A history of Greek mathematics - Wilbourhall.org

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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,