1. * Number the simplest and most universal idea.*
Amongst all the
ideas we have, as there is none suggested to the mind by more ways, so
there is none more simple, than that of unity, or one: it has no
shadow of variety or composition in it: every object our senses are
employed about; every idea in our understandings; every thought of our
minds, brings this idea along with it. And therefore it is the most
intimate to our thoughts, as well as it is, in its agreement to all
other things, the most universal idea we have. For number applies
itself to men, angels, actions, thoughts; everything that either
doth exist, or can be imagined.

2. * Its modes made by addition.*
By repeating this idea in our
minds, and adding the repetitions together, we come by the complex
ideas of the modes of it. Thus, by adding one to one, we have the
complex idea of a couple; by putting twelve units together, we have
the complex idea of a dozen; and so of a score, or a million, or any
other number.

3. * Each mode distinct.*
The simple modes of number are of all other
the most distinct; every the least variation, which is an unit, making
each combination as clearly different from that which approacheth
nearest to it, as the most remote; two being as distinct from one,
as two hundred; and the idea of two as distinct from the idea of
three, as the magnitude of the whole earth is from that of a mite.
This is not so in other simple modes, in which it is not so easy,
nor perhaps possible for us to distinguish betwixt two approaching
ideas, which yet are really different. For who will undertake to
find a difference between the white of this paper and that of the next
degree to it: or can form distinct ideas of every the least excess
in extension?

4. * Therefore demonstrations in numbers the most precise.*
The
clearness and distinctness of each mode of number from all others,
even those that approach nearest, makes me apt to think that
demonstrations in numbers, if they are not more evident and exact than
in extension, yet they are more general in their use, and more
determinate in their application. Because the ideas of numbers are
more precise and distinguishable than in extension; where every
equality and excess are not so easy to be observed or measured;
because our thoughts cannot in space arrive at any determined
smallness beyond which it cannot go, as an unit; and therefore the
quantity or proportion of any the least excess cannot be discovered;
which is clear otherwise in number, where, as has been said, 91 is
as distinguishable from go as from 9000, though 91 be the next
immediate excess to 90. But it is not so in extension, where,
whatsoever is more than just a foot or an inch, is not distinguishable
from the standard of a foot or an inch; and in lines which appear of
an equal length, one may be longer than the other by innumerable
parts: nor can any one assign an angle, which shall be the next
biggest to a right one.

5. * Names necessary to numbers.*
By the repeating, as has been said,
the idea of an unit, and joining it to another unit, we make thereof
one collective idea, marked by the name two. And whosoever can do
this, and proceed on, still adding one more to the last collective
idea which he had of any number, and gave a name to it, may count,
or have ideas, for several collections of units, distinguished one
from another, as far as he hath a series of names for following
numbers, and a memory to retain that series, with their several names:
all numeration being but still the adding of one unit more, and giving
to the whole together, as comprehended in one idea, a new or
distinct name or sign, whereby to know it from those before and after,
and distinguish it from every smaller or greater multitude of units.
So that he that can add one to one, and so to two, and so go on with
his tale, taking still with him the distinct names belonging to
every progression; and so again, by subtracting an unit from each
collection, retreat and lessen them, is capable of all the ideas of
numbers within the compass of his language, or for which he hath
names, though not perhaps of more. For, the several simple modes of
numbers being in our minds but so many combinations of units, which
have no variety, nor are capable of any other difference but more or
less, names or marks for each distinct combination seem more necessary
than in any other sort of ideas. For, without such names or marks,
we can hardly well make use of numbers in reckoning, especially
where the combination is made up of any great multitude of units;
which put together, without a name or mark to distinguish that precise
collection, will hardly be kept from being a heap in confusion.

