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There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031214558 ALGEBRA MADE EASY Being a clear explanation of the Mathematical FormulcB found in Prof. Thompson's Dynamo- Electric Machinery and Polyphase Electric Currents BY EDWIN J. HOUSTON, Ph. D., AND ARTHUR E. KENNELLY, So. D. NEW YORK AMERICAN TECHNICAL BOOK CO. 45 Vesby Street 1898 /N.WOHHB A.'- ^cJ COPTBIQHT, 1898, BY EDWIN J. UOUSTON and ARTHUR E. KENNELLT' BRAUNWORTH, MUNN & BARBER, PRINTERS AND BINDERS, BROOKLYN, N. Y. PREFACE. This little volunie has been prepared by the authors for the purpose of elucidating the mathematical formulae appearing in the pages of Professor Silvanus P. Thomp- son's " Dynamo-Electric Machinery," and " Polyphase Electric Currents." While the authors do not profess to make their read- ers mathematicians, they believe that their readers will be fully capable of understand- ing th* meaning of all the mathematical formulae appearing in the above-mentioned works, after a careful perusal of this little book. Philadelphia, December, 1897. CONTENTS. CHAFTEB PAGE I. Introduction, 1 11. The Symbols Commonly Em- ployed IN Algebra: Ad- dition, Subtraction, Mul- tiplication AND Division, 5 III. Powers anu Koots, 25 IV. Equations and Their Solu- tion, .... 37 V. Logarithms, 43 VI. Trigonometry, 56 VII. Differential Calculus, 72 VIII. Integral Calculus, 85 ALGEBRA MADE EASY. CHAPTER I. INTRODUCTION. The essential difference between pure and applied mathematics lies in the fact that symbols are employed in pure mathe- matics for the purpose of conveniently studying the relations between the quan- tities they represent, entirely independently of arithmetical or practical applications ; whereas, in applied mathematics the S3'm- bols are employed especially for the pur- pose of enabling practical and arithmetical solutions and applications to be obtained from the expi^essions ofi the laws control- ling such quantities. Just as there is no limit to infinite truth, a ALGEBRA MADE EAST. SO there is no limit to tlie range, extent, and complexity of pure mathematics ; but applied mathematics is limited in range, in order to be capable of ready applica- tion and utilization. When a formula or analysis in the department of applied mathematics becomes so complex, difficult, or intricate, as to render its solution and arithmetical computation more laborious than the object to be attained deserves, it thereby places itself beyond the pale of applied mathematics. Consequently, applied mathematics is relatively simple mathematics. The mathematics vphich the engineer employs must be relatively simple, because his duties compel him to adopt methods of computation that shall be I'eadily sus- ceptible of being checked and corrobo- rated, and shall not be so intricate as to demand undue shai-e of his time and thought. Anyone who can master arith- metic can master all the processes of applied mathematics, such as the engineer ALGEBRA MADE EASY. 3 has to use, since such mathematics has to be thought out and worked out in arithmetic. There never has been, and, proverbially, there never can be, a royal road to kno\vl- edge, the pathway to which is only found on the highway of labor. It is neither the intention nor the claim of the authors, in the following pages, to make their readers competent mathematicians. But it is their intention and claim to make them able to grasp and understand the meaning of the formulae and equations which are scattered throughout technolog- ical literature. This symbolic language, which so largely pervades scientific tech- nology, is the natural and beautiful lan- guage of exact quantitative expression. It is essentially a simple language, sliorn, by long and wearisome evolution, of almost every vestige of unnecessary or superfluous appendage, and which, Avhen properly enunciated, cari'ies a meaning to the student as clear and perspicuous as its expression is brief and direct. ALGEBRA MADE EASY. To handle and manipulate algebraic expressions, to solve equations and reduce them to their simplest forms, is an art attained only by study and practice, and with which the following pages do not deal. It has no essential pai't in the xmder- standing of mathematical expressions. CHAPTEE II. THE SYMBOLS COMMONLY EMPLOYED IN AL- GEBEA : ADDITION, SUBTRACTION, MULTI- PLICATION, AND DIVISION. + (Plus). The sign of addition. As 7 + 5, meaning the sum of five and seven ; *'. e., seven added to five, or five added to seven. = (Equality). The sign of equality. As 7 + 5 = 12 ; meaning that the sum of seven and five is equal to twelve. — (Minus). The sign of subtraction. As 7 — 5 = 2; meaning that five sub- tracted from seven is equal to 2. X (Multiplication). The sign of multi- plication. As 7 X 5 = 35 ; meaning that 7 multiplied by 5, or 5 multiplied by 7, is equal to 35. -H (Division). The sign of division. 5 6 ALGEBRA MADE EAST. As 7 -4- 5 = 1.4 ; meaning that seven divided by 5 is equal to 1^. . • . (Therefore). A sign used in mathe- matical reasoning as a mere symbol for the word " therefore." :: : (Ratio). Signs of proportion. As 7 ■■ 5 :■■ 14 : 10; meaning seven is to five as is 14 to 10. ( ) (Brackets). Various forms of par- entheses or brackets, employed foi- grouping into one mass a compound r- -| quantity. Thus 5 x (7 + 5) = 5 X ^^ 12. ^: \ \ \ ^^''■' (Symbols). Letters A, B, C, D, etc. ^ •' ^ of the alphabet representing quantities ; usually, but not necessarily, known constant quantities. Thus ^, is a symbol commonly used to represent the gravita- tional force which the earth exerts upon a gramme mass. In scientific units g = 980.07 dynes at the sea-level in Washing- ton, D. C; consequently, g, is more than a mere number — it stands for a certain ALGEBRA MADE EASY. 7 force having a magnitude of 980.07 units, the unit being the dyne. Again, n (Greek letter Pi) is a symbol commonly used to represent the ratio be- tween the circumference of a circle and the diameter, so that, Circumference of a circle = n x Diam- eter of the circle. Here n = 3.1416, approximately; or, roughly, 