130 THE EARLIEST GREEK GEOMETRY. THALES 
M. m 
considerable number of cases at a time when he found the (, 
length of the shadow of one object to be equal to its height. the 
But, even if Thales used the more general method indicated (wl: 
by Plutarch, that method does not, any more than the Egyptian °P^ 
se-qet calculations, imply any general theory of similar tri- c> j 
angles or proportions; the solution is itself a se-qet calculation, 
just like that in No. 57 of Ahmes’s handbook. In the latter ^ 
problem the base and the se-qet are given, and we have to that 
find the height. So in Thales’s problem we get a certain 111 a 
se-qet by dividing the measured length of the shadow of the 1 
stick by the length of the stick itself; we then only require 
to know the distance between the point of the shadow corre- a C1 
spending to the apex of the pyramid and the centre of the the 
base of the pyramid in order to determine the height; the W1 ^ 
only difficulty would be to measure or estimate the distance °PP' 
from the apex of the shadow to the centre of the base. too 
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(3) Geometrical theorems attributed to Thales. ^ ^ 
obse 
The following are the general theorems in elementary anc | 
geometry attributed to Thales. gu 
(1) He is said to have been the first to demonstrate that gg U 
a circle is bisected by its diameter. 1 0 £ 
(2) Tradition credited him with the first statement of the 
theorem (Eucl. I. 5) that the angles at the base of any men 
isosceles triangle are equal, although he used the more archaic py 
term ‘ similar ’ instead of ‘ equal ’. 2 qy n . 
(3) The proposition (Eucl. I. 15) that, if two straight lines * jq 
cut one another, the vertical and opposite angles are equal q esc 
was discovered, though not scientifically proved, by Thales. qq ia 
Eudemus is quoted as the authority for this. 3 as a 
(4) Eudemus in his History of Geometry referred to Thales c j ogt 
the theorem, of Eucl. I. 26 that, if two triangles have two ma ]j 
angles and one side respectively equal, the triangles are equal ¿he f 
in all respects. ^ 
‘ For he (Eudemus) says that the method by which Thales 
showed how to find the distances of ships from the shore kno^ 
necessarily involves the use of this theorem.’ 4 detei 
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1 Proclus on Eucl. I, p. 157. 10. 2 lb., pp. 250. 20-251. 2. 
3 lb., p. 299. 1-5. 4 lb., p. 852. 14-18. 1 I 
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