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There is a difficulty in the text of the enunciation of this pr The Greek runs τὸ ὑπὸ ῥητῶν μήκει συμμέτρων κατά τινα τῶν πρ τρόπων εὐθειῶν περιεχόμενον ὀρθογώνιον ῥητόν ἐστιν, where the re said to be contained by "rational straight lines commensurable in any of the aforesaid ways." Now straight lines can only be comm in length in one way, the degrees of commensurability being comme in length and commensurability in square only. But a straight li rational in two ways in relation to a given rational straight line, si be either commensurable in length, or commensurable in square on latter. Hence Billingsley takes κατά τινα τῶν προειρημένων τρόπων translating "straight lines commensurable in length and rational in aforesaid ways," and this agrees with the expression in the next "a straight line once more rational in any of the aforesaid ways order of words in the Greek seems to be fatal to this way of the passage.

The best solution of the difficulty seems to be to reject the any of the aforesaid ways" altogether. They have reference to which immediately precedes and which is itself open to the graves It is very prolix, and cannot be called necessary; it appears n connexion with an addition clearly spurious and therefore re Heiberg to the Appendix. The addition does not even pretend to for it begins with the words "for he calls rational straight line Hence we should no doubt relegate the Lemma itself to the August does so and leaves out the suspected words in the enun have done.

Exactly the same arguments apply to the Lemma added ( heading "Lemma") to x. 23 and the same words "in any of t ways" used with "medial straight lines commensurable in len enunciation of x. 24. The said Lemma must stand or fall with question, since it refers to it in terms: "And in the same way as w in the case of rationals...."

Hence I have bracketed the Lemma added to x. 23 and objectionable words in the enunciation of x. 24.

If P be one of the given rational straight lines (rational of c sense of x. Def. 3), the other can be denoted by kp, where k is the form m/n (where m, n are integers). Thus the rectangle is obviously rational since it is commensurable with p2. [x. Def. 4

A rational rectangle may have any of the forms ab, kď2, kA a, b are commensurable with the unit of length, and A with the u Since Euclid is not able to use kp as a symbol for a commensurable in length with p, he has to put his proof in a sponding to

p2: kp2 = p : kp,

whence, p, kp being commensurable, p2, kp" are so also.

X. 20, 21]

PROPOSITIONS 19-21

49

PROPOSITION 20.

If a rational area be applied to a rational straight line, it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied.

For let the rational area AC be applied to AB, a straight line once more rational in any of the aforesaid ways, producing BC as breadth;

I say that BC is rational and commensurable in length with BA.

For on AB let the square AD be described; therefore AD is rational.

A

[x. Def. 4]

But AC is also rational;

therefore DA is commensurable with AC.

And, as DA is to AC, so is DB to BC.

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therefore BC is also rational and commensurable in length with AB.

Therefore etc.

The converse of the last. If p is a rational straight line, any rational area is of the form kp2. If this be "applied" to p, the breadth is kp commensurable in length with p and therefore rational. We should reach the same result if we applied the area to another rational straight line σ. The breadth is then kp2 kp2 σ = k.σ or ko, say.

σ

=

m

n

PROPOSITION 21.

The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial.

For let the rectangle AC be contained by the rational straight lines AB, BC commensurable in square only ;

H. E. III.

4

And, since AB is incommensurable in length with BC,

for by hypothesis they are commensurable in square only,

while AB is equal to BD,

therefore DB is also incommensurable in length wit And, as DB is to BC, so is AD to AC; therefore DA is incommensurable with AC.

But DA is rational;

therefore AC is irrational,

so that the side of the square equal to AC is also ir

And let the latter be called medial.

Q.

A medial straight line, now defined for the first time, is so ca it is a mean proportional between two rational straight lines co in square only. Such straight lines can be denoted by p, p straight line is therefore of the form p√k or ktp. Euclid's pro irrational is equivalent to the following. Take p, pk comm square only, so that they are incommensurable in length.

Now

2

p: p√k = p2: p2√k,

whence [x. 11] p2k is incommensurable with p2 and theref [x. Def. 4], so that √p2√k is also irrational [ibid.].

2

A medial straight line may evidently take either of the for NAB, where of course B is. not of the form kA.

LEMMA.

If there be two straight lines, then, as the firs second, so is the square on the first to the rectangle contained by the two straight lines.

Let FE, EG be two straight

lines.

F

E

I say that, as FE is to EG, so is the square the rectangle FE, EG.

rectangle FE, EG.

Similarly also, as the rectangle GE, EF is to the on EF, that is, as GD is to FD, so is GE to EF.

Q. E

If a, b be two straight lines,

a: b = a2: ab.

PROPOSITION 22.

The square on a medial straight line, if appli rational straight line, produces as breadth a stra rational and incommensurable in length with that to is applied.

Let A be medial and CB rational,

and let a rectangular area BD equal to the square applied to BC, producing CD as breadth;

I say that CD is rational and incommensurable in length with CB.

For, since A is medial, the square on it is equal to a rectangular area contained by rational straight lines commensurable in square only.

[x. 21]

A

B

Let the square on it be equal to GF. But the square on it is also equal to BD; therefore BD is equal to GF

But it is also equiangular with it;

and in equal and equiangular parallelograms the sid the equal angles are reciprocally proportional; therefore, proportionally, as BC is to EG, so is EF to Therefore also, as the square on BC is to the so EG, so is the square on EF to the square on CD.

--- -1

therefore the square on CD is also rational;

therefore CD is rational.

And, since EF is incommensurable in length with for they are commensurable in square only,

[

and, as EF is to EG, so is the square on EF to the re FE, EG,

therefore the square.on EF is incommensurable w rectangle FE, EG.

But the square on CD is commensurable with the on EF, for the straight lines are rational in square; and the rectangle DC, CB is commensurable with angle FE, EG, for they are equal to the square on A therefore the square on CD is also incommensurable rectangle DC, CB.

But, as the square on CD is to the rectangle DC is DC to CB; therefore DC is incommensurable in length with CB. Therefore CD is rational and incommensurable i with CB.

Q.

Our algebraical notation makes the result of this proposition evident. We have seen that the square of a medial straight line is k. p. If we "apply" this area to another rational straight breadth is

√k. p2.

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straight line, which we may express, if we please, in the form √k'. commensurable with σ in square only, and therefore rational mensurable in length with σ.

Euclid's proof, necessarily longer, is in two parts.

Suppose that the rectangle √k. p2 = σ. x.

Then (1)

whence

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But o p2, and therefore kp2 ~ x2.

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