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word "side" for duvaμévy and "to be the side of" for dúvaob "side" will in such expressions be a short way of expressing th a square equal to (an area)." In this particular passage it is not q cable to use the words "side of" or "straight line the square on whi to," for these expressions occur just afterwards for two alternatives word dvvauern covers. I have therefore exceptionally translated "t lines which produce them" (i.e. if squares are described upon them αἱ ἴσα αὐτοῖς τετράγωνα αναγράφουσαι, literally " the (straight li describe squares equal to them": a peculiar use of the active of the meaning being of course "the straight lines on which are de squares" which are equal to the rectilineal figures.

66

BOOK X. PROPOSITIONS.

PROPOSITION I.

Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. Let AB, C be two unequal magnitudes of which AB is the greater:

I

say that, if from AB there be Asubtracted a magnitude greater

than its half, and from that which

KH

D

F

с

E

is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the magnitude C.

For C if multiplied will sometime be greater than AB.

[cf. v. Def. 4] Let it be multiplied, and let DE be a multiple of C, and greater than AB; let DE be divided into the parts DF, FG, GE equal to C, from AB let there be subtracted BH greater than its half, and, from AH, HK greater than its half,

and let this process be repeated continually until the divisions in AB are equal in multitude with the divisions in DE.

Let, then, AK, KH, HB be divisions which are equal in multitude with DF, FG, GE.

Now, since DE is greater than AB,

and from DE there has been subtracted EG less than its half,

and, from AB, BH greater than its half,

therefore the remainder GD is greater than the remainder HA.

X. 1]

PROPOSITION 1

15

And, since GD is greater than HA,

and there has been subtracted, from GD, the half GF,

and, from HA, HK greater than its half,

therefore the remainder DF is greater than the remainder AK. But DF is equal to C;

therefore C is also greater than AK.

Therefore AK is less than C.

Therefore there is left of the magnitude AB the magnitude AK which is less than the lesser magnitude set out, namely C.

Q. E. D.

And the theorem can be similarly proved even if the subtracted be halves.

parts

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. I might just as well have been deferred till the beginning of Book XII. Even Cantor (Gesch. d. Math. 1, 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. I comes exactly where it should be.

Euclid uses x. I 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

16

BOOK X

[x. I in XII. 2 (On the Sphere and Cylinder, 1. 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. I 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. 111. 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 (quoted by G. Vitali in Questioni riguardanti la geometria elementare, pp. 91-2) 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 » such that n . 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. AB < AC; and we have to prove that this assumption leads to an absurdity.

H M K
A X Y

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. AH> AC, (2) points K for which an integer n does exist such that n. 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. AY> AC, it follows that 2n. AX> AC: which is contrary to the hypothesis.

ror, there being two unequal magnitudes AD, CA AB being the less, when the less is continually sub in turn from the greater, let that which is left over measure the one before it;

I say that the magnitudes AB, CD are incommensurab

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For, if they are commensurable, some magnitud measure them.

Let a magnitude measure them, if possible, and let it let AB, measuring FD, leave CF less than itself, let CF measuring BG, leave AG less than itself, and let this process be repeated continually, until there some magnitude which is less than E.

Suppose this done, and let there be left AG less tha
Then, since E measures AB,

while AB measures DF,

therefore E will also measure FD.

But it measures the whole CD also ;

therefore it will also measure the remainder CF.

But CF measures BG;

therefore E also measures BG.

But it measures the whole AB also;

therefore it will also measure the remainder AG, the g the less:

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