A history of Greek mathematics - Wilbourhall.org

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202 SUCCESSORS OF -THE GREAT GEOMETERS
Let PN be the ordinate from P
;
to the axis meeting PT in Y, and join SY.
Now
PN*=AL.AN
= 4 AS. AN
But PlY = 2AY (since JJV= JT)
therefore A Y 2 = TA .
draw A Y at right angles
= 4AS.AT (since A# = AT).
AS,
and the angle TYS is right.
The triangles SYT, SYP being right-angled, and TY being
equal to YP, it follows that SP = ST.
With the same figure, let BP be a ray parallel to AN
impinging on the curve at P. It is required to prove that
the angles of incidence and reflection (to S) are equal.
We have SP = ST, so that the angles at the points T, P
'
are equal. So ', says the author, ' are the angles TPA, KPR
[the angles between the tangent and the curve on each side of
the point of contact]. Let the difference between the angles
be taken ; therefore the angles SPA, RPB which remain
[again mixed ' ' angles] are equal. Similarly we shall show
that all the lines drawn parallel to .4$ will be reflected at
equal angles to the point S.'
The author then proceeds :
'
Thus burning-mirrors constructed
with the surface of impact (in the form) of the
section of a right-angled cone may easily, in the manner