at these sounds, or their signs,
were the substitutes or intended representatives of the objects in
nature, either individually or collectively; for he would find that men,
by the instrument of speech, had contrived, by a term, equally to
express collections as well as individuals; as a man, or an army, which
latter might consist of many thousands of the same beings. When he had
arrived at this knowledge, he would be persuaded of the importance of
these terms, and feel the necessity of their precise and uniform
signification, as the representatives of the particular objects or
collections they professed to describe:--because, if different
significations were affixed to the same term, those who employed it
could not mean the same thing. These prefatory observations appear to be
proper, and it is important that the reader should bear them in mind;
but it will be evident that the most correct description of objects does
not constitute the process of reasoning, however indispensable it may be
as its foundation.

Reason, as the term itself shows, implies _ratio_, estimate, proportion,
or admeasurement; and in all the instances of reasoning that can be
adduced, this interpretation will apply in the strictest sense. But
_ratio_, estimate, &c. involve numbers, by which they can alone be
characterised or defined. Thus, by way of illustration, the estimate for
a building implies the number of the different materials, with their
_cost_, which is the number of pounds, shillings, and pence; also the
number of requisite workmen to be employed for such time, or number of
weeks, days, &c. at a certain stipend: admeasurement also consists of
numbers, whether it be employed on solids, fluids, or designate the
succession of our perceptions, called time[17]: and ratio or proportion
is equally the creature of numbers. In a preceding part of these
contributions, the importance of numbers has been considered, and a
confident belief expressed that no animal is capable of numeration; and
that the comprehension of addition and subtraction, the basis of all
calculation is exclusively the province of the human intellect. This
subject, however, requires a more extended investigation; and the
research would doubtless reward the toil of the inquirer.

It is generally acknowledged, that arithmetic, or the combination and
separation of numbers, is the purest and most certain system of
reasoning, and liable, when properly conducted, to no difference of