Let the similar figures similarly described on A and B be called U and V respectively.

Let the similar figures similarly described on C and D be called W and X respectively.

Art. 158. PROPOSITION XLV. (ii). (Euc. VI. 22, 2nd Part.)

ENUNCIATION. Let there be four straight lines A, B, C, D.
Let two similar figures be similarly described on A and B.
Let two similar figures be similarly described on C and D.
Let it be given that

the figure on A: the figure on B
= the figure on C: the figure on D.

It is required to prove that

:

A B C D.

Let the similar figures on A and B be called U and V respectively.
Then U: V = the square on A: the square on B.

[Art. 156.

Let the similar figures on C and D be called W and I respectively.
Then W : X = the square on C: the square on D.

[Art. 156.

.. the duplicate ratio of A: B
the duplicate ratio of C: E,

.. the square on A: the square on B the square on C: the square on E, .. the square on C: the square on D the square on C: the square on E, the square on D= the square on E. But if two squares are equal, their sides must be equal.

Art. 159. PROPOSITION XLVI. (Euc. VI. 31.)*

[Prop. 14.

[Prop. 37.

[Art. 147 (2).

[Prop. 21.

ENUNCIATION. In any right-angled triangle, any rectilineal figure described on the hypotenuse is equal to the sum of the two similar and similarly described figures on the sides.

Let ABC be a triangle right-angled at C.

On AB let any rectilineal figure X be described.

On BC let a rectilineal figure Y be described similar to X so that the side BC of Y corresponds to the side AB of X; and on AC let a rectilineal figure Z be described similar to X so that the side AC of Z corresponds to the side AB of X.

It is required to prove that

X = Y + Z.

* I am indebted to Mr H. M. Taylor, the author of the Pitt Press Euclid, and to the Syndicate of the Pitt Press for their kind permission to use this proof, which is substantially the same as that given, in the Pitt Press Euclid.

Since X and Y are similar figures, and AB, BC are corresponding sides, therefore by the corollary to Proposition 44

Z:

.. Y: square on BC = Y + Z : square on BC + square on AC

.. X: square on AB=Y+Z: square on AB.

[Prop. 43.

Y+Z: square on AB.

... X=Y+Z.

[Prop. 21.

Art. 160. EXAMPLE 72.

In an acute-angled triangle similar figures are similarly described on the sides, show that the sum of any two of them is greater than the third.

Art. 161. PROPOSITION XLVII. (Euc. VI. 25.)

ENUNCIATION. To describe a rectilineal figure similar to one given rectilineal figure and equal in area to another given rectilineal figure.

(In ordinary language to describe a figure having the shape of one given figure and the size of another.)

Let it be required to describe a figure similar to the figure ABCDE and equal to the figure FGHK.

On AB describe a rectangle ABLM equal to ABCDE.

On BL describe a rectangle BLNO equal to FGHK.
Take PQ a mean proportional between AB and BO.

[Prop. 33.

On PQ describe a figure PQRST similar to ABCDE, so that PQ may correspond to AB.

It will be shewn that PQRST is the figure required.
Since

AB: PQ = PQ : BO,

.. AB: BO is the duplicate ratio of AB: PQ.

Now

AB: BO ABLM BLNO
=

:

· ABCDE : FGHK.

Also the duplicate ratio of AB : PQ = ABCDE : PQRST. .. ABCDE: FGHK = ABCDE : PQRST.

Hence PQRST is equal to FGHK and similar to ABCDE.

[Prop. 30.

[Prop. 36.

[Prop. 17.

[Prop. 44.

[Prop. 21.

It is therefore the figure required.

Art. 162. Def. 26. FIGURES WITH SIDES RECIPROCALLY

PROPORTIONAL.

A figure is said to have the two sides about one angle reciprocally proportional to the two sides about an angle of another figure when these four sides are proportional in the following manner:

a side of the first figure: a side of the second figure

the other side of the second figure: the other side of the first figure.

Art. 163. PROPOSITION XLVIII. (i). (Euc. VI. 14, 1st Part.)

ENUNCIATION. Parallelograms having equal areas and having one angle of the one equal to one angle of the other have the sides about the equal angles reciprocally proportional.

Let the two parallelograms be placed so that the equal angles have the same vertex, and the sides of one at that vertex lie on the sides of the other at that vertex produced.

H. E.

13

When the parallelograms have been so placed let them be ABCD, BEFG, having the angles ABC, EBG equal; and let BE be on AB produced, and BG on CB produced.

Hence the two sides of ABCD meeting at B are reciprocally proportional to the two sides of BEFG meeting at B.

Art. 164. PROPOSITION XLVIII. (ii). (Euc. VI. 14, 2nd Part.)

ENUNCIATION. Parallelograms having one angle of the one equal to one angle of the other, and the sides about the equal angles reciprocally proportional are equal in area.

With the same figure as that in the first part of the proposition, let the parallelograms ABCD, BEFG have

The proof of the second part of Proposition 48 is obtained by writing the steps of the proof of the first part in reverse order.