SIR ISAAC NEWTON'S DE GRAVITATIONE ET AEQUIPONDIO FLUIDORUM
translated by
W. B. Allen*
It is agreed
that there are two methods to teach the science of gravitation and of the balance
of fluids and solids in fluids. However far it belongs to the science of mathematics,
it is plain that it is being exceedingly detached from physical considerations.
And here, accordingly, by the singular reason of its propositions from detached
[abstracted] principles, and they are sufficient to the attentive, by
geometrical manners, I conclude to demonstrate. Then with these doctrines from
natural philosophy somehow is the relationship to be estimated, to which extent
for the most part the explaining its phenomena in detail is to be adapted, and
thus when its use thereafter would be particularly manifest and the certainty
of the principles is probably confirmed, I will not regard it a burden yet to
illustrate with propositions from abundant experiments: so that, notwithstanding
this more open kind of judging being set forth in scholia, is not to be
confused with the former that I teach through lemmas, propositions, and
corollaries.
The
fundamentals from which this science is demonstrated are either called
definitions somehow; or axioms and postulates (that) are by no one not conceded.
And these I treat forthwith.
Definitions
The names,
quantity, duration. and space, are known for as much as they are able to be
defined by other names.
Def. 1: Place is part of space which things fill evenly.
Def. 2: Body
is that which fills place.
Def.3: Rest is remaining in the same place.
Def. 4: Motion is
change of place.
NOTE: I have said that body fills place; that is
that it so fills it that another thing of the same kind or some other body is
completely excluded, forasmuch as it is impenetrable. Let it be, however, that
place is called the part of space in which things inhere evenly, but so far as
here bodies and not penetrable things are being examined, I have chosen to
define (that to be) a part of space which something fills.
Moreover,
with the body being examined here to be set forth not insofar as it is provided
with the physical substance of sensible qualities but only insofar as it is
some moveable and impenetrable extension [magnitude]; accordingly, it has not
been defined by a philosophical manner, but I detached [abstracted] the
sensible qualities (which also the philosophers ought to abstract, unless I err
and claim to assign being excited [moved] only to modes of thinking different
from the motions of bodies); I have posited only as many of the properties as
are required for local motion. Thus far, as if in the place of physical bodies,
you can understand the abstract forms in the same manner they are contemplated
by geometers when they assign motion to them, as that done in Propositions 4
& 8, Book I of Euclid’s Elements. And in demonstrating the tenth of
the definitions in Book XI, this ought to be done; since indeed it would be
barely counted among the definitions and rather ought to be demonstrated among
the propositions,
unless perhaps it be
taken for an axiom.
I have earlier
defined motion to be change of place, because those motions, passage,
transference, migration, etc., are seen to be synonymous names. But if it be
preferred, let motion be passage or transference of a body from place to place.
For the rest in these
definitions, when I would have supposed space the distinction from a body is
given, and motion relative to a part of space therein, not however relative to
the position of a contiguous body have I determined it [lest this be assumed as
gratuitously against Cartesians, I shall endeavor to I do away with his
fictions].
I can explain his
doctrine in the following three propositions: 1. That for each body only
one proper motion fits from the truth of things (Article 28, 31, & 32; Part
2, Principiae Philosopher), in which transference is defined to be of
the one part of matter or one body from the vicinity of those bodies which
immediately are co-tangent with it, and inasmuch as it is seen to be at rest, to
the vicinity of others. (Article 25, Part 2 & Article 28, Part 3; Principiae) 2. That through a body’s own motion,
according to this definition, transference not only will be understood of any
particle of matter, or a body formed of parts being at rest among themselves,
but that whole which is simultaneously transferred; albeit reciprocally this
itself is able to consist with many parts which themselves may have another
motion. (Art 25, Part 2, Prin.)
3. That in addition to this motion proper to each body, there can yet be
motion actually in it through participating in innumerable others (or to the extent
that it is a part of other bodies having a different motion): appertaining
(Art. 31, Part 2, Prin.): which nevertheless are not motions in the
philosophical sense both when speaking rationally (Art. 29, Part 3) and
according to the truth of things (Art. 25, Part 2 and Art. 28, Part 3), but
only improperly and according to the sense of the vulgar. (Art. 24, 25, 28,
& 31; Part 2 and Art. 29. Part 3.) That kind he seems to describe (Art. 24,
Part 2 & Art. 28, Part 3) is the action by which some body may migrate from
one place to another.
And just as motion
consists of two kinds, obviously the proper and the derived, so two kinds of
place are assigned from which this motion may be accomplished, which are the
surfaces of the immediately surrounding bodies (Art. 15, Part 2), and the
situation within other bodies wherever (Art. 13, Part 2 & Art. 29, Part 3).
Now, just how
confused and discordant with reason this doctrine is, not only the absurd
consequences convince, but also Descartes himself, by contradicting himself,
seems to acknowledge. He has said, for instance, (that) the earth and the rest
of the planets properly, and speaking according to the philosophical sense, are
not moved, and without reason and by speaking only vulgarly, he has said that
it is itself moved on account of being transferred relative to the fixed stars
(Art. 26, 27, 28, 29; Part 3). But afterwards, nevertheless, he posits in the
earth and the planets an endeavor of receding from the sun as if from the
center of a circle that is moved, by which through a like endeavor in the
whirling vortices they are balanced in themselves at a distance from the sun.
(Art. 140, Part 3). What then? either this endeavor of the planets according to
Descartes is truly and philosophically (derived) from rest, or is it rather
derived from vulgar and non-philosophic motion? But Descartes seeks further
that the comet endeavors less to recede from the sun when it is advanced into
the first vortex and the position among the fixed stars entirely remaining, it
does not yet obey the pull of the vortices but is transferred relative to it
from the vicinity of the contiguous ether, and thus far, philosophically
speaking would whirl around the sun; as much as afterwards the comet together
with the matter of the vortex has torn away and does so as, according to the
philosophical nature of things. And when the whirling of the comets around the
sun in his philosophical sense should not effect [produce] an endeavor of receding
from the center, which whirling in the vulgar sense can effect [produce],
surely he ought to acknowledge motion in the vulgar more than the philosophical
sense.
Secondly, he is seen
to contradict himself when he posits a unique motion to be suitable for each
body according to the truth of things, and yet he determines to consider that
motion from our imagination, defining those being transferred from the vicinity
of bodies to be not those which are at rest but those which, as if being at
rest, are only looked upon as (if) accidentally
moving, just as, in Article 29 & 30, Part 2, is
explained more broadly. And from this he reckoned himself to be able to avoid
the difficulties around the mutual transference of bodies, whereby evidently
one more than another is said to be at rest when (its) position between the
banks would not be changed. (Art. 15, Part 2). But just as be manifest,
supposing that the matter of vortices is only seen by man, if you will, as being
at rest and (then) the earth, philosophically speaking, is simultaneously at
rest; supposing also that anyone else at the same time looks upon the same
matter of vortices as (if) moving circularly and (then) the earth
philosophically speaking is not at rest. In the same way a ship on a sea
simultaneously will be moved and not moved. And that I am not assuming motion
in the wider sense of the vulgar, and by which there are innumerable motions of
these bodies, but in his philosophical sense by which he has said there is only
one (motion) in body in any place, and it is proper to itself and suitable from
the nature of things (not our imaginations).
Thirdly, he is seen
by no means to agree with himself when he posits a single motion to be suitable
to every body according to the nature of things, and nevertheless (Art. 31,
Part 2) innumerable motions to appertain to each single body actually. For
motions which actually belong to some body are actually natural motions, and
thus motions in the philosophical sense and according to the truth of things,
although he insisted (that) it was motion in the vulgar sense alone. Add that,
when any whole is moved, all the parts of which once transferred will actually
be established as resting, unless it be granted as truly to be moved by
participating in the motion of the whole, and hence it has innumerable motions
according to the truth of things.
