act of his mind in each process as an addition, subtraction, multiplication, division, commutation, association, or distribution of numbers, under the definitions and laws set forth in the preceding chapters. To take a very simple

example: (a3 b2 cá) (a5 b6 c11) ÷ (a1 b3 c15)

=

=

(a3 a5 b2 b6 c5 c11) ÷ (a1 b3 c15) . . . by association and commutation.

...

(a3 b3 c16) ÷ (a1 b3 c15) by three partial multiplications by law of indices.

= a

commutation.

a4 b5 c... by three divisions by law of indices. 74. Explain how a multiplying machine, which can do no more at one time than multiply a number of ten places by another of ten places, may be used to multiply 13693456783231 by 46381239245932.

75. The involution of primary numbers may be accomplished merely by repeated multiplication. As soon, however, as one investigates logarithmic series, and the construction and use of Tables of Logarithms, he learns command of a more facile way of performing this laborious operation. Before learning the use of logarithms, one ought to demand good wages for the toil it would cost him to find 982; afterwards it becomes the work of a few minutes.

76. Evolution, as we have seen, is only occasionally possible under the primary concept of number; but even in the simplest of these possible cases the device of calculation familiarly used by the high-school pupil is rarely understood, else he would be able to find (however laboriously) the fifth root as well as the third. Of course evolution is too laborious to be carried to any extent until Logarithmic

Tables are comprehended, when it becomes easy. But if one understood how his device for extracting a second or third root was invented, he could on occasion make his own rule for finding a fifth root. Let us investigate. Properly distributing and associating, it is seen that—

(a + b)2 = a2 + b ( 2 a + b).

Also (a + b + c)2 = (a + b)2 + c { 2 (a + b) + c}, etc.

Here is declared a rule for the evolution of a second root of a number; for a specific composition of the power is displayed in a way to make decomposition easy. Likewise the formulæ for the evolution of a cube root are

(a + b)3 = a3 + b (3 a2 + 3 ab + b2),

and (a + b + c)3 = (a + b)3 + c { 3 (a + b)2 + 3 (a + b) c + c2}, etc.

In exactly the same way the formula for the evolution of a fifth root is

(a + b)5 = a5 + b (5 aa + 10 a3b + 10 a2b2 + 5 ab3 + l1), etc. Suppose the fifth root of 33554432 is required.

Now the preceding formulæ show that, if the root be considered as the sum of three numbers, the corresponding power of the sum of the first two is to be taken away, and the remainder decomposed to reveal the third summand of the root, and so on. Therefore we could not go wrong even by choosing parts of the root at random. But a consideration of the arithmetical notation may save much trouble; for it is plain that a fifth root of the number before us has two digit figures, that is, it is to be regarded as the sum of a number of tens and a number of ones. We compute as follows:

77. Now let the student compute again, taking 20 for a and 12 for b. Also let him prove 12 a cube root of 1728, taking 6, then 4, then 2, as summands of the root.

IX. FIRST EXTENSION OF THE NUMBER-CONCEPT.

78. The first extension of the concept of number is the identification of the ratio of any two magnitudes of the same kind, and without qualitative distinction for the purposes of the comparison, as a number.

79. This step was taken long ago (Cf. Introduction, p. 11), and is now universally accepted as a dictum, even where not clearly discerned as a matter of insight.

80. This development of the number-concept was no doubt occasioned in the history of human experience by problems of practical measurement. (Cf. Introduction, p. 13.)

Thought must have operated as follows: If the numerical relation of a yard to a foot is 3, surely there is a number denoting the relation of a yard to two feet, and of a

foot to a yard. That is, numbers which are fractions (vide § 83) of primary number were discerned. This advance still leaves number discrete, that is, increasing per saltum. But again, as a second step, if there is a numerical relation between two magnitudes, one of which is a fraction of the other, surely there must be a numerical relation between any two magnitudes of the same kind, even though neither be a fraction (vide § 83) of the other. Thus, when it is proved that the diagonal and side of a square are absolutely incommensurable (Euclid, Book X, 117), the mind cannot tolerate the thought that a numerical relation would exist, provided the diagonal were just the least bit shorter, yet, de facto, does not exist. This thought, I repeat, is intolerable. Moreover, since the ratio of a yard to a foot is an exact number, surely the ratio of a metre to a foot is exactly whatever it is. It is, of course, well known that the metre and foot are incommensurable

81. The connotation of all ratio (fractional and surd) as number evidently makes number continuous one way, to use a space metaphor on account of the exigencies of language. Thus, under this concept, number begins with a ratio smaller than any assignable fraction of 1, increases continuously, passing through all the discrete stages of primary number, to a ratio greater than any assignable primary number.

82. To illustrate: Start with the ratio of the weight of these pages to the weight of a granite bowlder. We begin either with a very small fraction of 1, or a surd smaller than a very small fraction of 1 (as the weights are commensurable or not, probability being vastly in favor of the latter case). Now, by gradual abrasion of the bowlder, decrease its mass; the ratios of the weights increase con

tinuously until they reach 1. Continue the abrasion, and the ratios increase continuously, passing through 2, 3, 4, etc. At length when the bowlder has been reduced to a grain of sand, the ratio will be greater than some high primary number.

83. The foregoing discourse presumes sufficient familiarity with the subject to insure the reception of the terms employed in their precise meaning; yet it may be serviceable to set forth the following definitions (Cf. § 205):

(1) MULTIPLE. One magnitude is a multiple of another when the former may be separated into equal parts, each equal to the latter. (Of course "multiple" includes the limiting case where the "part" is the whole, i.e., multiplication by 1. It is merely an imperfection of language which might seem to exclude this case.)

(2) SUBMULTIPLE. ple of the "former."

In (1) the "latter" is a submulti

(3) FRACTION. Any multiple of a submultiple is a fraction. (Of course if a is a fraction of b, b is a fraction of a; also a multiple of a submultiple may reduce either to submultiple or multiple.)

(4) COMMENSUurable. Two magnitudes are commensurable if either is a fraction of the other;

(6) RATIO. That definite (exact) numerical relation (Cf. § 80) of two magnitudes of the same kind, in virtue of which one is either a fraction of the other, or greater than one and less than another fraction of the other, which differ as little as we please, is called the ratio of the former to the latter.

Of course, from the very concept of ratios, and the