INTRODUCTORY NOTE

3

This fact indicates clearly that the general theorem in Eucl. x. 9 that squares which have not to one another the ratio of a square number to a square number have their sides incommensurable in length was not arrived at all at once, but was, in the manner of the time, developed out of the separate consideration of special cases (Hankel, p. 103).

The proposition x. 9 of Euclid is definitely ascribed by the scholiast to Theaetetus. Theaetetus was a pupil of Theodorus, and it would seem clear that the theorem was not known to Theodorus. Moreover the Platonic passage itself (Theaet. 147D sqq.) represents the young Theaetetus as striving after a general conception of what we call a surd. "The idea occurred to me, seeing that square roots (dvvápμes) appeared to be unlimited in multitude, to try to arrive at one collective term by which we could designate all these square roots.... I divided number in general into two classes. The number which can be expressed as equal multiplied by equal (iσov loákis) I likened to a square in form, and I called it square and equilateral....The intermediate number, such as three, five, and any number which cannot be expressed as equal multiplied by equal, but is either less times more or more times less, so that it is always contained by a greater and less side, I likened to an oblong figure and called an oblong number.... Such straight lines then as square the equilateral and plane number I defined as length (μñκos), and such as square the oblong square roots (ovváμeis), as not being commensurable with the others in length but only in the plane areas to which their squares are equal."

There is further evidence of the contributions of Theaetetus to the theory of incommensurables in a commentary on Eucl. x. discovered, in an Arabic translation, by Woepcke (Mémoires présentés à l'Académie des Sciences, XIV., 1856, pp. 658—720). It is certain that this commentary is of Greek origin. Woepcke conjectures that it was by Vettius Valens, an astronomer, apparently of Antioch, and a contemporary of Claudius Ptolemy (2nd cent. A.D.). Heiberg, with greater probability, thinks that we have here a fragment of the commentary of Pappus (Euklid-studien, pp. 169-71), and this is rendered practically certain by Suter (Die Mathematiker und Astronomen der Araber und ihre Werke, pp. 49 and 211). This commentary states that the theory of irrational magnitudes "had its origin in the school of Pythagoras. It was considerably developed by Theaetetus the Athenian, who gave proof, in this part of mathematics, as in others, of ability which has been justly admired. He was one of the most happily endowed of men, and gave himself up, with a fine enthusiasm, to the investigation of the truths contained in these sciences, as Plato bears witness for him in the work which he called after his name. As for the exact distinctions of the above-named magnitudes and the rigorous demonstrations of the propositions to which this theory gives rise, I believe that they were chiefly established by this mathematician; and, later, the great Apollonius, whose genius touched the highest point of excellence in mathematics, added to these discoveries a number of remarkable theories after many efforts and much labour.

"For Theaetetus had distinguished square roots [puissances must be the dvvapes of the Platonic passage] commensurable in length from those which are incommensurable, and had divided the well-known species of irrational lines after the different means, assigning the medial to geometry, the binomial to arithmetic, and the apotome to harmony, as is stated by Eudemus the Peripatetic.

"As for Euclid, he set himself to give rigorous rules, which he established,

relative to commensurability and incommensurability in general; he made precise the definitions and the distinctions between rational and irrational magnitudes, he set out a great number of orders of irrational magnitudes, and finally he clearly showed their whole extent."

The allusion in the last words must be apparently to x. 115, where it is proved that from the medial straight line an unlimited number of other irrationals can be derived all different from it and from one another.

The connexion between the medial straight line and the geometric mean is obvious, because it is in fact the mean proportional between two rational straight lines "commensurable in square only." Since (x + y) is the arithmetic mean between x, y, the reference to it of the binomial can be understood. The connexion between the apotome and the harmonic mean is explained by some propositions in the second book of the Arabic commentary. The 2xy harmonic mean between x, y is and propositions of which Woepcke x+y' quotes the enunciations prove that, if a rational or a medial area has for one of its sides a binomial straight line, the other side will be an apotome of corresponding order (these propositions are generalised from Eucl. x. 111—4); the 2xy 2xy fact is that x+y= x2-12⋅ (x − y). y2·

=

One other predecessor of Euclid appears to have written on irrationals, though we know no more of the work than its title as handed down by Diogenes Laertius'. According to this tradition, Democritus wrote πepì ἀλόγων γραμμῶν καὶ ναστῶν β', two Books on irrational straight lines and solids (apparently). Hultsch (Neue Jahrbücher für Philologie und Pädagogik, 1881, pp. 578-9) conjectures that the true reading may be πepì ảλóywv γραμμῶν κλαστῶν, on irrational broken lines." Hultsch seems to have in mind straight lines divided into two parts one of which is rational and the other irrational ("Aus einer Art von Umkehr des Pythagoreischen Lehrsatzes über das rechtwinklige Dreieck gieng zunächst mit Leichtigkeit hervor, dass man eine Linie construiren könne, welche als irrational zu bezeichnen ist, aber durch Brechung sich darstellen lässt als die Summe einer rationalen und einer irrationalen Linie"). But I doubt the use of Kλαστós in the sense of breaking one straight line into parts; it should properly mean a bent line, i.e. two straight lines forming an angle or broken short off at their point of meeting. It is also to be observed that vaσróv is quoted as a Democritean word (opposite to Kevóv) in a fragment of Aristotle (202). I see therefore no reason for questioning the correctness of the title of Democritus' book as above quoted.