But we see,
moreover, from the consequences how absurd this Cartesian doctrine is. And from
the first just as he insisted bitterly (that) the earth is not moved because it
is not transferred from the vicinity of the contiguous ether; just so will it
follow from the same principles that the internal particles of two bodies, when
they are not transferred from the immediate vicinity of the co-tangent
particles, have no motion properly (so) called, but are moved inasmuch as
participating in the motion of the external particles: on the contrary, in
respect of which external (particles) the inner portions would not be moved by
a proper motion, because they would not be transferred from the vicinity of the
internal parts: and thus in respect of which only the external surfaces of each
body are moved by a proper motion, and that the whole internal substance, that
is, the whole body is moved through participating in the motion of the external
surface. He erred, therefore, in defining motion fundamentally, as he assigned
it to bodies what is only suited to surfaces, and he did it so that nothing is
able to be the proper motion of any body whatever.
Secondly. What
if we look only at Article 25, Part 2?
Every body will have not only one but innumerable motions proper to
itself, whenever properly and according to the truth of things that of which
the whole is properly moved will be said to be moved. And because he defines
that motion by means of this body, he understands all that which simultaneously
is transferred, and yet himself agrees that another motion is possible from the
parts possessing (it) among themselves; suppose the vortex at one with all of
the planets or a floating ship at one with the whole sea of which they are part
[to which they appertain], or a man in a ship at one with the things which are
carried about with himself, or the revolving metal of a watch at one with the
particles of which they are made. Now, unless you say that he does not posit
the motion of the whole aggregation, motion properly and according to the truth
of things suiting the parts, he will be acknowledging that all these motions of
the clock’s revolutions, the men, the ship, and the vortex actually and
philosophically speaking will appertain to the revolving particles.
And whether
from these consequences it is besides manifest that nothing absolutely true and
proper can be said about the motions on account of anything else, but that all,
whether with respect to the contiguous bodies or detached bodies, are the same
philosophically, than which we are able to imagine nothing more absurd. Unless
it be granted (that) one physical motion is given of every body, and further
(that) changing relation and position among other bodies, they are only being
designated externally: it will follow that the earth, for example, endeavors to
recede from the center of the sun on account of the motion relative to the
fixed stars, and endeavors the less to recede on account of the lesser motion
relative to Saturn and the ethereal orbit in which it is carried, as well as
additionally less relative to Jupiter and the surrounding ether of which its
orbit is made up, and again less relative to Mars and its ethereal orbit, and
much less relative to the other orbits of ethereal matter which, carrying no
planet with them, are properly the annual orbits of earth; truly, relative to
its own orbit it does not endeavor, since within it is not moved. All which
endeavors and non-endeavors, when they can not absolutely be suitable, it is
rather said that one only absolute and natural motion of earth is suited, the
example of which is to endeavor to recede from the sun, and that its
transferences relative to external bodies are only being designated as
external.
Thirdly. It
will follow from Cartesian doctrine (that) motion can be generated where no
force is being impressed. For example, if God should so effect that the
whirling of our vortex would suddenly stand still, no forces being impressed on
earth which would simultaneously stand (it) still: Descartes would say that the
earth would then be moved in a philosophical sense on account of being
transferred from the vicinity of the co-tangent fluid, just as before he had
said it to be at rest in the same philosophical sense.
Fourthly. From
the same doctrine it will follow also that God himself cannot generate motion
in others though he should press the maximum force. For example, if God should
ever press the maximum force so that he might cause the heavenly star at one
with the most distant part of creation to revolve around the earth (namely, a
diurnal motion); nevertheless, according to Descartes, from this not the heaven
but the earth alone would actually be said to be moved (Art. 38, Part 3.) Just
as it had been similarly whether he had brought it about that the heaven had
been reversed from east to west by means of inborn force, or if by a small
force the earth might be reversed in opposing parts. But will anyone judge that
a part of the earth is drawn to recede from its center on account of a force in
heaven alone being impressed in some manner? Or, is it not more rational, to
agree that a force imparted from heaven so produced that to endeavor to recede
from the center of the whirling is caused thereby, and by that alone is it
properly and absolutely moved; and the force of earth being impressed, that
made its parts endeavor to recede from the center of the whirling, is caused
thereby, and by that alone is properly and absolutely moved: Even if it is
similar in either case to the transference of bodies among themselves.
Therefore both physical and absolute motion is being designed by something
other than by this transference, for dwelling on this transference is only for
designating externals.
Fifthly. It is
seen as alien to reason that bodies, apart from physical motion, distances, and
positions would change among themselves: But Descartes agrees that the earth
and the rest of the planets and fixed stars are properly speaking at rest, and
nevertheless change positions among themselves.
Sixthly. And,
on the contrary, it is seen to be no less alien to reason, that many bodies
preserve among themselves the same positions with respect to which another is
physically moved, while the others are at rest. But if God were to stop some
planet and make it so that the same position among the fixed stars would be
preserved continually, Descartes would not say that with respect to the stars
not being moved, the planets then would be moved on account of being
transferred from the matter of the vortex.
Seventhly. I ask by
what reason any body is said properly to be moved when other bodies from whose
vicinity it is transferred are not seen as being at rest, or rather when they
cannot be seen as being at rest. For example, in the same manner, our vortex,
by reason of the transference of matter following the circumference, can be
said to be moved entirely from the vicinity of the other matter surrounding the
vortex by revolving, if indeed the matter of the surrounding vortex might not
be able to be seen as being at rest, and that not only relative to our vortex,
but also insofar as these vortices are not at rest among themselves. Now, if
this philosopher assigns being transferred not to the numerable bodily
particles of the vortex, but to the generic space (as he says himself) in which
these vortices exist, we agree finally, for he would recognize that motion
ought to be assigned to space insofar as it is distinguished from body.
Lastly, that the
absurdity of his position be exceedingly manifest, I say that from it it will
follow (that) nothing is determining the velocity of another moved thing and
nothing is defining the line along which it is moved: And much more, that the
velocity of a body in motion without impediment can not be said to be uniform,
and neither (can there) be a straight line in which it will be carried out. On
the contrary, that nothing is able to be moved if, indeed, nothing is able to
be so without velocity and determination (destination].
But that these
things may be manifest, the impression having been laid out is that, after any
motion having been accomplished, no one can assign the place according to
Descartes at which the body was in the beginning of the accomplished motion, or
rather he has not said whence it is possible a body will be moved. And the
reason is that according to Descartes it is not possible to define and assign
the place except from the position of the surrounding bodies, and that after
any motion having been accomplished the position of the surrounding bodies remains
no more the same as it was before. For example, if the place of the planet
Jupiter were where (it was) the year before, then having been accomplished it
would be at rest; by what reasoning, I pray, will the philosopher, Descartes,
describe it? Not by means of the positions of particles of the fluid matter, if
indeed he would have the positions which those particles had the year before,
for they will have changed exceedingly. And neither will he describe (it) by
means of the positions of the sun and the fixed
stars, because of the unequal flux of the subtle matter around the poles of the
vortex in the central stars, (Art. 104, Part 3), the swaying, inflation, and absorption
of the vortex, and other truer causes, such as the whirling properly around the
center of the sun and the stars, the generation of spots, and the trajectory of
comets through the heavens, sufficiently changing both the magnitudes and
positions of the stars so that by straying they would hardly suffice for the
place sought without designating some milestone, and much less that by aid of
these it might be possible to determine and thus have described the place
accurately, in the same manner as would be postulated for describing the geometry.
