### Section II. Of the infinite divisibility of space and time.

WHEREVER ideas are adequate representations of objects, the relations, contradictions and agreements of the ideas are all applicable to the objects; and this we may in general observe to be the foundation of all human knowledge. But our ideas are adequate representations of the most minute parts of extension; and through whatever divisions and subdivisions we may suppose these parts to be arrived at, they can never become inferior to some ideas, which we form. The plain consequence is, that whatever appears impossible and contradictory upon the comparison of these ideas, must be really impossible and contradictory, without any farther excuse or evasion.

Every thing capable of being infinitely divided contains an infinite
number of parts; otherwise the division would be stopped short by the
indivisible parts, which we should immediately arrive at. If therefore any
finite extension be infinitely divisible, it can be no contradiction to
suppose, that a finite extension contains an infinite number of parts: And
vice versa, if it be a contradiction to
suppose, that a finite extension contains an infinite number of parts, no
finite extension can be infinitely divisible. But that this latter
supposition is absurd, I easily convince myself by the consideration of my
clear ideas. I first take the least idea I can form of a part of extension,
and being certain that there is nothing more minute than this idea, I
conclude, that whatever I discover by its means must be a real quality of
extension. I then repeat this idea once, twice, thrice, etc. and find the
compound idea of extension, arising from its repetition, always to augment,
and become double, triple, quadruple, etc. till at last it swells up to a
considerable bulk, greater or smaller, in proportion as I repeat more or
less the same idea. When I stop in the addition of parts, the idea of
extension ceases to augment; and were I to carry on the addition in infinitum, I clearly perceive, that the idea of
extension must also become infinite. Upon the whole, I conclude, that the
idea of all infinite number of parts is individually the same idea with that
of an infinite extension; that no finite extension is capable of containing
an infinite number of parts; and consequently that no finite extension is
infinitely divisible^{1}.

I may subjoin another argument proposed by a noted author^{2}, which seems to me very strong and beautiful.
'Tis evident, that existence in itself belongs only to unity, and is never
applicable to number, but on account of the unites, of which the number is
composed. Twenty men may be said to exist; but 'tis only because one, two,
three, four, etc. are existent; and if you deny the existence of the latter,
that of the former falls of course. 'Tis therefore utterly absurd to suppose
any number to exist, and yet deny the existence of unites; and as extension
is always a number, according to the common sentiment of metaphysicians, and
never resolves itself into any unite or indivisible quantity, it follows,
that extension can never at all exist. 'Tis in vain to reply, that any
determinate quantity of extension is an unite; but such-a-one as admits of
an infinite number of fractions, and is inexhaustible in its sub-divisions.
For by the same rule these twenty men may be
considered as a unite. The whole globe of the earth, nay the whole
universe may be considered as an unite. That
term of unity is merely a fictitious denomination, which the mind may apply
to any quantity of objects it collects together; nor can such an unity any
more exist alone than number can, as being in reality a true number. But the
unity, which can exist alone, and whose existence is necessary to that of
all number, is of another kind, and must be perfectly indivisible, and
incapable of being resolved into any lesser unity.

1. It has been objected to me, that infinite divisibility supposes only an infinite number of proportional not of aliqiot parts, and that an infinite number of proportional parts does not form an infinite extension. But this distinction is entirely frivolous. Whether these parts be called aliquot or proportional, they cannot be inferior to those minute parts we conceive; and therefore cannot form a less extension by their conjunction.

2. Mons. Malezieu.