Mr. T. S. Davies on Geometric and Geometers. 505 



contrary of this is manifest, for instance in the 4*'' Prop. 5 he refers 

 to (or makes use of ) 5 Def. 5 without ever having mentioned it, or 

 proved anything about it, after he had given it, D"^. Barrow^ has 

 sufficiently shewn the weakness of such objections ; see his Lect. 

 Mathem : P. 299 near the end, and the sentence which begins in 

 the last line of Page 303. nam si &c." 



,*■ Dr. Simson must have been " hard driven " for an objec- 

 tion to Simpson, to have quoted this case: inasmuch as it so 

 far furnishes a new argument against ihe petjection of Euclid's 

 work ; and thereby affixes a stigma to the '* 4**^ Edition," 

 which is as yet unremoved from any subsequent one. That 

 the canon enforced by Simpson is a safe, a sound, and an 

 essential one, who will deny? Nay, more: — not only should 

 the consistency of the definition be shown, but its completeness, 

 and xia freedom from redundancy^ likewise. There is perhaps 

 no instance in Euclid of incompleteness of defining conditions: 

 but there are several of redundancy — such as the square, and 

 similar rectilineal figures, so often quoted by editors and com- 

 mentators. The mischief done in such cases as these is rather 

 done to the logic than to the geometrical facts : but then, on 

 the other hand, these superfluous conditions might be contra- 

 dictory to the essential ones, just as easily as consistent with 

 them, or consequential from them. 



Instead of collecting all the definitions at the beginning of 

 a book er treatise, many judicious writers prefer to give them 

 pari passu, as the occasion for them, as well as their justifica- 

 tion, shall present itself. Probably the fifth definition of the 

 fifth book and those conhected essentially with it, may present 

 a good argument in favour of such a mode. In more elabo- 

 rate researches, and especially in the theories of modern geo- 

 metry, this becomes almost indispensable; as for instance in 

 the doctrine of " radical axes," " similitude," " poles and 

 polars," " anharmonic ratios " and " involution." Many 

 points, lines, circles, i-atios, and relations of different kinds 

 here dema7id speci^c names, which till their fiecessaty existefice 

 has been proved, it would be a manifest absurdity to confer 

 upon them. Under all circumstances, at any rate, whatever 

 arrangement may be given to the definitions in a printed book, 

 it is essential to observe the canon enforced by Simpson, as 

 above quoted, in giving the composition and logical place of 

 every definition. 



It would require more space than I have at my disposal to 

 enter upon the question of Euclid's fifth definition of his fifth 

 book ; and as I cannot treat it adequately, it must be passed 

 over altogether, till some future time. For the same reason, I 

 shall here offer no remark upon the portions of this paper gf 

 Simson's, which bear upon the fifth book. , 



