ontent with holding that finite lines may be divided into
an infinite number of parts, do yet farther maintain that each of those
infinitesimals is itself subdivisible into an infinity of other parts or
infinitesimals of a second order, and so on ad infinitum. These, I say,
assert there are infinitesimals of infinitesimals of infinitesimals, &c.,
without ever coming to an end; so that according to them an inch does not
barely contain an infinite number of parts, but an infinity of an
infinity of an infinity ad infinitum of parts. Others there be who hold
all orders of infinitesimals below the first to be nothing at all;
thinking it with good reason absurd to imagine there is any positive
quantity or part of extension which, though multiplied infinitely, can
never equal the smallest given extension. And yet on the other hand it
seems no less absurd to think the square, cube or other power of a
positive real root, should itself be nothing at all; which they who hold
infinitesimals of the first order, denying all of the subsequent orders,
are obliged to maintain.

131. OBJECTION OF MATHEMATICIANS.--ANSWER.--Have we not therefore
reason to conclude they are both in the wrong, and that there is
in effect no such thing as parts infinitely small, or an infinite
number of parts contained in any finite quantity? But you will
say that if this doctrine obtains it will follow the very foundations
of Geometry are destroyed, and those great men who have raised
that science to so astonishing a height, have been all the while
building a castle in the air. To this it may be replied that whatever is
useful in geometry, and promotes the benefit of human life, does still
remain firm and unshaken on our principles; that science considered as
practical will rather receive advantage than any prejudice from what has
been said. But to set this in a due light may be the proper business of
another place. For the rest, though it should follow that some of the
more intricate and subtle parts of Speculative Mathematics may be pared
off without any prejudice to truth, yet I do not see what damage will be
thence derived to mankind. On the contrary, I think it were highly to be
wished that men of great abilities and obstinate application would draw
off their thoughts from those amusements, and employ them in the study of
such things as lie nearer the concerns of life, or have a more direct
influence on the manners.

132. SECOND OBJECTION OF MATHEMATIC