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THE UNIVERSITY 
 OF ILLINOIS 
 
 LIBRARY 
 
 Sect 
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 Latest Date stamped below. A 
 charge is made on all overdue 
 
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 8057-S 
 

 
THE INTERPRETATION 
 
 OF 
 
 MATHEMATICAL FORMULA 
 
 BY 
 EDWIN J. HOUSTON, Pu. D. (Princeton), 
 AND 
 
 ARTHUR E. KENNELLY, Sc. D. 
 
 NEW YORK 
 AMERICAN TECHNICAL BOOK CO. 
 45 Vesty STREET 
 
 1898 
 

 
 EDWIN J. HOUSTON anp ARTHUR E. 
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 Maths res. 
 
 PREFACE. 
 
 
 
 Ir is wonderful how much is capable 
 of being expressed by a mathematical 
 formula, and how little mathematical 
 proficiency is necessary for its inter. 
 pretation. Nevertheless, it 1s commonly 
 believed that a long course of mathe- 
 matical training is essential to an inter. 
 pretation of such formule as are found in 
 ordinary technological text-books, The 
 authors have endeavored, in this little 
 book, to show that this is fallacious, 
 that, on the contrary, a mere knowledge 
 of arithmetic, as a preparatory training 
 to a perusal of this book, will give to 
 the student all the insight that is needed 
 
 ili 
 
 344820 
 
lv PREFACE. 
 
 to understand applied mathematical 
 formule. The authors of course do not 
 claim, however, that those who have read 
 this book thereby become expert mathe- 
 
 maticians. 
 
 PHILADELPHIA, January, 1898. 
 
CONTENTS. 
 
 CHAPTER PAGE 
 I. ADDITION, ; : ‘ : : 1 
 II. Susrraction, : ‘ , 14 
 II. Muvtrirrication, . ‘ ; BAe gat 
 LVowIVvISIion, —. A A : : 34 
 V. Invontutrion. Powers, . : Stes 
 Wie voturion. Roors, . ‘ ; 61 
 VII. Equations, ; ‘ : é ans (6 
 VIII. Loagarirunms, ; ; : i 78 
 IX. TRIGONOMETRY, : : : mes Yi 
 X. Hypersonic TRIGONOMETRICAL 
 Functions, . ; i A 128 
 XI. DirrereEntTIAL CaLcutus, 5 kon 
 XII. Inrrerart Carcutys, ‘ : 165 
 XIII. DeErtTERMINANTs, d : ; . 193 
 
 XIV. Synopsis or Sympots CoMMONLY 
 Founp IN MATHEMATICAL For- 
 
 MULA, ‘ r ; : - 2038 
 

 
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THE INTERPRETATION OF MATHE- 
 MATICAL FORMULA, 
 
 
 
 ALGHBRA. 
 
 CHAPTER I. 
 
 ADDITION. 
 
 Algebra is that branch of mathematics 
 which treats of the properties of numbers 
 and their general relations by means of 
 symbols. It may, therefore, be regarded 
 as a system of generalized arithmetic. 
 The principal operations dealt with in 
 algebra are addition, subtraction, “multi- 
 plication, division, involution, evolution, 
 
2, THE INTERPRETATION OF 
 
 and the solution of the equations involv- 
 ing these operations. We shall proceed 
 to consider these operations in order. 
 
 In arithmetic, if we add two numbers 
 together, we .obtain their sum; as, for 
 example, when we say that 5, added to 
 7, gives a total of 12; or, as we may 
 express it, the sum of 5 and 7 is equal 
 to 12; which, again, may be expressed 
 symbolically,5 +7 = 12. Here the sign 
 (+) is called the plus sign, and indicates 
 the operation of addition, while the sign 
 (=) 1s called the equality sign, and in- 
 dicates the condition of equality between 
 the two things which it connects. 
 
 The expression 
 
 5+7=19 (1) 
 
 is called an equation, in which the left- 
 hand member consists of two terms; 
 namely, 5 and 7, while the right-hand 
 
MATHEMATICAL FORMULA. 3 
 
 member consists of a single term; namely, 
 12. The equation is read thus: 
 
 Five plus seven is equal to twelve. 
 
 We may extend the terms of an equation 
 to any desirable extent. Thus we may 
 write the following equations which are 
 obviously true, 
 
 $8+3+6= 12, (2) 
 or o+t38+6=9- 3, (3) 
 or 4+34+2+1+4=103 (4) 
 
 The number of terms on either side of 
 the equation may be any whatever. All 
 that is necessary is that the sum of the 
 terms in the left-hand member should be 
 equal to the sum of the terms in the 
 right-hand member. 
 
 The four preceding equations deal with 
 terms. which are all simple numbers or 
 definite numerical quantities. We may, 
 
4 THE INTERPRETATION OF 
 
 however, extend the same reasoning sym- 
 bolically. Thus, in the equation 
 
 atb=ce (5) 
 
 we have a statement that the sum of two 
 quantities on the left-hand side, which are 
 represented respectively by the letters a 
 and 6, is equal to the quantity on the 
 right-hand side, represented by the letter 
 c. If we make a, equal to 5, and 6, equal 
 to 7, we reproduce equation (1), and we 
 are compelled to make ¢, equal to 12. In 
 other words, giving definite numerical 
 values to the terms on the left-hand side 
 of the equation determines the value of 
 the symbol on the right-hand side. There 
 is this difference, however, that in an 
 arithmetical equation such as appears 1n 
 equation (1), 
 
 5+7 = 19, 
 
MATHEMATICAL FORMULA. 5 
 
 we are dealing with numbers only, while 
 in the case of a corresponding algebraic 
 equation, 
 
 atb=<e, 
 
 we are not restricted to numerical quan- 
 tities, because although a, b, and c, may 
 stand for the numbers 5, 7, and 12, respec- 
 tively, and thus reproduce the arithmetical 
 equation, they may also stand for other 
 magnitudes which are not merely numeri- 
 cal. For example, if a man, weighing a 
 pounds, is suspended from a rope, and the 
 man holds in his hand a weight of 6 
 pounds, then the total weight which the 
 rope has to sustain is @ + 6 pounds, and 
 we may express this in the form of the 
 equation 
 
 c=a+b pounds, (6) 
 which is the same as the equation 
 
 a+tb=e pounds. (7) 
 
6 THE INTERPRETATION OF 
 
 Thus, if the man weighs 150 pounds, then 
 a = 150, and if the weight he carries in 
 his hand is 10 pounds, then 6 = 10, and 
 the total weight on the rope 
 
 G2 100 =e) 
 =yt Hf 8, pounds. 
 
 Here the quantities which are considered 
 in the equation are the magnitudes of cer- 
 tain weights, and these magnitudes are 
 treated numerically, through the use of the 
 unit of weight, or a pound, by which they 
 are compared numerically. Consequently, 
 while an arithmetical equation considers 
 equality between numbers, an algebraic 
 equation may consider equality between 
 any kind of magnitude, from the brilliiancy 
 of a fixed star to the area of a piece of 
 land. In all cases, however, the same 
 kind of magnitude only can be compared 
 in an equation; so that if the left-hand 
 
MATHEMATICAL FORMULZ. 7 
 
 member a + 6, of equation (7) represents 
 two magnitudes expressed in pounds, the 
 right-hand member c, must also represent 
 'a magnitude expressed in pounds. 
 Moreover, in an arithmetical equation, 
 such as 
 5 i nae, 
 
 the terms are all fixed in value, and the 
 equation is susceptible of but one inter- . 
 pretation; namely, that the sum of 5 and 
 7 is equal to 12; but in an algebraic 
 equation, 
 
 at+b=e, 
 
 the fact that the terms are symbolical, 
 enables them to represent any desired 
 numerical values; the only requisite 
 condition being that c¢, shall be equal to 
 the sum of the values of @ and 4, or that 
 the equality expressed by the equation 
 shall subsist, Thus, we have seen: that 
 
8 THE INTERPRETATION OF 
 
 if the weight a, of the man suspended by 
 the rope was 150 pounds, and the weight 4, 
 which he carried, was 10 pounds, then ¢, 
 the tension in pounds weight, supported 
 by the rope, 1s 160 pounds; but the 
 equation would also correctly represent 
 a case in which the man weighed 230 
 pounds, and the weight he carried 50 
 pounds; but in this case the total tension 
 on the rope would be 
 
 230 + 50 = 280 pounds. 
 
 It is evident, therefore, that the algebraic 
 equation, 
 
 at+tb=e, 
 
 is susceptible of an infinite variety of 
 interpretations, according to the values 
 which are given to the terms, and is, 
 therefore, of a much more general nature 
 
MATHEMATICAL FORMULA. 9 
 
 than the simple type of arithmetical 
 equation, 
 5+7=12, 
 
 to which it corresponds. 
 
 An algebraic equation involving the 
 operation of addition may extend to any 
 number of terms, just as we have seen an 
 arithmetical equation may similarly extend. 
 Thus, suppose that, at a railway station, a 
 truck is loaded with a number of boxes, 
 and that their individual weights are repre- 
 sented by the symbols a, 4, ¢, d, e, f and g, 
 respectively. Then, if we expressed the 
 total weight upon the truck by the letter w, 
 we obviously obtain the equation 
 
 w=at+b+e+d+e+ f+ pounds. (8) 
 
 This is a symbolic method of making the 
 following statement: The total weight 
 upon the truck of the seven boxes with 
 
10 THE INTERPRETATION OF 
 
 which it is loaded, is equal to the sum of 
 the weights of all these boxes. 
 
 It is evident that equation (8) may repre- 
 sent an infinite variety of arithmetical equa- 
 tions, each of which would be true in its own 
 particular case. We have only to substitute 
 for the symbols a, 4, ¢, d, é, f and g, their 
 proper numerical values, and the value of 
 w, their total weight, then follows from 
 the equation. If the weights of the indi- 
 vidual boxes are expressed in pounds, 2, 
 will be determined in pounds. If the 
 weights of the individual boxes are ex- 
 pressed in kilogrammes, w, will be deter- 
 mined in kilogrammes, and so on for any 
 unit of weight. Equation (8) is, there- 
 fore, an equation connecting gravitational 
 magnitudes, but it is evident that this 
 equation is capable of representing any 
 summation relation between any kind of 
 magnitude whatever, For example, if we 
 
MATHEMATICAL FORMUL. dah 
 
 form an equation for the total amount of 
 rainfall m a week, and represent this total 
 by the letter w, where w,is a quantity 
 expressed in inches of water evenly dis- 
 tributed over a level surface; then if 
 a, b,c, d, 6, f and g, represent the rainfall in 
 inches, occurring on each of the seven days 
 of the week respectively, the equation (8) 
 is no longer an equation between gravita- 
 tional magnitudes, but an equation between 
 rainfall magnitudes, and is susceptible 
 again of infinite variety of interpretations. 
 It is evident that the same algebraic equa- 
 tion would be true for any week in the 
 year, but the actual numerical solution of 
 the equation, obtained by substitution in 
 the terms on the right-hand side, would, 
 probably, vary considerably from week to 
 week. 
 
 Similarly, if instead of taking the weekly 
 rainfall, we desired to take the monthly 
 
12 THE INTERPRETATION OF 
 
 rainfall, we might construct a similar equa- 
 tion of 30 or 31 terms, according to the 
 month selected, and make their sum still 
 equal to w; but the quantity represented 
 by w, would no longer be a weekly rain- 
 fall, but a monthly rainfall. Again, we 
 might make an equation of 365 terms, each 
 term representing the rainfall during one 
 of the days of the year, and equate the 
 whole series to w. Here w, would be no 
 longer a weekly rainfall, but a yearly or 
 annual rainfall. A number of the terms 
 on the right-hand side might be numerically 
 equal to zero, representing the fact that on 
 those days there was no rainfall; the other 
 terms would be numerically expressed in 
 inches and fractions of an inch. In such 
 eases, if the letters of the alphabet are 
 not sufficiently numerous to supply all the 
 terms, the letters of the Greek alphabet, or 
 the German alphabet, may be used, or 
 
MATHEMATICAL FORMULA. icy 
 
 capital letters may be employed, or 
 suffixes may be used. Thus, equation (8) 
 might be written if desired 
 
 Pera Ps te A eB boa? +O}. 
 
 Here the first two terms on the right-hand 
 side are the letters Alpha and Beta, of the 
 Greek alphabet, the third term is the letter 
 d, of the German alphabet, the fourth and 
 fifth terms are English capitals, while the 
 sixth and seventh terms are @ prime, and 0 
 prime. It is not, of course, usual to adopt a 
 heterogeneous symbolism of this character, 
 and, as a rule, letters of the English alpha- 
 bet, either small, capital, or suffixed, are 
 the most generally employed. 
 
CHAPTER IL. 
 SUBTRACTION. 
 
 Ir we diminish the value of a certain 
 number, say 12, by another similar number, 
 say 5; or, as it is generally called, if we sub- 
 tract 5 from 12, we obtain the difference 
 7. This may be expressed in language by 
 the following equation: 
 
 Twelve less five is equal to seven. 
 Or, in numbers, 
 12-5 =%. (1) 
 
 Here the sign (—) is called the minus sign, 
 and indicates that the quantity which fol- 
 lows it is to be subtracted from the quan- 
 tity with which it is associated. 
 
 14 
 
MATHEMATICAL FORMULA. 15 
 
 This equation may be generalized alge- 
 braically in the following manner : 
 
 a—b=e, (2) 
 
 which makes the following statement: If 
 from the quantity a, we subtract the quan- 
 tity 6, the remainder will be the quantity c. 
 
 In the early stages of arithmetical 
 science, subtraction could only be per- 
 formed when the number to be subtracted 
 was less than the number from which the 
 subtraction was to be made. ‘Thus, it was 
 considered impossible to subtract six from 
 five. Later development, however, showed 
 ‘that by an extension in meaning of the 
 term subtraction, such an operation could 
 be readily performed, if it was considered 
 that the remainder was negative, so that the 
 equation 
 
 5-6=-1 
 
16 THE INTERPRETATION OF 
 
 represents that if six be taken from five, 
 the remainder is minus one. Or, equation 
 (2) holds good whether 4, is greater than, 
 equal to, or less than a. For example, sup- 
 pose that a, represents a bank account of 
 five hundred dollars, and suppose that 8, 
 represents a draft made on the account. 
 If 4, 1s less than a, the remainder ec, will be 
 a positive balance in dollars at the bank. 
 For example if @ = $500, and 6 = $300, ¢, 
 will be $200, but if 4, is greater than a, say 
 $550, then the account at the bank will be 
 overdrawn, and will leave a negative bal- 
 ance. Thus, 
 
 500 — 550 = — 50, 
 
 so that the account will owe $50 to the bank. 
 
 Any quantity to which the minus sign 
 is prefixed may, therefore, be considered as 
 a negative quantity, and from this point of 
 view subtraction is only an extension of 
 
MATHEMATICAL FORMULA. ce 
 
 the operation of addition, except that, 
 whereas simple addition only adds positive 
 terms, subtraction, included in addition, 
 considers the addition of terms which are 
 both positive and negative. 
 
 For example, if we consider the equation 
 of the preceding chapter, 
 
 c=a+), 
 
 where a, is the weight of a man who is 
 suspended from a rope; 4, 1s the weight of 
 an object which he holds in his hand; and 
 c,1s the total weight on the rope. Here, 
 b, instead of being a weight, may be a nega- 
 tive weight, or a sustaining power.. For 
 example, if he held in his hand a balloon 
 which exerted an ascensional pull of say 
 50 pounds, then 6, would be — 50, and if 
 his own weight were 150 pounds, 
 Gu pO 50 
 100 pounds, 
 
 I| 
 
18 ' THE INTERPRETATION OF 
 
 Here 0, is a negative quantity. Again, 
 if the balloon exerted an ascensional pull 
 greater than the weight of the man, say 
 200 pounds, then the weight which the 
 rope has to sustain would be 
 
 = 150 = 200 
 = — 50 
 
 o 
 
 or, c, would be a negative, or ascensional 
 pull, so that the man and rope would be 
 pulled upward. Consequently, in any 
 equation such as 
 
 Ci Ot) Gare 
 
 any of the terms a, 0, or c, may happen to 
 have negative values, and the addition 
 would then have to be made with these 
 considerations in view. For the negative 
 terms would have to be subtracted from 
 the positive terms, and if they exceeded 
 
MATHEMATICAL FORMULA. 19 
 
 the latter, the sum a, would remain nega- 
 tive. Thus, suppose the equation to 
 represent an account at the bank, and that 
 - four payments have been made to the 
 account. Then the equation is equivalent 
 to the statement that the total payment is 
 equal to the sum of the separate payments. 
 Suppose, however, that @ and 6, are pay- 
 ments which have been made to the credit 
 of the account, and that ¢ and d, are nega- 
 tive payments, or payments which have 
 been made at the expense of the account. 
 
 Then if 
 
 a = $100, 56 = $200, c = $300, and d 
 = $400, 
 
 the equation becomes : 
 
 ] 
 
 100 + 200 — 300 — 400 
 300 — 700 
 — $400. 
 
 a 
 
20 MATHEMATICAL FORMULA. 
 
 So that the total payment is a negative 
 or debit payment of $400. With this 
 view of negative quantities, it 1s evident 
 that simple addition and subtraction fall 
 under one set of operations. 
 
CHAPTER III. 
 MULTIPLICATION. 
 
 In arithmetic, when two numbers are 
 multiplied together the result is called 
 their product. ‘Thus, if we multiply 5 by 
 7, we obtain 35, as the product. ‘This is 
 expressed symbolically as follows: 
 
 bi X 7 = 35. 
 Here the sign (x) is called the multipli- 
 cation sign, and is read: “multiplied by,” 
 so that the equation reads: 
 
 Five multiphed by seven is equal to 
 
 thirty-five. 
 Expressed in algebraic symbols: 
 enU) 10 
 or, el pnp eae 
 
 21 
 
34 THE INTERPRETATION OF 
 
 In these equations c, stands for the prod- 
 uct of a and 6, whatever those symbols 
 may represent. Tor example, if a, repre- 
 sents in dollars and cents the wages of a 
 laborer per day, and 6, is the number of 
 days’ work done by the laborer at this pay, 
 then ¢, is the total wages due to him at the 
 end of ddays. Thus,ifa = $1.60 and 6 = 
 6 days, then the amount due to the laborer 
 is 
 c = 1.60 X 6 
 = 9.60 dollars. 
 
 It is evident that the equation 
 grit it 
 
 is susceptible of an infinite variety of inter- 
 pretations, according to the values which 
 may be assigned to a and 6. It is to be 
 noticed that if @ and 0, are mere numbers, 
 or numerical quantities, their product is 
 
MATHEMATICAL FORMULA. pay 
 
 also a number, but if @ and 8, are both 
 physical magnitudes, then their product ¢, 
 will not be the same kind of quantity as 
 either of the components. Thus, suppose 
 that a weight of a pounds, is raised through 
 a vertical distance of > feet, then we know 
 that the product ax 4, is equal to the 
 work done in the process of lifting, and is 
 expressed in units of work called foot- 
 pounds; one foot-pound being the amount 
 of work required to raise one pound through 
 a vertical distance of one foot against ravi- 
 tational force. Consequently, this equation 
 would read : 
 
 @ (pounds) x 6 (feet) = ¢ (foot-pounds). 
 
 Here it will be seen that the kind of 
 magnitude on the right-hand side of the 
 equation, namely, foot-pounds or work, 
 is different from either of the two magni- 
 
 d 
 
24 THE INTERPRETATION OF 
 
 tudes appearing on the left-hand side; 
 namely, weight and distance. 
 
 The multiplication sign is frequently 
 omitted between two symbols which are 
 to be multiphed together. Thus, the 
 above equation may be written 
 
 c= ab. 
 
 Here the period takes the place of the 
 multiplication sign. Or, we may write 
 
 C=", 
 
 both multiplication sign and period being 
 omitted; the symbols being merely 
 written after each other, and, since the 
 order of the two quantities to be multi- 
 plied is indifferent, we have also 
 
 Gu Ud, 
 
 where the multiplication sign is likewise 
 omitted. The two quantities a@ and 8, 
 
MATHEMATICAL FORMULA. 25 
 
 being written side by side indicate that 
 their product is to be taken. Similarly, 
 any number of quantities written in suc- 
 _ cession indicate that their continued prod- 
 uct has to be taken. For example, 
 
 w= gac 
 
 ig an equation which means that the 
 quantity w, 1s equal to the product of 
 the three quantities represented by the 
 symbols g, a@ and ¢, respectively. This 
 equation could be written: 
 
 C190 
 or, Gee. C, 
 Thus, if 
 g—2,¢ — 4, and¢ = 6 
 we = 48. 
 
 It will be evident that in any equation, 
 
 such as 
 e=atb—-~e 
 
26 THE INTERPRETATION OF 
 
 in which the three terms on the right- 
 hand side are represented as simple quan- 
 tities, any of these terms may stand for 
 products. Thus a, may be the product 
 of e and f, or 
 
 a=; 
 6, may be the product of h, & and 4, or, 
 b= hig; 
 
 while ¢, may be the product of p, g, 7 and 
 8, OF, 
 
 C = pgrs. 
 
 Consequently, such an equation may be 
 written 
 
 e= eof + hky — pars. 
 
 If the values of ¢, f, h, 4,7, p, q, 7, & are 
 all given, the value of a, can be deter- 
 mined by first multiplying e and f, 
 together, then multiplying h, & and 9, 
 
MATHEMATICAL FORMULA. 27 
 
 together, then multiplying p, g, 7 and s, 
 together and, finally, adding the three 
 products so obtained, the last quantity 
 being considered as negative. 
 
 In the arithmetical equation 
 
 Boe 60 ocr ex 1, 
 
 in which each of the terms on the right- 
 hand side is a product of a number and 
 number 7, it is clear that the same result 
 -will be obtained if we add the numbers 
 by which seven is to be multiplied before 
 making the product, or we may write 
 the equation: 
 
 63 = 7 X (5 +8 41) 
 = 1X9 
 = 63. 
 
