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[Illustration]
INTRODUCTION.
THE manner, in which Sir ~ISAAC NEWTON~ has published his philosophical
discoveries, occasions them to lie very much concealed from all, who
have not made the mathematics particularly their study. He once,
indeed, intended to deliver, in a more familiar way, that part of
his inventions, which relates to the system of the world; but upon
farther consideration he altered his design. For as the nature of
those discoveries made it impossible to prove them upon any other than
geometrical principles; he apprehended, that those, who should not
fully perceive the force of his arguments, would hardly be prevailed
on to exchange their former sentiments for new opinions, so very
different from what were commonly received[1]. He therefore chose
rather to explain himself only to mathematical readers; and declined
the attempting to instruct such in any of his principles, who, by
not comprehending his method of reasoning, could not, at the first
appearance of his discoveries, have been persuaded of their truth. But
now, since Sir ~ISAAC NEWTON~’s doctrine has been fully established
by the unanimous approbation of all, who are qualified to understand
the same; it is without doubt to be wished, that the whole of his
improvements in philosophy might be universally known. For this purpose
therefore I drew up the following papers, to give a general notion
of our great philosopher’s inventions to such, as are not prepared
to read his own works, and yet might desire to be informed of the
progress, he has made in natural knowledge; not doubting but there were
many, besides those, whose turn of mind had led them into a course of
mathematical studies, that would take great pleasure in tasting of this
delightful fountain of science.
2. IT is a just remark, which has been made upon the human mind, that
nothing is more suitable to it, than the contemplation of truth; and
that all men are moved with a strong desire after knowledge; esteeming
it honourable to excel therein; and holding it, on the contrary,
disgraceful to mistake, err, or be in any way deceived. And this
sentiment is by nothing more fully illustrated, than by the inclination
of men to gain an acquaintance with the operations of nature; which
disposition to enquire after the causes of things is so general, that
all men of letters, I believe, find themselves influenced by it. Nor
is it difficult to assign a reason for this, if we consider only, that
our desire after knowledge is an effect of that taste for the sublime
and the beautiful in things, which chiefly constitutes the difference
between the human life, and the life of brutes. These inferior animals
partake with us of the pleasures, that immediately flow from the bodily
senses and appetites; but our minds are furnished with a superior
sense, by which we are capable of receiving various degrees of delight,
where the creatures below us perceive no difference. Hence arises
that pursuit of grace and elegance in our thoughts and actions, and
in all things belonging to us, which principally creates imployment
for the active mind of man. The thoughts of the human mind are too
extensive to be confined only to the providing and enjoying of what
is necessary for the support of our being. It is this taste, which
has given rise to poetry, oratory, and every branch of literature and
science. From hence we feel great pleasure in conceiving strongly, and
in apprehending clearly, even where the passions are not concerned.
Perspicuous reasoning appears not only beautiful; but, when set forth
in its full strength and dignity, it partakes of the sublime, and not
only pleases, but warms and elevates the soul. This is the source of
our strong desire of knowledge; and the same taste for the sublime
and the beautiful directs us to chuse particularly the productions of
nature for the subject of our contemplation: our creator having so
adapted our minds to the condition, wherein he has placed us, that all
his visible works, before we inquire into their make, strike us with
the most lively ideas of beauty and magnificence.
3. BUT if there be so strong a passion in contemplative minds for
natural philosophy; all such must certainly receive a particular
pleasure in being informed of Sir ~ISAAC NEWTON~’s discoveries, who
alone has been able to make any great advancements in the true course
leading to natural knowledge: whereas this important subject had before
been usually attempted with that negligence, as cannot be reflected
on without surprize. Excepting a very few, who, by pursuing a more
rational method, had gained a little true knowledge in some particular
parts of nature; the writers in this science had generally treated of
it after such a manner, as if they thought, that no degree of certainty
was ever to be hoped for. The custom was to frame conjectures; and if
upon comparing them with things, there appeared some kind of agreement,
though very imperfect, it was held sufficient. Yet at the same time
nothing less was undertaken than intire systems, and fathoming at once
the greatest depths of nature; as if the secret causes of natural
effects, contrived and framed by infinite wisdom, could be searched
out by the slightest endeavours of our weak understandings. Whereas
the only method, that can afford us any prospect of success in this
difficult work, is to make our enquiries with the utmost caution, and
by very slow degrees. And after our most diligent labour, the greatest
part of nature will, no doubt, for ever remain beyond our reach.
4. THIS neglect of the proper means to enlarge our knowledge, joined
with the presumption to attempt, what was quite out of the power of our
limited faculties, the Lord BACON judiciously observes to be the great
obstruction to the progress of science[2]. Indeed that excellent person
was the first, who expresly writ against this way of philosophizing;
and he has laid open at large the absurdity of it in his admirable
treatise, intitled NOVUM ORGANON SCIENTIARUM; and has there likewise
described the true method, which ought to be followed.
5. THERE are, saith he, but two methods, that can be taken in the
pursuit of natural knowledge. One is to make a hasty transition
from our first and slight observations on things to general axioms,
and then to proceed upon those axioms, as certain and uncontestable
principles, without farther examination. The other method; (which he
observes to be the only true one, but to his time unattempted;) is to
proceed cautiously, to advance step by step, reserving the most general
principles for the last result of our inquiries[3]. Concerning the
first of these two methods; where objections, which happen to appear
against any such axioms taken up in haste, are evaded by some frivolous
distinction, when the axiom it self ought rather to be corrected[4];
he affirms, that the united endeavours of all ages cannot make it
successful; because this original error in the first digestion of
the mind (as he expresses himself) cannot afterwards be remedied[5]:
whereby he would signify to us, that if we set out in a wrong way; no
diligence or art, we can use, while we follow so erroneous a course,
will ever bring us to our designed end. And doubtless it cannot prove
otherwise; for in this spacious field of nature, if once we forsake
the true path, we shall immediately lose our selves, and must for ever
wander with uncertainty.
6. THE impossibility of succeeding in so faulty a method of
philosophizing his Lordship endeavours to prove from the many false
notions and prejudices, to which the mind of man is exposed[6]. And
since this judicious writer apprehends, that men are so exceeding
liable to fall into these wrong tracts of thinking, as to incur great
danger of being misled by them, even while they enter on the true
course in pursuit of nature[7]; I trust, I shall be excused, if, by
insisting a little particularly upon this argument, I endeavour to
remove whatever prejudice of this kind, might possibly entangle the
mind of any of my readers.
7. HIS Lordship has reduced these prejudices and false modes of
conception under four distinct heads[8].
8. THE first head contains such, as we are subject to from the very
condition of humanity, through the weakness both of our senses, and of
the faculties of the mind[9]; seeing, as this author well observes, the
subtilty of nature far exceeds the greatest subtilty of our senses or
acutest reasonings[10]. One of the false modes of conception, which
he mentions under this head, is the forming to our selves a fanciful
simplicity and regularity in natural things. This he illustrates by
the following instances; the conceiving the planets to move in perfect
circles; the adding an orb of fire to the other three elements, and
the supposing each of these to exceed the other in rarity, just in a
decuple proportion[11]. And of the same nature is the assertion of
~DES CARTES~, without any proof, that all things are made up
of three kinds of matter only[12]. As also this opinion of another
philosopher; that light, in passing through different mediums, was
refracted, so as to proceed by that way, through which it would move
more speedily, than through any other[13]. The second erroneous turn
of mind, taken notice of by his Lordship under this head, is, that
all men are in some degree prone to a fondness for any notions, which
they have once imbibed; whereby they often wrest things to reconcile
them to those notions, and neglect the consideration of whatever will
not be brought to an agreement with them; just as those do, who are
addicted to judicial astrology, to the observation of dreams, and to
such-like superstitions; who carefully preserve the memory of every
incident, which serves to confirm their prejudices, and let slip out of
their minds all instances, that make against them[14]. There is also
a farther impediment to true knowledge, mentioned under the same head
by this noble writer, which is; that whereas, through the weakness
and imperfection of our senses, many things are concealed. from us,
which have the greatest effect in producing natural appearances; our
minds are ordinarily most affected by that, which makes the strongest
impression on our organs of sense; whereby we are apt to judge of
the real importance of things in nature by a wrong measure[15]. So,
because the figuration and the motion of bodies strike our senses more
immediately than most of their other properties, DES CARTES and his
followers will not allow any other explication of natural appearances,
than from the figure and motion of the parts of matter. By which
example we see how justly his Lordship observes this cause of error to
be the greatest of any[16]; since it has given rise to a fundamental
principle in a system of philosophy, that not long ago obtained almost
an universal reputation.
9. THESE are the chief branches of those obstructions to knowledge,
which this author has reduced under his first head of false
conceptions. The second head contains the errors, to which particular
persons are more especially obnoxious[17]. One of these is the
consequence of a preceding observation: that as we are exposed to be
captivated by any opinions, which have once taken possession of our
minds; so in particular, natural knowledge has been much corrupted by
the strong attachment of men to some one part of science, of which
they reputed themselves the inventers, or about which they have spent
much of their time; and hence have been apt to conceive it to be of
greater use in the study of natural philosophy than it was: like
ARISTOTLE, who reduced his physics to logical disputations; and the
chymists, who thought, that nature could be laid open only by the
force of their fires[18]. Some again are wholly carried away by an
excessive veneration for antiquity; others, by too great fondness
for the moderns; few having their minds so well balanced, as neither
to depreciate the merit of the ancients, nor yet to despise the real
improvements of later times[19]. To this is added by his Lordship a
difference in the genius of men, that some are most fitted to observe
the similitude, there is in things, while others are more qualified to
discern the particulars, wherein they disagree; both which dispositions
of mind are useful: but to the prejudice of philosophy men are apt to
run into excess in each; while one sort of genius dwells too much upon
the gross and sum of things, and the other upon trifling minutenesses
and shadowy distinctions[20].
10. UNDER the third head of prejudices and false notions this writer
considers such, as follow from the lax and indefinite use of words in
ordinary discourse; which occasions great ambiguities and uncertainties
in philosophical debates (as another eminent philosopher has since
shewn more at large[21];) insomuch that this our author thinks a strict
defining of terms to be scarce an infallible remedy against this
inconvenience[22]. And perhaps he has no small reason on his side: for
the common inaccurate sense of words, notwithstanding the limitations
given them by definitions, will offer it self so constantly to the
mind, as to require great caution and circumspection for us not to be
deceived thereby. Of this we have a very eminent instance in the great
disputes, that have been raised about the use of the word attraction in
philosophy; of which we shall be obliged hereafter to make particular
mention[23]. Words thus to be guarded against are of two kinds. Some
are names of things, that are only imaginary[24]; such words are wholly
to be rejected. But there are other terms, that allude to what is real,
though their signification is confused[25]. And these latter must of
necessity be continued in use; but their sense cleared up, and freed,
as much as possible, from obscurity.
11. THE last general head of these errors comprehends such, as follow
from the various sects of false philosophies; which this author divides
into three sorts, the sophistical, empirical, and superstitious[26]. By
the first of these he means a philosophy built upon speculations only
without experiments[27]; by the second, where experiments are blindly
adhered to, without proper reasoning upon them[28]; and by the third,
wrong opinions of nature fixed in mens minds either through false
religions, or from misunderstanding the declarations of the true[29].
12. THESE are the four principal canals, by which this judicious author
thinks, that philosophical errors have flowed in upon us. And he
rightly observes, that the faulty method of proceeding in philosophy,
against which he writes[30], is so far from assisting us towards
overcoming these prejudices; that he apprehends it rather suited to
rivet them more firmly to the mind[31]. How great reason then has his
Lordship to call this way of philosophizing the parent of error, and
the bane of all knowledge[32]? For, indeed, what else but mistakes can
so bold and presumptuous a treatment of nature produce? have we the
wisdom necessary to frame a world, that we should think so easily,
and with so slight a search to enter into the most secret springs of
nature, and discover the original causes of things? what chimeras, what
monsters has not this preposterous method brought forth? what schemes,
or what hypothesis’s of the subtilest wits has not a stricter enquiry
into nature not only overthrown, but manifested to be ridiculous and
absurd? Every new improvement, which we make in this science, lets us
see more and more the weakness of our guesses. Dr. HARVEY, by that
one discovery of the circulation of the blood, has dissipated all the
speculations and reasonings of many ages upon the animal oeconomy.
