7.03. 5. 565. 11. 40.2. 6. 230. 12. 768. 118. Interpolation. The logarithm of 4683 cannot be found in an ordinary logarithm table, but the logarithms 46S0 and 4690 can be found. The logarithm of 4683 must be determined by interpolation between these two logarithms. The work can be arranged as follows: log 4680 = 3. 67025 log 4690 = 3. 67 117 92 (Tabular difference) Take .3 of this difference and add to the logarithm of 4680, thus .3 X 92 = 28, that is, 28 in the last two places. log 4680 = 3. 67025 +28 3. 67053 = log 4683 In general, if the fourth figure of a number is different than 3, as 7, take .7 of the tabular difference and add to the 136 LOGARITHMS logarithm of the smaller number. In interpolation do not carry the results beyond the fifth place. The above method is too elaborate for rapid efficient use of logarithms, and for efficiency, interpolation must be done mentally. The following steps and suggestions should prove helpful: Example 1. Find the logarithm of 753.4. I. Write the characteristic. II. Look for the mantissas of 753 and 754; they will be found side by side, thus: 87679 and 87737. III. Subtract mentally, getting the difference of 58 (in the last two places). IV. .4X58=23.2 or 23 in the last two places. V. 87679+23 = 87702. VI. Write the mantissa 87702 after the characteristic. Note. Tables in this book will give results correct to three or four figures. Students desiring more accurate results can obtain more com- plete tables in a small compact book at a moderate price. EXERCISE 3 Find the logarithms of the following: 1. 6346. Ans. 3.80250. 2. 8437. 3.92619. 3. 245.2. 2.38952. 4. 3.657. 0.56312. 5. 5637. 1.75105. 119. Multiplication by Logarithms. Recall the funda- mental part of the definition of a logarithm, namely, that " A logarithm is an exponent," and recall the law of expo- nents for multiplication whereby x 3 Xx 4 = x~. It will be seen that the sum of the logarithms of two numbers is the ANTILOGARITHMS 137 logarithm of their product. Therefore the following rule can be stated: Rule. To multiply two numbers add their logarithms and find the number corresponding to the sum. 120. Antilogarithms. The number which corresponds to a given logarithm is called its antilogarithm. Example 1. Find the antilogarithm of 2.94576. Look for the mantissa (94596) in the main part of the table. The first two figures of the number (88) will be found in the column marked N at the left side of the page, in the row in which 94596 is found. The third figure (3) is found at the top of the column in which 94596 is found. There- fore the figures corresponding to the mantissa 94596 are 883. The position of the decimal point is determined by the characteristic. The method is the opposite of the method of finding the characteristic. For example, in the above problem the characteristic is 2 and the figures are 883. Therefore, count 2 from the 8 thus: 12 883 The next figure is units thus: 'a 12 3 883 Rule. Count, from the first figure of the number obtained, as many places as indicated by the characteristic, the next figure will be units. If the characteristic is positive, count to the right, and if the characteristic is negative, count to the left. Zeros will have to be supplied in this case. If the mantissa cannot be found exactly in the tables, interpolate between the two numbers corresponding to the two nearest logarithms. 138 LOGARITHMS Example. Find the antilog of 3 . 56505. 56467 = log 367 56585 = log 368 (Tabular difference) 118 56467 = log 367 56505 = log required number Difference 38 Diff. = J38 = Tab.diff. 118 + Therefore 56506 = log 3673 T \% = log 3673 + The figures of the antilogarithm are 3673, the charac- teristic is 3, therefore count as follows: 3321 0.003673. Ans. In interpolation results are not carried more than one place beyond the figures given in the table. EXERCISE 4 Find the antilogarithms of the following logarithms: 1. 2.58286. Ans. 382.7. 2. 4.59106. 39000. 3. 0.76384. 5.8055. 4. 2.84365. .069767. 5. 1.08458. 12.15. 6. I. 23615. .17225. 7. 0.29336. 1.965. 8. 5.25527. 180000. MULTIPLICATION 139 121. Multiplication. The work of multiplication logarithms should be arranged as follows: Multiply 3675 by 78 . 56. log 3675 = 3.56526 log 78.56= 1.89520 by Ans. 5.46046 (Adding) 288710. (Antilog) EXERCISE 5 Perform the following multiplications by means of logarithms : 1. 243X375. 2. 3984X5.6 3. .024X7685. 4. . 654 X. 4379. 5. 76X5. 6. 24765X3.49. 7. .0024X76587. 8. .653X378X55.15. 9. 273X68.49X23.69. 10. . 002 X. 0347X56. 11. 1 27000 X. 5634. 12. 653.14X5.7823. Ans. 91126. 22310. 184.44. .28639. 380. 86432. 183.81. 13613. 442960. .0038865. 71552. 3776.7. 122. Division. Since sy*0 ■ /yd — /wi •*/ • •*• — ~~ •*/ • and since a logarithm is an exponent, the logarithm of the quotient of two numbers is the difference of the logarithms of the numbers. The logarithm of the divisor is to be subtracted from the logarithm of the dividend. The anti- logarithm of the difference is the quotient. 140 LOGARITHMS 123. Cologarithms. The above method of division is too inefficient when several numbers are to be multiplied and divided. For example: 254X78 1 53X125 Add: Add: log 254 = 2.40483 log 781 = 2.89265 5.29748 log 53=1.72428 log 125 = 2.09691 3.82119 Subtract: Therefore : 5.29748 3.82119 1.47629 = log 29.943 254X781 53X125 29.943. 9^4- V7R1 But „*/° can be written 254X781 X^Xy^, 53 X l .054. 0.427X167' 4 . i 0256 * 68 - 715 . .000051085. 762X45.19 3.1416X258.9 275X.02 " 6543X289 21. 32X1. 318' 147.88. 67293. 35X24.32 0.59525. 65X22 8 J65X694J 223 Q4 23.4X4.675X46.75 124. Powers of Numbers : | 52 = 5x5 = 25 Therefore, log 5 2 = log 5+log 5 = 2 log 5 POWERS AND ROOTS 143 Similarly, log 5 3 = log 5+log 5+log 5 = 3 log 5 From which the following rule can be stated: Rule. To raise a number to a power multiply the logarithm of the number by the exponent of the number and find the anti- logarithm. Example. Find 37 3 . log 37= 1.56820 3 4.70460 37 3 = 50652 EXERCISE 7 Find by means of logarithms: 1. 6 5 . Ans. 7776. 2. .37 4 . .018741. 3. 5.37 3 . 154.85. 4. 1.257 3 . 1.9862. 5. .1254 3 . 0.001972. 125. Roots of Numbers. 25 = 5 2 log 25 = 2 log 5 V25 = 5 log V25 = log 5 2 log V25 = 2 1og5 144 LOGARITHMS But Therefore Similarly, log 25 = 2 log 5 2 log \ / 25 = log25 log V25 = | log 25 log \/Qi= I log 64 From the above the following rule can be stated: Rule. To find the square root of a number divide the logarithm of the number by 2 and find the antilogarithm. To find the cube root of a number divide the logarithm by 3 and find the antilogarithm. To find any root of a number divide the logarithm of the number by the index of the root and find the antilogarithm. Note. The index of a root is the small number placed above and to the left of the radical sign, and indicates which root is to be taken. For example, in \^75, 3 is the index of the root. Example 1. Find \/lb. log 75=1.87506 ^3= .62502 _4.2172=-^75 Example 2. Find \Z.05. log .05 = 2.69897 2.69897 cannot be divided by 3 as in the previous problem, since 3 will not go in 2 evenly, and a negative remainder cannot be carried over into the positive mantissa. But 2.69897 can be written -2+ .69897, UNKNOWN EXPONENTS 145 or Then or hence -3 + 1.69897. -3 + 1.69897 -1+ .56632 1 . 56632 ^.05 = .3684 EXERCISE 8 Find the following: 1. ^50. 2. ^3775. 3. V73. 4. V8700. 3 6 I( 34.2 8X57.5 V \\48. 08X5. 962/ Ans. 3.684. 5.1614. .54773. 93.274. 3.6161. 126. Equations Containing an Unknown Exponent. Axiom. If two quantities are equal, their logarithms are equal. Therefore if 3 X = 81 log3 z = log81 x log 3 = log 81 xX. 47712 =1.90849 1.90849 x = .17712 z = 4 146 I LOGARITHMS EXERCISE 9 Solve for x: 1. 5* =625. 2. 3 J =243. 3. 2 J =19. 4. 4 J + 1 = 256. Ans. 4. 5. 4.2479 3. exercise 10 Miscellaneous Problems 1. Given F = f7ir 3 as the formula for the volume of a sphere, where r is the radius. Find the volume of a sphere having a radius of 13.34 in. Ans. 9944.3. 2. Find the diameter of a steel ball weighing 16 lbs. if steel weighs .283 lb. per cubic inch. Ans. 4.762. 3. Evalute the formula C=— -' (Capacity of aerial wire) when s=24 and r = \. Ans. .0019574. 4. The formula for compound interest, compounded annually, is ^4.=a(l + r)", where A = the amount at the end of n years, a is the original investment, r is the rate, and n the number of years. Find the amount of $150.00 at the end of 20 years, interest compounded annually at 4 per cent. Ans. $328.62. REVIEW PROBLEMS 147 5. How many years will it take $100.00 to amount to $240.66? Interest compounded annually at 5 per cent. Ans. 18 years. 6. Evaluate the formula, L= .0017XZXlog p (Inductance in a trans- mission line) when I = 25, d = 18, and R = .25. Ans. .07894. 7. Evaluate the formula, 2 41 3 Kl C = — ^ (Capacity of a submarine cable) lonog^ whenX = 6, Z = 1000, D=2.5, d = 1.5. Ans. 0.00666. 8. Evaluate the formula for the capacity of a submarine cable when C = . 25, I = 35,000, D = 3, d = 1 . 75. Ans. 6.929. CHAPTER XIII RIGHT TRIANGULATION 127. Constant Ratios in Right Triangles. The right triangles shown in Fig. 98 are similar since they have an Fig. 98. acute angle of the one equal to an acute angle of the other. CL\ _ 0,2 _ as C\ Co C3 (why?) Therefore all right triangles having one acute angle equal to 23° are similar, and therefore the ratio of the side opposite 23° to the hypotenuse in each triangle is constant (always the same). By actual measurement of the sides this ratio reduced to a decimal correct to .00001 can be calculated to be 0.39073. 148 CONSTANT RATIOS 149 In the same manner: bi _ 62 ci c 2 C3 0.92050 £l = ^ = ^ = . 42447 01 02 03 2 . 3558.5 b\ _ bo _ 63 d\ 0,2 0,3 *** 1.0884 Oi 62 03 *=.*=* = 2.5fiB3 «i do as Example 1. In the right triangle of Fig. 99, ™ = 0.39073 bO a = 23. 4438 ^r= 0.9205 bO 60. 23 90 b Fig. 99. 6 = 55.230 Example 2. In the right triangle of Fig. 100, a 42 a=? = 0.42477 ts= 1.0864 42 /> =? Observe that: a z's the side opposite 23°, b is the side adjacent to 23°, and c is he hypo enuse of he right triangle. 