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

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
186 AP0LL0N1US OF PERGA
of a line. 1
The second classification is the familiar division into
'plane, solid, and linear loci, plane loci being straight lines
and circles only, solid loci conic sections only, and linear loci
those which are not straight lines nor circles nor any of the
conic sections. The loci dealt with in our treatise are accordingly
all straight lines or circles. The proof of the propositions
is of course enormously facilitated by the use of
Cartesian coordinates, and many of the loci are really the
geometrical equivalent of fundamental theorems in analytical
or algebraical geometry. Pappus begins with a composite
enunciation, including a number of propositions, in these
terms, which, though apparently confused, are not difficult
to follow out:
4ft
If two straight lines be drawn, from one given point or from
1
two, which are (a) in a straight line or (b) parallel or
(c) include a given angle, and either (a) bear a given ratio to
one another or (/?) contain a given rectangle, then, if the locus
of the extremity of one of the lines is a plane locus given in
position, the locus of the extremity of the other will also be a
plane locus given in position, which will sometimes be of the
same kind as the former, sometimes of the other kind, and
will sometimes be similarly situated with reference to the
straight line, and sometimes contrarily, according to the
particular differences in the suppositions.' 2
'
(The words with reference to the straight line are obscure, but
'
the straight line is presumably some obvious straight line in
each figure, e. g., when there are two given points, the straight
line joining them.) After quoting three obvious loci added
'
by Charmandrus ', Pappus gives three loci which, though containing
an unnecessary restriction in the third case, amount
to the statement that any equation of the first degree between
coordinates inclined at fixed angles to (a) two axes perpendicular
or oblique, (h) to any number of axes, represents a
straight line. The enunciations (5-7) are as follows. 3
5. ' If, when a straight line is given in magnitude and is
moved so as always to be parallel to a certain straight line
given in position, one of the extremities (of the moving
straight line), lies on a straight line given in position, the
1
Pappus, vii, pp. 660. 18-662. 5.
3
lb., pp. 664. 20-666. 6.
2 16,'vii, pp. 662. 25-664. 7.