There are not bodies situated similarly in the world, whose position for length
of time are not changed among themselves, and much less those which are not
moved in a Cartesian sense, that is, either inasmuch as they are parts of other
such transferred bodies. And hence nothing fundamental is given by which the
place which was at an earlier time should be able to be designated now in the
present, or whence we are able to say such a place has now been more fully situated
in the nature of things. Now, when according to Descartes place would be
nothing other than the surfaces of surrounding bodies, or the position, as you
please, among other more distant bodies: it is impossible in the nature of
things that by this doctrine there might exist a longer time than these
positions of bodies remain the same from
which one would assume individual names [denominations]. And hence, about the
place of Jupiter, which it kept the year before, and with equal reason, about
the prior place of a moving body anywhere, according to the doctrine of
Descartes, it is manifest that not even God himself (standing newly established
with things) could accurately and in a geometrical sense describe (it),
especially when, on account of the changed positions of bodies, it would no
longer exist in the nature of things.
Now, according as any
motion having been completed, when the place in which it will have begun—that
is, the beginning of the spatial trajectory—cannot be assigned (as) no longer
being, nothing keeping the beginning of this spatial trajectory, there cannot
be length; hence, when velocity derives from the length of space traversed in
the time given, it will follow that there cannot be any velocity of something
moving, just as I wished at first to point out. Moreover, what could be said
about the beginning of the space traversed ought similarly to be understood
about all intermediate places; and thus, when space has neither a beginning nor
intermediate parts, it will follow (that) there would not be any space
traversed and hence not any motion being determined, the which I wished
secondly to indicate. In fact, on the contrary, it will follow (that) Cartesian
motion is not motion, since nothing of it is velocity, nothing determinate, and
no space anywhere, no distance will be traced. It is accordingly necessary that
the determination of places and thus of local motions is represented in some
unmoved being of which sort space or extension alone is as that which is seen
(as) distinct from body. And this the Cartesian philosopher would more
willingly recognize if he would heed the manner by which Descartes himself had
an idea of his extension as distinct from body, which he wished to discriminate
from bodily extension [magnitude] by naming it generically. (Article 10, 12,
& 18; Part 2). And what he had deduced from the whirling of the vortex, by
which the force of the ether was receding from the center, and thus the whole
of his mechanical, philosophy, is silently represented by this generic extension.
For the rest
when Descartes in Articles 4 & 11, Part 2, Principiae, is seen to
have demonstrated that body nowise differs from extension, (he is) abstracting
no doubt from hardness, color, gravity, cold, heat and the remaining qualities
of which body can do without, as if the one thing only remain of it, extension
in length, breadth, and depth, which hence alone belongs to its essence. And
when this is held along with the many things demonstrated before, and it is
alone that to be supposed a cause on account of which faith can be restricted
to this opinion; and by that not any doubt about the nature of motion would
remain; I will respond to this argument by saying what extension would be, what
body, and the manner by which they differ by turns. When, for example, by the
distinction of substances into thinking and extended, or rather into being
thought and being extended, would be the special foundation of Cartesian
philosophy, or what he contends is a notion of mathematical demonstration, its
overturning from the side of extension, as if it were built upon the truer
foundations of mechanical sciences, I would suppose a small thing by no means.
About extension,
then, it is probably expected that it is being defined either as substance or
accidents or nothing at all. But by no means nothing, surely, therefore it has
some mode of existence proper to itself, by of which it fits neither to
substance nor to accident. It is not substance, then, because not absolute in
itself, but only the productive [emanative] effect of God, and some affect must
subsist in every being; because, then, it would not subsist through the modes
of its proper affections which are being denominated substance, that is,
through actions inasmuch as they are thinking in the mind and motions in the
body. For, although philosophers do not define substance to be the being that
is able somehow to act, nevertheless all silently understand this about
substance, in the same manner that, for that reason, it is manifest they easily
agree (that) extension is substance in the form of body if it can be moved in
the manner and bears
the fruit of the actions of bodies. And, on the other hand, they by no means
agree [concede] (that) body is substance if it can neither be moved nor arouse
sensations or any perceptions somewhere in the mind. Moreover, since we can
clearly conceive extension alone without existing in some subject, as when any
extra-mundane spaces or places are imagined empty of bodies; and we believe
(it) to exist anywhere we imagine (that) there is no body, and we cannot believe
(that) it will perish along with body if God would someway destroy some one, it
will follow that this does not exist by means of the mode of accident inhering
in some subject. And thus it is no accident. And much less could it be called
nothing, since what is something more than accident then approaches to the
nature of substance. No idea is given of nothing, and neither are there any
properties, but we have the clearest idea of all of extension by abstracting, no
doubt, the affections and properties of body so that uniformity of space in
length, breadth, and depth, and unlimited by division, alone would remain. And
moreover there are several of its properties concomitant with this idea, which
I shall now enumerate not only as if laying out something (that) is but
simultaneously what should be.
l. Space can be
distinguished in every way into the parts the common limits of which we are
accustomed to call surfaces; and these surfaces can be distinguished in every
way into parts, the common limits of
which we name lines; and in turn these lines can be distinguished in every way
into the parts which we call points. Now [And] these surfaces have no depth,
nor lines breadth, nor the point any kind of dimension; unless you would say
that spaces are self-limiting mutually to the constant depth of piercing in the
surface they penetrate, as being what I have said is the limit of each without
common extremity: and thus, too, for lines and points. Further, spaces are
everywhere contiguous to spaces, and extension posited everywhere by the side
of extension, and thus of co-tangent parts everywhere there are common limits;
that is, from this, then, surfaces everywhere being divided by solids, and
everywhere lines in which the parts of surfaces are themselves co-tangent, and
everywhere points in which the parts of lines are continuously connected. And
from this there are all kinds of figures everywhere, spheres everywhere,
circles everywhere, ellipses, parabolas, and all the rest, as also every shape
and magnitude, although not delineated to sight. For the delineation of some
figure is not materially of that figure while space (is) newly produced, but is
only its bodily representation so that would now appear to sense which before was
insensible in space. Thus, for example, we believe all space in that way to be
spherical through which any sphere ever traverses progressively moving in a
single moment, albeit of this very sphere there no longer remain any sensible
vestiges. Indeed, we believe the space were spherical earlier than the sphere
had occupied (it), such that it could itself hold it [the sphere]; and thence,
when there are spaces everywhere which can adequately hold any material sphere
whatever, it is manifest (that) space is spherical everywhere. And so too about
the other figures. In the same manner, within clear water also, we see no
material figures, nevertheless several belong (to it) as only some color being
inserted by many ways into its differing parts
would make to appear. Color, however, if it were inserted, would not
constitute the material figures but would only cause (them) to be visible.
2. Space is
evenly extended everywhere to infinity. We cannot, for example, imagine
anywhere a limit such that simultaneously we are understanding what is given
beyond space. And from this all straight lines, paraboloid, hyperboloid, and
all cones and cylinders, and the remaining figures of this kind are evenly
advanced to infinity, and no one is limited albeit they are intercepted
randomly by all kinds of lines and surfaces continuing on crosswise, and among
themselves are everywhere constituted the segments of figures. That you might
have some true specimen of infinity; suppose some triangle with its base along
with one leg [side] at rest and the other leg [side] around the end of its
co-tangent base so that it whirls in the plane of the triangle as if the
triangle would gradually be opened at the vertex: and meanwhile let the point
turn in your mind, where the two legs [sides] come together if in this way they
would be produced thus far, and it is manifest that all these points would be
situated on the straight line in which the leg [side] at rest lies, and that
from this the distance by which the moving leg is continually revolved about it
at a perpetually greater reach, the other legs would traverse as far as the
parallel and can no longer come together with it anywhere. I ask next how much
was the distance of the last point at which the legs came together? Certainly
it was more than any (that) can be assigned, or rather not any of the points
was last, and hence the straight line in which all these points are found
combining is an achievement more than finite. Nor is it what some would say,
(that) this is by imagination only and not an achievement of infinity; for if
producing a triangle would be an achievement, its legs always being laid out
stretched towards some common point (would be) an achievement, if in this way
they were produced to where they both come together, and hence the point where
the products come together will always be an achievement of such kind, even if
it be supposed to be outside of the world of bodily limits; and thus the line
which all these points designate will actually be, however much it be advanced
beyond all distance.