 Herethesymbol( __), called the brackets 
 or parentheses, indicates that all of the 
 quantities within them are to be operated 
 
98 THE INTERPRETATION OF 
 
 upon by the multiplication sign. In other 
 words, the three quantities within the 
 brackets are to be grouped together and 
 considered as a single quantity. 
 
 From the equation 
 
 68=7x (6+34+1) 
 we also derive the equation 
 
 6= 385+ 214+7 
 =" 63, 
 
 The same rules apply algebraically. Thus, 
 
 from the equation: 
 w=a(b+e4+d), 
 
 which means that the quantity a, 1s equal 
 to the product of the quantity a, multi- 
 plied by the compound quantity which 
 is the sum of 0,¢ and d, we derive the 
 
 equation 
 «= ab+ac-4 ad, 
 
“MATHEMATICAL FORMUL2. 29 
 
 from which we see that when a factor a, 
 appears outside of the bracket containing 
 several terms, the factor may be considered 
 - to apply to each of these terms in suc- 
 cession. 
 
 Thus, suppose that a passenger steamer 
 has accommodation for / first-class passen- 
 gers, § second-class passengers and g 
 steerage passengers, and that the ship 
 makes ten journeys with every berth 
 filled. Then if a, be the total number of 
 people transported in all, in the 10 trips, 
 we have the equation 
 
 10f + 10g + 10s 
 or, Di =A) ire geste 8), 
 
 | 
 
 where the compound term or quantity in 
 the bracket, namely, + 7 + s, is the total 
 number of passengers in one trip. Simi- 
 
 Jarly, in n, trips where m, is any number, 
 
30 THE INTERPRETATION OF 
 
 the total number of passengers carried will 
 be . 
 ew=n(f+g+t+S8). 
 
 Sometimes a straight line or vinculum is 
 used to connect a number of terms into a 
 single group. With the aid of the vincu- 
 lum the last equation would be written 
 
 xew=nxftgt+s. 
 
 In a similar manner, several compound 
 terms may be associated together into a 
 product. Thus: 
 
 w=(a+b+c)d+e4+/) 
 
 means that the quantity a, is the product 
 of two compound terms, the first term 
 being the sum of a, 0 and e, while the 
 second is the sum of d,e and f. Here if 
 
 Arar sorte 
 and Moyes Uber toa wd 
 
MATHEMATICAL FORMUL®. 31 
 
 then, from the equation, 
 es eS 
 
 For example, if an elevator rises through 
 three stages, the first of which is a, 
 feet, the second 6, feet, and the third «, 
 feet, and carries three passengers, the first 
 of whom weighs d, pounds, the second e, 
 pounds, and the third f#, pounds, then the 
 total work done in lifting the three passen- 
 gers through the three stages is repre- 
 sented by the above equation. 
 
 iia 30 feet, > — 20 feet, and c = 20 
 feet, while d = 150 pounds, ¢ = 100. 
 pounds and f = 120 pounds, we have 
 
 8 
 | 
 
 (30 + 20 + 20) (150 + 100 + 120) 
 foot-pounds 
 
 = 70 X 370 
 
 25,900 foot-pounds. 
 
32 THE INTERPRETATION OF 
 
 As an example of multiplication in 
 aleebra, we may consider the rule for 
 determining the sum of an arithmetical 
 series; 7. @., a Series whose successive terms 
 have a constant difference, such as the series 
 
 or 8,16. 24) 625 (40) oeeme 
 
 This formula for determining the sum of 
 
 an arithmetical series is 
 S =n (21 — (Nie 
 
 where S,is the sum required, V is the 
 -number of terms summed, d is the com- 
 mon difference between any pair of suc- 
 cessive terms, /, is the last term, and 2, is 
 half the number of terms. 
 
 Thus, considering the first series 
 
 i: 3 D 7 9 Lis 
 
MATHEMATICAL FORMULA. 38 
 
 = 8 [2 X11 (621) 2) 
 3 [22 — 10] 
 
 3x 19 
 
 at 
 
 The sum required is therefore 36, and 
 we find by actual summation that 
 
 Peto, 10 ee eae — 86. 
 
CHAPTER IV. 
 DIVISION. 
 
 In arithmetic if one number is divided 
 by another number, the result is called 
 the quotient. This, if 15, be divided by 5, 
 the quotient is 8. Or, in symbols, 
 
 15 
 
 sae: 
 
 The sign of division is also employed to 
 represent the operation. Thus, 
 16+ 5 = 3, 
 where the sign (+) is called the sign of 
 division and is read: “ divided by.” 
 Similarly in algebra, the equation 
 
 C=b>.4a 
 34 
 
MATHEMATICAL FORMULA. 35 
 
 means that ¢, is a quantity which is equal 
 to the quotient of the quantity 6, divided 
 by the quantity a This equation might 
 _ be written either 
 
 b 
 ¢= — 
 a 
 or 
 c = b/a. 
 In one case the division bar is used to 
 separate the numerator 0, from the denom1- 
 
 nator a, of the fraction ace while in the 
 | a 
 
 second case, the line is written diagonally, 
 or as a solidus. Thus, if a= 10 and 6 
 
 = 17, 
 
 aaa hyenas: 
 
 The process of division may be extended 
 to compound terms which are contained in 
 
 brackets. Thus, 
 e©=(a+6—c)+(d+e) 
 
36 THE INTERPRETATION OF 
 
 means that the first compound term, which 
 is the sum of + a, + J and — «, is divided 
 by the sum of the quantities d and ¢, or 
 _at+b—-e 
 el i 
 
 If a+ b —«, be represented by A, and d 
 + e, by B, then 
 pe os or A= ah. 
 
 The operations of multiplication and 
 division may be readily associated in an 
 equation. For example, 
 
 Re? (wire). 
 c+td 
 
 Here the quantity a, 1s stated to be equal 
 to a quantity e, multiplied by a fraction, 
 the numerator of which is the sum of 
 three products; namely, the sum of ay, cd 
 and ef, while the denominator is the sum 
 
MATHEMATICAL FORMULA. of 
 
 of the quantities cand d. If we suppose 
 that the values of a, ¢, d, e, f and y, are all 
 given, we may proceed to simplify the 
 - equation by forming the numerator and 
 expressing it by the quantity A, and then 
 forming the denominator and expressing 1t 
 by the quantity 6. The equation will 
 then be 
 
 A 
 Gb gman ty kee 
 
 Ls 
 eA 
 =: 
 
 It will be found that a great number of 
 formule or algebraic rules for the determi- 
 nation of unknown quantities only involve 
 operations of addition, subtraction, multi- 
 plication and division. For example, in 
 calculating the expansion of a gas, refer- 
 ence is usually made to the following 
 formula: 
 
 V,= V, 1 + at), 
 
88 THE INTERPRETATION OF 
 
 where Vp, is the volume occupied by a gas 
 at the temperature of melting ice, or zero 
 degrees Centigrade, V,, 1s the volume 
 occupied at some other temperature ¢ de- 
 orees Centigrade, and a, is a coefficient or 
 constant numerical multiplier which is 
 0.00366. The equation is, therefore, equiv- 
 alent to the following statement: “ ‘The 
 volume of a gas at a temperature 7° C. is 
 equal to its volume at zero Centigrade, 
 multiphed by a quantity which is the sum 
 of unity and the product of the temper- 
 ature ¢ C. and 0.00366. Thus, if the 
 volume V, = 300 cubic feet and ¢ = 15° 
 C., then the volume of the gas at 15° C. is 
 
 V, = 300 (1 + 0.00366 X15) 
 
 300 (1 + 0.0549) 
 
 300 X 1.0549 
 
 316.47 or 316 1/2 cubic feet, 
 
 approximately. 
 
MATHEMATICAL FORMULA. 39 
 
 Again, in the discussion of electric cir- 
 cuits, in which two resistances of 7, ohms, 
 and 7, ohms, are connected in parallel, then 
 -it may be shown that the joint resistance 
 Le, of the pair is 
 
 "1 Ve 
 R= - — ohms. 
 Tiss he 
 
 
 
 Thus, if 7, = 5.5 and 7, = 3.7, then ZR, the 
 joint resistance, 1s 
 
 D,0 ee our 
 
 O-O i eet 
 
 20.35 
 
 2 
 
 te 
 
 
 
 v. 
 = 2.212 ohms approximately, 
 or 
 212 
 
 1000 
 
 
 
 ohms. 
 
 In the theory of the conduction of heat, 
 the following equation occurs : 
 
 Q=K'=t ar 
 
40 THE INTERPRETATION OF 
 
 Here, Q, stands for the quantity of heat 
 which passes in a given time, Z’seconds, 
 through a slab of material whose thermal 
 conductivity is A, and having a surface 
 area of A square centimetres, and a thick- 
 ness of ¢d centimetres, one face of the slab 
 being maintained at a temperature ¢° C. 
 and the other at a lower temperature ¢° C. 
 Then the equation asserts that the flow of 
 heat 1s equal to the product of the conduc- 
 tivity “XY, the difference of temperature 
 t’ — t, between the faces, the surface area 
 A, and the time 7} divided by the thick- 
 ness d. If we suppose a slab of copper 
 whose conductivity A = 1.08, the surface 
 area of the slab being A = 750 square 
 centimetres, its thickness d = 2 centi- 
 metres, and the difference of temperature 
 between the faces of the slab (¢ — ¢) = 
 5° C. then the flow of heat in the time 
 Z’= 10 seconds will be 
 
MATHEMATICAL FORMULA. 41 
 
 mpl:03:x.5 x 9750 X 10 
 = 
 
 fe 
 
 thermal units. 
 
 @ 
 
 = 19,312.5 
 
 It is important to observe that much 
 may be learned from the form of an equa- 
 tion concerning the nature of the quanti- 
 ties with which it deals, without actually 
 computing or solving it. .Thus, in the 
 preceding equation, we observe that the 
 flow of heat passing through a slab of 
 uniform material, increases directly with 
 the time 7} because the symbol 7} appears 
 as a factor, so that if we double the value 
 of Z} we necessarily double the value of Q. 
 Again, the value of @Q, increases directly 
 with the active area through which the 
 conduction of heat takes place, because 4, 
 appears also as a factor in the product. 
 Similarly, the value of @, increases directly 
 with the difference of temperature ¢'—4¢, 
 between the faces of the slab. On the 
 
49 THE INTERPRETATION OF 
 
 contrary, however, it will be noticed that 
 the thicker the slab, or the greater the 
 value of d, the smaller will be the value 
 of Y; or, in other words, that Q, varies 
 inversely with the thickness d. 
 
 In the mensuration of solids we find a for- 
 mula for obtaining the surface of a right 
 cylinder in terms of its radius and height 
 
 S = Inr(h + 1) 
 
 where Sj is the surface of the cylinder in 
 square inches, including base and top; 7, 1s 
 the radius of the cylinder in inches; A, is 
 the height of the cylinder in inches; and z, 
 is the numerical ratio of the circumference 
 of a circle to its diameter, or, approxti- 
 mately, 3.1416. 
 
 Consequently, if 7 = 2 and 2 = 10, 
 
 § = 2 X 38.1416 x 2 (10 + 2) 
 
 2 X 8.1416 x 2 X 12 
 
 = 150.797 square inches, approximately. 
 
MATHEMATICAL FORMULA. 43 
 
 The formula for determining the horse- 
 power of a single-cylinder engine from an 
 indicator diagram is as follows: 
 
 A 
 P= ane horse-power. 
 
 Here, the horse-power /, exerted upon one 
 end of the piston, is equal to a fraction 
 whose numerator is the continued product 
 of the active area <A, of the piston in 
 square inches, the mean effective pressure 7, 
 during one cycle or revolution, in pounds 
 per square inch, the number of revolutions 
 per minute /?, and Z, the length of the 
 stroke in feet and decimals of a foot. 
 Thus, if a diagram taken from a cylinder 
 shows that the mean pressure at the back 
 of the cylinder is say 25 pounds per square 
 inch, the length of the stroke Z = 3 feet, 
 the number of revolutions per minute 
 Ft = 160, and the active area of the back 
 
44 THE INTERPRETATION OF 
 
 of the piston was 120 square inches, then 
 the horse-power of the back of the cylinder 
 Is: 
 
 120 X 25 X 140 X38 
 
 P= 33,000 
 _ 1,440,000 
 ~ ~ 33,000 
 = 43.63 horse-power, 
 approximately. 
 
 It is shown in treatises on the theory of 
 probabilities that if the probabilities of the 
 truth of a statement made by witnesses 
 are respectively 7, Pe, 3, etc., the probabil- 
 ity of the truth of a statement made by all 
 the witnesses concurrently is 
 
 fee Pi Pe «+++ Pn 
 Pr Pe Pa +(1—p,) (1—py) (1 ps) ee 
 
 This equation states that the probability 
 that the event actually occurred, 1s a 
 
MATHEMATICAL FORMULA. 45 
 
 fraction, of which the numerator is the 
 product of the probabilities of true declara- 
 tion of the respective witnesses, and whose 
 denominator is the sum of two compound 
 terms. The first is the product of the 
 probabilities of true declaration as in the 
 numerator, and the second term is the 
 product of each probability subtracted 
 from unity into all the others similarly 
 subtracted. ‘Thus, if the probability that 
 
 A’s statements are reliable is 3 in 4 or 2 = p, : 
 that B’s as 33 TET SAID ety 1 
 “ Og “ & 5“6% 5 =p 
 “ D’s 66 “ LS Ry date & at yd 
 
 then, if A, b, Cand JL, all independently 
 concur in stating that a certain event 
 happened, the probability that it actually 
 happened, so far as it can be gauged from 
 this circumstance, is 
 
46 THE INTERPRETATION OF 
 
 PX 4X #x4 
 
 EXExX 4+ (1-9 0-9 0) 0-® 
 0 
 0 
 
 
 
 
 
 The probability is, therefore, 360 in 361, 
 or 3860 to 1 that the event actually 
 occurred. 
 
 If cannon balls are piled in the form of 
 a square pyramid, the number in one edge 
 of the base being x, the total number of 
 balls 4, in the completed pile is 
 
 N = % (n+ 1) (2n + 1). 
 hus; if 7,18 
 NW = ¥¥ (19) (37) 
 tr Ge Oe 
 = 4218, 
 
 If along rod whose weight is very small 
 be loaded with weights m,, mg, ms, etc., at 
 
MATHEMATICAL FORMULA. 47 
 
 distances measured from one end of the rod, 
 equal to a, @, #3, etc., then it 1s shown in 
 treatises on statics that the centre of 
 gravity of the loaded rod is situated ata 
 distance #, from the end of the rod, 
 expressed by the formula 
 
 _ My % + Me A% + Ms %y + 
 m + m,+ m+ .... 
 
 
 
 or #, 1s a fraction whose numerator is the 
 sum of the products formed by each weight 
 into its distance; while the denominator 
 is the sum of the weights or the total weight 
 of the loaded rod. 
 
 This is sometimes written 
 
 MA 
 eae 
 
 
 
 where Sm means the sum of all terms of 
 the type m, #, and Sm means the sum of 
 all terms of the type m. 
 
48 THE INTERPRETATION OF 
 
 
 
 UTA Cb wes 107, , = 5 inches 
 ein Co = hee 
 Ms = 4 OZ. i, =A 
 Mg = IEO0Z: C,; = ie 
 Then 
 ae 3X5 + 2X10 4+ 4x12 eee 
 Bie mad jt em 
 210 +h 20.443 ee 
 8 Die ee 
 99 
 
 0 a 9.9 inches from the end. 
 
 oT 
 
 It may be shown that the probability p, 
 of dealing all the 13 cards of one suit 
 to one player from a whist pack of 52 
 shuffled cards among four players is 
 
 1381 x 398 
 52! 
 
 Here p, the probability is equal to a frac- 
 tion, the denominator of which is the- 
 factorial of 52; 2. ¢, the product 52, 51. 
 50, 49, ete., 5, 4, 8, 2,1; or the continued - 
 
MATHEMATICAL FORMULA. 49 
 
 product of all the numbers between that 
 number and unity inclusive, while the 
 numerator is the product of the factorial 
 . of 13 and the factorial of 39. A factorial 
 is represented by the note of admiration 
 following the number. In this case 
 
 aa 
 P = 635,013,559,600 ’ 
 
 or one chance in about 635 billions. 
 
CHAPTER V. 
 INVOLUTION. POWERS. 
 
 Ir we multiply any number, say 5, by 
 itself, we obtain what is called its sguarz 
 
 or 
 5 xX 5 = 25, 
 
 so that 25, is the square, or the second 
 power of 5. 
 
 Similarly, if we multiply a number by 
 itself twice in succession, we obtain what is 
 called its cwbe, or 
 
 5X5 X 5 = 195, 
 
 so that 125, is the cube, or the third power 
 of 5. 
 
MATHEMATICAL FORMULZ. 51 
 In the same way 
 exe XD iees Oe = OUD 
 
 is the fourth power of 5, and 5X 5...” 
 times in succession is the nth power of 5. 
 A. special notation is employed to repre- 
 sent powers briefly and conveniently. 
 
 Thus 
 5? = 25, 
 where the small number appended is 
 called the index, or the exponent, or the 
 power of 5, and the equation is read, 5 
 squared, or raised to the second power, is 
 equal to 25. 
 In the same way 
 
 7 = 49, because 7 X 7 = 49. 
 Similarly, 
 5% = 125, because 5 X 5 X 5 = 125, 
 
 and the equation is read, 5, raised to the 
 third power, or cubed, is equal to 125. 
 
52 THE INTERPRETATION OF 
 
 Similarly, 16? = 4,096, or 16, raised to 
 the third power is equal to 4,096. 
 Sunilarly, 
 
 i) 
 125, 
 and so on for all powers of 5, 
 
 or 5" = nth power of 5. 
 
 This equation may be generalized alge- 
 braically as follows: 
 
 x = a° = nth power of a. 
 Thus, if 
 
 a@ = 2 and’ = 3 tome 
 
 If 
 = 256 and n= 2; w = 256*/==30ogem 
 
 If we multiply two powers together, as, 
 for example, when we multiply 
 
 5? X 5% or 25 X 125, 
 the product is 5° or 3,125. 
 
MATHEMATICAL FORMULA. 53 
 
 Here the exponent, or index of the prod- 
 uct, is obtained by adding the exponents or 
 indices 2 and 3, 
 
 or 5? X 53 = FC +9 = 5°, 
 
 This rule is of general application. The 
 index or exponent of a product of two 
 powers is always equal to the sum of their 
 individual indices or exponents. This 
 rule is expressed algebraically as follows: 
 
 Mb x ge x ae ros gee + b +o) 
 
 where «a, is any number; a, 0, and ¢, are 
 the powers of a, while their product is a, 
 raised to the sum of those powers. For 
 example, | 
 
 10! x 102 x 10° = 10¢+2+9 = 10° = 
 1,000,000, 
 
 or, 10 X 100 X 1,000 = 1,000,000. 
 
54 THE INTERPRETATION OF 
 It is evident that 
 10' = 10, 10* = 1,000, 
 107 = 100, 10* = 10,000, 
 
 and generally 
 10° = 1 followed by x. zeros. 
 
 From the law of addition of indices in 
 the formation of products, 1t follows that 
 
 10? X 10° = 10¢T® = 10° 
 
 so that if we multiply 10’, or 100 by 10°, 
 we obtain the same quantity, or 100, but 
 this is what we would obtain if we mul- 
 tiply 10° by unity. Consequently, we 
 infer that 10° = 1. This is a general rule, 
 expressed algebraically as follows : 
 
 w° = 1, whatever number 2, may be. 
 
 By the foregoing rule of the summation 
 
MATHEMATICAL FORMULE. 5D 
 
 of indices in the formation of products, it 
 follows that 
 
 eS hee bee Ii 
 
 Consequently, if we multiply 5? by 5-? we 
 obtain unity as the product. But if we 
 
 multiply 5° by ale v7. @., the reciprocal of 
 x 
 
 53, we obtain unity as the product. Con- 
 sequently, 
 
 This rule is of general application, and 
 
 1 i: 
 10-* = reciprocal of IO*.= 10° = 100’ 
 
 and, generally, 
 
 whatever numbers @ and nm, may be. As 
 
56 THE INTERPRETATION OF 
 
 an example of the use of positive and nega- 
 tive indices, we may take the following: 
 
 The frequency of oscillation correspond: 
 ing to the limits of the visible spectrum; 
 z.é., the number of vibrations per second 
 in the ether producing visible light, he 
 between, approximately, 894,000,000,000, 
 000 and 762,000,000,000,000 double vibra- 
 tions per second. .These may be ex: 
 pressed as 
 
 8.94 x 10“ and 7.62 xX 10%; 
 
 a. é@., between 3.94. (10 xX 10° (eee 
 times in all, or 1 followed by 14 zeros, 
 and 7.62 (10 x 10 . . .) 14 times in all, 
 or 1 followed by 14 zeros. 
 
 Again, the wave-lengths in free ether cor- 
 responding to these frequencies are con- 
 tained between the limits 
 
 0.000076 and 0.00004 centimetres, 
 
MATHEMATICAL FORMULZ. 57 
 which may be expressed 7.6 xX 10-> and 
 4 X 10~° centimetres ; 7. ¢,, 
 
 1 1 7.6 
 
 7.6 x 10° — 7.6 
 
 * 700,000 ~ 100,000 
 centimetres, and 
 
 1 1 + 
 
 4X 793 = * X 700,000 = 100,000 
 
 centimetres. 
 
