20 
Remarks on a Passage in the 
proposition, BC : AB : : 10 : 23 
BC 2 : AB 2 : : 100 : 529 
BC 2 + AB 2 : AB 2 : : 100 -f- 529 : 529, that is, since ABC is a 
a right angle, AC 2 : AB 2 : : 629 : 529. 
Now, the square root of 629 is 25.08. or 25 , as Comman- 
dine writes it, which not being a whole number, the proportion is 
left in the expression of the squares. All this is very clear 
and plain ; but, if we invert the order of reasoning, and suppose 
629, 526 to be the original numbers, we shall have 
AC 2 — AB 2 : AB 2 : : 629 - 526 : 526, that is, 
BC 2 : AB 2 : : 103 : 526, and 
BC : AB • : VB33 : Jffli. 
Now, there appears no reason why our author should, in the 
first instance, have assumed quantities, of which he was after- 
wards to take the square roots ; and it is not likely, if he had 
done so, that he would have fixed on such as these, of which the 
roots are not to be found in whole numbers. Besides, the dia- 
meters of the circles were all that he had occasion for ; and if 
he had them, there was no need of the ratio of BC to AB. All 
this is the more evident from the wide range which his object 
afforded him. The conclusion which he draws from his premis- 
ses is, that the diameter of the sphere is less than double the 
diameter of the tropic ; and this would be true for an obliquity 
of any magnitude which was less than 60° ; consequently it was 
of no importance whether he made it a few minutes, or even a 
few degrees too little or too great. 
The ratio of 10 to 23 gives an obliquity so very near to 23° 30', 
that it seems to be derived from that quantity. Indeed, Com- 
mandine, in his first note, has proceeded on this idea. 23° 30', 
likewise, forms a kind of round number, which, as Johnson 
somewhere remarks, will almost always be inaccurate, and there- 
fore might have given rise to suspicion : but I can see no other 
reason why it should have been fixed on ; and I am inclined to 
believe that it is an accidental consequence of the ratio which 
was adopted. Pappus, in all probability, as M. Lalande re- 
marks, 44 n’etoit pas autant observateur qu’ Erastothene, Ilip- 
parque et Ptolemee it may, therefore, be fairly supposed that 
he took his original quantity from Ptolemy, and he was the more 
