374 
EUCLID 
different kinds of triangle under their twofold classification 
(a) with reference to their sides (equilateral, isosceles and 
scalene), and (b) with reference to their angles (right-angled, 
obtuse-angled and acute-angled); such definitions are pro 
visional pending the proof of existence by means of actual con 
struction. A parallelogram is not defined; its existence is 
first proved in I. 33, and in the next proposition it is called a 
‘ parallelogrammic area ’, meaning an area contained by parallel 
lines, in preparation for the use of the simple word ‘ parallelo 
gram’ from I. 35 onwards. The definition of a diameter 
of a circle (17) includes a theorem; for Euclid adds that ‘such 
a straight line also bisects the circle’, which is one of the 
theorems attributed to Thales; but this addition was really 
necessary in view of the next definition (18), for, without 
this explanation, Euclid would not have been justified in 
describing a sewi-circle as a portion of a circle cut off' by 
a diameter. 
More important by far are the five Postulates, for it is in 
them that Euclid lays down the real principles of Euclidean 
geometry; and nothing shows more clearly his determination 
to reduce his original assumptions to the very minimum. 
The first three Postulates are commonly regarded as the 
postulates of construction, since they assert the possibility 
(1) of drawing the straight line joining two points, (2) of 
producing a straight line in either direction, and (3) of describ 
ing a circle with a given centre and ‘ distance ’. But they 
imply much more than this. In Postulates 1 and 3 Euclid 
postulates the existence of straight lines and circles, and 
implicitly answers the objections of those who might say that, 
as a matter of fact, the straight lines and circles which we 
can draw are not mathematical straight lines and circles; 
Euclid may be supposed to assert that we can nevertheless 
assume our straight lines and circles to be such for the purpose 
of our proofs, since they are only illustrations enabling us to 
imagine the real things which they imperfectly represent. 
But, again, Postulates 1 and 2 further imply that the straight 
line drawn in the first case and the produced portion of the 
straight line in the second case are unique; in other words, 
Postulate 1 implies that two straight lines cannot enclose a 
space, and so renders unnecessary the ‘ axiom ’ to that effect