376 
EUCLID 
Of the Common Notions there is good reason to believe 
that only five (at the most) are genuine, the first three and 
two others, namely ‘ Things which coincide when applied to 
one another are equal to one another ’ (4), and ‘ The whole 
is greater than the part ’ (5). The objection to (4) is that 
it is incontestably geometrical, and therefore, on Aristotle’s 
principles, should not be classed as an * axiom ’; it is a more 
or less sufficient definition of geometrical equality, but not 
a real axiom. Euclid evidently disliked the method of super 
position for proving equality, no doubt because it assumes the 
possibility of motion without deformation. But he could not 
dispense with it altogether. Thus in I. 4 he practically had 
to choose between using the method and assuming the whole 
proposition as a postulate. But he does not there quote 
Common Notion 4; he says ‘ the base BG will coincide with 
the base EF and will be equal to it’. Similarly in I. 6 he 
does not quote Common Notion 5, but says ‘the triangle 
DBG will be equal to the triangle A CB, the less to the greater, 
which is absurd ’. It seems probable, therefore, that even 
these two Common Notions, though apparently recognized 
by Proclus, were generalizations from particular inferences 
found in Euclid and were inserted after his time. 
The propositions of Book I fall into three distinct groups. 
The first group consists of Propositions 1-26, dealing mainly 
with triangles (without the use of parallels) but also with 
perpendiculars (11, 12), two intersecting straight lines (15), 
and one straight line standing on another but not cutting it, 
and making ‘adjacent’ or supplementary angles (13, 14). 
Proposition 1 gives the construction of an equilateral triangle 
on a given straight line* as base; this is placed here not so 
much on its own account as because it is at once required for 
constructions (in 2, 9, 10, 11). The construction in 2 is a 
direct continuation of the minimum constructions assumed 
in Postulates 1-3, and enables us (as the Postulates do not) to 
transfer a given length of straight line from one place to 
another; it leads in 3 to the operation so often required of 
cutting off* from one given straight line a length equal to 
another. 9 and 10 are the problems of bisecting a given angle 
and a given straight line respectively, and 11 shows how 
to erect a perpendicular to a given straight line from a given