Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/298

274 nothing in the demonstration which assumes that the circles cut one another, but the enunciation refers to this case only because it is shewn in III. 13 that if two circles touch one another, their circumferences cannot have more than one common point.

III. II, III. 12. The enunciations as given by Simson and others speak of the point of contact; it is however not shewn until III. 13 that there is only one point of contact. It should be observed that the demonstration in III. 1 1 will hold even if D and H be supposed to coincide, and that the demonstration in III. 12 will hold even if C and D be supposed to coincide. We may combine III. 11 and III. 12 in one enunciation thus.

If two circles touch one another their circumferences cannot have a common point out of the direction of the straight line which joins the centres.

III. II may be deduced from III. 7. For GH is the least line that can be drawn from G to the circumference of the circle whose centre is F, by III. 7. Therefore GH is less than GD, that is, less than GD; which is absurd. Similarly III. 12 may be deduced from III. 8.

III. 13. Simson observes, "As it is much easier to imagine that two circles may touch one another within in more points than one, upon the same side, than upon opposite sides, the figure of that case ought not to have been omitted; but the construction in the Greek text would not have suited with this figure so well, because the centres of the circles must have been placed near to the circumferences; on which account another construction and demonstration is given, which is the same with the second part of that which Campanus has translated from the Arabic, where, without any reason, the demonstration is divided into two parts."

It would not be obvious from this note which figure Simson himself supplied, because it is uncertain what he means by the "same side" and "opposite sides." It is the left-hand figure in the first part of the demonstration. Euclid, however, seems to be quite correct in omitting this figure, because he has shewn in III. 1 1 that if two circles touch internally there cannot be a point of contact out of the direction of the straight line which joins the centres. Thus, in order to shew that there is only one point of contact, it is sufficient to put the second supposed point of contact on the direction of the straight line which joins the