THE COLLECTION. BOOK VII 
401 
successive consequences, taking them as true, up to something 
admitted: if then (a) what is admitted is possible and obtain 
able, that is, what mathematicians call given, what was 
originally proposed will also be possible, and the proof will 
again correspond in the reverse order to the analysis, but if (h) 
we come upon something admittedly impossible, the problem 
-will also be impossible.’ 
This statement could hardly be improved upon except that 
it ought to be added that each step in the chain of inference 
in the analysis must be unconditionally convertible; that is, 
when in the analysis we say that, if A is true, B is true, 
we must be sure that each statement is a necessary conse 
quence of the other, so that the truth of A equally follows 
from the truth of B. This, however, is almost implied by 
Pappus when he says that we inquire, not what it is (namely 
B) which follows from A, but what it is (B) from which A 
follows, and so on. 
List of works in the ‘ Treasury of Analysis ’. 
Pappus adds a list, in order, of the books forming the 
’Ava\v6y.evos, namely : 
‘ Euclid’s Data, one Book, Apollonius’s Cutting-off of a ratio, 
two Books, Cutting-off of an area, two Books, Determinate 
Section, two Books, Contacts, two Books, Euclid’s Porisms, 
three Books, Apollonius’s Inclinations or Vergings (vevaeis), 
two Books, the same author’s Plane Loci, two Books, and 
Conics, eight Books, Aristaeus’s Solid Loci, five Books, Euclid’s 
Surface-Loci, two Books, Eratosthenes’s On means, two Books. 
There are in all thirty-three Books, the contents of which up 
to the Conics of Apollonius I have set out for your considera 
tion, including not only the number of the propositions, the 
diorismi and the cases dealt with in each Book, but also the 
lemmas which are required; indeed I have not, to the best 
of my belief, omitted any question arising in the study of the 
Books in question.’ 
Description of the trecdises. 
Then follows the short description of the contents of the 
various Books down to Apollonius’s Conics; no account is 
given of Aristaeus’s Solid Loci, Euclid’s Surf dee-Loci and 
1B8S.2 J} d