THE ELEMENTS. BOOK I 
375 
interpolated in Proposition 4, while Postulate 2 similarly im 
plies the theorem that two straight lines cannot have a 
common segment, which Simson gave as a corollary to I. 11. 
At first sight the Postulates 4 (that all right angles are 
equal) and 5 (the Parallel-Postulate) might seem to be of 
an altogether different character, since they are rather of the 
nature of theorems unproved. But Postulate 5 is easily seen 
to be connected with constructions, because so many con 
structions depend on the existence and use of points in which 
straight lines intersect; it is therefore absolutely necessary to 
lay down some criterion by which we can judge whether two 
straight lines in a figure will or will not meet if produced. 
Postulate 5 serves this purpose as well as that of providing 
a basis for the theory of parallel lines. Strictly speaking, 
Euclid ought to have gone further and given criteria for 
judging whether other pairs of lines, e. g. a straight line and 
a circle, or two circles, in a particular figure will or will not 
intersect one another. But this would have necessitated a 
considerable series of propositions, which it would have been 
difficult to frame at so early a stage, and Euclid preferred 
to assume such intersections provisionally in certain cases, 
e. g. in I. 1. 
Postulate 4 is often classed as a theorem. But it had in any 
case to be placed before Postulate 5 for the simple reason that 
Postulate 5 would be no criterion at all unless right angles 
were determinate magnitudes; Postulate 4 then declares them 
to be such. But this is not all. If Postulate 4 were to be 
proved as a theorem, it could only be proved by applying one 
pair of ‘ adjacent ’ right angles to another pair. This method 
would not be valid unless on the assumption of the invaria 
bility of figures, which would therefore have to be asserted as 
an antecedent postulate. Euclid preferred to assert as a 
postulate, directly, the fact that all right angles are equal; 
hence his postulate may be taken as equivalent to the prin 
ciple of the invariability of figures, or, what is the same thing, 
the homogeneity of space. 
For reasons which I have given above (pp. 339, 358), I think 
that the great Postulate 5 is due to Euclid himself; and it 
seems probable that Postdate 4 is also his, if not Postulates 
1-3 as well.