*0i Mr. T. S. Davies on Geometry and Geometers. 



it to the 14"^ axiom of Booli 1. (he means in Barrow's Euclid, for it 

 is the 10*^ in the Greek), hut all that in this case follows from that 

 axiom is that if two straight lines 

 AB, CD could meet with each 

 other in two points E, F, the parts 

 of them betwen E F must coincide, 

 and so A B, C D would have the 

 segment common to both ; but this 

 does not prove that they cannot 

 meet in two points, from which 

 their not having a common seg- 

 ment is deduced in the Greek Edition, but because they cannot have 

 a common segment, as is shewn in Cor. of 1 1 Prop. 1. 4*" Edition it 

 follows they cannot meet in two points." 



As regards a logical system, Dr. Simson is right in this 

 view; but in all that relates to the line and plane, especially 

 the first steps of deduction, there exists in most minds some 

 degree of confusion, arising from the vagueness of the defi- 

 nitions of those objects of our contemplation. In discussing 

 Euclid's method, it appears to be almost always forgotten that 

 his metaphysical creed was Platonism ; and questions arising 

 out of such antecedent creeds always must present themselves 

 in the initial steps of a science. 



It cannot be denied that Simpson might have retorted, had 

 he seen this, " who is so dull as not to perceive that two 

 straight lines cannot meet in two points?" It would be an 

 assumption far less removed from our first conceptions than 

 some that Euclid makes and Simson justifies. 



" 6. P. 265. The observation at the end of the note which begins 

 in p. 264, viz : that in the Corollary (of Prop 11 Book 1.) the lines 

 A B, B D, B C are supposed to be all in the same plane, which can- 

 not be assumed in 1 Prop. 11. is very just. Soon after the 4*° Edi- 

 tion was published I observed this error and corrected it in the way 

 Mr. Simpson has mentioned in this note, he is mistaken in thinking 

 the 10'** axiom he mentions here, to be Euclid's, the Axiom he means 

 is That two straight lines cannot have a common segment, which is 

 none of Euclid's, but is the 10*''. in Dr. Barrow's Edition, who took 

 it from Herigon's Cursus vol. 1. and to supply this, is the design of 

 Corollary 11 Prop. 1. 



" 7. Page 267. and the following contain several objections against 

 Euclid's doctrine of Proportionals, most of which with many others, 

 have been long ago distinctly and fully answered by the learned Dr. 

 Barrow in the 7*^ and 8"* of his Mathematical Lectures. Some of 

 which relate to the notes on the 4*° Euclid are as follows. 



" 8. At the bottom of P. 270 he says that ' Euclid or Eudoxus 

 does never (that I know of) refer to any definition till he has proved 

 either by actual construction , or by some Demonstration previous to 

 that in hand, that such definition involves no absurdity &c.' the 



