u, v may

(1) a, √a2 – B, (2) √A, NA-B, (3) JA, JA−6, have any of the forms

To find two straight lines incommensurable in squar make the sum of the squares on them medial but ther contained by them rational.

Let there be set out two medial straight lines A commensurable in square only, such that the rectang they contain is rational, and the square on AB is grea the square on BC by the square on a straight line mensurable with AB; [x. 3

let the semicircle ADB be described on AB, let BC be bisected at E,

let there be applied to AB a parallelogram equal to th on BE and deficient by a square figure, namely the AF, FB;

therefore AF is incommensurable in length with FB.

Let FD be drawn from Fat right angles to AB, and let AD, DB be joined.

x. 34]

PROPOSITIONS 33, 34

79

Since AF is incommensurable in length with FB, therefore the rectangle BA, AF is also incommensurable with the rectangle AB, BF.

[x. 11]

But the rectangle BA, AF is equal to the square on AD, and the rectangle AB, BF to the square on DB;

therefore the square on AD is also incommensurable with the square on DB.

And, since the square on AB is medial,

therefore the sum of the squares on AD, DB is also medial.

And, since BC is double of DF,

[III. 31, I. 47]

therefore the rectangle AB, BC is also double of the rectangle AB, FD.

But the rectangle AB, BC is rational;

therefore the rectangle AB, FD is also rational.

[x. 6]

But the rectangle AB, FD is equal to the rectangle AD, ᎠᏴ ; so that the rectangle AD, DB is also rational.

[Lemma]

Therefore two straight lines AD, DB incommensurable in square have been found which make the sum of the squares on them medial, but the rectangle contained by them rational.

Q. E. D.

In this case we take [x. 31, 2nd part] the medial straight lines

Take u, v such that, if x, y be the result of the solution,

and u, v are straight lines satisfying the given conditions. Euclid's proof is similar to the preceding.

(a) From (1) it follows [x. 18] that

whence

xy,

u2 v2,

and u, v are thus incommensurable in square.

.(1).

.(2),

Therefore uv is rational.

To find the actual form of u, v, we have, by solving the e

(if xy),

Bearing in mind the forms which

(1 + k2)4 ' ( 1 + k2)*

may tal

on x. 31), we shall find that u, v may have any of the forms

2

PROPOSITION 35.

To find two straight lines incommensurable in squ make the sum of the squares on them medial and the contained by them medial and moreover incommensur the sum of the squares on them.

Let there be set out two medial straight lines commensurable in square only, containing a medial and such that the square on AB is greater than the BC by the square on a straight line incommensu AB;

[x

F B

Then, since AF is incommensurable in length wi

AD is also incommensurable in square with DB. And, since the square on AB is medial, therefore the sum of the squares on AD, DB is also

And, since the rectangle AF, FB is equal to th on each of the straight lines BE, DF,

therefore BE is equal to DF;

therefore BC is double of FD,

so that the rectangle AB, BC is also double of the AB, FD.

But the rectangle AB, BC is medial; therefore the rectangle AB, FD is also medial. And it is equal to the rectangle AD, DB;

[Lemma

therefore the rectangle AD, DB is also medial.

E

And, since AB is incommensurable in length wit

while CB is commensurable with BE,

therefore AB is also incommensurable in length with

so that the square on AB is also incommensurable rectangle AB, BE.

But the squares on AD, DB are equal to the s AB,

and the rectangle AB, FD, that is, the rectangle AL equal to the rectangle AB, BE;

therefore the sum of the squares on AD, DB is ind surable with the rectangle AD, DB.

H. E. III.

82

BOOK X

[x. 35

squares

Therefore two straight lines AD, DB incommensurable in square have been found which make the sum of the on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them.

Q. E. D.

Take the medial straight lines found in x. 32 (2nd part), viz.

where x, y are the ascertained values of x, y.

Then u, v are straight lines satisfying the given conditions.
Euclid proves this as follows.

(a) From (1) it follows [x. 18] that xy.

..(1),

(2),

Therefore

and

(B)

(y)

u2 v3,

therefore uv is medial.

(8)

whence

That is, by (3) and (4),

(u2 + v2) ▼ uv.

The actual values are found thus. Solving the equations (1), we have