3t. In this case, tt, is the symbol of a mere numerical magnitude, or ratio between two geometrical quantities. A symbol, therefore, may stand for a number considered solely as such; or for a number representing any particular quantity, physical, astronomical, chemical, etc., stated in reference to a particular unit. The symbols x, y, z, X, Y, Z, are commonly, but not necessarily, used in the mathematical statement of relations, for quantities whose values are unknown, and which may or may not be determined fi'om the relationship given. A formula is a rule mathematically ex- 8 ALGEBRA MADE EASY. pressed, for determining the value of any quantity. Thus the equation : The circumference of a circle = tt X Diameter, is a simple formula from which the cir- cumference of a circle becomes known as soon as its diameter is given. ADDITION. In the equation c = a + h, we have a symbolic form for the following statement : c, is equal to the sum of a and h. If a, b and c, are mere numbers, and a = 5, while 5=7; then c = 12, because 12 = 5 + 7. If a and b, are symbols which represent electromotive forces acting in a circuit, or weights lying in a scale pan, then c, is either a total electromotive force, or a total weight, accoi'dingly. Consequently, a simple equation involving the process of addition may express a mere relation be- ALGEBRA MADE EAST. 9 tween ordinary numbers, or between num- bers which represent physical quantities expressed in terms of units. Thus, the equation, where a = 1, b = 2^, c = S^, d = i, gives the relation a> = 11; or, the unknown quantity x, in this equation, is known to be equal to 11, because the sum of the terms a, b, o and d, on the right-hand side of the equation, is known. Again, on page 214, of Thompson's " Dynamo-Electric Machinery," appears the following equation : Where, C^, termed " C sub a," is the symbol expressing the current strength in the armature of a shunt-wound dynamo ; O, is the current strength supplied to the main external circuit ; and C^^ termed " C sub s," is the current supplied to the shunt field. Here, the subscripts a and s, are used to distinguish between the current 10 ALGEBRA MADE EAST. strengths in the different portions of the circuit, and the equation malies in sym- bolic form the following statement : The current in the armature of the ma- chine is equal to the sum of the currents in the main circuit and the shunt field ; or, if C = 100 amperes, and C^, the shunt- field current, is one ampere, then the arma- ture current 6a = 100 + 1 = 101 amperes. Again, on page 188, of Thompson's "Polyphase Electric Currents," occurs the equation : ^a + l^qr + Kp = 0. This equation has reference to Fig. 130, on the preceding page. The equation is a symbolic method of concisely stating the following : The sum of the three electromotive forces which occur in the three branches pgi, qr, and I'p, represented symbolically by the symbols Fpq, Fq„ and V^, is always equal to zero. Consequently, if any pair of these electromotive ALGEBRA MADE EAST. 11 forces is, say 50 volts, then the remain- ing E. M. F. must be equal to — 50 volts ; or, if any particular E. M. F. is, say 25 volts, then the sum of the remaining pair of E. M. F.'s must be equal to — 25 volts. In other words, when the three E. M. F.'s expressed in volts, or other units, are added together, the sum total is zero. Similar considerations apply to any num- ber of added simple terms, such as are found in the equation : x = a + b + c + d+e. SUBTEACTIOIT. As already pointed out, the minus sign prefixed to a quantity indicates that the quantity has to be taken negatively, and has to be subtracted from the quantity with which it is associated. Thus in the equation : X = a — b, the right-hand side contains two terms, the first a, which is -|- or positive, and the 12 ALGEBRA MADE EASY. second — h^ which is negative ; b, must, therefore, be suV)tracted from a, or a must have its value diminished to the extent of the value of h. It may happen that h, is greater than a; as, for example, if a = 3 and 5=5; but the result is intei'preted by giving to the difference a negative sign ; or, in this case, a; — — 2. Conse- quently, in an equation containing simple terms, some of which are positive and some negative, it suffices to add all the positive terms for a positive sum, and all the negative terms for a negative sum, and then subtract' the latter from the former. Thus : ^ = 5 + 7-3 + 10-2, or, a = 5 + 7 + 10 - 3 - 2, or, cc = 22 — 5, or, a? = 17. On page 510, of Thompson's "Dynamo- Electric Machinery " appears the equation, Ca = C — O3. Here C, represents the total current ALGEBRA MADE EASY. l3 supplied to a slmnt motor; C^ the cur- rent supplied to the armature ; and C^, the current supplied to the shunt field. The equation is equivalent to the fol- lowina; statement : The current supplied to the armature is equal to the current supplied to the machine, less the current supplied to the shunt field ; so that if the total current (7, is 100 amperes, and the shunt-field cur- rent Cg, 1 ampere ; then the armature cur- rent Ca, would be 100 — 1 = 99 amperes. BEACKETS OR PARENTHESES. It is often desirable, in expressing an equation, to separate some of the terms into groups by placing them within a bracket. Thus, the equation xz=a — h-^-G-^-d — e, states that x, is the sum of all the quan- tities on the right-hand side, after due allowance has been made for their sign ; 14 ALGEBRA MADE EASY. i. e., after the proper additions and sub- tractions have been effected. It is often convenient to separate the positive qiian- tities from the negative quantities. The fij'st step is to bring them together in two groups ; thus, x = a + c + d— b — e. These groups may be included in brackets to give each the appearance of a single term. Thus, X = (a + c + d) + (- I) - e). Here the compound term -\- (— b — e) may be wi'itten — (b + e) ; because adding the sum of two negative quantities is the same as subtracting their positive sum, so that the sum of b and e, is to be subtracted from the first compound terra (a + c + d). Consequently, a negative sign