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 content 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. 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. It is be said that several theorems undoubtedly true are
discovered by methods in which infinitesimals are made use of, which
could never have been if their existence included a contradiction in
it; I answer that upon a thorough examination it will not be found
that in any instance it is necessary to make use of or conceive
infinitesimal parts of finite lines, or even quantities less than
the minimum sensible; nay, it will be evident this is never done, it
being impossible.

  133. By what we have premised, it is plain that very numerous and
important errors have taken their rise from those false Principles
which were impugned in the foregoing parts of this treatise; and the
opposites of those erroneous tenets at the same time appear to be most
fruitful Principles, from whence do flow innumerable consequences
highly advantageous to true philosophy. as well as to religion.
Particularly Matter, or the absolute existence of corporeal objects,
hath been shewn to be that wherein the most avowed and pernicious
enemies of all knowledge, whether human or divine, have ever placed
their chief strength and confidence. And surely, if by
distinguishing the real existence of unthinking things from their
being perceived, and allowing them a subsistance of their own out of
the minds of spirits, no one thing is explained in nature, but on
the contrary a great many inexplicable difficulties arise; if the
supposition of Matter is barely precarious, as not being grounded on
so much as one single reason; if its consequences cannot endure the
light of examination and free inquiry, but screen themselves under the
dark and general pretence of "infinites being incomprehensible"; if
withal the removal of this Matter be not attended with the least
evil consequence; if it be not even missed in the world, but
everything as well, nay much easier conceived without it; if,
lastly, both Sceptics and Atheists are for ever silenced upon
supposing only spirits and ideas, and this scheme of things is
perfectly agreeable both to Reason and Religion: methinks we may
expect it should be admitted and firmly embraced, though it were
proposed only as an hypothesis, and the existence of Matter had been
allowed possible, which yet I think we have evidently demonstrated
that it is not.

  134. True it is that, in consequence of the foregoing principles,
several disputes and speculations which are esteemed no mean parts
of learning, are rejected as useless. But, how great a prejudice
soever against our notions this may give to those who have already
been deeply engaged, and make large advances in studies of that
nature, yet by others we hope it will not be thought any just ground
of dislike to the principles and tenets herein laid down, that they
abridge the labour of study, and make human sciences far more clear,
compendious and attainable than they were before.

  135. Having despatched what we intended to say concerning the
knowledge of IDEAS, the method we proposed leads us in the next
place to treat of SPIRITS- with regard to which, perhaps, human
knowledge is not so deficient as is vulgarly imagined. The great
reason that is assigned for our being thought ignorant of the nature
of spirits is our not having an idea of it. But, surely it ought not
to be looked on as a defect in a human understanding that it does
not perceive the idea of spirit, if it is manifestly impossible