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

PLANE LOCI 187
other extremity will also lie on a straight line given in
position.'
(That is, x = a or y = b in Cartesian coordinates
straight line.)
represents a
6. If from any point straight lines be drawn to meet at given
'
angles two straight lines either parallel or intersecting, and if
the straight lines so drawn have a given ratio to one another
or if the sum of one of them and a line to which the other has
a given ratio be given (in length), then the point will lie on a
straight line given in position.'
(This includes the equivalent of saying that, if x, y be the
coordinates of the point, each of the equations x = my,
x + my = c represents a straight line.)
7. If any number of straight lines be given in position, and
'
straight lines be drawn from a point to meet them at given
angles, and if the straight lines so drawn be such that the
rectangle contained by one of them and a given straight line
added to the rectangle contained by another of them and
(another) given straight line is equal to the rectangle contained
by a third and a (third) given straight line, and similarly
with the others, the point will lie on a straight line given
in position.'
(Here we have trilinear or multilinear coordinates proportional
to the distances of the variable point from each of the
three or more fixed lines. When there are three fixed lines,
the statement is that ax + by = cz represents a straight line.
The precise meaning of the words 'and similarly with the
the others ' or ' of the others ' kolI tw \olttS)v 6/ioico?— -is
uncertain ; the words seem to imply that, when there were
more than three rectangles ax, by, cz . . , two of them were
.
taken to be equal to the sum of all the others ; but it is quite
possible that Pappus meant that any linear equation between
these rectangles represented a straight line. Precisely how
far Apollonius went in generality we are not in a position to
judge.)
The last enunciation (8) of Pappus referring to Book I
states that,
'
If from any point (two) straight lines be drawn to meet (two)
parallel straight lines given in position at given angles, and
188 APOLLONIUS OF PERGA
cut off from the parallels straight lines measured from given
points on them such that (a) they have a given ratio or
(b) they contain a given rectangle or (c) the sum or difference
of figures of given species described on them respectively is
equal to a given area, the point will lie on a straight line
given in position.' 1
The contents of Book II are equally interesting.
Some of
the enunciations shall for brevity be given by means of letters
instead of in general terms. If from two given points A, B
two straight lines be ' inflected ' (KXacrOaxriu) to a point P, then
(1), if AP 2 ^ BP 2 is given, the locus of P is a straight line;
(2) if AP, BP are in a given ratio, the locus is a straight line
or a circle [this is the proposition quoted by Eutocius in his
commentary on the Conies, but already known to Aristotle]
(4) if AP 2 is greater ' b}^ a given area than in a given ratio '
to BP 2 , i.e. if AP = 2 a 2 + m . BP 2 , the locus is a circle given in
position. An interesting proposition is (5) that, If from any
'
number of given points whatever straight lines be inflected to
one point, and the figures (given in species) described on all of
them be together equal to a given area, the point will lie on
a circumference (circle) given in position ; '
that is to say, if
a . AP 2 + fi
. BP 2 +
.
y CP 2 + ... = a given area (where a, ft,
.
y
.
are constants), the locus of P is a circle. (3) states that, if
AN be a fixed straight line and A a fixed point on it, and if
AP be any straight line drawn to a point P such that, if PN
is perpendicular to AN, AP 2 — a . AN
or a .
BN, where a- is a
given length and B is another fixed point on AN, then the
locus of P is a circle given in position ;
this is equivalent
to the fact that, if A be the origin, AN the axis of x, and
x = AN,y = PN be the coordinates of P, the locus x 2 + y
2
= ax
or x 2 + y
2
= a(x — b) is a circle. (6)
enunciated :
is somewhat obscurely
'
If from two given points straight lines be inflected
(to a point), and from the point (of concourse) a straight
line be drawn parallel to a straight line given in position and
cutting off from another straight line given in position an
intercept measured from a given point on it, and if the sum of
figures (given in species) described on the two inflected lines
be equal to the rectangle contained by a given straight line
and the intercept, the point at which the straight lines are
1
Pappus, vii, p. 666. 7-13.