THE TWO INTELLECTUAL METHODS 291 
With the above quotations we should read a passage of 
Proclus. 
‘ Nevertheless certain methods have been handed down. The 
finest is the method which by means of analysis carries 
the thing sought up to an acknowledged principle; a method 
which Plato, as they say, communicated to Leodamas, and by 
which the latter too is said to have discovered many things 
in geometry. The second is the method of division, which 
divides into its parts the genus proposed for consideration, 
and gives a starting-point for the demonstration by means of 
the elimination of the other elements in the construction 
of what is proposed, which method also Plato extolled as 
being of assistance to all sciences.’ 1 
The first part of this passage, with a like dictum in Diogenes 
Laertius that Plato ‘explained to Leodamas of Thasos the 
method of inquiry by analysis ’, 2 has commonly been under 
stood as attributing to Plato the invention of the method 
of mathematical analysis. But, analysis being according to 
the ancient view nothing more than a series of successive 
reductions of a theorem or problem till it is finalty reduced 
to a theorem or problem already known, it is difficult to 
see in what Plato’s supposed discovery could have consisted; 
for analysis in this sense must have been frequently used 
in earlier investigations. Not only did Hippocrates of Chios 
reduce the problem of duplicating the cube to that of finding 
two mean proportionals, but it is clear that the method of 
analysis in the sense of reduction must have been in use by 
the Pythagoreans. On the other hand, Proclus’s language 
suggests that what he had in mind was the philosophical 
method described in the passage of the Republic, which of 
'course does not refer to mathematical analysis at all; it may 
therefore well be that the idea that Plato discovered the 
method of analysis is due to a misapprehension. But analysis 
and synthesis following each other are related in the same 
way as the upward and downward progressions in the dialec 
tician’s intellectual method. It has been suggested, therefore, 
that Plato’s achievement was to observe the importance 
from the point of view of logical rigour, of the confirma 
tory synthesis following analysis. The method of division 
1 Proclus, Comm, on End. I, pp. 211. 18-212. 1. 
2 Diog. L. iii. 24, p. 74, Cobet. 
u 2