NEWTON S PRINCIPIA. THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY, BY SIR ISAAC NEWTON *prt 2
one
which hath its beginning and its end in the knowledge of Him whose glory the
heavens declare, and whose handiwork the firmament showeth forth. The
introduction of the pure and lofty doctrines of the PRINCIPIA was perseveringly
resisted. Descartes, with his system of vortices, had sown plausibly to the
imagination, and error had struck down deeply, and shot up luxuriantly, not
only in the popular, but in the scientific mind. Besides the idea in itself so
simple and so grand that the great masses of the planets were 38 LIFE OF SIR
ISAAC NEWTON. suspended in empty space, and retained in their orbits by an in
visible influence residing in the sun was to the ignorant a thing
inconceivable, and to the learned a revival of the occult qualities of the
ancient physics. This remark applies particularly to the continent. Leibnitz
misapprehended ; Huygens in part rejected ; John Bernouilli opposed ; and
Fontenelle never received the doc trines of the PRINCIPIA. So that, the saying
of Voltaire is prob ably true, that though Newton survived the publication of
his great work more than forty years, yet, at the time of his death, lie had
not above twenty followers out of England. But in England, the reception of our
author s philosophy was rapid and triumphant. His own labours, while Lucasian
Pro fessor ; those of his successors in that Chair Whiston and Saunderson ;
those of Dr. Samuel Clarke, Dr. Laughton, Roger Cotes, and Dr. Bentley ; the
experimental lectures of Dr. Keill and Desaguliers ; the early and powerful
exertions of David Gregory at Edinburgh, and of his brother James Gregory at
St. Andrew s, tended to diffuse widely in England and Scotland a knowledge of,
and taste for the truths of the PRINCIPIA. Indeed, its mathematical doctrines
constituted, from the first, a regular part of academical instruction ; while
its physical truths, given to the public in popular lectures, illustrated by
experiments, had, before the lapse of twenty ) ( ar.s, become familiar to, and
adopted by the general mind. Pemberton s popular " View of Sir Isaac
Newton s Philosophy" was published, in 1728 ; and the year after ward,
an English translation of the PRINCIPIA, and System of the World, by Andrew
Motte. And since that period, the labours of Le Seur and Jacquier, of Thorpe,
of Jebb, of Wright and others have greatly contributed to display the most
hidden treasures of the PRINCIPIA. About the time of the publication of the
Principia, James II., bent on re-establishing the Romish Faith, had, among
other ille gal acts, ordered by mandamus, the University of Cambridge to confer
the degree of Master of Arts upon an ignorant monk. Obedience to this mandate
was resolutely refused. Newton was one of the nine delegates chosen to defend
the independence of the University. They appeared before the High Court ; and
LIFE OF SIR ISAAC NEWTON. 39 successfully : the king abandoned his design. The
prominent part which our author took in these proceedings, and his eminence in
the scientific world, induced his proposal as one of the parlia mentary
representatives of the University. He was elected, in 1688, and sat in the
Convention Parliament till its dissolution. After the first year, however, he
seems to have given little or no attention to his parliamentary duties, being
seldom absent from the University till his appointment in the Mint, in 1695.
Newton began his theological researches sometime previous to 1691 ; in the
prime of his years, and in the matured vigour of his intellectual powers. From
his youth, as we have seen, he had devoted himself with an activity the most
unceasing, and an energy almost superhuman to the discovery of physical truth ;
giving to Philosophy a new foundation, and to Science a new temple. To pass on,
then, from the consideration of the material, more directly to that of the
spiritual, was a natural, nay, with so large and devout a soul, a necessary
advance. The Bible was to him of inestimable worth. In the elastic freedom,
which a pure and unswerving faith in Him of Nazareth gives, his mighty facul
ties enjoyed the only completest scope for development. His original endowment,
however great, combined with a studious application, however profound, would
never, without this libera tion from the dominion of passion and sense, have
enabled him to attain to that wondrous concentration and grasp of intellect,
for which Fame has as yet assigned him no equal. Gratefully he owned,
therefore, the same Author in the Book of Nature and the Book of Revelation.
These were to him as drops of the same unfathomable ocean ; as outrayings of
the same inner splendour ; as tones of the same ineffable voice ; as segments
of the same infinite curve. "With great joy he had found himself
enabled to proclaim, as an interpreter, from the hieroglyphs of Creation, the
existence of a God : and now, with greater joy, and in the fulness of his
knowledge, and in the fulness of his strength, he laboured to make clear, from
the utterances of the inspired Word, the far mightier confirmations of a Supreme
Good, in all its glorious amplitude of Being and of Attribute ; and to bring
the infallible workings thereof plainly home to the understandings and the 40
LIFE OF SIR ISAAC NEWTON. affections of his fellow-men ; and finally to add the
weight of his own testimony in favour of that Religion, whose truth is now. in
deed, " girded with the iron and the rock of a ponderous and co lossal
demonstration." His work, entitled, OBSERVATIONS UPON THE PROPHECIES
OF HOLY WRIT, PARTICULARLY THE PROPHECIES OF DANIEL AND THE APOCALYPSE OF ST.
JOHN, first published in London, in 1733 4to. consists of two parts : the one
devoted to the Prophecies oi Daniel, and the other to the Apocalypse of St.
John. In the first part, he treats concerning the compilers of the books of the
Old Testament ; of the prophetic language ; of the vision of the four beasts ;
of the kingdoms represented by the feet of the image composed of iron and clay
; of the ten kingdoms repre sented by the ten horns of the beast ; of the
eleventh horn of Daniel s fourth beast ; of the power which should change times
and laws ; of the kingdoms represented in Daniel by the ram and he-goat ; of
the prophecy of the seventy weeks ; of the times of the birth and passion of
Christ ; of the prophecy of the Scripture of Truth ; of the king who doeth
according to his will, and magnified himself above every god, and honoured
Mahuzzims, and regarded not the desire of women ; of the Mahuzzim, hon oured by
the king who doeth according to his will. In the sec ond part, he treats of the
time when the Apocalypse was written , of the scene of the vision, and the
relation which the Apocalypse has to the book of the law of Moses, and to the
worship of God in the temple ; of the relation which the Apocalypse has to the prophecies
of Daniel, and of the subject of the prophecy itself Newton regards the
prophecies as given, not for the gratification of man s curiosity, by enabling
him to foreknow ; but for his con viction that the world is governed by
Providence, by witnessing their fulfilment. Enough of prophecy, he thinks, has
already been fulfilled to afford the diligent seeker abundant evidence of God s
providence. The whole work is marked by profound erudition, sagacity and
argument. And not less learning, penetration and masterly reasoning are
conspicuous in his HISTORICAL ACCOUNT OF Two NOTABLE CORRUPTIONS OF SCRIPTURES
IN A LETTER TO A FRIEND. This LIFE OF SIR ISAAC NEWTON. 41 Treatise, first
accurately published in Dr. Horsley s edition of his works, relates to two texts
: the one, 1 Epistle of St. John v. 7 ; the other, 1 Epistle of St. Paul to
Timothy iii. 16. As this work had the effect to deprive the advocates of the
doctrine of the Trinity of two leading texts, Newton has been looked upon as an
Arian ; but there is absolutely nothing in his writings to warrant such a
conclusion. His regaining theological works consist of the LEXICON PROPHETICUM,
which was left incomplete ; a Latin Dissertation on the sacred cubit of the
Jews, which was translated into English, and published, in 1737. among the
Miscellaneous Works of John Greaves ; and FOUR LETTERS addressed to Dr.
Bentlty, contain ing some arguments in proof of a Deity. These Letters were
dated respectively : 10th December, 1692 ; 17th January, 1693 ; 25th February, 1693;
and llth February, 1693 the fourth bearing an earlier date than the third. The
best faculties and the profoundest acquirements of our author are convincingly
manifest in these lucid and powerful compositions. They were published in 1756,
and reviewed by Dr. Samuel Johnson. Newton s religious writings are
distinguished by their absolute freedom from prejudice. Everywhere, throughout
them, there glows the genuine nobleness of soul. To his whole life, indeed, we
may here fitly extend the same observation. He was most richly imbued with the
very spirit of the Scriptures which he so delighted to study and to meditate
upon. His was a piety, so fervent, so sincere and practical, that it rose up
like a holy incense from every thought and act. His a benevolence that not only
willed, but endeavoured the best for all. His a philanthropy that held in the
embracings of its love every brother-man. His a toleration of the largest and
the truest ; condemning per secution in every, even its mildest form ; and
kindly encouraging each striving after excellence : .1 toleration that came not
of indifference for the immoral and the impious met with their quick rebuke but
a toleration that came of the wise humbleness and the Christian charity, which
see, in the nothingness of self and the almightiness of TRUTH, no praise for
the ablest, and no blame for th^ feeblest in their strugglings upward to light
and life. 42 LIFE OF SIR ISAAC NEWTON, Tn the winter of 1691-2, on returning
from chapel, one morn ing, Newton foima tnat a favourite little dog, called
Diamond, had overturned a lighted taper on his desk, and that several pa pers
containing the results of certain optical experiments, were nearly consumed.
His only exclamation, on perceiving his loss, was, " Oh Diamond, Diamond,
little knowest thou the mischiel thou hast done," Dr. Brewster, in his
life of our author, gives the following extract from the manuscript Diary of
Mr. Abraham De La Pryme. a student in the University at the time of this oc
currence. " 1692. February, 3. What I heard to-day I must relate.
There is one Mr. Newton (whom I have very oft seen), Fellow of Trinity College,
that is mighty famous for his learning, being a most excellent mathematician,
philosopher, divine, &c. He has been Fellow of the Royal Society these many
years ; and among other very learned books and tracts, he : s written one upon
the mathe matical principles of philosophy, which has given him a mighty name,
he having received, especially from Scotland, abundance of congratulatory
letters for the same ; but of all the books he ever wrote, there was one of
colours and light, established upon thou sands of experiments which he had been
twenty years of making, and which had cost him many hundreds of pounds. This
book which he vaiued so much, and which was so much talked of, had the ill luck
to perish, and be utterly lost just when the learned author was almost at
pitting a conclusion at the same, after this manner : In a winter s morning,
leaving it among his other papers on his study table while he went to chapel,
the candle, which he had unfortunately left burning there, too, catched hold by
some means of other papers, and they fired the aforesaid book, and ut terly
consumed it and several other valuable writings ; arid which is most wonderful
did no further mischief. But when Mr. New ton came from chapel, and had seen
what was done, every one thought he would have run mad, he was so troubled
thereat that he was not himself for a month after. A long account of this his
system of colours you may find in the Transactions of the Royal Society, which
he had sent up to them long before this sad mis chance happened unto
him." LIFE OF SIR ISAAC NEWTON. 43 It will be borne in mind that all
of Newton s theological wri tings, with the exception of the Letters to Dr.
Bentley, were composed before this event which, we must conclude, from Pryme s
words, produced a serious impression upon our author for about a month. But M.
Biot, in his Life of Newton, relying on a memorandum contained in a small
manuscript Journal of Huygens, declares this occurrence to have caused a
deran-gement of New ton s intellect. M. Blot s opinions and deductions,
however, as well as those of La Place, upon this subject, were based upon
erroneous data, and have been overthrown by the clearest proof. There is not,
in fact, the least evidence that Newton s reason was, for a single moment,
dethroned ; on the contrary, the testimony is conclusive that he was, at all
times, perfectly capable of carry ing on his mathematical, metaphysical and
astronomical inquiries. Loss of sleep, loss of appetite, and irritated nerves
will disturb somewhat the equanimity of the most serene ; and an act done, or
language employed, under such temporary discomposure, is not a just criterion
of the general tone and strength of a man s mind. As to the accident itself, we
may suppose, whatever might have been its precise nature, that it greatly
distressed him, and, still further, that its shock may have originated the
train of nervous derangements, which afflicted him, more or less, for two years
afterward. Yet, during this very period of ill health, we find him putting
forth his highest powers. In 1692, he prepared for, and transmitted to Dr.
