to hinder its progress into this endless expansion; of that it can
neither find nor conceive any end. Nor let any one say, that beyond the
bounds of body, there is nothing at al ; unless he wil confine God
within the limits of matter. Solomon, whose understanding was filled
and enlarged with wisdom, seems to have other thoughts when he says,
'Heaven, and the heaven of heavens, cannot contain thee.' And he,
I think, very much magnifies to himself the capacity of his own
understanding, who persuades himself that he can extend his thoughts
further than God exists, or imagine any expansion where He is not.
3. Nor Duration by Motion.
Just so is it in duration. The mind having got the idea of any length of
duration, CAN double, multiply, and enlarge it, not only beyond its own,
but beyond the existence of all corporeal beings, and all the measures
of time, taken from the great bodies of al the world and their
motions. But yet every one easily admits, that, though we make duration
boundless, as certainly it is, we cannot yet extend it beyond all being.
God, every one easily allows, fil s eternity; and it is hard to find a
reason why any one should doubt that he likewise fil s immensity.
His infinite being is certainly as boundless one way as another; and
methinks it ascribes a little too much to matter to say, where there is
no body, there is nothing.
4. Why Men more easily admit infinite Duration than infinite Expansion.
Hence I think we may learn the reason why every one familiarly and
without the least hesitation speaks of and supposes Eternity, and sticks
not to ascribe INFINITY to DURATION; but it is with more doubting and
reserve that many admit or suppose the INFINITY OF SPACE. The reason
whereof seems to me to be this,--That duration and extension being used
as names of affections belonging to other beings, we easily conceive
in God infinite duration, and we cannot avoid doing so: but, not
attributing to him extension, but only to matter, which is finite, we
are apter to doubt of the existence of expansion without matter; of
which alone we commonly suppose it an attribute. And, therefore, when
men pursue their thoughts of space, they are apt to stop at the confines
of body: as if space were there at an end too, and reached no further.
Or if their ideas, upon consideration, carry them further, yet they term
what is beyond the limits of the universe, imaginary space: as if IT
were nothing, because there is no body existing in it. Whereas duration,
antecedent to all body, and to the motions which it is measured by, they
never term imaginary: because it is never supposed void of some other
real existence. And if the names of things may at al direct our
thoughts towards the original of men's ideas, (as I am apt to think they
may very much,) one may have occasion to think by the name DURATION,
that the continuation of existence, with a kind of resistance to any
destructive force, and the continuation of solidity (which is apt to be
confounded with, and if we will look into the minute anatomical parts of
matter, is little different from, hardness) were thought to have some
analogy, and gave occasion to words so near of kin as durare and durum
esse. And that durare is applied to the idea of hardness, as wel as
that of existence, we see in Horace, Epod. xvi. ferro duravit secula.
But, be that as it wil , this is certain, that whoever pursues his own
thoughts, will find them sometimes launch out beyond the extent of body,
into the infinity of space or expansion; the idea whereof is distinct
and separate from body and al other things: which may, (to those who
please,) be a subject of further meditation.
5. Time to Duration is as Place to Expansion.
Time in general is to duration as place to expansion. They are so much
of those boundless oceans of eternity and immensity as is set out and
distinguished from the rest, as it were by landmarks; and so are made
use of to denote the position of FINITE real beings, in respect one to
another, in those uniform infinite oceans of duration and space. These,
rightly considered, are only ideas of determinate distances from certain
known points, fixed in distinguishable sensible things, and supposed
to keep the same distance one from another. From such points fixed in
sensible beings we reckon, and from them we measure our portions of
those infinite quantities; which, so considered, are that which we call
TIME and PLACE. For duration and space being in themselves uniform and
boundless, the order and position of things, without such known settled
points, would be lost in them; and al things would lie jumbled in an
incurable confusion.
6. Time and Place are taken for so much of either as are set out by the
Existence and Motion of Bodies.
Time and place, taken thus for determinate distinguishable portions of
those infinite abysses of space and duration, set out or supposed to be
distinguished from the rest, by marks and known boundaries, have each of
them a twofold acceptation.
FIRST, Time in general is commonly taken for so much of infinite
duration as is measured by, and co-existent with, the existence and
motions of the great bodies of the universe, as far as we know anything
of them: and in this sense time begins and ends with the frame of this
sensible world, as in these phrases before mentioned, 'Before al time,'
or, 'When time shall be no more.' Place likewise is taken sometimes for
that portion of infinite space which is possessed by and comprehended
within the material world; and is thereby distinguished from the rest of
expansion; though this may be more properly cal ed extension than place.
