Mr. T. S. Davies on Geometty and Geometers. 501 



perlies of the circle in the third book, the two circles being 

 left (both in the enunciation and demonstration), can only be 

 accounted for on the hypothesis that the proofs we now have 

 were modified from proofs by transfer. This (as well as the 

 fact that most of them would be more neatly exhibited in re- 

 ference to a single circle than to two) would seem to point 

 out the change as an immediately recent one — most probably 

 Euclid's own change. I cannot, however, further enlarge 

 here upon this question. 



" 2. P. 261- The reasoning here is such that it is difficult to 

 know what his meaning is. ' he says the question here, is, whether 

 a triangle, under certain specified conditions, can, or cannot be 

 formed ? and therefor [e] to conclude any thing from the properties of 

 Triangles would be ridiculous, and nothing less than begging the 

 question.' does he mean that because the triangle is to be made, 

 and not yet formed that therefore it is not allowable to conclude 

 anything from the properties of triangles which have been already 

 demonstrated ? Were this true, he would, indeed, be in the right 

 to affirm as he does that the problem has a limitation, yet it would be 

 absurd to urge it in this case : he makes use of his own limitation 

 in this Problem, and why may not Euclid make use of his which is 

 easier to be understood, and the consequence from it that the circles 

 will meet sufficiently plain. But really this deserves no serious 

 reply." 



That the conclusion of the intersection of the circles can 

 be obtained from Euclid's limitation (viz. that any two sides 

 of a triangle are greater than the third), is true enough : but 

 that it is actually made to follow by any reason beyond " in- 

 spection " no one surely will say. A process tantamount to 

 Simpson's is essential', and whilst many may think that the 

 objection does " deserve a serious reply," no one will, I think, 

 consider what is given above as any reply at all. 



"3. In the same note he observes that the point F ought to have 

 been shown to fall below the line 

 EG (or rather, because the point G 

 is found by construction, that the 

 point G falls above EF.) this pro- 

 bably Euclid omitted, as it is easy to 

 see that DF being greater than DE 

 the angle DFE is less than a right 

 angle, and producing EF to H that 

 therefore DFH is an obtuse angle, 

 and consequently any straight line DH drawn to EF produced must 

 be greater than DE, and that DG which is equal to it must fall 

 above EH." 



That Simpson's objection was a valid one is now unques- 