Wallis the first proposition of the Treatise on Quadratures, with examples of
it in first, second and third flux ions. He investigated, in the same year, the
subject of haloes ; making and recording numerous and important observations
rela tive thereto. Those profound and beautiful Letters to Dr. Bentley were written
at the close of this and the beginning of the next year. In October, 1693,
Locke, who was then about publishing a second edition of his work on the Human
Understanding, request ed Newton to reconsider his opinions on innate ideas.
And in 1694, he was zealously occupied in perfecting his lunar theory ;
visiting Flamstead, at the Royal Observatory of Greenwich, in September, and
obtaining a series of lunar observations ; and 14 LIFE OF SIR ISAAC NEWTON.
commencing, in October, a correspondence with that distinguished practical
Astronomer, which continued till 1698. We now arrive at the period when Newton
permanently with drew from the seclusion of a collegiate, and entered upon a
more active and public life. He was appointed Warden of the Mint, in 1695, through
the influence of Charles Montague, Chancellor of the Exchequer, and afterward
Earl of Halifax. The current roin of the nation had been adulterated and
debased, and Mon tague undertook a re-coinage. Our author s mathematical and
chemical knowledge proved eminently useful in accomplishing this difficult and
most salutary reform. In 1699, he was pro moted to the Mastership of the Mint
an office worth twelve or fifteen hundred pounds per annum, and which he held
during the remainder of his life. He wrote, in this capacity, an official Re
port on the Coinage, which has been published ; he also prepared a Table of
Assays of Foreign Coins, which was printed at the end of Dr. Arbuthnot s Tables
of Ancient Coins, Weights, and Measures, in 1727. Newton retained his
Professorship at Cambridge till 1703. But he had, on receiving the appointment
of Master of the Mint, in 1699, made Mr. Whiston his deputy, with all the
emoluments of the office ; and, on finally resigning, procured his nomination
to the vacant Chair. In January 1697, John Bernouilli proposed to the most
distin guished mathematicians of Europe two problems for solution. Leibnitz,
admiring the beauty of one of them, requested the time for solving it to be
extended to twelve months twice the period originally named. The delay was
readily granted. Newton, how ever, sent in, the day after he received the
problems, a solution of them to the President of the Royal Society. Bernouilli
obtained solutions from Newton, Leibinitz and the Marquis De L Hopital ; but Newton
s though anonymous, he immediately recognised " tanquam ungue
leonem" as the lion is known by his claw. We may mention here the
famous problem of the trajectories proposed by Leibnitz, in 1716, for the
purpose of "feeling the pulse of the English Analysts."
Newton received the problem about five o clock in the afternoon, as he was
returning from the LIFE OF SIR ISAAC NEWTON. 45 Mint ; and though it was
extremely difficult and he himself much fatigued, yet he completed its solution,
the same evening before he went to bed. The history of these problems affords,
by direct comparison, a striking illustration of Newton s vast superiority of
mind. That amazing concentration and grasp of intellect, of which we have
spoken, enabled him to master speedily, and, as it were, by a single effort,
those things, for the achievement of which, the many would essay utterly in
vain, and the very, very few attain only after long and renewed striving. And
yet, with a modesty as unparalleled as his power, he attributed his successes,
not to any extraordinary sagacity, but solely to industry and patient thought.
Mr- kept the subject of consideration constantly before him, and waited till
the first dawning opened gradually into a full and clear light ; never
quitting, if possible, the mental process till the object of it were wholly
gained. He never allowed this habit of meditation to appear in his intercourse
with society ; but in the privacy of his own chamber, or in the midst of his
own family, he gave himself up to the deepest abstraction. Occupied with some
interesting investigation, he would often sit down on his bedside, after he
rose, and remain there, for hours, partially dressed. Meal-time would
frequently come and pass unheeded ; so that, unless urgently reminded, he would
neglect to take the re quisite quantity of nourishment. But notwithstanding his
anx iety to be left undisturbed, he would, when occasion required, turn aside
his thoughts, though bent upon the most intricate re search, and then, when
leisure served, again direct them to the very point where they ceased to act :
and this he seemed to ac complish not so much by the force of his memory, as by
the force of his inventive faculty, before the vigorous intensity of which, no
subject, however abstruse, remained long unexplored. Me was elected a member of
the Royal Academy of Sciences at Paris, in 1699, when that distinguished Body
were empowered, by a new charter, to admit a small number of foreign
associates. In 1700, he communicated to Dr. Halley a description of his re
flecting instrument for observing the moon s distance from the fixed stars.
This description was published in the Philosophical 46 LIFE OF SIR ISAAC
NEWTON, Transactions, in 1742. The instrument was the same as that produced by
Mr. Hadley, in 1731, and which, under the name of Hadley s Quadrant, has been
of so great use in navigation. On the assembling of the new Parliament, in
1701, Newton was re- elected one of the members for the University of
Cambridge. In 1703, he was chosen President of the Royal Society of London, to
which office he was annually re-elected till the period of his decease about
twenty-five years afterward. Our author unquestionably devoted more labour to,
and, in many respects, took a greater pride in his Optical, than his other
discoveries. This science he had placed on a new and indestruc tible basis ;
and he wished not only to build, but to perfect the costly and glowing
structure. He had communicated, before the publication of the PRINCIPIA, his
most important researches on light to the Royal Society, in detached papers
which were inserted in successive numbers of the Transactions ; but he did not
pub lish a connected view of these labours till 1704, when they appeared under
the title of OPTICS : OR, A TREATISE ON THE REFLEXIONS, REFRACTIONS, INFLEXIONS
AND COLOURS OF LIGHT. To this, but to no subsequent edition, were added two
Mathematical Trea tises, entitled, TRACTATUS DUO DE SPECIEBUS ET MAGNITUDINE
FIGURARUM cuRViLiNEARUM ; the one bearing the title TRACTATUS DE QUADRATURA
CuRVARUM ; and the other, that of ENUMERATIO LINEARUM TERTII ORDiNis. The
publication of these Mathemati cal Treatises was made necessary in consequence
of plagiarisms from the manuscripts of them loaned by the author to his friends.
Dr. Samuel Clarke published a Latin translation of the Optics, in in 1706 ;
whereupon he was presented by Newton, as a mark of his grateful approbation,
with five hundred pounds, or one hun dred pounds for each of his children. The
work was afterward translated into French. It had a remarkably wide
circulation, and appeared, in several successive editions, both in England and
on the Continent. There is displayed, particularly on this Opti cal Treatise,
the author s talent for simplifying and communica ting the profoundest
speculations. It is a faculty rarely united to that of the highest invention.
Newton possessed both ; and thus that mental perfectness which enabled him to
create, to combine, LIFE OF SIR ISAAC NEWTON. 47 and to teach, and so render himself,
not the "ornament" cnly; but inconceivably more, the
pre-eminent benefactor of his species. The honour of knighthood v/as conferred
on our author in 1705. Soon afterward, he was a candidate again for the Repre
sentation of the University, but was defeated by a large majority. It is
thought that a more pliant man was preferred by both min isters and electors.
Newton was always remarkable for simplicity of dress, and his only known
departure from it was on this oc casion, when he is said to have appeared in a
suit of laced clothes. The Algebraical Lectures which he had, Juring nine
years, delivered at Cambridge, were published by Whiston, in 1707, under the
title of ARITHMETICS UNIVERSALIS, SINE DE COMPOSI TIONS ET RESOLUTIONS
ARITHMETICA LIBER. This publication is said to have been a breach of confidence
on Whiston s part. Mr. Ralphson, not long afterward, translated the work into
English ; and a second edition of it, with improvements by the author, was
issued at London, 1712, by Dr. Machin. Subsequent editions, both in English and
Latin, with commentaries, have been published. In June, 1709, Newton intrusted
the superintendence of a second edition of the PRINCIPIA to Roger Cotes,
Plumian Pro fessor of Astronomy at Cambridge. The first edition had been sold
off for some time. Copies of the work had become very rare, and could only be
obtained at several times their original cost. A great number of letters passed
oetween the author and Mr. Cotes during the preparation of the edition, which
finally appeared in May, 1713. It had many alterations and improve ments, and
was accompanied by an admirable Preface from the pen of Cotes. Our author s
early Treatise, entitled, ANALYSIS PER EQUATIONES NUMERO TERMINORUM INFINITAS,
as well as a small Tract, Gearing the title of METHODUS DIFFERENTIALS, was
published, witn nis consent, in 1711. The former of these, and the Treatise De
Quadratura Curvarum, translated into Englisn, witn a .arge com mentary,
appeared in 1745. His work, entitled. ARTIS ANA LYTICS SPECIMINA, VEL GEOMETRIA
ANALYTICA, was iirs; given to the world in the edition of Dr. Horsley, 1779. 48
LIFE OF SIR ISAAC NEWTON. It is a notable fact, in Newton s history, that he
never volun* tarily published any one of his purely mathematical writings The cause
of this unwillingness in some, and, in other instances, of his indifference,
or, at least, want of solicitude to put forth his works may be confidently
sought for in his repugnance to every thing like contest or dispute. But, going
deeper than this aver sion, we find, underlying his whole character and running
parallel with all his discoveries, that extraordinary humility which always
preserved him in a position so relatively just to the behests of time and
eternity, that the infinite value of truth, and the utter worthlessness of
fame, were alike constantly present to him. Judging of his course, however, in
its more temporary aspect, as bearing upon his immediate quiet, it seemed the
most unfortunate. For an early publication, especially in the case of his
Method of Fluxions, would have anticipated all rivalry, and secured him from
the contentious claims of Leibnitz. Still each one will solve the problem of
his existence in his own way, and, with a manlike Newton, his own, as we
conceive, could be no other than the best way. The conduct of Leibnitz in this
affair is quite irreconcilable with the stature and strength of the man ;
giant-like, and doing nobly, in many ways, a giant s work, yet cringing himself
into the dimensions and performances of a common calumniator. Opening in 1699,
the discussion in question continued till the close of Leibnitz s life, in
1716. We give the summary of the case as contained in the Report of the
Committee of the Royal Society, the deliberately weighed opinion of which has
been adopted as an authoritative decision in all countries. " We have
consulted the letters and letter books in the custody of the Royal Society, and
those found among the papers of Mr. John Collins, dated between the years 1669
and 1677, inclusive ; and showed them to such as knew and avouched the hands of
Mr. Barrow, Mr. Collins, Mr. Oldenburg, and Mr. Leibnitz ; and compared those
of Mr. Gregory with one another, and with copies of some of them taken in the
hand of Mr. Collins ; and have extracted from them what relates to the matter
referred to us : all which extracts, herewith delivered to you, we believe to
be genuine and authentic. And by these letters and papers wf find: LIFE OF SIR
ISAAC NEWTON. 49 " I. Mr. Leibnitz was in London in the beginning of
the year 1673 ; and went thence in or about March, to Paris, where he kept a
correspondence with Mr. Collins, by means of Mr. Olden burg, till about
September, 1676, and then returned, by London and Amsterdam, to Hanover: and
that Mr. Collins was very free in communicating to able mathematicians what he
had received from Mr, Newton and Mr. Gregory. " II. That when Mr.