I will here quote a valuable remark of Zeuthen's relating to the classification of irrationals. He says (Geschichte der Mathematik im Altertum und Mittelalter, p. 56) "Since such roots of equations of the second degree as are incommensurable with the given magnitudes cannot be expressed by means of the latter and of numbers, it is conceivable that the Greeks, in exact investigations, introduced no approximate values but worked on with the magnitudes they had found, which were represented by straight lines obtained by the construction corresponding to the solution of the equation. That is exactly the same thing which happens when we do not evaluate roots but content ourselves with expressing them by radical signs and other algebraical symbols. But, inasmuch as one straight line looks like another, the Greeks did not get

1 Diog. Laert. IX. 47, p. 239 (ed. Cobet).

INTRODUCTORY NOTE

5

the same clear view of what they denoted (i.e. by simple inspection) as our system of symbols assures to us. For this reason it was necessary to undertake a classification of the irrational magnitudes which had been arrived at by successive solution of equations of the second degree." To much the same effect Tannery wrote in 1882 (De la solution géométrique des problèmes du second degré avant Euclide in Mémoires de la Société des sciences physiques et naturelles de Bordeaux, 2o Série, IV. pp. 395-416). Accordingly Book x. formed a repository of results to which could be referred problems which depended on the solution of certain types of equations, quadratic and biquadratic but reducible to quadratics.

Consider the quadratic equations

x2+2ax. p + B . p2 = 0,

where p is a rational straight line, and a, ẞ are coefficients. Our quadratic equations in algebra leave out the p; but I put it in, because it has always to be remembered that Euclid's x is a straight line, not an algebraical quantity, and is therefore to be found in terms of, or in relation to, a certain assumed rational straight line, and also because with Euclid p may be not only of the

n

a,

form a, where a represents a units of length, but also of the form which represents a length "commensurable in square only" with the unit of length, or A where A represents a number (not square) of units of area. The use therefore of p in our equations makes it unnecessary to multiply different cases according to the relation of p to the unit of length, and has the further advantage that, e.g., the expression p±√k.p is just as general as the expression k.pt. p, since p covers the form k. p, both expressions covering a length either commensurable in length, or commensurable in square only," with the unit of length.

Now the positive roots of the quadratic equations

66

}.

The negative roots do not come in, since x must be a straight line. The omission however to bring in negative roots constitutes no loss of generality, since the Greeks would write the equation leading to negative roots in another form so as to make them positive, i.e. they would change the sign of x in the equation.

Now the positive roots x1, x1, x2, x may be classified according to the character of the coefficents a, ẞ and their relation to one another.

I. Suppose that a, ẞ do not contain any surds, i.e. are either integers or of the form m/n, where m, n are integers.

Now in the expressions for x1, x, it may be that

Euclid expresses this by saying that the square on ap exceeds the square on p√ -ẞ by the square on a straight line commensurable in length with ap. In this case x, is, in Euclid's terminology, a first binomial straight line,

and x a first apotome.

Next, in the expressions for x, x' it may be that

(a2 + B), where m, n are integers, i.e. ẞ is of the form

Euclid expresses this by saying that the square on p√a2 + ß exceeds the square on ap by the square on a straight line commensurable in length with p√√a2 + B.

In this case x, is, in Euclid's terminology, a second binomial,

x1 = p(√λ + √λ −ẞ), xj'=p(√λ— √λ − ß),

X2 = p(√λ+B+ √λ), x2' = p(√λ+ß − √λ).

Thus x, x are of the same form as x, x.

If √λ – ß in x1, x, is not surd but of the form m/n, and if √λ+ß in x2, X2 is not surd but of the form m/n, the roots are comprised among the forms already shown, the first, second, fourth and fifth binomials and apotomes.

x is a third binomial straight line,
xa third apotome;

(2) in general, ẞ not being of the form

m2

λ,

no

x, is a sixth binomial straight line,

x a sixth apotome.

With the expressions for x, x, the distinction between the third and sixth binomials and apotomes is of course the distinction between the cases

m2
na

(1) in which ẞ= (A+B), or ẞ is of the form

and (2) in which ẞ is not of this form.

If we take the square root of the product of p and each of the six binomials and six apotomes just classified, i.e.

p2 (a ± √a2 − ß), p2(√a2 + ß ±a),

INTRODUCTORY NOTE

7

in the six different forms that each may take, we find six new irrationals with a positive sign separating the two terms, and six corresponding irrationals with a negative sign. These are of course roots of the equations

x+2ax2. p2±ß. p* = o.

These irrationals really come before the others in Euclid's order (x. 36—— 41 for the positive sign and x. 73-78 for the negative sign). As we shall see in due course, the straight lines actually found by Euclid are

I.

pk. p, the binomial ( ex duo ovoμáτwv)

and the apotome (ἀποτομή),

which are the positive roots of the biquadratic (reducible to a quadratic) x 2 (1 + k) p2. x2 + (1 − k)2 p1 = 0.

and the first apotome of a medial (μέσης ἀποτομὴ πρώτη),

which are the positive roots of

and the second apotome of a medial (μέσης ἀποτομὴ δευτέρα), which are the positive roots of the equation

the major (irrational straight line) (μeíčwv)

and the minor (irrational straight line) (¿λáoσwv),

which are the positive roots of the equation

5.

k2 I +

p* 0.

Ρ

√2 (1 + k2)

√ √1+k2 + k ± √2 (1 + k2)

+

Ρ

the "side" of a rational plus a medial (area) (pŋtòv kaì péσov dvvaμévn) and the "side" of a medial minus a rational area (in the Greek ʼn μerà pηtoû μέσον τὸ ὅλον ποιοῦσα),

which are the positive roots of the equation

the "side" of the sum of two medial areas (ỷ dúo μéσa dvvaμévn)

and the "side" of a medial minus a medial area (in the Greek ý μerà péσov μέσον τὸ ὅλον ποιοῦσα),

which are the positive roots of the equation

k2