ASELLIUS, by detecting the lacteal veins, shewed how little ground all
physicians and philosophers had in conjecturing, that the nutritive
part of the aliment was absorbed by the mouths of the veins spread
upon the bowels: and then PECQUET, by finding out the thoracic duct,
as evidently proved the vanity of the opinion, which was persisted in
after the lacteal vessels were known, that the alimental juice was
conveyed immediately to the liver, and there converted into blood.
13. AS these things set forth the great absurdity of proceeding in
philosophy on conjectures, by informing us how far the operations
of nature are above our low conceptions; so on the other hand, such
instances of success from a more judicious method shew us, that our
bountiful maker has not left us wholly without means of delighting
our selves in the contemplation of his wisdom. That by a just way of
inquiry into nature, we could not fail of arriving at discoveries
very remote from our apprehensions; the Lord ~BACON~ himself
argues from the experience of mankind. If, says he, the force of guns
should be described to any one ignorant of them, by their effects only,
he might reasonably suppose, that those engines of destruction were
only a more artificial composition, than he knew, of wheels and other
mechanical powers: but it could never enter his thoughts, that their
immense force should be owing to a peculiar substance, which would
enkindle into so violent an explosion, as we experience in gunpowder:
since he would no where see the least example of any such operation;
except perhaps in earthquakes and thunder, which he would doubtless
look upon as exalted powers of nature, greatly surpassing any art of
man to imitate. In the same manner, if a stranger to the original
of silk were shewn a garment made of it, he would be very far from
imagining so strong a substance to be spun out of the bowels of a small
worm; but must certainly believe it either a vegetable substance, like
flax or cotton; or the natural covering of some animal, as wool is
of sheep. Or had we been told, before the invention of the magnetic
needle among us, that another people was in possession of a certain
contrivance, by which they were inabled to discover the position of
the heavens, with vastly more ease, than we could do; what could have
been imagined more, than that they were provided with some fitter
astronomical instrument for this purpose than we? That any stone should
have so amazing a property, as we find in the magnet, must have been
the remotest from our thoughts[33].
14. BUT what surprizing advancements in the knowledge of nature may be
made by pursuing the true course in philosophical inquiries; when those
searches are conducted by a genius equal to so divine a work, will be
best understood by considering Sir ~ISAAC NEWTON~ discoveries.
That my’s reader may apprehend as just a notion of these, as can be
conveyed to him, by the brief account, which I intend to lay before
him; I have set apart this introduction for explaining, in the fullest
manner I am able, the principles, whereon Sir ~ISAAC NEWTON~ proceeds.
For without a clear conception of these, it is impossible to form any
true idea of the singular excellence of the inventions of this great
philosopher.
15. THE principles then of this philosophy are; upon no consideration
to indulge conjectures concerning the powers and laws of nature, but
to make it our endeavour with all diligence to search out the real
and true laws, by which the constitution of things is regulated. The
philosopher’s first care must be to distinguish, what he sees to be
within his power, from what is beyond his reach; to assume no greater
degree of knowledge, than what he finds himself possessed of; but to
advance by slow and cautious steps; to search gradually into natural
causes; to secure to himself the knowledge of the most immediate cause
of each appearance, before he extends his views farther to causes
more remote. This is the method, in which philosophy ought to be
cultivated; which does not pretend to so great things, as the more
airy speculations; but will perform abundantly more: we shall not
perhaps seem to the unskilful to know so much, but our real knowledge
will be greater. And certainly it is no objection against this method,
that some others promise, what is nearer to the extent of our wishes:
since this, if it will not teach us all we could desire to be informed
of, will however give us some true light into nature; which no other
can do. Nor has the philosopher any reason to think his labour lost,
when he finds himself stopt at the cause first discovered by him, or
at any other more remote cause, short of the original: for if he has
but sufficiently proved any one cause, he has entered so far into the
real constitution of things, has laid a safe foundation for others to
work upon, and has facilitated their endeavours in the search after
yet more distant causes; and besides, in the mean time he may apply
the knowledge of these intermediate causes to many useful purposes.
Indeed the being able to make practical deductions from natural causes,
constitutes the great distinction between the true philosophy and the
false. Causes assumed upon conjecture, must be so loose and undefined,
that nothing particular can be collected from them. But those causes,
which are brought to light by a strict examination of things, will be
more distinct. Hence it appears to have been no unuseful discovery,
that the ascent of water in pumps is owing to the pressure of the
air by its weight or spring; though the causes, which make the air
gravitate, and render it elastic, be unknown: for notwithstanding
we are ignorant of the original, whence these powers of the air are
derived; yet we may receive much advantage from the bare knowledge of
these powers. If we are but certain of the degree of force, wherewith
they act, we shall know the extent of what is to be expected from
them; we shall know the greatest height, to which it is possible by
pumps to raise water; and shall thereby be prevented from making any
useless efforts towards improving these instruments beyond the limits
prescribed to them by nature; whereas without so much knowledge as
this, we might probably have wasted in attempts of this kind much time
and labour. How long did philosophers busy themselves to no purpose
in endeavouring to perfect telescopes, by forming the glasses into
some new figure; till Sir ~ISAAC NEWTON~ demonstrated, that
the effects of telescopes were limited from another cause, than was
supposed; which no alteration in the figure of the glasses could
remedy? What method Sir ~ISAAC NEWTON~ himself has found for
the improvement of telescopes shall be explained hereafter[34]. But
at present I shall proceed to illustrate, by some farther instances,
this distinguishing character of the true philosophy, which we have now
under consideration. It was no trifling discovery, that the contraction
of the muscles of animals puts their limbs in motion, though the
original cause of that contraction remains a secret, and perhaps may
always do so; for the knowledge of thus much only has given rise to
many speculations upon the force and artificial disposition of the
muscles, and has opened no narrow prospect into the animal fabrick.
The finding out, that the nerves are great agents in this action,
leads us yet nearer to the original cause, and yields us a wider
view of the subject. And each of these steps affords us assistance
towards restoring this animal motion, when impaired in our selves,
by pointing out the seats of the injuries, to which it is obnoxious.
To neglect all this, because we can hitherto advance no farther, is
plainly ridiculous. It is confessed by all, that ~GALILEO~
greatly improved philosophy, by shewing, as we shall relate hereafter,
that the power in bodies, which we call gravity, occasions them to
move downwards with a velocity equably accelerated[35]; and that when
any body is thrown forwards, the same power obliges it to describe in
its motion that line, which is called by geometers a parabola[36]:
yet we are ignorant of the cause, which makes bodies gravitate. But
although we are unacquainted with the spring, whence this power in
nature is derived, nevertheless we can estimate its effects. When a
body falls perpendicularly, it is known, how long time it takes in
descending from any height whatever: and if it be thrown forwards,
we know the real path, which it describes; we can determine in what
direction, and with what degree of swiftness it must be projected,
in order to its striking against any object desired; and we can also
ascertain the very force, wherewith it will strike. Sir ~ISAAC
NEWTON~ has farther taught, that this power of gravitation extends
up to the moon, and causes that planet to gravitate as much towards
the earth, as any of the bodies, which are familiar to us, would, if
placed at the same distance[37]: he has proved likewise, that all
the planets gravitate towards the sun, and towards one another; and
that their respective motions follow from this gravitation. All this
he has demonstrated upon indisputable geometrical principles, which
cannot be rendered precarious for want of knowing what it is, which
causes these bodies thus mutually to gravitate: any more than we can
doubt of the propensity in all the bodies about us, to descend towards
the earth; or can call in question the forementioned propositions
of ~GALILEO~, which are built upon that principle. And as
~GALILEO~ has shewn more fully, than was known before, what
effects were produced in the motion of bodies by their gravitation
towards the earth; so Sir ~ISAAC NEWTON~, by this his
invention, has much advanced our knowledge in the celestial motions.
By discovering that the moon gravitates towards the sun, as well as
towards the earth; he has laid open those intricacies in the moon’s
motion, which no astronomer, from observations only, could ever find
out[38]: and one kind of heavenly bodies, the comets, have their motion
now clearly ascertained; whereof we had before no true knowledge at
all[39].
16. DOUBTLESS it might be expected, that such surprizing success should
have silenced, at once, every cavil. But we have seen the contrary.
For because this philosophy professes modestly to keep within the
extent of our faculties, and is ready to confess its imperfections,
rather than to make any fruitless attempts to conceal them, by seeking
to cover the defects in our knowledge with the vain ostentation of
rash and groundless conjectures; hence has been taken an occasion
to insinuate that we are led to miraculous causes, and the occult
qualities of the schools.
17. BUT the first of these accusations is very extraordinary. If by
calling these causes miraculous nothing more is meant than only, that
they often appear to us wonderful and surprizing, it is not easy
to see what difficulty can be raised from thence; for the works of
nature discover every where such proofs of the unbounded power, and
the consummate wisdom of their author, that the more they are known,
the more they will excite our admiration: and it is too manifest to
be insisted on, that the common sense of the word miraculous can have
no place here, when it implies what is above the ordinary course of
things. The other imputation, that these causes are occult upon the
account of our not perceiving what produces them, contains in it great
ambiguity. That something relating to them lies hid, the followers
of this philosophy are ready to acknowledge, nay desire it should be
carefully remarked, as pointing out proper subjects for future inquiry.
But this is very different from the proceeding of the schoolmen in
the causes called by them occult. For as their occult qualities were
understood to operate in a manner occult, and not apprehended by us; so
they were obtruded upon us for such original and essential properties
in bodies, as made it vain to seek any farther cause; and a greater
power was attributed to them, than any natural appearances authorized.
For instance, the rise of water in pumps was ascribed to a certain
abhorrence of a vacuum, which they thought fit to assign to nature. And
this was so far a true observation, that the water does move, contrary
to its usual course, into the space, which otherwise would be left
void of any sensible matter; and, that the procuring such a vacuity
was the apparent cause of the water’s ascent. But while we were not in
the least informed how this power, called an abhorrence of a vacuum,
produced the visible effects; instead of making any advancement in the
knowledge of nature, we only gave an artificial name to one of her
operations: and when the speculation was pushed so beyond what any
appearances required, as to have it concluded, that this abhorrence
of a vacuum was a power inherent in all matter, and so unlimited as
to render it impossible for a vacuum to exist at all; it then became
a much greater absurdity, in being made the foundation of a most
ridiculous manner of reasoning; as at length evidently appeared, when
it came to be discovered, that this rise of the water followed only
from the pressure of the air, and extended it self no farther, than
the power of that cause. The scholastic stile in discoursing of these
occult qualities, as if they were essential differences in the very
substances, of which bodies consisted, was certainly very absurd; by
reason it tended to discourage all farther inquiry. But no such ill
consequences can follow from the considering of any natural causes,
which confessedly are not traced up to their first original. How
shall we ever come to the knowledge of the several original causes of
things, otherwise than by storing up all intermediate causes which we
can discover? Are all the original and essential properties of matter
so very obvious, that none of them can escape our first view? This is
not probable. It is much more likely, that, if some of the essential
properties are discovered by our first observations, a stricter
examination should bring more to light.
18. BUT in order to clear up this point concerning the essential
properties of matter, let us consider the subject a little distinctly.
We are to conceive, that the matter, out of which the universe of
things is formed, is furnished with certain qualities and powers,
whereby it is rendered fit to answer the purposes, for which it was
created. But every property, of which any particle of this matter is
in it self possessed, and which is not barely the consequence of the
union of this particle with other portions of matter, we may call an
essential property: whereas all other qualities or attributes belonging
to bodies, which depend on their particular frame and composition, are
not essential to the matter, whereof such bodies are made; because
the matter of these bodies will be deprived of those qualities, only
by the dissolution of the body, without working any change in the
original constitution of one single particle of this mass of matter.