150 RIGHT TRIANGULATION For ihe purpose of abulat'ng these constant ratios it is con- venient to refer to them by names Referring to Fig. 101, Fig. 101. a . is called the s*ne of 23°, written sin 23 c b . is called the cosine of 23°, written cos 23 c a . T is called the tangent of 23°, written tan 23 c o - is called the cotangent of 23°, written cot 23°. a is called the secant of 23°, written sec 23°. c . is called the cosecant of 23°, written esc 23 c sin cos tan cot sec esc 23° 0.39073 0.92050 0.42447 2.3559 I 1.0864 2.5593 These ratios can be calculated for any angle and are called functions of the angle. In general, for any value of angle A, FUNCTIONS DEFINED 151 . A a opposite side sin A = - = - — c hypotenuse . b adjacent side COS A = — = c hypotenuse . _ a _ opposite side b adjacent side , . 6 adjacent side COt A = - = ; — a opposite side A c hypotenuse sec A = - = ~^- — 6 adjacent side . _ c _ hypotenuse CSC rx ■ opposite side By using a table of these ratios for all angles, if one side and one acute angle of a right triangle are known all the remaining parts can be calculated. Table IV contains these ratios for all angles. 128. Use of the Tables. Example 1. Find the sine of 37° 20'. Locate the sine page (Table IV). Find 37° in the column at the left. The sine of 37° 20' is found in the row marked 37° and in the column marked 20' at the top. Example 2. Find the cosine of 52° 50'. Find the cosine page (Table IV). (Cosine is written at the bottom of the page.) Find the degrees in the column at the right and the minutes at the bottom of the page. Observe that: To find the sin or tan, locate the angle in degrees at the left of the page and the minutes in the column at the top. To find the cos or cot, locate the angle in degrees at the right of the page and the minutes in the column at the bottom. 152 RIGHT TRIANGULATION Example 3. Find a and 6 in Fig. 102. 65/S a — = sin 37°. b5 /ri° £ = .60181. b5 b Fig. 102. a = 39.1176. To find 6: £= = cos 37°. 65 £=.79863. b5 6 = 51.9109 To find ZB: ZA+ZB = 90°. 37< '4-/5 = 90°. ZJ3=53°. Check. c 2 =a 2 +6 2 65 2 = = 39.1176 2 +51.9109 2 4225 = = 1530.2+2694.8+ 4225 = = 4225 + Note. Tables correct to five figures will give results correct to five figures. USE OF THE FUNCTIONS 153 Example 4. Find 6, c, and Z A in the triangle of Fig. 103. To find ZA: ZA+Z£=90°. ZA+Z59° = 90°. ZA=31°. To find b: r wt31 °- A= 1.6643. 6 =133.144. ?° = sin31°. c -=.51504. c 80 = . 51504c c= 155.328. Observe: To find b, T = tan A might have been used, which would give SO ^ = .60090. o , 80 .60090* To find c: 6=133.1 + 154 RIGHT TRIANGULATION By selecting the ratio which has the unknown side for numerator, the result is obtained by multiplying by a decimal rather than by dividing by a decimal. EXERCISE 1 Find x in the triangles of Fig. 104: (a) (a) Ans. 58.084. (&) (b) Ans. 90.308. 42 (c) Ans. 31.649. 340 (d) (d) Ans. 349.18. (e) Ans. 236.84. Fig. 104. 2. The diagonal of a rectangle is 48" and the angle between it and the base is 28°. Find the dimensions of the rectangle. Ans. 22.535; 42.361. TWO SIDES GIVEN 155 3. The altitude of an isosceles triangle is 15" and the base angles are 37° 5'. Find the sides of the triangle. Ans. 24.876; 24.876; 39.690. a/ and E. Call this rate e. Ans. e=E sin <£. 145. Voltage Generated by a Rotating Loop. A voltage will be generated in the loop OCDM, Fig. 144 proportional to the rate at which CD cuts the lines of force. The maxi- mum voltage is obtained in the position OC (0 = 90°) and the minimum voltage in the position 0C"( as abscissas and the instantaneous 180 ELECTRICAL APPLICATIONS voltages as ordinates. (This is a sine or voltage curve. Save it. for future use.) 3. From the curve of Problem 4 read the value of the voltage at 45°, 135°, 225° and 315°. Ans. 7.1, 7.1, -7.1, -7.1. 146. Phase Angle. The value of the instantaneous volt- age can be shown also by revolving a line, of a length repre- senting the maximum voltage, about a point. The angle between the line and the X-axis is the angular position , and is called the phase angle. The projection upon the Y-axis is the instantaneous voltage ( = Esm4>). The diagram is arranged as Fig. 145. Fig. 145. EXERCISE 4 1 With scale and protractor compute the instantaneous voltage if E= 10, when = 25°, 80°, 120° and 325°. Ans. 4.2, 9.8, 8.7, -5.7. 2. On the same axis and scale as used in Problem 2, Exercise 3, plot another curve where # = 8 and is when the 4> of the other curve is 30°. (This curve will represent a voltage curve of a loop similar to OCDM but 30° behind it and not as large a loop.) 3. If the two loops are connected in series and rotated, the combined voltage at any time will be the sum of the two PHASE ANGLE 181 300-Kilowatt Three-Phase Turbo-Alternator. 35,000-Kilowatt Three-Phase Turbo-Alternator. 182 ELECTRICAL APPLICATIONS instantaneous voltages. Plot a curve of such sums on the same scale and axis as used in Problem 2. (Use the com- pass or dividers to lay off the sums, and save reading the numerical values.) (This curve is the resultant curve of the sum of the other two curves.) Compare the maximum of the resultant curve with the sum of the maximum of the other two curves. Can you see from the two curves why these two quantities are not equal? 147. Vectors. The method of Problem 3 above is one- of the methods of adding two A.C. voltages. The two voltages can also be added by vector addition. Some physical quantities can be represented completely by means of a straight line. The length of the line can be used to represent the numerical value of the quantity and the direction of the line to represent the sense of the quantity. For example, a distance of 3 miles can be repre- sented by a line 3" long, and, if it is a distance to the east, that fact can be represented by making the line point to to the right by means of an arrow head thus: Then a line 5" long and pointing in the opposite direction thus 4- will represent a distance of 5 miles to the west. Such quantities are called vector quantities. A straight line representing a vector quantity is called a vector. 148. Scalar Quantities. Quantities which can be ex- pressed completely by stating their magnitude are called scalar quantities. An example of a scalar quantity is 5 bushels or 7 ohms, etc. THE USE OF VECTORS 183 149. The Use of Vectors. Many quantities in electricity- can be expressed by vectors much more clearly than by words. Vectors form a valuable method for adding and subtracting certain quantities, and are used in all texts on electricity. Therefore it is necessary to study methods of expressing quantities by vectors and interpreting the meaning of quantities expressed by vectors. Example. A man travels 4 miles east and then 3 miles north. How far and in what direction is he from the starting point. Solve by vectors. Solution. The line AC, Fig. 146 represents 4 miles to the east and the line CB represents 3 miles to the north. The line AB represents the single equivalent journey. AB is called the vector sum of AC and CB. If AB is measured to the same scale as AC and CB, it will be found to represent 5 miles, and if the angle A is measured with a pro- tractor it will be found to be about 37°. Therefore the vector AB is interpreted as represent- ing a distance of 5 miles in the direction of 37° to the north of east. EXERCISE 5 Dolve by vectors : 1. A man travels 10 miles south then 12 miles east. How far and in what direction is he from the starting point? 2. A man travels 10 miles west then 2 miles north and 8 miles southeast. How far and in what direction is he from the starting point. . 184 PRACTICAL APPLICATIONS 150. The Composition of Forces. Another and more important use made of vectors is to study the effect of two forces acting upon the same object. Example. An object is acted upon by two forces, one of 6 lbs. and one of 8 lbs. acting at an angle of 90°, Fig. 147. What is the value and direction of the single force that would produce the same effect? Fig. 147. Fig. 148. By a law of mechanics a force produces the same effect upon an object whether it acts independently or in con- nection with other forces. Therefore, A B and CD acting together would produce the same effect as if AC acted and then AB acted. This could be represented as in Fig. 148. Then AB represents the value and direction of the single resultant force. Or taking the forces as shown in Fig. 147 and completing a parallelogram as Fig. 149 the diagonal AD also represents the resultant. AB of Fig. 148 = AD of Fig. 149, since CD = AB (opposite sides of a parallelogram) . Fig. 149. SUBTRACTION OF VECTORS 185 EXERCISE 6 1. Add a force of 5 lbs. to a force of 12 lbs. acting at right angles to it. 2. Add a force of 8 lbs. to a force of 12 lbs. acting at an angle of 60°. Hint. Draw the vectors at an angle of 60° and complete the parallelogram, etc. 3. Add a force of 3 lbs. acting horizontally and a force of 4 lbs acting at an angle of 20° above horizontal, and a force of 2 lbs. acting at an angle of 50° above the horizontal. Hint. Three or more vectors can be added by adding two and then the third one to that sum, etc., or they may be added by being placed end on end in the proper direction as Fig. 150. jg0?^ /so" ""^^ ^^^c > ^-"-20° 3 Fig. 150. 4. Add a force of 5 lbs. acting horizontally, and a force of 6 lbs. acting at an angle of 37° above the horizontal and a force of 4.5 lbs. acting at an angle of 60° above the hori- zontal. 151. Subtraction of Vectors. From algebra it is known that subtracting a number is the same as adding an equal number with the opposite sign, thus: 4-(+2)=4+(-2). 