 The following table will still further 
 illustrate the subject of positive and nega- 
 tive indices: 
 
 10°=1; 10'= 10; 10? = 100; 10°?= 1,000, 
 etc. 
 10-? = 0.1; 10-* = 0.01; 10-* = 0.008, 
 
 etc. 
 It is important to observe that the 
 property of the summation of indices in the 
 
58 THE INTERPRETATION OF 
 
 formation of the product only applies to 
 powers of the same base. Thus, 
 
 T= ie 
 
 because the indices 5, 8 and 2, in this 
 equation are the indices of a common base ; 
 
 z.é, 7; but it would obviously not be true 
 of 
 
 PX Ds = ore 
 
 because here the powers are of different 
 bases ; namely, 5 and 7 
 
 If we allow a stone to fall from the 
 hand toward the earth, the formula which 
 expresses the distance, through which it 
 will fall in a given time ¢ seconds, is 
 
 feet. 
 This equation is equivalent to the state- 
 ment: The vertical distance in feet 
 
MATHEMATICAL FORMULA. 59 
 
 through which the stone will descend after 
 ¢ seconds have elapsed from the moment of 
 release of the stone, will be the product of 
 ‘the quantity g, into the square of the 
 time ¢ seconds, divided by 2; g, 1s known 
 to be, approximately, 32.2 feet per second 
 per second, so that the equation becomes 
 32.90 
 
 noe 
 
 = 16.17 feet. 
 
 — 
 —= 
 
 If ¢=1,2 = 1x 1=1 ands = 16.1 feet. 
 
 Ifitz=2,f=2 x2=4 ands = 644 
 feet. 
 
 lee — 1.5, = 1.5 X 1.5 = 2.25 and 
 gs = 86.225 feet. 
 
 Again, the volume of a sphere of radius 
 
 7 feet, is known to be 
 
 3 
 V= 28 cubic feet ; 
 

 
 
 
 60 MATHEMATICAL FORMULA. 
 
 where V,is the volume in cubic feet and 
 z, is the ratio of the circumference to t 
 diameter, or 3.1416, approximates 
 Consequently, 
 4X 8.1416 7 
 3 4 
 4.1888 7°. 
 
 jis 
 
 vier Lhe ater V = 4.1888 (1 Xa 1) 
 = 4.1888 cubic feet. 
 
 If 7 = 2, V = 4.1888 (2 X 2 « 2) = 
 33.5104 cubic feet. 
 
 Pa > 
 
 a2 
 If r = 1.75, V = 4.1888 (1.75 x 1.75 x 
 1.75) = 4.1888 Xx 5.3594 = 22.45 cubic 
 feet. 
 

 

 
CHAPTER VI. 
 EVOLUTION. ROOTS. 
 
 WE have seen that ¢nvolution consists 
 - inraising a quantity, say a, to some power, 
 or performing the operation 
 
 1 6 mee EF Pe 
 
 - Hvolution consists in reversing this, or is 
 the inverse of the above operation. Thus, 
 if we know that 5? = 25, or that 25, is the 
 square of 5, we determine by the process 
 of evolution that 5, is the number whose 
 square is 25; 5, is then said to be the 
 square root of 25. In the same way, 
 having given the relation by involution, 
 
 5? = 195, 
 
 61 
 
62 THE INTERPRETATION OF 
 
 evolution shows that the number 5, is the 
 cube root or the third root of 125. In the 
 same way 2, is the square root of 4, be- 
 cause 27 = 4; 2 is the cube root of 8, 
 because 2? = 8; 3 1s the fourth root of 81, 
 because 3* = 81, and so on. 
 
 A root of a number is represented sym- 
 
 bolically by a radical sign ¥. Thus 
 a= VA n 
 
 means that a, is the cube root of the quan- 
 tity n, so that 
 ee, 
 Similarly, 
 Aiain 
 
 means that a, is the mth root of the quan- 
 tity n. The equation 
 
 Oh) Te, 
 
MATHEMATICAL FORMUL®. 63 
 
 or a, is equal to the square root of n, is 
 often written 
 
 Gi Vales 
 
 that is, the superscript 2, 1s omitted in 
 the radical sign. Thus, the equation 
 
 a= / 64 
 
 means that a, is the number whose square 
 is 64, and, consequently @ = 8. When 
 the expression whose root is to be ex- 
 tracted 1s a compound term, a line or 
 vinculum is placed over it, or brackets 
 areemployed. Thus, 
 
 @= V 32 + 32, ora = V(82 + 32) 
 
 are equivalent to a = 8. 
 
 As an example in evolution, or the ex- 
 traction of roots, the following case may 
 be considered. The formula which gives 
 the period of time of complete vibration 
 
64 THE INTERPRETATION OF 
 
 of a simple pendulum, making extremely 
 small oscillations, is 
 
 Vee 
 iY q seconds. 
 
 This equation is equivalent to the follow- 
 ing statement: The time Z} occupied by 
 a pendulum in making one complete to- 
 and-fro motion, of indefinitely small amph- 
 tude, is the product of 27, or 6.2832, and 
 the square root of a ‘fraction whose numer- 
 ator is the length of the pendulum and 
 whose denominator is the intensity of 
 gravity at the location considered. Thus, 
 if the length of the pendulum be 8.05 
 feet, and g, the intensity or acceleration 
 of gravity, be 32.2 feet per second! per 
 second, then 
 
 8.05 
 32.2 
 
MATHEMATICAL FORMULA. 65 
 
 = 9 xX 3.1416 tie 
 4 
 
 ox Bt 6 ee 
 = 3.1416 seconds. 
 
 It is evident that 
 4=VE 
 
 because 
 
 @P=axgah 
 
 Again, from a known relation, whose dis- 
 covery 18 ascribed to Pythagoras, between 
 the lengths of the sides of a right-angled 
 triangle; namely, 
 
 Ilypothenuse = Y (Perpendicular)? + (Base)? , 
 if we have a triangle OAD, Fig. 1, which 
 contains a right angle or 90° at A, then 
 
 OB =v(UA)* + (AB). 
 If OA = 4 feet and AL = 3 feet 
 
C60 THE INTERPRETATION OF 
 
 OB = v4 + 3? 
 =Vv1l6+9 
 = V25 
 = 5 
 because 5, is evidently the square root of 
 25, since 5? = 25, 
 
 
 
 Fig. 1. 
 
 Another convenient method of express- 
 ing roots or radicals consists in the em- 
 ployment of fractional indices. ‘Tlrus, 
 from the general law of the addition of 
 indices or exponents in the formation of 
 powers, we have: 
 
 10? x 10! = 10@+ = 10! = 10, 
 
MATHEMATICAL FORMULE. 67 
 so that 10!, multiplied by itself, or squared, 
 gives 10; or, 
 
 (10?)* = 10, or 103, is the square root of 10. 
 Similarly, 
 
 10! X 10! x 10! = 10¢+!+) = 10! = 10, 
 so that 10! = V10. 
 
 This is capable of generalization, algebra- 
 ically, in the formula 
 
 1 n ,—— 
 ax = Va = nth root of the number 2. 
 
 As an example, 9? = 3 because 3 X 3 = 9. 
 
 We have hitherto considered powers 
 which were formed with indices which are 
 whole numbers or integers, but it 1s now 
 easy to see what a fractional power means. 
 For example, by the law of the addition of 
 indices, 
 
 TOS ic OEE 2 = 71 08 
 
68 THE INTERPRETATION OF 
 so that 
 10? = V 108 
 
 or, the square root of the cube of ten; 
 
 10? = 10! x 10! x 10! 
 
 10! = (V10)’; 
 or, the cube of the square root of ten. 
 Consequently, 
 
 10''= (V10) = vier 
 
 or, the cube of the square root of ten is 
 equal to the square root of the cube of ten, 
 aud generally, 
 
 m 
 m 
 
 A (S} n _— 
 Asay ta ean 
 (4) n/\m 
 = ge = (Va)”. 
 If then we take a power a*, and divide its 
 
 index by some quantity, say 3, we obtain 
 a, which is the cube root of a*, and again 
 
MATHEMATICAL FORMULA. 69 
 
 if we multiply its index by any number, 
 say 4, we obtain a, which is the fourth 
 power of a = (a*)*. | 
 
 Again, if we have an equation 
 
 m= git 
 
 the equation means that a, is a quantity 
 which is equal to the 15th root of the 
 11th power of a, or to the 11th power of 
 the 15th root of a; 7. e., 
 
 C= (ate)"! — V (ay 
 = Ya" = @y 
 
CHAPTER VII. 
 EQUATIONS. 
 
 Aw equation is an algebraic expression 
 of equality between two quantities, em- 
 ploying the sign =. Equations differ 
 almost infinitely in nature, complexity, and 
 length. It is impossible, within the limits 
 of this little work, to devote sufficient 
 space to the subject of the treatment or 
 manipulations of equations, so that the 
 solution of any given equation may be 
 obtained. It will be sufficient if the 
 student obtains from this book a clear 
 understanding of the meaning of the state- 
 ment contained in any equation, so as to 
 be able to interpret its signification. There 
 
 are, however, a few general and simple 
 70 
 
MATHEMATICAL FORMULA. a | 
 
 rules which may be set down as a guide 
 for the student in dealing with equations. 
 
 Equations may be divided into the fol- 
 lowing classes : 
 
 (1) Simple equations, or equations of 
 the first degree, are those which involve 
 the first power of an unknown quantity. 
 Thus, 
 
 w= @ 
 a+tb=et+d 
 
 e=a(p+7+*) 
 
 are forms of simple equations, because the 
 unknown quantity represented by a, does 
 not appear except in the first power; @. @,, 
 there are no powers of w of the type a 
 and no roots of 2, of the type Vz. 
 
 (2) Quadratic equations, or equations of 
 the second degree, are those in which occur 
 the second power of an unknown quantity. 
 
fe: THE INTERPRETATION OF 
 
 Thus 
 ray 
 avet+ be+te=O0 
 
 are examples of quadratic equations. 
 
 (3) Lquations of the third degree, or those 
 involving third powers of an unknown 
 quantity: Thus, 
 
 
 
 Tigi t 
 aw + ba? +ca+d=0 
 3 | 
 Slee |. age 
 C ad 
 
 are examples of equations of the third 
 degree. 
 
 Similarly, equations may be formed of 
 any degree. Equations of the first and 
 second may be solved by definite rules; 
 many of those of the third degree may be 
 solved; but equations of the fourth, or 
 higher degrees, can only be solved rigor- 
 
MATHEMATICAL FORMULZ. 73 
 
 ously in special cases, although arithmet- 
 ical approximations to their solution can 
 in all cases be obtained to any desired 
 _ degree of accuracy. 
 
 If the same algebraic operation be per- 
 formed upon both sides or members of an 
 equation, the equality remains unaltered, 
 although the form of the equation may be 
 ereatly changed. Lor example, if 
 
 med 
 then 
 etb=at+b 
 
 because a certain quantity 0, is added to 
 both # and a, and if these latter are equal, 
 they must remain equal when increased by 
 the quantity 0. 
 
 Again e=a it ec=a 
 or (c+ 6) = (@ + a) 
 or Vz —Va 
 
 or Mx = MA 
 
14 TILE INTERPRETATION OF 
 
 or — — ae 
 
 In all these cases, the same operation is 
 performed on each side of the equation. 
 Many transformations may be effected by 
 
 this process. Jor example, in the equa- 
 tion 
 e+az= 10, 
 
 if we subtract the quantity a, from each 
 side of the equation, we obtain 
 et+a—-a=10-a4. 
 
 On the left-hand side we now have a, 
 added to a, and then a subtracted from 
 the result. Consequently, the as cancel, 
 or may be removed from the left-hand side, 
 and the equation becomes 
 
 42 Haat WS ee 2 
 
 We thus see that a quantity may be shifted 
 from one side of an equation to the other 
 
MATHEMATICAL FORMULA. 1D 
 
 by changing its sign, because, in the orig- 
 inal equation a, appeared on the left-hand 
 side under the positive sign, while in the 
 transformed equation it appears on the 
 right-hand side, under the negative sign. 
 
 As an example of algebraic equations 
 employed in physics, we may take the fol- 
 lowing : | 
 
 If V,, be the volume of a liquid at tem- 
 perature 0° C., and V,, be the volume at a 
 temperature ?° C., 
 
 V.= Vi(l t+ at + pf + yt). 
 
 The equation states that the volume at tem- 
 perature # C.,is equal to the volume at 
 0° C., multiplied by a compound term; 7. ¢., 
 the term within the brackets. This term is 
 the sum of unity, and @ times the tempera- 
 ture elevation ¢, 6 times the square of the 
 temperature elevation, and » times the 
 cube of the temperature elevation, 
 
76 THE INTERPRETATION OF 
 
 | 
 
 For example, if V, = 100 cubic centime- 
 tres, ¢ = 10° C.; a. = 0.000002 
 0.00000083889, yv = 0.00000007173, or 
 a = 9.53 X 10-5 6 = 8389 x 10% 
 y = 7.173 X 10~*. 
 
 Then 
 
 V,= 100 (1 + 10 X 2:53 X 10° Ose 
 8.389 X 10-7 + 108 x 7.173 x 10°) 
 100 (1 + 2.53 X 107° +:8:389 4 
 await ie Wess ce ATs 8) 
 100 (1 + 0.0000253 + 0.00008389 
 + 0.00007173) 
 AROLOIS GPs 0.00018092) 
 100 (1.00018092) 
 100.018092 cubie centimetres. 
 
 As an example of an equation involving 
 an infinite series, the following may be 
 considered : 
 
 2 
 
 7 1 1 ] <I ? ae 
 eo pletgt ett ae 
 
MATHEMATICAL FORMULZ. Pig 
 
 This shews that the square of the quantity 
 x, Which as far as four decimal places 
 is 8.1416, divided by 6, is equal to the sum 
 of the reciprocals of the squares of the suc- 
 cessive natural numbers carried to infinity. 
 It is evident that the value of z, might be 
 computed by the aid of such a series. It 
 could never be determined with absolute 
 accuracy because an infinite number of 
 terms would be required, but it could be - 
 computed to any desired number of 
 decimal places. 
 
CHAPTER VIII. 
 LOGARITHMS, 
 
 In the equation 
 
 the power 2, to which the base 5, is raised 
 in order to be equal to 25, is called the 
 logarithm of 25, to the dase 5. Similarly, ' 
 
 Oe 
 
 may be stated by saying that n, is the 
 logarithm of the number a, to the base a; 
 ad, may be any positive number, and n, 
 may be any number, positive or negative, 
 and integral or fractional; 2. @, a whole 
 number or a fraction, or a whole number 
 and a fraction. | 
 8 
 
MATHEMATICAL FORMULA. "9 
 
 In applied mathematics, logarithms are 
 of considerable importance in performing 
 calculations, as well as in theoretical dis: 
 cussions. The base which is_ usually 
 selected is 10, and logarithms to this base 
 are called common logarithms. 
 
 From the general law of the addition 
 of exponents in forming a product, we 
 know that 
 
 10? X 10? = 10¢*® = 10°. 
 
 On the left-hand side of this equation the 
 logarithms of the two factors or quantities 
 to be multiplied are, respectively, 2 and 3, 
 while the logarithm of the product is their 
 sum, 5. This rule is of general applica- 
 tion, and may be expressed as follows: 
 
 If we sum or add the logarithms of two 
 numbers, we obtain the logarithm of their 
 product. 
 
80 THE INTERPRETATION OF 
 
 Thus the logarithm of 100 = 2, because 
 10? = 100; the logarithm of 1,000 = 3, 
 because 10? = 1,000; and the logarithm of 
 100 X 1,000 = 5, because 10° X 10? = 10°. 
 
 If we suppose that a table of logarithms 
 to the base 10, is prepared for all numbers 
 between 0, and infinity, not merely for 
 whole numbers, but for all decimal parts 
 as well, we obtain what is called a table 
 of logarithms. All numbers which He be- 
 tween 1 and 10, will have a logarithm 
 lying between 0 and 1, because 10° = 1 
 and 10° = 10. 
 
 Similarly, all numbers lying between 10 
 and 100, will have a logarithm lying be- 
 tween 1 and 2, because 10! = 10, and 10? = 
 100. Similarly, any number lying be- 
 tween 10* and 10°*?, will have a logarithm 
 lying between a and a + 1. 
 
 If we take up a table of common log- 
 arithms and look for the logarithm of a 
 
MATHEMATICAL FORMULA. 81 
 
 number, say 15, we find the decimals 
 1760913. This represents the decimal 
 part, or mantissa, of the logarithm, and the 
 characteristic, or whole number, is to be 
 supplied by the student. Since 15, lies 
 between 10 and 100, its logarithm hes 
 between 1 and 2, or is 1 and some frac- 
 tion; the decimal fraction being .1760913, 
 the complete logarithm is 1.1760913, which 
 may be stated symbolically as follows: 
 
 , 1 (01-160918 — Lb: 
 or, 
 log,, 15 = 1.1760918. 
 
 Here the subscript 10, denotes that 10, is 
 the base employed. Strictly speaking, an 
 indefinitely great number of decimal places 
 in the fractional part or mantissa would 
 have to be employed to obtain the abso- 
 lutely true logarithm, but for all practical 
 purposes it is found that seven places of 
 
82 THE INTERPRETATION OF 
 
 decimals are sufficient, and in many cases 
 even five places are employed for the ordi- 
 nary degree of accuracy required. We 
 have seen that the logarithm of 15, is 
 1.17609138, but the logarithm of 150, will 
 only differ from the logarithm of 15, in the 
 characteristic or whole number. This 
 characteristic must lie between 2 and 8, 
 which are the logarithms of 100 and 1,000, 
 respectively, so that the complete logarithy 
 of 150 is 2.1760913. 
 This may be obtained in another way 
 since | 
 15 = 101-1600 
 and 
 10 = 10! 
 10 x 15 = 10! x 1 1-2460918 = 1 O@- 1760918) 
 
 In the same way the logarithm of 1,500, 
 ‘or 
 log 1,500 = 3.1760913, 
 
MATHEMATICAL FORMULA. 83 
 log 15,000 = 4.1760913, 
 
 log 1.5 0.1760913, 
 log 0.15 = 1.1760913, 
 
 or has a characteristic of — 1 
 
 log 0.015 = 2.1760913, ete. 
 
 Similarly, the logarithm of the number 
 16, is found from the tables to be .2041200 
 for the mantissa, and the characteristic 
 is 1, because 16 lies between 10 and 100; 
 therefore, the complete logarithm is 
 1.2041200, or 16 = 1017041200, 
 
 If now we multiply 15 by 16, or make 
 
 — 1.1760913 4 
 ee 913 x 10! Sawees 
 we obtain as their product 
 
 he 1 Q4.1760918 + 1.2041200) — 1 0 @.3802118) 
 
 Here az, is the number whose logarithm 
 9.3802113 is greater than 2 and less than 
 
84 THE INTERPRETATION OF 
 
 8. Consequently, a, lies between 10° = 
 100, and 10? = 1,000. Entering the table 
 of logarithms for the number whose 
 mantissa is .8802118, we find that this 
 corresponds to 24, so that the answer 
 is 240, or 
 
 15 X 16 = 240, 
 
 Here we have performed the computation 
 without the aid of arithmetical multipli- 
 cation, by the summation of logarithms. 
 This rule is of general application. If we 
 desire to form a product of two or more 
 numbers, we find their logarithms and add 
 them together; the sum is the logarithm 
 of the product sought. As an example 
 of the application of logarithms, we may 
 consider the following: 
 
 What is the number of feet in the cir- 
 cumference of the earth considered as a 
 sphere of diameter 7,918 miles of 5,280 feet 
 
MATHEMATICAL FORMULA. 85 
 
 each? Here the circumference may be 
 
 written 
 O=n X 7,918 X 5,280, 
 
 or, taking z as 3.1416, 
 CO = 3.1416 X 7,918 X 5,280. 
 
 If we obtain tbe logarithms of 3.1416, 
 7,918 and 5,280, and add them together, 
 their sum will be the logarithm of the 
 quantity C, required. The logarithm of 
 8.1416 will lie between 0 and 1, or will 
 have a characteristic of 0. The logarithm 
 of 7,918, will lie between 8 and 4, or will 
 have a characteristic of 8. The logarithm 
 of 5,280, will lie between 8 and 4, or will 
 have a characteristic of 8. Consequently, 
 we find by reference to a table of loga- 
 
 rithms 
 log 3.1416 = 0.4971509 
 “7918 = 3.8986155 
 “ 5280 = 3.7226339 
 8.1184008. 
 
86 THE INTERPRETATION OF 
 
 Consequently | 
 C! = 108:118#003 
 
 and C, lies between 108 and 10% The 
 number corresponding to the mantissa 
 0.1184008 is found from the tables to be 
 1.31341, so that 
 
 C= 1.81341 X 108 or 131,841,000, 
 
 as far as six places of decimals. If we 
 make the computation we find 
 
 3.1416 X 7,918 X 5,280 = 131,340,996,864, 
 
 so that the arithmetically computed result 
 differs from the result obtained from 
 ‘ seven-place logarithms by less than 4 parts 
 in 131 millions. 
 
 If now it should be required to divide 16 
 by 15, or to compute by logarithms 
 
 pete 
 152 
 
MATHEMATICAL FORMULAE. 87 
 
 we have already seen that the logarithm of 
 16, 1s 1.2041200, and of 15, is 1.1760913. 
 Consequently, 
 
 1 (12041200 
 
 ~ 7 ()ii760078 
 — 1 (12081200 y¢ 4 Q— 1-1760018 
 = 1 ()4.7041200 — 1.1760913) 
 
 1 ()(.0280287) 
 
 Here @ is a number lying between 1 and 
 10, because the logarithm has a characteris- 
 tic lying between 0 and 1, and the mantissa 
 0.0280287 is found in the tables to corre- 
 spond to the number 1.06667. The com- 
 puted quotient is found to be 1.0666666 ; 
 or, as this is sometimes written, 1.06. 
 