Extension we apprehend to be one of these essential properties, and
impenetrability another. These two belong universally to all matter;
and are the principal ingredients in the idea, which this word matter
usually excites in the mind. Yet as the idea, marked by this name,
is not purely the creature of our own understandings, but is taken
for the representation of a certain substance without us; if we
should discover, that every part of the substance, in which we find
these two properties, should likewise be endowed universally with any
other essential qualities; all these, from the time they come to our
notice, must be united under our general idea of matter. How many
such properties there are actually in all matter we know not; those,
of which we are at present apprized, have been found out only by our
observations on things; how many more a farther search may bring to
light, no one can say; nor are we certain, that we are provided with
sufficient methods of perception to discern them all. Therefore, since
we have no other way of making discoveries in nature, but by gradual
inquiries into the properties of bodies; our first step must be to
admit without distinction all the properties, which we observe; and
afterwards we must endeavour, as far as we are able, to distinguish
between the qualities, wherewith the very substances themselves are
indued, and those appearances, which result from the structure only of
compound bodies. Some of the properties, which we observe in things,
are the attributes of particular bodies only; others universally belong
to all, that fall under our notice. Whether some of the qualities and
powers of particular bodies, be derived from different kinds of matter
entring their composition, cannot, in the present imperfect state
of our knowledge, absolutely be decided; though we have not yet any
reason to conclude, but that all the bodies, with which we converse,
are framed out of the very same kind of matter, and that their
distinct qualities are occasioned only by their structure; through
the variety whereof the general powers of matter are caused to produce
different effects. On the other hand, we should not hastily conclude,
that whatever is found to appertain to all matter, which falls under
our examination, must for that reason only be an essential property
thereof, and not be derived from some unseen disposition in the frame
of nature. Sir ~ISAAC NEWTON~ has found reason to conclude,
that gravity is a property universally belonging to all the perceptible
bodies in the universe, and to every particle of matter, whereof they
are composed. But yet he no where asserts this property to be essential
to matter. And he was so far from having any design of establishing
it as such, that, on the contrary, he has given some hints worthy of
himself at a cause for it[40]; and expresly says, that he proposed
those hints to shew, that he had no such intention[41].
19. IT appears from hence, that it is not easy to determine, what
properties of bodies are essentially inherent in the matter, out of
which they are made, and what depend upon their frame and composition.
But certainly whatever properties are found to belong either to any
particular systems of matter, or universally to all, must be considered
in philosophy; because philosophy will be otherwise imperfect. Whether
those properties can be deduced from some other appertaining to
matter, either among those, which are already known, or among such as
can be discovered by us, is afterwards to be sought for the farther
improvement of our knowledge. But this inquiry cannot properly have
place in the deliberation about admitting any property of matter or
bodies into philosophy; for that purpose it is only to be considered,
whether the existence of such a property has been justly proved or not.
Therefore to decide what causes of things are rightly received into
natural philosophy, requires only a distinct and clear conception of
what kind of reasoning is to be allowed of as convincing, when we argue
upon the works of nature.
20. THE proofs in natural philosophy cannot be so absolutely
conclusive, as in the mathematics. For the subjects of that science
are purely the ideas of our own minds. They may be represented to our
senses by material objects, but they are themselves the arbitrary
productions of our own thoughts; so that as the mind can have a full
and adequate knowledge of its own ideas, the reasoning in geometry
can be rendered perfect. But in natural knowledge the subject of
our contemplation is without us, and not so compleatly to be known:
therefore our method of arguing must fall a little short of absolute
perfection. It is only here required to steer a just course between
the conjectural method of proceeding, against which I have so largely
spoke; and demanding so rigorous a proof, as will reduce all philosophy
to mere scepticism, and exclude all prospect of making any progress in
the knowledge of nature.
21. THE concessions, which are to be allowed in this science, are by
Sir ~ISAAC NEWTON~ included under a very few simple precepts.
22. THE first is, that more causes are not to be received into
philosophy, than are sufficient to explain the appearances of nature.
That this rule is approved of unanimously, is evident from those
expressions so frequent among all philosophers, that nature does
nothing in vain; and that a variety of means, where fewer would
suffice, is needless. And certainly there is the highest reason
for complying with this rule. For should we indulge the liberty of
multiplying, without necessity, the causes of things, it would reduce
all philosophy to mere uncertainty; since the only proof, which we can
have, of the existence of a cause, is the necessity of it for producing
known effects. Therefore where one cause is sufficient, if there really
should in nature be two, which is in the last degree improbable, we can
have no possible means of knowing it, and consequently ought not to
take the liberty of imagining, that there are more than one.
23. THE second precept is the direct consequence of the first, that to
like effects are to be ascribed the same causes. For instance, that
respiration in men and in brutes is brought about by the same means;
that bodies descend to the earth here in EUROPE, and in AMERICA from
the same principle; that the light of a culinary fire, and of the sun
have the same manner of production; that the reflection of light is
effected in the earth, and in the planets by the same power; and the
like.
24. THE third of these precepts has equally evident reason for it.
It is only, that those qualities, which in the same body can neither
be lessened nor increased, and which belong to all bodies that are
in our power to make trial upon, ought to be accounted the universal
properties of all bodies whatever.
25. IN this precept is founded that method of arguing by induction,
without which no progress could be made in natural philosophy. For as
the qualities of bodies become known to us by experiments only; we
have no other way of finding the properties of such bodies, as are
out of our reach to experiment upon, but by drawing conclusions from
those which fall under our examination. The only caution here required
is, that the observations and experiments, we argue upon, be numerous
enough, and that due regard be paid to all objections, that occur, as
the Lord BACON very judiciously directs[42]. And this admonition is
sufficiently complied with, when by virtue of this rule we ascribe
impenetrability and extension to all bodies, though we have no sensible
experiment, that affords a direct proof of any of the celestial bodies
being impenetrable; nor that the fixed stars are so much as extended.
For the more perfect our instruments are, whereby we attempt to find
their visible magnitude, the less they appear; insomuch that all the
sensible magnitude, which we observe in them, seems only to be an
optical deception by the scattering of their light. However, I suppose
no one will imagine they are without any magnitude, though their
immense distance makes it undiscernable by us. After the same manner,
if it can be proved, that all bodies here gravitate towards the earth,
in proportion to the quantity of solid matter in each; and that the
moon gravitates to the earth likewise, in proportion to the quantity
of matter in it; and that the sea gravitates towards the moon, and
all the planets towards each other; and that the very comets have the
same gravitating faculty; we shall have as great reason to conclude by
this rule, that all bodies gravitate towards each other. For indeed
this rule will more strongly hold in this case, than in that of the
impenetrability of bodies; because there will more instances be had of
bodies gravitating, than of their being impenetrable.
25. THIS is that method of induction, whereon all philosophy is
founded; which our author farther inforces by this additional precept,
that whatever is collected from this induction, ought to be received,
notwithstanding any conjectural hypothesis to the contrary, till such
times as it shall be contradicted or limited by farther observations on
nature.
[Illustration]
[Illustration]
~BOOK I.~
CONCERNING THE
MOTION of BODIES
IN GENERAL.
CHAP. I.
Of the LAWS of MOTION.
HAVING thus explained Sir ~ISAAC NEWTON’s~ method of reasoning
in philosophy, I shall now proceed to my intended account of his
discoveries. These are contained in two treatises. In one of them, the
MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY, his chief design is to
shew by what laws the heavenly motions are regulated; in the other,
his OPTICS, he discourses of the nature of light and colours, and of
the action between light and bodies. This second treatise is wholly
confined to the subject of light: except some conjectures proposed
at the end concerning other parts of nature, which lie hitherto more
concealed. In the other treatise our author was obliged to smooth the
way to his principal intention, by explaining many things of a more
general nature: for even some of the most simple properties of matter
were scarce well established at that time. We may therefore reduce Sir
~ISAAC NEWTON~’s doctrine under three general heads; and I
shall accordingly divide my account into three books. In the first I
shall speak of what he has delivered concerning the motion of bodies,
without regard to any particular system of matter; in the second I
shall treat of the heavenly motions; and the third shall be employed
upon light.
2. IN the first part of my design, we must begin with an account of the
general laws of motion.
3. THESE laws are some universal affections and properties of matter
drawn from experience, which are made use of as axioms and evident
principles in all our arguings upon the motion of bodies. For as it
is the custom of geometers to assume in their demonstrations some
propositions, without exhibiting the proof of them; so in philosophy,
all our reasoning must be built upon some properties of matter, first
admitted as principles whereon to argue. In geometry these axioms are
thus assumed, on account of their being so evident as to make any
proof in form needless. But in philosophy no properties of bodies can
be in this manner received for self-evident; since it has been observed
above, that we can conclude nothing concerning matter by any reasonings
upon its nature and essence, but that we owe all the knowledge, we
have thereof, to experience. Yet when our observations on matter have
inform’d us of some of its properties, we may securely reason upon them
in our farther inquiries into nature. And these laws of motion, of
which I am here to speak, are found so universally to belong to bodies,
that there is no motion known, which is not regulated by them. These
are by Sir ~ISAAC NEWTON~ reduced to three[43].
4. THE first law is, that all bodies have such an indifference to rest,
or motion, that if once at rest they remain so, till disturbed by some
power acting upon them: but if once put in motion, they persist in
it; continuing to move right forwards perpetually, after the power,
which gave the motion, is removed; and also preserving the same degree
of velocity or quickness, as was first communicated, not stopping or
remitting their course, till interrupted or otherwise disturbed by some
new power impressed.
5. THE second law of motion is, that the alteration of the state of
any body, whether from rest to motion, or from motion to rest, or
from one degree of motion to another, is always proportional to the
force impressed. A body at rest, when acted upon by any power, yields
to that power, moving in the same line, in which the power applied
is directed; and moves with a less or greater degree of velocity,
according to the degree of the power; so that twice the power shall
communicate a double velocity, and three times the power a threefold
velocity. If the body be moving, and the power impressed act upon the
body in the direction of its motion, the body shall receive an addition
to its motion, as great as the motion, into which that power would
have put it from a state of rest; but if the power impressed upon a
moving body act directly opposite to its former motion, that power
shall then take away from the body’s motion, as much as in the other
case it would have added to it. Lastly, if the power be impressed
obliquely, there will arise an oblique motion differing more or less
from the former direction, according as the new impression is greater
or less. For example, if the body A (in fig. 1.) be moving in the
direction A B, and when it is at the point A, a power be impressed upon
it in the direction A C, the body shall from henceforth neither move
in its first direction A B, nor in the direction of the adventitious
power, but shall take a course as A D between them: and if the power
last impressed be just equal to that, which first gave to the body
its motion; the line A D shall pass in the middle between A B and A
C, dividing the angle under B A C into two equal parts; but if the
power last impressed be greater than the first, the line A D shall
incline most to A C; whereas if the last impression be less than the
first, the line A D shall incline most to A B. To be more particular,
the situation of the line A D is always to be determined after this
manner. Let A E be the space, which the body would have moved through
in the line A B during any certain portion of time; provided that body,
when at A, had received no second impulse. Suppose likewise, that A F
is the part of the line A C, through which the body would have moved
during an equal portion of time, if it had been at rest in A, when it
received the impulse in the direction A C: then if from E be drawn a
line parallel to, or equidistant from A C, and from F another line
parallel to A B, those two lines will meet in the line A D.
6. THE third and last of these laws of motion is, that when any body
acts upon another, the action of that body upon the other is equalled
by the contrary reaction of that other body upon the first.
7. THESE laws of motion are abundantly confirmed by this, that all the
deductions made from them, in relation to the motion of bodies, how
complicated soever, are found to agree perfectly with observation. This
shall be shewn more at large in the next chapter. But before we proceed
to so diffusive a proof; I chuse here to point out those appearances of
bodies, whereby the laws of motion are first suggested to us.