Leibnitz was the first time in London, he contended for the invention of
another differential method, properly so called ; and, notwithstanding he was
shown by Dr. Pell that it was Newton? s method, persisted in maintaining it to
be his own invention, by reason that he had found it by himself without knowing
what Newton had done before, and had much improved it. And we find no mention
of his having any other differential method than Newton s before his letter of
the 21st of June, 1677, which was a year after a copy of Mr. Newton s letter of
the 10th of December, 1672, had been sent to Paris to be communicated to him ;
and above four years after Mr. Collins began to communicate that letter to his
correspondents ; in which letter the method of fluxions was sufficiently
described to any intelligent person. "III. That by Mr. Newton s
letter, of the 13th of June, 1676 it appears that he had the method of fluxions
above five years before the writing of that letter. And by his Analysis per
^Equationes numero Terminorum Infmitas, communicated by Dr. Barrow to Mr.
Collins, in July, 1669, we find that he had invented the method before that
time. "IV. That the differential method is one and the same with the
method of fluxions, excepting the name and mode of notation ; Mr. Leibnitz
calling those quantities differences which Mr. Newton calls moments, or
fluxions ; and marking them with a letter d a mark not used by Mr. Newton.
" And, therefore, we take the proper question to be, not who invented
this or that method, but, who was the first inventor of the method ? And we
believe that those who have reputed Mr. Leibnitz the first inventor knew little
or nothing of his correspond ence with Mr. Collins and Mr. Oldenburg long
before, nor of Mr. 50 LIFE OP SIR ISAAC NEWTON. Newton s hiving that method
above fifteen years before Mr Leibnitz began to publish it in the Acta
Eruditorum of Leipsic. " For which reason we reckon Mr. Newton the
first inventor ; and are of opinion that Mr. Keill, in asserting the same, has
been no ways injurious to Mr. Leibnitz. And we submit to the judg ment of the
Society, whether the extract and papers, now pre sented to you, together with
what is extant, to the same pur pose, in Dr. Wallis s third volume, may not
deserve to be made public." This Report, with the collection of
letters and manuscripts, under the title of COMMERCIUM EPISTOLICUM D. JOHANNIS
COLLINS ET ALIORUM DE ANALYSI PROMOTA JuSSU SoCIETATIS REGIES EDITUM, appeared
accordingly in the early part of 1713. Its publication seemed to infuse
additional bitterness into the feelings of Leibnitz, who descended to unfounded
charges and empty threats. He had been privy counsellor to the Elector of Han
over, before that prince was elevated to the British throne ; and in his
correspondence, in 1715 and 1716, with the Abbe Conti, then at the court of
George L, and with Caroline, Princess of Wales, he attacked the doctrines of
the PRINCIPIA, and indirectly its author, in a manner very discreditable to
himself, both as a learned and as an honourable man. His assaults, however,
were triumphantly met; and, to the complete overthrow of his rival pretensions,
Newton was induced to give the finishing blow. The verdict is universal and
irreversible that the English preceded the German philosopher, by at least ten
years, in the invention of fluxions. Newton could not have borrowed from
Leibnitz ; but Leibnitz might have borrowed from Newton. A new edition of the
Commercium Epistolicum was published in 1722-5 (?) ; but neither in this, nor
in the former edition, did our author take any part. The disciples,
enthusiastic, capable and ready, effectually shielded, with the buckler of
Truth, the character of the Master, whose own conduct throughout was replete
with delicacy, dignity and justice. He kept aloof from the controversy in which
Dr. Keill stood forth as the chief representative of the Newtonian side till the
very last, when, for the satisfaction of the King, George L. rather than for
his own, he consented to put forth his LIFE OF SI| L^.-vJ NEWTON. 5i hand and
firmly secure his rights upon a certain and impregnable, basis. A petition to
have inventions for promoting the discovery of the longitude at sea, suitably
rewarded, was presented to the House of Commons, in 1714. A committee, having
been appointed to investigate the subject, called upon Newton and others for
their opinions. That of our author was given in writing, A report, favourable
to the desired measure, was then taken up, and a bill for its adoption
subsequently passed. On the ascension of George I., in 1714, Newton became an
object of profound interest at court. His position under govern ment, his surpassing
fame, his spotless character, and. above all, his deep and consistent piety,
attracted the reverent regard of the Princess of Wales, afterward queen
-consort to George II. She was a woman of a highly cultivated mind, and derived
the greatest pleasure from conversing with Newton and corresponding with
Leibnitz. One day, in conversation with her, our author men tioned and
explained a new system of chronology, which he had composed at Cambridge, where
he had been in the habit " of refreshing himself with history and
chronology, when he wac weary with other studies." Subsequently, in
the year 1718, she requested a copy of this interesting and ingenious work
Newton, accordingly, drew up an abstract of the system from the separate papers
in which it existed, and gave it to her on condition that it should riot be
communicated to any other person. Sometime afterward she requested that the
Abbe Conti might be allowed to have a copy of it The author consented: and the
abbe received a copy of the manuscript, under the like injunction and promise
of secrecy. This manuscript bore the title of " A short Chronicle,
from the First Memory of Tilings in Europe, to the Conquest of Persia, by
Alexander the Great." After Newton took up his residence in London, he
lived in a style suited to his elevated position and rank. He kept his car
riage, with an establishment of three male and three female serv ants. But to
everything like vain show and luxury he was utterly averse. His household
affairs, for the last twenty years of his life, were under the charge of his
niece, Mrs. Catherine Barton, 52 LIFE OF SIR ISAAC NEWTON. wife and widow of
Colonel Barton a woman of great beauty and accomplishment and subsequently
married to John Conduit, Esq. At home Newton was distinguished by that
dignified and gentle hospitality which springs alone from true nobleness. On
all pro per occasions, he gave splendid entertainments, though without
ostentation. In society, whether of the palace or the cottage, his manner was self-possessed
and urbane ; his look benign and affable ; his speech candid and modest ; his
whole air undisturb edly serene. He had none of what are usually called the
singu larities of genius ; suiting himself easily to every company except that
of the vicious and wicked ; and speaking of himself and others, naturally, so
as never even to be suspected of vanity. There was in him, if we may be allowed
the expression, a WHOLE NESS of nature, which did not admit of such
imperfections and weakness the circle was too perfect, the law too constant,
and the disturbing forces too slight to suffer scarcely any of those
eccentricities which so interrupt and mar the movements of many bright spirits,
rendering their course through the world more like that of the blazing meteor
than that of the light and life-impart ing sun. In brief, the words GREATNESS
and GOODNESS could not, humanly speaking, be more fitly employed than when
applied as the pre-eminent characteristics of this pure, meek and vene rable
sage. In the eightieth year of his age, Newton was seized with symptoms of
stone in the bladder. His disease was pronounced incurable. He succeeded,
however, by means of a strict regimen, and other precautions, in alleviating
his complaint, and procuring long intervals of ease. His diet, always frugai,
was now extremely temperate, consisting chiefly of broth, vegetables, and
fruit, with, now and then, a little butcher meat. He gave up the use of his
carriage, and employed, in its stead, when he went out, a chair. All invitations
to dinner were declined ; and only small parties were received, occasionally,
at his own house. In 1724 he wrote to the Lord Provost of Edinburgh, offering
to contribute twenty pounds yearly toward the salary of Mr. Maclaurin, provided
he accepted the assistant Professorship of Mathematics in the University of
that place. Not only in the LIFE OP SIR ISAAC NEWTON. 53 cause of ingenuity and
learning, but in that of religion in relieving the poor and .assisting his
relations, Newton annually expended large sums. He was generous and charitable
almost to a fault. Those, he would often remark, who gave away nothing till
they died, never gave at all. His wealth had become considerable by a prudent
economy ; but he regarded money in no other light than as one of the means
wherewith he had been intrusted to do good, and he faithfully employed it
accordingly. He experienced, in spite of all his precautionary measures, a
return of his complaint in the month of August, of the same year, 1 724, when
he passed a stone the size of pea ; it came from him in two pieces, the one at
the distance of two day.s from the other. Tolerable good health then followed
for some months. In Janu ary, 1725, however, he was taken with a violent cough
and inflam mation of the lungs. In consequence of this attack, he was pre
vailed upon to remove to Kensington, where his health greatly improved. In
February following, he was attacked in both feet with the gout, of the approach
of which he had received, a few years before, a slight warning, and the
presence of which now produced a very beneficial change in his general health.
Mr. Conduit, his nephew, has recorded a curious conversation which took place,
at or near this time, between himself and Sir Isaac. "I was, on Sunday
night, the 7th March, 1724-5, at Kensing ton, with Sir Isaac Newton, in his
lodgings, just after he was out of a fit of the gout, which he had had in both
of his feet, for the first time, in the eighty-third year of his age. He was
better after it, and his head clearer and memory stronger than I had known them
for some time. He then repeated to me, by way of dis course, very distinctly,
though rather in answer to my queries, than in one continued narration, what he
had often hinted to me before, viz. : that it was his conjecture (he would
affirm nothing) that there was a sort of revolution in the heavenly bodies ;
that the vapours and light, emitted by the sun, which had their sedi ment, as
water and other matter, had gathered themselves, by degrees, into a body, and
attracted more matter from the planets, and at last made a secondary planet
(viz. : one of those that go round another planet), and then, by gathering to
them, and 54 LIFE OF SIR ISAAC NEWTON. attracting more matter, became a primary
planet ; and then, bf increasing still, became a comet, which, after certain
revolutions, by coming nearer and nearer to the sun, had all its volatile parts
condensed, and became a matter tit to recruit and replenish the sun (which must
waste by the constant heat and light it emitted), as a faggot would this fire
if put into it (we were sitting by a wood fire), and that that would probably
be the effect of the comet of 1680, sooner or later ; for, by the observations
made upon it, it appeared, before it came near the sun, with a tail only two or
three degrees long ; but, by the heat it contracted, in going so near the sun,
it seemed to have a tail of thirty or forty degrees when it went frpm it ; that
he could not say when this comet would drop into the sun ; it might perhaps
have five or six revo lutions more first, but whenever it did it would so much
increase the heat of the sun that this earth would be burned, and no ani mals
in it could live. That he took the three phenomena, seen by Hipparchus, Tycho
Brahe, and Kepler s disciples, to have been of this kind, for he could not
otherwise account for an extraor dinary light, as those were, appearing, all at
once, among the the fixed stars (all which he took to be suns, enlightening
other planets, as our sun does ours), as big as Mercury or Venus seems to us,
and gradually diminishing, for sixteen months, and then sinking into nothing.
He seemed to doubt whether there were not intelligent beings, superior to us,
who superintended these revolutions of the heavenly bodies, by the direction of
the Supreme Being. He appeared also to be very clearly of opinion that the
inhabitants of this world were of short date, and alledged, as one reason for
that opinion, that all arts, as letters, ships, printing, needle, &c., were
discovered within the memory of history, which could not have happened if the
world had been eternal ; and that there were visible marks of ruin upon it
which could not be effected by flood only. When I asked him how this earth
could have been repeopled if ever it had undergone the same fate it was
threatened with hereafter, by the comet of 1680, he answered, that required the
power of a Creator. He said he took all the planets to be composed of the same
matter with this earth, viz. : earth, water, stones, &c.3 but variously concocted.
