A history of Greek mathematics - Wilbourhall.org

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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