 As a further example of the use of loga- 
 rithms, consider the following problem. 
 The force of gravitation between two 
 
 homogeneous spheres of masses m, and meg, 
 
88 THE INTERPRETATION OF 
 
 grammes, respectively, is expressed by the 
 equation, 
 
 
 
 where d, is the distance between the cen- 
 tres of the spheres in centimetres, and 
 a, the gravitation constant; a, 1s known 
 to be 6.48 X 10-8, approximately. What 
 then will be the attractive force be- 
 tween the earth and the moon expressed 
 in dynes, if the earth’s mass is 6.02 x 10” 
 grammes, the moon’s mass is 7.525 X 10” 
 erammes, and the mean distance between 
 their centres 3.8444 Xx 10” centimetres? 
 The preceding equation may, therefore, be 
 written : 
 
 6.02 * 107 KT S25e eno 
 3.8444 X 10” x 3.8444 x 10” 
 _ 6.48 X 6.02 X 7.525  10* 
 ~ §8,8444 X 3.8444 «x 10” 
 6.48 6.02. % 7.695 xaos 
 3.8444 x 3,8444 . 
 
 FE = 6.48 X 10° x 
 
 
 
MATHEMATICAL FORMULZ. 89 
 
 Here the logarithms of the three quantities 
 in the numerator, and also of the two 
 quantities in the denominator, have all a 
 characteristic of zero, since the numbers 
 lie between 1 and 10, Consequently, by 
 reference to tables 
 
 log 6.48 = 0.8115750 log 3.8444 = 0.5848286 
 
 log 6.02 = 0.7795965 log 3.8444 = 0.5848286 
 
 log 7.525 = 0.8765065 1.1696572 
 2.4676780 
 
 The numerator of the fraction is, therefore, 
 10747680 While the denominator is 10170, 
 Dividing the numerator by the denomi- 
 nator, or subtracting the logarithm of the 
 denominator from the logarithm of the 
 numerator, we have 
 
 / 
 
 2.4676780 
 1.1696572 
 1.2980208, 
 
90 THE INTERPRETATION OF 
 
 which is the logarithm of 19.8619. Conse- 
 
 quently, 
 Hr = 19.8619 xX 10” 
 
 = 1.98619 x 10” dynes. 
 
 Another important use of logarithms is 
 in involution and evolution. or example, 
 suppose that it be required to find the 
 cube of 5,280, corresponding to the num- 
 ber of cubic feet ina cubic mile. This 
 will be equivalent to solving by log- 
 arithms the equation : 
 
 Vico 2oue 
 If 
 
 5,280 = 10% 
 then 
 
 we = (10°)? = 10™. 
 
 Here a, is the logarithm of 5,280, and 3a, is 
 three times this logarithm, as indeed is 
 evident from the fact that the operation of 
 cubing is multiplication to the number by 
 
 itself twice in succession, or equivalent of 
 
MATHEMATICAL FORMULA. 91 
 
 adding its logarithm to itself twice in suc- 
 cession. The logarithm of 5,280, will le 
 between 3 and 4, and is found from a 
 table to be 
 
 3.7226339. 
 
 Multiplying this by 3, we obtain 
 
 11.1679017, 
 or 
 
 aaa 11.1679017 — 11 0.1679017 
 aw = LOM — 19 x 1 0160017, 
 
 and the number corresponding to the man- 
 tissa, .1679017, is 
 
 1.47198. 
 Consequently, 
 v= 1.47198 X 10" = 147,198,000,000. 
 The true answer is 
 
 aw = 1.47197952 X 104% = 147,197,952,000, 
 
92 THE INTERPRETATION OF 
 
 In the same way, if it should be required 
 to determine the 15th root of the number 
 576, or to compute by logarithms, 
 
 ices 576%, 
 
 we obtain the logarithm of 576, and then 
 divide this by 15. 
 
 2.760422 
 log @ = ee = 0,1840281. 
 
 CO 
 
 which mantissa corresponds to the number 
 1.52767; or, 
 
 eID 2T BT. 
 
 as far as five places of decimals. 
 As an example of the use of logarithms 
 
 in theoretical physics, we may consider 
 the following equation : 
 
 h = (1+ 0.003662) x 1,839,300 log centimetres, 
 
 2 
 
MATHEMATICAL FORMULZE. 93 
 
 where h, is the vertical difference of eleva- 
 tion in centimetres between two stations at 
 which the barometric pressures of p, and 
 . px, are respectively observed, ¢, being the 
 temperature of the air in °C. 
 
 Thus, if 
 ee—o0.20) andy, — 360, ¢=— 15° C,, 
 
 then 
 
 30.25 
 30 
 
 = (1+ 0,549) x 1,839,300 x log 1.008333 
 = 1.0549 X 1,839,300 x log 1.008333 
 = 1.0549 X 1,839,300 X 0.0036040. 
 
 
 
 h =(1+ 0.00866 X 15) X 1,839,300 log 
 
 This triple product, now cleared of loga- 
 rithms, may be computed either arithmet- 
 ically, or by logarithms; the result will be 
 found to be 
 
 h = 6992.7 centimetres. 
 
 All the logarithms we have hitherto con- 
 sidered have been common logarithms with 
 
94 THE INTERPRETATION OF 
 
 10, for the base. In mathematical theory, 
 however, and frequently in applied mathe- 
 inatics, another and a more natural base sug- 
 gests itself. ‘This base, as far as seven deci- 
 mal places, is 2.7182818, and 1s often repre- 
 sented by the symbol « Logarithms to 
 this base are called natwral logarithms, 
 Naperian logarithms, or hyperbolic loga- 
 rithms. For all practical purposes it 1s suf. 
 ficient to remember that a natural logarithm 
 of a number is always greater than the 
 common logarithm of the same number in 
 a definite ratio, which is 2.3026, approx- 
 imately, or 
 
 2.71828" * 2-0 = 10m 
 
 If, therefore, m, is a logarithm in the com- 
 mon system, 2.30262 will be the approx- 
 imate logarithm in the Naperian system. 
 Thus the logarithm of 10 to the base ¢, or 
 
MATHEMATICAL FORMULA. 95 
 
 log, 10 = 1 x 2.3026 = 2.3026, approxi- 
 mately. 
 
 As an example of the application of 
 _ hyperbolic logarithms in apphed mathe- 
 matics, we may consider the following 
 equation: 
 
 W = p, % logs (+). 
 
 Here W, is the work done by a volume of 
 gas expanding at constant temperature 
 from a volume of 2, to a volume 2. Sup- 
 pose, for example, that a volume of 5,000 
 cubic centimetres of a gas under a pres- 
 sure of 2,000 grammes per square centi- 
 metre, expands to a volume of 10,000 cubic 
 centimetres. Then the work done in the 
 process will be 
 
 10,000 
 
 W = 2,000 x 5,000 x log, =m 
 ? 
 
96 
 
 | 
 
 MATHEMATICAL FORMULA. 
 
 10° log, 2 
 
 107 X 2.3026 logy 2 
 
 2.3026 X 10° logy, 2 
 
 2.3026 xX 10° X 0.8010300 
 
 6.93 X 10° centimetre-grammes, 
 
CHAPTER IX. 
 TRIGONOMETRY. 
 
 Trigonometry is the science relating to 
 the measurement of angles in a plane, with 
 particular reference to triangles, and also 
 to figures bounded by straight lines. An 
 angle is measured by the amount of open- 
 ing between two straight lines meeting at a 
 point. In all practical applications, when 
 this opening completes one revolution, it is 
 divided into 860 equal parts called degrees, 
 which in their turn are each divided into 60 
 equal parts called mnutes, and each of these 
 into 60 equal parts called seconds. Con- 
 sequently, a complete revolution is divided 
 into 1,296,000 seconds, or 21,600 minutes, 
 
 or 3860 degrees. Almost all trigonomet- 
 97 
 
98 THE INTERPRETATION OF 
 
 rical tables in practical use refer to these 
 angular units. For theoretical purposes it 
 is found useful to employ a different unit 
 called a radian. Thus in Fig. 2, suppose a 
 
 fixed line OA, and a movable line OD, 
 
 Fie. 2. 
 
 capable of rotating about the fixed point 
 QO, inthe plane AOL. Then, if the point 
 B, traces out an are AB, the length of the 
 arc AJ, is the measure of the angle a, 
 included between OA and OL. If the 
 angle a, be such that the are AS, is equal 
 in length to the moving radius, or vadius 
 vector OB, then the angle a, is 1 radian. 
 
MATHEMATICAL FORMUL®. 99 
 
 This angle expressed in degrees is 57° 17’ 
 44.8", approximately. Ifthe length of the 
 radius vector is taken as unity, then the 
 _magnitude of the angle a, will be the 
 length of the are AL. It will be evident 
 that if we trace out a complete revolution 
 or circle with the radius vector, the length 
 of this circumference will be 27 units, 
 and, therefore, the angle corresponding to 
 a complete revolution is 360°, or 2z 
 radians. Similarly, half a revolution is 
 180°, or z radians; a quarter of a revolu- 
 
 tion is a right angle of 90°, or a radians ; 
 and, generally, an angle of m degrees is an 
 
 angle of 0 x m radians. 
 
 An angle may extend beyond 360°. 
 Thus two complete revolutions represent 
 720°, or 4 radians, and 8 complete revolu- 
 
 tions 1,080°, or 67 radians, and so on. In 
 
100 Titi INTERPRETATION OF 
 
 most practical applications, however, 
 angles of less than 360° are considered, 
 and, in the majority of cases, angles less 
 than 90°. 
 
 An angle is considered positive when the 
 direction of rotation of the radius vector is 
 counter-clockwise; that is, the opposite direc- 
 tion to the rotation of the hands of a clock, 
 when viewed from in front of the clock. 
 If the rotation be made in the opposite 
 direction, or clockwise, the angle is nega- 
 
 tive. Thus the angle AOS, Fig. 3, may 
 
 e v7 
 be regarded either as — 45°, or — ee 
 (7 
 dians; or, as + 315°, or + or radians. 
 
 When the angle considered has the radius 
 vector between A and -A,, Fig. 3, it is said 
 to lie in the first quadrant. When the 
 radius vector lies between A, and A,, it is 
 said to lie in the second quadrant. When 
 
MATHEMATICAL FoRMULA. 101 
 between A, and A,, in the third quadrant, 
 and when between A, and A, in the 
 
 fourth quadrant. 
 Besides the magnitude of an angle itself, 
 
 
 
 as determined in degrees or in radians, 
 there are several important magnitudes 
 connected with it which are called the 
 trigonometrical functions, and which must 
 be memorized since they constantly occur 
 
102 THE INTERPRETATION OF 
 
 in all trigonometrical writings. These 
 trigonometrical functions or ratios, some 
 times called the circular functions, are as 
 follows : 
 
 The sine ; the cosine. 
 The tangent ; the cotangent. 
 The secant, and the cosecant. 
 
 Occasionally the versed sine and coversed 
 sine are added, but they are rarely used. 
 In Fig. 4, the angle a, is represented as 
 being contained between the fixed line 
 OA, called the znztial line, and the radius 
 vector OL. If we let fall a perpendicular 
 DBC, from the point SL, on to the initial line 
 we obtain a right-angled triangle OLC 
 Then the ratio of the length of the per- 
 pendicular LC; to the length of the hypoth- 
 
 enuse, or radius vector OJ, or the frac- 
 
 tion BE. is called the szne of the angle a, 
 
 OL 
 
MATHEMATICAL FORMULE. 103 
 
 and is writtenin abbreviation sin a. Conse: 
 
 i per licular 
 sequently, sin a = BC _ perpendicular 
 
 OB hypothenuse ° 
 
 o--—~------—----y@ 
 
 
 
 Fie. 4. 
 If the hypothenuse or radius vector be 
 chosen of unit length, then the length of 
 the perpendicular is the sine of the angle 
 a,or sn a = BC. Thus, if a = 60° and 
 OB, is unity, it will be found that the 
 length of BC, is approximately, 0.866, so 
 that sin 60° = 0.866, as far as three decimal 
 places. By reference to trigonometrical 
 
104 TIE INTERPRETATION OF 
 
 tables it will be found that sin 60° = 
 
 0.8660254, as far as seven decimal places. 
 The ratio of the length of the base OC, 
 
 to the hypothenuse or radius vector OB, is 
 
 called the cosine of the angle a, so that 
 
 0 IC = cosine a, which is abbreviated to 
 
 OL 
 cosa. If OL, be unity this ratio becomes 
 simply OC, and OC'= cosa. Foranangle 
 of 60° as shown, the length of OC, will be 
 found to be exactly half that of OJ, or 
 
 i 
 cos 60° Sr eee 0.50. 
 
 The eit of the length of the perpen- 
 dicular BC, to the length of the base OC, 
 
 or the fraction ae is called the tangent 
 
 of the angle a, and is abbreviated tan a. 
 
 , BG 
 Thus, for an angle of 60° the ratio OO 
 
 will be found to be 1.732, approximately, 
 and by reference to trigonometrical tables 
 
MATHEMATICAL FORMUL. 105 
 
 the tangent of 60° is given as 1.7320508, as 
 far as seven places of decimals. 
 
 Similarly, the ratio of the length of the 
 base OC, to the length of the perpendic- 
 OC 
 BO? 
 
 the reciprocal of the fraction representing 
 
 wlar LC, or the fraction (which is 
 
 the tangent) is called the cotangent of the 
 angle a, and is abbreviated cot a. Conse- 
 
 OU 
 quently, cot a =—,, the value of the co- 
 
 BO? 
 tangent of an angle, will evidently be the 
 reciprocal of the tangent, so that, since we 
 have seen that the tangent of 60°, or tan 
 60° = 1.732, we know that the cotangent of 
 
 | 1 
 the same angle, or cot 60° = Toa ae 
 
 0.5773503, approximately, as far as seven 
 places of decimals. 
 
 The ratio of the length of the radius 
 vector, or hypothenuse LV, to the length of 
 
106 THE INTERPRETATION OF 
 
 the base OC, or the fraction represented 
 
 
 
 by ea is called the secant of the angle 
 
 a, and is abbreviated see a. Consequently, 
 BC 
 OC” 
 
 be written sec a = 
 
 sec a = 
 
 With OS, unity, this may 
 
 1 
 aE 
 evidently the reciprocal of the cosine of 
 1 
 cos a’ 
 since we have seen that the cosine of 60° = 
 
 0.5, the secant of 60°, or sec 60° = = 
 = 50, 
 
 The ratio of the length of the radius 
 vector or hypothenuse OJ, to the length 
 
 This ratio is 
 
 
 
 
 
 the same angle, or sec a = and 
 
 of the perpendicular BC, or the fraction 
 
 represented by oan is called the cosecant 
 
 of the angle a, and is abbreviated cosec a. 
 
 Consequently, cosec a = =~. With 
 
MATHEMATICAL FORMULA. 107 
 
 OB, unity, this may be written cosec a = 
 
 ia This ratio is evidently the recip- 
 
 BOC 
 
 rocal of the sine of the same angle or 
 
 and, since we have seen 
 
 
 
 cosec a = 
 SIN @ 
 
 that the sine of 60° is, approximately, 
 
 (0.866, the cosecant of 60°, or cosec 60° = . 
 1 
 0.866 
 
 imal places. 
 It is important to remember that 
 
 = 1.1547005, as far as seven dec- 
 
 
 
 
 
 iL 
 Cot we = == 
 tan @ 
 I 
 sec a= 
 COS @ 
 1 
 coset a = — ° 
 sin @ 
 
 If we study the increase of an angle in 
 the first quadrant, it will be seen that the 
 length of the perpendicular LC, which 
 
108 THE INTERPRETATION OF 
 
 with unit radius is the sine, increases 
 steadily, although not proportionately, from 
 0, when the angle is zero, to unity at 90°, 
 when /C, coincides with the radius vector. 
 The sine of an angle may have any 
 numerical value between 0 and 1, but can- 
 not exceed unity. The same is true as 
 regards the cosine, but in the case of the 
 tangent its value commences at zero, when 
 the angle is zero, but increases indefinitely 
 as the angle approaches 90°, because, while 
 the perpendicular “£C’ then approaches 
 unity, the base OC’ becomes indefinitely 
 small and, therefore, the ratio of the per- 
 pendicular to the base becomes indefinitely 
 great. This is expressed by saying that 
 tan 90° = infinity; or, as it is usually 
 written, tan 90 = o. 
 
 Similarly, the cotangent of an angle 
 commences with an indefinitely great 
 value and diminishes as the angle is in- 
 
MATHEMATICAL FORMULA. 109 
 
 creased, until it becomes 0 when a = 90°. 
 The cosecant commences at infinity when 
 a = 0, and diminishes to unity when a 
 -= 90°. The secant commences at unity 
 when a = 0 and increases to infinity 
 when a = 90°. 
 
 As we increase the angle beyond the 
 first quadrant as represented in Fig. 4, we 
 cause the values of the trigonometrical 
 functions to repeat themselves cyclically. 
 Thus, in Fig. 5, the length of the perpen- 
 dicular let fall from the end of the radius 
 vector of unit length upon the initial line 
 is always the sine of the corresponding 
 angle. Thus, when OS, reaches 60° at B,, 
 L, CU; = 0.866, approximately, or sin 60° = 
 0.866. When, for example, OD reaches 
 150° at 6, the length B, C, is 0.5, and 
 sin 150° = 0.5. When OB, passes into 
 the third quadrant, the perpendicular CB, 
 is below the initial line, or is negative in 
 
110 THE INTERPRETATION OF 
 
 value; consequently, at such a point, for ex- 
 ample as OS, corresponding to a = 240°, 
 B, C; = —0.866 and sin 240° = —0.866. 
 
 
 
 
 oO 
 
 
 
 ¥ a 
 Fie. 5. 
 
 Again, when the angle passes through the 
 fourth quadrant the perpendicular LC, is 
 still below the initial line, and is, there- 
 
MATHEMATICAL FORMULA. ria fs 
 
 fore, still negative, so that at the position 
 OB,, corresponding to a = 350°, or —10°, 
 the length L, C,, is 0.1736482, approxi- 
 mately, and sin 350° = sin —10° = 
 —0.1736482 as far as seven decimal places. 
 The sine of the angle is, therefore, positive 
 in the first two quadrants, and is negative 
 in the third and fourth quadrants. 
 
 The cosine of an angle, when described 
 with unit radius vector, is equal to the 
 length of the base OC. In the first quad- 
 rant this base is always positive. ‘Thus, in 
 Fie. 5, cos 60° = OC, = 0.5. In the sec- 
 ond quadrant the base OC, lies on the 
 left-hand side of the origin O, and is, 
 therefore, reckoned as negative. Conse- 
 quently, the cosine of all angles in the 
 second quadrant is negative. At the posi- 
 tion indicated, OL,, the cosine OC, 1s, 
 approximately, —0.866, and cos 150° = 
 —0.866, approximately. Similarly, in the 
 
be TILE INTERPRETATION OF 
 
 third quadrant, the base OC, still lies to 
 the left-hand side of the origin, and is 
 negative, so that the cosine of the angle 
 240° is OC; = —0.5. In the fourth quad- 
 rant, the base falls on the right-hand side 
 of the ongin and is positive. Conse- 
 quently, cos 850° = cos —10° = OG, = 
 0.9848079, approximately. The cosines 
 of angles in the first and fourth quadrants 
 are, therefore, positive, and in the second 
 and third quadrants, negative. 
 
 In the same way the tangent, cotangent, 
 secant and cosecant could be followed out 
 through the different quadrants, always 
 remembering that a base is negative when 
 drawn on the left-hand side of the origin, 
 and a perpendicular is negative when 
 drawn below the origin, while the radius 
 vector remains positive throughout the 
 revolution. It is sufficient to observe, 
 however, that the tangent of the angle is 
 
MATHEMATICAL FORMULA. i og BS’ 
 
 the ratio of the sine to the cosine. Thus, 
 
 m Hig, 5, 
 
 sin a = BC 
 OB 
 
 eh OG 
 we 
 
 ee BCs BO) OB 
 oe” (OC OB. OC 
 
 
 
 
 
 
 
 Consequently, if the sine and cosine of an 
 angle are found, we obtain the tangent by 
 dividing the latter into the former, while 
 the cosecant, secant and cotangent are the 
 reciprocals of the sine, cosine and tangent, 
 respectively. Thus, we have seen that 
 sin 240° = — 0.866, approximately, and 
 
114 THE INTERPRETATION OF 
 
 
 
 that the cos 240° = — 0.5. Therefore, 
 — 0.866 
 tan 240° = = + 1.7382. 
 east) 
 1 
 ) 9AI)O — Ra “ 
 Then cosec 240° = Sn 1.1547 
 E af 
 sec 240 mary 
 cot 240° = Sue = + 0.57735 
 ee yey ; 
 
 The foregoing rules will enable the stu- 
 dent to understand how the trigonometri- 
 cal ratios or functions are defined or 
 measured for all angles. The science of 
 trigonometry deals largely with the appli- 
 cations of these ratios. There are a num- 
 ber of important relations connecting the 
 trigonometrical ratios. Thus, whatever 
 the angle a may be, the following equation 
 holds true, 
 sin? a + cos*a = 1; 
 
 or the square of the sine of the angle a, 
 
MATHEMATICAL FORMULZ. 115 
 
 added to the square of the cosine of that 
 angle, gives unity. 
 Thus, in the case of Fig. 3, 
 
 sin 60° = 0.866 and cos 60° = 0.5. 
 
 Therefore, sin? 60° = (0.866)? = 0.750 
 and cos* 50° = (0.5)? = 0.250. 
 Therefore sin? 60° + cos? 60° = 1.000. 
 
 
 
 From these relations it will be evident 
 that if we know the sine of an angle we 
 ean obtain its cosine, and from these the 
 tangent and the reciprocal values, the’ 
 secant, the cosecant and the cotangent. 
 
 It is not the intention of this book to 
 develop the theory of trigonometry, but 
 merely to enable a student intelligently to 
 comprehend or interpret the meaning of 
 any trigonometrical formula. We must 
 refer him to treatises on trigonometry for 
 such developments as will enable him to 
 solve problems in trigonometry. We take 
 
116 THE INTERPRETATION OF 
 
 the following cases, however, as examples 
 of trigonometrical equations: 
 
 Suppose that a tall object, such as a flag- 
 staff AL, Fig. 6, has an unknown height 
 which it is desired to measure, and that 
 the observer at O, on the same level as the 
 bottom A of the staff, observes the magni- 
 
 
 
 Fig. 6. 
 
 tude of the angle AOL = a, which the staff 
 subtends at hiseye. Then, if the observer 
 measures the length of the base OA, which 
 separates him from the staff, he is at once 
 enabled to compute the height of the staff 
 
 AB 
 AB, because he knows that OA = tan a. 
 