J LIFE OP SIR ISAAC NEWTON. 55 asked him why he would not publish his
conjectures, as conjec tures, and instanced that Kepler had communicated his ;
and though he had not gone near so far as Kepler, yet Kepler s guesses were so
just and happy that they had been proved and demonstrated by him. His answer
was, " I do not deal in con jectures." But, on my talking to
him about the four observations that had been made of the comet of 1680, at 574
years distance, and asking him the particular times, he opened his Principia,
which laid on the table, and showed me the particular periods, viz.: 1st. The
Julium Sidus, in the time of Justinian, in 1106, in 1680. " And I,
observing that he said there of that comet, incidet in corpus solis, and in the
next paragraph adds, stellae fixae refici possunt, told him I thought he owned
there what we had been talking about, viz. : that the comet would drop into the
sun, and that fixed stars were recruited and replenished by comets when they
dropped into them ; and, consequently, that the sun would be recruited too ;
and asked him why he would not own as fully what he thought of the sun as well
as what he thought of the fixed stars. He said, that concerned us more; and,
laugh ing, added, that he had said enough for people to know his
meaning." In the summer of 1725, a French translation of the chronolo
gical MS., of which the Abbe Conti had been permitted, some time previous, to
have a copy, was published at Paris, in violation of all good faith. The Punic
Abbe had continued true to his promise of secrecy while he remained in England
; but no sooner did he reach Paris than he placed the manuscript into the hands
of M. Freret, a learned antiquarian, who translated the work, and accompanied
it with an attempted refutation of the leading points of the system. In
November, of the same year, Newton received a presentation copy of this
publication, which bore the title of ABREGE DE CHRONOLOGIE DE M. LE CHEVALIER
NEWTON, FAIT PAR LUI-MEME, ET TRADUIT SUR LE MANUSCRIPT ANGLAIS. Soon afterward
a paper entitled, REMARKS ON TFE OBERVATIONS MADE ON A CHRONOLOGICAL INDEX OF
SIR ISAAC NE.WTON, TRANSLATED INTO FRENCH BY THE OBSERVATOR, ANL PUBLISHED AT
PARIS, 56 LIFE OF SIR ISAAC NEWTON, was drawn up by our author, and printed in
the Philosophical Transactions for 1725. It contained a history of the whole
matter, and a triumphant reply to the objections of M. Freret. This answer
called into the field a fresh antagonist, Father Soueiet, whose five
dissertations on this subject were chiefly remarkable for the want of knowledge
and want of decorum, which they displayed. In consequence of these discussions,
Newton was in duced to prepare his larger work for the press, and had nearly
completed it at the time of his death. It was published in 1728, under the
title of THE CHRONOLOGY OF THE ANCIENT KINGDOMS AMENDED, TO WHICH is PREFIXED A
SHORT CHRONICLE FROM THE FIRST MEMORY OF THINGS IN EUROPE TO THE CONQUEST OF
PERSIA BY ALEXANDER THE GREAT. It consists of six chap ters: 1. On the
Chronology of the Greeks; according to Whiston, our author wrote out eighteen
copies of this chapter with his own hand, differing little from one another. 2.
Of the Empire of Egypt; 3. Of the Assyrian Empire; 4. Of the two contempo rary
Empires of the Babylonians and Medes ; 5. A Description of the Temple of
Solomon ; 6. Of the Empire of the Persians ; this chapter was not found copied
with the other five, but as it was discovered among his papers, arid appeared
to be a continu ation of the same work, the Editor thought proper to add it
thereto. Newton s LETTER TO A PERSON OF DISTINCTION WHO HAD DESIRED HIS OPINION
OF THE LEARNED BlSHO^ LLOYD S HYPOTHESIS CONCERNING THE FORM OF THE MOST
ANCIENT ^EAR, closes this enumeration of his Chronological Writings. A ihird
edition of the PRINCIPIA appeared in 1726, with many changes and additions.
About four years were consumed in its preparation and publication, which were
under the superintend- ance of Dr. Henry Pemberton, an accomplished
mathematician, and the author of "A VIEW OF SIR ISAAC NEWTON S PHILO
SOPHY." 1728. This gentleman enjoyed numerous opportunities of
conversing with the aged and illustrious author. " I found,"
says Pemberton, " he had read fewer of the modern mathemati cians than
one could have expected; but his own prodigious invention readily supplied him
with what he might have an occa sion for in the pursuit of any subject he
undertook. I have often LIFE OF SIR ISAAC NEWTON. 57 heard him censure the
handling geometrical subjects ly algebraic calculations ; and his book of
Algebra he called by the name of Universal Arithmetic, in opposition to the
injudicious title of Geometry, which Descartes had given to the treatise,
wherein he shows how the geometer may assist his invention by such kind of
computations. He thought Huygens the most elegant of any mathematical writer of
modern times, and the most just imitator of the ancients. Of their taste and
form of demonstration, Sir Isaac always professed himself a great admirer. I have
heard him even censure himself for not following them yet more closely than he
did ; and speak with regret of his mistake at the begin ning of his
mathematical studies, in applying himself to the works of Descartes and other
algebraic writers, before he had considered the elements of Euclid with that
attention which so excellent a writer deserves." " Though his
memory was much decayed," continues Dr. Pemberton, "he
perfectly understood his own writings." And even this failure of memory,
we would suggest, might have been more apparent than real, or, in medical
terms, more the result of func tional weakness than organic decay. Newton seems
never to have confided largely to his memory : and as this faculty mani fests
the most susceptibility to cultivation ; so, in the neglect of due exercise, it
more readily and plainly shows a diminution of its powers. Equanimity and
temperance had, indeed, preserved Newton singularly free from all mental and
bodily ailment. His hair was, to the last, quite thick, though as white as
silver. He never made use of spectacles, and lost but one tooth to the day of
his death. He was of middle stature, well-knit, and, in the latter part of his
life, somewhat inclined to be corpulent. Mr. Conduit says, " he had a
very lively and piercing eye, a comely and gra cious aspect, with a fine head
of hair, white as silver, without any baldness, and when his peruke was off was
a venerable sight." According to Bishop Atterbury, "in the
whole air of his face and make there was nothing of that penetrating sagacity
which appears in his compositions. He had something rather languid in his look
and manner which did not raise any great expectation 58 LIFE OF SIR ISAAC
NEWTON. in those who did not know him." Hearne remarks, " Sir
Isaac was a man of no very promising aspect. He was a short, wellset man. He
was full of thought, and spoke very little in com pany, so that his
conversation was not agreeable. When he rode in his coach, one arm would be out
of his coach on one side and the other on the other." These different
accounts we deem easily reconcilable. In the rooms of the Royal Society, in the
street, or in mixed assemblages, Newton s demeanour always courteous,
unassuming and kindly still had in it the overawings of a profound repose and
reticency, out of which the communica tive spirit, and the "lively and
piercing eye" would only gleam in the quiet and unrestrained freedom
of his own fire-side. " But this I immediately discovered in
him," adds Pemberton, still further, "which at once both
surprised and charmed me. Neither his extreme great age, nor his universal
reputation had rendered him stiff in opinion, or in any degree elated. Of this
I had occasion to have almost daily experience. The remarks I continually sent
him by letters on his Principia, were received with the utmost goodness. These
were so far from being any ways displeasing to him, that, on the contrary, it
occasioned him to speak many kind things of me to my friends, and to honour me
with a public testimony of his good opinion." A modesty, open ness,
and generosity, peculiar to the noble and comprehensive spirit of Newton.
" Full of wisdom and perfect in beauty," yet not lifted up by
pride nor corrupted by ambition. None, how ever, knew so well as himself the
stupendousness of his discoveries in comparison with all that had been
previously achieved ; and none realized so thoroughly as himself the littleness
thereof in comparison with the vast region still unexplored. A short time
before his death he uttered this memorable sentiment: " I do not know
what I may appear to the world ; but to myself I seem to have been only like a
boy playing on the sea-shore, and diverting myself in now and then finding a
smoother pebble or a prettier shell than ordinary, while the great ocean of
truth lay all undis covered before me." How few ever reach the shore
even, much less find "a smoother pebble or a prettier shell!"
Newton had now resided about two years at Kensington ; and LIFE OF SIR ISAAC
NEWTON. 59 the air which he enjoyed there, and the state of absolute rest,
proved of great benefit to him. Nevertheless he would occasion ally go to town.
And on Tuesday, the 28th of February, 1727, he proceeded to London, for the
purpose of presiding at a meeting of the Royal Society. At this time his health
was considered, by Mr. Conduit, better than it had been for many years. But the
unusual fatigue he was obliged to suffer, in attending the meeting, and in
paying and receiving visits, speedily produced a violent return of the
affection in the bladder. He returned to Kensington on Saturday, the 4th of
March. Dr. Mead and Dr. Cheselden attended him ; they pronounced his disease to
be the stone, and held out no hopes of recovery. On Wednesday, the 15th of
March, he seemed a little better; and slight, though groundless, encouragement
was felt that he might survive the attack. From the very first of it, his
sufferings had been intense. Paroxysm followed paroxysm, in quick succession :
large drops )f sweat rolled down his face ; but not a groan, not a complaint,
not the least mark of peevishness or impatience escaped him : and during the
short intervals of relief, he even smiled and con versed with his usual
composure and cheerfulness. The flesh quivered, but the heart quaked not ; the
impenetrable gloom was settling down : the Destroyer near ; the portals of the
tomb opening, still, arnid this utter wreck and dissolution of the mortal, the
immortal remained serene, unconquerable : the radiant light broke through the
gathering darkness ; and Death yielded up its sting, and the grave its victory.
On Saturday morning, 18th, he read the newspapers, and carried on a pretty long
conversation with Dr. Mead. His senses and faculties were then strong and
vigorous ; but at six o clock, the same evening, he became insen sible ; and in
this state he continued during the whole of Sunday, and till Monday, the 20th,
when he expired, between one and two o clock in the morning, in the
eighty-fifth year of his age. And these were the last days of Isaac Newton.
Thus closed the career of one of earth s greatest and best men. His mission was
fulfilled. Unto the Giver, in many-fold addition, the talents were returned.
While it was yet day he had worked ; and for the night that quickly cometh he
was not unprepared. Full of 60 LIFE OF SIR ISAAC NEWTON. years, ind full of
honours, the heaven-sent was recalled ; and, in the confidence of a "
certain hope," peacefully he passed awa} into the silent depths of
Eternity. His body was placed in Westminster Abbey, with the state and
ceremonial that usually attended the interment of the most distinguished. In
1731, his relatives, the inheritors of his personal estate, erected a monument
to his memory in the most conspicu ous part of the Abbey, which had often been
refused by the dean and chapter to the greatest of England s nobility. During
the same year a medal was struck at the Tower in his honour ; arid, in 1755, a
full-length statue of him, in white marble, admirably executed, by Roubiliac,
at the expense of Dr. Robert Smith, was erected in the ante-chamber of Trinity
College, Cambridge. There is a painting executed in the glass of one of the
windows of the same college, made pursuant to the will of Dr. Smith, who left
five hundred pounds for that purpose. Newton left a personal estate of about
thirty-two thousand pounds. It was divided among his four nephews and four
nieces of the half blood, the grand-children of his mother, by the Reve rend
Mr. Smith. The family estates of Woolsthorpe arid Sustern fell to John Newton,
the heir-at-law, whose great grand-father was Sir Isaac s uncle. Before his
death he made an equitable distribution of his two other estates : the one in
Berkshire to the sons and daughter of a brother of Mrs. Conduit ; and the
other, at Kensington, to Catharine, the only daughter of Mr. Conduit, and who
afterward became Viscountess Lymington. Mr. Con duit succeeded to the offices
of the Mint, the duties of which he had discharged during the last two years of
Sir Isaac s life. Our author s works are found in the collection of Castilion,
Berlin, 1744, 4to. 8 torn.; in Bishop Horsley s Edition, London, 1779, 4to. 5
vol.; in the Biographia Brittannica, &c. Newton also published Bern.
