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(a + b) Xc=axc+ bxc=c× (a+b), which is the law of distribution in multiplication and addition.

The construction and proof are not essentially different when some or all the magnitudes involved are affected with the negative instead of the positive sign.

The second factor may also consist of two or more terms; for

(a + b) x (c + d) = a × (c + d) + b × (c + d),
by the distributive law,

=(c + d) Xa + (c + d) × b,

by the commutative law,

and again, by the distributive and commutative laws, (c+a) Xa+(c+d) × b=cXa+da+c x b + d x b = axc+ad+b×c+bxd.

Hence, replacing the sign + by the double sign ±, the completed formula for the distributive law in multiplication is

(±a+b)X(±c±d)=(±a)×(±c)+(±a)X(±d)

21.

+(±b)X(±c)+(±b)×(±d).

With the Sign of Reciprocation. Since a, b, c, d, in the formula last written, are any real magnitudes whatever, they may be replaced by their reciprocals, | a, |b, c, d. For the same reason, in the formula

(±a±b)X(±c)=(±a)×(±c)+(±b)×(±c), (c) may be replaced by /(±c), giving the corresponding distributive law with the sign of reciprocation: (±a ±b) / (±c) == ( ± a) / ( ± c) + (±b) / (±c).

But while the sign X is distributive over the successive terms of a sum (Art. 20), that is:

X(±a±b±...)=X(±a)+× (±b) + × ...,

the sign is not, as may be readily seen by constructing the product X / (a+b), and the sum of products X/a +X/b, and comparing the results, which will be found. to differ.

22. Agenda: Theorems in Proportion, Arithmetical Multiplication and Division.

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prove that

and that

(3). If

prove that

(4). If

prove that

A+BBC+D: D (Componendo.)

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(5). If A: B :: Q : R and B : C :: P : Q,

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(6). Show that corresponding to

cx (Xa + X b) = X (X c × a + Xe X b)

there is the analogous formula

(7).

result is 6.

c! (/a + /b) = | ( | c | a + |c|b).

Construct the product 3 X 2 and show that the

(8). Construct the quotient 1 2 / 3 and show that the result is 4.

(9). Construct 5/3 and 2 X 5/3.

VII.

EXPONENTIALS AND LOGARITHMS.

23. Definitions.* Suppose P, Q to be two points moving in a straight line, the former with a velocity, more strictly a speed, ** proportional to its distance from a fixed origin O, the latter with a constant speed. Let x denote

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the variable distance of P, y that of Q from the origin, and let λ be the speed of P when x = OJI, μ the constant speed of Q. As it arrives at the positions J, P', P'' successively, P is moving at the rates:

λ, x'λ, x''λ, respectively,

the corresponding values of x and y are:

x=0J= 1, x'= 1 + JP', x''=1+JP",

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* Napier's definition of a logarithm. Napier: Mirifici logarithmorum canonis descriptio (Lond. 1620), Defs. 1-6, pp. 1-3, and the Construction of the Wonderful Canon of Logarithms (Macdonald's translation, 1889), Def. 26, p. 19. Also MacLaurin: Treatise of Fluxions, vol. i, chap. vi, p. 158, and Montucla: Histoire des Mathematiques, t. ii, pp. 16-17, 97.

**The speed of a moving point is the amount of its rate of change of position regardless of direction. Velocity takes account of change of direction as well as amount of motion. See Macgregor: Elementary Treatise on Kinematics and Dynamics, pp. 22, 23 and 55.

The speed of P obviously known By means of this

and Q is supposed to pass the origin at the instant when P passes, at which point x=0J=1. relatively to that of Q, or vice versa, is as soon as the ratio of μ to λ is given. construction the terms modulus, base, exponential, and logarithm are defined as follows:

(i). μλ, a given value of which, say m, determines a system of corresponding distances x', x'', . . . and y', y'',... is called the modulus of the system.

(ii).

The modulus having been assigned, the value of x, corresponding to y=0J=1, is determined as a fixed magnitude, and is called the base of the system. Let it be denoted by b.

(iii). x, x', x'', . . . are called the exponentials of y, y', y'', . . . respectively, with reference either to the modulus m or the base b, and the relation between x and y is written

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x=expm J', or xby.**

(iv). y, y', y'', ... are called the logarithms of x, x', respectively, with reference to the modulus m, or the base b, and the symbolic statement of this definition is either

y=log, x, or y= 'log x. *

The convention that P shall be at duces the convenient relations

when Q is at O intro

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logm I=0, and expm 0= 1, or b2 = and because (by definition) y : I when x = = b,

*

logm b

I,

I, and expm I= = b, or b1 — b.

blog is the German notation. English and American usage has hitherto favored writing the base as a subscript to log thus, log; but Mr. Cathcart in his translation of Harnack's Differential u. Integral Rechnung has retained the German form, which is here adopted as preferable.

**See Appendix, page 139.

Exponentials and logarithms are said to be inverse to each other.

(v). The logarithms whose modulus is unity are called natural logarithms, and the corresponding base is called natural base, the special symbols for which are ln and e respectively; thus

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y = log, x = ln x, and x ey = exp y

represent the logarithm and its inverse, the exponential, in the natural system.

24. Relations between Base and Modulus. Let the speed of P remain equal to Xx as in Art. 23 while the speed of Q is changed from μ to kμ. The modulus, μ μ/λ=m, will then be changed to kμ/λ=km, and the distance of Q from the origin, corresponding to the distance x, will become ky. Hence

μ

ky=logkm x = k logm x;

that is: To multiply the modulus of a logarithm by any real quantity has the effect of multiplying the logarithm itself by the same quantity.

In particular we may write

logm x = m ln x.

Corresponding to this relation between logarithms in the systems whose moduli are m and km, the inverse, or exponential relation is

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* These are also sometimes called Napierian logarithms, but it is well known that the numbers of Napier's original tables are not natural logarithms. The relation between them is expressed by the formula

Napierian log of x=107 ln (107 | x).

Napier's system was not defined with reference to a base or modulus. See the article on Napier by J. W. L. Glaisher in the Encyclopædia Britannica, ninth edition.

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