186 ELECTRICAL APPLICATIONS Similarly, in the subtraction of vectors, to subtract vector 1 from vector 2, add to vector 2 a vector (3) equal to vector 1 but of opposite direction, as Fig. 151. Fig. 151. Vector 4 then is the difference between vectors 1 and 2. Subtraction of vectors can be arranged more con- veniently as in Fig. 150. Complete the parallelogram for adding vectors 1 and 2. Fig. 152. Vector 5 represents the sum of 1 and 2. Vector 6 is equal and parallel to vector 4, and therefore represents the difference between 1 and 2. SUBTRACTION OF VECTORS 187 Proof. Vector 3 = Vector 1 (Construction) Vector 3 = line 7 (Opposite sides of a parallelo- gram) Therefore, Vector 1 = line 7 Vector 1 is parallel to line 7 (Const.) Therefore lines 1, 4, 7, 6 form a parallelogram. Therefore vector 6 is equal to, and parallel to vector 4, hence vector 6 represents the vector difference between 1 and 2. Rule. When two vectors are drawn from a point and the parallelogram is completed, the diagonal of the parallelogram, drawn from the point is the vector sum of the two vectors and the other diagonal is the vector difference. Note. The vector difference may be directed in either direction, depending upon the order in which the vectors are subtracted. EXERCISE 7 1 . Add by vectors a force of 40 lbs. and a force of 60 lbs. acting at an angle of 30°. Note. Alternating current voltages may differ in phase, that is in the time of reaching maximum or minimum value. Phase is ex- pressed as an angle which corresponds to the difference in position of two coils on the armature. Alternating current voltages can bo rep- resented by vectors, the phase angle is generally measured from a horizontal line to the vector. 2. Add by vectors a voltage of 8 volts, phase angle 15°, to a voltage of 10 volts, phase angle 45°. Compare the sum with the maximum resultant from the curves of Problem 3, Exercise 4. 3. Check the result of Problem 2 by solving the tri- angle by trigonometry. 188 ELECTRICAL APPLICATIONS 4. Add by vectors 12 volts, phase angle 10°, 15 volts, phase angle 35° and 8 volts, phase angle 20°. Measure magnitude, phase angle, and instantaneous value of the resultant. (See page 180.) 5. Solve No. 4 by trigonometry. 6. Study the vector diagram of Problem 4 to see what can be read from it in regard to instantaneous values of the separate voltages, etc. 7. Circuits of 12 volts, phase angle 20° and 20 volts, phase angle 45° are connected in series. Find the vector sum and difference. 8. A circuit has 110 volts, phase angle 36°. It is desired to make the circuit contain 150 volts, phase angle 2.8°. How many volts must be added and in what phase. 9. Solve No. 8 by trigonometry. 152. Current. Current is proportional to voltage, there- fore, the current curve is similar to the voltage curve and is expressed by the formula i=I sin , where i is the instantaneous current and / the maximum current. If the circuit contains no inductance or capacity, the current will be in phase with the voltage. That is, the cur- rent will be zero when the voltage is zero, and maximum when the voltage is maximum. If the circuit contains inductance only, the current will lag 90° behind the vo tage. If the circuit contains capacity only, the current will lead the voltage by 90°. Most alternating current circuits contain resistance, inductance and capacity. The current is not ordinarily in phase with the voltage. POWER 189 EXERCISE 8 1. Draw roughly a current curve in phase with its volt- age curve. 2. Draw roughly a current curve leading its voltage curve by 90°. 3. Draw roughly a current curve lagging behind its voltage curve by 90°. 4. Draw roughly a current curve lagging behind its voltage curve by 30°. The above exercises can be represented vectorially as in Fig. 153. (a) (c) (b) Fig. 153. (d) Fig. 153 (a) represents current in phase with the voltage. Fig. 153 (6) represents the current leading the voltage by 90°. Fig. 153 (c) represents the current lagging 90° behind the voltage. Fig. 153 (d) represents the current lagging 30° behind the voltage. 153. Power. From the curves of Exercise 8, it is evident that, at any instant, the power is equal to the instantaneous current times the instantaneous voltage. 190 ELECTRICAL APPLICATIONS . ■_ 7^>r * i * t "V ! ^r *4l I ifi 1 '3 y it's JL ■BaiL Wm~ yy • ; t . ~^L^ V - ■?** V •* % ■■}£} (j^HL 4^ "«& M^^ -'*V ^WSrll Wt$. , ~ v ' : ^Liv»lS Double Circuit Three-Phase Transmission Line with Pin Type Insulators. POWER 191 The current and voltage do not reach a maximum at the same time, hence maximum power is not maximum current times maximum volt- age. Similarly, effective power - // \ is not equal to effective cur- J^ rent times effective voltage, / ^ j but, the power at any instant Ecos<£ i is equal to the current times Fig. 154. the component of the voltage that is in the same phase with the current. Graphically, this can be represented as in Fig. 154. Power, then, is represented by the formula P = EI cos , where P is the maximum of effective power, E is the maxi- mum or effective voltage and 7 is maximum or effective current. 4> is the phase angle between the current and volt- age, and cos is called the power factor. EXERCISE 9 1. If r=10.5, #=125, and = 40° find the power. 2. In Problem 1 what is the power factor? Ans. .766a. 3. In an A.C. circuit the current lags 40°, the volt- meter indicates 120 volts, the ammeter 16 amperes. What would a wattmeter read? 4. An ammeter in a 120 volt A.C. circuit indicates 8 amperes. A wattmeter in the same circuit reads 500 watts. Find the power factor and phase difference. 5. A 110 volt A.C. generator is to deliver 2100 watts to a circuit having a power factor of 0.84. What current is necessary? 192 ELECTRICAL APPLICATIONS 154. Force on a Conductor Carrying Current in a Mag- netic Field. The force exerted on a conductor carrying current and lying in a magnetic field is expressed by the equation IIH sin 4> F = 10 where F = force in dynes on the conductor ; / = current through the conductor; 1 = length of the conductor in centimeters; H = strength of the magnetic field in gausses, = angle between the conductor and the magnetic field. EXERCISE 10 1. Find the force on a conductor 45 cm. long carrying a current of 12 amperes in a field of 16,000 gausses, if the angle between the conductor and the lines of magnetic forcois80°. Ans. 850,860. 2. How great a magnetic flux will be required to exert a maximum force (0 = 90°) of 1,250,000 dynes on a conductor 18" long carrying a current of 5 amperes? Ans. 54,680. 3. Find when F = 2,350,000, J =8, 1=2', H= 125,000. Ans. 22° 40'. 155. Impedance. Impedance in an alternating current circuit may be computed from the formula Z^+^L-^f, where Z = impedance ; R = resistance of circuit in ohms; /= number of cycles per second; L = inductance in henries; C = capacity in farads. FORCE ON A CONDUCTOR 193 Double Circuit Three-Phase Transmission Line with Suspension Insulators. 194 ELECTRICAL APPLICATIONS EXERCISE 11 1 Compute Z when R = 230, /= 60, L = .15, C = .0003. Ans. 235.1. 2. When L = .122 and C = . 00024, what value of / will make (2ir/L- ^) = ° ? Ans - 29 " 4L 3. When C = .0004 farad and /=500, what value of L will make (tofL - ^) = ? Ans. .00025. Note. High frequency currents are used in radio work. 156. Computation of Current. Current in an alter- nating current circuit may be found from the equation E 1 = where J = current ; E = voltage. ^ 2 +( 2 ^-2?c) 2 ' EXERCISE 12 1. Find the current that will flow through a circuit of 125 ohms, .02 henry, and .00003 farad, when a 60-cycle 120- volt alternating-current voltage is applied. Ans. .806. MISCELLANEOUS PROBLEMS The problems which follow are selected to give to the student, as nearly as possible, practical experience in com- putation. The problems are not classified but cover a wide and general field. Logarithms should be used in the computations. 1. A distance AB of 154.59', Fig. 155, is measured along the bank of a river. A tree C is sighted on the opposite bank, making the angle at A equal to 34° 28', and the angle at B equal to 90°. Find the width of the river BC. Ans. 106.11 ft. 34°28' Fig. 155. Fig. 156. 2. To find the distance between A and B separated by a lake, Fig. 156, readings were made as shown. Find the distance AB. Ans. 952.4. 195 196 MISCELLANEOUS PROBLEMS 3. Two sides of a triangular lot are found to be 173 rods and 194 rods. The included angle is 56° 15'. Find the area. • Ans. 87 . 2 acres. 4. Eight holes are to be drilled in a circle of radius 5 in. How far apart are the holes? Ans. 3.8267. 5. To drill the holes in the above problem it will be necessary to move the table, as in Fig. 157. D F Fig. 157. Find Ans. AB. 1.464 BC. 3.536 AM. 5. MN. 5. AD. 8.537 DE. 3.536 AF. 10. 6. The latitude of Washington, D. C, is 38° 55' N. How many miles east and west on the earth's surface make a difference in time of 1 hour in the latitude of Washington? Radius of the earth is 3957 miles. Ans. 806 . 02. 7. Find x, Fig. 158. - Ans. 0.0625. Fig. 158. MISCELLANEOUS PROBLEMS 197 8. Find x, Fig. 159 (angles are 90°). Ans. 4.0588. Fig. 159. 9. The angle of elevation of the top of a mountain, observed from a point directly south of it is 60°. From a point 1 mile directly east of the first point and on the same level with it the angle of elevation is 45°. Find the height of the mountain. Ans. 6466 . 8. 10. A light-house was observed to bear directly east from a ship. After the ship sailed 4 miles north the light-house bore 55° 30' east of south. How far was the ship from the light-house at the time of each observation? Ans. 5.8202, 7.062. 11. One side of a hexagon is 2 in. Find the area. Ans. 10.3926. 12. Find the radius of the circle, Fig. 160. Ans. .5429. Fig. 160. 320 lbs 410 lbs. Fig 161. 13. If two forces of 410 lbs. and 320 lbs. pull at an angle of 51° 37', as shown in Fig. 161, the line AB, which is the di- agonal of the paral- 198 MISCELLANEOUS PROBLEMS lelogram of which AM and AN are the sides, represents the value and direction of the resultant force. Find the value of the resultant force and the angle which it makes with the 410-lb. force. Ans. 658.36 lbs., 22° 23' 47". 