Varcnii Geographia, &c., 1681, 8vo. There are, however, numerous
manuscripts, letters, and other papers, which have never been given to the
world: these are preserved, in various collections, namely, in the library of
Trinity College, Cambridge ; in the library of Corpus Christi College, Oxford ;
in the library of Lord Macclesfield : and, lastly arid LIFE OF SIR ISAAC
NEWTON. 61 chiefly, in the possession of the family of the Earl of Portsmouth,
through the Viscountess Lymington. Everything appertaining to Newton has been
kept and che rished with peculiar veneration. Different memorials of him are
preserved in Trinity College, Cambridge ; in the rooms of the Royal Society, of
London : and in the Museum of the Royal Society of Edinburgh. The manor-house,
at Woolsthorpe, was visited by Dr. Stuke ley, in October, 1721, who, in a
letter to Dr. Mead, written in 1727, gave the following description of it:
" Tis built of stone, as is the way of the country hereabouts, and a
reasonably good one. They led me up stairs and showed me Sir Isaac s stud}-,
where I supposed he studied, when in the country, in his younger days, or
perhaps when he visited his mother from the University. I observed the shelves
were of his own making, being pieces of deal boxes, which probably he sent his
books and clothes down in on those occasions. There were, some years ago, two
or threr hundred books in it of his father-in-law, Mr. Smith, which Sir Isaac
gave to Dr. Newton, of our town." The celebrated appletree, the fall
of one of the apples of which is said to have turned the attention of Newton to
the subject of gravity, was destroyed by the wind about twenty years ago ; but
it has been preserved in the form of a chair. The house itself has been
protected with religious care. It was repaired in 1798, and a tablet of white
marble put up in the room where our author was born, with the follow, ng
inscription : " Sir Isaac Newton, son of John Newton, Lord of the
Manor of Woolsthorpe, was born in this room, on the 25th of December,
1642." Nature and Nature s Laws wei-e hid in night, God said,
" Let NEWTON be," and all was light. THE PEINCIPIA. THE
AUTHOR S PREFACE SINCE the ancients (as we are told by Pappus), made great
account oi the science of mechanics in the investigation of natural things :
and the moderns, laying aside substantial forms and occult qualities, have
endeav oured to subject the phenomena of nature to the laws of mathematics, I
have in this treatise cultivated mathematics so far as it regards philosophy.
The ancients considered mechanics in a twofold respect ; as rational, which
proceeds accurately by demonstration ; and practical. To practical me chanics
all the manual arts belong, from which mechanics took its name. Rut as
artificers do not work with perfect accuracy, it comes to pass that mechanics
is so distinguished from geometry, that what is perfectly accu rate is called
geometrical , what is less so, is called mechanical. But the errors are not in
the art, but in the artificers. He that works with less accuracy is an
imperfect mechanic ; and if any could work with perfect accuracy, he would be
the most perfect mechanic of all ; for the description if right lines and
circles, upon which geometry is founded, belongs to me chanics. Geometry does
not teach us to draw these lines, but requires them to be drawn ; for it
requires that the learner should f.rst be taught to describe these accurately,
before he enters upon geometry ; then it shows how by these operations problems
may be solved. To describe right lines and circles are problems, but not
geometrical problems. The solution of these problems is required from mechanics
; and by geometry the use of them, when so solved, is shown ; and it is the
glory of geometry that from those few principles, brought from without, it is
able to produce so many things. Therefore geometry is founded in mechanical
practice, and is nothing but that part of universal mechanics which accurately
proposes and demonstrates the art of measuring. But since the manual arts are
chiefly conversant in the moving of bodies, it comes to pass that geometry is
commonly referred to their magnitudes, and mechanics to their motion. In this
sense rational mechanics will be the science of motions resulting from any
forces whatsoever, and of the forces required to produce any mo tions,
accurately proposed and demonstrated. This part of mechanics was i:;vm THE
AUTHOR & PREFACE. cultivated by the ancients in the five powers which
relate to manual arts, who considered gravity (it not being a manual power), ho
Otherwise than as it moved weights by those powers. Our design not respecting
arts, but philosophy, and our subject not manual but natural powers, we
consider chiefly those things which relate to gravity, levity, elastic force,
the resist ance of fluids, and the like forces, whether attractive or impulsive
; and therefore we offer this work as the mathematical principles :f philosophy
; for all the difficulty of philosophy seems to consist in this from the phenom
ena of motions to investigate the forces of nature, and then from these forces
to demonstrate the other phenomena ; and to this end the general propositions
in the first and second book are directed. In the third book we give an example
of this in the explication of the System of the World : for by the propositions
mathematically demonstrated in the former books, we in the third derive from
the celestial phenomena the forces of gravity with which bodies tend to the sun
and the several planets. Then from these forces, by other propositions which
are also mathematical, we deduce the mo tions of the planets, the comets, the
moon, and the sea. I wish we could do- rive the rest of the phenomena of nature
by the same kind of reasoning from mechanical principles; for I am induced by
many reasons to suspect that they may all depend upon certain forces by which
the particles of bodies. by some causes hitherto unknown, are either mutually
impelled towards each other, and cohere in regular figures, or are repelled and
recede from each other; which forces being unknown, philosophers have hitherto
at tempted the search of nature in vain ; but I hope the principles here laid
down will afford some light either to this or some truer method of philosophy.
In the publication of this work the most acute and universally learned Mr.
Edmund Halley not only assisted me with his pains in correcting the press and
taking care of the schemes, but it was to his solicitations that its becoming
public is owing ; for when he had obtained of me my demonstra tions of the
figure of the celestial orbits, he continually pressed me to com municate the
same to the Royal Societ//, who afterwards, by their kind en couragement and
entreaties, engaged me to think of publishing them. But after I had begun to
consider the inequalities of the lunar motions, and had entered upon some other
things relating to the laws and measures oi gravity, and other forces : and the
figures that would be described by bodies attracted according to given laws ;
and the motion of several bodies moving among themselves; the motion of bodies
in resisting mediums; the forces, densities, and motions, of rn( Hums ; the
orbits of the comets, and such like ; Ixix deferred that publication till I had
made a searcli into those matters, and could put forth the whole together. What
relates to the lunar motions (be ing imperfect), I have put all together in the
corollaries of Prop. 66, to avoid being obliged to propose and distinctly
demonstrate the several things there contained in a method more prolix than the
subject deserved, and in terrupt the series of the several propositions. Some
things, found out after the rest, I chose to insert in places less suitable,
rather than change the number of the propositions and the citations. I heartily
beg that what 1 have here done may be read with candour; and that the defects
in a subject so difficult be not so much reprehended as kindly supplied, and in
vestigated by new endeavours of mv readers. ISAAC NEWTON. Cambridge, Trinity
Coupge May 8, liHB. In the second edition the second section of the first book
was enlarged. In the seventh section of the second book the theory of the
resistances of fluids was more accurately investigated, and confirmed by new
experiments. In the third book the moon s theory and the profession of the
equinoxes were more fully deduced from their principles ; and the theory of the
comets was confirmed by more examples of the calculati >n of their
orbits, done also with greater accuracy. In this third edition the resistance
of mediums is somewhat more largely handled than before; and new experiments of
the resistance of heavy bodies falling in air are added. In the third book, the
argument to prove that the moon is retained in its orbit by the force of
gravity is enlarged on ; and there are added new observations of Mr. Pound s of
the proportion of the diameters of Ju.piter to each other : there are, besides,
added Mr. Kirk s observations of the comet in 16SO ; the orbit of that comet
com puted in an ellipsis by Dr. Halley ; and the ortit of the comet in computed
by Mr. Bradley, OOK I. THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY
DEFINITIONS. DEFINITION I. 77w? quantity of matter is the measure of the same,
arising from its density and hulk conjutictly. THUS air of a double density, in
a double space, is quadruple in quanti ty ; in a triple space, sextuple in
quantity. The same thing is to be un derstood of snow, and fine dust or
powders, that are condensed by compres sion or liquefaction and of all bodies
that are by any causes whatever differently condensed. I have no regard in this
place to a medium, if any such there is, that freely pervades the interstices
between the parts oi bodies. It is this quantity that I mean hereafter
everywhere under the name of body or mass. And the same is known by the weight
of each body ; for it is proportional to the weight, as I have found by
experiments on pendulums, very accurately made, which shall be shewn hereafter.
DEFINITION II. The quantity of motion is the measure nf tlie same, arising from
the velocity and quantity of matter corjunctly. The motion of the whole
i<! the sum of the motions of all the parts ; and therefore in a
body double in quantity, with equal velocity, the motion is iouble ; with twice
the velocity, it is quadruple, DEFINITION III. The vis insita, or innate force
of matter, is a power of resisting, hy which every body, as much as in it lies,
endeavours to persevere in its present stale, whether it be of rest, or of
moving uniformly forward in a right line. This force is ever proportional to
the body whose force it is ; and differs nothing from the inactivity of the
mass, but in our manner of conceiving T4 THE MATHEMATICAL PRINCIPLES it. A
body, from the inactivity of matter, is not without difficulty put out of its state
of rest or motion. Upon which account, this vis insita, may, by a most
significant name, be called vis inertia, or force of inactivity. Hut a body
exerts this force only, when another force, impressed upon it, endeavours to
change its condition ; and the exercise of this force may bo considered both as
resistance and impulse ; it is resistance, in so far as the body, for
maintaining its present state, withstands the force impressed; it is impulse,
in so far as the body, by not easily giving way to the impressed force of
another, endeavours to change the state of that other. Resistance is usually
ascribed to bodies at rest, and impulse to those in motion; but motion and
rest, as commonly conceived, are only relatively distin guished ; nor are those
bodies always truly at rest, which commonly are taken to be so. DKFLMTIOX IV.
Ait impressed force is an action exerted upon a body, in order to change its
state, either of rest, or of moving uniformly forward in a right line. This
force consists in the action only; and remains no longer in the body, when the
action is over. For a body maintains every new state it acquires, by its vis
inertice only. Impressed forces are of differe.it origins as from percussion,
from pressure, from centripetal force. DEFINITION V. A centripetal force is
that by which bodies are drawn or impelled, or any way tend, towards a point as
to a centre. Of this sort is gravity, by which bodies tend to the centre of the
earth magnetism, by which iron tends to the loadstone ; and that force, what
ever it is, by which the planets are perpetually drawn aside from the rec
tilinear motions, which otherwise they would pursue, and made to revolve in
curvilinear orbits. A stone, whirled about in a sling, endeavours to re cede
from the hand that turns it ; and by that endeavour, distends the sling, and
that with so much the greater force, as it is revolved with the greater
velocity, and as soon as ever it is let go, flies away. That force which
opposes itself to this endeavour, and by which the sling perpetually draws back
the stone towards the hand, and retains it in its orbit, because it is directed
to the hand as the centre of the orbit, I call the centripetal force. And the
same thing is to be understood of all bodies, revolved in any orbits. They all
endeavour to recede from the centres of their orbits ; and wore it not for the
opposition of a contrary force which restrains them to, and detains them in
their orbits, which I therefore call centripetal, would tiy off in right lines,
with an uniform motion. A projectile, if it was not for the force of gravity,
would not deviate towards the earth, tut would OF NATUJIAL PHILOSOPHY. 7fl go
off from it in a right line, and that with an uniform motion,, if the re
sistance of the air was taken away. It is by its gravity that it is drawn aside
perpetually from its rectilinear course, and made to deviate towards the earth,
more or less, according to the force of its gravity, and the velo city of its
motion. The less its gravity is, for the quantity of its matter, or the greater
the velocity with which it is projected, the less will it devi ate from a
rectilinear course, and the farther it will go. If a leaden balJ, projected
from the top of a mountain by the force of gunpowder with a given velocity, and
in a direction parallel to the horizon, is carried in a curve line to the
distance of two miles before it falls to the ground ; the same, if the
resistance of the air were taken away, with a double or decuple velocity, would
fly twice or ten times as far. And by increasing the velo city, we may at
pleasure increase the distance to which it might be pro jected, and diminish
the curvature of the line, which it might describe, till at last it should fall
at the distance of 10, 30, or 90 degrees, or even might go quite round the
whole earth before it falls ; or lastly, so that it might never fall to the
earth, but go forward into the celestial spaces, and pro ceed in its motion in
iiifiuitum. And after the same manner that a pro jectile, by the force of
gravity, may be made to revolve in an orbit, and go round the whole earth, the
moon also, either by the force of gravity, if it is endued with gravity, or by
any other force, that impels it towards the earth, may be perpetually drawn
aside towards the earth, out of the r&tilinear way, which by its innate
force it would pursue; and would be made to revolve in the orbit which it now
describes ; nor could the moon with out some such force, be retained in its
orbit. If this force was too small, it would not sufficiently turn the moon out
of a rectilinear course : if it was too great, it would turn it too much, arid
draw down the moon from its orbit towards the earth. It is necessary, that the
force be of a just quantity, and it belongs to the mathematicians to find the
force, that may serve exactly to retain a body in a given orbit, with a given
velocity ; and vice versa, to determine the curvilinear way, into which a body
projected from a given place, with a given velocity, may be made to deviate
from its natural rectilinear way, by means of a given force. The quantity of
any centripetal force may be considered as of three kinds; aboolu e,
accelerative, and motive. DEFINITION VI. The absolute quantity of a centripetal
force is the measure f >f the same proportional to the efficacy of
the cause that propagates it from the cen tre, through the spaces round about.
