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Definition 1

Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional.

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This definition is not quite complete. It is apparent from its use that the notion of similarity assumes a correspondence of consecutive vertices and sides. Consider, for instance, pentagons.
The two pentagons ABCDE and FGHKL have equal corresponding angles: A = F, B = G, C = H, D = K, and E = L, and the sides about their equal angles are proportional in the appropriate order:

EA:AB | = | LF:FG,
AB:BC | = | FG:GH,
BC:CD | = | GH:HK,
CD:EF | = | HK:KL, and
EA:AB | = | KL:LF.

It wouldn't be allowed, for instance, if the angles of one figure equalled the angles of the other, but in some haphazard order. And it wouldn't be allowed for the orders of the terms in the proportions to be permuted, for instance, the second proportion could not be AB:BC = GH:FG.

Book VI Introduction - Definition 2.

© 1996
D.E.Joyce
Clark University