A history of Greek mathematics - Wilbourhall.org

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ARISTARCHUS OF SAMOS 5
the Greek is even remarkably attractive. The content from
the mathematical point of view is no less interesting, for we
have here the first specimen extant of pure geometry used
with a trigonometrical object, in which respect it is a sort of
forerunner of Archimedes's Measurement of a Circle.
Aristarchus
does not actually evaluate the trigonometrical ratios
on which the ratios of the sizes and distances to be obtained
depend ; he finds limits between which they lie, and that by
means of certain propositions which he assumes without proof,
and which therefore must have been generally known to
mathematicians of his day. These propositions are the equivalents
of the statements that,
(1) if oc is what we call the circular measure of an angle
and oc is less than \ it, then the ratio sin oc/oc decreases, and the
ratio tan oc/oc increases, as a increases from to J it ;
(2) if /3 be the circular measure of another angle less than
\ it, and oc > /3, then
sin a oc tan oc
sin ft (3 tan fi
Aristarchus of course deals, not with actual circular measures,
sines and tangents, but with angles
(expressed not in degrees
but as fractions of right angles), arcs of circles and their
chords. Particular results obtained by Aristarchus are the
equivalent of the following :
^ > sin 3° > fa
[Prop. 7]
is >sinl°>^, [Prop. 11]
1 > cosl° > §§, [Prop. 12]
1 >cos 2 l° > |f. [Prop. 13]
The book consists of eighteen propositions.
Beginning with
six hypotheses to the effect already indicated, Aristarchus
declares that he is now in a position to prove
(1) that the distance of the sun from the earth is greater than
eighteen times, but less than twenty times, the distance of the
moon from the earth
(2) that the diameter of the sun has the same ratio as aforesaid
to the diameter of the moon