Thus the magnetic force is greater in one load-stone and less in another
according to their sizes and strength of intensity. 76 THE MATHEMATICAL
PRINCIPLES DEFINITION VII. The accelerative quantity of a centripetal force is
the measure, of tht same, proportional to the velocity which it generates in a
given time. Thus the force of the same load-stone is greater at a less
distance, and less at a greater : also the force of gravity is greater in
valleys, less on tops of exceeding high mountains ; and yet less (as shall
hereafter be shown), at greater distances from the body of the earth ; but at
equal distan ces, it is the same everywhere ; because (taking away, or allowing
for, the resistance of the air), it equally accelerates all falling bodies,
whether heavy or light, great or small. DEFINITION VIII. TJie motive quantity
of a centripetal force, is the measure of the samt\ proportional to the motion
which it generates in a given twip. Thus the weight is greater in a greater
body, less in a less body ; and. in the same body, it is greater near to the
earth, and less at remoter dis tances. This sort of quantity is the
centripetency, or propension of the whole body towards the centre, or, as I may
say, its weight ; and it is al ways known by the quantity of an equal and
contrary force just sufficient to Ifinder the descent of the body. These
quantities of forces, we may, for brevity s sake, call by the names of motive,
accelerative, and absolute forces ; and, for distinction s sake, con sider
them, with respect to the bodies that tend to the centre ; to the places of
those bodies ; and to the centre of force towards which they tend ; that is to
say, I refer the motive force to the body as an endeavour and propen sity of
the whole towards a centre, arising from the propensities of the several parts
taken together ; the accelerative force to the place of the body, as a certain
power or energy diffused from the centre to all places around to move the
bodies that are in them : and the absolute force to the centre, as endued with
some cause, without which those motive forces would not be propagated through
the spaces round about ; whether that cause be some central body (siuh as is
the load-stone, in the centre of the magnetic force, or the earth in the centre
of the gravitating force), or anything else that does not yet appear. For I
here design only to give a mathematical notion of those forces, without
considering their physical causes and seats. Wherefore the accelerative force
will stand in the same relation to the motive, as celerity does to motion. For
the quantity of motion arises from the celerity drawn into the quantity of
matter : and the motive force arises from the accelerative force drawn into the
same quantity of matter. For the sum of the actions of the accelerative force,
upon the several ; articles of the body, is the motive force of the whole.
Hence it is, that near the OF NATURAL PHILOSOPHY. 77 surface of the earth,
where the accelerative gravity, or force productive of gravity, in all bodies
is the same, the motive gravity or the weight is as the body : but if we should
ascend to higher regions, where the accelerative gravity is less, the weight
would be equally diminished, and would always be as the product of the body, by
the accelerative gravity. So in those re gions, where the accelerative gravity
is diminished into one half, the weight of a body two or three times less, will
be four or six times less. I likewise call attractions and impulses, in the
same sense, accelerative, and motive ; and use the words attraction, impulse or
propensity of any sort towards a centre, promiscuously, and indifferently, one
for another ; considering those forces not physically, but mathematically :
wherefore, the reader is not to imagine, that by those words, I anywhere take
upon me to define the kind, or the manner of any action, the causes or the
physical reason thereof, or that I attribute forces, in a true and physical
sense, to certain centres (which are only mathematical points) ; when at any
time I happen to speak of centres as attracting, or as endued with attractive
powers. SCHOLIUM. Hitherto I have laid down the definitions of such words as
are less known, and explained the sense in which I would have them to be under
stood in the following discourse. I do not define time, space, place and
motion, as being well known to all. Only I must observe, that the vulgar
conceive those quantities under no other notions but from the relation they
bear to sensible objects. And thence arise certain prejudices, for the re
moving of which, it will be convenient to distinguish them into absolute and
relative, true and apparent, mathematical and common. I. Absolute, true, and
mathematical time, of itself, and from its own na ture flows equably without
regard to anything external, and by another name is called duration : relative,
apparent, and common time, is some sen sible and external (whether accurate or
unequable) measure of duration by the means of motion, which is commonly used
instead of true time ; such as an hour, a day, a month, a year. II. Absolute
space, in its own nature, without regard to anything exter nal, remains always
similar and immovable. Relative space is some mo vable dimension or measure of
the absolute spaces ; which our senses de termine by its position to bodies ;
and which is vulgarly taken for immo vable space ; such is the dimension of a
subterraneous, an aereal, or celestial space, determined by its position in respect
of the earth. Absolute and relative space, are the same in figure and magnitude
; but they do not re main always numerically the same. For if the earth, for
instance, moves, a space of our air, which relatively and in respect of the
earth remains al ways the same, will at one time be one part of the absolute
space into which TS THE MATHEMATICAL PRINCIPLES the air passes ; at another
time it will be another part of the same, and so. absolutely understood, it
will be perpetually mutable. III. Place is a part of space which a body takes
up, and is according to the space, either absolute or relative. I say, a part
of space ; not the situation, nor the external surface of the body. For the
places of equal solids are always equal ; but their superfices, by reason of
their dissimilar figures, are often unequal. Positions properly have no
quantity, nor are they so much the places themselves, as the properties of
places. The motion of the whole is the same thing with the sum of the motions
of the parts ; that is, the translation of the whole, out of its place, is the
same thing with the sum of the translations of the parts out of their places ;
and therefore the place of the whole is the same thing with the sum of the
places of the parts, and for that reason, it is internal, and in the whole
body. IV. Absolute motion is the translation of a body from one absolute place
into another ; and relative motion, the translation from one relative place
into another. Thus in a ship under sail, the relative place of a body is that
part of the ship which the body possesses ; or that part of its cavity which
the body fills, and which therefore moves together with the ship : and relative
rest is the continuance of the body in the same part of the ship, or of its
cavity. But real, absolute rest, is the continuance of the body in the same
part of that immovable space, in which the ship itself, its cavity, and all
that it contains, is moved. Wherefore, if the earth is really at rest, the
body, which relatively rests in the ship, will really and absolutely move with
the same velocity which the ship has on the earth. But if the earth also moves,
the true and absolute motion of the body will arise, partly from the true
motion of the earth, in immovable space ; partly from the relative motion of
the ship on the earth ; and if the body moves also relatively in the ship ; its
true motion will arise, partly from the true motion of the earth, in immovable
space, and partly from the relative mo tions as well of the ship on the earth,
as of the body in the ship ; and from these relative motions will arise the
relative motion of the body on the earth. As if that part of the earth, where
the ship is, was truly moved toward the east, with a velocity of 10010 parts;
while the ship itself, with a fresh gale, and full sails, is carried towards
the west, with a velocity ex pressed by 10 of those parts ; but a sailor walks
in the ship towards the east, with 1 part of the said velocity ; then the
sailor will be moved truly in immovable space towards the east, with a velocity
of 10001 parts, and relatively on the earth towards the west, with a velocity
of 9 of those parts. Absolute time, in astronomy, is distinguished from
relative, by the equa tion or correction of the vulgar time. For the natural
days are tr^y un equal, though they are commonly considered as equal, and used
for a meas ure of time ; astronomers correct this inequality for their more
accurate deducing of the celestial motions. It may be, that there is no such
thing as an equable motion, whereby time may H accurately measured. All mo OF
NATURAL PHILOSOPHY. 79 tions may be accelerated and retarded; but the true, or
equable, progress of absolute time is liable to no change. The duration or
perseverance of the existence of things remains the same, whether the motions
are swift or slow, or none at all : and therefore it ought to be distinguished
from what are only sensible measures thereof ; and out of which we collect it,
by means of the astronomical equation. The necessity of which equation, for deter
mining the times of a phamomenon, is evinced as well from the experiments of
the pendulum clock, as by eclipses of the satellites of Jupiter. As the order
of the parts of time is immutable, so also is the order of the parts of space.
