of a mile or diameter of the earth, which may be
signified by that inch. When therefore I delineate a triangle on paper,
and take one side not above an inch, for example, in length to be the
radius, this I consider as divided into 10,000 or 100,000 parts or more;
for, though the ten-thousandth part of that line considered in itself is
nothing at all, and consequently may be neglected without an error or
inconveniency, yet these described lines, being only marks standing for
greater quantities, whereof it may be the ten--thousandth part is very
considerable, it follows that, to prevent notable errors in practice, the
radius must be taken of 10,000 parts or more.

128. LINES WHICH ARE INFINITELY DIVISIBLE.--From what has been said
the reason is plain why, to the end any theorem become universal in
its use, it is necessary we speak of the lines described on paper
as though they contained parts which really they do not. In doing
of which, if we examine the matter thoroughly, we shall perhaps
discover that we cannot conceive an inch itself as consisting of,
or being divisible into, a thousand parts, but only some other line which
is far greater than an inch, and represented by it; and that when we say
a line is infinitely divisible, we must mean a line which is infinitely
great. What we have here observed seems to be the chief cause why, to
suppose the infinite divisibility of finite extension has been thought
necessary in geometry.

129. The several absurdities and contradictions which flowed from this
false principle might, one would think, have been esteemed so many
demonstrations against it. But, by I know not what logic, it is held that
proofs a posteriori are not to be admitted against propositions relating
to infinity, as though it were not impossible even for an infinite mind
to reconcile contradictions; or as if anything absurd and repugnant could
have a necessary connexion with truth or flow from it. But, whoever
considers the weakness of this pretence will think it was contrived on
purpose to humour the laziness of the mind which had rather acquiesce in
an indolent scepticism than be at the pains to go through with a severe
examination of those principles it has ever embraced for true.

130. Of late the speculations about Infinities have run so high, and
grown to such strange notions, as have occasioned no small scruples and
disputes among the geometers of the present age. Some there are of great
note who, not c