14. An unknown force combined with a force of 128 lbs. produces a resultant of 200 lbs. and this resultant makes an angle of 18° 24' with the 128-lb. force. Find the inten- sity and direction of the unknown force. Ans. 88.326 lbs., 54° 37' 16". 15. A. Find the tension in the cable AB, Fig. 162. Ans. 3901 lbs. Q 5000 lbs. Fig. 162. Note. The parallelograms of forces, Fig. 163, can be used. 16. Find the distance x between the plugs, Fig. 164. Ans. 2.2930 in. Fig. 164. MISCELLANEOUS PROBLEMS 199 17. Find the distance Ans. 3.9065. Fig. 165. 18. From the top of a hill the angles of depression of two objects 5280 ft. apart on a straight level road leading to the hill are 5° and 15°. Find the height of the hill. Ans. 685.9 ft, APPENDIX DICTIONARY OF THE TERMS USED IN ELEMENTARY MATHEMATICS Abscissa. The horizontal or X coordinate of a point- Acute Angle. An angle of less than 90°. Adjacent. Next to or adjoining. Adjacent Angles. Angles having a common side and a common vertex. Altitude. Height measured perpendicular to the base. Angle. The difference in direction of two lines that meet at a point- Angle of Depression. The angle made by the line of sight to an object below the horizontal plane of the observer, and the plane. Angle of Elevation. The angle made by the line of sight to an object above the horizontal plane of the observer, and the plane. Antecedent. The first or third term of a proportion. Antilogarithm. The number corresponding to a given loga- rithm. Apothem. The perpendicular distance from the center of a regular polygon to one of the sides. Arc. Part of the circumference of a circle. Axiom. A truth that is self-evident- Base. The side or surface upon which a geometrical figure appears to rest. Base. An expression which is to be raised to a power. Binomial. An expression having two terms. Bisector. A line or plane which divides a geometrical figure into two equal parts. 201 202 DICTIONARY OF THE TERMS USED Central Angle. An angle which has its vertex at the center of a circle. Characteristic. The whole number part of a logarithm. Check. Proof of the solution of a problem. Chord. A straight line connecting two points on the cir- cumference of a circle. Circle. A plane figure bounded by a curved line every point of which is equidistant from a point called the center. Circular Mil. A unit of area equal to the area of a circle one mil in diameter. Circumference. The curved line forming a circle. Circumscribed. Drawn about, as a polygon circumscribed about a circle. Coefficient. The coefficient of a factor is the product of all the remaining factors. Cologarithm. The logarithm of the reciprocal of a number. Commensurable Quantities. Quantities whose common unit of measure is a rational quantity. Complement. The angle which will add to a given angle to make 90°. Concentric. Having the same centers. Cone. A solid generated by the rotation of a right triangle about one of its legs. Congruent. Equal in all respects. Consequent. The second or fourth term of a proportion. Constant. A quantity whose value does not change. Continued Proportion. A proportion of three or more ratios. Converse. Reversed in the order of relation. Coordinate. The distance of a point from the coordinate axes. Coordinate Axes. The reference lines by which a graph is plotted. DICTIONARY OF THE TERMS USED 203 Corollary. A truth which follows naturally from another truth. Corresponding Parts. Parts similarly placed in congruent or similar geometric figures. Cube. A rectangular solid having all of its faces squares. Cube. The third power of a number. Cutting Speed. The speed at which work passes the point of a cutting tool. Cylinder. The solid generated by revolving a rectangle about one of its sides. Definite Number. A number having always the same value. Degree. 3^0 of the angular space about a point. Diagonal. A line connecting any two non-adjacent vertices of polygon or polyhedron. Diameter. A line passing through the center of a circle and ending at the circumference. Dihedral Angle. The angle formed by two intersecting planes. Ellipse. Section of a cone made by a plane cutting the cone not parallel to or cutting the base. Equation. A statement that two expressions are equal. Equiangular. A plane figure having equal angles. Equilateral. A plane figure having equal sides. Equivalent. Equal in area or volume. Evaluation. The process of substituting definite numbers for general numbers and performing the operations indicated. Exponent. A small number written above and to the right of a number (called the base) to show how many times the base is to be used as a factor. Exterior Angle. The angle between one side of a polygon and an adjacent side extended. Extremes. The first and fourth terms of a proportion. 204 DICTIONARY OF THE TERMS USED Factor. One of the quantities which, multiplied together, form a given product. Formula. Statement of a rule or principle in terms of general numbers. Frustum. The part of a cone or a pyramid next the base, formed by cutting off the top. Fulcrum. The point about which a lever turns. Function. A quantity that depends upon another quantity for its value. Function of an Angle. Sin, cos,, or tan, etc., of the angle. General Number. A number that has different values in different problems. Graph. A curve representing the relation between two variables. Great Circle. The largest circle that can be drawn on the surface of a sphere. Hexagon. A plane figure having six sides. Hypotenuse. The side of a right triangle opposite the right angle. Incommensurable Quantities. Quantities whose common unit of measure is an irrational quantity. Index. The number above and to the left of the radical sign which indicates what root is to be taken. Index. The end of a slide rule scale. Inscribed. Drawn within; a polygon is inscribed in a circle when all the vertices of the polygon lie on the circum- ference of the circle. Inscribed Angle. An angle whose vertex lies on circumfer- ence of a circle and whose sides are chords of the circle. Intercept. To cut. Intercept. The part included between two points. Interpolation. The process of finding intermediate terms from a table by means of the two nearest terms given. DICTIONARY OF THE TERMS USED 205 Irrational Quantity. A quantity whose value cannot be expressed exactly by decimals or fractions, as y/2. Isosceles Triangle. A triangle having two sides equal. Least Common Denominator. The smallest number that will contain two or more denominators evenly. Lever. A rigid piece capable of turning about a point called a fulcrum. Leverage. Tendency to turn, turning moment. Lever Arm. Distance from the fulcrum of a lever to a force acting upon the lever, measured perpendicular to the direction of the force. Locus. The path of a point or curve moving according to some law. Logarithm. The exponent of that power of a base (usually 10) which equals a given number. Lune. A portion of the surface of a sphere bounded by the semicircumferences of two great circles. Mantissa. The decimal part of a logarithm. Mean Proportion. When the second and third terms of a proportion are the same the proportion is a mean pro- portion. Mean Proportional. The second (or third) term of a mean proportion. Means. The second and third terms of a proportion. Median. A straight line drawn from the vertex of a triangle to the midpoint of the opposite side. Member. One of the parts of an equation separated by the equal sign. Mil. One thousandth of an inch. Monomial. An expression having only one term. Negative Number. A number to be subtracted, or a number on the number scale to the left of the zero. 206 DICTIONARY OF THE TERMS USED Oblique Angle. Any plane angle not a right angle, straight angle, or perigon. Oblique Triangle. A triangle having all oblique angles. Obtuse Angle. An angle greater than a right angle and less than a straight angle. Octagon. A plane figure having eight sides. Ordinate. The vertical or Y coordinate of a point. Parallel Lines. Straight lines in the same plane that will not meet if extended. Parallelogram. A quadrilateral having its opposite sides parallel. Parallelopiped. A six-sided solid all of whose faces are parallelograms. Pentagon. A plane figure having five sides. Perigon. An angle of 360°. Perimeter. The distance around a plane figure. Perpendicular. At right angles to a line or surface. Phase. Position in a cycle of rotation or oscillation. Place. The position of a figure in a number. Plane. A surface without curvature. Polygon. A plane figure bounded by straight lines. Polyhedral Angle. The solid angle formed by three cr more planes meeting at a point. Polyhedron. A solid bounded by planes. Polynomial. An expression having two or more terms. Positive Number. Ordinary or arithmetical number. Power. Product of equal factors. Prism. A solid whose bases are congruent and parallel and whose sides are parallelograms. Projection. (Of a point on a line.) The foot of the per- pendicular from that point to the line. (Of a line or surface on a line or surface.) The line or surface" join- DICTIONARY OF THE TERMS USED 207 ing the projection of all points of that line or surface, on the second line or surface. Proportion. When two ratios are equal, they form a propor- tion. Protractor. An instrument for drawing and measuring angles- Pyramid. A solid having for its base a polygon and for its sides triangles meeting at a common point. Quadrant. One-fourth of a circle or perigon. Quadrilateral. A plane figure having four sides. Radius. A straight line from the center of a circle to the circumference, or from the center of a sphere to the surface. Ratio. The relation of one quantity to another quantity of the same kind. Rational Quantity. A quantity whose value can be ex- pressed exactly by decimals or fractions. Reciprocal. One divided by a given number. Rectangle. A parallelogram having right angles. Rectilinear. Having straight lines. Reflex Angle. An angle greater than 180°. Regular Polygon. A polygon that has its sides equal and its