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

DIOCLES 201
unless we have a fragment from it in the Fragmentum
mathematicum Bobieme. But Moslem writers regarded Diocles
as the discoverer of the parabolic burning-mirror; 'the ancients',
says al Singari (Sachawi, Ansarl), made mirrors of plane
'
surfaces. Some made them concave (i.e. spherical) until
Diocles (Diiiklis) showed and proved that, if the surface of
these mirrors has its curvature in the form of a parabola, they
then have the greatest power and burn most strongly. There
is a work on this subject composed by Ibn al-Haitham.' This
work survives in Arabic and in Latin translations, and is
reproduced by Heiberg and Wiedemann 1 ; it does not, however,
mention the name of Diocles, but only those of Archimedes
and Anthemius. Ibn al-Haitham says that famous
men like Archimedes and Anthemius had used mirrors made
up of a number of spherical rings ; afterwards, he adds, they
considered the form of curves which would reflect rays to one
point, and found that the concave surface of
a paraboloid of
revolution has this property. It is curious to find Ibn al-
Haitham saying that the ancients had not set out the proofs
sufficiently, nor the method by which they discovered them,
words which almost exactly recall those of Anthemius himself.
Nevertheless the whole course of Ibn al-Haitham' s proofs is
on the Greek model, Apollonius being actually quoted by name
in the proof of the main property of the parabola required,
namely that the tangent at any point of the curve makes
equal angles with the focal distance of the point and the
straight line drawn through it parallel to the axis. A proof
of the property actually survives in the Greek Fragmentum
mathematicum Bobiense, which evidently came from some
treatise on the parabolic burning-mirror ; but Ibn al-Haitham
does not seem to have had even this fragment at his disposal,
since his proof takes a different course, distinguishing three
different cases, reducing the property by analysis to the
known property AN = AT, and then working out the synthesis.
The proof in the Fragmentum is worth giving. It is
substantially as follows, beginning with the preliminary lemma
that, if the tangent at any point P, meets the axis at T FT} x
and if AS be measured along the axis from the vertex A
equal to \AL where AL is the parameter, then SF = ST
y
1
Bibliotheca mathematica, x 3 , 1910, pp. 201-37*
202 SUCCESSORS OF -THE GREAT GEOMETERS
Let PN be the ordinate from P ;
to the axis meeting PT in Y, and join SY.
Now
But PlY = 2AY (since JJV= JT)
therefore A Y 2 = TA .
draw A Y at right angles
PN*=AL.AN
= 4 AS. AN
= 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