lLES 
THE ANGLE IN A SEMICIRCLE 137 
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would be found, could be drawn in practice), and to put in the 
two diagonals. The equality of the opposite sides would 
doubtless, in the first beginnings of geometry, be assumed as 
obvious, or verified by measurement. If then it was assumed 
that a rectangle is a figure with all its angles right angles and 
each side equal to its opposite, it would be easy to deduce 
certain consequences. Take first the two triangles ABC, BCD. 
Since by hypothesis AB — BG and CB is common, the two 
triangles have the sides AB, BG respectively equal to the sides 
BG, GD, and the included angles, being right angles, are equal; 
therefore the triangles ABC, BCB are equal in all respects 
(ef. Eucl, I. 4), and accordingly the angles ACB (i.e. OGB) and 
BBC (i.e. OBG) are equal, whence (by the converse of Eucl. I. 5, 
known to Thales) OB = OG. Similarly by means of the 
equality of AB, GB we prove the equality of OB,OG. Conse 
quently OB, OG, OB (and OA) are all equal. It follows that 
a circle with centre 0 and radius OA passes through B, G, B 
also ; since AO, OG are in a straight line, AG is a diameter of 
the circle, and the angle ABC, by hypothesis a right angle, is 
an ‘ angle in a semicircle It would then appear that, given 
any right angle as ABC standing on AC as base, it was only 
necessary to bisect AG at 0, and 0 would then be the centre of 
a semicircle on AC as diameter and passing through B. The 
construction indicated would be the construction of a circle 
about the right-angled triangle ABC, which seems to corre 
spond well enough to Pamphile’s phrase about ‘ describing on 
(i. e. in) a circle a triangle (which shall be) right angled ’. 
(y) Thales as astronomer. 
Thales was also the first Greek astronomer. Every one 
knows the story of his falling into a well when star-gazing, 
and being rallied by ‘a clever and pretty maidservant from 
Thrace ’ for being so eager to know what goes on in the 
heavens that he could not see what was straight in front 
of him, nay, at his very feet. But he was not merely a star 
gazer. There is good evidence that he predicted a solar eclipse 
which took place on May 28, 585 B. C. We can conjecture 
the basis of this prediction. The Babylonians, as the result 
of observations continued through centuries, had discovered 
the period of 223 lunations after which eclipses recur; and