Suppose those parts to be moved out of their places, and they will be moved (if
the expression may be allowed) out of themselves. For times and spaces are, as
it were, the places as well of themselves as of all other things. All things
are placed in time as to order of succession ; and in space as _to order of
situation. It is from their essence or nature that they are places ; and that
the primary places of things should be moveable, is absurd. These are therefore
the absolute places ; and trans lations out of those places, are the only
absolute motions. But because the parts of space cannot be seen, or
distinguished from one another by our senses, therefore in their stead we use
sensible measures of them. For from the positions and distances of things from
any body con sidered as immovable, we define all places ; and then with respect
to such places, we estimate all motions, considering bodies as transferred from
some of those places into others. And so, instead of absolute places and
motions, we use relative ones; and that without any inconvenience in common af
fairs ; but in philosophical disquisitions, we ought to abstract from our
senses, and consider things themselves, distinct from what are only sensible
measures of them. For it may be that there is no body really at rest, to which
the places and motions of others may be referred. But we may distinguish rest
and motion, absolute and relative, one from the other by their properties,
causes and effects. It is a property of rest, that bodies really at rest do
rest in respect to one another. And therefore as it is possible, that in the
remote regions of the fixed stars, or perhaps far beyond them, there may be
some body absolutely at rest ; but impossi ble to know, from the position of
bodies to one another in our regions whether any of these do keep the same
position to that remote body; it follows that absolute rest cannot be
determined from the position of bodies in our regions. It is a property of
motion, that the parts, which retain given positions to their wholes, do partake
of the motions of those wholes. For all the parts of revolving bodies endeavour
to recede from the axis of motion ; and the impetus of bodies moving forward,
arises from the joint impetus of all the parts. Therefore, if surrounding
bodies are moved, those that are relatively at rest within them, will partake
of their motion. Upon which account, the true and absolute motion of a body
cannot be Jeter- 8C THE MATHEMATICAL PRINCIPLES mined by the translation of it
from those which only seem to rest ; for the external bodies ought not only to
appear at rest, but to be really at rest. For otherwise, all included bodies,
beside their translation from near the surrounding ones, partake likewise of
their true motions ; and though that translation were not made they would not
be really at rest, but only seem to be so. For the surrounding bodies stand in
the like relation to the surrounded as the exterior part of a whole does to the
interior, or as the shell does to the kernel ; but, if the shell moves, the kernel
will also move, as being part of the whole, without any removal from near the
shell. A property, near akin to the preceding, is this, that if a place is
moved, whatever is placed therein moves along with it ; and therefore a body,
which is moved from a place in motion, partakes also of the motion of its
place. Upon which account, all motions, from places in motion, are no other
than parts of entire and absolute motions ; and every entire motion is composed
of the motion of the body out of its first place, and the motion of this place
out of its place ; and so on, until we come to some immovable place, as in the
before-mentioned example of the sailor. Where fore, entire and absolute motions
can be no otherwise determined than by immovable places : and for that reason I
did before refer those absolute motions to immovable places, but relative ones
to movable places. Now no other places are immovable but those that, from
infinity to infinity, do all retain the same given position one to another ;
and upon this account must ever remain unmoved ; and do thereby constitute
immovable space. The causes by which true and relative motions are
distinguished, one from the other, are the forces impressed upon bodies to
generate motion. True motion is neither generated nor altered, but by some
force impressed upon the body moved : but relative motion may be generated or
altered without any force impressed upon the body. For it is sufficient only to
impress some force on other bodies with which the former is compared, that by
their giving way, that relation may be changed, in which the re lative rest or
motion of this other body did consist. Again, true motion suffers always some
change from any force impressed upon the moving body ; but relative motion docs
not necessarily undergo any change by such forces. For if the same forces are
likewise impressed on those other bodies, with which the comparison is made,
that the relative position may be pre served, then that condition will be
preserved in which the relative motion consists. And therefore any relative
motion may be changed when the true motion remains unaltered, and the relative
may be preserved when the true suffers some change. Upon which accounts; true
motion does by no means consist in such relations. The effects whicli
distinguish absolute from relative motion arc, the forces of receding from the
axis of circular motion. For there are no such forces in a circular motion
purely relative, but in a true and absolute cir cular motion., they are greater
or less, according t the quantity of the OF NATURAL PHILOSOPHY. 1 motion. If a
vessel, hung: by & }ong cord, is so often turned ubout that the cord is
strongly twisted, then filled with water, and held at rest together with the
water ; after, by the sudden action of another force, it is whirled about the
contrary way, and while the cord is untwisting itself, the vessel continues for
some time in this motion ; the surface of the water will at first be plain, as
before the vessel began to move : but the vessel; by grad ually communicating
its motion to the water, will make it begin sensibly ^to revolve, and recede by
little and little from the middle, and ascend to the sides of the vessel,
forming itself into a concave figure (as I have experi enced), and the swifter
the motion becomes, the higher will the water rise, till at last, performing
its revolutions in the same times with the vessel, it becomes relatively at
rest in it. This ascent of the water shows its en deavour to recede from the
axis of its motion ; and the true and absolute circular motion of the water,
which is here directly contrary to the relativej discovers itself, and may be
measured by this endeavour. At first, when the relative motion of the water in
the vessel was greatest, it pro duced no endeavour to recede from the axis ;
the water showed no tendency to the circumference, nor any ascent towards the
sides of the vessel, but remained of a plain surface, and therefore its true
circular motion had not yet begun. But afterwards, when the relative motion of
the water had decreased, the ascent thereof towards the sides of the vessel
proved its en deavour to recede from the axis ; and this endeavour showed the
real cir cular motion of the water perpetually increasing, till it had acquired
its greatest quantity, when the water rested relatively in the vessel. And
therefore this endeavour does not depend upon any translation of the water in
respect of the ambient bodies, nor can true circular motion be defined by such
translation. There is only one real circular motion of any one revolving body,
corresponding to only one power of endeavouring to recede from its axis of
motion, as its proper and adequate effect ; but relative motions, in one and
the same body, are innumerable, according to the various relations it bears to
external bodies, and like other relations, arc altogether destitute of any real
effect, any otherwise than they may perhaps par take of that one only true
motion. And therefore in their system who suppose that our heavens, revolving
below the sphere of the fixed stars, carry the planets along with them ; the
several parts of those heavens, and the planets, which are indeed relatively at
rest in their heavens, do yet really move. For they change their position one
to another (which never happens to bodies truly at rest), and being carried
together with their heavens, partake of their motions, and as parts of
revolving wholes, endeavour to recede from the axis of their motions. Wherefore
relative quantities are not the quantities themselves, whose names they bear,
but those sensible measures of them (either accurate cr inaccurate), which arc
commonly used instead of the measured quantities themselves. And if the meaning
of words is to he determined bv their 82 THE MATHEMATICAL PRINCIPLES use, then
by the names time, space, place and motion, their measures arv properly to be
understood ; and the expression will be unusual, and purely mathematical, if
the measured quantities themselves are meant. Upon which account, they do
strain the sacred writings, who there interpret those words for the measured
quantities. Nor do those less defile the purity of mathematical and
philosophical truths, who confound real quan tities themselves with their
relations and vulgar measures. It is indeed a matter of great difficulty to
discover, and effectually to distinguish, the true motions of particular bodies
from the apparent ; be cause the parts of that immovable space, in which those
motions are per formed, do by no means come under the observation of our
senses. Yet the thing is not altogether desperate : for we have some arguments
to guide us, partly from the apparent motions, which are the differences of the
true motions ; partly from the forces, which are the causes and effects of the
true motions. For instance, if tAvo globes, kept at a given distance one from
the other by means of a cord that connects them, were revolved about their
common centre of gravity, we might, from the tension of the cord, discover the
endeavour of the globes to recede from the axis of their motion, and from
thence we might compute the quantity of their circular motions. And then if any
equal forces should be impressed at once on the alternate faces of the globes
to augment or diminish their circular motions, from the increase or decr ase of
the tensicn of 1 le cord, we might infer the increment or decrement of their
motions : and thence would be found on what faces those forces ought to be
impressed, that the motions of the globes might be most augmented ; that is, we
might discover their hindermost faces, or those which, in the circular motion,
do follow. But the faces which follow being known, and consequently the
opposite ones that precede, we should likewise know the determination of their
motions. And thus we might find both the quantity and the determination of this
circu lar motion, even in an immense vacuum, where there was nothing external
or sensible with which the globes could be compared. But now, if in that space
some remote bodies were placed that kept always a given position one to another,
as the fixed stars do in our regions, we could not indeed determine from the
relative translation of the globes among those bodies, whether the motion did
belong to the globes or to the bodies. But if we observed the cord, and found
that its tension was that very tension which the motions of the globes
required, we might conclude the motion to be in the globes, and the bodies to
be at rest ; and then, lastly, from the trans lation of the globes among the
bodies, we should find the determination oi their motions. But how we are to
collect the true motions from their causes, effects, and apparent differences ;
and, vice versa, how from the mo tions, either true or apparent, we may come to
the knowledge of theii causes and effects, shall be explained more at large in
the following tra<;t For to this end it was that I composed it. OF
NATURAL PHILOSOPHY. AXIOMS, OR LAWS OF MOTION. LAW I. Hvery body perseveres in
its state of rest, or of uniform motion in a ri^ht line, unless it is compelled
to change that state by forces impressed thereon. PROJECTILES persevere in
their motions, so far as they are not retarded by the resistance of the air, or
impelled downwards by the force of gravity A top, whose parts by their cohesion
are perpetually drawn aside from rectilinear motions, does not cease its
rotation, otherwise than as it is re tarded by the air. The greater bodies of
the planets and comets, meeting with less resistance in more free spaces,
preserve then jDotions both pro gressive and circular for a much longer time.
LAW II. The alteration of motion is ever proportional to the motive force
impreus ed ; and is made in the direction of the right line in. which that
force is impressed. If any force generates a motion, a double force will
generate double the motion, a triple force triple the motion, whether that
force be impressed altogether and at once, or gradually and successively. And
this motion (being always directed the same way with the generating force), if
the body moved before, is added to or subducted from the former motion,
according as they directly conspire with or are directly contrary to each other
; or obliquely joined, when they are oblique, so as to produce a new motion
compounded from the determination of both. LAW III. To every action there is
always opposed an equal reaction : or the mu tual actions of two bodies upon
each other are always equal, and di rected to contrary parts. Whatever draws or
presses another is as much drawn or pressed by that other. If you press a stone
with your finger, the finger is also pressed by the stone. If a horse draws a
stone tied to a rope, the horse (if I may so say) will be equally drawn back
towards the stone: for the distended rope, by the same endeavour to relax or
unbend itself, will draw the horse as much towards the stone, as it does the
stone towards the horse, and will obstruct the progress of the one as much as
it advances that of the other. 84 THE MATHEMATICAL PRINCIPLES If a body impinge
upon another, and by its force change the motion of (It* other, that body also
(because of the equality of the mutual pressure) will undergo an equal change,
in its own motion, towards the contrary part. The changes made by these actions
are equal, not in the velocities but in the motions of bodies ; that is to say,
if the bodies are not hindered by any other impediments. For, because the
motions are equally changed, the changes of the velocities made towards
contrary parts are reciprocally pro portional to the bodies. This law takes
place also in attractions, as will be proved in the next scholium. COROLLARY I.
A body by two forces conjoined will describe the diagonal of a parallelo gram,
in the same time that it wovld describe the sides, by those forces apart. If a
body in a given time, by the force M impressed apart in the place A, should
with an uniform motion / be carried from A to B ; and by the force N impressed
apart in the same place, should be carried from A to c ~\) C ; complete the
parallelogram ABCD, and, by both forces acting together, it will in the same
time be carried in the diagonal from A to D. For since the force N acts in the
direction of the line AC, parallel to BD, this force (by the second law) will
not at all alter the velocity generated by the other force M, by which the body
is carried towards the line BD. The body therefore will arrive at the line BD
in the same time, whether the rorce N be impressed or not ; and therefore at
the end of that time it will he found somewhere in the line BD. By the same
argument, at the end of the same time it AY ill be found somewhere in the line
CD. Therefore it will be found in the point D, where both lines meet. But it
will move in ;i right line from A to D, by Law I. COROLLARY II. And hence is
explained the composition of any one direct force AD, out of any two oblique
forces AC and CD ; and, on the contrary, the re solution of any one direct
force AD into two oblique forces AC and CD : which composition and resolution
are abundantly confirmed from, mechanics. As if the unequal radii OM and ON
drawn from the centre O of any wheel, should sustain the weights A and P by the
cords MA and NP ; and the forces of those weights to move the wheel were
required. Through the rentre O draw the right line KOL, meeting the cords
perpendicularly in A and L; and from the centre O, with OL the greater of the
distances OF NATURAL PHILOSOPHY. OK arid OL, describe a circle, meeting the
cord MA in D : and drawing OD, make AC paral- "^ lei and DC
perpendicular thereto. Now, it being indifferent whether the points K, L, D, of
the cords be lixed to the plane of the wheel or not, the weights will have the
same effect whether they are suspended from the points K and L, or from D and
L. Let the whole force of the weight A be represented by the line AD, and let
it be resolved into the forces AC and CD ; of which the force AC, drawing the
radius OD directly from the centre, will have no effect to move the wheel : but
the other force DC, drawing the radius DO perpendicularly, will have the same
effect as if it drew perpendicularly the radius OL equal to OD ; that is, it w
ill have the same effect as the weight P, if that weight is to the weight A as
the force DC is to the force DA ; that is (because of the sim ilar triangles
ADC, DOK), as OK to OD or OL. Therefore the weights A and P, which are
reciprocally as the radii OK and OL that lie in the same right line, will be
equipollent, and so remain in equilibrio ; which is the well known property of
the balance, the lever, and the wheel. If either weight is greater than in this
ratio, its force to move the wheel will be so much greater. If the weight p,
equal to the weight P, is partly suspended by the cord NJO, partly sustained by
the oblique plane pG ; draw p}i, NH, the former perpendicular to the horizon,
the latter to the plane pG ; and if the force of the weight p tending downwards
is represented by the line /?H, it may be resolved into the forces joN, HN. If
there was any plane /?Q, perpendicular to the cord y?N, cutting the other plane
pG in a line parallel to the horizon, and the weight p was supported only by
those planes pQ, pG, it would press those planes perpendicularly with the
forces pN, HN; to wit, the plane joQ, with the force joN, and the plane pG with
the force HN. And therefore if the plane pQ was taken away, so thnt the weight
might stretch the cord, because the cord, now sustaining the weight, supplies
the place of the plane that was removed, it will be strained by the same force
joN which pressed upon the plane before. Therefore, the tension of this oblique
cord joN will be to that of the other perpendic ular cord PN as jt?N to joH.
