CHARACTER OF TREATISES 
21 
d and brought to 
, Fermat, Leibniz 
, is equivalent to 
c segment, and of 
and a segment of 
3 of the solids of 
etic he calculated 
urse of which cal- 
te to the value of 
nbers; he further 
ogy by which he 
to that which we 
90 million million 
it the principles of 
! centre of gravity 
me, a hemisphere, 
a paraboloid and 
! we shall see, has 
geometry since the 
hich was formerly 
e whole science of 
as to give a most 
st and stability of 
tion floating in a 
wards, but so that 
low the surface of 
lematical achieve- 
orld’s history. 
mments of mathe- 
n of the plan of 
Dsitions, the stern 
y relevant to the 
npressive in their 
we in the mind of 
Dossible to find in 
uestions or proofs 
1 There is at the 
same time a certain mystery veiling the way in which he 
arrived at his results. For it is clear that they were not 
discovered by the steps which lead up to them in the finished 
treatises. If the geometrical treatises stood alone, Archi 
medes might seem, as Wallis said, ‘as it were of set purpose 
to have covered up the traces of his investigation, as if he had 
grudged posterity the secret of his method of inquiry, while 
he wished to extort from them assent to his results And 
indeed (again in the words of Wallis) ‘ not only Archimedes 
but nearly all the ancients so hid from posterity their method 
of Analysis (though it is clear that they had one) that more 
modern mathematicians found it easier to invent a new 
Analysis than to seek out the old’. A partial exception is 
now furnished by The Method of Archimedes, so happily dis 
covered by Heiberg. In this book Archimedes tells us how 
he discovered certain theorems in quadrature and cubature, 
namely by the use of mechanics, weighing elements of a 
figure against elements of another simpler figure the mensura 
tion of which was already known. At the same time he is 
careful to insist on the difference between (1) the means 
which may be sufficient to suggest the truth of theorems, 
although not furnishing scientific proofs of them, and (2) the 
rigorous demonstrations of them by orthodox geometrical 
methods which must follow before they can be finally accepted 
as established: 
‘certain things’, he says, ‘first became clear to me by a 
mechanical method, although they had to be demonstrated by 
geometry afterwards because their investigation by the said 
method did not furnish an actual demonstration. But it is 
of course easier, when we have previously acquired, by the 
method, some knowledge of the questions, to supply the proof 
than it is to find it without any previous knowledge.’ ‘ This ’, 
he adds, ‘ is a reason why, in the case of the theorems that 
the volumes of a cone and a pyramid are one-third of the 
volumes of the cylinder and prism respectively having the 
same base and equal height, the proofs of which Eudoxus was 
the first to discover, no small share of the credit should be 
given to Democritus who was the first to state the fact, 
though without proof.’ 
Finally, he says that the very first theorem which he found 
out by means of mechanics was that of the separate treatise