6. * Another reason for the necessity of names to numbers.*
This I
think to be the reason why some Americans I have spoken with, (who
were otherwise of quick and rational parts enough,) could not, as we
do, by any means count to 1000; nor had any distinct idea of that
number, though they could reckon very well to 20. Because their
language being scanty, and accommodated only to the few necessaries of
a needy, simple life, unacquainted either with trade or mathematics,
had no words in it to stand for 1000; so that when they were
discoursed with of those greater numbers, they would show the hairs of
their head, to express a great multitude, which they could not number;
which inability, I suppose, proceeded from their want of names. The
Tououpinambos had no names for numbers above 5; any number beyond that
they made out by showing their fingers, and the fingers of others
who were present. And I doubt not but we ourselves might distinctly
number in words a great deal further than we usually do, would we find
out but some fit denominations to signify them by; whereas, in the way
we take now to name them, by millions of millions of millions, &c., it
is hard to go beyond eighteen, or at most, four and twenty, decimal
progressions, without confusion. But to show how much distinct names
conduce to our well reckoning, or having useful ideas of numbers,
let us see all these following figures in one continued line, as the
marks of one number: v. g.

Nonillions Octillions Septillions Sextillions Quintrillions

857324 162486 345896 437918 423147

Quartrillions Trillions Billions Millions Units

248106 235421 261734 368149 623137

The ordinary way of naming this number in English, will be the often repeating of millions, of millions, of millions, of millions, of millions, of millions, of millions, of millions, (which is the denomination of the second six figures). In which way, it will be very hard to have any distinguishing notions of this number. But whether, by giving every six figures a new and orderly denomination, these, and perhaps a great many more figures in progression, might not easily be counted distinctly, and ideas of them both got more easily to ourselves, and more plainly signified to others, I leave it to be considered. This I mention only to show how necessary distinct names are to numbering, without pretending to introduce new ones of my invention.

7. * Why children number not earlier.*
Thus children, either for want
of names to mark the several progressions of numbers, or not having
yet the faculty to collect scattered ideas into complex ones, and
range them in a regular order, and so retain them in their memories,
as is necessary to reckoning, do not begin to number very early, nor
proceed in it very far or steadily, till a good while after they are
well furnished with good store of other ideas: and one may often
observe them discourse and reason pretty well, and have very clear
conceptions of several other things, before they can tell twenty.
And some, through the default of their memories, who cannot retain the
several combinations of numbers, with their names, annexed in their
distinct orders, and the dependence of so long a train of numeral
progressions, and their relation one to another, are not able all
their lifetime to reckon, or regularly go over any moderate series
of numbers. For he that will count twenty, or have any idea of that
number, must know that nineteen went before, with the distinct name or
sign of every one of them, as they stand marked in their order; for
wherever this fails, a gap is made, the chain breaks, and the progress
in numbering can go no further. So that to reckon right, it is
required, (1) That the mind distinguish carefully two ideas, which are
different one from another only by the addition or subtraction of
one unit: (2) That it retain in memory the names or marks of the
several combinations, from an unit to that number; and that not
confusedly, and at random, but in that exact order that the numbers
follow one another. In either of which, if it trips, the whole
business of numbering will be disturbed, and there will remain only
the confused idea of multitude, but the ideas necessary to distinct
numeration will not be attained to.

8. * Number measures all measureables.*
This further is observable in
number, that it is that which the mind makes use of in measuring all
things that by us are measurable, which principally are expansion
and duration; and our idea of infinity, even when applied to those,
seems to be nothing but the infinity of number. For what else are
our ideas of Eternity and Immensity, but the repeated additions of
certain ideas of imagined parts of duration and expansion, with the
infinity of number; in which we can come to no end of addition? For
such an inexhaustible stock, number (of all other our ideas) most
clearly furnishes us with, as is obvious to every one. For let a man
collect into one sum as great a number as he pleases, this
multitude, how great soever, lessens not one jot the power of adding
to it, or brings him any nearer the end of the inexhaustible stock
of number; where still there remains as much to be added, as if none
were taken out. And this endless addition or addibility (if any one
like the word better) of numbers, so apparent to the mind, is that,
I think, which gives us the clearest and most distinct idea of
infinity: of which more in the following chapter.