Within these two are confined, and by the observable parts of them
are measured and determined, the particular time or duration, and the
particular extension and place, of all corporeal beings.
7. Sometimes for so much of either as we design by Measures taken from
the Bulk or Motion of Bodies.
SECONDLY, sometimes the word time is used in a larger sense, and is
applied to parts of that infinite duration, not that were really
distinguished and measured out by this real existence, and periodical
motions of bodies, that were appointed from the beginning to be for
signs and for seasons and for days and years, and are accordingly our
measures of time; but such other portions too of that infinite uniform
duration, which we upon any occasion do suppose equal to certain lengths
of measured time; and so consider them as bounded and determined. For,
if we should suppose the creation, or fall of the angels, was at the
beginning of the Julian period, we should speak properly enough, and
should be understood if we said, it is a longer time since the creation
of angels than the creation of the world, by 7640 years: whereby we
would mark out so much of that undistinguished duration as we suppose
equal to, and would have admitted, 7640 annual revolutions of the sun,
moving at the rate it now does. And thus likewise we sometimes speak of
place, distance, or bulk, in the great INANE, beyond the confines of the
world, when we consider so much of that space as is equal to, or capable
to receive, a body of any assigned dimensions, as a cubic foot; or do
suppose a point in it, at such a certain distance from any part of the
universe.
8. They belong to all finite beings.
WHERE and WHEN are questions belonging to al finite existences, and are
by us always reckoned from some known parts of this sensible world, and
from some certain epochs marked out to us by the motions observable in
it. Without some such fixed parts or periods, the order of things would
be lost, to our finite understandings, in the boundless invariable
oceans of duration and expansion, which comprehend in them al finite
beings, and in their full extent belong only to the Deity. And therefore
we are not to wonder that we comprehend them not, and do so often find
our thoughts at a loss, when we would consider them, either abstractly
in themselves, or as any way attributed to the first incomprehensible
Being. But when applied to any particular finite beings, the extension
of any body is so much of that infinite space as the bulk of the body
takes up. And place is the position of any body, when considered at a
certain distance from some other. As the idea of the particular duration
of anything is, an idea of that portion of infinite duration which
passes during the existence of that thing; so the time when the thing
existed is, the idea of that space of duration which passed between some
known and fixed period of duration, and the being of that thing. One
shows the distance of the extremities of the bulk or existence of the
same thing, as that it is a foot square, or lasted two years; the other
shows the distance of it in place, or existence from other fixed points
of space or duration, as that it was in the middle of Lincoln's Inn
Fields, or the first degree of Taurus, and in the year of our Lord 1671,
or the 1000th year of the Julian period. Al which distances we measure
by preconceived ideas of certain lengths of space and duration,--as
inches, feet, miles, and degrees, and in the other, minutes, days, and
years, &c.
9. All the Parts of Extension are Extension, and all the Parts of
Duration are Duration.
There is one thing more wherein space and duration have a great
conformity, and that is, though they are justly reckoned amongst our
SIMPLE IDEAS, yet none of the distinct ideas we have of either is
without all manner of composition: it is the very nature of both of them
to consist of parts: but their parts being al of the same kind, and
without the mixture of any other idea, hinder them not from having a
place amongst simple ideas. Could the mind, as in number, come to so
small a part of extension or duration as excluded divisibility, THAT
would be, as it were, the indivisible unit or idea; by repetition of
which, it would make its more enlarged ideas of extension and duration.
But, since the mind is not able to frame an idea of ANY space without
parts, instead thereof it makes use of the common measures, which, by
familiar use in each country, have imprinted themselves on the memory
(as inches and feet; or cubits and parasangs; and so seconds, minutes,
hours, days, and years in duration);--the mind makes use, I say, of
such ideas as these, as simple ones: and these are the component parts
of larger ideas, which the mind upon occasion makes by the addition of
such known lengths which it is acquainted with. On the other side, the
ordinary smallest measure we have of either is looked on as an unit in
number, when the mind by division would reduce them into less fractions.
Though on both sides, both in addition and division, either of space or
duration, when the idea under consideration becomes very big or very
small, its precise bulk becomes very obscure and confused; and it is the
NUMBER of its repeated additions or divisions that alone remains clear
and distinct; as wil easily appear to any one who will let his thoughts
loose in the vast expansion of space, or divisibility of matter. Every
part of duration is duration too; and every part of extension is
extension, both of them capable of addition or division in infinitum.
