ABE has to the triangle F GL the duplicate ratio of that which the side A B has to the homologous side FG (VI. 19). Therefore also the polygon A B C D E has to the polygon F G H KL the duplicate ratio of that which AB has to the homologous side F G. Wherefore, similar polygons, &c. Q. E. D.

COROLLARY 1.-In like manner it may be proved, that similar foursided figures, or figures of any number of sides, are, one to another in the duplicate ratio of their homologous sides; and this has been proved of triangles (VI. 19). Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.

COROLLARY 2.-If to A B and FG, two of the homologous sides of the polygon, a third proportional M be taken (VI. 11), AB has to M the duplicate ratio of that which AB has to FG (V. Def. 10). But the foursided figure or polygon upon AB, has to the four-sided figure or polygon upon F G likewise the duplicate ratio of that which A B has to FG (VI. 20, Cor. 1). Therefore AB is to M, as the figure upon AB to the figure upon FG (V. 11); and this has been proved in triangles (VI. 19, Cor.). Therefore, universally, it is manifest, that if three straight lines be proportionals, the first is to the third, as any rectilineal figure upon the first is to a similar and similarly described rectilineal figure upon the second. Corollary 3.-Because all squares are similar figures, the ratio of any two squares to one another, is the same as the duplicate ratio of their sides, for all their sides are homologouз. Therefore two similar triangles are to one another as the squares of their homologous sides (V. 11). Therefore, also generally, any two similar rectilineal figures or polygons are to one another as the squares of their homologous sides.

Corollary 4.-The perimeters (that is, the sums of all their sides) of similar rectilineal figures or polygons are to one another as their homologous sides, homologous diagonals, or homologous altitudes.

Corollary 5.-Similar rectilineal figures or polygons are to one another as the squares of homologous diagonals, or homologous altitudes; or, as the squares of their perimeters.

Corollary 6.-Rectilineal figures or polygons are similar, which are divisible into the same number of similar and similarly situated triangles.

Corollary 7.-If similar rectilineal figures similarly described upon straight lines be equal, these straight lines are also equal.

Exercise. To construct a rectilineal figure similar to a given rectilineal figure and having a given ratio to it.

Rectilineal figures which are similar tʊ she same rectilineal figure, are similar to one another.

Let each of the rectilineal figures A and B be similar to the rectilineal figure C. The figure A is similar to the figure B.

Because the figure A is similar to the figure C, they are equiangular, and have their sides about the equal angles proportional (VI. Def. 1). Because the figure B is similar to the figure C, they are equiangular, and have their sides about the equal angles proportionals (VI. Def. 1).

AA

B

Therefore the figures A and B are each of them equiangular to the figure C, and have the sides about their equal angles proportional.

33ind fourth proportional 2023 VL 12). Upon PR A similarly situated to either

s73 onst.), and on AB and vstuated rectilineal figures KAB te sumilar and similarly situated Therefore the rectilineal

e proportionals; that is, KAB As LCD as MF is to NH Se ratio to each of the rectilineal

3. Where are equal to each other (V. 9),

Therefore the

and they are similar and similarly situated (Const.). straight line GH is equal to the straight line PR (VI. 20, Cor. 7). Because A B is to CD, as EF is to PR, and PR is equal to GH. Therefore A B is to CD, as E F is to G H (V. 7). Q. E. D.

Corollary.-If any number of straight lines be continual proportionals, the similar rectilineal figures similarly described upon them, are also continual proportionals; and conversely.

iangular parallelograms have to one another the ratio which is compounded of the ratios of their sides

Let AC and CF be equiangular parallelograms, having the angle BCD equal to the angle ECG. The ratio of the parallelogram AC to the parallelogram CF, is the same with the ratio which is compounded of the ratios of their sides.

Place BC and CG in a straight line; and D C and A CE shall also be in a straight line (I. 14). Complete the parallelogram DG (I. 31). Take any straight line K, and find L a fourth proportional to BC, CG, and B K (VI. 12); and M a fourth proportional to D C, CE, and L.

