continuity of possible ratios, the ratio of the first of two magnitudes to the second is greater than the ratio to the second of any magnitude less than the first. Also two ratios are equal if every numerical fraction greater than either is greater than the other, and less than either is less than the other.

A ratio is often spoken of as "incommensurable," of course as an abbreviated expression, since it takes two things to be incommensurable. You might as well say, 66 х is equal," as to say "x is incommensurable." The abbreviation is for incommensurable with 1. Incommensurable ratios may be called surds.

Let it be clearly noted that a multiple, a submultiple, or a fraction of any magnitude, is another of the same kind; but that the ratio of two is a number. Thus a fraction of a time is a time, of a surface a surface, of a solid a solid. But the ratio of one solid to another is a number, in this case called the volume of the former with respect to the latter.

Note also, any number may be regarded as its ratio to 1, and that all numerical fractions are ratios, but not all ratios are numerical fractions.

In illustration of the definition of a ratio, and its notation, if of incommensurables, consider the yard and the metre. Measurement (vide § 203) not excessively refined, gives the number 0.9143+ for the ratio of a yard to a metre. This is to be understood to mean that a yard is greater than of a metre and less than 1. Measurement more refined would yield a numerical fraction still more closely approximating the ratio. The ratio in question has been found to be greater than 0.914392, and less than 0.914393.

143

(7) SURD. Of the one-way continuous Number, the concept of which we have now attained, those numbers which are incommensurable with 1 may be called surds. It is matter of discovery that the √2 is incommensurable or a surd.

The term surd is sometimes exclusively referred to the results of such operations as √2; but Newton's use is a philosophical one. For the √2 is found out to be 1.41421 +, that is a number, no fraction of one, but greater than 1,41421, and less than 1.41422, which is surely a number of precisely the same kind as the ratio of a yard to a metre, or of a circle to its diameter (0.914392 + and 3.14159 respectively). Incommensurable numbers resulting from evolution may be distinguished as radicalsurds, or simply radicals. (Vide § 145.)

X. SIGNIFICANCE OF OPERATIONS, AND SPECIAL OPERA~ TIONAL DEVICES, APPROPRIATE ΤΟ THE FIRST EXTENSION OF THE NUMBER-CONCEPT.

84. Euclid probably never clearly unified his concepts of ratio and number; but following Euclid (q.v., and ef. Halsted's Elements of Geometry), it may be shown that there is a combination of ratios which obeys the same laws that govern the addition of primary numbers, or of fractions of concrete magnitudes, an inverse operation corresponding exactly to subtraction; another operation ("composition of ratios"), which obeys the same laws as the multiplication of primary numbers, and an inverse ("altering" a magnitude in a given ratio), corresponding to division.

But, from the very definition of a submultiple of any

magnitude, the finding of a submultiple is identified as an operation of division, since the problem is to find a magnitude which multiplied produces the given magnitude. Now, when number has been discerned as a magnitude, these reflections make it plain that a fraction of a number is the number resulting from the division of that number by another, that one-half of 1 is 12, etc. Also, when ratios have been identified as numbers, and number thus becomes one-way continuous, the operational significance of the principles, established in Chapter VII. for Addition and Multiplication and their inverses, extends to all number (primary, fractional, and surd) thus far conceived.

*

Finally, inasmuch as a fractional number is the result of dividing one primary number by another, it may be represented most conveniently by the notation already established for division. (Vide § 49.)

It would be impracticable to invent individual symbols, since an unending number of different symbols would be demanded to designate even the fractional numbers lying between two consecutive primary numbers; nor could any such symbol be used otherwise than as a record, since in any calculation with fractions it is the generating numbers which are utilized, and not the fractional number itself.

85. It seems to me that there is no way substantially different from the lines of thought I have followed, whereby one can really understand what he is doing in the operation 7/8 x 9/5 for instance. Teachers of arithmetic would do well to ponder their methods at this point.

* The only explanation (?) of such conclusions to be found even in the splendid Text Book of Algebra by Professor Chrystal, is "the statement that % X2 is 1⁄2 of % is merely a matter of some interpretation, arithmetical or other, that is given to a symbolical result demonstrably in accordance with the laws of symbolical operation." Vol. i., p. 13.

86. It remains to investigate devices for performing the seven numerical operations in this extended region of number, and in two cases to discover the effect, the meaning, of an operational combination; viz., in involution, if the exponent be a fraction or a surd. In the first place, it is to be borne in mind that it is one thing to conceive an operation, and another to perform it. For example, at the conclusion of these introductory lectures, it will be plain to all (if now obscure) that such operations as involving 10 under the exponent, or finding the logarithm of 5 to the base 12, are perfectly intelligible, even though ignorance of logarithmic series, or of the use of a table of logarithms, should leave one without devices adequate to the performance of the calculations.

87. It should be observed that the terms numerator and denominator applied to the numbers involved in a numerical fraction, or even to the "terms" of a ratio of incommensurables (e.g., √2/6) may be used as convenience suggests; but conceived operationally they are to be thought as dividend and divisor. The numerical symbols in the algebra of this chapter are still to be understood as representing primary numbers.

88. The "rules" for the operations of addition, multiplication, and division of fractions follow immediately from the definition of a fractional number, which is merely the recognition that the inverse of multiplication is always possible, that the result of the division of any primary number by any other is a number.

Substraction remains refractory, and meaningless unless the minuend be greater than the subtrahend.

The rules are only the generalization of Sections 51 and 55, q.v., yet it may be serviceable to discuss them.

a> b; therefore the common rule.

But how shall we perform a + b/c, or a / b + c /d, if a/b, b/c, and e/d are fractions? The operation is distinctly conceivable; but the device for performing it requires an intermediary step of multiplication, which must therefore be investigated. Consider

(1) a/b c = ac/b = a ÷ b/c = c÷b/a. ·(2) a / b ÷ c = a/bc = ac÷b.

(3) a/b × c/d = ac /bd, etc. (Cf. § 51, (3)). ad/be,

(4) ab÷c/dad /bc, etc. (Cf. § 51, (4)).

(5) a/b = a/b × c/c = ac /bc, also a / b = (a / b ÷ c) a/c xc= b/c

(6) ax b/c = ab/c = = a ÷ c/b,

all by the laws of division and multiplication (vide § 51). Therefore the common rules: From (1), To multiply a fraction, multiply the numerator or divide the denominator; from (1), to multiply by a fraction, multiply by the numerator and divide by the denominator; from (2), to divide a fraction multiply the denominator or divide the numerator; from (1), to divide by a fraction divide by the numerator and multiply by the denominator, etc.; from (3) and (4) for cases where both terms of the operation are fractions. Also from (5) it is obvious that to multiply or divide both terms of a fraction by the same number neither increases nor diminishes it; and from (6), the result is