8. DAILY observation makes it appear to us, that any body, which we
once see at rest, never puts it self into fresh motion; but continues
always in the same place, till removed by some power applied to it.
9. AGAIN, whenever a body is once in motion, it continues in that
motion some time after the moving power has quitted it, and it is left
to it self. Now if the body continue to move but a single moment, after
the moving power has left it, there can no reason be assigned, why it
should ever stop without some external force. For it is plain, that
this continuance of the motion is caused only by the body’s having
already moved, the sole operation of the power upon the body being the
putting it in motion; therefore that motion continued will equally be
the cause of its farther motion, and so on without end. The only doubt
that can remain, is, whether this motion communicated continues intire,
after the power, that caused it, ceases to act; or whether it does not
gradually languish and decrease. And this suspicion cannot be removed
by a transient and slight observation on bodies, but will be fully
cleared up by those more accurate proofs of the laws of motion, which
are to be considered in the next chapter.
10. LASTLY, bodies in motion appear to affect a straight course without
any deviation, unless when disturbed by some adventitious power acting
upon them. If a body be thrown perpendicularly upwards or downwards,
it appears to continue in the same straight line during the whole time
of its motion. If a body be thrown in any other direction, it is found
to deviate from the line, in which it began to move, more and more
continually towards the earth, whither it is directed by its weight:
but since, when the weight of a body does not alter the direction of
its motion, it always moves in a straight line, without doubt in this
other case the body’s, declining from its first course is no more,
than what is caused by its weight alone. As this appears at first
sight to be unquestionable, so we shall have a very distinct proof
thereof in the next chapter, where the oblique motion of bodies will be
particularly considered.
11. THUS we see how the first of the laws of motion agrees with
what appears to us in moving bodies. But here occurs this farther
consideration, that the real and absolute motion of any body is not
visible to us: for we are our selves also in constant motion along with
the earth whereon we dwell; insomuch that we perceive bodies to move
so far only, as their motion is different from our own. When a body
appears to us to lie at rest, in reality it only continues the motion,
it has received, without putting forth any power to change that motion.
If we throw a body in the course or direction, wherein we are carried
our selves; so much motion as we seem to have given to the body, so
much we have truly added to the motion, it had, while it appeared to us
to be at rest. But if we impel a body the contrary way, although the
body appears to us to have received by such an impulse as much motion,
as when impelled the other way; yet in this case we have taken from the
body so much real motion, as we seem to have given it. Thus the motion,
which we see in bodies, is not their real motion, but only relative
with respect to us; and the forementioned observations only shew us,
that this first law of motion has place in this relative or apparent
motion. However, though we cannot make any observation immediately on
the absolute motion of bodies, yet by reasoning upon what we observe
in visible motion, we can discover the properties and effects of real
motion.
12. WITH regard to this first law of motion, which is now under
consideration, we may from the foregoing observations most truly
collect, that bodies are disposed to continue in the absolute motion,
which they have once received, without increasing or diminishing their
velocity. When a body appears to us to lie at rest, it really preserves
without change the motion, which it has in common with our selves:
and when we put it into visible motion, and we see it continue that
motion; this proves, that the body retains that degree of its absolute
motion, into which it is put by our acting upon it: if we give it such
an apparent motion, which adds to its real motion, it preserves that
addition; and if our acting on the body takes off from its real motion,
it continues afterwards to move with no more real motion, than we have
left it.
13. AGAIN, we do not observe in bodies any disposition or power within
themselves to change the direction of their motion; and if they had any
such power, it would easily be discovered. For suppose a body by the
structure or disposition of its parts, or by any other circumstance in
its make, was indued with a power of moving it self; this self-moving
principle, which should be thus inherent in the body, and not depend on
any thing external, must change the direction wherein it would act, as
often as the position of the body was changed: so that for instance,
if a body was lying before me in such a position, that the direction,
wherein this principle disposes the body to move, was pointed directly
from me; if I then gradually turned the body about, the direction
of this self-moving principle would no longer be pointed directly
from me, but would turn about along with the body. Now if any body,
which appears to us at rest, were furnished with any such self-moving
principle; from the body’s appearing without motion we must conclude,
that this self-moving principle lies directed the same way as the
earth is carrying the body; and such a body might immediately be put
into visible motion only by turning it about in any degree, that this
self-moving principle might receive a different direction.
14. FROM these considerations it very plainly follows, that if a body
were once absolutely at rest; not being furnished with any principle,
whereby it could put it self into motion, it must for ever continue in
the same place, till acted upon by something external: and also that
when a body is put into motion, it has no power within it self to make
any change in the direction of that motion; and consequently that the
body must move on straight forward without declining any way whatever.
But it has before been shewn, that bodies do not appear to have in
themselves any power to change the velocity of their motion: therefore
this first law of motion has been illustrated and confirmed, as much as
can be from the transient observations, which have here been discoursed
upon; and in the next chapter all this will be farther established by
more correct observations.
15. BUT I shall now pass to the second law of motion; wherein, when it
is asserted, that the velocity, with which any body is moved by the
action of a power upon it, is proportional to that power; the degree of
power is supposed to be measured by the greatness of the body, which
it can move with a given celerity. So that the sense of this law is,
that if any body were put into motion with that degree of swiftness, as
to pass in one hour the length of a thousand yards; the power, which
would give the same degree of velocity to a body twice as great, would
give this lesser body twice the velocity, causing it to describe in the
same space of an hour two thousand yards. But by a body twice as great
as another, I do not here mean simply of twice the bulk, but one that
contains a double quantity of solid matter.
16. WHY the power, which can move a body twice as great as another with
the same degree of velocity, should be called twice as great as the
power, which can give the lesser body the same velocity, is evident.
For if we should suppose the greater body to be divided into two equal
parts, each equal to the lesser body, each of these halves will require
the same degree of power to move them with the velocity of the lesser
body, as the lesser body it self requires; and therefore both those
halves, or the whole greater body, will require the moving power to be
doubled.
17. THAT the moving power being in this sense doubled, should just
double likewise the velocity of the same body, seems near as evident,
if we consider, that the effect of the power applied must needs be
the same, whether that power be applied to the body at once, or in
parts. Suppose then the double power not applied to the body at
once, but half of it first, and afterwards the other half; it is not
conceivable for what reason the half last applied should come to have
a different effect upon the body, from that which is applied first;
as it must have, if the velocity of the body was not just doubled by
the application of it. So far as experience can determine, we see
nothing to favour such a supposition. We cannot indeed (by reason of
the constant motion of the earth) make trial upon any body perfectly at
rest, whereby to see whether a power applied in that case would have a
different effect, from what it has, when the body is already moving;
but we find no alteration in the effect of the same power on account of
any difference there may be in the motion of the body, when the power
is applied. The earth does not always carry bodies with the same degree
of velocity; yet we find the visible effects of any power applied to
the same body to be, at all times the very same: and a bale of goods,
or other moveable body lying in a ship is as easily removed from place
to place, while the ship is under sail, if its motion be steady, as
when it is fixed at anchor.
18. NOW this experience is alone sufficient to shew to us the whole of
this law of motion.
19. SINCE we find, that the same power will always produce the same
change in the motion of any body, whether that body were before moving
with a swifter or slower motion; the change wrought in the motion of
a body depends only on the power applied to it, without any regard to
the body’s former motion: and therefore the degree of motion, which the
body already possesses, having no influence on the power applied to
disturb its operation, the effects of the same power will not only be
the same in all degrees of motion of the body; but we have likewise no
reason to doubt, but that a body perfectly at rest would receive from
any power as much motion, as would be equivalent to the effect of the
same power applied to that body already in motion.
20. AGAIN, suppose a body being at rest, any number of equal powers
should be successively applied to it; pushing it forward from time to
time in the same course or direction. Upon the application of the first
power the body would begin to move; when the second power was applied,
it appears from what has been said, that the motion of the body would
become double; the third power would treble the motion of the body; and
so on, till after the operation of the last power the motion of the
body would be as many times the motion, which the first power gave it,
as there are powers in number. and the effect of this number of powers
will be always the same, without any regard to the space of time taken
up in applying them: so that greater or lesser intervals between the
application of each of these powers will produce no difference at all
in their effects. Since therefore the distance of time between the
action of each power is of no consequence; without doubt the effect
will still be the same, though the powers should all be applied at
the very same instant; or although a single power should be applied
equal in strength to the collective force of all these powers. Hence
it plainly follows, that the degree of motion, into which any body
will be put out of a state of rest by any power, will be proportional
to that power. A double power will give twice the velocity, a treble
power three times the velocity, and so on. The foregoing reasoning will
equally take place, though the body were not supposed to be at rest,
when the powers began to be applied to it; provided the direction, in
which the powers were applied, either conspired with the action of
the body, or was directly opposite to it. Therefore if any power be
applied to a moving body, and act upon the body either in the direction
wherewith the body moves, so as to accelerate the body; or if it act
directly opposite to the motion of the body, so as to retard it: in
both these cases the change of motion will be proportional to the
power applied; nay, the augmentation of the motion in one case, and
the diminution thereof in the other, will be equal to that degree of
motion, into which the same power would put the body, had it been at
rest, when the power was applied.
21. FARTHER, a power may be so applied to a moving body, as to act
obliquely to the motion of the body. And the effects of such an oblique
motion may be deduced from this observation; that as all bodies are
continually moving along with the earth, we see that the visible
effects of the same power are always the same, in whatever direction
the power acts: and therefore the visible effects of any power upon
a body, which seems only to be at rest, is always to appearance the
same as the real effect would be upon a body truly at rest. Now
suppose a body were moving along the line A B (in fig. 2.) and the eye
accompanied it with an equal motion in the line C D equidistant from A
B; so that when the body is at A, the eye shall be at C, and when the
body is advanced to E in the line A B, the eye shall be advanced to F
in the line C D, the distances A E and C F being equal. It is evident,
that here the body will appear to the eye to be at rest; and the line
F E G drawn from the eye through the body shall seem to the eye to be
immoveable; though as the body and eye move forward together, this
line shall really also move; so that when the body shall be advanced
to H and the eye to K, the line F E G shall be transferred into the
situation K H L, this line K H L being equidistant from F E G. Now
if the body when at E were to receive an impulse in the direction of
the line F E G; while the eye is moving on from F to K and carrying
along with it the line F E G, the body will appear to the eye to move
along this line F E G: for this is what has just now been said; that
while bodies are moving along with the earth, and the spectator’s eye
partakes of the same motion, the effect of any power upon the body
will appear to be what it would really have been, had the body been
truly at rest, when the power was applied. From hence it follows, that
when the eye is advanced to K, the body will appear somewhere in the
line K H L. Suppose it appear in M; then it is manifest, from what has
been premised at the beginning of this paragraph, that the distance H
M is equal to what the body would have run upon the line E G, during
the time, wherein the eye has passed from F to K, provided that the
body had been at rest, when acted upon in E. If it be farther asked,
after what manner the body has moved from E to M? I answer, through a
straight line; for it has been shewn above in the explication of the
first law of motion, that a moving body, from the time it is left to it
self, will proceed on in one continued straight line.
22. IF E N be taken equal to H M and N M be drawn; since H M is
equidistant from E N, N M will be equidistant from E H. Therefore the
effect of any power upon a moving body, when that power acts obliquely
to the motion of the body, is to be determined in this manner. Suppose
the body is moving along the straight line A E B, if when the body is
come to E, a power gives it an impulse in the direction of the line E
G, to find what course the body will afterwards take we must proceed
thus. Take in E B any length E H, and in E G take such a length E N,
that if the body had been at rest in E, the power applied to it would
have caused it to move over E N in the same space of time, as it would
have employed in passing over E H, if the power had not acted at all
upon it. Then draw H L equidistant from E G, and N M equidistant from
E B. After this, if a line be drawn from E to the point M, where these
two lines meet, the line E M will be the course into which the body
will be put by the action of the power upon it at E.
23. A MATHEMATICAL reader would here expect in some particulars more
regular demonstrations; but as I do not at present address my self to
such, so I hope, what I have now written will render my meaning evident
enough to those, who are unacquainted with that kind of reasoning.
