Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/284

260 from the centre F shall not fall entirely within the circle described from the centre; the condition that A and C are greater than B, ensures that one of these circles shall not fall entirely without the other. Hence the circles must meet. It is easy to see this as Simson says, but there is something arbitrary in Euclid's selection of what is to be demonstrated and what is to be seen, and Simson's language suggests that he was really conscious of this.

I. 24. In the construction, the condition that DE is to be the side which is not greater than the other, was added by Simson; unless this condition be added there will be three cases to consider, for F may fall on EG, or ahove EG, or below EG. It may be objected that even if Simson's condition be added, it ought to be shewn that F will fall below EG. Simson accordingly says "...it is very easy to perceive, that DG being equal to DF, the point G is in the circumference of a circle described from the centre D at the distance DF, and must be in that part of it which is above the straight line EF, because DG falls above DF, the angle EDG being greater than the angle EDF." Or we may shew it in the following manner. Let H denote the point of intersection of DF and EG. Then, the angle DHG is greater than the angle DEG, by I. 16; the angle DEG is not less than the angle DGE, by I. 19; therefore the angle DHG is greater than the angle DGH. Therefore DH is less than DG, by I. 20. Therefore DH is less than DF.

If Simson's condition be omitted, we shall have two other cases to consider besides that in Euclid. If F falls on EG, it is obvious that EF is less than EG. If F falls above EG, the sum of DF and EF is less than the sum of DG and EG, by I. 21; and therefore EF is less than EG.

I. 26. It will appear after I. 32 that two triangles which have two angles of the one equal to two angles of the other, each to each, have also their third angles equal. Hence we are able to include the two cases of I. 26 in one enunciation thus; if two triangles have all the angles of the one respectively equal to all the angles of the other, each to each, and have also a side of the one, opposite to any angle, equal to the side opposite to the equal angle in the other, the triangles shall he equal in all respects.

The first twenty-six propositions constitute a distinct section