before a bracket or parenthesis reverses the sign of all the terms within it. Thus, in the above equation let a = 1, b — 2, c = 3, d — 4, and e = 5. ALGEBRA MADE EASY. 15 Then i2! = l-2 + 3 + 4-5, = (1 + 3 + 4) + (- 2 - 5), = (1 + 3 + 4) - (2 + 5), = 8-7, = 1. The equation might also be written a;=l + 3 + 4-2-5. Where the line serves as a pair of par- entheses to group 1, 3, and 4, into a com- pound term, or, a; = [1 + 3 + 4] - [2 + 5 ], or, a? = -11 + 3 + 41- { 2 + 5 I That is to say, any form of bracket or parenthesis might be used to separate the two groups of terms from each other, MULTIPLICATIOIS". An equation, 00 = a X h, means that the quantity w, is the prod- 16 ALGEBRA MADE EASY. uct of a and h ; so tliat if a = 5 and h = 10, x = 50. Where simple terms are employed, as in this case, the multi- plication sign may be omitted, and the equation is written, X = a h, meaning that x, is equal to the product of a and h. In general, when two symbols fol- low each other without any sign between them, their product is thus indicated ; for example, X = abed. Here x, is the product of a, multiplied by h, multiplied by c, and multiplied by d. If a = 1, Zi = 2, c = 3, and d = ^-j x = 24:. In some cases a point or period takes the place of the multiplication sign. Thus, X = a.i.c.d. On page 168, of Thompson's "Dynamo- Electric Machinery," occurs the equation : (average) ^ = n Z N, where E, is the average E. M. F. generated ALGEBRA MADE EASY. 17 by a dynamo-electric machine; n, is the number of revolutions of its armature per second ; Z, is a certain number of conduct- ors on the armature surface ; and N, the total number of magnetic lines from one field pole that traverse the armature. Consequently, the equation is equivalent to the following statement: The average E. M. F. is the product of the number of turns made by the armature per second, a certain number of conductors lying upon the surface of the armature, and the total number of magnetic lines travers- ing the armature. Compound terms formed of the products of simple terms or factors may be subjected to addition or subtraction like simple terms. Thus, on page 189, of Thompson's " Polyphase Electric Currents," occurs the equation : where W, is the power in watts supplied by a triphase star-wound armature, as rep- 18 ALGEBRA MADE EAST. resented in Fig. 54, on page 46 ; T^n, is the effective pressure between the termi- nals m and n; and F"^o, the pressure between the terminals m and o ; the cur- rent i, being in the branch %, and the cur- rent c, in the branch o. The equation is equivalent to the following statement : The total electric power supplied by a machine is equal to the sum of two com- pound terras, the first of which is the product of the current h, and pressure F^n) s-nd the second, the product of the current c, and the pressure V^^- In some cases a factor appears outside a bracket. For example, on page 189, of Thompson's " Dynamo-Electric Ma- chineiy," appears the equation : l£={R + r, + r^) a This is given in relation to series dyna- mos. Jl, is the E. M. F. generated by the machine ; R, the external i-esistance of the circuit ; 'i\, the resistance of the armature ; and r^, the resistance of the magnets as ALGEBRA MADE EAST. 19 shown in Fig. 125. C, is the current strength through the circuit in amperes. This equation expresses the fact that the product of the .current sti-ength in the circuit in amperes multiplied by a com- pound term, which is the sum of all the resistances in the circuit, is equal to the total E. M. F. in the circuit. This is only another way of stating Ohm's law for the circuit. Since an equation x = ah, is the same as x—ha\ or, since the oi'der of factors in an equation is indifferent, the equation on page 189, of Thompson's "Dynamo-Elec- tric Machinery," may be written : E= C'(i2 + r, + r^,) or E= CR-\- Ci\ + (7;v This shows that a factor placed ontside a bracket is to be multiplied by each term Avithin the bracket. Suppose, for example, that the current strength G, in the cii-cuit, is 100 amperes ; that the external resistance 20 ALGEiBRA MADE EASY. R, is 1 olim ; tliat tlie armature resistance T„ is T7^^,tl3 ohm ; and the field magnet re- 2 sistance r^, is tka^^ ohm — then the equa- tion may be written : E =■■ 100 (1 + 0.01 + 0.02), E = 100 (1.03) = 103 volts ; or, multiplying each term within the bracket by the factor outside, ^ = 100 + 1 + 2 = 103 volts. DIVISIOIS'. The operation of division is represented in algebra either by the division har -4-, or, more commonly, by the fraction har employed in ordinary arithmetic. Thus X = a -^ h, may be written a If a = 3 and 5 = 4, this becomes a? = 3 -j- 4, or a? = |. = 0.75. Sometimes the ALGEBRA MADK EASY. 21 bar talies an inclined form, wlien it is called a solidiis, thus, x — a/h. Take, for example, the expression for Ohm's law, Avhich is written in the Inter- national system of notation 1=1. R If E, the E. M. F., is 100 volts, and R, the resistance, is 5 ohms, the current strength I = -^r- = 20 amperes, and the equation is equivalent in this case to the following statement : The current strength in amperes, in a circuit having an E. M. F. of 100 volts, and a resistance of 5 ohms, is equal to 100 divided by 5, or to 20 amperes. Thus, on page 341, of Thompson's "Dynamo-Electric Machinery," occui's the equation which is referred to as the funda- mental equation of the continuous-current dynamo : E^nZN-. 108, where 7^, is the number of revolutions of 22 ALGEBRA MADE EAST. the armature per second, Z, is the num- ber of armature conductors employed, N, is the number of ma2;netic lines throus;h the armature, and 10^, is 10 raised to the 8th power, or 1 followed by 8 zeros. The equation may be written : ^ nZN nZN ^ = TO^ «^ 100,000,000 '''°^*'- This is solved by multiplying together theproper numerical values of «, Z, and N, and dividing the product by 100,000,000. The quotient is the E. M. F. of the ma- chine in volts. On page 493, of Thompson's " Dynamo- Electric Machinery," there occur the equa- tions : Jx where w, is the power utilized in a motor, E, is the C. E. M. F. of the motor, C, is the current through the armature ; i2, is the resistance of the motoi', and S, is the pressure at the motor terminals. This ALGEBRA MADE EAST. 23 is an expression Avlnch practically contains three equations ; namely, w - ^O (1) ^C=.