24. NOW as we have been shewing, that some actual force is necessary
either to put bodies out of a state of rest into motion, or to change
the motion, which they have once received; it is proper here to
observe, that this quality in bodies, whereby they preserve their
present state, with regard to motion or rest, till some active force
disturb them, is called the ~VIS INERTIAE~ of matter: and
by this property, matter, sluggish and unactive of it self, retains
all the power impressed upon it, and cannot be made to cease from
action, but by the opposition of as great a power, as that which first
moved it. By the degree of this ~VIS INERTIAE~, or power of
inactivity, as we shall henceforth call it, we primarily judge of the
quantity of solid matter in each body; for as this quality is inherent
in all the bodies, upon which we can make any trial, we conclude it to
be a property essential to all matter; and as we yet know no reason
to suppose, that bodies are composed of different kinds of matter, we
rather presume, that the matter of all bodies is the same; and that
the degree of this power of inactivity is in every body proportional
to the quantity of the solid matter in it. But although we have no
absolute proof, that all the matter in the universe is uniform, and
possesses this power of inactivity in the same degree; yet we can with
certainty compare together the different degrees of this power of
inactivity in different bodies. Particularly this power is proportional
to the weight of bodies, as Sir ~ISAAC NEWTON~ has demonstrated[44].
However, notwithstanding that this power of inactivity in any body can
be more certainly known, than the quantity of solid matter in it; yet
since there is no reason to suspect that one is not proportional to
the other, we shall hereafter speak without hesitation of the quantity
of matter in bodies, as the measure of the degree of their power of
inactivity.
25. THIS being established, we may now compare the effects of the
same power upon different bodies, as hitherto we have shewn the
effects of different powers upon the same body. And here if we limit
the word motion to the peculiar sense given to it in philosophy, we
may comprehend all that is to be said upon this head under one short
precept; that the same power, to whatever body it is applied, will
always produce the same degree of motion. But here motion does not
signify the degree of celerity or velocity with which a body moves,
in which sense only we have hitherto used it; but it is made use of
particularly in philosophy to signify the force with which a body
moves: as if two bodies A and B being in motion, twice the force
would be required to stop A as to stop B, the motion of A would be
esteemed double the motion of B. In moving bodies, these two things are
carefully to be distinguished; their velocity or celerity, which is
measured by the space they pass through during any determinate portion
of time; and the quantity of their motion, or the force, with which
they will press against any resistance. Which force, when different
bodies move with the same velocity, is proportional to the quantity of
solid matter in the bodies; but if the bodies are equal, this force is
proportional to their respective velocities, and in other cases it is
proportional both to the quantity of solid matter in the body, and also
to its velocity. To instance in two bodies A and B: if A be twice as
great as B, and they have both the same velocity, the motion of A shall
be double the motion of B; and if the bodies be equal, and the velocity
of A be twice that of B, the motion of A shall likewise be double that
of B; but if A be twice as large as B, and move twice as swift, the
motion of A will be four times the motion of B; and lastly, if A be
twice as large as B, and move but half as fast, the degree of their
motion shall be the same.
26. THIS is the particular sense given to the word motion by
philosophers, and in this sense of the word the same power always
produces the same quantity or degree of motion. If the same power act
upon two bodies A and B, the velocities, it shall give to each of them,
shall be so adjusted to the respective bodies, that the same degree
of motion shall be produced in each. If A be twice as great as B, its
velocity shall be half that of B; if A has three times as much solid
matter as B, the velocity of A shall be one third of the velocity of B;
and generally the velocity given to A shall bear the same proportion to
the velocity given to B, as the quantity of solid matter contained in
the body B bears to the quantity of solid matter contained in A.
27. THE reason of all this is evident from what has gone before. If
a power were applied to B, which should bear the same proportion to
the power applied to A, as the body B bears to A, the bodies B and
A would both receive the same velocity; and the velocity, which B
will receive from this power, will bear the same proportion to the
velocity, which it would receive from the action of the power applied
to A, as the former of these powers bears to the latter: that is, the
velocity, which A receives from the power applied to it, will bear
to the velocity, which B would receive from the same power, the same
proportion as the body B bears to A.
28. FROM hence we may now pass to the third law of motion, where
this distinction between the velocity of a body and its whole motion
is farther necessary to be regarded, as shall immediately be shewn;
after having first illustrated the meaning of this law by a familiar
instance. If a stone or other load be drawn by a horse; the load
re-acts upon the horse, as much as the horse acts upon the load; for
the harness, which is strained between them, presses against the horse
as much as against the load; and the progressive motion of the horse
forward is hindred as much by the load, as the motion of the load is
promoted by the endeavour of the horse: that is, if the horse put forth
the same strength, when loosened from the load, he would move himself
forwards with greater swiftness in proportion to the difference between
the weight of his own body and the weight of himself and load together.
29. THIS instance will afford some general notion of the meaning of
this law. But to proceed to a more philosophical explication: if a body
in motion strike against another at rest, let the body striking be
ever so small, yet shall it communicate some degree of motion to the
body it strikes against, though the less that body be in comparison
of that it impinges upon, and the less the velocity is, with which
it moves, the smaller will be the motion communicated. But whatever
degree of motion it gives to the resting body, the same it shall lose
it self. This is the necessary consequence of the forementioned power
of inactivity in matter. For suppose the two bodies equal, it is
evident from the time they meet, both the bodies are to be moved by the
single motion of the first; therefore the body in motion by means of
its power of inactivity retaining the motion first given it, strikes
upon the other with the same force, wherewith it was acted upon it
self: but now both the bodies being to be moved by that force, which
before moved one only, the ensuing velocity will be the same, as if the
power, which was applied to one of the bodies, and put it into motion,
had been applied to both; whence it appears, that they will proceed
forwards, with half the velocity, which the body first in motion had:
that is, the body first moved will have lost half its motion, and the
other will have gained exactly as much. This rule is just, provided
the bodies keep contiguous after meeting; as they would always do, if
it were not for a certain cause that often intervenes, and which must
now be explained. Bodies upon striking against each other, suffer an
alteration in their figure, having their parts pressed inwards by the
stroke, which for the most part recoil again afterwards, the bodies
endeavouring to recover their former shape. This power, whereby bodies
are inabled to regain their first figure, is usually called their
elasticity, and when it acts, it forces the bodies from each other,
and causes them to separate. Now the effect of this elasticity in the
present case is such, that if the bodies are perfectly elastic, so
as to recoil with as great a force as they are bent with, that they
recover their figure in the same space of time, as has been taken up
in the alteration made in it by their compression together; then this
power will separate the bodies as swiftly, as they before approached,
and acting upon both equally, upon the body first in motion contrary
to the direction in which it moves, and upon the other as much in the
direction of its motion, it will take from the first, and add to the
other equal degrees of velocity: so that the power being strong enough
to separate them with as great a velocity, as they approached with, the
first will be quite stopt, and that which was at rest, will receive all
the motion of the other. If the bodies are elastic in a less degree,
the first will not lose all its motion, nor will the other acquire
the motion of the first, but fall as much short of it, as the other
retains. For this rule is never deviated from, that though the degree
of elasticity determines how much more than half its velocity the body
first in motion shall lose; yet in every case the loss in the motion
of this body shall be transferred to the other, that other body always
receiving by the stroke as much motion, as is taken from the first.
30. This is the case of a body striking directly against an equal body
at rest, and the reasoning here used is fully confirmed by experience.
There are many other cases of bodies impinging against one another: but
the mention of these shall be reserved to the next chapter, where we
intend to be more particular and diffusive in the proof of these laws
of motion, than we have been here.
CHAP. II.
Farther proofs of the LAWS OF MOTION.
HAVING in the preceding chapter deduced the three laws of motion,
delivered by our great philosopher, from the most obvious observations,
that suggest them to us; I now intend to give more particular proofs
of them, by recounting some of the discoveries which have been made in
philosophy before Sir Isaac Newton. For as they were all collected by
reasoning upon those laws; so the conformity of these discoveries to
experience makes them so many proofs of the truth of the principles,
from which they were derived.
2. LET us begin with the subject, which concluded the last chapter.
Although the body in motion be not equal to the body at rest, on which
it strikes; yet the motion after the stroke is to be estimated in the
same manner as above. Let A (in fig. 3.) be a body in motion towards
another body B lying at rest. When A is arrived at B, it cannot proceed
farther without putting B into motion; and what motion it gives to
B, it must lose it self, that the whole degree of motion of A and B
together, if neither of the bodies be elastic, shall be equal, after
the meeting of the bodies, to the single motion of A before the stroke.
Therefore, from what has been said above, it is manifest, that as soon
as the two bodies are met, they will move on together with a velocity,
which will bear the same proportion to the original velocity of A, as
the body A bears to the sum of both the bodies.
3. IF the bodies are elastic, so that they shall separate after the
stroke, A must lose a greater part of its motion, and the subsequent
motion of B will be augmented by this elasticity, as much as the motion
of A is diminished by it. The elasticity acting equally between both
the bodies, it will communicate to each the same degree of motion; that
is, it will separate the bodies by taking from the body A and adding
to the body B different degrees of velocity, so proportioned to their
respective quantities of matter, that the degree of motion, wherewith
A separates from B, shall be equal to the degree of motion, wherewith
B separates from A. It follows therefore, that the velocity taken from
A by the elasticity bears to the velocity, which the same elasticity
adds to B, the same proportion, as B bears to A: consequently the
velocity, which the elasticity takes from A, will bear the same
proportion to the whole velocity, wherewith this elasticity causes the
two bodies to separate from each other, as the body B bears to the sum
of the two bodies A and B; and the velocity, which is added to B by the
elasticity, bears to the velocity, wherewith the bodies separate, the
same proportion, as the body A bears to the sum of the two bodies A
and B. Thus is found, how much the elasticity takes from the velocity
of A, and adds to the velocity of B; provided the degree of elasticity
be known, whereby to determine the whole velocity wherewith the bodies
separate from each other after the stroke[45].
4. AFTER this manner is determined in every case the result of a body
in motion striking against another at rest. The same principles will
also determine the effects, when both bodies are in motion.
5. LET two equal bodies move against each other with equal swiftness.
Then the force, with which each of them presses forwards, being equal
when they strike; each pressing in its own direction with the same
energy, neither shall surmount the other, but both be stopt, if they
be not elastic: for if they be elastic, they shall from thence recover
new motion, and recede from each other, as swiftly as they met, if they
be perfectly elastic; but more slowly, if less so. In the same manner,
if two bodies of unequal bigness strike against each other, and their
velocities be so related, that the velocity of the lesser body shall
exceed the velocity of the greater in the same proportion, as the
greater body exceeds the lesser (for instance, if one body contains
twice the solid matter as the other, and moves but half as fast) two
such bodies will entirely suppress each other’s motion, and remain from
the time of their meeting fixed; if, as before, they are not elastic:
but, if they are so in the highest degree, they shall recede again,
each with the same velocity, wherewith they met. For this elastic
power, as in the preceding case, shall renew their motion, and pressing
equally upon both, shall give the same motion to both; that is, shall
cause the velocity, which the lesser body receives, to bear the same
proportion to the velocity, which the greater receives, as the greater
body bears to the lesser: so that the velocities shall bear the same
proportion to each other after the stroke, as before. Therefore if the
bodies, by being perfectly elastic, have the sum of their velocities
after the stroke equal to the sum of their velocities before the
stroke, each body after the stroke will receive its first velocity.
And the same proportion will hold likewise between the velocities,
wherewith they go off, though they are elastic but in a less degree;
only then the velocity of each will be less in proportion to the defect
of elasticity.