angles equal. Resultant. The single force replacing two or more forces. Also applies to other vectors. Rhomboid. A parallelogram having its angles oblique and its adjacent sides unequal. Rhombus. A parallelogram having its angles oblique and its sides equal. Right Angle. An angle of 90°. Right Prism. A prism whose sides are perpendicular to the base. 208 DICTIONARY OF THE TERMS USED Right Triangle. A triangle having one right angle. Rim Speed. The speed of a point on the surface of a revolv- ing object. Runner. The sliding glass piece on a slide rule. Scalar Quantity. A quantity which can be expressed com- pletely by giving its magnitude. Sometimes called a scalar. Scalene Triangle. A triangle having no two sides equal. Secant. A straight line passing through a circle. Sector. The portion of a circle between two radii. Segment. The portion of a circle between a chord and the circumference. Similar. Of the same shape. Slide. The central portion of a slide rule. Slide Rule. A calculating instrument. Solid. Any object having definite shape. Sphere. A solid bounded by a curved surface every point of which is equidistant from the center. Square. The second power of a number. Square. A rectangle having equal sides. Square Root. The number which when multiplied by itself equals a given number. Straight Angle. An angle of 180°. Substitution. The process of replacing one quantity by another. Subtend. Cut off. Supplement. The angle which added to a given angle will make 180°. Symmetrical. The same on both sides of a point line, or plane. Tangent. Touching but not cutting. Tetrahedron. A polyhedron having four equal faces. DICTIONARY OF THE TERMS USED 209 Term. An expression whose parts are not separated by the plus or minus sign. Theorem. A statement to be proved. Torque. Forces tending to produce rotation. Transpose. To change a term from one member of an equation to the other. Transversal. A line cutting other lines. Trapezium. A quadrilateral having no two sides parallel. Trapezoid. A quadrilateral having one pair of parallel sides. Triangle. A plane figure having three sides. Variable. A quantity that changes in value. Vector. A line representing a vector quantity. Vector Quantity. A quantity having magnitude and direc- tion or sense. Vertical Angles. If two lines intersect, the opposite angles are vertical angles. RELATIONS BETWEEN THE TRIGONOMETRIC FUNCTIONS Certain relations exist between the functions of angles. These relations are used in finding one function from other functions. The following table is given for reference. The proofs of the relations are given in books more com- plete in the theory of trigonometry: 1. sin 2 z+cos 2 x=l. „ sin x = tan x. cos a: cos x = cot x. sin x 4. sec 2 a;=l+tan 2 x, 5. esc 2 £=1-1- cot 2 x. 6. 1 sin x= -. 7. cos x = 8. tan x = cscx 1 sees" 1 cot X 9. sin (x-\-y) =sin x cos y+cos x sin y. 10. cos (x+y) = cos x cos y— sin x sin y. tan rc+tan y 11. tan (x+?/) = 1 — tanx tan ?/' 211 212 RELATIONS OF TRIGONOMETRIC FUNCTIONS . cot x cot y — 1 12. cot (x-\-y) = — i — • v J cota;+cot y 13. sin (x — y) = sin x cos y —cos a: sin ?/. 14. cos (x— y)= cos z cos 2/+sin x sin y„ . . tan re— tan i/ 15. tan (x — y) = ,— rr — * • v *" 1+tanrc tany . . cot x cot ?y — 1 16. cot (*-v)«-^+^t7' 17. sin 2a; = 2 sin a; cos a:. 18. cos 2a: = cos 2 x — sin 2 jr. 2 tan a; 19. tan 2a; = y^Fx cot 2 a;— 1 20. cot 2a; = -75 — t — • 2 cot x . re /l — cos re 21. si n?2 =±^ 2^~- I +cos re 22. cos re /l — cos a; 23. tang-i^jq^^. re /l+cosre 24. cot 2 =± Vl^cos"x- FORMULAS The following list of formulas have a wide and general use. The student should know each formula to have a useful knowledge of mathematics. 1. tt = 3.1416. 2. C = 2irr. C = circumference of a circle, r = radius. 3. C = — ~ — . C = circumference of ellipse, d\ and (h = long and short diameters. Areas: 4. Square A = a 2 . a = side. 5. Rectangle A = ab. a and b are the dimensions. 6. Parallelogram A = bh. b = base, h = height. 7. Trapezoid A = %h(b+b') . /i = height. b and b' are the parallel sides. 8. Triangle A = \bh. 6 = base, h = height. 9. Triangle A = Vs(s — a)(s— b)(s— c). a, b and c are the sides, s = %(a+b+c). 10. Hexagon A = 2.598s 2 . s = side. 11. Circle A=rr 2 . r = radius. 12. Ellipse A=ir—T— . d\ and rf2 = long and short diam- eters. 13. Surface of sphere A =4tt2. r = radius. Volumes: 14. Cube V = a 3 . a = edge. 15. Rectangular solid V = lwh, I, w, and h are the dimen- sions. 213 214 FORMULAS 16. Rectangular solid, Parallelopiped, Prism or Cylinder. V = bh, 6 = area of the base, ft = height. 17. Cone or Pyramid V = \bh, b = area of base, ft = height. 18. Frustum of a Cone or Pyramid. y = xj l ( -\-b' + Vbb'), 6 and b' are the areas of the bases and ft = height. 19. Sphere F = ^rr\ r = radius. 20. Spherical segment V= \ira [ r 2 + — I . a = altitude, r = radius of the base. Right Triangle: 21. c 2 = a 2 -\-lr. c = hypotenuse, a and b the sides. Q uadratic Equation : -b±\ / b 2 +4ac 22 - X = 2aT -' Trigonometry : <• ■ a b c 23. Law oi sines: sin A sin B sin C 24. Law of cosines a 2 = b 2 + c 2 — 2bc cos A. TABLE 1.— DECIMAL EQUIVALENTS Of Eighths, Sixteenths, Thirty-seconds \\n Sixty-fourths of an Inch .015625 .03*125 .046875 .0625 .078125 •09375 •109375 .1250 . 140625 ■15625 •171875 • 1875 .203125 .21X75 • 234375 .2500 .265625 .28125 .296875 ■ 3125 .328125 •34375 •359375 •3750 ■390625 .40925 .421S75 ■4375 •453125 .46875 .4£4375 .5000 515625 53125 546875 5625 57^125 5<)375 609375 6250 640625 65625 671875 6875 703125 7>875 734375 7500 765625 78125 796875 8125 828125 84375 859375 8750 890625 90625 921875 9375 953125 96875 984375 215 216 TABLE II.— COMMON LOGARITHMS Common Logarithms n OI2 3456789 10 00000 00432 00860 01284 01703 02119 02531 02938 03342 03743 ii 04139 o453 2 04922 05308 05690 06070 06446 06819 0718S o7555 12 07918 08279 08636 0S991 09342 09691 10037 10380 10721 11059 13 "394 11727 12057 123S5 12710 13033 *3354 13672 13988 1430 1 14 14613 14922 15229 15534 15836 16137 1643S 16732 17026 173*9 IS 17609 17898 18184 18469 18752 19033 19312 i959o 19866 20140 iG 20412 206S3 20952 21219 214S4 2174S 22011 22272 22531 22789 17 2304S 23300 23553 23805 24055 24304 2455 1 24797 25042 252S5 18 255 2 7 25768 26007 26245 26482 26717 26951 27184 27416 27646 19 27S75 28103 28330 28556 2S7S0 29003 29226 29447 29667 298SS 20 30103 303=0 30535 30750 30963 3**75 3i3 2 7 3*597 31806 3- OI 5 21 32222 3242S 32634 32S3S 33041 33244 33445 33646 33846 34044 22 34242 34439 34635 34S30 35025 352i3 354i 1 35603 35793 35984 = 3 36173 36361 36549 36736 36922 37107 37291 37475 3765S 37840 24 3S021 3S202 3S382 38561 38739 3S917 39094 39270 39445 39620 25 39794 39967 40140 40312 40483 40654 40S24 40993 41162 4*33° 26 41497 41664 41830 41996 42160 42325 4248S 42651 42813 42975 27 43*3 6 43 2 97 43457 43616 43775 43933 44091 4424S 44404 44560 28 44716 44S71 45025 45179 45332 45484 45637 45788 45939 46090 29 46240 463S9 4653S 466S7 46835 46982 47129 47276 47422 47567 30 47712 47857 48001 48144 48287 4S430 4S572 4S714 48855 48996 31 49*36 49276 49415 49554 49693 49S3 1 49969 50106 5 243 50379 32 50515 50651 50786 50920 51055 51188 51322 5*455 5*587 51720 33 SlSSi 51983 52114 52244 52375 52504 52634 52763 52892 53020 34 53148 53275 53403 53529 53656 53782 53908 54033 54158 54283 35 54407 54531 54654 54777 549oo 55023 55145 55267 55388 55509 36 55630 55751 55871 5599i 56110 56229 56348 56467 56585 56703 37 56S20 56937 57054 57*7* 57287 57403 57519 57634 57749 57864 38 57978 5S092 58206 5 8 32o 58433 58546 58659 5S77 1 588S3 58995 39 59106 59218 59329 59439 5955o 59660 5977o 59879 59988 60097 40 60206 60314 60423 60531 60638 60746 60853 6o959 61066 61172 4i 61278 61384 61490 6iS95 61700 61S05 61909 62014 62118 62221 42 62325 62428 62531 62634 62737 62S39 62941 63043 63*44 63246 43 63347 63448 63548 63649 63749 63849 63949 64048 64147 64246 44 64345 64444 64542 64640 64738 64836 64933 65031 65128 65225 45 653 21 65418 65514 65610 65706 65801 65896 65992 66087 66181 46 66276 66370 66464 66558 66652 66745 66839 66932 67025 67117 47 67210 67302 67394 67486 67578 67669 67761 67852 67943 6S034 48 68124 68215 6S30S 6S395 68485 68574 6S664 6S753 6S842 68931 49 69020 6910S 69197 69285 59373 69461 69548 69636 69723 69S10 50 69897 69984 70070 70157 70243 70329 70415 70501 705S6 70672 51 70757 70842 70927 71012 71096 71181 71265 7*349 7*433 7*5*7 52 71600 716S4 71767 71850 7*933 72016 72099 72181 72263 72346 53 72428 72509 72S9I 72673 72754 72835 72916 72997 73078 73*59 54 73239 7332o 73400 7348o 7356o 73640 737*9 73799 73878 73957 01234 56789 TABLE II.