^(^3 (2) w=.E^^) (3) These equations may be interpreted as follows : (1) The useful power in watts is equal to the pi'oduct of the C. E. M. F. of the motor in volts, and the curi-ent strength passing through its armature in amperes. (2) The product of the C. E. M. F. in volts and current in amperes in the arma- ture, is equal to the product of the C. E. M. F., multiplied by the difference between the pressure in volts at terminals and the C. E. M. F. in volts, divided by the resist- ance of the machine in ohms. (3) The useful power of the motor in watts is equal to the last-named quantity. Thus, if ©, the pressure at terminals, = 24 ALGEBRA MADE EAST. 120 volts, R = 0.05 ohm, C = 100 am- peres, ^ = 115 volts ; then, by (3), ,,. (120 - 115) w = 115 ^ ^ 0.05 = 115 (AJ 0.05 = 115 X 100 = 11,500 watts. CHAPTER III. POWEES AND ROOTS. If we multiply a number by itself, the number is said to be squared, and is repre- sented by an exponent, index, or super- script 2, placed close to and above the number. Thus, ay. a = a^; or, 10 x 10 = 10^ = 100. Similarlj^, if a number be multiplied by itself and again by itself, it is said to be cubed. Thus a Y, a Y. a = c^ Similarly, axaxaxa = a''', and so on, This general rule is expressed symbolically by a X a ... to n terms = a" ; 25 26 ALGEBRA MADE EASY. where n, is any whole number or integer. Thus, 108 ^ 100,000,000, or 1 followed by 8 zeros. Thus on page 493, of Thompson's " Dy- namo-Electric Machinery," occurs the equa- tion: W=w-\- C^^B. Where W, is the power in watts supplied to the motor ; w, is the power in watts utilized in the motor; O, the current strength in amperes passing through the motor ; and R, the resistance of the motor in ohms. This means that the total power supplied is eqnal to the sum of two terms, one being the utilized power, and the other the product of the square of the current and tlie resistance of the motor. Thus, if w = 10 kilowatts, or 10,000 watts; C ^ ^Q amperes, R = 0.1 ohm, then, W= 10,000 -f 50 X 50 X 0.1 =10,250 watts. ALGEBRA MADE EAST. 27 On page 144, of Thompson's " Polyphase Electric Currents," there is an equation : Here T, which is the torque of the motor, is expressed as the product of q, and a frac- tion. The numerator of the fraction is sR, or the product of s and R ; the denomina- tor of the fraction is the sum of two terms : namely, the square of R, and the square of Ic, multiplied by the square of s. This equation might be written : just as 5 (3/4) may be written 15/4. On page 152, of Thompson's "Polyphase Electric Currents," appears the equation : The rotor heat H= K{fl- oof. Here £1 (Capital Omega) is an angular velocity of a rotary magnetic field ex- pressed in unit angles per second, and oo, (small Omega) is the angular velocity of 28 ALGEBRA MADE EASY. the rotor, or revolving member of the motor. Consequently, (il — oo) is the dif- ference between two angular velocities, or the angular velocity of the field relatively to the moving armature or rotor, so that if the armature revolves at exactlj'' the same speed as the field (p. — go) — 0. Then the equation states that the heat H, is the product of JK^, and the square of the quan- tity (n — £»). I^, itself is stated at the top of the page to be : In the same \vay, any combination of terms may be raised to any power. Thus, (a + b - c + dy means that the quantit}^ (a, + h — c + d) must be multiplied by itself four times in succession. The first product M'ould be the square, the second product would be the cube, the third product would be the fourth power, and the fourth product would be the fifth power. ALGEBRA MADE EASY. 29 If we multiplj^ 10^ or 100, by 10^ or 1000, we know that the product is 10^ or 100,000. Shuilai'ly, if we multiply a^ by a?j the product ahi!^ = cv", and not a'^. This rale, which is of general application, shows that when products are formed of the powers of a quantity, the indices or expo- nents of the powers are added together to form the product. According to this rule, 10^ X 10° = ]^Q(3 + o) JJei-e the index is the sum of 3 and 0, or simply 3, representing 10^ From this it is evident that if we multiply 10^ by 10°, we leave it unaltered, just as it Avould be if multiiDlied by unity, because 103 X I = 203_ Consequently, 10° = 1. This relation is generally true and is ex- pressed by the equation a°= 1, or aP = I, always, Avhatever a or x may be. Again, 10^ X 10-3 = iq(z-z, = jq" = 1 ; 30 ALGEBRA MADE EASY. . ■ . dividing both sides of tlie equation, 103 X 10-3 = 1 by 10^ we obtain 10^ X 10-3 1 103 103- Here, on the left-hand side, the 10^, in the numerator cancels the 10^, in the denomi- nator ; so that we obtain 10- =103- Similarly, ■' or, is the Teciprocal of 1 0^ =■ 0.1. 1 n-^ = 10 10^, or, is the reciprocal of 10^ = 0.02. 1 0-8 = 10 108 or, is the reciprocal of 100,000,000. = 0.000,- 000,01, and generally. ALGEBRA MADE BAST. 31 a~ or, is the reciprocal of a°, or of a.a.a. to n times. , EADIOALS OE KOOTS. By the genei'al law of the summation of exponents, a^ = a} X a} z= a'-^^^'^ a = a^ = a* X (X^ = a '* + ^'' Here the fractional index or exponent, rep- resents what is called a root. For, a^ is obviously, by the last equa- tion, .that quantity which multiplied by itself gives a, or is the square root of a. Thus, if (X = 9, a* = 3. a* is often writ- ten va or "J a. Again, a = «! = a* X a* X a^ = a*^* + * + '^• Here a* is the quantity which, cubed, gives a, or is the cube root of a, and may be written V a. Similarly, the square root of any quantity, such as ah, is written 32 ALGEBRA MADE EASY. * "Jlih. or \/ (ab). Thus, on page 194, of Thompson's " Dynamo-Electric Machinery," there is an equation, This means that the resistance R, is equal to the square root of the product of two resistances ; one of which is the resistance of the armature, and the othei- the resistance of the shunt field-magnets. If the product, r^ o\, is represented by the symbol r ; or, if we assume that r^ r^ = r, then it would follow that si ^ \J r. The square root of the product of two quantities is called their geometrical mean, so that the equation [XII] declares that R, is the geometrical mean of the two resistances, r^ and ')\ In a similar manner, rn is the quantity whose «-th power gives r, and may be written V q\ Thus, ALGEBRA MADE EASY. 33 125* = V' 1^5 = 5, because 5X5X5 = 125 ; or, 5, is tlie cube root of 125. We see, therefore, that uO ^ Uj /\ vO J or, is the square of x. W /\ sc the reciprocal of the square of x. or, is the square root of x. In a similar manner we may have any fraction for the exponent of x. Thus, we know by the law of summation of indices that x^ = x^ y. x^ X x^ or (cc*)' so that 0?