Lest someone might
now object that we cannot imagine infinite extension, I agree: But meanwhile I
contend that we can understand (it). We can imagine a greater extension and
also a still greater, but we understand a greater extension than any we can
imagine to exist. And from this, incidentally, the faculty of understanding is
clearly distinguished from (that) of imagining.
If, however, one
would say, moreover, that we do not understand what infinite being would be
except by means of negating the limit of the end, and that this is, to that
extent, falsifying the conception negatively: I deny it. For the path or limit
is the check or the negation of several realities or existences to being by
means of limitation, and by which we the less conceive (as) something being
restrained being in the limits, wherefore we discover liquid better by positing
(it) in itself; that is, for that reason we conceive (it) better positively.
And hence by negating all limits the conception turns out altogether positively.
End is a name to the extent of the negative sense, and thus far infinity, when
it would be the negation of a negation (that is, the end), will be a name
entirely positive to the extent of our sense and conception (of it), albeit
grammatically it is seen negatively. Add that the finite quantities of several
infinite surfaces are known accurately lengthwise by geometry. And thus, I am
able to determine positively and exactly the solid quantities of infinite
solids now by length, now by width, and to compare them with given finite solids. But this is not
the place for that.
What if
Descartes, now, would say (that) extension were not infinite but only
indefinite: by reason of grammar he is correct. For the name indefinite may not
at all be said about that which is an achievement but always with respect to
the possibility of the future, only denoting something (that) has not yet been
determined as definite. Thus, before God established something by creating the
world, (if he at anytime had not established (it)), the quantity of matter, the
number of stars, and all the rest would have been indefinite, which therefore
will have become defined by the creation of the world. Thus matter is
indefinitely divisible, but always either finitely or infinitely divided.
(Article 26, Part I; Article 34, Part II) Thus, the indefinite line is that the
future length of which would not yet be determined. And so is that space
indefinite whose future magnitude would not yet be determined; therefore what
is truly an achievement is not to be defined but either does or does not have a
limit, and thus is either finite or infinite. It is no objection that he says
(it) is indefinite as far as we (know), that is, we are only ignorant of its
ends and do not positively know there to be none, (Article 27, Part I):
allowing for our ignorance, then, not even God understands there to be none
only indefinitely but certainly and positively; on the other hand, because
however much we still imagine (it) negatively, we nevertheless positively and
certainly understand (that) this transcends all limits. But I see what
Descartes apprehended, doubtlessly should space be posited (as) infinite, it
would surely become God on account of the perfection of infinity. But by no
means, for perfection is not infinity except insofar as it is allotted to
perfection. Infinity of intellect, power, happiness, etc., is entirely perfect;
infinity of ignorance, impotence, misery, etc., is entirely imperfection; and
infinity of extension of whatever kind is the perfection of whatever is
extended.
3. The parts of space
are immobile. If they be moved, either it is saying that their motion would be
transference from the vicinity of other co-tangent parts, just as Descartes has
defined the motion of bodies, and that I have sufficiently set forth to be
absurd; or it is saying that it would be transference from space to space, that
is, out of itself, unless perchance it be said that two spaces coincide
everywhere, the mobile and the immobile. The rest of space will best be shown
to be immobile by means of duration. In the same manner that, for example, the
parts of duration are individuated by means of the order, as if therefore (for
the sake of example) the order of yesterday’s day could be exchanged with this
day, and turn out to be afterwards, the individuation would be lost and there
would no longer be yesterday’s day but this day: thus, the parts of space are
so individuated by means of their positions that if two could exchange
positions, however much, they would simultaneously exchange individuation, and
each would be numerically converted into the other. On which account, among themselves, the parts of duration and
space may be understood to be only the same order and positions themselves
which actually are; they have no other principle of individuation beyond these
order and positions, which hence are unable to change.
4. Space is
being to the extent of being a relation. No being would exist or can exist that
is not in some manner attributed to space. God is everywhere, created minds are
somewhere and body in the space that it fills; and whatever is neither
everywhere nor somewhere, that is not. And hence it follows that space would be
the productive [emanative] effect of the existence of being primarily, because
by positing being anywhere space is posited. And similarities
can be affirmed about duration: obviously both are
affections or attributes of being which are denominated according to its
quantity of existence in the individual to the extent of the amount [size] of
the presence and perseverance in itself. Thus, God’s quantity of existence
according to duration has been eternal, and according to the space in which he
is present, infinite; and the quantity of existence of created things according
to duration has been as much as the quantity of duration from the beginning of
existence, and according to the size [amount] of the presence as much as also
the space in which it is present.
For the rest, then,
it cannot be imagined that God corresponds to the image of the extended body
and divisible parts: knowing (that) spaces themselves are not a divisible
achievement, and beyond any being whatever has a mode proper to itself by which
it is present in space. Thus, for example, the relation of duration by length is
different to space than to body. For we do not ascribe differences of duration
to the differences of parts, but we say all simultaneously endure. The same is
the moment of duration at Rome and at London, the same of earth, the heavenly
stars, and the universe. And in the same way we understand every single moment
of duration thus, by means of universal space, by its own manner, without any
part of its concept dispersed: therefore no more does one contradict that mind
also could be dispersed conceptually through space in its own manner, without
any part.
5. The positions,
distances, and local motions of bodies are assigned to parts of space. And that
is manifest from the first and fourth surveys of the properties of space, and
it will be more manifest if you conceive vacuums to be dispersed among the
corpuscles, or you attend to that which I have said first about motion. Let
there be subjoined to this, moreover, that there does not appertain to space
any force of impeding or advancing or whatever by reason of
the changing motions of bodies. And from this
projectile bodies describe straight lines in a uniform motion if not being counteracted
by a different one impeding 9them). But more about this later.
6. Lastly, space is
of eternal duration and immutable nature, just as that productive [emanative]
effect of being would be eternal and immutable. If ever there were not space,
God would then be present nowhere, and hence he created space somewhere after
he himself had not been present, or, what is not less discordant with reason,
he created his own ubiquity. Again however much we can perhaps imagine nothing
to be in space, nevertheless we can not think space not to be; in the same
manner we can not think duration not to be, although it would be possible to suppose
nothing at all to endure. And that is manifest by means of extra-mundane
spaces, which (when we imagine the world to be finite) we can not think not to
be, however much they are neither revealed to us by God, nor become known
through our senses, nor from spaces within the world to the extent of existing
dependently. But it is customarily believed about these spaces that they are
nothing. Indeed, they are truly spaces. Space, even if it would be the vacuum
of body, nevertheless is not in itself vacuum. And that is something however
much spaces are nothing besides. Fifthly, it is acknowledged that spaces are no
more spaces where the world exists than where nothing is, unless perhaps you
say that God, when he created the world in this space, he simultaneously
repeated space in himself. Whatever is thus of more reality in one space than
in another, that is in body and not in space; in the same manner it will be
more clearly manifest if in this manner that puerile and misleading judgment
from childhood be deposed, that extension inheres in bodies only as accidents
in the subject, without which it cannot actually exist.
