Chapter 12
The Fraction

89. Primitive Fractions. Of the artificial forms of number—as we may call the fraction, the irrational, the negative, and the imaginary in contradistinction to the positive integer—all but the fraction are creations of the mathematicians. They were devised to meet purely mathematical rather than practical needs. The fraction, on the other hand, is already present in the oldest numerical records—those of Egypt and Babylonia—was reckoned with by the Romans, who were no mathematicians, and by Greek merchants long before Greek mathematicians would tolerate it in arithmetic.

The primitive fraction was a concrete thing, merely an aliquot part of some larger thing. When a unit of measure was found too large for certain uses, it was subdivided, and one of these subdivisions, generally with a name of its own, made a new unit. Thus there arose fractional units of measure, and in like manner fractional coins.

In time the relation of the sub-unit to the corresponding principal unit came to be abstracted with greater or less completeness from the particular kind of things to which the units belonged, and was recognized when existing between things of other kinds. The relation was generalized, and a pure numerical expression found for it.

90. Roman Fractions. Sometimes, however, the relation was never completely enough separated from the sub-units in which it was first recognized to be generalized. The Romans, for instance, never got beyond expressing all their fractions in terms of the uncia, sicilicus, etc., names originally of subdivisions of the old unit coin, the as.

91. Egyptian Fractions. Races of better mathematical endowments than the Romans, however, had sufficient appreciation of the fractional relation to generalize it and give it an arithmetical symbolism.

The ancient Egyptians had a very complete symbolism of this sort. They represented any fraction whose numerator is 1 by the denominator simply, written as an integer with a dot over it, and resolved all other fractions into sums of such unit fractions. The oldest mathematical treatise known,—a papyrus36 roll entitled “Directions for Attaining to the Knowledge of All Dark Things,” written by a scribe named Ahmes in the reign of Ra-ä-us (therefore before 1700 b. c.), after the model, as he says, of a more ancient work,—opens with a table which expresses in this manner the quotient of 2 by each odd number from 5 to 99. Thus the quotient of 2 by 5 is written 3̇ 15̇, by which is meant 1 3 + 1 15; and the quotient of 2 by 13, 8̇ 52̇ 104̇. Only 2 3, among the fractions having numerators which differ from 1, gets recognition as a distinct fraction and receives a symbol of its own.

92. Babylonian or Sexagesimal Fractions. The fractional notation of the Babylonian astronomers is of great interest intrinsically and historically. Like their notation of integers it is a sexagesimal positional notation. The denominator is always 60 or some power of 60 indicated by the position of the numerator, which alone is written. The fraction 3 8, for instance, which is equal to 22 60 + 30 602, would in this notation be written 22 30. Thus the ability to represent fractions by a single integer or a sequence of integers, which the Egyptians secured by the use of fractions having a common numerator, 1, the Babylonians found in fractions having common denominators and the principle of position. The Egyptian system is superior in that it gives an exact expression of every quotient, which the Babylonian can in general do only approximately. As regards practical usefulness, however, the Babylonian is beyond comparison the better system. Supply the 0-symbol and substitute 10 for 60, and this notation becomes that of the modern decimal fraction, in whose distinctive merits it thus shares.

As in their origin, so also in their subsequent history, the sexagesimal fractions are intimately associated with astronomy. The astronomers of Greece, India, and Arabia all employ them in reckonings of any complexity, in those involving the lengths of lines as well as in those involving the measures of angles. So the Greek astronomer, Ptolemy (150 a. d.), in the Almagest (μ𝜖γάλη σύνταξις) measures chords as well as arcs in degrees, minutes, and seconds—the degree of chord being the 60th part of the radius as the degree of arc is the 60th part of the arc subtended by a chord equal to the radius.

The sexagesimal fraction held its own as the fraction par excellence for scientific computation until the 16th century, when it was displaced by the decimal fraction in all uses except the measurement of angles.

93. Greek Fractions. Fractions occur in Greek writings—both mathematical and non-mathematical—much earlier than Ptolemy, but not in arithmetic.37 The Greeks drew as sharp a distinction between pure arithmetic, ὰρι𝜃μητική, and the art of reckoning, λoγιστική, as between pure and metrical geometry. The fraction was relegated to λoγιστική. There is no place in a pure science for artificial concepts, no place, therefore, for the fraction in άρι𝜃μητική; such was the Greek position. Thus, while the metrical geometers—as Archimedes (250 b. c.), in his “Measure of the Circle” (κύκλoυ μ𝜖́τρησις), and Hero (120 b. c.)—employ fractions, neither of the treatises on Greek arithmetic before Diophantus (300 a. d. ) which have come down to us—the 7th, 8th, 9th books of Euclid’s “Elements” (300 b. c.), and the “Introduction to Arithmetic” ([𝜖ισαγωγήαρι𝜃μητικὴ]) of Nicomachus (100 a. d.)—recognizes the fraction. They do, it is true, recognize the fractional relation. Euclid, for instance, expressly declares that any number is either a multiple, a part, or parts ([μ𝜖̀ρη]), i. e. multiple of a part, of every other number (Euc. VII, 4), and he demonstrates such theorems as these:

If A be the same parts of B that C is of D, then the sum or difference of A and C is the same parts of the sum or difference of B and D that A is of B (VII, 6 and 8).

If A be the same parts of B that C is of D, then, alternately, A is the same parts of C that B is of D (VII, 10).

But the relation is expressed by two integers, that which indicates the part and that which indicates the multiple. It is a ratio, and Euclid has no more thought of expressing it except by two numbers than he has of expressing the ratio of two geometric magnitudes except by two magnitudes. There is no conception of a single number, the fraction proper, the quotient of one of these integers by the other.

In the αρι𝜃μητικὰ of Diophantus, on the other hand, the last and transcendently the greatest achievement of the Greeks in the science of number, the fraction is granted the position in elementary arithmetic which it has held ever since.