DH

KLM E F

Because the ratios of K to L, and L to M, are the same with the ratios of the sides; that is, of BC to CG, and DC to CE (Const.): but the ratio of K to M is that which is said to be compounded of the ratios of K to L, and L to M (V. Def. A). Therefore K has to M the ratio compounded of the ratios of the sides. Because BC is to CG, as the parallelogram AC is to the parallelogram CH (VI. 1). But BC is to CG, as K is to L (Const.). Therefore K is to L, as the parallelogram AC is to the parallelogram CH (V. 11). Again, because DC is to CE, as the parallelogram CH is to the parallelogram CF (VI. 1). But DC is to CE, as L is to M (Const.). Therefore L is to M, as the parallelogram CH is to the parallelogram CF (V. 11). Wherefore, ex æquali, K is to M, as the parallelogram A C is to the parallelogram CF (V. 22). But K has to M the ratio which is compounded of the ratios of the sides. Therefore also the parallelogram AC has to the parallelogram CF, the ratio which is compounded of the ratios of the sides. Wherefore, equiangular parallelograms, &c. Q. E. D.

Otherwise.-Because there are three parallelograms A C, CH, and CF. The first AC has to the third CF the ratio which is compounded of the ratio of the first AC to the second CH, and of the ratio of the second CH to the third CF (V. Def. A). But AC is to CH, as B C is to CG; and CH is to CF, as CD is to CE (VI. 1). Therefore A C has to C F the ratio which is compounded of the ratios which are the same with the ratios of the sides.

Corollary 1.-Triangles which have an angle of the one equal to an angle of the other, are to one another as the rectangles contained by the sides about those angles.

Corollary 2.-Equiangular parallelograms are to one another as the rectangles contained by their adjacent sides.

Corollary 3.-Equiangular triangles and parallelograms are to one another as the rectangles contained by their bases and altitudes.

(VI. 13). Upon G H describe the rectilineal figure K GH similar and similarly situated to the figure A B C (VI. 18).

Because B C is to GH, as G H is CF; and if three straight lines be proportionals, as the first is to the third, so is the figure described upon the first to the similar and similarly described figure upon the second (VI. 20, Cor. 2). Therefore BC is to CF, as the rectilineal figure A B C is to the rectilineal figure KG H. But BC is to C F, as the parallelogram BE is to the parallelogram EF (VI. 1). Therefore the rectilineal figure A B C is to the rectilineal figure K GH, as the parallelogram B E is to the parallelogram EF (V. 11). But the rectilineal figure A B C is equal to the parallelogram BE (Const.). Therefore the rectilineal figure KGH is equal to the parallelogram EF (V. 14). But the parallelogram EF is equal to the rectilineal figure D (Const.), Therefore also the rectilineal figure KGH is equal to the rectilineal figure D; and it is similar to the rectilineal figure A B C. Therefore the rectilineal figure KGH has been described similar to the rectilineal figure A B C, and equal to the rectilineal figure D. Q.E.F.

The general proposition intended here may be enunciated in the following terms: To construct a rectilineal figure of given species and magnitude. The construction amounts to this: On the side of the figure given in species construct an equal rectangle, and on one side of this rectangle construct another rectangle equal to the figure given in magnitude. The sides of these figures being placed in the same straight line, and a mean proportional between them being found, will give the side of the figure required. By this proposition, while a superficies may be preserved as to quantity, its form may be changed.

PROP. XXVI. THEOREM.

If two similar parallelograms have a common angle, and be similarly situated; they are about the same diagonal.

A G

D

Let the parallelograms BD and EG be similar and similarly situated, and have the angle D A B common. They are about the same diagonal. For, if not, let, if possible, the parallelogram BD have its diagonal ALC in a different straight line from AF, the diagonal of the parallelogram EG. Let GF meet ALC in L. Through I draw LK parallel to A D or B C.

K

E

B

Because the parallelograms BD and KG are about the same diagonal, they are similar to one another (VI. 24). Therefore D A is to A B, as GA is to AK (VI. Def. 1). Because B D and E G are similar parallelograms (Hyp.), DA is to AB, as GA is to AE. Therefore, GA is to A E, as GA is to AK (V. 11). Because GA has the same ratio to each of the straight lines A E and AK. Therefore, AK is equal to AE (V. 9); the less to the greater, which is impossible. Therefore the parallelogram BD cannot have its diagonal in ALC a different straight line from AF. Wherefore BD and EG are about the same diameter AF C. Therefore, if two similar, &c. Q. E. D.

This proposition is a sort of converse of Prop. XXIV. of this Book, and Prop. XXV. seems to be awkwardly placed between them.

L