Commentary on Proposition X.1

Back to Proposition X.1.

This proposition will be remembered because it is the lemma required in Euclid’s proof of XII. 2 to the effect that circles are to one another as the squares on their diameters. Some writers appear to be under the impression that XII. 2 and the other propositions in Book XII. in which the method of exhaustion is used are the only places where Euclid makes use of X. 1; and it is commonly remarked that X. 1 might just as well have been deferred till the beginning of Book XII. Even Cantor (Gesch. d. Math. I3, p. 269) remarks that “Euclid draws no inference from it [X. 1], not even that which we should more than anything else expect, namely that, if two magnitudes are incommensurable, we can always form a magnitude commensurable with the first which shall differ from the second magnitude by as little as we please.” But, so far from making no use of X. 1 before XII. 2, Euclid actually uses it in the very next proposition, X. 2. This being so, as the next note will show, it follows that, since X. 2 gives the criterion for the incommensurability of two magnitudes (a very necessary preliminary to the study of incommensurables), X. 1 comes exactly where it should be.

Euclid uses X. 1 to prove not only XII. 2 but XII. 5 (that pyramids with the same height and triangular bases are to one another as their bases), by means of which he proves (XII. 7 and Por.) that any pyramid is a third part of the prism which has the same base and equal height, and XII. 10 (that any cone is a third part of the cylinder which has the same base and equal height), besides other similar propositions. Now XII. 7 Por. and XII. 10 are theorems specifically attributed to Eudoxus by Archimedes (On the Sphere and Cylinder, Preface), who says in another place (Quadrature of the Parabola, Preface) that the first of the two, and the theorem that circles are to one another as the squares on their diameters, were proved by means of a certain lemma which he states as follows: “Of unequal lines, unequal surfaces, or unequal solids, the greater exceeds the less by such a magnitude as is capable, if added [continually] to itself, of exceeding any magnitude of those which are comparable with one another,” i.e. of magnitudes of the same kind as the original magnitudes. Archimedes also says (loc. cit.) that the second of the two theorems which he attributes to Eudoxus (Eucl. XII. 10) was proved by means of “a lemma similar to the aforesaid.” The lemma stated thus by Archimedes is decidedly different from X. 1, which, however, Archimedes himself uses several times, while he refers to the use of it in XII. 2 (On the Sphere and Cylinder, I. 6). As I have before suggested (The Works of Archimedes, p. xlviii), the apparent difficulty caused by the mention of two lemmas in connexion with the theorem of Eucl. XII. 2 may be explained by reference to the proof of X. 1. Euclid there takes the lesser magnitude and says that it is possible, by multiplying it, to make it some time exceed the greater, and this statement he clearly bases on the 4th definition of Book V., to the effect that “magnitudes are said to bear a ratio to one another which can, if multiplied, exceed one another.” Since then the smaller magnitude in X. 1 may be regarded as the difference between some two unequal magnitudes, it is clear that the lemma stated by Archimedes is in substance used to prove the lemma in X. 1, which appears to play so much larger a part in the investigations of quadrature and cubature which have come down to us.

Besides being employed in Eucl. X. 1, the “Axiom of Archimedes” appears in Aristotle, who also practically quotes the result of X. 1 itself. Thus he says, Physics VIII. 10, 266 b 2, “By continually adding to a finite (magnitude) I shall exceed any definite (magnitude), and similarly by ‘continually subtracting from it I shall arrive at something less than it,” and ibid. III. 7, 207 b 10 “For bisections of a magnitude are endless.” It is thus somewhat misleading to use the term “Archimedes’ Axiom” for the “lemma” quoted by him, since he makes no claim to be the discoverer of it, and it was obviously much earlier.

Stolz (see G. Vitali in Questioni riguardanti le matematiche elementari, I., pp. 129—30) showed how to prove the so-called Axiom or Postulate of Archimedes by means of the Postulate of Dedekind, thus. Suppose the two magnitudes to be straight lines. It is required to prove that, given two straight lines, there always exists a multiple of the smaller which is greater than the other.

Let the straight lines be so placed that they have a common extremity and the smaller lies along the other on the same side of the common extremity.

If AC be the greater and AB the smaller, we have to prove that there exists an integral number $n$ such that $n \cdot$ AB $>$ AC.

Suppose that this is not true but that there are some points, like B, not coincident with the extremity A, and such that, $n$ being any integer however great, $n \cdot$ AB $<$ AC; and we have to prove that this assumption leads to an absurdity.

A H X M Y K B C

The points of AC may be regarded as distributed into two “parts,” namely

  1. points H for which there exists no integer $n$ such that $n \cdot$ AH $>$ AC,

  2. points K for which an integer $n$ does exist such that $n \cdot$ AK $>$ AC.

This division into parts satisfies the conditions for the application of Dedekind’s Postulate, and therefore there exists a point M such that the points of AM belong to the first part and those of MC to the second part.

Take now a point Y on MC such that MY $<$ AM. The middle point (X) of AY will fall between A and M and will therefore belong to the first part; but, since there exists an integer $n$ such that $n \cdot$ AY $>$ AC, it follows that $2n \cdot$ AX $>$ AC: which is contrary to the hypothesis.