and the expressions in (2), (3) with b in place of JB.

PROPOSITION 36.

If two rational straight lines commensurable in only be added together, the whole is irrational; and called binomial.

For let two rational straight lines AB, BC 5 surable in square only be added

together;

I say that the whole AC is irrational.

For, since AB is incommensurable in length with 10 for they are commensurable in square onlyand, as AB is to BC, so is the rectangle AB, B square on BC,

15

therefore the rectangle AB, BC is incommensurable square on BC.

But twice the rectangle AB, BC is commensura the rectangle AB, BC [x. 6], and the squares on AB commensurable with the square on BC-for AB, rational straight lines commensurable in square onlytherefore twice the rectangle AB, BC is incomme 20 with the squares on AB, BC.

And, componendo, twice the rectangle AB, BC with the squares on AB, BC, that is, the square on is incommensurable with the sum of the squares on

But the sum of the squares on AB, BC is rationa 25 therefore the square on AC is irrational,

so that AC is also irrational.

And let it be called binomial.

Q. I

84

BOOK X

[x. 36, 37

Here begins the first hexad of propositions relating to compound irrational straight lines. The six compound irrational straight lines are formed by adding two parts, as the corresponding six in Props. 73-78 are formed by subtraction. The relation between the six irrational straight lines in this and the next five propositions with those described in Definitions II. and the Props. 48-53 following thereon (the first, second, third, fourth, fifth and sixth binomials) will be seen when we come to Props. 54-59; but it may be stated here that the six compound irrationals in Props. 36-41 can be found by means of the equivalent of extracting the square root of the compound irrationals in x. 48-53 (the process being, strictly speaking, the finding of the sides of the squares equal to the rectangles contained by the latter irrationals respectively and a rational straight line as the other side), and it is therefore the further removed compound irrational, so to speak, which is treated first.

In reproducing the proofs of the propositions, I shall for the sake of simplicity call the two parts of the compound irrational straight line x, y, explaining at the outset the forms which x, y really have in each case; x will always be supposed to be the greater segment.

In this proposition x, y are of the form p, √k. p, and (x+y) is proved to be irrational thus.

therefore (x + y)2, and therefore (x + y), is irrational.

This irrational straight line, p+k. p, is called a binomial straight line. This and the corresponding apotome (p-k.p) found in x. 73 are the positive roots of the equation

If two medial straight lines commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial straight line.

For let two medial straight lines AB, BC commensurable in square only and containing

a rational rectangle be added A

together;

I

say that the whole AC is irrational.

For, since AB is incommensurable in length with BC,

therefore the squares on AB, BC are also incommensurable with twice the rectangle AB, BC ;

[cf. x. 36, 11. 9-20]

there fore AC is irrational.

And let it be called a first bimedial straight li

Q

Here x, y have the forms ktp, kp respectively, as found in x
Exactly as in the last case we prove that

whence

x2 + y2 2xy,

(x + y)2

2xy.

But xy is rational;

5

10

therefore (x+y), and consequently (x + y), is irrational.

ρ

The irrational straight line k1p + kip is called a first bimedial This and the corresponding first apotome of a medial (k3p – X. 74 are the positive roots of the equation

If two medial straight lines commensurable in s and containing a medial rectangle be added together is irrational; and let it be called a second bimedi line.

For let two medial straight lines AB, BC com in square only and containing

a medial rectangle be added together;

I say

that AC is irrational.

For let a rational straight line DE be set out, and let the parallelogram DF equal to the square on AC be applied to DE, producing DG as breadth.

E

A

B

+

H

15 Then, since the square on AC is equal to the AB, BC and twice the rectangle AB, BC,

let EH, equal to the squares on AB, BC, be appli

But, by hypothesis, twice the rectangle AB, BC medial.

And EH is equal to the squares on AB, BC, 25 while FH is equal to twice the rectangle AB, BC; therefore each of the rectangles EH, HF is medial.

30

And they are applied to the rational straight line therefore each of the straight lines DH, HG is ratio incommensurable in length with DE.

Since then AB is incommensurable in length with and, as AB is to BC, so is the square on AB to the r AB, BC,

therefore the square on AB is incommensurable with angle AB, BC.

35 But the sum of the squares on AB, BC is comme with the square on AB,

and twice the rectangle AB, BC is commensurable rectangle AB, BC.

Therefore the sum of the squares on AB, BC i 40 mensurable with twice the rectangle AB, BC.

45

But EH is equal to the squares on AB, BC, and HF is equal to twice the rectangle AB, BC.

Therefore EH is incommensurable with HF, so that DH is also incommensurable in length with

Therefore DH, HG are rational straight lines surable in square only;

so that DG is irrational.

But DE is rational;

and the rectangle contained by an irrational and a 50 straight line is irrational;

therefore the area DF is irrational,

and the side of the square equal to it is irrational.

medial area, but does not explain why. It is because, by hy squares on AB, BC are commensurable, so that the sum of th commensurable with either [x. 15] and is therefore a medial area

In this case [x. 28, note] x, y are of the forms kp, λp/k res Apply each of the areas (x+y) and 2xy to a rational straig

suppose

x2 + y2 = σu,

2xy = σv.

Now it follows from the hypothesis, x. 15 and x. 23, Por. th a medial area; and so is 2xy, by hypothesis;

therefore ou, ov are medial areas.

Therefore each of the straight lines u, v is rational and ~ σ

Therefore, by (1), (2), u, v are rational and ~.

It follows, by x. 36, that (u + v) is irrational.

Therefore (u+v) σ is an irrational area [this can be deduce

by reductio ad absurdum],

whence (x + y)2, and consequently (x + y), is irrational.

This and the corresponding second apotome of a medial found in x. 75 are the positive roots of the equation

If two straight lines incommensurable in squ make the sum of the squares on them rational, but th contained by them medial, be added together, the who line is irrational: and let it be called major.