On AB describe the equilateral triangle ABC. Produce BC to D, so that CD may be equal to CB. Join AD. Then AD shall be at right angles to AB. For, the angle CAD is equal to the angle CDA, and the angle CAB is equal to the angle CBA, by I. 5. Therefore the angle BAD is equal to the two angles ABD, BDA, by Axiom 2. Therefore the angle BAD is a right angle, by I. 32.
The propositions from I. 35 to I. 48 inclusive may be said to constitute the third section of the first Book of the Elements. They relate to equality of area in figures which are not necessarily identical in form.
I. 35. Here Simson has altered the demonstration given by Euclid, because, as he says, there would be three cases to consider in following Euclid's method. Simson however uses the third Axiom in a peculiar manner, when he first takes a triangle from a trapezium, and then another triangle from the same trapezium, and infers that the remainders are equal. If the demonstration is to be conducted strictly after Euclid's manner, three cases must be made, by dividing the latter part of the demonstration into two. In the left-hand figure we may suppose the point of intersection of BE and DC to be denoted by G. Then, the triangle ABE is equal to the triangle DCF; take away the triangle DGE from each; then the figure ABGD is equal to the figure EGCF; add the triangle GBC to each; then the parallelogram ABGD is equal to the parallelogram EBCF. In the right-hand figure we have the triangle AEB equal to the triangle DEC; add the figure BEDC to each; then the parallelogram ABCD is equal to the parallelogram EBCF.
The equality of the parallelograms in I. 35 is an equality of area, and not an identity of figure. Legendre proposed to use the word equivalent to express the equality of area, and to restrict the word equal to the case in which magnitudes admit of superposition and coincidence. This distinction, however, has not been generally adopted, probably because there are few cases in which any ambiguity can arise; in such cases we may say especially, equal in area, to prevent misconception.
Cresswell, in his Treatise of Geometry, has given a demonstration of I. 35 which shews that the parallelograms may be