— COMMON LOGARITHMS of Numbers from 000 to 999 217 ■ 01 2 34 56789 55 74°3 6 74i 1 5 74194 74273 7435 1 74429 74507 74586 74663 74741 56 74819 74896 74974 75051 75128 75205 75282 75358 75435 755** 57 75587 75664 75740 758i5 75S91 75967 76042 76118 76193 76268 58 76343 76418 76492 76567 76641 76716 76790 76864 76938 77012 59 77085 77IS9 77232 77305 77379 77452 77525 77597 77670 77743 60 778i5 77887 77960 78032 78104 7 S I7 6 78247 78319 78390 78462 61 78533 78604 78675 78746 78817 7SSSS 78958 79029 79099 79169 62 79 2 /'9 79309 79379 79449 795i8 79588 79657 79727 79796 79865 63 79934 80003 £0072 80140 80209 80277 80346 80414 804S2 80550 64 80618 806S6 80754 80821 80889 80956 81023 81090 81158 81224 65 81291 81358 81425 81491 81558 81624 81690 8*757 81823 81889 66 8i954 82020 82086 82151 82217 82282 82347 82413 82478 82543 67 82607 82672 82737 82802 82866 82930 82995 83059 83*23 83187 68 83251 83315 83378 83442 83506 83569 83632 83696 83759 83822 69 83885 83948 840 1 1 84073 84136 84198 84261 84323 84386 84448 70 84510 84572 84634 84696 84757 84819 84880 84942 85003 85065 71 85126 85187 85248 85309 8537o 85431 85491 85552 85612 85673 72 85733 85794 S5854 85914 85974 86034 86094 86153 £6213 86273 73 86332 863:12 86451 S6510 S6570 86629 86688 86747 86806 86864 74 86923 86982 87040 87099 87157 87216 87274 87332 87390 87448 75 87506 87564 87622 87679 87737 87795 87852 87910 87967 88024 76 8S0S1 S8138 S8195 88252 88309 88366 88423 S8480 S8536 88593 77 88649 8S705 88762 S8818 88874 88930 88986 89042 89098 89*54 78 89209 89265 89321 89376 89432 89487 89542 8 9597 89653 89708 79 89763 89818 89873 89927 89982 90037 90091 90146 90200 90255 80 90309 90363 90417 90472 90526 90580 90634 90687 90741 90795 81 90849 90902 90956 91009 91062 91116 91169 91222 9*275 91328 82 91381 9*434 91487 9*54© 9*593 9*645 91698 9*75* 91803 91855 83 91908 91960 92012 92065 92117 92169 92221 92273 92324 92376 84 92428 924S0 92531 925S3 92634 92686 92737 92788 92840 92891 85 92942 92993 93044 93095 93146 93*97 93247 93298 93349 93399 86 9345° 935oo 9355 1 93601 93651 93702 93752 93S02 93852 93902 87 93952 94002 94052 94101 94iSi 94201 94250 94300 94349 94399 88 94448 94498 94547 94596 94645 94694 94743 94792 94841 94890 89 94939 94988 95036 95oSs 95*34 95182 95231 95279 95328 95376 90 95424 95472 95521 95569 956i7 95665 95713 9576i 95809 95856 91 95904 95952 95999 96047 96095 96142 96190 96237 962S4 96332 92 96379 96426 96473 96520 96567 96614 96661 96708 96755 96802 93 96848 96895 96942 969S8 97035 97081 97128 97*74 97220 97267 94 973*3 97359 97405 974S 1 97497 97543 97589 97635 97681 97727 95 97772 97818 97864 97909 97955 98000 9S046 98091 98137 98182 96 98227 98272 98318 98363 98408 98453 98498 9 S 543 98588 98632 97 9S677 98722 98767 98811 98856 98900 9 8 94S 98989 99034 99078 98 99123 99167 99211 99255 99300 99344 99388 99432 99476 99520 99 99564 99607 99651 99695 99739 99782 99826 99870 999*3 99957 01-3 4 56 789 218 .TABLE III.— LOGARITHMS OP Common Logarithms JLOG TANOF.NT Angle 30 40 50 6o' 13 14 I5 C 16 17 18 i9 23 24 2S C 26 27 28 29 30° 31 32 33 34 35° 36 37 38 39 40' 4i 42 43 44 — 00 2.24192 2.54308 2.71940 2.S4464 2.94195 T. 02162 T. 08914 1. 14780 T.19971 T. 24632 T.28S65 i-3 2 747 i-3 6 336 1.39677 T. 42805 i-4575° I-48534 1.51178 I-53697 T. 56107 1. 58418 1. 60641 1-62785 1.64858 T. 66867 T. 68818 T. 70717 T. 72567 1-74375 1. 76144 1.77877 f- 79579 1. 81252 I.82S99 i.845 2 3 1. 86126 1.87711 T. 89281 T. 90837 1. 92381 T. 93916 1-95444 1.96966 3-46373 2.30888 2.577S8 2.74292 2.S6243 2.95627 1. 03361 1.09947 1. 15688 T. 20782 1-25365 1-29535 I-33365 1.36909 1. 40212 1.4330S 1.46224 T.489S4 T. 51606 I. 54106 T. 56498 I.58794 1. 61004 1-63135 1.65197 1. 67196 1. 69138 r. 71028 1. 72872 f • 74673 t- 76435 f. 78163 1. 79860 f.81528 1.83171 T. 84791 1.86392 1.87974 1. 89541 i^ogs i. 92638 F.94171 f. 95698 1 .97219 1.98737 3.76476 2 .36689 2 . 61009 2.76525 2-87953 2.970:; 1.045 8 T. 10956 f- 16577 1. 21578 T. 26086 i-3 OI 95 1-33974 1-37476 T. 40742 T. 43806 T. 46694 1.49430 T. 52031 I.545 12 1.56887 1. 59168 1. 61364 1.63484 1-65535 1.67524 1.69457 i-7 I 339 L73I75 1.74969 T. 76725 T.7844S I . 80140 1. 81803 I.83442 1-85059 1.86656 1.88236 1. 89801 I-9I353 3.94086 2.41807 2. 64009 2. 78649 2.06581 2.46385 2.66816 2 . S0674 2.91185 9835 8 2.99662 .92894 .94426 •9595 2 .97472 .98989 i". 05666 1.11943 1. 17450 T. 22361 T. 26^97 1.30846 1-3457'' ^•38035 I. 41266 1.44299 1. 47160 1.49872 i-5 2 452 I.549I5 1.57274 I-59540 1. 61722 1.63830 1.65870 1.67850 1.69774 1. 71648 1.73476 1.75264 1. 77015 1.78732 1. 80419 1.82078 1-83713 1-85327 1.06775 1. 12909 T. 18306 I. 23130 T. 27496 I. 31489 i-35!7o I-385S9 T. 41 784 1.44787 1 .47622 i-5°3" 1.52870 I.553I5 1.57658 I.59909 1.62079 1. 64175 1.66204 T. 68174 r . 700S9 f-71955 1-73777 1-75558 2. 16273 2.50527 2.69453 2.82610 2.92716 1 .00930 T. 07858 1-13854 1 . 19146 1.23887 T. 28186 T. 32122 f-35757 £•39136 1.42297, I. 45271 1.45750 r .48080 1 . 48534 2. 24192 2.54308 2.71940 2 . 84464 2.94195 T. 02162 T.0S914 I. 14780 T.19971 1.24632 r. 28065 3-32747 1-36336 1.39677 1.42S05 1. 90061 T.91610 i-93 I 5° T. 94681 r . 96205 1.97725 1.99242 I.77303 1. 79015 1.80697 T. 82352 1.83984 1.85594 T. 87185 1.88759 1.90320 I. 91868 1.93406 1-94935 I-96459 1.97978 1-99495 1.50746 1-53285 1-557" i. 58039 1.60276 3- 62 433 1-64517 1-66537 1.68497 1 . 70404 c . 72262 c. 74077 c- 75852 i- 77591 i. 79297 i. 80975 ".82626 1.51178 f-53697 1. 56107 1. 58418 1 . 60641 T. 62785 T. 64858 T. 66867 i. 68818 i. 70717 1.72567 1-74375 1. 76144 f. 77877 t- 79579 t. 81252 3.82899 1. 84254 I 1.84523 1.85860 1. 86126 1.87448 T.87711 1. 89020 3.90578 1.92125 1. 9366i i-95'9° 1. 96712 1. 98231 1.99747 1.89281 T. 90837 1. 92381 1.93916 1-95444 1 .96966 T. 98484 0.00000 89 88 87 86 8S 84 83 82 81 8o e 79 78 77 76 75° 74 73 72 7i 7o° 69 68 67 66 65° 64 63 62 61 6o° 59 58 57 56 55 ; 54 53 5 2 51 go e 49 48 47 46 45° 60' So' 40 30' Angle L.OU COXA^GliJSX TRIGONOMETRIC FUNCTIONS of Sines and Cosines LOG SINE 219 Angl e o' 10' 20' 30' 40' 50' 60' 45° I.84949 1.85074 1. 85200 1.85324 1.85448 I. 85571 1-85693 44 46 1-85693 T. 85815 1-85936 T. 86056 T. 86176 T. 86295 1.S6413 43 47 1. 86413 1.86530 T. 86647 T. 86763 1.86879 1.86993 1. 87107 42 48 f .87107 T. 87221 J- 87334 1.87446 1-87557 T. 87668 1.87778 41 49 1.S777S 1.87887 1.87996 I. 88105 1. 88212 T. 88319 1.88425 40° 5o° 188425 T. 88531 1.88636 T. 88741 T. 88844 T. 88948 1.89050 39 Si T. 89050 T. 89152 1.89254 I.89354 i- s 9455 I.89554 1-89653 38 52 1-89653 1.89752 1. 89S49 1.89947 1 . 90043 1. 90139 1.90235 37 53 1.90235 i-9°33° T. 90424 1.90518 1 .90611 1.90704 1.90796 36 54 1.90796 I.90887 T. 90978 I. 91069 1.91158 1. 9 1 248 i-9 x 336 35° 55° i-9 J 336 1-91425 T.91512 I-9I599 T. 91686 T. 91772 1-9^857 34 56 1-91857 1. 91942 1.92027 1 .92111 T. 92194 1 .92277 I.92359 33 57 i-9 2 3S9 T. 92441 1 . 92522 1 . 92603 1.92683 L92763 1 .92842 32 58 T. 92842 T. 92921 I-92999 1.93077 I -93 I 54 1.93230 i-933°7 31 59 i-933°7 1.93382 1-93457 !• 93532 1 . 93606 1.93680 1-93753 30° 6o° 1 -93753 1.93826 T- 93898 T. 93970 1. 94041 1.94112 T. 94182 29 61 1. 94182 1.94252 1. 94321 1.94390 T. 94458 1.94526 1-94593 28 62 1-94593 1 .94660 1.94727 1-94793 T. 94858 1.94923 1 .94988 27 63 T.949S8 i-95°5 2 T.95116 I-95I79 1-95242 I - 95304 I-95366 26 64 1-95366 I-95427 1.95488 1-95549 1.95609 1.95668 1.95728 25° 65° i-957 2 8 1.95786 T. 95844 T. 95902 1.95960 T. 96017 T. 96073 24 66 T. 96073 1. 96129 T. 96185 1 .96240 1 .96294 1.96349 1 . 96403 23 67 1.96403 T. 96456 T. 96509 1.96562 T. 96614 1 .96665 1 .96717 22 68 1. 96717 1.96767 T.9681C T.96S6S T. 96917 T. 96966 T. 97015 21 69 1. 97015 T. 97063 1.97111 1-97*59 1.97206 1.97252 1.97299 20 J 7o° 1.97299 1-97344 1 -9739° 1-97435 T. 97479 T.97523 T. 97567 19 7i I-97567 1 . 97610 I-97653 1 .97696 I-9773 8 1.97779 1. 97821 18 72 1. 97821 T. 97861 1.97902 T. 97942 1 . 97982 1 .98021 1. 98060 17 73 T.9S060 T.9S09S 1.98136 1. 98174 T. 982 1 1 1.98248 1.98284 16 74 1.98284 T. 98320 1-98356 i.9 8 39i 1.9S426 T. 98460 1.98494 15° 75° 1.98494 T. 98528 T. 98561 I.98594 1.98627 T. 98659 1.98690 14 76 1.98690 T. 98722 I-98753 1.98783 1-98813 J-y ' ; 4.s T. 98872 13 77 I.98S72 1.98901 1.98930 1.98958 1 .9S9S6 1. 99013 T. 99040 12 78 1.99040 T. 99067 I-99093 1.99119 I-99M5 1. 99170 i-99 J 95 11 79 1.99195 T. 99219 1.99243 1.99267 1 .99290 1-99313 1-99335 IO° 8o° ^•99335 1-99357 1-99379 1.99400 T. 99421 T. 99442 T. 99462 9 81 1.99462 1.99482 1. 99501 T. 99520 I -99539 1-99557 1-99575 8 82 1-99575 1-9959.1 1 .99610 1.99627 T. 99643 T. 99659 T. 99675 7 83 I-99675 T. 99690 1.99705 T. 99720 1-99734 1.99748 1. 99761 6 84 T. 99761 1-99775 I.99787 1.99S00 T. 99812 T. 99823 T. 99834 5° 85° T. 99834 T. 99845 T. 99856 T. 99866 T. 99876 T. 99885 T. 99894 4 86 T. 99894 T. 99903 1.99911 T. 99919 T. 99926 1-99934 1.99940 3 87 1.999+0 1.99947 1-99953 1-99959 T. 99964 T. 99969 1.99974 2 88 1.99974 T. 99978 1.99982 1.99985 T. 99988 T. 99991 1-99993 1 89 1-99993 1-99995 1-99997 1.99998 1-99999 0.00000 . 00000 0° 60' 50' 40' 30' 20' 10' 0' I ^ngle JUOO C05.J.NJS 220 TABLE III.— LOGARITHMS OF Common Logarithms tOG SINE Angle o' 10' 20' 3°' 40' 50' 60' 0° . — °o 3-46373 3.7647S 2 ■ 94° s 4 2.06578 2. 16268 2. 24186 §9 1 2.24186 2.30879 2.3667S 2.41792 2.46366 2.50504 2.54282 88 2 2.54282 2.57757 '2.60973 2 .63968 2.66769 2 . 69400 2. 71880 87 3 3. 71880 2 . 74226 2.76451 2.7S568 2.80585 2.S2513 2.84358 86 4 2.84358 2.86128 2.S7S29 2.89464 2 .91040 2.92561 2 . 94030 85° 5° 2.94030 2.95450 2.96825 2.93157 2.99450 T. 00704 T. 01923 84 6 1. 01923 1. 03109 T. 04262 1.05386 1. 0648 1 1.07548 T. 08589 83 7 1.08589 T. 09606 T. 10599 1.11570 T.12519 I-I3447 1. 14356 82 8 I.I435 6 1. I5-M5 1. 16116 T. 16970 1.17! 37 T.1S628 I-I9433 81 9 T- 19433 1.20223 1.20999 1. 21761 1.22509 1.23244 1.23967 8o° IO° T. 23967 1.24677 T. 25376 1 . 26063 T. 26739 T. 27405 T. 28060 79 11 1. 28060 T. 28705 1.29340 T. 29966 1.30582 1:31189 1. 31788 78 12 1.317SS i-3 2 37 8 1.32960 1-33534 1. 34100 1.3465S. 1.35209 77 13 T. 35209 T. 35752 T. 36289 T. 36819 1 -37341 I-37858 1.38368 76 14 1.38368 I.38871 I.39369 T. 39860 1.40346 1.40S25 1. 41300 75° 15° 1. 41300 1. 41768 T. 42232 T. 42690 i-4.'i43 I-4359I 1.44034 74 16 1.44034 T. 44472 T. 44905 7.45334 I.45758 I.4617S 1.46594 73 17 T. 46594 1.47005 T.47411 T. 47814 T. 48213 1.48607 1.48998 72 18 T. 48998 i-493 s 5 1.49768 T. 50148 1-50523 I.50S96 T. 51264 7i 19 T. 51264 I. 51629 i-5 I 99i i-5 2 35o 1-52705 I-53056 1 • 53405 70° 20° I-53405 I.5375I I.54093 1-54433 T. 54769 i-55 102 1-55433 69 21 1-55433 i.5576i 1.56085 1 . 56408 1.56727 1.57044 1-57358 68 22 7-5735S 1.57669 I-5797 8 1.58284 1.58588 T.5S889 T.5918S 67 23 T. 59188 1.59484 1-59778 1.60070 1.60359 i . 60646 1. 60931 66 24 I. 60931 1.61214 T. 61494 T. 61773 T. 62049 1.62323 1-62595 65° 25° ^.62595 T. 62865 1-63133 1.63398 T. 63662 1.63924 T. 64184 64 26 1. 64184 T. 64442 T. 64698 1.64953 1.65205 1-65456 I-65705 63 27 i.657 5 1.65952 T. 66197 1. 66441 T.666S2 1.66922 1.67161 62 28 T.67161 1-67398 1-67633 T.67S66 I.6S098 T. 68328 1-68557 61 29 T.6S557 T. 68784 1. 69010 1.69234 1.69456 1.69677 1.69897 60 ° 3o° T. 69897 T.70115 T. 70332 I.70547 T. 70761 T. 70973 1.71184 59 3i T.71184 i-7 I 393 1. 71602 1. 71809 T. 72014 T. 72218 1. 72421 58 32 1. 72421 T. 72622 1.72823 T. 73022 T. 73219 T. 73416 1.73611 S7 33 1.73611 1.73805 1-73997 I. 74189 1-74379 1 . 