 ~ 
 
MATHEMATICAL FORMULA. ala Be 
 
 If we multiply both sides of this equation 
 by OA, we have 
 
 AB 
 Giaees OA = OA tana 
 
 or ibe A tanto. 
 
 Consequently, the unknown height AB, is 
 equal to the product of the measured base 
 OA, and the tangent of the observed 
 angle. ‘Thus, if OA, were 100 feet and the 
 angle a, was observed to be 30° 35’, then 
 
 AB = 100 X tan 80° 35’; 
 = 100 Xx 0.5910 = 59.1 feet, 
 approximately. 
 
 The attraction of gravitation 1s known 
 to be different at different points of the 
 earth’s surface. Its value depends upon 
 the latitude, and upon the elevation of the 
 place at which the observation is made. 
 The formula is usually expressed as follows: 
 
 g = 930.6056 — 2.5028 cos 27 — 0.000008 h, dynes ; 
 
118 THE INTERPRETATION OF 
 
 where g, is the gravitational force in dynes 
 or C. G. S. (centimetre-gramme-second) 
 units of force, 7, 1s the latitude of the 
 locality, and f, is its height in centt- 
 metres above the level of the sea. 
 
 Thus, at the latitude of New York, which 
 is, approximately, 40° 43’, and at an eleva- 
 tion of 10 metres, or 1000 centimetres 
 above the mean tide level, the gravitational 
 force would be: 
 
 g = 980.6056 — 2.5028 cos 2 (40° 43°) — 
 0.000003 x 1,000; 
 = 980.6056 — 2.5028 cos (81° 26) — 
 0.008 ; 
 = 980.6056 — 2.5028 x 0.14896 — 0.008 ; 
 = 980.6056 — 0.3728 — 0.003; 
 = 980.2298 dynes. 
 
 As another example of a trigonometrical 
 equation we may consider the following: 
 
 t= ZL cos a, 
 
MATHEMATICAL FORMULA. 119 
 
 Here JZ, represents the length of a degree 
 of longitude on the earth’s surface measured 
 at the equator, a is the latitude of any 
 _ place on the earth’s surface and / the 
 length of a degree of longitude at that 
 place. ‘The equation states that the length 
 of the degree varies as the cosine of the 
 latitude, and since the cosine of an angle 
 diminishes from 1 at 0°, to 0 at 90°, it is 
 evident that at the pole the length of the 
 degree of longitude would be indefinitely 
 small, while at the equator itis, approxt- 
 mately, 69.6 geographical miles. The 
 length of a degree longitude at New York, 
 or any place having the latitude of New 
 York, is 
 1 = 69.6 X cos 40° 43°; 
 = 69.6 X 0.75794; 
 = 52.81 miles. 
 
 In the problem of determining the apparent 
 time at sea, and, therefore, with the aid of 
 
120 THE INTERPRETATION OF 
 
 the chronometer, the longitude of a ship, 
 the following equation is employed : 
 
 H SRR 
 cos (=) = sin S sin (S— Z)cosec P cosec C. 
 
 Here Z, is the angular distance of the sun’s 
 centre from the zenith, or point immedi- 
 ately overhead at the ship, as determined 
 by the sextant; P, is the co-declination, or 
 polar distance, of the sun; 2. e., the angular 
 distance of the sun’s centre from the pole 
 of the earth, as it would be seen by an 
 observer at the centre of the earth, if the 
 earth’s mass were transparent; C; is the 
 co-latitude, or the latitude of the ship sub- 
 tracted from 90°; 7. @, the angular dis- 
 tance of the ship from the earth’s pole. S, 
 is the half sum of P, Cand Z; 
 
 aticgeit Oren 72% 
 
 2 
 
 or an angle equal to half the sum of the 
 three angles above defined ; and 7, is the 
 
 or S = 
 
MATHEMATICAL FORMULAE. 121 
 
 hour angle, or the apparent time at the 
 ship, expressed in degrees. The equation 
 states that the cosine of half the hour angle 
 is equal to the square root of the product 
 of four quantities; the first is the sine of 
 the half sum; the second is the sine of the 
 half sum reduced by the angle Z, the 
 third is the cosecant of the polar distance ; 
 and the fourth is the cosecant of the co- 
 latitude. ‘These quantities may be found 
 from trigonometrical tables when the three 
 angles , C’ and Z, are known. It is 
 usual, however, to make the computation 
 with the aid of logarithms as_ already 
 described, since the multiplication of the 
 four trigonometrical functions is performed 
 logarithmetically by adding four loga- 
 rithms, and the square root is then readily 
 obtained by dividing the logarithmic sum 
 by two. For this reason it is customary to 
 seek in the logarithmic tables, not the 
 
122 TIE INTERPRETATION OF 
 
 simple sine or cosecant, but the logarithm of 
 that sine or cosecant. In practice this opera- 
 tion 1s carried out by the mere addition of 
 logarithms by the navigator, without the ne- 
 cessity of writing down the above equation. 
 _ Thus, suppose aship, at 52° 12° 42" north 
 latitude, observes, with a sextant, that the 
 altitude of the sun’s centre above the 
 horizon is 89°5'28"; the declination of 
 the sun, or its angular distance above the 
 earth’s equator, as it might be observed 
 from the centre of the earth, obtamed by 
 tables, being 15° 8'10" north. Required 
 the apparent time at the ship the moment 
 the observation was made. 
 Here Z, the zenith distance 
 of the sun’s centre, is 
 90° — (39° 5’ 28") = 50% 54° 82": 
 P, the polar distance of the 
 sun’s centre, 1s 
 90° — (15° 8’ 10°) = (4 es 
 
MATHEMATICAL FORMULZ. 123 
 
 C, the co-latitude of the ship, 
 is 
 90° — (52° 12’ 49") = 37° 47’ 18" 
 The sum of these angles is 163° 33' 40" 
 and S, the half sum, is 81° 46’ 50”. 
 
 Since S = 81° 46’ 50”, sin S = 0.989726. 
 “« S—Z= 81° 46’ 50" — (50° 5432") = 
 B0° 52’ 18’, sin (S — Z) = 0.51312. 
 “ P = 74° 51' 50’, cosee P = 1.086. 
 “ C= 87° 47 18’, cosec C = 1.682. 
 
 These values are found by reference to 
 trigonometrical tables. 
 The equation becomes, therefore, 
 
 H ane OEY pee gi hf eA) SSeS ee Ee Se hes I , oie eee 
 cos (5) = 4/ 0.989726 x 0.51312 X 1.036 X 1.632. 
 
 = 0.9266 
 = cos 22° 5', approximately. 
 “. a = 99° 5’, 
 
 or HH = 44° 10’. 
 
124 THE INTERPRETATION OF 
 
 Every 15° represents one hour of time, 
 and every 15’ represents 4 minutes of time. 
 
 Consequently, /Z = 2 hours, 56 minutes, 
 40 seconds, approximately. 
 
 Arithmetical computation in this case 
 is seen to be lengthy and tedious, but it 
 is greatly facilitated by employing log- 
 arithms as follows: 
 
 Since S’= 81° 46’ 50" the 
 logarithm of the sine of S, by 
 
 tables, is = 9.99552 
 (S — Z) = 30° 52' 18’, log sin 
 (S— Z)orLsin(S—Z) = 9.71020 
 P = 74° 51’ 50", log cosec P, 
 or L cosee P = 10.01535 
 C’= 87° 47' 18", log cosee C; 
 or LZ cosec C = 1021273 
 
 The logarithm of the product 39.93380. 
 Dividing this sum by 2, to | 
 extract the square root, = 19.96690. 
 
MATHEMATICAL FORMULZ. 125 
 
 This corresponds to the logarithm of 
 
 ~/ 
 
 the cosine of 22° 5 
 
 .. = 22° 5’ 00’ or H = 44° 10’ 00’, 
 and this angle expressed in time, allowing 
 15° to the hour, 15’ to the minute, and 15” 
 to the second, gives 2h."56m. 40s. 
 
 It will be observed that in trigonomet- 
 rical tables, 10 is always added to the 
 characteristic of a logarithm, so as to 
 avoid the use of negative characteristics. 
 The logarithm 10.01535, therefore, really 
 represents 0.01535, while 9.01535 repre- 
 sents 1.01535, and so on. 
 
 The area of a triangle two of whose 
 sides have lengths @ and c, while their 
 included angle is #, is expressed by the 
 equation 
 
 A = 5 aesin B 
 
 This equation states that the area is 
 equal to half the product of the two sides 
 
126 THE INTERPRETATION OF 
 
 into the sine of the included angle. Thus, 
 if @ = 50 feet, c = 60 feet and JS, the 
 included angle = 45°, sin L = 0.7071068, 
 and 
 
 A =4-x 50 x 60 x 0,70710Gee 
 1,500 x 0.7071068: 
 = 1,060.66 square feet. 
 
 It is sometimes convenient to employ in 
 trigonometrical equations what is called 
 inverse notation, with the meaning of 
 which, therefore, the student should be 
 familiar. 
 
 The ordinary expressions sin a, cos a, 
 tan a, sec a, cosec a, cot a, refer to the 
 trigonometrical ratios of the angle a. The 
 inverse expressions sin~' a, cos~’ a, tan! a, 
 sec”’ a, cosec! a, and cot~’ a express the 
 angles of which the sine, cosine, tangent, 
 secant, cosecant, and cotangent are respec- 
 tively equal to the quantity a. 
 
MATHEMATICAL FORMUL. LO 
 
 Thus tan-* 1 is an angle whose tangent 
 is-1, or is 45°. 
 cos“ 1 is an angle whose cosine is 
 L, ons. 
 sin-' (4) is an angle whose sine is $ 
 or 30°. 
 The equation 
 
 tan7! 9=—ea 
 
 means that a, is an angle whose tangent is 
 2,or 63° 26’. This will be evident on tak- 
 ing the tangent of both sides of the 
 
 equation : 
 tan (tan 2) = tan a, 
 
 Here the two operations tan, and tan—', are 
 inverse to each other or cancel, leaving 
 
 = tan a 
 
 from which, by trigonometrical tables, 
 a = 63° 26, approximately. 
 
CHAPTER X. 
 
 HYPERBOLIC TRIGONOMETRICAL FUNCTIONS. 
 
 We have seen that plane trigonometry 
 deals with the trigonometrical or circular 
 functions: 7. é., those in which the radius 
 vector describes a circle, as shown in Figs. 
 2, 3,4 and 5. There are, however, certain 
 mathematical applications in which the 
 functions are hyperbolic; 7. e, those in 
 which the radius vector describes a hyper- 
 bola. In Fig. 7, the curve A LCis part of 
 a rectangular hyperbola ; 7%. e., part of a sec- 
 tion of a right cone whose vertical angle is 
 90°, when cut by a plane parallel to the 
 axis of the cone. If the distance OA, 
 which is called the semd-axis of the curve, 
 
 is taken as unity, and the radius vector OB, 
 128 
 
MATHEMATICAL FORMULA. 129 
 
 turning around the fixed point O, moves 
 over the hyperbola ABC; then twice the 
 area comprised between OA, OB, and the 
 curve ABC, is the hyperbolic angle de- 
 
 
 
 Fig. 7. 
 
 scribed by the radius vector. This hyper- 
 bolic angle is not equal to the angle A OB, 
 between the semi-axis OA and the radius 
 vector OF. Thus, the shaded area 
 marked OAS, in the figure is, approxi- 
 
130 THE INTERPRETATION OF 
 
 mately, 0.44 square inches, if the base or 
 semi-axis (A, is one inch. Consequently, 
 the hyperbolic angle traced out in this 
 case by the radius vector OB, is 2 X 0.44 = 
 0.88, approximately. As the radius vector 
 advances along the hyperbola ALC, the 
 shaded area rapidly increases, and with 
 it the hyperbolic angle, but even when the 
 radius vector is infinitely long, or extends 
 over an infinite length of the curve ADC, 
 the circular angle between OA and OB, 
 cannot exceed the angle AOJF, since OF, is 
 what is called the asymptote of the curve, 
 or the line which continually approaches 
 the curve ALC, but only meets it at 
 infinity. 
 
 If we let fall a perpendicular BD, from 
 the end 5, of the radius vector upon the 
 base line OY, the length LD, is called 
 the hyperbolic sine of the hyperbolic angle. 
 Thus, in the case represented in Fig. 7, 
 
MATHEMATICAL FORMULA. 131 
 
 BD =1,if OA = 1. Consequently, the 
 hyperbolic sine of the hyperbolic angle 
 0.88 is 1. This is represented sym- 
 bolically by the equation: 
 
 sinh 0.88 = 1, 
 
 which may be read “the h-sine of 0.88 is 
 unity.” 
 
 Similarly, the length OD, from the 
 origin to the foot of the perpendicular LY, 
 is the hyperbolic cosine of the hyperbolic 
 angle; or 
 
 OD = cosh a, 
 
 which is read the length OP (when OA 
 = 1) isthe h-cosine of the hyperbolic 
 angle a In the case represented in Fig. 
 
 (, a = 0.88, approximately, and OD = 
 1.41, approximately, Consequently, 
 
 1.41 = cosh 0.88. 
 
 From these two quantities all the remain- 
 
13D THE INTERPRETATION OF 
 
 ing hyperbolic functions may be immedi: 
 ately deduced. | 
 
 
 
 sinh 
 Thus, tant w= re 
 : cosh a 
 3. 
 seth “aw. = see 
 % cosh a’ 
 1 
 cosech a = ———= 
 sinh a’ 
 ecoth a = oe 
 tanh a 
 
 These relations are precisely similar to 
 those which apply in the case of the cir- 
 cular functions, that is to say, the tangent 
 is the ratio of the sine to the cosine, while 
 the secant, cosecant and cotangent are 
 respectively reciprocals of the cosine, sine 
 and tangent. In the case of Fig. 7, 
 
 tanh 0.88 = a5 = 0.707, approximately ; 
 
 il 
 
 sech 0.88 1.414 
 
 = 0.707, approximately ; 
 
MATHEMATICAL FORMULA. 133 
 
 cosech 0.88 = = 1.0, approximately ; 
 
 tek] 
 
 coth 0.88 = O70% 1.414, approximately. 
 
 As the radius vector advances, the 
 hyperbolic sine and cosine both increase 
 indefinitely, while their ratio tanh a, 
 approximates more and more nearly to 
 unity. 
 
 Tables are published of the numerical 
 values of the hyperbolic functions, and 
 by means of these tables it will always 
 be possible to apply numerically such 
 formule as appear in technical works. 
 
 As an example of the use of hyperbolic 
 functions we may consider the following 
 formula, which gives the deflection at the 
 end of a horizontal beam of uniform cross- 
 section built into a solid wall or support 
 at one end, and having a weight or load 
 
134 MATHEMATICAL FORMULA. 
 
 FP, attached to the free end, accompanied 
 by a horizontal tension Q. 
 Pate ap (7 _. tanh at) 
 d n 
 where /, is the length of the beam, 
 n=4/ @_ 
 WME 
 where /, is a co-efficient of elasticity, Z, the 
 moment of inertia of the section. ‘Then 
 the formula states that the deflection at 
 the free end of the beam is equal to the 
 quotient of P by Q, into a compound term 
 within the bracket, the first term of which 
 is the length of the beam, and the second, 
 which is negative, is one nth of the hyper- 
 bolic tangent of the product of the length 
 and ». It, for example, n/ = 2.0 say, then 
 a table of hyperbolic tangents would show 
 that tanh n/ = tanh 2 = 0.964, approxi- 
 mately. 
 
CHAPTER XI. 
 DIFFERENTIAL CALCULUS. 
 
 Suppose an observer is carried on 
 a railway train and that he wishes to 
 determine by observations the speed at 
 which the train is moving from time to 
 time. It is, of course, possible that he 
 might connect a centrifugal indicator with 
 the car axle and cause the instrument to 
 automatically indicate or record the speed 
 from moment to moment, and such instru- 
 ments are actually used for the purpose, 
 but without such an instrument he would 
 be obliged to make an observation of the 
 distance through which the train ran in a 
 given time taken from a_ stop-watch. 
 
 Thus, if in one minute by the watch the 
 135 
 
136 THE INTERPRETATION OF 
 
 train ran exactly three-quarters of a 
 mile, then the speed of the train is evi- 
 dently 3/4ths of a mile per minute, or 45 
 miles per hour. This is not stating that 
 the train actually maintained 45 miles per 
 hour throughout the minute under obser- 
 vation, because the speed might actually 
 have been, say 50 miles an hour during 
 some portion of the minute, and, perhaps, 
 40 miles an hour at some other portion. 
 It is only a statement that the mean speed 
 during the minute was 45 miles per hour. 
 If, however, the observer could measure 
 10 seconds of time accurately, by a stop- 
 watch, and observed that the train ran 
 650 feet in this time, then the speed of 
 the train during 10 seconds would be an 
 average of 65 feet per second; or, 65 x 
 225,000: 
 5280 
 = 42.61 miles per hour. In this case the 
 
 3600 = 225,000 feet per hour = 
 
MATHEMATICAL FORMULZE. 137 
 
 computed speed would more nearly repre- 
 sent the actual speed of the train at the 
 time of observation, because during 10 
 seconds there would be less time for the 
 speed to vary than during 60 seconds, 
 but still the observer could not say posi- 
 tively that the speed did not vary during 
 10 seconds, and he would be still some- 
 what uncertain of the exact speed when 
 he commenced the observation. Now 
 supposing that it was possible for him to | 
 make a measurement during a single 
 second, however difficult this might be in 
 practice, then if the distance run-through by 
 the train was exactly 66 feet in this second, 
 the speed would be 66 feet per second, or 
 45 miles per hour. The observer would 
 now be satisfied that he had determined 
 the speed of the train at the moment when 
 the observation was made with a greater 
 degree of accuracy, because there was so 
 
138 THE INTERPRETATION OF 
 
 little time for variation to take place. 
 But if the train had its brakes applied at 
 the instant when the second of time was 
 observed, it is quite possible that its speed 
 would be rapidly falling, and that the 
 speed at the commencement of the second 
 would be appreciably greater than at the 
 end of the second, so that even here, in the 
 case of a rapidly retarded train, the time 
 of observation could be too great. Reason- 
 ing, however, in this way we might assume 
 that supposing it were possible to make an 
 observation of the distance run through by 
 the train in an exceedingly small interval 
 of time which we may represent by 44, 
 (pronounced Delta 7¢), say the 1/1000th 
 part of a second, the corresponding short 
 distance which we may represent by 4s, 
 (pronounced Delta s) would enable the 
 velocity of the train to be computed from 
 
 the quotient ore with a very high degree 
 
MATHEMATICAL FORMULA. 139 
 
 of accuracy; and that, theoretically, if ¢, 
 were made indefinitely small, as repre- 
 sented symbolically by d¢, the correspond- 
 ing vanishingly small distance ds, would 
 enable the true velocity to be computed 
 from the equation 
 os 
 
 dt i 
 In this equation ds, taken by itself, has no 
 meaning, because it is the space run 
 through by the train in an indefinitely 
 small interval of time; and so too, df, 
 taken by itself, has no meaning, because it 
 stands for an indefinitely small interval of 
 
 time, but the fraction a can be consid- 
 
 ered as the limit of = as ¢, is reduced 
 
 indefinitely, standing for a_ perfectly 
 definite notion ; namely, the instantaneous 
 velocity of the train when the time chosen 
 
140 THE INTERPRETATION OF 
 
 for observing it is so extremely short that 
 there is no possibility of error due to 
 variation of speed in that time. 
 
 In the language of the differential cal- 
 culus ds, is the differential of the space 
 run through by the train; dé, is the differ- 
 ential of the time during which the obser- 
 
 vation of distance 1s measured, and 
 
 “ 
 the differential coefficient of the space s, 
 run through with respect to the time t. A 
 differential coefficient has, therefore, 
 always the meaning of a ratio between 
 the variation of two quantities carried 
 to a limit. Thus the differential coeft- 
 
 cient , 1s the limit of the ratio = when 
 At, is made indefinitely small. 
 
 The object of the differential calculus is 
 to determine from a known relation be- 
 
 tween two connected quantities the rate 
 
MATHEMATICAL FORMULZ. 141 
 
 at which one varies when a variation 1s 
 given to the other. Suppose, for example, 
 that we consider the case of a stone falling 
 to the ground from a given elevation. 
 We know by observation, in connection 
 with the theory of the subject, that the 
 Jaw which expresses the distance through 
 which the stone will fall in a given time 
 ¢ seconds, neglecting air-friction, 1s 
 
 coi 596 feet, 
 
 where g, is the constant representing the 
 earth’s gravitational acceleration, or 32.2 
 feet per-second-per-second, approximately. 
 Consequently, the formula becomes 
 
 Salo le feet, 
 approximately. 
 Having given this relation between the 
 time ¢, and the distance fallen through s, 
 the velocity at any instant is evidently 
 
142 THE INTERPRETATION OF 
 
 defined in some way by this relation. We 
 might imagine that the observation was 
 made at any particular instant, say ¢ = 3, 
 or three seconds after the stone com- 
 menced to fall. At this time s = 16.1 
 
 xX 3 
 
 = "16.1 K29° S144 Steer 
 
 By taking a small interval of time, say the 
 one-tenth part of a second, thereby mak- 
 ing ¢ = 3.1, we find that the space fallen 
 through up to this time is 
 
 s = 16.1 x (8.1)? = 154.72 feet, 
 
 so that, in the interval of time 0.1 second, 
 the space fallen through will have been 
 154.72 — 144.9 
 
 = 9.82 feet 
 
 or 4s = 9.82 feet, and 4¢, the interval 
 of time, 0.1 second; so that the velocity, 
 
MATHEMATICAL FORMULA. 148 
 
 as gauged from this interval, will be 
 4s 9.82 
 ea 0,1 
 But it will be evident that this is only a 
 mean velocity during the time 4¢, since 
 
 = 98.2 feet per second. 
 
 the stone is constantly accelerated, and it 
 will be somewhat in excess of the actual 
 velocity of the stone at the moment ¢ = 3. 
 
