446 
EUCLID 
I 
5 
V 
property assigned by the author to bodies of the same kind is 
quite different from what we attribute to bodies of the same 
specific gravity; he purports to prove that bodies of the 
same kind have 'power proportional to their size, and the effect 
of this, combined with the definitions, is that they move at 
speeds proportional to their volumes. Thus the tract is the 
most precise statement that we possess of the principle of 
Aristotle’s dynamics, a principle which persisted until Bene 
detti (1530-90) and Galilei (1564-1642) proved its falsity. 
There are yet other fragments on mechanics associated with 
the name of Euclid. One is a tract translated by Woepcke 
from the Arabic in 1851 under the title ‘Le livre d’Euclide 
sur la balance ’, a work which, although spoiled by some com 
mentator, seems to go back to a Greek original and to have 
been an attempt to establish a theory of the lever, not from a 
general principle of dynamics like that of Aristotle, but from 
a few simple axioms such as the experience of daily life might 
suggest. The original work may have been earlier than 
Archimedes and may have been written by a contemporary of 
Euclid. A third fragment, unearthed by Duhem from manu 
scripts in the Bibliothèque Nationale in Paris, contains four 
propositions purporting to be ‘liber Euclidis de ponderibus 
secundum terminorum circumferentiam ’. The first of the 
propositions, connecting the law of the lever with the size of 
the circles described by its ends, recalls the similar demon 
stration in the Aristotelian Mechanica ; the others attempt to 
give a theory of the balance, taking account of the weight of 
the lever itself, and assuming that a portion of it (regarded as 
cylindrical) may be supposed to be detached and replaced by 
an equal weight suspended from its middle point. The three 
fragments supplement each other in a curious way, and it is a 
question whether they belonged to one treatise or were due to 
different authors. In any case there seems to be no indepen 
dent evidence that Euclid was the author of any of the 
fragments, or that he wrote on mechanics at all. 1 
1 For further details about these mechanical fragments see P. Duhem, 
Les origines de la statique, 1905. esp. vol. i, pp. 61-97. 
I 
I 
I 
ï 
♦ 
* 
I