-, represents the cube of the square root of x, and may be written Xi^ (n/"^)' For example, if a; = 4, «« = ( v^)' = 2' = 8. 34 ALGEBKA MADK EASY. Again, a^ = iff = \l'^. For example, if a; = 4, 4« = VT^ = v^~64 = 8. So that the square root of the cube of a?, is equal to the cube of the square root of 2. In general cc^ = {x^^ = iofy = v/^ = (Va;)". On page 137, of Thompson's " Dynamo- Electric Machinery," there is the following equation : W = 0.0033 X 10-^ X « X B'\ This expresses the fact that the power W, in watts, expended per cubic centimetre of iron, by hysteresis, is the product of four quantities : The first of these is the 33 numerical constant 0.0033, or— ; the ' 10,000 ' second is 10"'' or — =■ = ; the 10^ 10,000,000 ' ALGEBRA MADE EASY. 35 third is n, or the number of magnetic cycles executed in the iron per second, while the fourth is B^^, or the lAth power of B, the magnetic density in the iron, expi'essed in C. G. S. units per square centimetre. ^1.6 ^ ^H so that if we form the 16th power of B, and then take the tenth I'oot of this quantity ; that is, if we perform the operation B^^, and then v -fi'", we obtain the quantity B^'^. Or, if we take the tenth root of B, written v' B, and then take its sixteenth power, we shall obtain the same result. This quantity will obviously be greater than B, itself, or B^, and will be less than B x B, or B^, since 1.6 is inter- mediate between 1 and 2. On page 160, of Thompson's "Poly- phase Electric Currents," occurs the fol- lowing formula : i; = \/ r''4:Tt^ {n + mf B\ This is equivalent to the following state- ment: The impedance in ohms I^, is 36 ALGEBRA MADE EASY. equal to the square root of a compound quantity. This compound quantity is the sum of two terms. The fii'st term is the square of the resistance in ohms, while the second tei'm is the product of 4 into the square of the quantity n, which is 3.1416, approximately, or the ratio of a circumference to its diameter, into the square of the sum (pi + m) of two fre- quencies, into the square of the inductance L, in henrys. This equation might be written : or i; CHAPTER IV. EQUATIONS AND THEIR SOLUTION. The solution of equations is only to be successfully attained by pi'actice, but the elementary rules for their operation are very simply expressed. When two things are equal to each other, which is the condition expressed by an equation, the same opei-ation performed upon each will leave the equality un- changed. Thus, if a ■ = b (1) 2 ■ __ h 2 (2) a^- = Jf (3) a + - G - = b + G (4) In equation (2) we have divided both sides by 2, and th e quotients must remain 37 38 ALGEBRA MADE EASY. equal. In equation (3) we have squared both sides of the equation, and the result must be equality. In equation (4) we liav^e added the same quantity c, to both sides of the equation and equality must still subsist. If a^h^c (5) then a^l -I = c -h (6) or, a = c — h (7) From this it is evident that we may carry over a term from one side of an equation to the other by changing its sign ; for, in equation (5) the positive quantity /;, is on the left-hand side of the equation, whereas in equation (7), which is dei'ived from equation (5) by subtracting h, from both sides, the quantity 1), appears on the right band with the negative sign. An equation x = a-\'h-\-G-\-d etc. is an equation of the first degree, because X, appears as of the first power, or x^. ALGEBEA MADE EASY. 39 The equations a? = h + G or a? -{- ax = b + c are quadratic eqioations, or equations of the second degree, because there occurs a second power of x, in the equation. Similarly, such an equation as a^ + 3a^ + 3a; = 5 is an equation of the third degree; and, generally, an equation involving a;", is an equation of the nth degree. An equation can always be solved, when of the first, second, or third degree, by definite rules. That is to say, it can be so manipulated, by suitable operations up >n both sides, that the value of x, can be obtained. Tliere is, however, no known way of generally solving equations of higher degrees than the third ; i. e., equations of the fourth, fifth, sixth, etc., degrees. But the numerical values of the unknown 40 ALGEBRA MADE EASY. quantity can be obtained by approximation with all desired accuracy by a definite procedure. Every algebraic expression is referable to the preceding rules ; that is to say, to combinations of additions, subtractions, multiplications, divisions, powers, and roots ; and, however difficult it may be to solve or manipulate equations, the fore- going explanations will always enable any algebraic equation to be understood, or to be arithmetically solved, Avhen all the symbols are replaced by their proper numerical values. Thus, we may take what is, perhaps, the longest and most complex formula appearing in Thompson's "Polyphase Electric Currents." This ap- pears on page 163, as Tor ue = r \ ^ - m n ■\- 7n -| 1^ + ^n^ 1} in -\- mfl This equation is equivalent to the follow- ing statement : ALGEBRA MADE EASY. 41 The torque of tlie motor is the product of three quantities, the first quantity q, is expressed immediately below as that is to say, the qtiantity q, is one quarter of the product of Z, into the square of A, into the square of B^ into 3.1416. The second quantity is the resistance r. The thii'd quantity is contained within a pair of brackets and consists of the sum of two fractions. The first fraction has as numerator the difference between two frequencies n and m, I'espectively. The denominator of the fraction is the square of r, added to the product of 4, into the square of 3.1416 into the square of the inductance L, into the square of the difference {n — m).