74568 L74756 56 34 1.74756 1-74943 1.75128 I.753I3 1.75496 1.75678 1.75859 55° 35° 1.75859 1.70039 T. 76218 1.76395 T. 76572 T. 76747 I.76922 54 36 1. 76922 T. 77095 1.77268 1.77439 1.77609 1.7777S 1.77946 53 37 1.77946 T.78113 1.7S2S0 1.78445 1 . 78609 1.78772 I.78934 52 38 1.78934 T. 79095 1.79256 I-794I5 1-79573 I-7973I 1.79887 5i 39 1.79887 1 . 80043 T. 80197 1. 80351 1.80504 T. 80656 1.80807 50° 40 T. 80807 T. 80957 T.81106 T. 81254 I. 81402 T. 81549 T. 81694 49 41 1 .81694 1.81839 T. 81983 T. 82126 1.82269 1. 82410 1. 82551 48 42 1. 82551 1. 82691 T. 82830 T. 82968 1. S3 106 1.83242 1.83378 47 43 1.83378 i-835 1 3 T. 83648 T.837S1 7. 83914 1 . 84046 T. 84177 46 44 1. 84177 1.84308 1.84437 1.84566 T. 84694 I.84822 1 . 84949 45° 60' 50' 4c' 30' 20' io' o' . \ngle u )G COS! NE TRIGONOMETRIC FUNCTIONS ol Tangents and Cotangents EOG TANGENT 221 Angl S <>' IO' 20' 30' 40' 50' 60' 45° O.OOOOO 0.00253 0.00505 0.00758 O.OIOII 0.01263 0.01516 44 46 O.OI516 0.01769 0.02022 0.02275 0.02528 0.02781 0.03034 43 47 O.O3O34 0.03288 0.03541 0.03795 0.0404S 0.04302 0.04556 42 48 0.0455 6 0.04810 0.05065 0.05319 0.05574 0.05S29 c. 06084 41 49 0.06084 0.06339 0.06594 0.06S50 0.07106 0.07362 0.07619 40° 5o° 0.07619 0.07875 0.08132 0.0S390 0.08647 0.08905 0.09163 39 5i 0.09163 0.09422 0.096S0 0.09939 0. 10199 0.10459 0. 10719 38 52 0. 10719 0. I09S0 0. 11241 0. 11502 0. 11764 0. 12026 0. 12289 37 53 0. 12289 0.12552 0. 12815 0.13079 0.13344 0. 13608 0.13874 36 54 0.13874 0. 14140 0. 14406 0.14673 0. 14941 0. 15209 O.I5477 35° 55° O.I5477 0.15746 0. 16016 0. 16287 0.16558 0. 16829 0. 17101 34 56 0. 17101 0.17374 0. 1764S 0. 17922 0. 18197 0. 18472 0.18748 33 57 0.18748 0. 19025 0.19303 0.195S1 0. 19860 0. 20140 0. 20421 32 58 0. 20421 0. 20703 2.20985 0. 21268 0.21552 0.21837 0. 22123 31 59 0.22123 0.22409 0.22697 0.22985 0.23275 0.23565 0.23856 30° 60 ° 0.23856 0. 24148 0.24442 0.24736 0.25031 0.25327 0.25625 29 61 0.25625 0.25923 0. 26223 0. 26524 0. 26825 0. 27128 0.27433 28 62 0.27433 0.27738 0.28045 0.28352 0. 28661 0.28972 0.29283 27 63 0. 292S3 0.29596 0.29911 0.30226 0.30543 0.30S62 0.31182 26 64 0.31182 o.3 I S03 0.31826 0.32150 0.32476 0.32804 o.33i33 25° 65° o.33i33 0.33463 0.33796 0.34130 0.34465 0.34803 o.35M2 24 66 0.35142 0.35483 0.35825 0.36170 0.36516 0.36865 0.37215 23 67 0.37215 0.37567 0.37921 0.3827S 0.38636 0.38996 o.39359 22 68 o.39359 0.39724 0.40091 0.40460 0.40832 0.41206 0.41582 21 69 0.41582 0.41961 0.42342 O.42726 o.43 rl 3 0.43502 0.43893 20° 7o° 0.43893 0.44288 0.44685 0.45085 0.45488 0.45894 0.46303 19 7i 0.46303 0.46715 0.47130 0.4754S 0.47969 0.48394 0.48822 l8 72 0.48822 0.49254 0.49689 0. 50128 0.50570 0. 51016 0.51466 17 73 0. 51466 0. 51920 0.52378 0.52840 0.53306 o.53776 0.54250 16 74 0.54250 0.54729 0.55213 0.5S70I 0.56194 0.56692 o.57i95 15° 75° o.57i95 0.57703 0. 58216 0.58734 0.59258 o.59788 0.60323 14 76 0.60323 0.60864 0. 61411 0.61965 0.62524 0.63091 0.63664 13 77 0. 63664 0.64243 0.64830 0.65424 0.66026 0.66635 0.67253 12 78 0.67253 0.67878 0.68511 0.69154 0.69805 0.70465 o.7"35 II 79 0.7H3S 0. 71814 0.72504 0.73203 °-739 I 4 0.74635 0.75368 10° 8o° o.75368 0.76113 0.76870 0.77639 0. 78422 0. 79218 0. 80029 9 81 0.80029 0.80854 0. 81694 0.82550 0.83423 0.84312 0.85220 8 82 0. 85220 0.86146 0. 87091 0.88057 0.89044 0.90053 0.91086 7 83 0. 91086 0.92142 0.93225 o.94334 0.95472 0.96639 0.97838 6 84 0.97838 0.99070 1.00338 1 .01642 1.02987 1.04373 1.05805 5° 85° 1.05805 1.07284 1. 08815 1. 10402 1. 12047 1. 13757 I.I5S36 4 86 I-I5536 1. 17390 1. 19326 1.21351 1-23475 1.25708 1. 28060 3 87 1. 28060 1.30547 i.33 l8 4 I-3599 1 1. 38991 1. 42212 1.45692 2 88 1.45692 1.49473 I.536I5 1. 58193 1.63311 1.69112 1.75808 I 89 1.75808 1.83727 1. 93419 2.05914 2.23524 2.53627 00 1 ° 6o' 50' 40' 30' 20' 10' 0' 4ngle LOG COTAN GENT 222 TABLE IV.— TRIGONOMETRIC FUNCTIONS Natural Sines SINE Angle 0' 10' 20' 30' 4o' So' 60' o° 0.00000 0.00291 0.00582 0.00873 0.01164 0.01454 0.01745 89 z O.01745 0.02036 0.02327 0.02618 0.02908 0.03199 0.03490 88 2 O.03490 0.03781 0.04071 0.04362 0.04653 0.04943 0.05234 87 3 O.05234 0.05524 0.05814 0.06105 0.06395 0.06685 0.06976 86 4 O.06976 0.07266 0.07556 0.0784c 0.08136 0.08426 0.08716 85 5° 0.08716 0.09005 0.09295 0.09^85 0.09874 0.10164 0.10453 84 6 O.IO453 0.10742 0.11031 0. 11320 0.1 1609 0.11898 0.12187 83 7 0.12187 0.12476 0.12764 0.13053 O.I334I 0.13629 0.13917 82 8 O.I39 1 ? 0.14205 O.I4493 0. 14781 0.15069 O.I5356 0.15643 81 9 0.15643 0.1593 1 0. 16218 0.16505 0.16792 0.17078 O.I7365 8o° IO° 0.1736S 0.17651 o.i7937 0. 18224 0.18509 0.18795 0.19081 79 ii 0. 19081 0. 19366 0.19652 O.I9937 0. 20222 0. 20507 0.20791 78 12 0. 20791 0.21076 0. 21360 0.21644 0. 21928 0.22212 0.22495 77 13 0.22495 0.22778 0. 23062 0.23345 0.23627 0. 23910 0.24192 76 14 0.24192 0.24474 0.24756 0.25038 0.25320 0.25601 0.25882 75° 15° 0.25882 0.26163 0.26443 0.26724 0. 27004 0. 27284 0.27564 74 16 0.27564 0.27843 0.28123 0. 28402 0.28680 0.28959 0.29237 73 17 0.29237 0.29515 0.29793 0.30071 0.30348 0.30625 0.30902 72 18 0.30902 o.3"7 8 0.31454 0.31730 0.32006 0.32282 0.32557 71 i9 o.3 2 557 0.32832 0.33106 0.33381 0.33655 0.33929 0.34202 70° 20° 0.34202 o.34475 0.34748 0.35021 0.35293 0.35565 0.35837 69 21 o.35 8 37 0.36108 0.36379 0.36650 0.36921 0.37191 0.37461 68 22 0.37461 0.37730 o.37999 0.38268 0.38537 0.38805 0.39073 67 23 o.39 73 0.39341 0.3960S 0.39875 0.40142 0.40408 0.40674 66 24 0.40674 0.40939 0.41204 0.414C9 0.41734 0.41998 0.42262 65° 25° 0.42262 0.42525 0.42788 0.4305 1 0.43313 0.43575 0.43837 64 26 0.43837 0. 44098 o.44359 0.44620 0.448S0 0.45140 0.45399 63 27 o.45399 0.45658 0.45917 0.46175 0.46433 0.46690 0.46947 62 28 0.46947 0.47204 0.47460 0.47716 0.47971 0.48226 0.48481 61 29 0.48481 o.48735 0.48989 0.49242 0.49495 0.49748 0.50000 6o° 30° 0. 50000 0.50252 0.50503 0.50754 0. 51004 0.51254 O.51504 59 31 0.51504 o.5i753 0. 52002 0.52250 0.52498 0.52745 0.52992 58 32 0.52992 0.53238 0.53484 0.5373° 0.53975 0.54220 0.54464 57 33 0.54464 0.54708 0.5495 1 o.55 T 94 o.55436 0.55678 o.559i9 56 34 0.55919 0.56160 0.56401 0.56641 0.56880 0.57119 o.57358 55° 35° 0.57358 o.57596 0.57833 0.58070 0.58307 0.58543 0.58779 54 36 0.58779 0. 59014 0.59248 0.59482 0.59716 o.59949 0.60182 53 37 0.60182 0.60414 0.60645 0.60876 0.61107 0.61337 0.61566 52 38 0.61566 0.61795 0.62024 0.62251 0.62479 0.62706 0.62932 51 39 0.62932 0.63158 0.63383 0.63608 0.63832 0.64056 0.64279 50° 40° 0.64279 0.64501 0.64723 0.64945 0.65166 0.65386 0.65606 49 41 0. 65606 0.65825 0.66044 0.66262 0.66480 0.66697 0.66913 48 42 0.66913 0.67129 0.67344 0.67559 0.67773 0.67987 0.68200 47 43 0. 68200 0.68412 0.68624 0.68835 0.69046 0.69256 0. 69466 46 44 0.69466 0.69675 0.69883 0. 70091 0. 70298 0.70505 0. 7071 1 45 60' 50' 40' 30' 20' 10' 0' / Lngle CO&UN1 TABLE IV.— TRIGONOMETRIC FUNCTIONS and Cosines 223 SINE Angl e o' io' 20' 30' 40' 5o' 60' 44 45° 0.70711 0. 70916 0. 71121 0.71325 0.71529 0.71732 o.7 J 934 46 0-71934 0. 72136 o.7 2 337 0.72537 0.72737 0.72937 o.73i35 43 47 0-73135 o.73333 o.7353i 0.73728 0.73924 0. 74120 0.74314 42 48 0.743M 0.74509 0.74703 0. 74896 0.75088 0. 75280 o.7547i 4i 49 0.7S47I 0.75661 0-75851 0. 76041 0. 76229 0. 76417 0. 76604 40° go 0. 76604 0. 76791 0.76977 0.77162 0-77347 0.77S3I 0.77715 39 51 0.77715 0.77897 0.78070 0.78261 0. 78442 0.78622 0.78801 38 53 0.7SS01 0.78980 0.7915C 0.79335 0.79512 0. 7968S 0. 79S64 37 53 0.79864 0.80038 0.80212 0.80386 0.80558 0.80730 0. 80902 36 54 0.80902 0.81072 0.81242 0.81412 0.81580 0.81748 0.81915 35° 55° 0.81915 0.82082 0.82248 0.82413 0.82577 0.82741 0.82904 34 56 0.82904 0.83066 0.83228 0.83389 0.83549 0.83708 0.83867 33 37 0.83867 0.84025 0.84182 0.84339 0.84495 0.84650 0.84805 32 58 0.84805 0.84959 0.85112 0.85264 0.85416 0.85567 0.85717 31 59 0.85717 0.85866 0.86015 0.86163 0.S6310 0.86457 0.86603 30° 60 ° 0.86603 0.86748 0.86892 0.87036 0.87178 0.87321 0.87462 29 61 0.87462 0.87603 0.87743 0.878S2 0.88020 0.88158 0.88295 28 62 0.88295 0.88431 0.88566 0.88701 0.88835 0.88968 0. 89101 27 63 0.89101 0.89232 0.89363 0.89493 0.89623 0.89752 0.89879 26 64 0.89879 0.90007 0.90133 0.90259 0.90383 0.90507 0.90631 25° 65° 0.90631 0.90753 0.90875 0.90996 0.91116 0.91236 0.91355 24 66 0.91355 0.91472 0.91590 0. 91706 0.91822 0.91936 0.92050 23 67 0. 92050 0.92164 0.92276 0.92388 0.92499 0.92609 0.92718 22 68 0.92718 0.92827 0.92935 0.93042 0.93148 0.93253 0-93358 21 69 0.93358 0.93462 0.93565 0.93667 0.93769 0.93869 0.93969 20° 70° 0.93969 0.94068 0.94167 0.94264 0.94361 0.94457 0-9455 2 19 7i 0.9455 2 0.94646 0.94740 0.94832 0.94924 0.95015 0.95106 18 72 0.95106 0.95195 0.95284 0.95372 0.95459 0.95545 0.95630 17 73 0.95630 0.95715 0-95799 0.95882 0.95964 0. 96046 0. 96126 16 74 0.96126 0.96206 0.96285 0.96363 0.96440 0.96517 0.96593 15° 75° 0.96593 0.96667 0.96742 0.96815 0.96887 0.96959 0.97030 14 76 0.97030 0.97100 0.97169 0.97237 0.97304 0.97371 o.97437 13 77 o-97437 0.97502 0.97566 0.97630 0.97692 0.97754 0.97815 12 78 0.97815 0.97875 0.97934 0.97992 0.9S050 0.98107 0.98163 11 79 0.98163 0.98218 0.98272 0.98325 0.98378 0.98430 0.98481 10° 8o° 0.98481 0-9853 1 0.98580 0.98629 0.98676 0.98723 0.98769 9 81 0.98769 0.98814 0.9885S 0.98902 0.98944 0.98986 0.99027 8 82 0.99027 0.99067 0.' 99 106 0.99144 0. 99182 0.99219 0.99255 7 83 0.99255 0.99290 0.99324 0.99357 0.99390 0.99421 0.99452 6 84 0.99452 0.99482 0.99511 0.99540 0.99567 0-99594 0.99619 5° 85 0. 99619 0.99644 0.99668 0.99692 0.99714 0.99736 0.99756 4 86 0.99756 0.99776 0.99795 0.99813 0.99831 0.99847 0.99863 3 87 0.99863 0.99878 0.99892 0.99905 0.99917 0.99929 o.99939 2 88 0.99939 0,99949 0.99958 0.99966 0.99973 0.99979 0.99985 I 89 0.99985 0.99989 o.99993 0.99996 0.99998 1 . 00000 r . 00000 o° 60' 50' 40' 30' 20' 10' 0' A ogle COSJLNE 224 TABLE IV.— TRIGONOMETRIC FUNCTIONS Natural Tangents TANGENT Angle o' io' 20' 30' 40' 50' 6o' 89 0° 0.00000 0.00291 0.00582 0.00873 0.01164 0.01455 0.01746 i 0.01746 0.02036 0.02328 0.02619 0.02910 0.03201 0.03492 88 a 0.03492 0.03783 0.04075 0.04366 0.04658 0.04949 0.05241 87 3 0.05241 0.05533 0.05S24 0.06116 0.0640S 0.06700 0.06993 86 4 0.06993 0.07285 0.0757SJ 0.07870 0.08163 0.08456 0.08749 85° 5° 0.08749 0.09042 0.09335 0.09629 0.09923 0. 