therefore

x2 σx;

and, since x2: 0x = x:σ,

x σ.

PROPOSITION 23.

A straight line commensurable with a medial str is medial.

Let A be medial, and let B be commensurable v I say that B is also medial.

For let a rational straight line CD be set out,

and to CD let the rectangular area CE equal to the square on A be applied, producing ED as breadth;

therefore ED is rational and incommensurable in length with CD.

[x. 22]

And let the rectangular area CF equal to the square on B be applied to CD, producing DF as breadth.

Since then A is commensurable with B,

Α

E

the square on A is also commensurable with the squ But EC is equal to the square on A,

and CF is equal to the square on B; therefore EC is commensurable with CF.

And, as EC is to CF, so is ED to DF;

therefore ED is commensurable in length with DF. But ED is rational and incommensurable in le DC;

therefore DF is also rational [x. Def. 3] and incomm in length with DC.

Therefore CD, DF are rational and commens square only.

54

BOOK X

[x. 23

But the straight line the square on which is equal to the rectangle contained by rational straight lines commensurable in square only is medial;

[x. 21] therefore the side of the square equal to the rectangle CD, DF is medial.

And B is the side of the square equal to the rectangle CD, DF;

therefore B is medial.

PORISM. From this it is manifest that an area commensurable with a medial area is medial.

[And in the same way as was explained in the case of rationals [Lemma following x. 18] it follows, as regards medials, that a straight line commensurable in length with a medial straight line is called medial and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.

But, if any straight line be commensurable in square with a medial straight line, then, if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commensurable in square only.]

As explained in the bracketed passage following this proposition, a straight line commensurable with a medial straight line in square only, as well as a straight line commensurable with it in length, is medial.

Algebraical notation shows this easily.

If kp be the given straight line, Akp is a straight line commensurable in length with it and λ. kp a straight line commensurable with it in square only.

But Ap and A. p are both rational [x. Def. 3] and therefore can be expressed by p', and we thus arrive at k1p', which is clearly medial.

Euclid's proof amounts to the following.

Apply both the areas k.p2 and k.p2 (or Ak.p) to a rational straight line σ.

p2

The breadths √. and X2 √k. P2 (or λ√k. P2)

σ

σ

σ

are in the ratio of the

areas √k.p2 and λ2k. p2 (or λk.p2) themselves and are therefore commensurable.

Now [x. 22] √k.is rational but incommensurable with σ.

σ

Therefore X2√k.2 (or √k. ""))

is so also;

σ

The Porism states that Akp2 is a medial area, which is indeed

PROPOSITION 24.

The rectangle contained by medial straight lines surable in length is medial.

For let the rectangle AC be contained by th straight lines AB, BC which are commensurable in length;

I say that AC is medial.

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

And, since AB is commensurable in length with BC,

while AB is equal to BD,

therefore DB is also commensurable in length with BC;

so that DA is also commensurable with AC.

But DA is medial;

therefore AC is also medial.

Q.

[

There is the same difficulty in the text of this enunciation as X. 19. The Greek says "medial straight lines commensurable i any of the aforesaid ways"; but straight lines can only be comme length in one way, though they can be medial in two ways, as expla addition to the preceding proposition, i.e. they can be either com in length or commensurable in square only with a given medial s For the same reason as that explained in the note on x. 19 I ha "in any of the aforesaid ways" in the enunciation and bracketed t to x. 23 to which it refers.

kp and λk1p are medial straight lines commensurable in le rectangle contained by them is Ap2, which may be written kp a fore clearly medial.

Euclid's proof proceeds thus. Let x, Ax be the two medial st commensurable in length.

mAB, n AB.

PROPOSITION 25.

The rectangle contained by medial straight lines surable in square only is either rational or medial.

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

I

that AC is either rational

say or medial.

For on AB, BC let the squares AD, BE be described; therefore each of the squares AD, BE is medial.

Let a rational straight line

FG be set out,

B

A

E

to FG let there be applied the rectangular parallelog equal to AD, producing FH as breadth,

to HM let there be applied the rectangular parallelog equal to AC, producing HK as breadth,

and further to KN let there be similarly applied NL BE, producing KL as breadth;

therefore FH, HK, KL are in a straight line.

Since then each of the squares AD, BE is media and AD is equal to GH, and BE to NL,

therefore each of the rectangles GH, NL is also me

And they are applied to the rational straight line therefore each of the straight lines FH, KL is rat incommensurable in length with FG.

And, since AD is commensurable with BE, therefore GH is also commensurable with NL. And, as GH is to NL, so is FH to KL; therefore FH is commensurable in length with KL.

x. 25]

PROPOSITIONS 24, 25

57

Therefore FH, KL are rational straight lines commen

surable in length;

therefore the rectangle FH, KL is rational.

And, since DB is equal to BA, and OB to BC, therefore, as DB is to BC, so is AB to BO.

But, as DB is to BC, so is DA to AC,

and, as AB is to BO, so is AC to CO; therefore, as DA is to AC, so is AC to CO.

[x. 19]

[VI. I] [id.]

But AD is equal to GH, AC to MK and CÒ to NL ; therefore, as GH is to MK, so is MK to NL ;

therefore also, as FH is to HK, so is HK to KL; [vi. 1, V. 11] therefore the rectangle FH, KL is equal to the square on HK.

But the rectangle FH, KL is rational;

therefore the square on HK is also rational.

[VI. 17]

Therefore HK is rational.

And, if it is commensurable in length with FG,

HN is rational;

[x. 19]

but, if it is incommensurable in length with FG,

KH, HM are rational straight lines commensurable in square only, and therefore HN is medial.

Therefore HN is either rational or medial.

But HN is equal to AC;

therefore AC is either rational or medial.

Therefore etc.

[x. 21]

Two medial straight lines commensurable in square only are of the form ktp, √λ.kp

The rectangle contained by them is A. kp2. Now this is in general medial; but, ifλ=kk, the rectangle is kk'p2, which is rational.

Euclid's argument is as follows. Let us, for convenience, put x for k3p, so

that the medial straight lines are x, √λ. x.

Form the areas x, x. λ. x, λx2,

and let these be respectively equal to ou, ov, ow, where σ is a rational straight line.

Since x2, Ax2 are medial areas,

so are σu, ow',

whence u, w are respectively rational and ~ σ.