* The second fi'action, wliich is to be sub- tracted from the first, has, as its numer- ator, the sum of the two frequencies n + m, *By fimispi-int, tlie square of (;i — m) has been omitted in the text referred to. 42 ALGEBRA MADE EASY. and as its denominator the square of the resistance ;■, added to the product of 4, into the square of 3.1416, into the square of the inductance L, into the square of the sum of the frequencies {n -\- m). CHAPTER V. LOGAEITHMS. We have seen that by the law of the summation of indices and similarly If, then, the quantity a}-^ = a ; or, the i.3d power of the hme x is a, and x?^ = h; or, the 3.6th power of the base x is i, and xf'-^ — g; or, the 4.9th power of the base X = c, it follows that ah = c. Suppose that we had a table of indices of a given base, say 5, and that we found from this table that the number 25, was 5^, or had an index of 2, while the number 125, had the index 3, corresponding to 5^; then we should know that 25 X 125 = 52 X 5^ = 5^, 43 44 ALGEBRA MADK EASY. and, if the table informed us that the number corresponding to the power 5, was 8125, then we should know that : 25 X 125 = 3125, and we should have been saved the trouble of performing the multiplication. Here the indices 2, 3, and 5, are the re- spective logaritlims of the numbers 25, 125, and 3125, to the base 5. The ordinary tables of logarithms are usually employed for the purpose of en- abling multiplication and division to be effected quickly and conveniently without actual arithmetical computation. The base of the ordinary table of common logarithms, as they are called, is 10, so that, since 10^ = 10 10^ = 100 10^ = 1000 10°= 1 10-1 = QJ lQ-2 ^ Q(^^ ;^Q-3 = oQoi^ it follows that to the base 10, the loga- rithm of 10 is 1 ; the logarithm of 100, is ALGEBRA MADE EASY. 45 2 ; of 1000 is 3 ; of 1 is ; of 1/lOth, or 0.1 is -1 ; of 0.01 is -2 ; of 0.001 is -3, etc. All numbers lying between 10 and 100, will have logarithms lying between 1 and 2. All numbers lying between 100 and 1000, will have logarithms lying be- tween 2 and 3, and so on. Thus, if we want to multiply 15 by 16 ; or have to perform by logarithms the solution of a; = 15 X 16, we know that the logarithm of 15, lies be- tween 1 and 2, because 15, lies between 10 and 100, and 10' = 10 and 10^ = 100. By reference to a table of seven- place logarithms, or logarithms carried to seven decimal places, the logarithm of 15 is .1760913. This is the decimal part or mantissa. The complete logarithm is 1.1760913, because the characteristic is 1, and is supplied by the reader. The char- acteristic distinguishes the logarithm from that of 0.15, or 0.0015, or 1.5, or 1500, all of which have the same mantissa, but 46 ALGEBRA MADE EASY. differ in their characteristics, their loga- rithms being respectively — 1 + 0.1760913, -3 + 0.1760913, 0.1760913 and 3.1760913. Again, the logarithm of 16, is shown in the tables to be .2041200, and with the proper characteristic of 1, is written 1.2041200. We now have 15 = 10'-'™9i^ 1(3 := IQl. 2041200 or a; = 15 X 16 = io.''-3802ii3. Here the number x, has as its logarithm the number 2.3802113. Its characteristic is 2, and the corresponding number, there- fore, lies bet^veen 10* or 100, and 10^, or 1000. The decimal part, or mantissa, is .3802113. This is found in the logarithm tables to be the logarithm of 240, so that 240 = 10 ^-^^o^iis and a? = 15 X 16 = 240. ALGEBRA MADE EASY. 47 Again, if we look in the logarithm table for the logarithm of the nnuiber 5280, which is the number of feet in a mile, Ave should find that to seven places of decimals the mantissa is .7226339. This means that lO^-^'^ssg ^ gggo. The 3, or cliaracteristic of the logarithm, is not given in the table, but is known by the reader, because the number 5280 lies between 1000, for which the logarithm is 3, and 10,000, for which the logarithm is 4, so that he supplies the characteristic when he writes the logarithm down. Again, if we look for the logarithm of 24,900, which is, approximately, the number of miles around the earth at the equator, we should find the value 4.3961993. Here the characteristic 4, is known because the number falls between 10,000 and 100,000, whose logarithms are 4 and 5 respectively. If now, we add these two logarithms together, we per- form in fact the equation : X = 103.7226339 y^ -|A4.3961993 = -j^Q8.1188332 •48 ALGEBRA MADE EASY. Here the logarithm 8.1188332, consists of the characteristic 8, and the decimal part or mantissa 0.1188332, which in the logarithmic tables corresponds to the number 131472. Since the characteristic is 8, we know that the number x, lies be- tween 108 and 10", or between 100,000,000 and 1,000,000,000, so that the number is evidently 131,472,000 and is correct as far as 6 places of figures. This product X, is evidently the number of feet around the earth at the equator according to the above calculation. Actual multiplication or arithmetical solution of the equation, X = 5280 X 24,900 , gives X = 131,472,000 feet, which agrees exactly with the above loga- rithmic computation. Again, we know that 10* 1 10» -- 10" = j^ = 10" X iQb = 10" X 10-" - lo'"-". ALGEBRA MADE EAST. 49 Thus, the quotient of two numbers, the dividend of which is 10", and the divisor is 10'', is expressed as 10""'', so that just as tte sum of two logarithms gives the loga- rithm of their product, the difference of two logarithms gives the logarithm of their quotient. Thus, if we want to divide 170 by 26, by the aid of logarithms, or solve the equation 170 as = — , we proceed as follows : The logarithm of 170, lies between 2 and 3. By tables its mantissa is .2304489. The complete logarithm of 170 is, there- fore, 2.2304489. Similarly the logarithm of 26, lies be- tween 1 and 2. By tables its