A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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
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 Conies 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 1 ,p2 >Pa> 2h P5 or Pi Pi Pz » » > >
Pa > Ph
>
Pe
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
PiPzPz — ^PiP5 a (where X is a constant ratio and a a given
length), and in the second case p Y P2 Pz — ^P\P$P§i 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, ^.^§ *_» (
r -^^ ) = A. Pappus
p 2 2\ a V pn /
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 important
discoveries still remaining to be made. For himself, he
noticed that every one in his day was occupied with the elements,
the first principles and the natural origin of the subjectmatter
of investigation ; ashamed to pursue such topics, he had
himself proved propositions of much more importance and