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FROM PLATO TO EUCLID
Aristotle’s denial of even the potential existence of a sum
of magnitudes which shall exceed every definite magnitude
was, as he himself implies, inconsistent with the lemma or
assumption used by Eudoxus in his method of exhaustion.
We can, therefore, well understand why, a century later,
Archimedes felt it necessary to justify his own use of the
lemma :
e the earlier geometers too have used this lemma : for it is by
its help that they have proved that circles have to one another
the duplicate ratio of their diameters, that spheres have to
one another the triplicate ratio of their diameters,, and so on.
And, in the result, each of the said theorems has been accepted
no less than those proved without the aid of this lemma.’ 1
(¿j Mechanics.
An account of the mathematics in Aristotle would be incom
plete without a reference to his ideas in mechanics, where he
laid down principles which, even though partly erroneous,
held their ground till the time of Benedetti (1530-90) and
Galilei (1564-1642). The Mechanica included in the Aris-
tQtelian writings is not indeed Aristotle’s own work, but it is
very close in date, as we may conclude from its terminology ;
this shows more general agreement with the terminology of
Euclid than is found in Aristotle’s own writings, but certain
divergences from Euclid’s terms are common to the latter and
to the Mechanica ; the conclusion from which is that the
Mechanica was written before Euclid had made the termino-
logy of mathematics more uniform and convenient, or, in the
alternative, that it was composed after Euclid’s time by persons
who, though they had partly assimilated Euclid’s terminology,
were close enough to Aristotle’s date to be still influenced
by his usage. But the Aristotelian origin of many of the
ideas in the Mechanica is proved by their occurrence in
Aristotle’s genuine writings. Take, for example, the principle
of the lever. In the Mechanica we are told that,
, ‘ as the weight moved is to the moving weight, so is the
length (or distance) to the length inversely. In fact the mov
ing weight will more easily move (the system) the farther it
is away from the fulcrum. The reason is that aforesaid,
1 Archimedes, Quadrature of a Parabola, Preface.