mantissa is .4149733. The complete logarithm of 26 is, therefore, 1.4149733. We now have 50 ALGEBRA MADE EASY. 26 = 10i-"«''23 ITO . ' — 1 A(2.2304489-1.4149733) ••26 170 2o Here the logarithm of the numbei- x, lies between and 1, so that x, is between 1 and 10. The mantissa is .8154756. This is found in tables to correspond to 6.53846. If we divide 26 into 270, by the ordinary- arithmetical process, we find in fact that the quotient is 6.53846 as far as 5 decimal places. For example, on page 156, of Thomp- son's " Dynamo-Electric Machinery," oc- curs the equation : d" Permeance = 2.274 X a" X logjQ-p. This is equivalent to the statement that the pernieauce is the product of the con- stant numerical quantity 2.274, into a", ALGEBRA MADE EASY. 51 into the logarithm to the base 10 ; i. e., the common logarithm, of the quotient di ' The logarithmic quantity might be con- sidered as the common logarithm of the quotient 4" ^' obtained by first dividing d^' by d-^' arith- metically, and then obtaining the logarithm of the quotient by examining a table of logarithms. But the same result will be obtained if we subtract the logarithm of o^i" from the logarithm of d^'. Thus, sup- pose that c/g" = 48, and that r//' = 6. Then d^' logio^-, = logio8; or, the logarithm of 8 to the base 10, which, by reference to logarithm tables, is 0.9030900. In other words W-^^°^^'> = 8. But we may arrive at the same result by taking the logarithm of 48, or 1.6812413 52 ALGEBRA MADE EASY. and the logarithm of 6 = 0.7781513, and subtracting them. We then have — -10(0.9030900) Logarithms are also used to perform con- veniently and quickly involution or evolu- tion; i. e., to obtain powers, or to extract roots. Thus it would be a troublesome operation to obtain the 12th po^ver of say 15, or to solve the equation but with the aid of logarithms this is very simply performed, because the logarithm of 15, is found to be 1.1760913. • 15 = ;[()(1-1760913) and 2512= /JQ1.1760913\12 — 2Q(1.1760913) X 12 = 2Q14.1130956 = 10'" X 1.29746 since by reference to tables the number ALGEBRA MADE EASY. 63 1.29746 has tlie logaritlim 0.1130956, or X = 129,746,000,000,000, so far as six places of figures. The actual number is 129,746,337,890,625. Similarly, the 4tli root of 15, is obtained by dividing the logarithm of 15 by 4. Thus 1.17G09I3 15 = 10 4. — -_- J- 1.1760913 VI 5 = 154 - 10—^ = -YC^a.nm^ — 1.968, approximately, by reference to tables. Consequently, (1.968)* = 15 approxi- mately. For some calculations the base 10, is in- convenient, and a base is then adopted which is more natural. In the theory of numbers and their exponents, this base, as far as five decimal places, is the number 2.71828 and is called the Naperian base, and is usually represented by the symbol e. A logarithm of this base is usually called a natural .logarithm, a Naperian logarithm, or a hyperbolic logo- 54 ALGEBRA MADE EASY. rithm, to distinguisli it from the common logarithm to the base 10. It is written log, X. Thus, if (2.71828 )" = x then n = log, x. It can be readily shown that the Na- perian logarithm of a number is greater than the common logarithm of that num- ber in a fixed i-atio which is, approxi- mately, 2.3026, so that, if we multiply the common logarithm of a number by 2.3026, we obtain its approximate Naperian logarithm. Thus the common logarithm of 15, is 1.1760913 or 15 = lO'i™^^^ The Naperian logarithm of 15, is, there- fore, approximately, 2.3026 X 1.1760913 = 2.708, approxi- mately, or, 15 = e^™^, approximately, = (2.71828)^-^8. Thus on page 157, of Thompson's ALGBBEA MADE EAST. 55 "Dynamo-Electric Machinery," there oc- curs the expression : (L -y 1 dia (1 n This means — x 2.3026 logio -4 ap- proximately. CHAPTER VI. TKIGONOMETKY. TRiGOisrojiETRY is tlie science whicli deals with ambles and tlieir relations in geometrical figures. There are two ways of measuring angles in general use. The first consists in the ordinary- method of dividing a complete revolution into 360°, and measuring the angle in degrees, minutes, and seconds ; there being 60 minutes in a degree and 60 seconds in a minute. The second method, which is important in theoretical treatment as distinguished from practical treatment, measures an angle by the ratio of its arc to its radius. Thus, in Fig. 1, the ratio of the length a, of the arc of the angle a or AOB to the length of the radius r, of the circle on S6 ALGEBRA MADE EASY. 57 which it is drawn, is called the radian measure of the angle a. It is obvious that if the arc «, is the same length as the radius r, the ratio — will be unity, and this will be a unit angle in radian measure. Such an angle, which is called a radian, when expressed in degrees, is equal to 57° 17' 45", approximately. A complete cir- cumference, having a length which i-s 27r times, or 6.2832 times, the length of the radius, such a complete revolution of 360° is equal to 'in radians. Conse- quently, a right angle is — or — radians, and a single degree is = radians. ^ ^ 360 180 58 ALGEBRA MADE EAST. In dealing witli angles, certain ratios called the trigonornetrical ratios ov functions are constantly used, and it is> important to clearly understand their nature and mean- ing. Thus, let a, (Fig. 2) be an angle a = AOjB, included between the lines OA and 0J3. Then let fall a perpendicular J5C, from the point J^, upon the base OA. Then the fraction whose numerator is the length of the perpendicular JSC, and whose denominator is the length of the radius, or the fraction : — - is called the sifie of the OJJ angle a, and is Avritten sin a, as an abbre- viation of the term sine of angle a. Suppose, for example, that the length of the line £0, is 1 inch, while the length of the line OA, which is also the length of the line, or radius 0£, is unity, or li JSC 1 inches. Then the fraction -—-y = — = 0.8 (JO I4 is the sine of the angle a ; or, in this case, sin a = 0.8. The fraction whose numerator is the ALGEBRA MADE EASY. length between O and G, or the base OG, and whose denominator is OB ; that is the OG fraction 7yr,i is called the cosine of the Fie- S angle a, and is written cos a, as an abbre- viation for the term cosine of angle «-, The fraction whose numerator is the length of the perpendicular BG, and whose denominator is the length of the BG base OG^ or the fraction -?