A history of Greek mathematics - Wilbourhall.org

ON CONTACTS OR TANGENCIES 185
given circles be a, b, c and their centres A, B, C. Let D, E, F
be the external centres of similitude so that BD : DC— b : c, &c.
Suppose the problem solved, and let P, Q, R be the points
of contact. Let PQ produced meet the circles with centres
A, B again in K, L. Then, by the proposition (1) above, the
segments KGP, QHL are both similar to the segment PYQ
;
therefore they are similar to one another.
It follows that PQ
produced beyond L passes through F. Similarly QR, PR
produced pass respectively through D, E.
Let PE, QD meet the circle with centre C again in M, N.
Then, the segments PQR, RNM being similar, the angles
PQR, RNM are equal, and therefore MN is parallel to PQ.
Produce NM to meet EF in V.
Then
EV:EF = EM: EP = EC:EA = c:a;
therefore the point V is given.
Accordingly the problem reduces itself to this :
Given three
points V, E, D in a straight line, it is required to draw DR, ER
to a point R on the circle with centre C so that, if DR, ER meet
the circle again in N, M, NM produced shall pass through V.
This is the problem of Pappus just solved.
Thus R is found, and DR, ER produced meet the circles
with centres B and A in the other required points Q, P
respectively.
(e) Plane loci, two Books.
Pappus gives a pretty full account of the contents of this
work, which has sufficed to enable restorations of it to
be made by three distinguished geometers, Fermat, van
Schooten, and (most completely) by Robert Simson. Pappus
prefaces his account by a classification of loci on two
different plans. Under the first classification loci are of three
kinds: (1) efeKTiKoi, holding-in or fixed; in this case the
locus of a point is a point, of a line a line, and of a solid
a solid,
where presumably the line or solid can only move on
itself so that it does not change its position: (2) Siego-
Slkol, pasdng-along : this is the ordinary sense of a locus,
where the locus of a point is a line, and of a line a solid:
(3) dvao-Tpo(f)iKoi, moving backvjards and forwards, as it were,
in which sense a plane may be the locus of a point and a solid