And therefore if the weight p is to the weight A in a ratio compounded of the
reciprocal ratio of the least distances of the cords PN, AM, from the centre of
the wheel, and of the direct ratio of pH tojoN, the weights will have the same
effect towards moving the wheel, and will therefore sustain each other : as any
one may find by experiment. But the weight p pressing upon those two oblique
planes, may be con sidered as a wedge between the two internal surfaces of a
body split by it; and hence tlif ft IV.P* of th^ v, ^dge and the mallet may be
determined; foi 8G THE MATHEMATICAL PRINCIPLES because the force with which the
weight p presses the plane pQi is to the force with which the same, whether by
its own gravity, or by the blow of a mallet, is impelled in the direction of
the line joH towards both the planes, as joN to pH ; and to the force with
which it presses the other plane pG, as joN to NH. And thus the force of the
screw may be deduced from a like resolution of forces ; it being no other than
a wedge impelled with the force of a lever. Therefore the use of this Corollary
spreads far and wide, and by that diffusive extent the truth thereof is farther
con firmed. For on what has been said depends the whole doctrine of mechan ics
variously demonstrated by different authors. For from hence are easily deduced
the forces of machines, which are compounded of wheels, pullics, levers, cords,
and weights, ascending directly or obliquely, and other mechan ical powers ; as
also the force of the tendons to move the bones of animals. COROLLARY III. The
(/uaittity of motion, which is collected by taking the sum of the mo tions
directed towards the same parts, and the difference of those that are directed
to contrary parts, suffers no change from the action oj bodies among
themselves. For action and its opposite re-action are equal, by Law III, and
there fore, by Law II, they produce in the motions equal changes towards oppo
site parts. Therefore if the motions are directed towards the same parts.
whatever is added to the motion of the preceding body will be subducted from
the motion of that which follows ; so that the sum will be the same as before.
If the bodies meet, with contrary motions, there will be an equal deduction
from the motions of both ; and therefore the difference of the motions directed
towards opposite parts will remain the same. Thus if a spherical body A with
two parts of velocity is triple of a spherical body B which follows in the same
right line with ten parts of velocity, the motion of A will be to that of B as
6 to 10. Suppose, then, their motions to be of 6 parts and of 10 parts, and the
sum will be 16 parts. Therefore, upon the meeting of the bodies, if A acquire
3, 4, or 5 parts of motion, B will lose as many ; and therefore after reflexion
A will proceed With 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the sum
remaining always of 16 parts as before. If the body A acquire 9, 10, 11, or 12
parts of motion, and therefore after meeting proceed with 15, 16, 17, or 18
parts, the body B, losing so many parts as A has got, will either proceed with
1 part, having lost 9, or stop and remain at rest, as having lost its whole
progressive motion of 10 parts ; or it will go back with 1 part, having not
only lost its whole motion, but (if 1 may so say) one part more; or it will go
back with 2 parts, because a progressive mo tion of 12 parts is taken off. And
so the sums of the Conspiring motions 15 ,1, or 16-1-0, and the differences of
the contrary i otions 17 1 and OF NATURAL PHILOSOPHY. [S 2, will always be
equal to 16 parts, as they were before tie meeting and reflexion of the bodies.
But, the motions being known with whicli the bodies proceed after reflexion,
the velocity of either will be also known, by taking the velocity after to the
velocity before reflexion, as the motion after is to the motion before. As in
the last case, where the motion of tho body A was of parts before reflexion and
of IS parts after, and the velocity was of 2 parts before reflexion, the
velocity thereof after reflexion will be found to be of 6 parts ; by saying, as
the parts of motion before to 18 parts after, so are 2 parts of velocity before
reflexion to (5 parts after. But if the bodies are cither not spherical, or,
moving in different right lines, impinge obliquely one upon the other, and
their mot ons after re flexion are required, in those cases we are first to
determine the position of the plane that touches the concurring bodies in the
point of concourse , then the motion of each body (by Corol. II) is to be
resolved into two, one perpendicular to that plane, and the other parallel to
it. This done, be cause the bodies act upon each other in the direction of a
line perpendicu lar to this plane, the parallel motions are to be retained the
same after reflexion as before ; and to the perpendicular motions we are to
assign equal changes towards the contrary parts ; in such manner that the sum
of the conspiring and the difference of the contrary motions may remain the
same as before. From such kind of reflexions also sometimes arise the circular
motions of bodies about their own centres. But these are cases which I do not
consider in what follows ; and it would be too tedious to demonstrate every
particular that relates to this subject. COROLLARY IV. The common centre of
gravity of two or more bodies does not alter its state of motion or rest by the
actions of the bodies among themselves ; and therefore the common centre of
gravity of all bodies acting upon each other (excluding outward actions and
impediments) is either at rest, or moves uniformly in a right line. For if two
points proceed with an uniform motion in right lines, and their distance be
divided in a given ratio, the dividing point will be either at rest, or proceed
uniformly in a right line. This is demonstrated here after in Lem. XXIII and
its Corol., when the points are moved in the same plane ; and by a like way of
arguing, it may be demonstrated when the points are not moved in the same
plane. Therefore if any number of Kdies move uniformly in right lines, the
common centre of gravity of any two of them is either at rest, or proceeds
uniformly in a right line ; because the line which connects the centres of
those two bodies so moving is divided at that common centre in a given ratio.
In like manner the common centre of those two and that of a third body will be
either at rest or moving uni formly in aright line because at that centre the
distance 1 etween th? 88 THE MATHEMATICAL PRINCIPLES common centre of the two
bodies, and the centre of this last, is divided in a given ratio. In like
manner the common centre of these three, and of a fourth body, is either at
rest, or moves uniformly in a right line ; because the distance between the
common centre of the three bodies, and the centre of the fourth is there also
divided in a given ratio, and so on m itifinitum. Therefore, in a system of
bodies where there is neither any mutual action among themselves, nor any
foreign force impressed upon them from without, and which consequently move
uniformly in right lines, the common centre of gravity of them all is either at
rest or moves uniformly forward in a right line. Moreover, in a system of two
bodies mutually acting upon each other, since the distances between their
centres and the common centre of gravity of both are reciprocally as the
bodies, the relative motions of those bodies, whether of approaching to or of
receding from that centre, will be equal among themselves. Therefore since the
changes which happen to motions are equal and directed to contrary parts, the
common centre of those bodies, by their mutual action between themselves, is
neither promoted nor re tarded, nor suffers any change as to its state of
motion or rest. But in a system of several bodies, because the common centre of
gravity of any two acting mutually upon each other suffers no change in its
state by that ac tion : and much less the common centre of gravity of the
others with which that action does not intervene ; but the distance between
those two centres is divided by the common centre of gravity of all the bodies
into parts re ciprocally proportional to the total sums of those bodies whose
centres they are : and therefore while those two centres retain their state of
motion or rest, xhe common centre of all does also retain its state : it is
manifest that the common centre of all never suffers any change in the state of
its mo tion or rest from the actions of any two bodies between themselves. But
in such & system all the actions of the bodies among themselves either hap
pen between two bodies, or are composed of actions interchanged between some
two bodies ; and therefore they do never produce any alteration in the comrrv n
centre of alias to its state of motion or rest. Wherefore tiince that centre,
when the bodies do not act mutually one upon another, Oilier is nt rest or
moves uniformly forward in some right line, it will,
:v\>U7ithst?nding the mutual actions of the bodies among themselves,
always jAY-jevere in its state, either of rest, or of proceeding uniformly in a
right liiv,, unless it is forced out of this state by the action of some power
imprev^-d from without upon the whole system. And therefore the same law take*1
place in a system consisting of many bodies as in one single body, with wsgard
to their persevering in their state of motion or of rest. For the pi \\jressive
motion, whether of one single body, or of a whole system of bodies us always to
be estimated from the motion of the centre of gravity. COROLLARY V. The motions
cf bcdies included in a given space a ~e Ike same among OF NATURAL PHILOSOPHY.
89 themselves, whether that space is at rest, or moves uniformly forwards in a
right line without any circular motion. For the differences of the motions
tending towards the same parts, and the sums of those that tend towards
contrary parts, are, at first (by sup position), in both cases the same ; and
it is from those sums and differences that the collisions and impulses do arise
with which the bodies mutually impinge one upon another. Wherefore (by Law II),
the effects of those collisions will be equal in both cases ; and therefore the
mutual motions of the bodies among themselves in the one case will remain equal
to the mutual motions of the bodies among themselves in the other. A clear
proof of which we have from the experiment of a ship ; where all motions happen
after the same manner, whether the ship is at rest, or is carried uniformly
forwards in a right line. COROLLARY VI. If bodies, any how moved among themselves,
are urged in the direct-ton of parallel lines by equal accelerative forces,
they will all continue to move among themselves, after the same manner as if
they had been urged by no such forces. For these forces acting equally (with
respect to the quantities of the DO dies to be moved), and in the direction of
parallel lines, will (by Law II) move all the bodies equally (as to velocity),
and therefore will never pro duce any change in the positions or motions of the
bodies among themselves. SCHOLIUM. Hitherto I have laid down such principles as
have been received by math ematicians, and are confirmed by abundance of
experiments. By the first two Laws and the first two Corollaries, Galileo
discovered that the de scent of bodies observed the duplicate ratio of the
time, and that the mo tion of projectiles was in the curve of a parabola;
experience agreeing with both, unless so far as these motions are a little
retarded by the re sistance of the air. When a body is falling, the uniform
force of its gravity acting equally, impresses, in equal particles of time,
equal forces upon that body, and therefore generates equal velocities; and in
the whole time impresses a whole force, and generates a whole velocity
proportional to the time. And the spaces described in proportional times are as
the velocities and the times conjunctly ; that is, in a duplicate ratio of the
times. And when a body is thrown upwards, its uniform gravity im presses forces
and takes off velocities proportional to the times ; and the times of ascending
to the greatest heights are as the velocities to be taken off, and those
heights are as the velocities and the times conjunetly, or ir, the duplicate
ratio of the velocities. And if a body be projected in any direction, the
motion arising from its projection jS compounded with the 90 THE MATHEMATICAL
PRINCIPLES motion arising from its gravity. As if the body A by its motion of
piojection alone could describe in a given time the right line AB, and with its
motion of falling alone could describe in the same time the altitude AC ;
complete the paralellogram ABDC, and the body by that compounded motion will at
the end of the time be found in the place D ; and the curve line AED, which
that body describes, will be a parabola, to which the right line AB will be a
tangent in A ; and whose ordinate BD will be as the square of the line AB. On
the same Laws and Corollaries depend those things which have been demon strated
concerning the times of the vibration of pendulums, and are con firmed by the
daily experiments of pendulum clocks. By the same, to gether with the third
Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huvgens, the greatest geometers of
our times, did severally determine the rules of the congress and reflexion of
hard bodies, and much about the same time communicated their discoveries to the
Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis,
indeed, was something more early in the publication ; then followed Sir
Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed
the truth of the thing before the Royal Society by the experiment of pendulums,
which Mr. Mariottc soon after thought fit to explain in a treatise entirely
upon that subject. But to bring this experiment to an accurate agreement with
the theory, we are to have a due regard as well to the resistance of the air
bodies. Let the spherical bodies CD F II as to the clastic force of the
concurrin A, B be suspended by the parallel and equal strings AC, Bl), from the
centres C, D. About these centres, with those intervals, describe the
semicircles EAF, GBH, bisected by the radii CA, DB. Bring the body A to any
point R of the arc EAF, and (withdrawing the body B) let it go from thence, and
after one oscillation suppose it to return to the point V : then RV will be the
0 comments:
Post a comment