10216 0. 10510 84 6 0. 10510 0. 10805 0. 11099 0.11394 0.1168S 0.11983 0. 12278 83 7 0. 12278 0.12574 0. 12869 0.13165 0.13461 O.I3758 0. 14054 82 8 0.14054 0.1435* 0. 14648 O.I4945 0.15243 0.15540 0.15838 81 9 0.15838 0.16137 0.16435 0.16734 0.17033 0.17333 0.17633 8o° 10° o.i7 6 33 0.17933 0.18233 0.18534 0.18835 0.19136 0.19438 79 ii 0.19438 0. 19740 0. 20042 0.20345 0. 20648 0.20952 0.21256 78 12 0.2125' 0. 21560 0. 21S64 0. 22169 0.22475 0. 22781 0.23087 77 13 0.23087 0.23393 0. 23700 0. 24008 0.24316 0.24624 0.24933 76 14 0.24933 0. 25242 0.25552 0.25862 0.26172 0.26483 0.26795 75° 15° 0.26795 0.27107 0.27419 0.27732 0.28046 0.28360 0.28675 74 16 0.28675 0.28990 0.29305 0.29621 0.29938 0.30255 o.30573 73 i7 0.30573 0.30S91 0.31210 0. 3*530 0.31850 0.32171 0.32492 72 18 0.32492 0.32814 0. 33*36 0.33460 0.33783 0. 3410S o.34433 71 i9 0.34433 o.34758 0.350 8 5 0.354* 2 o.35740 0.36068 0.36397 70° 20° o.3 6 397 0.36727 0.37057 0.37388 0.37720 0.38053 0.38386 69 21 0.38386 0.38721 0.39055 o.3939i 0.39727 0.40065 0.40403 68 22 0.40403 0.40741 0.41081 0.41421 0.41763 0.42105 0.42447 67 23 0.42447 0.42791 0. 43*36 0.43481 0.43828 0.44I75 0.44523 66 24 o.445 2 3 0.44872 0.45222 0-45573 0.45924 0.46277 0.46631 6S° 25° 0.46631 0.46985 o.4734i 0.47698 0.48055 0.48414 0.48773 64 26 0.48773 0.49134 0.49495 0.49S5S 0. 50222 0.50587 0.50953 63 27 o.5 953 0.51320 0.516S8 0.52057 0.52427 0.52798 0.53*7* 62 28 0.53I7 1 o.53545 0.53920 0.54296 0.54673 o.5505* o.5543i 61 29 0.55431 0.55812 0.56194 0.56577 0.56962 0.57348 o-57735 6o° 30° o.57735 0.58124 0.58513 0.58905 0.59297 0.59691 0. 60086 59 31 0.600S6 0.604S3 0.60881 0.612S0 0.61681 0.62083 0.62487 58 32 0.62487 0.C2S92 0.63299 0.63707 0.64117 0.64528 0.64941 57 33 3.64941 0.65355 0.65771 0.66189 0.66608 0.67028 0.6745* 56 34 0.67451 0.67S75 0.6S301 0.68728 0.69157 0.695S8 0.70021 55° 35° 0. 70021 0.70455 0.70S91 0.7*329 0.71769 0. 72211 0.72654 54 36 0.72654 0. 73100 0.73547 0.73996 0.74447 0.74900 0.75355 53 37 o.75355 0.75S12 0.76272 0.76733 0.7719G 0. 77661 0. 78129 52 38 0.78129 0.78598 0. 79070 o.79544 0.80020 0. 80498 0.80978 5* 39 0.80978 0.81461 0.81946 0.S2434 0.82923 0.83415 0.83910 50° 40° 0.83910 0.84407 0. 34906 0.85408 0.85912 0.86419 0.86929 49 41 0.86929 0.S7441 0.87955 0.88473 0.88992 0.89515 0.90040 48 42 0.90040 0.90569 0.91099 0.91633 0. 92170 0.92709 0.93252 47 43 0.93252 0.93797 0.94345 0.94896 0-9545* 0.96008 0.96569 46 44 0.96569 0. 97*33 0.97700 0.98270 0.98843 0.99420 1 . 00000 45° 60' 50' 40' 30' 20' 10' o' / ingle COIANGENX TABLE IV.- and Cotangents -TRIGONOMETRIC FUNCTIONS 225 TANGENT Angle 0' io' 20' 30' 4°' 50' 60' 45° 1 1. 00c 00 1.00583 1.01170 1.01761 1-02355 1.02952 1-03553 44 46 1-03553 1. 04158 1.04766 i-°5378| 1.05994 1. 06613 1.07237 43 47 1.07237 1.07864 1.08496 I-09I3 1 ! 1.09770 1 . 10414 1 . 11061 42 48 1. 11061 1.11713 1. 12369 1. 13029 1. 13694 1. 14363 1-15037 41 49 I.I5037 I.IS7I5 1. 16398 1. 17085 i- J 7777 1. 18474 I.I9I75 40° 50° I.I9I7S 1. 19882 1-20593 1. 21310 1. 22031 1.22758 1.23490 39 5i 1.23490 1.24227 1.24969 1. 25717 1. 26471 1.27230 1.27994 38 52 1.27994 1.28764 1. 29541 1.30323 1.31110 1. 31904 1.32704 37 53 1.32704 1.335" 1.34323 I.35M2 L35968 1.36800 1-37638 36 54 1.37638 1.38484 I-3933 6 1. 40195 1.41061 I.4I934 1. 42815 35° 55° 1. 42815 L43703 1.44598 I.455 01 1. 4641 1 1-47330 1.48256 34 56 1.48256 1. 49190 i-5 OI 33 1. 51084 i.5 2 °43 1. 53010 I-53987 33 57 1.53987 I-5497 2 1.55966 1.56969 1. 57981 1.59002 1.60033 32 58 1.60033 1. 61074 1. 62125 1. 63185 1.64256 1.65337 1.66428 31 59 1.66428 I.67530 1.68643 1.69766 1 . 70901 1. 72047 1-73205 30 60 ° 1.73205 1-74375 1.75556 1.76749 1-77955 1. 79174 1 . 80405 29 61 1 . 80405 1. 81649 1.82906 1. 84177 1.85462 1.86760 1.88073 28 62 1.88073 1 . 89400 1. 90741 1.92098 1-9347° 1.94858 1. 96261 27 63 1. 96261 1.97680 1.99116 2.00569 2.02039 2.03526 2.05030 26 64 2.05030 2.06553 2.08094 2.09654 2.11233 2. 12832 2.14451 25° 65° 2.14451 2. 16090 2.17749 2.19430 2. 21132 2. 22857 2. 24604 24 66 2. 24604 2.26374 2.28167 2. 299S4 2.31826 2.33693 2-35585 23 67 2.355 8 5 2.37504 2-39449 2.41423 2.43422 2.4545 1 2.47509 22 68 2.47509 2-49597 2-5 I 7 I 5 2.53 C6 5 2. 56046 2. 5S261 2. 60509 21 69 2.60509 2.62791 2.65109 2.67462 2.69853 2. 72281 2.74748 20° 70° 2.74748 2.77 2 54 2. 79802 2.82391 2.85023 2.87700 2. 90421 19 71 2.90421 2.93189 2.96004 2.98S69 3.01783 3-04749 3-0776S 18 72 3.07768 3.10842 3- I 397 2 3- I 7 I 59 3. 20406 3-237I4 3.27085 17 73 3- 2 7oSs 3-30521 3-34023 3-37594 3-41236 3-4495 1 3-48741 16 74 3-48741 3.52609 3.56557 3.60588 3-64705 3.68909 3-73205 15° 75° 3.73205 3-77595 3.82083 3.86671 3-9 J 364 3.96165 4.01078 14 76 4.01078 4.06107 4. 11256 4-16530 4.21933 4.27471 4-33J48 13 77 4.33148 4-3 8 969 4-44942 4-5 IQ 7i 4.57363 4.63825 4.70463 12 78 4-70463 4. 77286 4.84300 4-9 I 5 l6 4.98940 5.06584 5-14455 11 79 5-14455 5.22566 5.30928 5-39552 5- 4845 J 5.57638 5.67128 10° 8o° 5.67128 5-76937 5.87080 5-97576 6.08444 6.19703 6.31375 9 81 6.313/5 6.43484 6.56055 6.69116 6. 82694 6.96823 7-"S37 8 82 7- "537 7.26873 7.42871 7-59575 7.77035 7-95302 8.14435 7 83 8.14435 8.34496 8-55555 8.77689 9.00983 9. 2 5530 9.5!436 6 84 9-5I436 9.78817 10.0780 10.3854 10.7119 11.0594 n.43 01 5° 85° n. 43 01 11.8262 12.2505 12.7062 13.1969 13.7267 14-3007 4 86 14.3007 14.9244 15.6048 16.3499 17.1693 18.0750 19.0811 3 87 19.081 1 20. 2056 21.4704 22.9038 24.5418 26.4316 28.6363 2 88 28.6363 31.2416 34.3678 38.1885 42.9641 49.1039 57.2900 I 69 57.2900 68.7501 8S-9398 114-589 171-885 343-774 00 0° 60' 50' 40' 30' 20' 10' 0' J ^ngle COTANGENT INDEX Abscissa, 114 Addition (algebraic), 50 — of vectors, 183 Angle, 119 — ■ complementary, 120 — from three sides given, 174 - phase, 180 - right, 119 - straight, 119 — supplementary, 120 — theorems, 120 Antilogarithm, 137 Area of a triangle, 175 Armature winding, 46 Axis, 110, 113 Base, 29 Characteristic, 132 Checking an equation, 3 Circle, 26, 125 Circular mil, 40 Coefficient, 30 Cologarithm, 140 Composition of forces, 184 Complementary angles, 120 Condenser, 37, 93 Conductor in magnetic field, 154 Congruent triangles, 121 Coordinates, 113 Cosecant, 150 Cosine, 150 — law of, 172 Cotangent, 150 Current, alternating, 188 Cutting speed, 79 Degree, 120 Depression, 161 Direct proportion, 76 Division, 64 — by slide rule, 18 Efficiency, 71 Electrical applications of trigo- nometry, 177 Elevation, 161 Equation, 1 — - checking of, 3 — fractional, 5 — graph of, 115 — quadratic, 95 — simultaneous, 104 Evaluation, 29, 31 — of formulas, 22, 32 Exponent, 31 — in division, 65 — in multiplication, 59 227 228 INDEX Exponent, unknown, 14.") Extremes, 71 Field intensity, 43 Flux, 45 Force on a magnet, 38, 87 Formulas, 32 — electrical, 33, 86 — evaluation of, 22, '■'>- — quadratic equation, 100 — square and square root, 25, 86 , — transposition of, 37 Fractional equations, 5 Fractions reduced to a given denominator, 75 Fulcrum, 56 Functions, 150 — as lines, 164 — logs of, 160 — of angles greater than 90°, 1(55 ( tears, 79, 83 General number, 29 Geometry, 119 Graph, 109 — of an equation, 115 — of simultaneous equations, 117 Grouping, 31 Heat in electric current, 30 Horse power, 88 Impedance, 192 Inductance coil, 92 Instantaneous voltage, 179 Interpolation, 135, 158 Inverse proportion, 76 Isosceles triangle, 122 Leverage, 56 — law of, 60 Lever arm, 56 Lines of force, 179 Location of p< ii ts, 114 Logarithms, 131 Magnetic circuit, 45 Mantissa, 132 Means, 74 Member, 7 Mil, 40 Minute, 120 Monomial, 60 - division of, 59 — multiplication of, 65 Multiplication, "<' — by slide rule, 19 Murray loop. 91 Negative numbers, 49 Numbers, negative, 49 positive, 49 Oblique triangle, 169 Ohm's law, 34 Ordinate, 114 Origin, 114 Parallel lines, 123 Parallelogram, 125 Parenthesis, 31 Percentage, 69 Perigon, 120 Phase angle, 180 Polynomials, 62, 63 Positive numbers, 49 Power, 30, 142 INDEX 229 Pow T factor, 191 - in A. C. circuit, 189 — in D. C. circuit, 86 Projection, 177 Proportion, 23, 68, 72, 127 direct, 76 — inverse, 76 Protractor, 120 Pulleys, 79 — speed of, 82 Quadratic equation, 95 — by completing square, 98 — by formula, 100 Quadrilaterals, 124 Ratio, 68, 127 — given, 71 Rectangle, 125 Resistances in parallel, 39 — in series, 33 Resultant, 184 Rhombus, 125 Right angle, 119 — triangle, 101, 128 — triangulation, 148 Rim speed, 79 Roots of numbers, 143 Rotating loop, 179 Scalar quantity, 182 Secant, 150 Second, 120 Separating in a given ratio, 71 Signs, 29 Similar triangles, 128 Simultaneous equations,. 104 — by addition and subtraction, 104 Simultaneous equation, by sub- stitution, 107 — graph of, 117 Sine, 150 - law of, 169 Slide rule, 13, 131 — division by, 18 — general suggestions for, 26 — locating decimal point, 20 — multiplication by, 19 — scales, 14, 24 — types of, 13 — use in special computations, 22, 23, 25, 26, 27 Square, 125 Square and square root formulas, 86 Straight angle, 119 Subtraction (algebraic), .50 — of vectors, 185 Supplementary angles, 120 Surface speed, 79 Tangent, 150 — law of, 174 Temperature change of resistance, 42 Term, 7 Transformation of formulas, 37 Transposition, 7 Trapezium, 124 Trapezoid, 124 Triangle, 120 — area of, 175 — - congruent, 121 — isosceles, 122 — oblique, 169 — right, 101, 128 — similar, 128 230 INDEX Triangulation, right, 148 Trinomial square, 97 Turning moment, 56 Unknown exponent, 145 Vectors, 182 — difference of, 185 Vectors, sum of, 18.3 Vertex, 119 Voltage of D. C. generator, 90 Wheatstone bridge, 71 Work done by an electric current, 35 UN ,VERS1TY OF CAL.FOKN.A LIBRARY Los Angeles Thisb ooK i sC,u e on, h e 1 a.,-a.e... m p.— • University of California. Los Anaeles L 005 839 144 2 AA 000 790 291 9 SC RN BRANCH, UNIVERSITY OF CALIFORNIA, LLBRARY, 'LOS ANGELES, CALIF.