Having
described extension, explaining bodily nature remains on the other side. Of
this, however, when it would exist not necessarily but by divine will, the
explanation will be more uncertain for the reason that it is by no means
permitted to know the limits of divine power; obviously matter can have been
created by one manner only or, in the opposite case, there are several by which
another and thus a different one has been allowed to produce beings most
similar to bodies. And however much it seem not at all credible, God can create
beings most similar to bodies, of which all their actions bring forth and
exhibit phenomena, and nevertheless they would not be bodies in essential and
metaphysical constitution: (even) when
nevertheless I will no way have a clear and distinct perception in this matter,
I would not dare to affirm the contrary, and hence I do not wish positively to
say just what would be bodily nature but rather, describing whatever kind of
being is in bodies by means of all the similarity of which, we are not able not
to perceive (that) the creation is in the power of God, and hence which we can
by no means certainly say not to be bodies.
Whenever any
man would be conscious in himself that he could move his own body with decision
and he also believes that the same power belongs to another man, by which
through thinking alone they similarly move their own bodies: the power of moving
with choice of anyone at all no wise being denied by God, his thinking is
infinitely more powerful and the capacity more prompt. And by an equal reason
it is conceded that God by the sole action of thinking or willing could embrace
any defined space by certain limits that some bodies not advance [penetrate]
into it.
What if he
were to exercise this very power, and he should cause that some space should
arise in the image of a mountain or whatever body being terminated above the
earth impervious to bodies, and thus light and all pressing things would stop
or would rebound; it seems impossible that with the aid of our senses (which
should be constituted judges in this matter only) we will disclose this space
not actually to be body; it were indeed tangible on account of the impenetrability,
and visibly opaque and colored on account of the reflection of light, and a
blow would resonate for the reason that the neighboring air would be moved by
the blow.
Let us suppose,
accordingly, various spaces to be disseminated throughout the world, some one
of which being defined by certain limits, it would turn out impervious to
bodies by divine power, and from the hypothesis it is manifest that this would
obstruct, and probably reflect, the motions of bodies, and all the properties
of bodily particles would be involved, unless the thing were immobile. Further
this impenetrability not always to be conserved in the same part of space but
able to be transferred hither and thither according to certain laws, so that
nevertheless as if the quantity and form of this impenetrable space were not
changed, there would be no property of body which were not suited to this. Let
there be formed, the thing tangible, and mobile, and able to be reflected, and
to reflect, and by joining things in some way not less constituting a part than
any other corpuscle, and I do not see why it can not equally act on our minds
and on the contrary be borne, when it were nothing other than the effect of the
divine mind within the definite quantity of space being evoked. For it is
certain (that) God is able by his own will to move our perceptions, and hence
attaches such kind of power to the effect of his will.
In the same manner if
several spaces in this manner, both in bodies and in themselves, became impervious,
they would sustain all the changes of corpuscles and would exhibit the same
phenomena. And thus, in order that if this world were entirely constituted from
this mode of being, its condition would hardly seem to be otherwise. And hence
these beings either were bodies or most similar to bodies. Now if they were
bodies, then bodies we can define to be determinate entities of extension
which God everywhere present affects with certain conditions: to the extent
they are (1) that they be mobile, and for that reason I have not said (they)
are the numerical parts of space which are absolutely immobile, but only the
definite quantities which are able to be transferred from space to space; (2)
that by this mode no two could coincide by any part, whether as if they be
impenetrable and hence as if being reflected, counter acting by mutual motions
and opposing by certain laws; (3) that they could arouse in created minds
various perceptions of the senses and dreams [fancies], and, on the other hand,
be moved by itself, not surprising when the description of the origin would be
based on this.
For the rest of the
explanation, then, it will serve to observe the following:
(1) That it is no aid
to the existence of these beings if we represent some unintelligible substance
to be given, in which the shape [form] would inhere substantially only by
attribution: extension and the achievement of divine will are enough. Extension
sustains the place of the substantial subject, in which the shape [form] of
body is conserved by means of divine will; and this effect of divine will is
the shape [form] or formal reason of body denominating every dimension of space
in which body is produced.
(2) These
beings were no less real than bodies, nor, I have said, less capable of substance.
Whatever, indeed, we believe to belong to the reality of bodies, that is done on account of their appearances
and sensible qualities. And hence these beings, when by this manner they were capable
of all qualities, and similarly able to exhibit all these phenomena, we should
judge (them) to be no less real by the manner they would exist. They would not
the less be substances, if indeed by means of God alone they should equally
subsist and endure through accidents.
(3) Between extension
and the shape [form] imposed on it (that) such-like bears is the analogy of the
sort the Aristotelians posited between prime matter and substantial form; to
the extent that they say, doubtlessly, the same matter is capable of every
shape [form], and derives the denomination of the number of body from the
shapes [form]. Thus, for example, I posit (that) shape [form], however many,
can be transferred through any space, and thus (can) denominate body everywhere.
(4) They differ,
however, (in) that extension (when it would be and what, and what kind, and how
much) has more reality than prime matter, and thus (in) that it can, also be
understood, in the same manner and shape [form] as I have assigned to body. If
indeed there is some difficulty in the conception, it is not the shapes [forms]
which God would insert into space, but in the manner by which he would insert
(them). But this would not hold up for a difficulty, if indeed the same thing
would occur in the manner by which we move our (bodily) parts, and nevertheless
we do not the less believe we can move. If this manner may be known within us,
we know by equal reason by what stipulation God, too, can move bodies both from
some given place expelling and impeding the limited form, not having been
expelled or otherwise, however much on the contrary they can be advanced into
[penetrated], that is, so to cause that this space were impenetrable and endued
with the shape [form] of body.
(5) Accordingly, I
have deduced the description of this bodily nature from the capacity of moving
our bodies so that every difficulty in the conception is at length removed by
this; and moreover so that (within our inmost conscious) it would be manifest
(that) God might have created the world by nothing other than willing the
action, in the same manner both we by only willing the action move our own
bodies; and above and beyond that the analogy being set forth between our
faculties and the Divine faculties is the greater than thus far observed by the
philosopher. In the imagining of God creating is our being witness to the holy
page. And his image shines more completely within us if by the manner he has
shadowed the power of creating and thus the rest of his attributes (it) is
equal to our conceded capacities, nor is it prejudicial that we are creatures
ourselves, and thus far the evidence of his attributes could not be similarly
conceded in us. For although by this reason the power of creating minds is not
delineated in any capacity of the created mind, nevertheless the creating mind
(when it would be the image of God) is of much greater nature than the body
that perhaps more prominently contains (it) within itself.[i]
But more overt by moving bodies we do not create something nor are we able to
create but we only shadow forth the power of creating. We are not able, for
example, so to cause that some spaces would be impervious to bodies, but we
only move bodies, and this not of any whatever but only our own to which we are
united by the divine constitution and not by our will, nor by any manner whatever
but according to whatever laws which God has imposed on us. If, however,
someone should prefer (that) I call this our power finite and the lowest degree
of power which God the creator has established this would no more derogate from
divine power than by his understanding he would derogate the understanding that
also befits us to a finite degree, (especially when it is not by our own and
independent power but by law imposed by God on us that we move our bodies). In
fact, if someone believe it is possible that God produced some understanding
creature so far perfect as by the aid of divine concurrence he could again
produce creatures of lesser order, that, therefore, would not derogate from
divine power, nor would he be positing the infinitely greater claim, by which
creatures may be elicited not only immediately but mediately from other
creatures. And thus anyone would probably prefer to posit a soul of the world
being created by God, to be (that) on which he imposes this law, that defined
spaces would describe bodies by properties, than to believe this service to be
immediately furnished by God. And for that reason the world is not to be called
the creature of this soul but of God alone, who might create by establishing a
soul of such nature that the world would arise necessarily. But I do not see
why God himself does not immediately inform space with bodies; provided that,
by formal reason we distinguish bodies from the achievement of divine will. It
has indeed been contradicted that it would itself be the achievement of
willing, or something other than the effect only which that achievement
produces in space. In whatever the effect differs no less from that achievement
than Cartesian space, or the substance of bodies according to the vulgar
conception, if by this mode it is created, we suppose that existence is to be
derived from the will or to be the being of divine reason.
