MENELAUS’S SPHABBICA 
373 
But, by III. 1 applied to the triangle BC X A X cut by the 
transversal PC.,A.,, 
sin A 1 A s _ sin G X C 3 sin PA X 
sin BA 3 sin BG 3 sin PG X ’ 
sin A X A 3 _ sin PA X sin BA., sinPJ, 1 sin PC 
sin G x G 3 sin PG X sin BG 3 sin PG X sin PG 3 
from above, 
_ Hr 
~~ r i r H 
Part 2 of the proposition proves that, if PG 2 A 2 be drawn 
such that sin 2 PC 2 = sin PA 2 . sin PG, or r 2 2 = Rr (where r 2 is 
the radius of the parallel circle through C 2 ), BC 2 —BA 2 is a 
maximum, while Parts 3, 4 discuss the limits to the value of 
the ratio between the arcs A X A 3 and G X G 3 . 
Nothing is known of the life of Claudius Ptolemy except 
that he was of Alexandria, made observations between the 
years A.D. 125 and 141 or perhaps 151, and therefore presum 
ably wrote his great work about the middle of the reign of 
Antoninus Pius (a.d. 138-61). A tradition handed down by 
the Byzantine scholar Theodorus Meliteniota (about 1361) 
states that he was born, not at Alexandria, but at Ptolemais 
fj 'Ep/ieiov. Arabian traditions, going back probably to 
Hunain b. Ishaq, say that he lived to the age of 78, and give 
a number of personal details to which too much weight must 
not be attached. 
The MaOrificcTiKT) avura^Ls (Arab. Almagest). 
Ptolemy’s great work, the definitive achievement of Greek 
astronomy, bore the title MaQrniaTiKfjs Swragecos (3i(3Xia ty, 
the Mathematical Collection in thirteen Books. By the time 
of the commentators who distinguished the lesser treatises on 
astronomy forming an introduction to Ptolemy’s work as 
/¿LKpos cia-Tpoi'o/j.ovfj.evos (tottos), the ‘Little Astronomy’, the 
book came to be called the ‘ Great Collection ’, geyaX-q avv- 
to.£ls. Later still the Arabs, combining the article Al with 
1523,2 X