6. IF the velocities, wherewith the bodies meet, are not in the
proportion here supposed; but if one of the bodies, as A, has a swifter
velocity in comparison to the velocity of the other; then the effect
of this excess of velocity in the body A must be joined to the effect
now mentioned, after the manner of this following example. Let A be
twice as great as B, and move with the same swiftness as B. Here A
moves with twice that degree of swiftness, which would answer to the
forementioned proportion. For A being double to B, if it moved but
with half the swiftness, wherewith B advances, it has been just now
shewn, that the two bodies upon meeting would stop, if they were not
elastic; and if they were elastic, that they would each recoil, so as
to cause A to return with half the velocity, wherewith B would return.
But it is evident from hence, that B by encountring A will annul half
its velocity, if the bodies be not elastic; and the future motion of
the bodies will be the same, as if A had advanced against B at rest
with half the velocity here assigned to it. If the bodies be elastic,
the velocity of A and B after the stroke may be thus discovered. As
the two bodies advance against each other, the velocity, with which
they meet, is made up of the velocities of both bodies added together.
After the stroke their elasticity will separate them again. The degree
of elasticity will determine what proportion the velocity, wherewith
they separate, must bear to that, wherewith they meet. Divide this
velocity, with which the bodies separate into two parts, that one of
the parts bear to the other the same proportion, as the body A bears to
B; and ascribe the lesser part to the greater body A, and the greater
part of the velocity to the lesser body B. Then take the part ascribed
to A from the common velocity, which A and B would have had after the
stroke, if they had not been elastic; and add the part ascribed to B to
the same common velocity. By this means the true velocities of A and B
after the stroke will be made known.
7. IF the bodies are perfectly elastic, the great ~HUYGENS~
has laid down this rule for finding their motion after concourse[46].
Any straight line C D (in fig. 4, 5.) being drawn, let it be divided
in E, that C E bear the same proportion to E D, as the swiftness of A
bore to the swiftness of B before the stroke. Let the same line C D be
also divided in F, that C F bear the same proportion to F D, as the
body B bears to the body A. Then F G being taken equal to F E, if the
point G falls within the line C D, both the bodies shall recoil after
the stroke, and the velocity, wherewith the body A shall return, will
bear the same proportion to the velocity, wherewith B shall return, as
G C bears to G D; but if the point G falls without the line C D, then
the bodies after their concourse shall both proceed to move the same
way, and the velocity of A shall bear to the velocity of B the same
proportion, that G C bears to G D, as before.
8. IF the body B had stood still, and received the impulse of the other
body A upon it; the effect has been already explained in the case, when
the bodies are not elastic. And when they are elastic, the result of
their collision is found by combining the effect of the elasticity with
the other effect, in the same manner as in the last case.
9. WHEN the bodies are perfectly elastic, the rule of
~HUYGENS~[47] here is to divide the line C D (fig. 6.) in E as
before, and to take E G equal to E D. And by these points thus found,
the motion of each body after the stroke is determined, as before.
10. IN the next place, suppose the bodies A and B were both moving the
same way, but A with a swifter motion, so as to overtake B, and strike
against it. The effect of the percussion or stroke, when the bodies are
not elastic, is discovered by finding the common motion, which the two
bodies would have after the stroke, if B were at rest, and A were to
advance against it with a velocity equal to the excess of the present
velocity of A above the velocity of B; and by adding to this common
velocity thus found the velocity of B.
11. IF the bodies are elastic, the effect of the elasticity is to be
united with this other, as in the former cases.
12. WHEN the bodies are perfectly elastic, the rule of HUYGENS[48] in
this case is to prolong C D (fig. 7.) and to take in it thus prolonged
C E in the same proportion to E D, as the greater velocity of A bears
to the lesser velocity of B; after which F G being taken equal to F E,
the velocities of the two bodies after the stroke will be determined,
as in the two preceding cases.
13. THUS I have given the sum of what has been written concerning the
effects of percussion, when two bodies freely in motion strike directly
against each other; and the results here set down, as the consequence
of our reasoning from the laws of motion, answer most exactly to
experience. A particular set of experiments has been invented to make
trial of these effects of percussion with the greatest exactness. But
I must defer these experiments, till I have explained the nature of
pendulums[49]. I shall therefore now proceed to describe some of the
appearances, which are caused in bodies from the influence of the power
of gravity united with the general laws of motion; among which the
motion of the pendulum will be included.
14. THE most simple of these appearances is, when bodies fall down
merely by their weight. In this case the body increases continually
its velocity, during the whole time of its fall, and that in the very
same proportion as the time increases. For the power of gravity acts
constantly on the body with the same degree of strength: and it has
been observed above in the first law of motion, that a body being once
in motion will perpetually preserve that motion without the continuance
of any external influence upon it: therefore, after a body has been
once put in motion by the force of gravity, the body would continue
that motion, though the power of gravity should cease to act any
farther upon it; but, if the power of gravity continues still to draw
the body down, fresh degrees of motion must continually be added to
the body; and the power of gravity acting at all times with the same
strength, equal degrees of motion will constantly be added in equal
portions of time.
15. THIS conclusion is not indeed absolutely true: for we shall find
hereafter[50], that the power of gravity is not of the same strength at
all distances from the center of the earth. But nothing of this is in
the least sensible in any distance, to which we can convey bodies. The
weight of bodies is the very same to sense upon the highest towers or
mountains, as upon the level ground; so that in all the observations
we can make, the forementioned proportion between the velocity of a
falling body and the time, in which it has been descending, obtains
without any the least perceptible difference.
16. FROM hence it follows, that the space, through which a body falls,
is not proportional to the time of the fall; for since the body
increases its velocity, a greater space will be passed over in the same
portion of time at the latter part of the fall, than at the beginning.
Suppose a body let fall from the point A (in fig. 8.) were to descend
from A to B in any portion of time; then if in an equal portion of time
it were to proceed from B to C; I say, the space B C is greater than A
B; so that the time of the fall from A to C being double the time of
the fall from A to B, A C shall be more than double of A B.
17. THE geometers have proved, that the spaces, through which bodies
fall thus by their weight, are just in a duplicate or two-fold
proportion of the times, in which the body has been falling. That is,
if we were to take the line D E in the same proportion to A B, as the
time, which the body has imployed in falling from A to C, bears to the
time of the fall from A to B; then A C will be to D E in the same
proportion. In particular, if the time of the fall through A C be twice
the time of the fall through A B; then D E will be twice A B, and A C
twice D E; or A C four times A B. But if the time of the fall through
A C had been thrice the time of the fall through A B; D E would have
been treble of A B, and A C treble of D E; that is, A C would have been
equal to nine times A B.
18. IF a body fall obliquely, it will approach the ground by slower
degrees, than when it falls perpendicularly. Suppose two lines A B, A
C (in fig. 9.) were drawn, one perpendicular, and the other oblique to
the ground D E: then if a body were to descend in the slanting line
A C; because the power of gravity draws the body directly downwards,
if the line A C supports the body from falling in that manner, it
must take off part of the effect of the power of gravity; so that
in the time, which would have been sufficient for the body to have
fallen through the whole perpendicular line A B, the body shall not
have passed in the line A C a length equal to A B; consequently the
line A C being longer than A B, the body shall most certainly take up
more time in passing through A C, than it would have done in falling
perpendicularly down through A B.
19. THE geometers demonstrate, that the time, in which the body
will descend through the oblique straight line A C, bears the same
proportion to the time of its descent through the perpendicular A B,
as the line it self A C bears to A B. And in respect to the velocity,
which the body will have acquired in the point C, they likewise
prove, that the length of the time imployed in the descent through A
C so compensates the diminution of the influence of gravity from the
obliquity of this line, that though the force of the power of gravity
on the body is opposed by the obliquity of the line A C, yet the time
of the body’s descent shall be so much prolonged, that the body shall
acquire the very same velocity in the point C, as it would have got at
the point B by falling perpendicularly down.
20. IF a body were to descend in a crooked line, the time of its
descent cannot be determined in so simple a manner; but the same
property, in relation to the velocity, is demonstrated to take place in
all cases: that is, in whatever line the body descends, the velocity
will always be answerable to the perpendicular height, from which the
body has fell. For instance, suppose the body A (in fig. 10.) were hung
by a string to the pin B. If this body were let fall, till it came to
the point C perpendicularly under B, it will have moved from A to C in
the arch of a circle. Then the horizontal line A D being drawn, the
velocity of the body in C will be the same, as if it had fallen from
the point D directly down to C.
21. IF a body be thrown perpendicularly upward with any force, the
velocity, wherewith the body ascends, shall continually diminish,
till at length it be wholly taken away; and from that time the body
will begin to fall down again, and pass over a second time in its
descent the line, wherein it ascended; falling through this line with
an increasing velocity in such a manner, that in every point thereof,
through which it falls, it shall have the very same velocity, as it
had in the same place, when it ascended; and consequently shall come
down into the place, whence it first ascended, with the velocity which
was at first given to it. Thus if a body were thrown perpendicularly
up in the line A B (in fig. II.) with such a force, as that it should
stop at the point B, and there begin to fall again; when it shall have
arrived in its descent to any point as C in this line, it shall there
have the same velocity, as that wherewith it passed by this point C
in its ascent; and at the point A it shall have gained as great a
velocity, as that wherewith it was first thrown upwards. As this is
demonstrated by the geometrical writers; so, I think, it will appear
evident, by considering only, that while the body descends, the power
of gravity must act over again, in an inverted order, all the influence
it had on the body in its ascent; so as to give again to the body the
same degrees of velocity, which it had taken away before.
22. AFTER the same manner, if the body were thrown upwards in the
oblique straight line C A (in fig. 9.) from the point C, with such a
degree of velocity as just to reach the point A; it shall by its own
weight return again through the line A C by the same degrees, as it
ascended.
23. AND lastly, if a body were thrown with any velocity in a line
continually incurvated upwards, the like effect will be produced upon
its return to the point, whence it was thrown. Suppose for instance,
the body A (in fig. 12.) were hung by a string A B. Then if this body
be impelled any way, it must move in the arch of a circle. Let it
receive such an impulse, as shall cause it to move in the arch A C; and
let this impulse be of such strength, that the body may be carried from
A as far as D, before its motion is overcome by its weight: I say here,
that the body forthwith returning from D, shall come again into the
point A with the same velocity, as that wherewith it began to move.
24. IT will be proper in this place to observe concerning the power of
gravity, that its force upon any body does not at all depend upon the
shape of the body; but that it continues constantly the same without
any variation in the same body, whatever change be made in the figure
of the body: and if the body be divided into any number of pieces,
all those pieces shall weigh just the same, as they did, when united
together in one body: and if the body be of a uniform contexture,
the weight of each piece will be proportional to its bulk. This has
given reason to conclude, that the power of gravity acts upon bodies
in proportion to the quantity of matter in them. Whence it should
follow, that all bodies must fall from equal heights in the same space
of time. And as we evidently see the contrary in feathers and such
like substances, which fall very slowly in comparison of more solid
bodies; it is reasonable to suppose, that some other cause concurs to
make so manifest a difference. This cause has been found by particular
experiments to be the air. The experiments for this purpose are made
thus. They set up a very tall hollow glass; within which near the top
they lodge a feather and some very ponderous body, usually a piece
of gold, this metal being the most weighty of any body known to us.
This glass they empty of the air contained within it, and by moving a
wire, which passes through the top of the glass, they let the feather
and the heavy body fall together; and it is always found, that as the
two bodies begin to descend at the same time, so they accompany each
other in the fall, and come to the bottom at the very same instant,
as near as the eye can judge. Thus, as far as this experiment can be
depended on, it is certain, that the effect of the power of gravity
upon each body is proportional to the quantity of solid matter, or
to the power of inactivity in each body. For in the limited sense,
which we have given above to the word motion, it has been shown, that
the same force gives to all bodies the same degree of motion, and
different forces communicate different degrees of motion proportional
to the respective powers[51]. In this case, if the power of gravity
were to act equally upon the feather, and upon the more solid body,
the solid body would descend so much slower than the feather, as to
have no greater degree of motion than the feather: but as both bodies
descend with equal swiftness, the degree of motion in the solid body is
greater than in the feather, bearing the same proportion to it, as the
quantity of matter in the solid body to the quantity of matter in the
feather. Therefore the effect of gravity on the solid body is greater
than on the feather, in proportion to the greater degree of motion
communicated; that is, the effect of the power of gravity on the solid
body bears the same proportion to its effect on the feather, as the
quantity of matter in the solid body bears to the quantity of matter
in the feather. Thus it is the proper deduction from this experiment,
that the power of gravity acts not on the surface of bodies only, but
penetrates the bodies themselves most intimately, and operates alike
on every particle of matter in them. But as the great quickness, with
which the bodies fall, leaves it something uncertain, whether they do
descend absolutely in the same time, or only so nearly together, that
the difference in their swift motion is not discernable to the eye;
this property of the power of gravity, which has here been deduced from
this experiment, is farther confirmed by pendulums, whose motion is
such, that a very minute difference would become sufficiently sensible.