^,is called the 60 ALGEBRA MADK EASY. tangent of the angle a, and is written tan a, which is an abbreviation, of the term, tangent of angle a. The fraction whose numerator is the length of the radius OB, and whose de- nominator is the leno-th of the base 00, OB or the fraction j-pj , is called the secant of the angle a, and is written sec a, as an abbreviation for the term, secant of angle a. The secant of the angle is the recipro- cal of the cosine or sec a = . Con- cos a sequently, Avhen the cosine of the angle is known the secant becomes immediately known. The fraction whose numerator is the length of the radius OB, and whose de- nominator is the length of the perpendicu- OB lar BC, or the fraction -y-~- , is called the ' BO cosecant of the angle a, and is written cosec a, as an abbreviation for the term, cosecant of angle a. The cosecant of an ALGEBRA MADE F.ASY. 61 angle is the reciprocal of the sine of the angle; or, cosec a = ^ sm a The fraction whose numerator is the length of the base OC, and whose denomi- nator is the length of the perpendicular BC, or the fraction -^-?^, is called the cotan- gent of the angle a, and is Avritten cot a, as an abbreviation for the term, cotangent of angle a. It is the reciprocal of the tan- gent of the angle, so that cot a = . tan a As an angle increases in value, that is to say, as the radius OB^ is carried further and farther aAvay from the initial line OA, these trigonometrical ratios — i. e., the sine, cosine, tangent, and their reciprocals, the cosecant, secant, and cotangent — undergo variation. Confiiring our atten- tion to the first right angle or first quad- rant / i. e., to the angle a, whose value is not greater than 90°, the sine increases from to 1 ; the cosine diminishes from 1 62 ALGEBRA MADE EASY. to ; and the tangent increases from to 00. In other words the sine of 90° is unity ; the cosine of 90° is zero ; and the tangent of 90°, is indefinitely great, or infinity. In Fig. 3, the radius at the angle 60°, has a perpendicular BCx', assuming that the radius OA^ or OB^ is of unit length, the length BC^, will be found to be 0.866, approximately ; and this is the ratio of BO, OB, or the sine of the angle 60°; or, sin 60° = 0.866. The length of the base OG, will be found by measurement to be half OB; or, if OB, is unit length its value will be 0.5, so that the cosine of the angle 60° is 0.5 ; or, cos 60° = 0.5. Similarly, the fraction represented by the BO, _ 0.866 _ , ^_ odr^^~ is the tangent of the angle «; so that tan 60° = 1.732. ALGEBRA MADE EASY. 63 Carrying the moving radius, or radius vector as it is called, past the perpendicular Oh, into the second quadrant, to a position Fig. 3 OB2, which is 150° angularly distant from OA, the length of the line -^362, which represents the sine of the angle B^OA, is 0.5, so that sin 150° = 0.5. The length of 64 ALGEBRA MADE EASY. the line OC'g wliicli measures the cosine of the angle B^OA, is 0.866 but being meas- ured backwai'd from O, or in the negative direction, it is written —0.866, so that cos 150° = -0.866. The tangent of the angle 150° is the fraction B,C, _ 0.5 _ _ .^ ~0C, =^0866 ~ -^^^^ approximately, so that tan 150° — —0.577, approximately. The cotangent would be the reciprocal of this or =W7 = -l-^^2- CaiTying the moving radius or radius vector into the third qtiadrant, to such a position, for example, as that represented by OB^, so as to include an angle A OB^, of 240°, the length C^B.^, is the same as -SgCi, but is now measured below the line OA, or negatively ; so that sin 240° = —0.866. The cosine OC^ — —0.5 so that cos 240 = -0.5. The tuns-ent i?,a -0.866 Oa -0.5 '■^'^"' ALGEBRA MADE EASY. 65 wliile the cosine, secant, and cotangent are the reciprocals of these three quantities respectively. Carrying the radius vector into the fourth quadrant, to such a position as is represented by the line OB^, so as to include an ancrle of 350°, between OA and OB^, the length -S464, which measures tlie sine of the anofle, will be found to be —0.174, approximately, the minus sign being attached, because B^Q/^ is below the line 0A\ the cosine, or the length OC^, is +0.985, approximately, so that cos 350° = 0.985. The tangent B,C^ _ -0.174 Oa 0.985 = -0.176, approximately, and the cosecant, secant, and tangent are found as the reciprocals of these three quantities, respectively. Trigonometrical tables give the numeri- cal values of the sine, cosine, tangent, cosecant, secant, and cotangent for all angles to any reasonable desired degree of 66 ALGEBRA MADE EASY. accuracy. Tables are also commonly given of the common logarithms of the trigono- metrical ratios for convenience in multiply- ing and dividing them. On page 648, of Thompson's "Dynamo- Electric Machinery," occurs the equation : ei -— El cos 6. This means that the E. M. F. induced in an adjacent coil of wire, at any instant by the revolving magnet, is equal to the prod- uct of a certain maximum E. M. F., denoted by the symbol JjJ-^ , and the cosine of the angle which is included between the position of the magnet, at the instant con- sidered, and the position of maximum E. M. F. If the angle 6 = 0, cos 6, will be found by reference to a table, or by exami- nation of Figs. 2 and 3, to be unity, and the equation becomes ej = ^i ; whereas, if the magnet has turned through an angle of 90°, it cannot at that instant induce any E. M. F. We find, correspondingly, that cos 90° = 0, so that e^ = U^ X = 0. ALGEBRA MADE EASY. 67 On page 160, of Thompson's "Poly- phase Electric Currents," the equation occurs : r cos cp-^ — y. Here ^Jj, is a certain angle ; namely, an angle of lag, or the angle between an E. M. F. and the current it is supposed to produce. The equation states that the cosine of this angle cp-^ is the quotient obtained by dividing the resistance r ohms, by the impedance I^ ohms, and if r = say, 3 ohms, and I^ — 5, then r 3 and the equation becomes cos