Lastly, having
described the greatest use of the idea of bodies has illuminated that it has
most clearly involved, and should confirm and explain, particular truths of
metaphysics. For example, we are not able to posit bodies by this manner,
except that we simultaneously posit God to exist, and he would have created
bodies in empty space ex nihilo, and it is the being distinguished from
minds, but able nevertheless to be united with mind. Say, pray, which opinion
of those now being published, therefore, might elucidate some one of these
truths or rather is not opposed to everyone and restores the perplexity. If we
say with Descartes (that) extension is body, do we not rather manifestly spread
the way to atheism, for then that extension is not being created but was from
eternity, whereupon we have an absolute idea of it without any relation to God,
and thus we are able to conceive existence for the time being as if at that
time we would suppose God not to be. And no distinction of mind from body,
according to this philosophy, is understandable, lest simultaneously we say
that mind is by no means extension, and thus is substantially present in no
extension, or is no place; and so too if we say it is not that by means of
which it is seen; however, I have plainly restored its minimum understandable
union with body, not saying (it is) impossible. Moreover, if the distinction of
substances into thinking and extended is lawful and perfect, then God does not
eminently contain, and hence he cannot create, extension within himself; but
God and extension are two substances
severally called absolutely complete and singular. On the other hand,
however, if extension is eminently contained within God or within all thinking
being, certainly the idea of extension will be eminently contained in the idea
of thinking, and hence the distinction of the ideas will be not insofar as
but as if {emphasis added} both could be suited to the same thing of
created substance, that is body to thinking or the thinking thing extended.
Now, if the vulgar idea, or rather non-idea of body is embraced, obviously that
bears in body some non-understandable reality, as they say the substance is
(that) in which their qualities inhere: This
(beyond that which is not understandable) as well as the Cartesian
arguments is accompanied by the same inconveniences. For when it cannot be
understood, it is impossible that his distinction from the substance of the
mind be understood. Nor indeed has assuming the separation from the shape
[form] of substance or the attributes of substance sufficed; for if exposed
substances have no essential difference, the same substantial forms or
attributes are able to be altered to suit and (as if by change, even if it not
be simultaneous) to bring about mind and body.
And thus if we do not understand this difference of the exposed
attributes of substances, neither can we knowingly affirm that mind and body differ.
Or if they do differ, we are not able
to discern any foundation of union. Moreover, in this the proved substances of
bodies without qualities and shapes [forms] are assigned somewhat less reality
in words, but not less in concept, than the substances of God abstracted from
his attributes. Both are similarly conceived tried [proved] nakedly, or rather
are not conceived except in common with whatever of intelligible reality they
do not confuse with apprehension. And from this it is not astonishing that
atheists are born [generated] ascribing to bodily substances that which ought
only to be of the divine. Fifthly, in looking about, nothing else of the
atheists has quite appeared, as a result of which this notion of bodies, as is
possessing complete reality absolutely and independently in themselves, such as
for the most parts we all are wont from childhood, lest I err through neglect,
to conceive mentally, such as that we verbally say to be created and dependent.
I believe it would have been both by reason and this prejudice, that the name
of substance is singularly attributed of God and of created beings in the
Schools, and that the idea being formulated of body would continue with the
philosopher and be indulged vaguely, since he endeavors at the same time to
formulate an independent idea of things dependent on God. For certainly
whatever can not be independently of God, cannot truly be understood
independent of the idea of God. God would no less sustain himself by created
beings than they are sustained by accidents, to the same degree that created
substance, whether you observe the degree of dependence or the degree of
reality, is an intermediate nature between God and accidents. And hence its
idea no less involves the conception of God than the idea of accident
(involves) the conception of created substance. Thus no reality other than the
derived and incomplete ought to be fulfilled in itself. Letting go of the
prejudice is accordingly premised and it is rather the imputing the substantial
reality of this mode of attributes, which are real by themselves, both
understandable and not needing a subject in which they inhere, than a subject
in which we can not conceive that as dependent nor formulate any idea of it.
And we will hardly do this with grudging if (beyond the idea of body exposed
above) we acknowledge in our souls (that) we can conceive space without any
subject existing, provided we think the vacuum. And hence something fits in
this reality of substance. But if, moreover, the mobility of the part (as
Descartes supposed) were involved in this idea, no one would soundly nor easily
concede substance to be body. In the same manner, if we have the idea of an
attribute or of this power by which God, by action of willing alone is able to
create being; perhaps we will conceive this attribute only in itself, without
any substantial subject subsisting and involving the rest of its attributes.
But meanwhile, while we are not able to formulate the idea not only of this
attribute but also of the proper power by which we move our own bodies; it
would be to chance to say what thing might be the foundation of mind substantially.
Thus much
about the nature of body: in explaining which I believe myself sufficiently to
have surpassed what corresponds most clearly to the kind I have revealed,
whereby creation is in the power of God, and out of which creation if this very
world should not be constituted, at least some other similar (world) by this
can be constituted. And there would be differences in matters to the extent of
the properties and nature, but only in the method by which God could create one
thing as well as another: surely the distinction of body from extension is
hereby sufficiently illuminated. For obviously extension would be eternal,
infinite, uncreated, uniform throughout, to no extent at all mobile, nor able
to induce motions in bodies or thinking in minds; body itself truly has an
opposed manner in all this, save if it had not pleased God to have created
always and everywhere. For I would not dare to deny in God this very power.
[And if anyone perceive otherwise, he should say where prime matter has been
able to be created and whence the power of creating, therefore, has been
conceded in God. Or if there were no beginning of this power, but it has kept
the same from eternity as it keeps now, then he has been able to have created
from eternity. For it is the same thing to say that there never was
powerlessness for creating in God, or that he always had the power of creating
and has always been able to be created. In the same manner is he assigned the
space in which matter at the beginning has not been able to be created, or it
should be conceded (that) God, then, has been able to have created everywhere.][ii]
As for the
rest of the Cartesian argument I now respond more strictly: we take from body
(just as he bids) gravity, duration, and all sensible qualities, so that
nothing finally would remain except what belongs to its essence. Will,
accordingly, extension only remain? Not at all. For we reject additionally that
capacity or power by which the perceptions of thinking things move. For when
the distinction is only between the ideas of thinking and extension so that
something would not be manifest to be the foundation of the connection or
relation unless that be caused by divine power; the capacities of bodies can be
rejected with this reserved extension, but it would not be rejected with the
reserved bodily nature. Obviously, the changes which can be induced in bodies
by natural causes are only accidental and not denoting the substance actually
to be changed. But if anything could induce the change which transcends natural
causes, it is more than accidentally and has radically attainted the substance.
According to the sense of the demonstration those only are being rejected of
which body, by force of nature, can be void and deprived. But no one would
object that bodies which are not united to minds cannot immediately move their
perceptions. And hence when bodies are given united to minds by nothing, it
will follow (that) this power is not among their essentials. The observation is
that this does not act by actual union but only by the capacity of bodies by
which they are capable of this union by force of nature. As by whatever
capacity belongs to all bodies, it is manifest from it that the parts of the
brain, especially the more subtle by which the mind is united, are in continual
flux, the new ones succeeding to those flying off. And it is not lesser to take
(off) this, whether regarding the divine achievement or bodily nature, than to
take (off) the other capacity by which bodies in themselves are able alternately
to transfer mutual actions, that is, than to force body back into empty space.