But THE LEAST PORTIONS OF EITHER OF THEM, WHEREOF WE HAVE CLEAR AND
DISTINCT IDEAS, may perhaps be fittest to be considered by us, as the
simple ideas of that kind out of which our complex modes of space,
extension, and duration are made up, and into which they can again be
distinctly resolved. Such a small part in duration may be called a
MOMENT, and is the time of one idea in our minds, in the train of their
ordinary succession there. The other, wanting a proper name, I know not
whether I may be al owed to cal a SENSIBLE POINT, meaning thereby the
least particle of matter or space we can discern, which is ordinarily
about a minute, and to the sharpest eyes seldom less than thirty seconds
of a circle, whereof the eye is the centre.
10. Their Parts inseparable.
Expansion and duration have this further agreement, that, though they
are both considered by us as having parts, yet their parts are not
separable one from another, no not even in thought: though the parts
of bodies from whence we take our MEASURE of the one; and the parts of
motion, or rather the succession of ideas in our minds, from whence we
take the MEASURE of the other, may be interrupted and separated; as the
one is often by rest, and the other is by sleep, which we call rest too.
11. Duration is as a Line, Expansion as a Solid.
But there is this manifest difference between them,--That the ideas
of length which we have of expansion are turned every way, and so make
figure, and breadth, and thickness; but duration is but as it were the
length of one straight line, extended in infinitum, not capable of
multiplicity, variation, or figure; but is one common measure of al
existence whatsoever, wherein al things, whilst they exist, equally
partake. For this present moment is common to all things that are now in
being, and equally comprehends that part of their existence, as much as
if they were al but one single being; and we may truly say, they al
exist in the SAME moment of time. Whether angels and spirits have any
analogy to this, in respect to expansion, is beyond my comprehension:
and perhaps for us, who have understandings and comprehensions suited
to our own preservation, and the ends of our own being, but not to the
reality and extent of all other beings, it is near as hard to conceive
any existence, or to have an idea of any real being, with a perfect
negation of all manner of expansion, as it is to have the idea of any
real existence with a perfect negation of al manner of duration. And
therefore, what spirits have to do with space, or how they communicate
in it, we know not. Al that we know is, that bodies do each singly
possess its proper portion of it, according to the extent of solid
parts; and thereby exclude al other bodies from having any share in
that particular portion of space, whilst it remains there.
12. Duration has never two Parts together, Expansion altogether.
DURATION, and TIME which is a part of it, is the idea we have of
PERISHING distance, of which no two parts exist together, but follow
each other in succession; an EXPANSION is the idea of LASTING distance,
al whose parts exist together and are not capable of succession. And
therefore, though we cannot conceive any duration without succession,
nor can put it together in our thoughts that any being does NOW exist
to-morrow, or possess at once more than the present moment of duration;
yet we can conceive the eternal duration of the Almighty far different
from that of man, or any other finite being. Because man comprehends not
in his knowledge or power al past and future things: his thoughts are
but of yesterday, and he knows not what to-morrow wil bring forth. What
is once past he can never recal; and what is yet to come he cannot make
present. What I say of man, I say of al finite beings; who, though they
may far exceed man in knowledge and power, yet are no more than the
meanest creature, in comparison with God himself. Finite or any
magnitude holds not any proportion to infinite. God's infinite duration,
being accompanied with infinite knowledge and infinite power, he sees
al things, past and to come; and they are no more distant from his
knowledge, no further removed from his sight, than the present: they al
lie under the same view: and there is nothing which he cannot make exist
each moment he pleases. For the existence of all things, depending upon
his good pleasure, all things exist every moment that he thinks fit to
have them exist. To conclude: expansion and duration do mutual y embrace
and comprehend each other; every part of space being in every part of
duration, and every part of duration in every part of expansion. Such a
combination of two distinct ideas is, I suppose, scarce to be found in
al that great variety we do or can conceive, and may afford matter to
further speculation.
CHAPTER XVI.
IDEA OF NUMBER.
1. Number the simplest and most universal Idea.
Amongst al 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 al
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 90 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 al 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, wil 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 wel 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. 857324
Octillions. 162486
Septillions. 345896
Sextillions. 437918
Quintrillions. 423147
Quartrillions. 248106
Trillions. 235421
Billions. 261734
Millions. 368149
Units. 623137
The ordinary way of naming this number in English, will be the often
repeating of millions, of mil ions, of millions, of millions, of
millions, of millions, of mil ions, of millions, (which is the
denomination of the second six figures). In which way, it wil 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 col ect 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, til a good while
after they are well furnished with good store of other ideas: and one
may often observe them discourse and reason pretty wel , and have very