 The differential calculus supplies rules 
 for determining the instantaneous velocity, 
 
 or, in this case, the limit of > when 4, 
 
 instead of being 0.1 second, is reduced 
 indefinitely, and which, therefore, enable 
 
 the instantaneous velocity V = to be 
 
 ds 
 
 dt’ 
 
 determined accurately. It might be 
 
 shown by rules of the differential calculus 
 
 that the true velocity is gt, or 32.2 4, so 
 ds 
 
 = 82. 
 that a 32.2 ¢, 
 
144 THE INTERPRETATION OF 
 
 so that the velocity of the stone at the end 
 of the third second, or when ¢ = 8, is 
 2 = 96.6 feet per second, 
 t 
 
 and we should find that the result ob- 
 tained above, of 98.2 feet-per-second grad- 
 ually approached, the instantaneous 
 velocity 96.6, as the interval of time 4¢, 
 was reduced from 0,1 second to a vanish- 
 ingly small quantity. 
 
 In the case of the moving train, the 
 differential calculus would be of no assist- 
 ance to the observer, unless he knew the 
 law which connects the distance traversed 
 by the train with respect to time. We 
 have only alluded to the case of a mov- 
 ing train in order to give a conception of 
 the limiting rate which is so constantly 
 dealt with in the theory of the differential 
 calculus, 
 
MATHEMATICAL FORMULA. 145 
 Whenever a differential coefficient such 
 
 a -/_,.is met with in a formula, it is to be 
 
 Bs 
 
 regarded as an actual quotient or fraction ; 
 namely, the limit which the fraction a 
 8 
 
 assumes when 4s, is indefinitely reduced. 
 
 It is always to be remembered that ds, 
 in the differential calculus is not the prod- 
 uct of the quantity d, into a quantity s, as 
 it would be in ordinary algebra, but an 
 abbreviation for “ differential of s.” 
 
 As another example, consider the case 
 of a reservoir of water, discharging 
 through a pipe, and let g, be the quantity 
 of water in the reservoir, and ¢, the time. 
 Then, during discharge, there will be a 
 certain flow of water through the pipe. 
 If g, be expressed in cubic centimetres 
 and ¢, in seconds, we may ascertain that 
 the rate of discharge in a given time Jt, 
 
146 THE INTERPRETATION OF 
 
 say 10 seconds, amounts to 4g = 15,000 
 cubic centimetres. Then the average floy 
 
 of liquid through the pipe will be 
 
 
 
 Aq _ 15,000 
 
 At Timer 
 
 cubic centimetres per second. In order to 
 obtain the true instantaneous value of the 
 flow, it would be necessary to consider 
 the interval of time 4¢, in which the 
 measurement was made, indefinitely re- 
 duced, as represented by the symbol di, 
 and the corresponding .quantity of escape 
 
 in that time dg, so that the instantaneour 
 flow will be 
 
 a 
 
 She ae 
 In practice the differential calculus could’ 
 not here be employed, unless the relation 
 
 between the quantity of water in the 
 reservoir and the time was known, but the 
 
MATHEMATICAL FORMULA. 147 
 dy 
 dt? 
 
 namely, the instantaneous rate of change 
 
 conception afforded by the symbol 
 
 in the quantity with respect to time, or 
 the instantaneous flow, would be perfectly 
 definite, whether it could be employed 
 practically or not. 
 
 Again, we may consider the quantity of 
 heat g, which must be given to a gramme 
 of a substance to raise its temperature 6°C. 
 
 26’ 
 js the mean thermal capacity of the sub- 
 
 If we take the quotient 47 this quotient 
 
 stance. Thus, if we take 46, as unity or 
 as 1° C., then 4Q, is the amount of 
 heat which must be given to a gramme of 
 the substance to raise its temperature 
 1°C. This quantity of heat is generally 
 not the same at different temperatures, 
 so that the quantity 4@Q, which would 
 have to be given to a substance to raise it 
 
148 THE INTERPRETATION OF 
 
 from 5° to 6° C. is not the same as that 
 required to raise it from say 31 to 32° C. 
 Consequently, the thermal capacity varies 
 with the temperature, and a strict defini- 
 tion is given by the equation 
 
 
 
 C= et or the limiting value of hs 
 when the variation 46, is made indefinitely 
 small. This may be regarded as_ the 
 instantaneous rate of change of heat with 
 respect to temperature, at any given tem- 
 perature. 
 
 It is shown in works on electricity that 
 if a loop of wire be subjected to variations 
 in the quantity of magnetic flux 6, which 
 passes through it, either by being moved 
 through magnetic flux, as by passing the 
 poles of a magnet, or by having magnetic 
 flux moved through it, as by moving a 
 magnet past it, the electromotive force 
 
MATHEMATICAL FORMULA. 149 
 
 (abbreviated E. M. F.), generated in the 
 loop, or the electric pressure tending to 
 cause an electric current to flow through 
 it is represented by the formula 
 
 dp 
 eet 3 
 
 where 6, is the magnetic flux, e, the 
 E. M. F. expressed in C. G. S. units and /, 
 is the time expressed in seconds. This 
 formula means that the E. M. F. is the 
 instantaneous time-rate-of-change of mag- 
 netic flux through the loop. If the loop 
 at a certain instant, ¢ = 10, contains, say 
 1,000,000 C. G. S. units of magnetic flux, 
 and if in the next half second or at ¢ = 
 10.5, the magnetic flux had increased to 
 1,100,000 C. GS. units, the increase would 
 amount to 100,000 in the time 0.5 second, 
 and the average rate of increase during the 
 
 half second would be 
 
150 THE INTERPRETATION OF 
 
 100,000 4d | 
 ye = op = 200,000 
 
 so that the average E. M. F. would be 
 200,000 C.G.S. units. If the change took 
 place uniformly, this would be the actual 
 iE. M. F. throughout the half second, but 
 if the change occurred irregularly, the 
 E. M. F. might have a value at some part 
 of the time greatly in excess of 200,000 
 units, and, at some other part of the 
 time, greatly in deficit. But at any 
 instant, if we conceived that the interval 
 of time 4¢, be reduced indefinitely, the 
 
 ratio so obtained of which would 
 
 i 
 thereby become iss is the true E. M. F. at 
 ad 
 
 40 
 At’ 
 
 that instant, no matter how rapidly the 
 rate-of-change in flux may be varying. 
 If there be any known law connecting the 
 variable 6, or flux, called the dependent 
 
MATHEMATICAL FORMULA. 151 
 
 variable, with the variable ¢, or time, called 
 the endependent variable, then, from this 
 law, it will be possible, by the rules of the 
 
 differential calculus, to determine how ao 
 
 dt’ 
 will vary from moment to moment, or what 
 the instantaneous I. M. I. in the loop will 
 be from moment to moment. Thus, if the 
 flux through the loop increases steadily 
 with time, so that say @ = at, then the rules 
 of the differential calculus show what is 
 evident on reflection, that since each second 
 adds an amount of flux to the loop equal to 
 a, the E. M. F. will be @ units steadily; or 
 
 I® 
 AT — i A 
 
 As another example, consider a simple 
 pendulum consisting of a fine thread at- 
 tached to a small heavy particle acting as 
 the bob, This pendulum may have a cer- 
 
152 THE INTERPRETATION OF 
 
 tain length / cms., and it will have a cer. 
 tain periodic time, or time of one complete 
 double swing, Z’seconds. Then, if we vary 
 the length /, in any manner, and study the 
 effect of this variation upon the periodic 
 time, we make /, the independent variable or 
 the variable whose change is arbitrary, and 
 make 7} the dependent variable, or the vari- 
 able whose changes are not arbitrary, but 
 are determined by the changes inf The 
 law connecting the period Z} with the 
 length 7 is expressed algebraically as 
 
 L = ony/ seconds, 
 
 where z, is the ratio 3.14159... of a cir- 
 
 follows: 
 
 cumference to its diameter, /, is the length 
 of the pendulum in centimetres, and g, the 
 accelerating force of gravity, at the loca- 
 tion considered, in centimetres-per-second- 
 
MATHEMATICAL FORMULA. 153 
 
 per-second; so that the periodic time is 
 6.2832 multiplied by the square root of 
 the length divided by the gravitational 
 constant. If now we make a change in /, 
 we make a corresponding change in 7} be- 
 cause as we lengthen the pendulum it 
 swings more slowly, but the rate of change 
 in 7; with respect to /, or the rate-of-change 
 of periodic time per centimetre increase in 
 pendulum length, is 
 
 AT’ 
 Ale 
 
 where 4/, is a given increase in /, and 47} 
 the corresponding increase in 7! This, 
 however, is only an average rate of increase 
 in periodic time, for the interval 4/, con- 
 sidered, and is not the true rate at that 
 length. Thus, suppose g = 981 centime- 
 tres-per-second-per-second, and 7 = 100 cen- 
 timetres or one metre, 
 
154. THE INTERPRETATION OF 
 
 100 
 981? 
 = 6.283 VU.1019368 ; 
 6.283 xX (0.8192); 
 
 = 2.0055 seconds. 
 
 = 
 | 
 
 2 X 3.1416 X 
 
 The half cycle, or single vibration, would, 
 therefore, take 1.00275 seconds, so that the 
 pendulum one metre long would be nearly a 
 second’s pendulum. If now we increase 
 the length by one centimetre or make 
 Al = 1, 
 
 T= 2 X 3.1416 x a 
 6.283 »/0.10294 ; 
 6.283 (0.32085) ; 
 
 9.0159 seconds. 
 
 [| 
 
 Consequently, 
 AT = 2.0159 — 2.0055 = 0.0104 seconds, 
 
 and 
 
MATHEMATICAL FORMULA. 155 
 
 meee 00.104.” é. 
 Maes 107 0.0104 second per cen- 
 
 
 
 timetre. 
 
 This is the average rate of change in the 
 period of the pendulum per centimetre of 
 increase in its length, but the actual rate 
 of change when /= 100, is this ratio when 
 Al, is so far reduced that the pendulum 
 does not sensibly alter in length; or, 
 
 al’ 
 
 dl © 
 This differential coefficient of Z} with re- 
 spect to /,is known, by the rules of the 
 differential calculus, to be, 
 
 Cie ae, 
 db Wig 
 
 When 7 = 100, and g = 981, this is 
 nade 3.1416 
 
 di ~ /981 x 100° 
 
156 THE INTERPRETATION OF 
 
 _ 8.1416 
 
 = 758100 
 
 81416» 
 
 TEP Ey: 
 
 = 0.01003 seconds-per-centimetre. 
 Consequently, the rate of increase of per- 
 iodic time per centimetre of length is 
 0.01003 seconds-per-centimetre, in a metre 
 pendulum, whereas the approximate process 
 of calculation, assuming the actual variation 
 of one centimetre, gave 0.0104 seconds-per- 
 centimetre. 
 
 If a body move in time 4, seconds, 
 through s, centimetres, the instantaneous 
 velocity is expressed by the differential 
 coefficient of the space described with re- 
 spect to the time, or 
 
 v= ds timetr d 
 = 7 centimetres per second. 
 
 But if we consider the body moving witha 
 velocity v, centimetres per second, in time 
 
MATHEMATICAL FORMULA. 157 
 
 t, seconds, this velocity may be increasing 
 or diminishing, and its time-rate-of-increase 
 is called acceleration. This instantaneous 
 
 : : ; et Av 
 time-rate-of-increase is the limit of Ti 
 
 and becomes 
 
 But if we substitute for v, in this equa- 
 tion its equal 
 
 we obtain 
 
 ds 
 _ Aa) 
 pre ae 
 
 centimetres-per-second-per-second, and this 
 is written 
 d*s 
 ie dp’ 
 so that 
 ds 
 dé 
 
158 THE INTERPRETATION OF 
 
 is called the second differential coefficient 
 of s, with respect to ¢, to distinguish it from 
 ds 
 dt’ 
 which is the first differential coefficient of 
 s, with respect to 7. In the same way 
 
 d*s 
 
 dt?’ 
 
 which is the third differential coefficient or 
 the differential coefficient of the differen- 
 tial coefficient of the differential coefficient 
 with respect to ¢, and so on. It would 
 mean the instantaneous time rate of accel- 
 eration. 
 
 A differential equation is an equation 
 which contains differentials or differential 
 cofficients. We will consider such an equa- 
 tion with a view of interpreting its mean- 
 ing. 
 
 In Helmholtz’s “Sensation of Tone,” 
 
MATHEMATICAL FORMULA. 159 
 
 App. IX., appears the following equation 
 in relation to the vibration of a tuning fork: 
 2 
 
 mM. ee = = an ye 2 + Asin ws, 
 where 2, is the excursion of a heavy vibrat- 
 ing point from its mean position of rest, @ 
 is a coefficient of elasticity, 4, is a coeffii- 
 cient of frictional opposition to motion, A, 
 is the amplitude or maximum value of an 
 impressed vibratory force, ¢, the time in 
 seconds starting from some particular 
 epoch, », a number measuring the fre- 
 quency of the variations of the impressed 
 force, and m, is the mass of the heavy body 
 -acted upon. The meaning of these sym- 
 bols is given in the text referred to. 
 
 The quantity on the left-hand side of the 
 equation is the product of the mass m, and 
 
 de 
 de’ 
 
160 THE INTERPRETATION OF 
 
 the second differential coefficient of the ex- 
 cursion 2, with respect to time. The first 
 differential coeificient, 
 
 dx 
 
 dt 
 would represent the instantaneous velocity 
 of the body at the time considered. The 
 second differential coefficient, 
 
 ad*e 
 
 dt? 
 is the instantaneous rate-of-change of ve- 
 locity, or the instantaneous acceleration. 
 The right-hand side of the equation has 
 three terms. The first is the product of 
 the excursion @ and the coefficient of elastic- 
 ity — a. This is a force tending to restore 
 the body to its initial condition of rest, and 
 increases as the excursion increases. The 
 second term is the product of the coefi- 
 cient — 0°, and the instantaneous velocity 
 
MATHEMATICAL FORMULA. 161 
 
 i) 
 dt 
 This is a force of friction tending to op- 
 pose the motion, and increasing directly 
 with the velocity. The third term is the 
 impressed vibratory force, acting on the 
 body, and is the product of the constant A, 
 and the trigonometrical expression sin (72). 
 As #, increases, the angle represented by nt, 
 increases proportionately, and passes suc- 
 cessively at a definite rate through all four 
 quadrants. The sine of this angle, repre- 
 sented by sin(vt) will, therefore, pass 
 successively through 0, + 1, 0, — 1, 0, +1, 
 ete., and all intermediate values. Conse- 
 quently, the third term will pass through 
 the values 0.4, 0 — A, 0, + A, etc; and 
 all intermediate values, which 1s equivalent 
 to the statement that the impressed force 
 alternates at a definite rate between the 
 values + A and — A, according to a sine 
 
162 THE INTERPRETATION OF 
 
 law. The equation, therefores, states that 
 at any and every instant the force of iner- 
 tia, which is the product of the mass and 
 acceleration, is equal to the sum of all the 
 forces acting on the body, the first being 
 the elastic force of restitution, the second 
 the frictional force of retardation, and the 
 third the impressed force. 
 
 It will thus be evident that even when 
 the student is not sufficiently acquainted 
 with the rules of the differential calculus to 
 deduce differential coefficients, or manipu- 
 late them in equations, he will, neverthe- 
 less, be able to interpret the meaning of 
 the equations which appear in_ technical 
 works employing differential calculus. 
 
 It is shown in treatises on dynamics that 
 if a rigid body be free to move about a 
 fixed axis and has a moment of inertia A, 
 gramme-cm’s; and is acted upon by any 
 force which at time ¢ seconds, exerts a mo- 
 
MATHEMATICAL FORMULA. 163 
 
 ment about the axis of d/ dyne-cms, the 
 following equation holds at any and every 
 instant 
 ee eit 
 de K 
 ao . a 
 
 Here 7g) 18 the second differential coef- 
 ficient of the angle 6, described by the body 
 about the axis with respect to the time ¢, or 
 is the instantaneous time-rate-of increase of 
 the instantaneous time-rate-of-increase of 
 angle. If we call the instantaneous time-rate- 
 of-increase of angle the instantaneous angu- 
 lar velocity, and if we call the instantaneous 
 time-rate-of-increase of angular velocity, the 
 instantaneous angular acceleration, then the 
 equation states that the angular acceleration 
 at any instant is the quotient of the mo- 
 ment of the applied force divided by the 
 moment of inertia of the body. It may be 
 possible to determine by the rules of the 
 
164 MATHEMATICAL FORMULA. 
 differential calculus and so make use of the 
 equation numerically, but even if the infor- 
 mation is not forthcoming by which the 
 equation may be practically applied, or if 
 the knowledge is not available by which 
 from the data of the problem the computa- 
 tion may be made, still the interpretation of 
 the meaning of this equation carries to the 
 mind of the student a definite and perfectly 
 intelligible law. 
 
CHAPTER XIL 
 INTEGRAL CALCULUS. 
 
 We have seen that the principal object 
 of the differential calculus is to determine 
 the limiting ratio of the variation in a de- 
 pendent variable, with respect to an indefi- 
 nitely small variation of an independent 
 variable with which it is connected. The 
 principal object of the integral calculus is to 
 effect the inverse operation; namely, to de- 
 termine from the limiting ratio of variation 
 between a dependent and an independent 
 variable, the fundamental relation between 
 these variables. ‘Thus, from the known 
 law that the distances fallen through by a 
 stone in time ¢ seconds, is expressed by the 
 
 formula 
 165 
 
166 THE INTERPRETATION OF 
 ' 1 2 
 Si = 397, 
 
 where ¢, 1s the independent variable, and 
 s, the dependent variable, the differential 
 calculus determines the limiting relation 
 between their variations, or gives the dif- 
 ferential coefficient 
 
 ae 
 dt it ie g ) 
 
 and states that the instantaneous velocity 
 atany moment is equal to the product of 
 the gravitational acceleration-constant g, 
 and the time of descent ¢, or that for any 
 very minute change in the independent 
 variable amounting to dt, the correspond- 
 ing change in the dependent variable is 
 
 ds = gt.dt. 
 
 The integral calculus would find its ap- 
 plication in the inverse relation: Having 
 given the observed fact that the velocity 
 
MATHEMATICAL FORMULA. 167 
 
 of a falling stone is éxpressed by the 
 formula, 
 
 what will be the relation between the 
 primitive variables s and ¢, corresponding 
 to this condition? We shall see that any 
 such inquiry naturally leads to the summa- 
 tion of an indefinitely great number of in- 
 definitely small terms. 
 
 Fig. 8, shows a straight line O V, drawn 
 from the origin O, to a scale such that in 
 each second of time, as measured along the 
 base or axis of abscisse O7' the elevation 
 of the line represents the velocity of the 
 falling stone, Thus, at ¢ = 1 second, the 
 elevation of the point on the line above 1 
 on the base, represents, to scale, 32.2 feet 
 per second; at 2 seconds it is 64.4, and so 
 on for all times, integral or fractional, in- 
 cluded between¢ = 0 and ¢ = 2.5 = 7: 
 
168 THE INTERPRETATION OF 
 
 This line, therefore, expresses graphically 
 the equation 
 
 — gt, 
 where g = 382.2, as far as ¢ = 2.5 seconds. 
 
 
 
 
 
 
 Joan 
 
 
 
 
 Ld nes ae OLA ac Be eS 
 
 _ a Pes a ens 
 a OE 8 SRT ee 
 
 34 
 Fie. 8. t 
 
 It will be evident that if we consider the 
 moving stone at some particular instant, 
 say at the time when ¢ = 1, and w = 82.2, 
 the space through which the stone will 
 move at this velocity in a small interval of 
 
MATHEMATICAL FORMULA. 169 
 
 time 4¢, will be 4s = v4é, so that, for 
 
 
 
 example, if 4¢ = ane second, the stone 
 
 32,2 
 1000 
 time. Strictly speaking, the stone will 
 have moved through more than this dis- 
 
 
 
 will move through 4s == feet in this 
 
 tance, because during this interval of time 
 it will have been accelerated or will be 
 moving faster at the end than at the begin- 
 ning of the period. But if we make the 
 interval of time 4¢, small enough, and theo- 
 retically, if we make it indefinitely small 
 or dt, the equation, 
 
 ds = vdt, 
 
 will become strictly true. At present, 
 however, we may accept for our purposes 
 the rough calculation 
 
 As = vat 
 
 where 4¢ = 1/4th second say, or 0.25. 
 
170 THE INTERPRETATION OF 
 
 Then we may divide the whole base line 
 OT of 2.5 seconds, into intervals of time Z¢ 
 each equal to 0.25 second, and there will 
 evidently be ten of such intervals. If 
 we take the first interval, when ¢ = 0, the 
 stone will evidently be just starting from 
 rest, and its initial velocity will be 0. 
 This may be represented by the making 
 % = 0. Consequently, 
 
 AS) = V4t = 0 X 0.25 = 0 feet. 
 
 And the space moved through by this 
 rough calculation will be 0 feet in the first 
 quarter second, although we know that 
 owing to taking 4¢ so long as 1/4th second, 
 the result is untrue since the stone has cer- 
 tainly started from rest in this time. At 
 the commencement of the second interval, 
 or when 
 
 t= 0.25 v, = 0.25 X 32.2 = 8.05 feet per second. 
 
