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

SERENUS 525
the base of the cone to the projection of the vertex on its
plane ; the areas of the axial triangles are therefore proportional
to the generators of the cone with the said circle as
base and the same vertex as the original cone.
Prop. 50 is to
the effect that, if the axis of the cone is equal to the radius of
the base, the least axial triangle is a mean proportional
between the greatest axial triangle and the isosceles triangular
section perpendicular to the base ;
that is, with the above notation,
if r = V(p 2 + d 2 ), then r \/{p 2 + d 2 ) :rp = rp:p b^.od,
then a + d < b + c (a,d are the sides other than the base of one
axial triangle, and b, c those of the other axial triangle compared
with it; and if ABC, ADEbe two axial triangles and
the centre of the base, BA 2 + AC 2 =DA 2 + AE 2
because each
of these sums is equal to 2 A 2 + 2 BO 2 ,
Prop. 1 7). This proposition
again depends on the lemma (Props. 52, 53) that, if
straight lines be ' inflected ' from the ends of the base of
a segment of a circle to the curve (i. e. if we join the ends
of the base to any point on the curve) the line (i. e. the sum of
the chords) is greatest when the point taken is the middle
point of the arc, and diminishes as the point is taken farther
and farther from that point.
Let B be the middle point of the
arc of the segment ABC, D, E any
other points on the curve towards
G\ I say that
AB + BC>AD + DG>AE+EC.
With B as centre and BA as radius
describe a circle, and produce AB,
AD, AE to meet this circle in F, G,
H. Join FG, GC, HG
Since AB = BG = BF, we have AF = AB + BG
Also the
angles BFC, BGF are equal, and each of them is half of
the angle ABG.
526 COMMENTATORS AND BYZANTINES
Again
lAGC = IAFC = \LABC = \LADC;
therefore the angles DGC, DCG are equal and DG — DC;
therefore
Similarly
AG = AD + DC.
EH = EC and All = AE+ EC.
But, by Eucl. III. 7 or 15, AF>AG >AH, and so on
;
therefore
AB + BC> AD + DC>AE+ EC, and so on.
In the particular case where the segment ABC is a semicircle
AB 2 + BC 2 = AC 2 = AD 2 + DC 2 , &c, and the result of
Prop. 57 follows.
Props. 58-69 are propositions of this sort: In equal right
cones the triangular sections through the axis are reciprocally
proportional to their bases and conversely (Props. 58, 59)
right cones of equal height have to one another the ratio
duplicate of that of their axial triangles (Prop. 62); right
cones which are reciprocally proportional to their bases have
axial triangles which are to one another reciprocally in the
triplicate ratio of their bases and conversely (Props. 66, 67);
and so on.
Theon of Alexandria lived towards the end of the fourth
century A.D. Suidas places him in the reign of Theodosius I
(379-95); he tells us himself that he observed a solar eclipse
at Alexandria in the year 365, and his notes on the chronological
tables of Ptolemy extend down to 372.
Commentary on the
Syntaxis.
We have already seen him as the author of a commentary
on Ptolemy's Syntaxis in eleven Books. This commentary is
not calculated to give us a very high opinion of Theon's
mathematical calibre, but it is valuable for several historical
notices that it gives, and we are indebted to it for a useful
account of the Greek method of operating with sexagesimal
fractions, which is illustrated by examples of multiplication,
division, and the extraction of the square root of a non-square
number by way of approximation. These illustrations of
numerical calculation have already been given above (vol. i,