When, however water
would resist the motions of solid trajectories less than quicksilver, and air
much less than water, and ethereal space still less than air, we further reject
every force of impeding the motion of trajectories, and [then] reasonably we
will reject bodily nature. In the same manner if the subtle matter be deprived
of every force of impeding the motion of globules, 1 would no longer believe
there to be subtle matter but a dispersed vacuum. Accordingly, if in this way
the aerial or ethereal space would be by its own manner so that the projectiles
of comets or whatever bodies you wish would yield to motions without any
resistance, I would believe (it) to be utterly empty. For it is impossible that bodily fluids would not resist the
motions of trajectories, namely, if it be not disposed to the motion in accord
with the velocity of its motion (Part 2, Epistle 96, To Mersenne),[iii]
just as I have assumed.
Also all this force
can be taken to be manifest throughout space, if by a manner space and body
alternately differ; and hence (that) I have taken to be possible is not to be
denied sooner than they are proved not to differ, lest we equate (them), it
being allowed by petitio principii.
But (that) not any doubt would
remain, it derives from the observation of the aforesaid that empty spaces are
given in the nature of things. For if fluid were entirely bodily without any
vacuous pores, that, by means of dividing the subtlest parts however, were
equally dense and thus any other fluid whatever, and it would not yield less to
the inertial motion of trajectories, on the contrary, much more, if the
projectile in this way were porous; for the reason that it would penetrate its inmost pores, and not by way
of the whole external surface but also it would come together and be by the
impeding of all the internal parts on the surfaces. But with the ether on the
other hand, forasmuch as there is a little resistance collected so that in resisting
quicksilver, it is seen to be more than ten or a hundred thousand less in
force: surely much the greatest part of space ought to be kept for vacuum,
being dispersed among the ethereal corpuscles. That same thing, moreover, would
be granted by collecting the differing gravity of these fluids, as is that of
their densities or that the contents (of their) material quantities show in
space, here the descending of the heavier, there the swaying of the pendulum.
But it is not now the place for explaining this.
Let it be seen
accordingly how fallacious and false Descarte’s very argument is, (as) if
someone rejecting the accidents of bodies not only would preserve extension as
if that could be supposed, but also the capacities which are able to move, now
perceptions of the mind, now other bodies. For if we reject, moreover, these
very capacities and every power of moving so that the precise conception of
uniform space would remain, is there any vortex, is there any world Descartes
will fabricate out of this extension? No one would reasonably invoke (it), lest
God first, who alone would be able (replacing these capacities or bodily nature
in the proportion he had before arranged) newly to beget bodies in this space.
Accordingly, in virtue of the foregoing, I have assigned bodily nature to
consist in the then enumerated capacities.
And so at last, when
spaces are not themselves bodies but only the places to which they belong and
in which they are moved, which I have defined sufficiently by local motion, I
esteem (them) to be established. I do not see what more could be required in
this matter, unless perhaps that for them this does not satisfy, I would teach
them that by means of space, the parts of which are filled with bodies, I have
defined places understanding the Cartesian generic space in which spaces
considered particularly, or Cartesian bodies, are moved, and surely they will
hardly keep refuting that in our definitions.
Now that I
have digressed enough, let us return to the proposition.
[What follows is the
translation of A. Rupert Hall and Marie Boas Hall:]
Definition 5.
Force is the causal principle of motion and rest. And it is either an external
one that generates or destroys or otherwise changes impressed motion in some
body; or it is an internal principle by which existing motion or rest is
conserved in a body, and by which any being endeavors to continue in its state
and opposes resistance.
Definition 6.
Conatus [endeavor] is resisted force, or force in so far as it is resisted.
Definition 7. Impetus
is force in so far as it is impressed on a thing.
Definition 8. Inertia
is force within a body, lest its state should be easily changed by an external
exciting force.
Definition 9.
Pressure is the endeavor of contiguous parts to penetrate into each others’
dimension. For if they could penetrate the pressure would cease. And pressure
is only between contiguous parts, which in turn press upon others contiguous to
them, until the pressure is transmitted to the most remote parts of any body,
whether hard, soft or fluid. And upon this action is based the communication of
motion by means of a point or surface of contact.
Definition 10. Gravity
is a force in a body impelling it to descend. Here, however, by descent is not
only meant a motion towards the center of the Earth but also towards any point
or region, or even from any point. In this way if the conatus of the anther
whirling about the Sun to recede from its centre be taken for gravity, the
aether in receding from the Sun could be said to descend. And so by analogy,
the plane is called horizontal that is directly opposed to the direction of
gravity or conatus.
Moreover, the
quantity of these powers, namely motion, force, conatus, impetus, inertia,
pressure and gravity may be reckoned in a double way: that is, according to
either intension or extension.
Definition 11.
The intension of any of the above-mentioned powers is the degree of its
quality.
Definition 12.
Its extension is the amount of space or time in which it operates.
Definition 13. Its
absolute quantity is the product of its intension and its extension. So, if the
quantity of intension is 2, and the quantity of extension 3, multiply the two
together and you will have the absolute quantity 6.
Moreover, it
will help to illustrate these definitions from individual powers. And thus
motion is either more intense or more remiss, as the space traversed in the
same time is greater or less, for which reason a body is usually said to move
more swiftly or more slowly. Again, motion is more or less in extension as the
body moved is greater or less, or as it is acting in a larger or smaller body.
And the absolute quantity of motion is composed of both the velocity and
the magnitude of the moving body. So force,
conatus, impetus or inertia are more intense as they are greater in the same or
an equivalent body: they have more extension when the body is larger, their
absolute quantity arises from both. So
the intension pressure is proportional to the increase of pressure upon the
surface-area; its extension proportional to the surface pressed. And the absolute
quantity results from the intension of the pressure and the area of the surface
pressed. So, lastly, the intension of gravity is proportional to the specific
gravity of the body; its extension is proportional to the size of the heavy
body, and absolutely speaking the quantity of gravity is the product of the
specific gravity and mass of the gravitating body. And whoever fails to
distinguish these clearly, necessarily falls into many errors concerning the
mechanical sciences.
In addition
the quantity of these powers may sometimes be reckoned from the period of
duration; for which reason there will be an absolute quantity which will be the
product of intension, extension and duration. In this way if a body [of size] 2
is moved with a velocity 3 for a time 4 the whole motion will be 2 x (3 X 4 or 12) [sic].
Definition 14. Velocity is the intension of motion,
slowness is remission.
Definition 15. Bodies are denser when their inertia is more
intense, and rarer when it is more remiss.
The rest of the
above-mentioned powers have no names.
It is however to be
noted that if, with Descartes or Epicurus, we suppose rarefaction and
condensation to be accomplished in the manner of relaxed or compressed sponges,
that is, by the dilation and contraction of pores which are either filled with
some most subtle matter or empty of matter, then we ought to estimate the size
of the whole body from the quantity of both its parts and its pores in
Definition 15; so that one may consider inertia to be remitted by the
increase of the pores and intensified by their
diminution, as though the pores, which offer no inertial resistance to change,
and whose mixtures with the truly corporeal parts give rise to all the various
degrees of inertia, bear some ratio to the parts.
But in order that you
may conceive of this composite body as a uniform one, imagine its parts to be
infinitely divided and dispersed everywhere throughout the pores, so that in
the whole composite body there is not the least particle of extension without
an absolutely perfect mixture of parts and pores thus infinitely divided.
Certainly it suits mathematicians to contemplate things in the light of such
reasoning, or if you prefer in the Peripatetic manner; but in physics things
seem otherwise.
