A history of Greek mathematics - Wilbourhall.org

THE COLLECTION. BOOK VII 401
successive consequences, taking them as true, up to something
admitted : if then (a) what is admitted is possible and obtainable,
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 (b)
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 consequence
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\v6fiei>o$, 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 (vtvoei?),
two Books, the same author's Plane Loci, two Books, and
Conies, 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 Conies of Apollonius I have set out for your consideration,
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
treatises.
Then follows the short description of the contents of the
various Books down to Apollonius's Conies; no account is
given of Aristaeus's Solid Loci, Euclid's Surface-Loci and
1688.2 X> d