402 
PAPPUS OF ALEXANDRIA 
Eratosthenes’s On means, nor are there any lemmas to these 
works except two on the Surface-Loci at the end of the Book, 
The contents of the various works, including those of the 
lost treatises so far as they can be gathered from Pappus, 
have been described in the chapters devoted to their authors, 
and need not be further referred to here, except for an 
addendum to the account of Apollonius’s Conics which is 
remarkable. Pappus has been speaking of the ‘ locus with 
respect to three or four lines ’ (which is a conic), and proceeds 
to say (p. 678. 26) that we may in like manner have loci with 
reference to five or six or even more lines; these had not up 
to his time become generally known, though the synthesis 
of one of them, not by any means the most obvious, had been 
worked out and its utility shown. Suppose that there are 
five or six lines, and that p x , p 2 , p z , p 4 , p 5 or p x , p 2 , p 3 , p 4 ,p 5 ,p G 
are the lengths of straight lines drawn from a point to meet 
the five or six at given angles, then, if in the first case 
Pi Pi 2h = ^PiP5 a (where A is a constant ratio and a a given 
length), and in the second case p x p % p z — \p 4 p. p 6 , the locus 
of the point is in each case a certain curve given in position. 
The relation could not be expressed in the same form if 
there were more lines than six, because there are only three 
dimensions in geometry, although certain recent writers had 
allowed themselves to speak of a rectangle multiplied by 
a square or a rectangle without giving any intelligible idea of 
what they meant by such a thing (is Pappus here alluding to 
Heron’s proof of the formula for the area of a triangle in 
terms of its sides given on pp. 322-3, above ?). But the system 
of compounded ratios enables it to be expressed for any 
number of lines thus, — • ——for = A. Pappus 
proceeds in language not very clear (p. 680. 30); but the gist 
seems to be that the investigation of these curves had not 
attracted men of light and leading, as, for instance, the old 
geometers and the best writers. Yet there were other impor 
tant discoveries still remaining to be made. For himself, he 
noticed that everyone in his day was occupied with the elements, 
the first principles and the natural origin of the subject- 
matter of investigation; ashamed to pursue such topics, he had 
himself proved propositions of much more importance and