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For, since the point F is the centre of the circle DKL,

FD is equal to FK.

But FD is equal to A;

therefore KF is also equal to A.

Again, since the point G is the centre of the circle LKH,

GH is equal to GK.

But GH is equal to C;

therefore KG is also equal to C. And FG is also equal to B;
therefore the three straight lines KF, FG, GK are equal to the three straight lines A, B, C.

Therefore out of the three straight lines KF, FG, GK, which are equal to the three given straight lines A, B, C, the triangle KFG has been constructed.

Q. E. F.

1 2 3

Proposition 23.

On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.

Let AB be the given straight line, A the point on it, and the angle DCE the given rectilineal angle;

thus it is required to construct on the given straight line AB, and at the point A on it, a rectilineal angle equal to the given rectilineal angle DCE.

On the straight lines CD, CE respectively let the points D, E be taken at random; let DE be joined, and out of three straight lines which are equal to the three

1 This is the first case in the Elements of a διορισμός to a problem in the sense of a statement of the conditions or limits of the possibility of a solution. The criterion is of course supplied by the preceding proposition.

2 This is usually translated (e.g. by Williamson and Simson) “But it is necessary,” which is however inaccurate, since the Greek is not δεῖ δέ but δεῖ δή. The words are the same as those used to introduce the διορισμός in the other sense of the “definition” or “particular statement” of a construction to be effected. Hence, as in the latter case we say “thus it is required” (e.g. to bisect the finite straight line AB, I. 10), we should here translate “thus it is necessary.”

3 To this enunciation all the MSS. and Boethius add, after the διορισμός, the words “because in any triangle two sides taken together in any manner are greater than the remaining one.” But this explanation has the appearance of a gloss, and it is omitted by Proclus and Campanus. Moreover there is no corresponding addition to the διορισμός of VI. 28.

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

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