 At this velocity, if maintained uniform, the 
 
MATHEMATICAL FORMULA. 171 
 
 stone would traverse in the succeeding 
 interval 4¢, a distance 
 
 Pie 8.05. 0.25.2 2:01 90: feet: 
 
 If we proceed in this way to determine at 
 each of the ten points on the base-line O7} 
 what will be the velocity at that moment, 
 and what will be the distance moved 
 through by the stone, assuming the velocity 
 uniform during the next interval, we obtain 
 the following equations: 
 
 FEET 
 
 At to=0 Vo 82.2 0 Aso= 0 <0.25= 0 
 
 4:=0.25 = 8.05 As,= 8.05X0.25= 2.0125 
 t2=0.50 =16.10 Asg=16.1 0.25= 4.0250 
 ts=0.75 3 =24.15 As;=24.15x0.25= 6.03875 
 t4=1.00 % =82.20 As,=32.2 X0.25= 8.0500 
 ts=1.25 vs +» =40.25 As,—40.25x0.25=—10.0625 
 ts=1.50 =48.30 As.=48.30X0.25=12.0750 
 t7=1.75 7 =66.35 As,;=55.35 x0.25=14.0875 
 ts=2.00 =64.40 As,=—64.400.25=16.1000 
 %9=2.25 Up =72.45 Ass =72.45 x 0.25=18.1125 
 
 At the end of the interval commenc- 
 ing with ¢), the time 7’ = 2.5 seconds, 
 will have been reached, and the total 
 computed fall, $ = 90.5625 feet, 
 
 
 
Lie THE INTERPRETATION OF 
 
 It will be evident that this result will 
 necessarily be too small, owing to the 
 manner in which the process has been con- 
 ducted, since we have taken the initial 
 velocity of the stone in each equation in 
 order to obtain the distance traversed in 
 the interval, thus ignoring the influence of 
 acceleration during the interval. If, how- 
 ever, instead of taking the interval of time 
 At as 1/4th second, we reduced it one-half, 
 or made it 1/8th second, each equation of 
 
 the type 
 As = vit 
 
 would be more nearly true, and the result- 
 ing sum 95.595 feet, would be also more 
 nearly true, but there would be 20 equa- 
 tions to sum up, instead of 10. It is evi- 
 dent that as we take 4¢ smaller and 
 smaller, we increase the number of equa- 
 tions, but we arrive closer to the truth, 
 
MATHEMATICAL FORMULA. Vis 
 
 Strictly speaking, therefore, we should 
 make 4¢ indefinitely small or equal to dz, 
 and sum an indefinitely great number of 
 equations of the type 
 
 ds = vdt. 
 
 The sum of this indefinitely great number 
 of eauations would give s, accurately; or 
 
 s = sum of all terms vdt 
 
 fanemirom ¢— 0 to\t = 2.5. This is ex- 
 pressed in the language of the integral 
 
 calculus by 
 TT 
 8 =| vdt, 
 0 
 
 where | stands for “the sum of all terms 
 
 of the type.” The superscript Z} and the 
 subscript 0, show that the upper and lower 
 limits of the variable ¢, between which the 
 summation is to be effected, are 7’ seconds 
 and 0 seconds, respectively. . 
 
 The rules and theory of the integral 
 
174 THE INTERPRETATION OF 
 calculus show that the operation indicated 
 by the integral | which it would be im- 
 
 possible to carry out arithmetically, since 
 it is impossible to write down an infinite 
 number of terms, is capable of being 
 directly computed without performing any 
 such process. In the case considered, the 
 rules of the integral calculus show that 
 
 Gut 
 
 s= 
 2 
 
 
 
 or, that the integral 
 
 2 
 (ea=S 
 0 
 
 9 e 
 
 
 
 Since 
 
 2 39.2 X 2.5 X 2.5 
 7-35, 7. ee 
 
 = 100.625 feet. 
 This would be the sum arrived at were it 
 possible to write down and sum up the 
 indefinitely great number of terms required 
 in the case considered, 
 
MATHEMATICAL FORMULZ. 175 
 
 By reference to Fig. 8 it will be seen 
 that the integral which we have considered 
 is capable of a simple geometrical interpre- 
 tation. Tor if we consider any interval of 
 time 4t = 1/4th second say that following - 
 the time ¢ = 1 second, the velocity at this 
 time is 32.2 feet per second, represented 
 by the ordinate 1v. The space traversed 
 at this velocity in the succeeding quarter 
 second is 
 
 1 
 As = vat = 82.2 X 7 = 8.05, 
 
 and is evidently the area of the vertical 
 strip luca, whose sides are v and Jf, re- 
 spectively. From this it will be seen that 
 the total space traversed in 2.5 seconds, as 
 summed from ten equations, is the sum of 
 the areas of the vertical strips indicated in 
 Fig. 8 by the dotted lines. The true dis- 
 tance traversed is, however, by the in- 
 tegral calculus, 
 
176 THE INTERPRETATION OF 
 1 
 &s = 9g L e) 
 In the figure, the ordinate 7’'V is equal to 
 
 gT; and the base OZ'is equal to Z} so that 
 
 1 
 
 CS Leet ee units of area 
 
 do] 
 
 or, the area of the triangle OZ’V = 100.625. 
 
 It is evident from the figure that as we 
 reduce the length of the intervals 4¢, we 
 make the vertical strips more numerous, 
 and their total area will more nearly agree 
 with the area of the whole triangle; and 
 finally when the intervals dé are indefinitely 
 small, and the steps indefinitely numerous, 
 they will exactly conform at their termina- 
 tions with the line Ov V, and their agere- 
 gate area will exactly coincide with the 
 area. of the triangle O7'V. 
 
 Any integral may, therefore, be re- 
 
 garded as the area of a certain geometrical 
 
MATHEMATICAL FORMULA. Was 
 
 figure and to be composed of an indefi- 
 nitely great number of elementary strips 
 which, finally, coincide completely with 
 the area. 
 
 An important practical application of 
 the process of integration is found in many 
 forms of meters. Consider, for example, 
 a gas meter. At any moment when the 
 gas jets ina building are lighted, there will 
 be a certain flow of gas into the building 
 through the meter. This flow may be 
 expressed by the symbol f ‘Then in any 
 brief interval of time 4¢, the quantity of 
 gas 4g, admitted to the building will be 
 
 Ag = f4t 
 and, if the number of gas jets remained 
 unaltered and burned steadily, the quan- 
 tity of gas consumed in any given number 
 of hours would be 
 
 Y = fi, 
 
178 THE INTERPRETATION OF 
 
 but if a number of gas jets are turned on 
 and off irregularly, or if the pressure in 
 the mains varies, so that the flow is not 
 uniform, then the total quantity of gas 
 supplied to the house is not proportional to 
 the time. At any moment, however, thz 
 following equation is true: 
 
 dq = fdt; 
 
 dt, being an indefinitely brief interval of 
 time, and dg, the corresponding minute 
 quantity of flow. If we sum all the equa- 
 tions which may be formed between the 
 time ¢ = 7, and ¢ = 7, we obtain a per- 
 fectly accurate theoretical statement for 
 the quantity of gas supplied ; namely, 
 
 Ts 
 g aoon | fat. 
 1, 
 The meter, if properly constructed, will 
 
 perform this integration or summation 
 mechanically, and will give between the 
 
MATHEMATICAL FORMULA. 179 
 
 time 7;, and the time Z}, the value of the 
 integral written on the right-hand side of 
 this equation. Meters of this type are 
 therefore, frequently called integrating 
 meters. Of course their operation does 
 not require the use of the integral calculus, 
 but the integral calculus enables their 
 operation to be very briefly and accurately 
 stated in symbolical language, and if there 
 is any known law connecting the flow 
 with the time, the integral is capable of 
 being evaluated arithmetically. 
 
 Fig. 9, represents the flow of gas 
 through a meter, during different success- 
 ive intervals. It will be seen that dur- 
 ing the daytime the flow falls to zero or 
 to a very small quantity. At night-time 
 it rises to a maximum and varies accord- 
 ing to the demand in the building. If the 
 ordinates are laid off to a proper scale of 
 flow, and the abscissz to a proper scale of 
 
180 THE INTERPRETATION OF 
 
 time, the area of the curve between the 
 ordinates Z; and 7), will give the integral 
 
 T. 
 g= | jdt. 
 
 Ts 
 
 which the meter mechanically registers. 
 
 
 
 
 ef 
 
 Rate of Supply 
 
 Fie. 9. 
 
 As another example of the meaning and 
 application of the processes of the integral 
 calculus, let us consider the determination 
 
 of a height by means of barometric obser- 
 vations. 
 
MATHEMATICAL FORMUL@. 181 
 
 Let us suppose that at a certain station, 
 say at the base of a mountain, and at some 
 height 2, metres above the mean sea level, 
 the barometer shows an atmospheric pres- 
 sure of p, millimetres of mercury. Then, 
 from the known volume of air at the 
 observed temperature and at this pressure 
 p, the atmosphere behaves as though it 
 were a layer of air of //, centimeters in 
 height having a uniform density equal to 
 the density at the point of observation, 
 whereas, in reality, its density continually 
 diminishes as we ascend, and its real 
 height is consequently far greater. The 
 height /7/, is called the height of the homo- 
 geneous atmosphere, or the virtual height 
 to which the atmosphere would extend if its 
 density remained constant at that observed. 
 
 If now we ascend with the barometer 
 through a very small distance 4h meters, 
 the virtual height //, of the atmosphere 
 
189 THE INTERPRETATION OF 
 
 above the original station will be reduced 
 from [7 to H — 4H, while the barometric 
 pressure will be reduced from p to p — 
 Ap; 4p, being the small fall in the bar- 
 ometric pressure due to the small differ- 
 ence in height 4h. The proportional 
 diminution in the virtual height of the 
 atmosphere ; or, 
 
 4H _ 4p 
 
 Fidee 
 and since the diminution in virtual height 
 Aff, is the same as the change in real 
 
 height 4h, then 
 
 Multiplying both sides of this equation by 
 Hf, we obtain 
 
MATHEMATICAL FORMULZ. 183 
 
 It would be possible to call the new pres- 
 sure p’, at the slightly elevated or second 
 station, which is p — 4p, and repeat the 
 equation for this new pressure and succes- 
 sively raise the barometer say a foot at a 
 time, and write down a new equation of 
 this type each time: 
 The elevation raised through 
 i sh SN ta 
 Pm 
 where 7,,, 1s the pressure at any station 
 in the upward series. The total height 
 through which the barometer had been 
 raised from the first station to the last 
 would then be found by summing up all 
 these equations in the following manner: 
 
 D1 
 Ah, = ee 
 
184. THE INTERPRETATION OF 
 
 th, = —HP, 
 Hs 
 
 where p’, stands for the pressure at the 
 last or nth station, The sum of all the 
 terms on the left-hand side of these equa- 
 tions 1s the total height (A’ — h) through 
 which the barometer has been moved from 
 the first station of elevation h, to the last 
 station of which the required elevation is 
 h'. The sum of the terms on the right- 
 hand sides of these equations is the sum 
 
 of all terms of the type —H = . ‘Phe 
 
 resulting summation equation will, how- 
 ever, not be quite correct, because each 
 individual equation contains a small error 
 due to the fact that in raising the barome- 
 ter say a foot, the density of the air at 
 the new elevation is somewhat less than 
 that at the last preceding elevation, and 
 the equation, is therefore vitiated; but if 
 
MATHEMATICAL FORMULA. 185 
 
 we suppose that, instead of raising the 
 barometer a foot at a time, we raised it 
 through an indefinitely small distance, 
 and make an indefinitely great number of 
 such stages of observation, each differing 
 by dh metres, the imaginary equations 
 become strictly accurate, and the sum total 
 becomes strictly accurate, even although it 
 would be practically impossible to perform 
 the experiment in this way. But by the 
 rules of the integral calculus we can deter- 
 mine what the sum of the indefinitely 
 great number of terms on the right-hand 
 side would be; for, it is expressed as the 
 integral 
 
 ep’ — Hdp 
 
 B-2= | 
 p P 
 
 This integral is known to be 
 
 H loges, 
 
186 ‘THE INTERPRETATION OF 
 and the total difference of elevation, or 
 
 (h' —h) = Hog, © 
 or the total elevation, between the first 
 and last station of the barometer is 
 the product of //, the virtual height of 
 the homogeneous atmosphere, and the 
 Naperian logarithm of the ratio between 
 the first and last pressures or readings of 
 the barometer. Thus, if » = 750 milli- 
 metres, and p’ = 375 millimetres of 
 mercury, and //, the apparent height of 
 the homogeneous atmosphere, 8 X 10° 
 centimetres. Then 
 
 30, 
 1b 
 8 X 10° log, 2 cms. 
 
 
 
 h = 8 X 10° log, 
 
 We may either look for the natural log- 
 arithm of the number 2 in tables of 
 Naperian logarithms, or we may look for 
 
MATHEMATICAL FORMULA. 187 
 
 the common logarithm of 2, and multiply 
 it by the constant 2.3026 so that 
 
 h = 8 X 10° X 2.8026 X logy 2 
 
 8 X 2.3026 x 0.3010300 x 105 
 5.546 X 10° centimeters. 
 
 5,546 X 10° metres. 
 
 5,546 metres or 3.446 miles. 
 
 If 
 
 Il 
 
 lI 
 
 It is of course assumed either that the 
 condition of the atmosphere remains uni- 
 form during the process of carrying the baro- 
 meter up the mountain; or, that the two 
 observations at the first and last stations are 
 made simultaneously by two observers. 
 
 We have hitherto considered single in- 
 tegration. Just as it is possible, and often 
 either necessary or convenient, in the dif- 
 ferential calculus to employ asecond differ- 
 ential co-efficient, or the differential of a 
 differential, so, in the inverse operation of 
 the integral calculus, it is often either neces- 
 
188 THE INTERPRETATION OF 
 
 sary or convenient to employ an integral of 
 an intégral, or a double integral as it 1s called, 
 
 and. which is indicated by the sign ||. 
 
 As an example of the natural intro- 
 duction of a double integral, let us consider 
 a water-pipe carrying a stream of water, 
 and suppose that it be required to deter- 
 tnine the total quantity of water g, which 
 flows through a pipe in a given time 4, 
 from observations which determine the 
 flow or velocity of movement of water, at 
 different points of the. cross-section of the 
 pipe. In other words, we are supposed to 
 know the velocity at different points of 
 the cross-section and not to know the 
 average velocity or total flow. [If the 
 velocity continued uniform at every point, 
 it would only be necessary to find, from the 
 velocity at each point, the average velocity, 
 and this would give us at once the total 
 
MATHEMATICAL FORMULA. 189 
 
 flow, because if A, be the area of cross- 
 section of the pipe, in square centimetres, 
 and v, the mean velocity in centimetres- 
 per-ssecond; then the volume of water 
 flowing in each second will be 
 
 V = Av cubic centimetres, or grammes, 
 of water per second. 
 
 It is evident that a single integration 
 will enable us to determine the average 
 velocity v, from the assumed knowledge 
 of the velocity at each point in the cross- | 
 section. But if we suppose that the 
 velocity is not uniform, but varies accord- 
 ing to an assigned law, not only at differ- 
 ent points of the cross-section but also at 
 different times, then we shall have to 
 determine not only the average velocity of 
 the cross-section at any moment by the 
 single integration, but also the average 
 velocity in time by a second integration, 
 
190 THE INTERPRETATION OF 
 
 and the solution of the problem may, 
 therefore, be capable of expression as a 
 double integration, in which one integra- 
 tion refers to space and the other to time. 
 
 When, therefore, such an equation is 
 presented as the following 
 
 F= fal A dw dy, 
 Yid X1 
 
 it means that the quantity /} is a double 
 integral, or the integral of an integral. 
 
 The equation may be represented as 
 follows: 
 
 ies es I A de | dy 
 Agee Xi 
 
 or, representing the quantity inside the 
 bracket by BS, 
 
 - F\” Bay. 
 Y1 
 
 If, therefore, we integrate the quantity A, 
 with respect to @ alone, that is consider: 
 
MATHEMATICAL FORMULA. 191 
 
 ine @, as an independent variable, and 
 denote by 4, the integral so obtained, and 
 then integrate the quantity 5, with re- 
 spect to y, alone, considering y, the inde- 
 pendent variable, the result of the second 
 integration will give the quantity / 
 
 In the same way triple, quadruple or 
 higher integrals may be regarded as a 
 succession of integrals, one being taken at 
 atime. ‘The area of a plane figure whose 
 sides conform to a definite geometrical 
 law can be usually expressed as a double 
 integral, the first integral being taken with 
 respect to a, a length parallel to one axis 
 of co-ordinates, and the second integral 
 being taken with respect to y, a length 
 parallel to the other axis of co-ordinates. 
 Similarly, the volume of a figure bounded 
 by outlines which are defined by any 
 geometrical law can usually be expressed 
 as a triple integral, the first integral being 
 
192 MATHEMATICAL FORMULA. 
 
 taken with respect to a, the second with 
 respect to y, and the third with respect 
 to z In other words, the volume is 
 expressed as equal to an indefinitely great 
 number of little elements of volume, each 
 of which has a cube of dimensions dx x 
 dy x dz. Thus the equation 
 
 V= \\| A dx dy dz 
 
 represents the simplest form of a volume 
 or triple integral. In practice, triple inte- 
 gration is, usually, as far as multiple 
 
 integration extends. 
 
CHAPTER XIII 
 DETERMINANTS. 
 
 Tue following pair of symmetrical equa- 
 tions, 
 Oa =D (A) 
 URS OY nO, 
 
 are called s¢multaneous equations involving 
 two unknown quantities; namely, w and 
 y. From these two equations both of the 
 unknown quantities may be computed. 
 
 Similarly, from the three following sim- 
 ultaneous equations, 
 
 4¢ + 2y + 82 = 17 
 LO 22 = 3 (D) 
 e+ Oy — 42 = %, 
 
 193 
 
194 THE INTERPRETATION OF 
 
 each of the three unknowns a@, y, and 2, 
 can be computed; and, in general, any set 
 of m, simultaneous equations, of the first 
 degree, all of which are independent of 
 each other; 2. ¢, are not reducible one to 
 another, so that each contains an inde- 
 pendent statement,—are sufficient for the 
 determination of x unknown quantities. 
 
 When only two simultaneous equations 
 have to be dealt with, as in the pair given 
 above, the computation is a simple matter, 
 and, in some cases, it can be carried out 
 by mere inspection. It is easy to see that 
 the pair of above equations are satisfied by 
 the results, 
 
 Drea nt sane 
 since 
 8X1+2=5 
 2 X13 XD ae 
 
 When, however, a set of simultaneous 
 
MATHEMATICAL FORMULA. 195 
 
 equations containing more than three un- 
 knowns is given, the computation, while 
 not necessarily difficult, is often very 
 tedious and lengthy. It has been found 
 that the process may be conducted in a 
 regular way, which can be formulated by a 
 sort of algebraic shorthand, which enables 
 the process to be carefully inspected, 
 checked, and often simplified. This proc- 
 ess has led to the introduction and use of 
 what are called determinants. 
 
 A. determinant consists of a symmetrical 
 assemblage of quantities in rows and col- 
 ums, bounded by a pair of vertical lines, 
 there being as many rows as there are 
 columns. Thus: 
 
 ag 
 
 fst] |e | [os 
 
 Beet 
 
 
 
 
 
 
 
 are determinants of the second order, 
 
196 THE INTERPRETATION OF 
 
 because each determinant has two rows 
 and two columns. Similarly, 
 
 4° 9" 2 h-o w 6) 4s 
 Denny Talreeo ao ae 
 uh pla: Sn henge 3° Om 
 
 are determinants of the third order, be- 
 cause each has three rows and three 
 columns. Again, 
 
 16 40058 2 a 
 O 22) 20) Lae 
 15° 46. 94) 23 aa, 
 24.18 12 10 1 
 GAD c2 15 ae 
 
 is a determinant of the fifth order, because 
 it has five rows and five columns. 
 The separate numbers, or symbols, in a 
 
MATHEMATICAL FORMULZ. 197 
 determinant ; or, as they are called, the ede- 
 ments, are to be multiplied according to a 
 definite rule. A determinant of the second 
 order naturally produces terms or products 
 having two elements each. A determinant 
 of the third order naturally produces terms 
 or products of three elements each ; and a 
 determinant of the mth order naturally 
 
 produces terms or products of 2 elements, 
 
 each. Thus: 
 
 abe 
 
 aef 
 ght 
 
 = aet + dhe + gbf — gec —hfa — db, 
 
 
 
 
 
 or the determinant is identical with the 
 product : 
 
 aev + dhe + gbf — gec — hfa — idb. 
 
 This determinant of the third order is, 
 
 therefore, a brief method of expressing 
 
198 THE INTERPRETATION OF 
 
 these six terms, each containing products 
 of three elements. ‘The terms are obtained 
 by taking one element in each horizontal 
 row, and combining with it one letter in 
 each of all the other rows. Thus, the first 
 term ae, takes the first element of the first 
 row, the second element of the second row, 
 and the third element of the third row, 
 affixing to the same the positive sign. 
 The fifth term has the first letter of the 
 first row, the third letter of the second 
 row, and the second letter of the third 
 row, prefixing to the product the negative 
 sign. In this way each term has one ele- 
 ment, and only one element, out of each 
 row, and takes the positive or negative 
 sign according to the way in which the 
 selection is made, following a definite rule. 
 
 A. complete determinant; t. €, a deter- 
 minant which has no zeros in it, of the 
 2d order, is identically equivalent to two 
 
MATHEMATICAL FORMULAE. 199 
 
 terms, each consisting of the product of 
 two elements, 
 
 A complete determinant of the third 
 order is identically equivalent to six terms, 
 each consisting of the product of three 
 elements. 
 
 A complete determinant of the fourth 
 order is identically equivalent to twenty- 
 four terms, each consisting of the product 
 of four elements. 
 
 A complete determinant of the fifth 
 order is identically equivalent to 120 
 terms, each consisting of the product of 
 five elements. 
 
 A. complete determinant of the nth 
 order is equivalent to n! terms, each con- 
 sisting of the product of m elements. 
 
 The application of determinants may 
 be illustrated sufficiently for our present 
 purposes by considering the three simulta- 
 neous equations (5). 
 