This will be farther discoursed on in another place[52]; but here I
shall make use of the principle now laid down to explain the nature of
what is called the center of gravity in bodies.
25. THE center of gravity is that point, by which if a body be
suspended, it shall hang at rest in any situation. In a globe of a
uniform texture the center of gravity is the same with the center of
the globe; for as the parts of the globe on every side of its center
are similarly disposed, and the power of gravity acts alike on every
part; it is evident, that the parts of the globe on each side of the
center are drawn with equal force, and therefore neither side can
yield to the other; but the globe, if supported at its center, must
of necessity hang at rest. In like manner, if two equal bodies A and
B (in fig. 13.) be hung at the extremities of an inflexible rod C D,
which should have no weight; these bodies, if the rod be supported at
its middle E, shall equiponderate; and the rod remain without motion.
For the bodies being equal and at the same distance from the point of
support E, the power of gravity will act upon each with equal strength,
and in all respects under the same circumstances; therefore the weight
of one cannot overcome the weight of the other. The weight of A can no
more surmount the weight of B, than the weight of B can surmount the
weight of A. Again, suppose a body as A B (in fig. 14.) of a uniform
texture in the form of a roller, or as it is more usually called a
cylinder, lying horizontally. If a straight line be drawn between C and
D, the centers of the extreme circles of this cylinder; and if this
straight line, commonly called the axis of the cylinder, be divided
into two equal parts in E: this point E will be the center of gravity
of the cylinder. The cylinder being a uniform figure, the parts on each
side of the point E are equal, and situated in a perfectly similar
manner; therefore this cylinder, if supported at the point E, must hang
at rest, for the same reason as the inflexible rod above-mentioned
will remain without motion, when suspended at its middle point. And it
is evident, that the force applied to the point E, which would uphold
the cylinder, must be equal to the cylinder’s weight. Now suppose two
cylinders of equal thickness A B and C D to be joined together at C B,
so that the two axis’s E F, and F G lie in one straight line. Let the
axis E F be divided into two equal parts at H, and the axis F G into
two equal parts at I. Then because the cylinder A B would be upheld
at rest by a power applied in H equal to the weight of this cylinder,
and the cylinder C D would likewise be upheld by a power applied in I
equal to the weight of this cylinder; the whole cylinder A D will be
supported by these two powers: but the whole cylinder may likewise be
supported by a power applied to K, the middle point of the whole axis
E G, provided that power be equal to the weight of the whole cylinder.
It is evident therefore, that this power applied in K will produce the
same effect, as the two other powers applied in H and I. It is farther
to be observed, that H K is equal to half F G, and K I equal to half
E F; for E K being equal to half E G, and E H equal to half E F, the
remainder H K must be equal to half the remainder F G; so likewise G
K being equal to half G E, and G I equal to half G F, the remainder I
K must be equal to half the remainder E F. It follows therefore, that
H K bears the same proportion to K I, as F G bears to E F. Besides,
I believe, my readers will perceive, and it is demonstrated in form
by the geometers, that the whole body of the cylinder C D bears the
same proportion to the whole body of the cylinder A B, as the axis F G
bears to the axis E F[53]. But hence it follows, that in the two powers
applied at H and I, the power applied at H bears the same proportion to
the power applied at I, as K I bears to K H. Now suppose two strings
H L and I M extended upwards, one from the point H and the other from
I, and to be laid hold on by two powers, one strong enough to hold up
the cylinder A B, and the other of strength sufficient to support the
cylinder C D. Here as these two powers uphold the whole cylinder, and
therefore produce an effect, equal to what would have been produced
by a power applied to the point K of sufficient force to sustain the
whole cylinder: it is manifest, that if the cylinder be taken away,
the axis only being left, and from the point K a string, as K N, be
extended, which shall be drawn down by a power equivalent to the weight
of the cylinder, this power shall act against the other two powers, as
much as the cylinder acted against them; and consequently these three
powers shall be upon a balance, and hold the axis H I fixed between
them. But if these three powers preserve a mutual balance, the two
powers applied to the strings H L and I M are a balance to each other;
the power applied to the string H L bearing the same proportion to
the power applied to the string I M, as the distance I K bears to the
distance K H. Hence it farther appears, that if an inflexible rod A B
(in fig. 15.) be suspended by any point C not in the middle thereof;
and if at A the end of the shorter arm be hung a weight, and at B
the end of the longer arm be also hung a weight less than the other,
and that the greater of these weights bears to the lesser the same
proportion, as the longer arm of the rod bears to the shorter; then
these two weights will equiponderate: for a power applied at C equal to
both these weights will support without motion the rod thus charged;
since here nothing is changed from the preceding case but the situation
of the powers, which are now placed on the contrary sides of the line,
to which they are fixed. Also for the same reason, if two weights A
and B (in fig. 16.) were connected together by an inflexible rod C D,
drawn from C the center of gravity of A to D the center of gravity of
B; and if the rod C D were to be so divided in E, that the part D E
bear the same proportion to the other part C E, as the weight A bears
to the weight B: then this rod being supported at E will uphold the
weights, and keep them at rest without motion. This point E, by which
the two bodies A and B will be supported, is called their common center
of gravity. And if a greater number of bodies were joined together,
the point, by which they could all be supported, is called the common
center of gravity of them all. Suppose (in fig. 17.) there were three
bodies A, B, C, whose respective centers of gravity were joined by the
three lines D E, D F, E F: the line D E being so divided in G, that D
G bear the same proportion to G E, as B bears to A; G is the center
of gravity common to the two bodies A and B; that is, a power equal
to the weight of both the bodies applied to G would support them, and
the point G is pressed as much by the two weights A and B, as it would
be, if they were both hung together at that point. Therefore, if a
line be drawn from G to F, and divided in H, so that G H bear the same
proportion to H F, as the weight C bears to both the weights A and
B, the point H will be the common center of gravity of all the three
weights; for H would be their common center of gravity, if both the
weights A and B were hung together at G, and the point G is pressed as
much by them in their present situation, as it would be in that case.
In the same manner from the common center of these three weights, you
might proceed to find the common center, if a fourth weight were added,
and by a gradual progress might find the common center of gravity
belonging to any number of weights whatever.
26. AS all this is the obvious consequence of the proposition laid down
for assigning the common center of gravity of any two weights, by the
same proposition the center of gravity of all figures is found. In a
triangle, as A B C (in fig. 18.) the center of gravity lies in the line
drawn from the middle point of any one of the sides to the opposite
angle, as the line B D is drawn from D the middle of the line A C to
the opposite angle B[54]; so that if from the middle of either of the
other sides, as from the point E in the side A B, a line be drawn, as
E C, to the opposite angle; the point F, where this line crosses the
other line B D, will be the center of gravity of the triangle[55].
Likewise D F is equal to half F B, and E F equal to half F C[56]. In a
hemisphere, as A B C (fig. 19.) if from D the center of the base the
line D B be erected perpendicular to that base, and this line be so
divided in E, that D E be equal to three fifths of B E, the point E is
the center of gravity of the hemisphere[57].
27. IT will be of use to observe concerning the center of gravity of
bodies; that since a power applied to this center alone can support
a body against the power of gravity, and hold it fixed at rest; the
effect of the power of gravity on a body is the same, as if that whole
power were to exert itself on the center of gravity only. Whence it
follows, that, when the power of gravity acts on a body suspended by
any point, if the body is so suspended, that the center of gravity
of the body can descend; the power of gravity will give motion to
that body, otherwise not: or if a number of bodies are so connected
together, that, when any one is put into motion, the rest shall, by
the manner of their being joined, receive such motion, as shall keep
their common center of gravity at rest; then the power of gravity
shall not be able to produce any motion in these bodies, but in all
other cases it will. Thus, if the body A B (in fig. 20, 21.) whose
center of gravity is C, be hung on the point A, and the center C be
perpendicularly under A (as in fig. 20.) the weight of the body will
hold it still without motion, because the center C cannot descend any
lower. But if the body be removed into any other situation, where the
center C is not perpendicularly under A (as in fig. 21.) the body by
its weight will be put into motion towards the perpendicular situation
of its center of gravity. Also if two bodies A, B (in fig. 22.) be
joined together by the rod C D lying in an horizontal situation, and
be supported at the point E; if this point be the center of gravity
common to the two bodies, their weight will not put them into motion;
but if this point E is not their common center of gravity, the bodies
will move; that part of the rod C D descending, in which the common
center of gravity is found. So in like manner, if these two bodies were
connected together by any more complex contrivance; yet if one of
the bodies cannot move without so moving the other, that their common
center of gravity shall rest, the weight of the bodies will not put
them in motion, otherwise it will.
28. I SHALL proceed in the next place to speak of the mechanical
powers. These are certain instruments or machines, contrived for the
moving great weights with small force; and their effects are all
deducible from the observation we have just been making. They are
usually reckoned in number five; the lever, the wheel and axis, the
pulley, the wedge, and the screw; to which some add the inclined
plane. As these instruments have been of very ancient use, so the
celebrated ~ARCHIMEDES~ seems to have been the first, who discovered
the true reason of their effects. This, I think, may be collected
from what is related of him, that some expressions, which he used to
denote the unlimited force of these instruments, were received as very
extraordinary paradoxes: whereas to those, who had understood the cause
of their great force, no expressions of that kind could have appeared
surprizing.
29. ALL the effects of these powers may be judged of by this one rule,
that, when two weights are applied to any of these instruments, the
weights will equiponderate, if, when put into motion, their velocities
will be reciprocally proportional to their respective weights. And what
is said of weights, must of necessity be equally understood of any
other forces equivalent to weights, such as the force of a man’s arm,
a stream of water, or the like.
30. BUT to comprehend the meaning of this rule, the reader must know,
what is to be understood by reciprocal proportion; which I shall now
endeavour to explain, as distinctly as I can; for I shall be obliged
very frequently to make use of this term. When any two things are so
related, that one increases in the same proportion as the other, they
are directly proportional. So if any number of men can perform in a
determined space of time a certain quantity of any work, suppose drain
a fish-pond, or the like; and twice the number of men can perform twice
the quantity of the same work, in the same time; and three times the
number of men can perform as soon thrice the work; here the number
of men and the quantity of the work are directly proportional. On
the other hand, when two things are so related, that one decreases
in the same proportion, as the other increases, they are said to be
reciprocally proportional. Thus if twice the number of men can perform
the same work in half the time, and three times the number of men can
finish the same in a third part of the time; then the number of men
and the time are reciprocally proportional. We shewed above[58] how to
find the common center of gravity of two bodies, there the distances of
that common center from the centers of gravity of the two bodies are
reciprocally proportional to the respective bodies. For C E in fig. 16.
being in the same proportion to E D, as B bears to A; C E is so much
greater in proportion than E D, as A is less in proportion than B.
31. NOW this being understood, the reason of the rule here stated will
easily appear. For if these two bodies were put in motion, while the
point E rested, the velocity, wherewith A would move, would bear the
same proportion to the velocity, wherewith B would move, as E C bears
to E D. The velocity therefore of each body, when the common center of
gravity rests, is reciprocally proportional to the body. But we have
shewn above[59], that if two bodies are so connected together, that the
putting them in motion will not move their common center of gravity;
the weight of those bodies will not produce in them any motion.