Definition 16. An
elastic body is one that can be condensed by pressure or compressed within
narrower limits; and a non-elastic body is one that cannot be condensed by that
force.
Definition 17. A hard
body is one whose parts do not yield to pressure.
Definition 18. A
fluid body is one whose parts yield to an overwhelming pressure.[iv]
Moreover, the pressures by which the fluid is driven in any direction
whatsoever (whether these are exerted on the external surface alone, or on the
internal parts by the action of gravity or any other cause), are said to be
balanced when the fluid rests in equilibrium. This is asserted on the
assumption that the pressure is exerted in some one direction and not towards
all at once.
Definition 19. The
limits defining the surface of the body (such as wood or glass) containing the
fluid, or defining the surface of the external part of the same fluid
containing some internal part, constitute the vessel of fluid [vas fluidi].
In these
definitions, however, I refer only to absolutely hard or fluid bodies, for one
cannot ratiocinate mathematically concerning ones partially so, on account of
the innumerable circumstances affecting the figures, motions and contexture of
the least particles. Thus I imagine that a fluid does not consist of hard
particles, but that it is of such a kind that it has no small portion or particle
which is not likewise fluid. And moreover, since the physical cause of fluidity
is not to be examined here, I define the parts not as being in motion among
themselves, but only as capable of motion, that is, as being everywhere so divided
one from another that, although they may be supposed to be in contact and at
rest with respect to one another, yet they do not cohere as though stuck
together, but can be moved separately by any impressed force and can change the
state of rest as easily as the state of motion if they move relatively. Indeed,
I suppose that the parts of hard bodies do not merely touch each other and
remain at relative rest, but that they do besides so strongly and firmly
cohere, and are so bound together, as it were by glue, that no one of them can
be moved without all the rest being drawn along with it; or rather that a hard
body is not made up of conglomerate parts but is a single undivided and uniform
body which preserves its shape most resolutely, whereas a fluid body is
uniformly divided at all points.
And thus I
accommodated these definitions not to physical things but to mathematical
reasoning, after the manner of the Geometers who do not accommodate their
definitions of figures to the irregularities of physical bodies. And just as
the dimensions of physical bodies are best determined from their geometry (as
the measurement of a field from plane geometry, although a field is not a true
plane; and the measurement of the Earth from the doctrine of the sphere even
though the Earth is not precisely spherical) so the properties of physical
fluids and solids are best known from this mathematical doctrine, even though
they are not perhaps absolutely nor uniformly fluid or solid as I have defined
them here.
Propositions
on Non-Elastic Fluids
Axioms
1. From like
postulates like consequences ensure.
2. Bodies in
contact press each other equally.
Proposition 1.
All the parts of a non-gravitating fluid, compressed with the same intention in
all directions, press each other equally or with equal intension.
Proposition 2. And compression does not cause
a relative motion of the parts.
Demonstration of Both
Let us first
suppose that the fluid is contained and uniformly compressed by the spherical
boundary AB whose center is K [Fig. B]. Any small portion of it CGEH is bounded
by the two spherical surfaces CD and EF described about the same center K and
by the conical surface GKH whose vertex is at K. And it is manifest that CGEH
cannot in any way approach the center K because all the matter between the
spherical surfaces CD and EF would everywhere approach the same center for the
same reason,a1 and so would penetrate
the volume of the fluid contained within the sphere EF.b1 Nor can CGER recede in any direction towards
the circumference AB because all that shell of fluid between DC and EF would
similarly recede for the same reasona1 and so
would penetrate the volume of fluid between the spherical surfaces AB and CD.b1
Nor can it be squeezed out sideways, say towards, H, since if we imagine
another little section Hy, terminated in every direction by the same spherical
surfaces and a similar conical surface and
contiguous to GH and H, this section Hy may for the same reason be
squeezed out towards H,a1 and so
effect a penetration of volume by the mutual approach of contiguous parts.b1 And so it is that no portion of fluid
CGEH can exceed its limits because of pressure. And hence all the parts remain
in equilibrium. Which is what I wished to demonstrate first.
I say also
that all parts press each other equally, and with the same intension of
pressure that the external surface is pressed. To show this, imagine that PSQR
is a part of the said fluid AB contained by similar spherical segments PRQ and
PSQ, and that its compression upon the internal surface PSQ is as great as that
upon the external surface PRQ [Fig. A ]. For I have already shown that this
part of the fluid remains in equilibrium, and so the effects of the pressures
acting on both of its surfaces are equal, and hence the pressures are equal.c1 d1
And thus since
spherical surfaces such as PSQ can be described anywhere in the fluid AB, and
can touch any other given surfaces in any points whatever, it follows that the
extension of the pressure of the parts along the surfaces, wherever placed, is
as great as the pressure on the external surface of the fluid. Which is the
second point I wished to demonstrate.
Moreover, as
the force of this argument is based on the equality of the surfaces PRQ and
PSQ, lest it should seem that there is some disparity, in that one is within
the fluid and the other is a segment of the external surface, it will help to
imagine that the whole sphere AB is a part of an indefinitely larger volume of
fluid, in which it is contained as within a vessel, and is everywhere
compressed just as its part PRQS is pressed upon the surface PSQ by another
part
PABQS. For the method
by which the sphere AB is compressed is of no significance, so long as its
compression is supposed to be equal everywhere.
Now that these things
have been demonstrated for a fluid sphere, I say lastly that all the parts of
the fluid D (bounded in any manner at all, and compressed with the same
intension in all directions) press each other equally and are not made to move
relatively by the compression. For let AB be an indefinitely greater fluid
sphere compressed with the same degree of intension; and let be some part of it
equal and similar to D. From what has already been demonstrated it follows that
this part is compressed with an equal intension in all directions and that the
intension of the pressure is the same as that of the sphere AB, that is (by
hypothesis) as that which compressed the fluid D. Thus the compression of the
similar and equal fluids, D and *, is equal; and hence the effects will be equal.a2
But all the parts of the sphere ABb2 and so of the fluid contained in it, press
each other equally, and the pressure does not cause a relative motion of the
parts. For which reason the same is true of
the fluid D.a2
Q.E.D.
Corollary 1. The internal parts of a fluid
press each other with the same intension as that by which the fluid is pressed
on its external surface.
Corollary 2. If the intension of the pressure
is not everywhere the same, the fluid does not remain in equilibrium. For since
it stays in equilibrium because the pressure is everywhere uniform, if the
pressure is anywhere increased, it will predominate there and cause the fluid
to recede from that region.c2
Corollary 3. If no
motion is caused in a fluid by pressure, the intension of the pressure is
everywhere the same. For if it is not the same, motion will be caused by the
predominant pressure.d2
Corollary 4. A
fluid presses on whatever bounds it with the same intension as the fluid is
pressed by whatever bounds it, and vice versa. Since the parts of a fluid are
certainly the bounds of contiguous parts and press each other with an equal
intension, conceive the aforesaid fluid to be part of a greater fluid, or
similar and equal to such a part, and similarly compressed, and the assertion
will be evident.a2
Corollary 5. A
fluid everywhere presses all its bounds if they are capable of withstanding the
pressure applied, with that intension with which it is itself pressed in any
place. For otherwise it would not be pressed everywhere with the same
intention.e2 On which assumption it yields to the more
intense pressure.d2 And so it will either be condensedf2, or it will break through the bounds
where the pressure is less.f2
Scholium. I
have proposed all this about fluids, not as contained in hard and rigid
vessels, but within soft and quite flexible bounds (say within the internal
surface of a homogeneous exterior fluid), so that I might more clearly show
that their equilibrium is caused only by an equal degree of pressure in all
directions. But once a fluid is put into equilibrium by an equal pressure, it
is all one whether you imagine it to be contained within rigid or yielding
bounds.