900 THE INTERPRETATION OF 
 
 The value of the unknown quantity a, 
 in these three equations, is obtained by 
 the quotient of two determinants; namely, 
 
 
 
 12 oe hes 
 8 1-2 
 pow 9 =4| 2 3 
 As aeaee 294 
 foi Wis -8 
 1 9-4 
 
 Similarly, the value of y, is the ratio of 
 another pair of determinants: 
 
 eT RR aT 
 | 
 cw 
 Or 
 oe) 
 co 
 
MATHEMATICAL FORMULE. 201 
 
 and, finally, the value of z, is the ratio of a 
 third set-of determinants ; namely, 
 
 
 
 Ae 17 
 feria 
 
 eo tel 59 SSE 
 A598 945: 
 eyelis 2 
 1 9-4 
 
 Each of the determinants in the nume- 
 rator and denominator may be worked out 
 by the equation already given for a deter- 
 minant of the 3d order. The student will 
 observe that the determinants forming the 
 * denominators of the above fractions are all 
 the same, and that the determinants form- 
 ing the numerators differ from each other 
 in a manner which is readily traced by 
 reference to the coefficients in the three 
 simultaneous equations (/). 
 
202 MATHEMATICAL FORMULA. 
 
 There are various rules for treating, 
 simplifying and reducing determinants, 
 but it will be sufficient for the student to 
 remember that a determinant is an abbrev1- 
 ated form of a symmetrical set of products 
 such as commonly presents itself in the 
 process of solving simultaneous equations, 
 
CHAPTER XIV. 
 
 SYNOPSIS OF SYMBOLS COMMONLY FOUND IN 
 MATHEMATICAL FORMULA. 
 
 qe Plus, or sign of addition. 5+ 7 =12. 
 
 — Minus, or sign of subtraction. 7 — 
 5 = 2. 
 
 + Plusorminus. 7+5=12 or 2. 
 
 ~ Difference sign. 5~%7=+2. The 
 difference between 5 and 7 is 2. 
 
 > Greater than. 7>5; seven is 
 
 greater than five. 
 < Less than. 5< 7; five is less than 
 
 seven. 
 = Hquality 5+7=12. 
 + Inequality. 5+ 7; five is not equal 
 
 to seven: or, 7 and 5 are unequal. (2). 
 203 
 
204 THE INTERPRETATION OF 
 
 = Identity. 2(a@+ 6) =2a+ 26; both 
 members invariably identically equal. 
 
 2 Equality or Superiority. f 2 80; f 
 greater than or equal to 30. 
 
 S Equality or Inferiority. f $30; f 
 less than or equal to 30. 
 
 ~ Nearly equal to. 0.667 ~ 4. 
 
 x Multiplication. 5 x 7 = 35. 
 
 . Multiplication. @.b=axX b= ab. 
 
 + Division. 3+ 4= 0.75. 
 
 ae 3 
 — Division bar. aes 83+4=0.75. 
 
 / Division solidus. 38/4=8+4=0.%5. 
 
 () Brackets’ or Parentheses. 5(6+ 7) 
 
 = 5{6+71=5[6+7]=5 X13, 
 
 | Vinculum lines 5 X6+7+8 
 =5(6+74+8)=5 X21 = 105, 
 _o, Infinity. 8 xX 38X8....ad infin 
 
 tum=o. 
 
MATHEMATICAL FORMUL. 205 
 
 «, Varies as. Pressure of a liquid col- 
 umn « the depth of liquid. 
 
 .*., Lherefore. Because 3+5=8 .°. 
 $+3+5=3+48. | 
 
 me Mnces ot ot oo + 8.68 TH 
 = 8. 
 
 - 3: :, Proportionality. 3: 5::6 : 10; 
 three is to five, as is six to ten. 
 
 *, Square. 3° = 9, 
 
 ® Cube. 3? = 2%. 
 
 ", Index or Exponent. 38"=38 X 8 X 8, 
 m times in all. 
 
 -*, Negative Index. 3-"= 
 
 *, Surd Exponent. 4? = square root 
 of 4 = 2. 
 
 m , Fractional Exponent. 4 = nth root 
 of 4. 
 
 Jor ¥, Radical or Surd. V4=42=2 
 = square root of 4. 
 
 veCube Root, V¥27 =3;3 V729 = 9, 
 
206 THE INTERPRETATION OF 
 
 Vv, nth Root. Va= ai = nth root of a. 
 !or L, Factorial T!=7 xX 6xX5x4xX 
 LD ace enn 
 
 =, Summation. (abe) = sum of all 
 
 terms of the type ade. 
 
 
 
 
 
 
 
 , Determinant. ; 7 = AX as eee 
 a . Circumference 
 7, Bi (Greek). Ratio ~ ‘Diamerenn = 
 Blt LOU I gee 
 log, 2, Common Logarithm of @. 
 
 logy) 100 = 2.000. 
 
 log. x, or hyp. log. v, Naperian Logarithm 
 of a, log, 100 = 4.6052. 
 
 Zor j, Sign of the imaginary. ¢@=7 = 
 V—1, 
 
 f(a) Funetion of « «*, V2, log a, am, 
 are functions of « denotable by 7(@). 
 
 sin a, Sine of angle a, Perpendicular + 
 
 hypothenuse, 
 
MATHEMATICAL FORMULA. YO7T 
 
 cos a, Cosine of angle a Base + hy- 
 pothenuse. 
 
 tan a, Tangent of angle a. Perpendicu- 
 lar + base. 
 
 cot a, Cotangent of angle a. Reciprocal 
 of tangent. ; 
 
 sec a, Secant of angle a Reciprocal of 
 cosine. 
 
 cosec a, Cosecant of angle a Recipro- 
 eal of sine. 
 
 vers a, Versed sine of anglea, 1 — cos a. 
 
 sin-’ a, Inverse sine. The angle whose 
 sine is alpha. 
 
 sinh a, Hyperbolic sine of angle a. 
 
 cosh a, Hyperbolic cosine of angle a. 
 
 tanh a, Hyperbolic tangent of angle a. 
 sinh a/cosh a. 
 
 coth a, Hyperbolic cotangent of angle a. 
 1/tanh a. 
 
 sech a, Hyperbolic secant of angle a. 
 1/cosh a. 
 
908 THE INTERPRETATION OF 
 
 cosech a, Hyperbolic cosecant of angle a. 
 1/sinh a. 
 Ay, Difference of ¥. 
 dy, Differential of y. Limit of sy. 
 gi Differential coefficient of y, with re- 
 wv 
 hehe ts , AY 
 spect to a Limiting ratio of rr when 4@ 
 = 0, 
 
 a ; : : 
 mae Second differential coefficient of y, 
 
 (a) 
 with respect to 2. da) - 
 di 
 
 d° eerie RAL 
 on) nth differential coefficient of y to @. 
 
 
 
 y, Differential of y, with respect to 
 dy 
 time. dt : 
 
 y, Differential of y, with respect to 
 
 d 
 space. - ; 
 
MATHEMATICAL FORMULA. 209 
 | Integration sign. 
 | J (@)da, Integral of f(@), with respect 
 
 to @. 
 
 || Double Integral. 
 
 | | i Triple Integral. 
 
F 
 ae! 
 w 
 
 in 
 
 i 
 
 
 
INDEX. 
 
 Algebra, Definition of, 1. 
 Algebraic and Arithmetical Equations, Difference 
 between, 5-8. 
 Equation, 5. 
 Angle, Cosecant of, 107, 
 , Cosine of, 104. 
 , Cotangent of, 105. 
 , First Quadrant of, 100. 
 , Fourth Quadrant of, 101, 
 , Hyperbolic, 129. 
 , Positive, 100. 
 , Secant of, 106. 
 , second Quadrant of, 101. - 
 , Sine of, 103. 
 , Pangent of, 104. 
 , Third Quadrant of, 101. 
 Apparent Time at Sea, Trigonometrical Formula 
 for, 119, 124. 
 Application of Differential Calculus to the Laws 
 of Falling Bodies, 141-148. 
 211 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
912 INDEX. 
 
 Application of Integral Calculus to the Laws of 
 Falling Bodies, 170-176. 
 
 Applications of Logarithms, 84-89. 
 
 of Logarithms in Involution and Evyolu- 
 tion, 90-93. . 
 
 of Second Differential Coefficient, 159- 
 164, . 
 
 Arithmetical Equation, 4. 
 
 : Series, Formula for, 32. 
 
 Asymptote of Curve, 130, 
 
 
 
 
 
 
 
 B 
 Bar, Division, 35. 
 Barometric Observations of Height, Application of 
 Calculus to, 181. 
 Base, Logarithmic, 78. 
 Bracket, 27. 
 
 C 
 
 Calculus, Differential, 135. 
 
 , Integral, 165. 
 Cancellation, 74. 
 
 Characteristic of Logarithm, 81. 
 Circular Arc, Degrees of, 96. 
 Arc, Minutes of, 96. 
 Arc, Seconds of, 96. 
 Functions, 102. 
 
 
 
 
 
 ' 
 
 
 
 
 
INDEX. 213 
 
 Coefficient, Differential, 140. 
 
 , Second Differential, 158. 
 , Lhird Differential, 158. 
 Common Logarithms, 79. 
 Complete Determinant, 198. 
 Compound Terms, 36. 
 Conduction of Heat, Formula for, 39, 40. 
 Cosecant of Angle, 107. 
 
 Cosine, Hyperbolic, 131. 
 
 Cosine of Angle, 104. 
 Cotangent of Angle, 105, 
 Counter-Clockwise Motion, 100. 
 Cube of a Number, 50. 
 
 Cube Root, 62. 
 
 Curve, Asymptote of, 130. 
 
 
 
 
 
 D 
 
 Decimal Part of Logarithm, 81. 
 
 Definition of Algebra, 1. 
 
 of Equation, 2. 
 
 Deflection of Horizontal Beam Supported at One 
 End, Formula for, 188, 134. 
 
 Degrees of Circular Are, 96. 
 
 Dependent Variables, 150. 
 
 Determinant, 198. 
 
 , Elements of, 196. 
 
 , Sign for, 206. 
 
 
 
 
 
 
 
914 INDEX. 
 
 Determinants, 193. 
 
 of the Second Order, 197. 
 
 of the Third Order, 197. 
 
 Difference between Algebraic and Arithmetical 
 Equations, 5-8, 
 
 Sign, 203. 
 
 Differential, 140. 
 
 Calculus, 135. 
 
 Calculus, Nature of, 135-139. 
 
 Coefficient, 140. 
 
 Distance Passed through by Freely Falling Body, 
 
 Formula for, 58. 
 
 Division, 34. 
 
 Bar, 35. 
 
 , sign of, 34. 
 
 Double Integral, Sign for, 209. 
 
 Integrals, 188. 
 
 — Integration, 188. 
 
 E 
 
 Elements of Determinant, 196. 
 Equality Sign, 2. 
 Equation, Algebraic, 5. 
 , Arithmetical, 4. 
 ———.,, Definition of, 2. 
 
 ——, Left-Hand Side of, 8, 
 , Number of Terms of, 9. 
 , Quadratic, 71. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
INDEX. 215 
 
 Equation, Right-Hand Side of, 3. 
 
 , Simple, 71. 
 
 Equations, 70. 
 
 of the Fourth Degree, 73, 74. 
 
 of the Third Degree, 72. 
 
 , Simultaneous, 193. 
 
 , Lrigonometrical, 117, 118. 
 
 Evolution, 61. 
 
 and Involution, Application of Logarithms 
 in, 90-93. 
 
 Expansion of a Gas, Formula for, 37, 38. 
 
 Exponent of Number of Quantity, 51. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 F 
 
 Factorial, 48. 
 
 —, Sign for, 206. 
 
 First Quadrant of Angle, 100. 
 
 Formula for Arithimetical Series, 32. 
 
 for Conduction of Heat, 39, 40. 
 
 for Deflection of Horizontal Beam Sup- 
 ported at One End, 133, 134. 
 
 for Distance Passed through by Freely 
 Falling Body, 58. 
 
 for Expansion of a Gas, 37, 38. 
 
 for Gravitational Force at Different Lati- 
 tudes, 117, 118. 
 
 for Horse-Power or Single-Cylinder 
 Engine, 43, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
216 INDEX. 
 
 Formula for Joint Resistance, 39. 
 
 for Period of Time of Complete Vibration 
 of Simple Pendulum, 63, 64. 
 
 for Surface of Right Cylinder, 42. 
 
 for Volume of Liquid at Given Tempera- 
 ture, 75. 
 
 for Work Done by Volume of Expanding 
 Gas at Constant Temperature, 95. 
 
 Relating to Theory of Probabilities, 44, 
 45. 
 
 , Trigonometrical, 115. 
 
 Fourth Degree, Equations of, 73, 74. 
 
 Power of a Number, 51. 
 
 — Quadrant of Angle, 101. 
 
 Root, 62. 
 
 Fractional Indices, 66-69. 
 
 Functions, Circular, 102. 
 
 , Hyperbolic, 132. 
 
 , Hyperbolic Trigonometrical, 126. 
 
 , Trigonometrical, 102. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 H 
 
 Horse-Power of Single Cylinder Engine, Formula 
 for, 43. 
 
 Hyperbola, Rectangular, 128, 
 
 Hyperbolic Angle, 129. 
 
 Cosine, 131. 
 
 
 
INDEX. yA bras 
 
 Hyperbolic Functions, 132. 
 Logarithms, 94. 
 
 Sine, 130. 
 
 Trigonometrical Functions, 126. 
 
 
 
 
 
 
 
 Identity, Sign of, 204. 
 
 Independent Variables, 151. 
 
 Index of a Number, 51. 
 
 Indices, Fractional, 66-69. 
 
 , Positive and Negative, 54-58. 
 
 Inequality, Sign of, 202. 
 
 Inferiority, Sign of, 204. 
 
 ‘Infinity, Sign of, 204. 
 
 Initial Line of Radius Vector, 102. 
 
 Instantaneous Velocity of Falling Body, 148. 
 
 Integral Calculus, 165. 
 
 Calculus, Application of, to the Laws of 
 Falling Bodies, 170-176. 
 
 Calculus, Scope of, 166-168. 
 
 Integrals, Double 188. 
 
 , Friple, 192. 
 
 Integrating Meter, 177-179. 
 
 Integration, Double, 188. 
 
 Sign, 209. 
 
 Inverse Notation, 126. 
 
 Inyolution, 50, 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 218 INDEX. 
 
 Involution and Evolution, Application of Loga- 
 rithms in, 90-93. 
 
 J 
 
 Joint Resistance, 39. 
 
 L 
 
 Law of Summation of Indices, 54. 
 
 Laws of Falling Bodies, Application of Differen- 
 tial Calculus to, 141-148. 
 
 Left-Hand Member of Equation, 3. 
 
 Logarithm, Characteristic of, 81. 
 
 , Decimal Part of, 81. 
 
 , Mantissa of, 81. 
 
 , Negative Characteristic of, 125. 
 
 Logarithmic Base, 78. 
 
 Logarithms, 78. 
 
 , Applications of, 84-89. 
 
 , Common, 79. 
 
 , Hyperbolic, 94. 
 
 ———., Naperian, 94. 
 
 , Natural, 94. 
 
 , Tables of, 80. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 M 
 
 Mantissa of Logarithm, 81. 
 Meter, Integrating, 177-179, 
 
INDEX. 219 
 
 Mathematical Formula for Number of Balls in 
 Square Pyramid of Cannon Ball, 46. 
 
 Formule, Symbols Commonly Found in, 
 203-209. 
 
 Minus Sign, 14. 
 
 Minutes of Circular Arce, 96. 
 
 Motion, Counter-Clockwise, 100. 
 
 , Negative, 100. 
 
 , Positive, 100. 
 
 Multiplication, 21. 
 
 Sign, 21. 
 
 Sign, Omission of, 24. 
 
 ——— Sign, Use of Period for, 25. 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 Naperian Logarithms, 94. 
 
 Natural Logarithms, 94. 
 
 Nature of Differential Calculus, 185-139. 
 Negative Characteristic of Logarithm, 125. 
 Index, Sign for, 205. 
 
 Motion, 100. 
 
 Remainder, 15. 
 
 7 Terms, 16-18. 
 
 Notation, Inverse, 126. 
 
 Nth Power of a Number, 51, 
 
 Nth root, 62. 
 
 Number, Cube of, 50. 
 
 
 
 
 
 
 
 
 
 
 
220 | INDEX. 
 
 Number, Fourth Power of, 51. 
 
 , Index of, 51. 
 
 ——., Nth Power of, 51. 
 
 of Terms of Equation, 9. 
 
 or Quantity, Exponent of, 51. 
 , Square of, 50. 
 
 , Powers of, 50. 
 
 
 
 
 
 
 
 
 
 
 
 O 
 
 Omission of Multiplication Sign, 24. 
 
 li 
 
 Parenthesis, 27. | 
 
 Plus Sign, 2. 
 
 Positive and Negative Indices, Use of, 54-58. 
 Angle, 100. 
 
 Motion, 100. 
 
 Power, Second, 50. 
 
 , Third, 50. 
 
 Powers of Numbers, 50. 
 
 Product, 21. 
 
 
 
 
 
 
 
 Q 
 
 Quadratic Equation, 71. 
 Quantity of Magnetic Flux, Application of Calcu- 
 lus to the Determination of, 148, 150. 
 
INDEX. 291 
 
 Quantity of Water Discharged from a Reservoir, 
 Application of Calculus to the Determi- 
 nation of, 145, 147. 
 
 Quotient, 34. 
 
 R 
 
 Radian, 96. 
 
 Radical Sign, 63. 
 
 Radius Vector, 99. 
 
 Vector, Initial Line of, 102. 
 Rectangular Hyperbola, 128. 
 Remainder, Negative, 15. 
 Right-Hand Member of Equation, 3. 
 Root, Cube, 62. 
 
 , Fourth, 62. 
 
 ——., Nth, 62. 
 
 , Square, 61. 
 
 , Third, 62. 
 
 Roots, 61. 
 
 
 
 
 
 
 
 
 
 S 
 
 Scope of Integral Calculus, 166-168. 
 
 Secant of Angle, 106. 
 
 Second Differential Coefficient, 158. 
 
 Differential Coefficient, Applications of, 
 159-164. 
 
 Power, 50. 
 
 Quadrant of Angle, 101. 
 
 Seconds of Circular Arc, 96. 
 
 
 
 
 
 
 
959 INDEX. 
 
 Simple Equation, 71. 
 
 Pendulum, Formula for Period of Time of 
 Complete Vibration of, 63, 64. 
 
 Sign, Difference, 203. 
 
 , Equality, 2. 
 
 for Determinant, 206. 
 for Double Integral, 209. 
 for Factorial, 206. 
 
 for Negative Index, 205. 
 for “ Since,” 205. 
 
 for Summation, 206. 
 
 for Surd, 205. 
 
 for “ Therefore,” 205. 
 for Triple Integral, 209. 
 for “ Varies as,” 204. 
 
 -—_——.,, Minus, 14. 
 
 , Multiplication, 21. 
 
 of Division, 34. 
 
 ef Identity, 204. 
 
 of Inequality, 202. 
 
 of Inferiority, 204. 
 
 of Infinity, 204. 
 
 of Integration, 209. 
 
 of Superiority, 204. 
 
 of the Imaginary, 206. 
 
 » Plus, 2. 
 
 , Radical, 63. 
 
 Simultaneous Equations, 193. 
 
 “ Since,” Sign for, 205. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
INDEX. Bee 
 
 Sine, Hyperbolic, 130. 
 
 -of Angle, 1038. 
 
 Solidus, 35. 
 
 Square of a Number, 50 
 
 Pyramid of Cannon Balls, Mathematical 
 Formula for Number of Balls in, 46, 
 
 Root, 61. 
 
 Subtraction, 14. 
 
 Sum of All the Terms, Symbol for, 47. 
 
 Summation of Indices, Law of, 54. 
 
 , Sign for, 206. 
 
 Superiority, Sign of, 204. 
 
 Surd, Sign for, 205. 
 
 Surface of Right Cylinder, Formula for, 42. 
 
 Symbol for Sum of All the Terms, 47. 
 
 Symbols, 10 
 
 Commonly Found in Mathematical For- 
 mul, 203-209. 
 
 T 
 
 Tables of Logarithms. 80. 
 
 Tangent of Angle, 104. 
 
 Terms, Compound, 36. 
 
 , Negative, 16-18. 
 
 of Equation, 9. 
 
 “S'herefore,” Sign for, 205. 
 
 Theory of Probabilities, Formule Relating to, 44, 
 4, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
os): INDEX. 
 
 Thermal Capacity of a Body, Applications of 
 Calculus to Determination of, 147-149. 
 
 Third Degree, Equations of, 73. 
 
 Differential Coefficient, 158. 
 
 Power, 50. 
 
 Quadrant of Angle, 101. 
 
 Root, 62. 
 
 Trigonometrical Equations, 117, 118. 
 
 Formula for Apparent Time at Sea, 119, 
 124, 
 
 Formule, 115. 
 
 Functions, 102. 
 
 Functions, Variations in Value of, with 
 Variations of Angle, 108-114. 
 
 Trigonometry, 97. 
 
 Triple Integral, Sign for, 209. 
 
 Integrals, 192. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 U 
 
 Use of Period for Multiplication Sign, 25. 
 of Vinculum, 30. 
 
 
 
 Vv 
 
 Variables, Dependent, 150. 
 ——, Independent, 151. | 
 Variations of Trigonometrical Functions with 
 Variations of Angle, 108-114. 
 
 % 
 
 
 
INDEX, 225 
 
 Vector, Radius, 99. 
 
 Velocity, Instantaneous, of Falling Body, 143. 
 
 Vinculum, Use of, 30. 
 
 Volume of Liquid at Given Temperature, Formula 
 Fors. 75; 
 
 W 
 
 Whole Number or Characteristic of Logarithm, 81. 
 Work Done by Volume of Expanding Gas at 
 Constant Temperature, Formula for, 95. 
 
 THE END. 
 

 

 

 
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