Therefore in any of these mechanical engines, if, when the bodies are
put into motion, their velocities are reciprocally proportional to
their respective weights, whereby the common center of gravity would
remain at rest; the bodies will not receive any motion from their
weight, that is, they will equiponderate. But this perhaps will be yet
more clearly conceived by the particular description of each mechanical
power.
32. THE lever was first named above. This is a bar made use of to
sustain and move great weights. The bar is applied in one part to
some strong support; as the bar A B (in fig. 23, 24.) is applied at
the point C to the support D. In some other part of the bar, as E, is
applied the weight to be sustained or moved; and in a third place, as
F, is applied another weight or equivalent force, which is to sustain
or move the weight at E. Now here, if, when the level should be put in
motion, and turned upon the point C, the velocity, wherewith the point
F would move, bears the same proportion to the velocity, wherewith the
point E would move, as the weight at E bears to the weight or force at
F; then the lever thus charged will have no propensity to move either
way. If the weight or other force at F be not so great as to bear this
proportion, the weight at E will not be sustained; but if the force at
F be greater than this, the weight at E will be surmounted. This is
evident from what has been said above[60], when the forces at E and F
are placed (as in fig. 23.) on different sides of the support D. It
will appear also equally manifest in the other case, by continuing
the bar B C in fig. 24. on the other side of the support D, till C G
be equal to C F, and by hanging at G a weight equivalent to the power
at F; for then, if the power at F were removed, the two weights at G
and E would counterpoize each other, as in the former case: and it is
evident, that the point F will be lifted up by the weight at G with the
same degree of force, as by the other power applied to F; since, if the
weight at E were removed, a weight hung at F equal to that at G would
balance the lever, the distances C G and C F being equal.
33. IF the two weights, or other powers, applied to the lever do
not counterbalance each other; a third power may be applied in any
place proposed of the lever, which shall hold the whole in a just
counterpoize. Suppose (in fig. 25.) the two powers at E and F did not
equiponderate, and it were required to apply a third power to the point
G, that might be sufficient to balance the lever. Find what power in
F would just counterbalance the power in E; then if the difference
between this power and that, which is actually applied at F, bear the
same proportion to the third power to be applied at G, as the distance
C G bears to C F; the lever will be counterpoized by the help of this
third power, if it be so applied as to act the same way with the power
in F, when that power is too small to counterbalance the power in E;
but otherwise the power in G must be so applied, as to act against the
power in F. In like manner, if a lever were charged with three, or any
greater number of weights or other powers, which did not counterpoize
each other, another power might be applied in any place proposed,
which should bring the whole to a just balance. And what is here said
concerning a plurality of powers, may be equally applied to all the
following cases.
34. IF the lever should consist of two arms making an angle at the
point C (as in fig. 26.) yet if the forces are applied perpendicularly
to each arm, the same proportion will hold between the forces applied,
and the distances of the center, whereon the lever rests, from the
points to which they are applied. That is, the weight at E will be to
the force in F in the same proportion, as C F bears to C E.
35. BUT whenever the forces applied to the lever act obliquely to the
arm, to which they are applied (as in fig. 27.) then the strength of
the forces is to be estimated by lines let fall from the center of
the lever to the directions, wherein the forces act. To balance the
levers in fig. 27, the weight or other force at F will bear the same
proportion to the weight at E, as the distance C E bears to C G the
perpendicular let fall from the point C upon the line, which denotes
the direction wherein the force applied to F acts: for here, if the
lever be put into motion, the power applied to F will begin to move in
the direction of the line F G; and therefore its first motion will be
the same, as the motion of the point G.
36. WHEN two weights hang upon a lever, and the point, by which the
lever is supported, is placed in the middle between the two weights,
that the arms of the lever are both of equal length; then this lever is
particularly called a balance; and equal weights equiponderate as in
common scales. When the point of support is not equally distant from
both weights, it constitutes that instrument for weighing, which is
called a steelyard. Though both in common scales, and the steelyard,
the point, on which the beam is hung, is not usually placed just in the
same straight line with the points, that hold the weights, but rather
a little above (as in fig. 28.) where the lines drawn from the point
C, whereon the beam is suspended, to the points E and F, on which the
weights are hung, do not make absolutely one continued line. If the
three points E, C, and F were in one straight line, those weights,
which equiponderated, when the beam hung horizontally, would also
equiponderate in any other situation.
[Illustration]
But we see in these instruments, when they are charged with weights,
which equiponderate with the beam hanging horizontally; that, if
the beam be inclined either way, the weight most elevated surmounts
the other, and descends, causing the beam to swing, till by degrees
it recovers its horizontal position. This effect arises from the
forementioned structure: for by this structure these instruments are
levers composed of two arms, which make an angle at the point of
support (as in fig. 29, 30.) the first of which represents the case
of the common balance, the second the case of the steelyard. In the
first, where C E and C F are equal, equal weights hung at E and F will
equiponderate, when the points E and F are in an horizontal situation.
Suppose the lines E G and F H to be perpendicular to the horizon, then
they will denote the directions, wherein the forces applied to E and
F act. Therefore the proportion between the weights at E and F, which
shall equiponderate, are to be judged of by perpendiculars, as C I, C
K, let fall from C upon E G and F H: so that the weights being equal,
the lines C I, C K, must be equal also, when the weights equiponderate.
But I believe my readers will easily see, that since C E and C F are
equal, the lines C I and C K will be equal, when the points E and F are
horizontally situated.
37. IF this lever be set into any other position (as in fig. 31.) then
the weight, which is raised highest, will outweigh the other. Here,
if the point F be raised higher than E, the perpendicular C K will be
longer than C I: and therefore the weights would equiponderate, if
the weight at F were less than the weight at E. But the weight at
F is equal to that at E; therefore is greater, than is necessary to
counterbalance the weight at E, and consequently will outweigh it, and
draw the beam of the lever down.
38. IN like manner in the case of the steelyard (fig. 32.) if the
weights at E and F are so proportioned, as to equiponderate, when the
points E and F are horizontally situated; then in any other situation
of this lever the weight, which is raised highest, will preponderate.
That is, if in the horizontal situation of the points E and F the
weight at F bears the same proportion to the weight at E, as C I bears
to C K; then, if the point F be raised higher than E (as in fig. 32.)
the weight at F shall bear a greater proportion to the weight at E,
than C I bears to C K.
39. FARTHER a lever may be hung upon an axis, and then the two arms
of the lever need not be continuous, but fixed to different parts of
this axis; as in fig. 33, where the axis A B is supported by its two
extremities A and B. To this axis one arm of the lever is fixed at the
point C, the other at the point D. Now here, if a weight be hung at E,
the extremity of that arm, which is fixed to the axis at the point C;
and another weight be hung at F, the extremity of the arm, which is
fixed on the axis at D; then these weights will equiponderate, when the
weight at E bears the same proportion to the weight at F, as the arm D
F bears to C E.
40. THIS is the case, if both the arms are perpendicular to the axis,
and lie (as the geometers express themselves) in the same plane; or, in
other words, if the arms are so fixed perpendicularly upon the axis,
that, when one of them lies horizontally, the other shall also be
horizontal. If either arm stand not perpendicular to the axis; then, in
determining the proportion between the weights, instead of the length
of that arm, you must use the perpendicular let fall upon the axis from
the extremity of that arm. If the arms are not so fixed as to become
horizontal, at the same time; the method of assigning the proportion
between the weights is analogous to that made use of above in levers,
which make an angle at the point, whereon they are supported.
41. FROM this case of the lever hung on an axis, it is easy to make a
transition to another mechanical power, the wheel and axis.
42. THIS instrument is a wheel fixed on a roller, the roller being
supported at each extremity so as to turn round freely with the wheel,
in the manner represented in fig. 34, where A B is the wheel, C D the
roller, and E F its two supports. Now suppose a weight G hung by a cord
wound round the roller, and another weight H hung by a cord wound about
the wheel the contrary way: that these weights may support each other,
the weight H must bear the same proportion to the weight G, as the
thickness of the roller bears to the diameter of the wheel.
43. SUPPOSE the line _k l_ to be drawn through the middle of the
roller; and from the place of the roller, where the cord, on which the
weight G hangs, begins to leave the roller, as at _m_, let the line_
m n_ be drawn perpendicularly to _k l_; and from the point, where
the cord holding the weight H begins to leave the wheel, as at _o_,
let the line _o p_ be drawn perpendicular to _k l_. This being done,
the two lines _o p_ and _m n_ represent two arms of a lever fixed on
the axis _k l_; consequently the weight H will bear to the weight G
the same proportion, as _m n_ bears to _o p_. But _m n_ bears the
same proportion to _o p_, as the thickness of the roller bears to the
diameter of the wheel; for _m n_ is half the thickness of the roller,
and _o p_ half the diameter of the wheel.
44. IF the wheel be put into motion, and turned once round, that the
cord, on which the weight G hangs, be wound once more round the axis;
then at the same time the cord, whereon the weight H hangs, will be
wound off from the wheel one circuit. Therefore the velocity of the
weight G will bear the same proportion to the velocity of the weight H,
as the circumference of the roller to the circumference of the wheel.
But the circumference of the roller bears the same proportion to the
circumference of the wheel, as the thickness of the roller bears to
the diameter of the wheel, consequently the velocity of the weight
G bears to the velocity of the weight H the same proportion, as the
thickness of the roller bears to the diameter of the wheel, which is
the proportion that the weight H bears to the weight G. Therefore as
before in the lever, so here also the general rule laid down above is
verified, that the weights equiponderate, when their velocities would
be reciprocally proportional to their respective weights.
45. IN like manner, if on the same axis two wheels of different sizes
are fixed (as in fig. 35.) and a weight hung on each; the weights will
equiponderate, if the weight hung on the greater wheel bear the same
proportion to the weight hung on the lesser, as the diameter of the
lesser wheel bears to the diameter of the greater.
46. IT is usual to join many wheels together in the same frame, which
by the means of certain teeth, formed in the circumference of each
wheel, shall communicate motion to each other. A machine of this nature
is represented in fig. 36. Here A B C is a winch, upon which is fixed
a small wheel D indented with teeth, which move in the like teeth of
a larger wheel E F fixed on the axis G H. Let this axis carry another
wheel I, which shall move in like manner a greater wheel K L fixed on
the axis M N. Let this axis carry another small wheel O, which after
the same manner shall turn about a larger wheel P Q fixed on the roller
R S, on which a cord shall be wound, that holds a weight, as T. Now
the proportion required between the weight T and a power applied to
the winch at A sufficient to support the weight, will most easily be
estimated, by computing the proportion, which the velocity of the point
A would bear to the velocity of the weight. If the winch be turned
round, the point A will describe a circle as A V. Suppose the wheel E F
to have ten times the number of teeth, as the wheel D; then the winch
must turn round ten times to carry the wheel E F once round. If wheel K
L has also ten times the number of teeth, as I, the wheel I must turn
round ten times to carry the wheel K L once round; and consequently
the winch A B C must turn round an hundred times to turn the wheel K L
once round. Lastly, if the wheel P Q has ten times the number of teeth,
as the wheel O, the winch must turn about one thousand times in order
to turn the wheel P Q, or the roller R S once round. Therefore here
the point A must have gone over the circle A V a thousand times, in
order to lift the weight T through a space equal to the circumference
of the roller R S: whence it follows, that the power applied at A will
balance the weight T, if it bear the same proportion to it, as the
circumference of the roller to one thousand times the circle A V; or
the same proportion as half the thickness of the roller bears to one
thousand times A B.
47. I SHALL now explain the effect of the pulley. Let a weight hang by
a pulley, as in fig. 37. Here it is evident, that the power A, by which
the weight B is supported, must be equal to the weight; for the cord C
D is equally strained between them; and if the weight B move, the power
A must move with equal velocity. The pulley E has no other effect, than
to permit the power A to act in another direction, than it must have
done, if it had been directly applied to support the weight without the
intervention of any such instrument.