4 
ARISTARCHUS OF SAMOS 
Archimedes also says that it was Aristarchus who dis 
covered that the apparent angular diameter of the sun is about 
l/720th part of the zodiac circle, that is to say, half a degree. 
We do not know how he arrived at this pretty accurate figure : 
but, as he is credited with the invention of the a-Kdcjir], he may 
have used this instrument for the purpose. But here again 
tlje discovery must apparently have been later than the trea 
tise On sizes and distances, for the value of the subtended 
angle is there assumed to be 2° (Hypothesis 6). How Aris 
tarchus came to assume a value so excessive is uncertain. As 
the mathematics of his treatise is not dependent on the actual 
value taken, 2° may have been assumed merely by way of 
illustration ; or it may have been a guess at the apparent 
diameter made before he had thought of attempting to mea 
sure it. Aristarchus assumed that the angular diameters of 
the sun and moon at the centre of the earth are equal. 
The method of the treatise depends on the just observation, 
which is Aristarchus’s third ‘ hypothesis ’, that ‘ when the moon 
appears to us halved, the great circle which divides the dark 
and the bright portions of the moon is in the direction of our 
eye ’ ; the effect of this (since the moon receives its light from 
the sun), is that at the time ,of the dichotomy the centres of 
the sun, moon and earth form a triangle right-angled at the 
centre of the moon. Two other assumptions were necessary ; 
first, an estimate of the size of the angle of the latter triangle 
at the centre of the earth at the moment of dichotomy : this 
Aristarchus assumed (Hypothesis 4) to be Hess than a quad 
rant by one-thirtieth of a quadrant’, i.e. 87°, again an inaccu 
rate estimate, the true value being 89° 50' ; secondly, an esti 
mate of the breadth of the earth’s shadow where the moon 
traverses it : this he assumed to be £ the breadth of two 
moons ’ (Hypothesis 5). 
The inaccuracy of the assumptions does not, however, detract 
from the mathematical interest of the succeeding investigation. 
Here we find the logical sequence of propositions and the abso 
lute rigour of demonstration characteristic of Greek geometry ; 
the only remaining drawback would be the practical difficulty 
of determining the exact moment when the moon ‘ appears to 
us halved ’. The form and -style of the book are thoroughly 
classical, as befits the period between Euclid and Archimedes ; 
the Greek is 
the mathema 
have here th 
with a trigon 
forerunner of 
chus does m 
on which the 
depend; he h 
means of cert 
and which t 
mathematicia 
valents of tht 
(1) if a is a 
and a is less t 
ratio tan oc/oc 
(2) if /3 be t 
\ it, and a > /5 
Aristarchus o 
sines and tan< 
but as fracti 
chords. Part 
equivalent of 
The book cc 
six hypothest 
declares that 1 
(1) that the d 
eighteen times 
moon from th 
(2) that the d 
said to the dia