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MATH.-STAT.
SIM ISAAC MIBWfOM
NEWTON S PRINCIPIA.
THE
MATHEMATICAL PRINCIPLES
OF
NATURAL PHILOSOPHY,
BY SIR ISAAC NEWTON;
TRANSLATED INTO ENGLISH BY ANDREW MOTTE.
TO WHICH IS ADDKTV
NEWTON S SYSTEM OF THE WORLD ;
With a Portrait taken from the Bust in the Royal Observatory at Greenwich.
FIRST AMERICAN EDITION, CAREFULLY REVISED AND CORRECTED,
WITH A LIFE OF THE AUTHOR, BY PI. W. CHITTENDEN, M. A., &e.
NEW-YORK
PUBLISHED BY DANIEL ADEE, 45 LIBERTY STREET.
p*-
Kntered according to Act of Congress, in the year 1846, by
DANIEL ADEE.
3!Ltht Clerk s Office ut tiie Southern Oisli:ct Court of New-York.
TWuey * Lockwoof, Stom
16 Spruce St. N. Y.
DEDICATION.
TO THE
TEACHERS OF THE NORMAL SCHOOL
OF THE STATE OF NEW-YORK.
GENTLEMEN !
A stirring freshness in the air, and ruddy streaks upon the
horizon of the moral world betoken the grateful dawning of a new
ora. The days of a drivelling instruction are departing. With
us is the opening promise of a better time, wherein genuine man
hood doing its noblest work shall have adequate reward.
TEACHER is the highest and most responsible office man can fill.
Its dignity is, and will yet be held commensurate with its duty
a duty boundless as man s intellectual capacity, and great as his
moral need a duty from the performance of which shall emanate
an influence not limited to the now and the here, but which surely
will, as time flows into eternity and space into infinity, roll up, a
measureless curse or a measureless blessing, in inconceivable
swellings along the infinite curve. It is an office that should be
esteemed of even sacred import in this country. Ere long a hun
dred millions, extending from the Atlantic to the Pacific, from
Baffin s Bay to that of Panama, shall call themselves American
citizens. What a field for those two master-passions of the hu
man soul the love of Rule, and the love of Gain ! How shall
our liberties continue to be preserved from the graspings of Am
bition and the corruptions of Gold ? Not by Bills of Rights
4 DEDICATION.
Constitutions, and Statute Books ; but alone by the rightly culti
vated hearts and heads of the PEOPLE. They must themselves
guard the Ark. It is yours to tit them for the consecrated
charge. Look well to it : for you appear clothed in the majesty
of great power ! It is yours to fashion, and to inform , to save,
and to perpetuate. You are the Educators of the People : you
are the prime Conservators of the public weal. Betray your
trust, and the sacred fires would go out, and the altars crumble
into dust : knowledge become lost in tradition, and Christian no
bleness a fable ! As you, therefore, are multiplied in number,
elevated in consideration, increased in means, and fulfill, well and
faithfully, all the requirements of true Teachers, so shall our fa
voured land lift up her head among the nations of the earth, and
call herself blessed.
In conclusion, Gentlemen, to you, as the conspicuous leaders
in the vast and honourable labour of Educational Helbrm, ana
Popular Teaching, the First American Edition of the PRINCIPIA ol
Newton the greatest wrork of the greatest Teacher is most
respectfully dedicated.
N. W. CHITTENDEN.
INTRODUCTION TO THE AMERICAN EDITION.
THAT the PRINCIPIA of Newton should have remained so gen
erally unknown in this country to the present day is a somewhat
remarkable fact ; because the name of the author, learned with
the very elements of science, is revered at every hearth-stone
where knowledge and virtue are of chief esteem, while, abroad,
in all the high places of the land, the character which that name
recalls is held up as the noblest illustration of what MAN may be,
and may do, in the possession and manifestation of pre-eminent
intellectual and moral worth ; because the work is celebrated, not
only in the history of one career and one mind, but in the history
of all achievement and human reason itself; because of the spirit
of inquiry, which has been aroused, and which, in pursuing its
searchings, is not always satisfied with stopping short of the foun
tain-head of any given truth ; and, finally, because of the earnest
endeavour that has been and is constantly going on, in many
sections of the Republic, to elevate the popular standard of edu
cation and give to scientific and other efforts a higher and a
better aim.
True, the PRINCIPIA has been hitherto inaccessible to popular
use. A few copies in Latin, and occasionally one in English may
be found in some of our larger libraries, or in the possession of
some ardent disciple of the great Master. But a d^ad language
in the one case, and an enormous price in both, particularly in
that of the English edition, have thus far opposed very sufficient
obstacles to the wide circulation of the work. It is now, how
ever, placed within the reach of all. And in performing this la
bour, the utmost care has been taken, by collation, revision, and
otherwise, to render the First American Edition the most accurate
and beautiful in our language.
u Le plus beau monument que
l
? on puisse clever a la gloire de Newton, c est une bonne edition
de ses ouvrages :" and a monument like unto that we would here
V: INTRODUCTION TO
set up. The PRINCIPIA, above all, glows with the immortality of
a transcendant mind. Marble and brass dissolve and pass away ;
but the true creations of genius endure, in time and beyond time,
forever : high upon the adamant of the indestructible, they send
forth afar and near, over the troublous waters of life, a pure, un
wavering, quenchless light whereby the myriad myriads of barques,
richly laden with reason, intelligence and various faculty, are
guided through the night and the storm, by the beetling shore
and the hidden rock, the breaker and the shoal, safely into havens
calm and secure.
To the teacher and the taught, the scholar and the student, the
devotee of Science and the worshipper of Truth, the PRINCIPIA
must ever continue to be of inestimable value. If to educate
means, not so much to store the memory with symbols and facts,
as to bring forth the faculties of the soul and develope them to the
full by healthy nurture and a hardy discipline, then, what so effec
tive to the accomplishment of that end as the study of Geometri
cal Synthesis ? The Calculus, in some shape or other, is, indeed,
necessary to the successful prosecution of researches in the higher
branches of philosophy. But has not the Analytical encroached
upon the Synthetical, and Algorithmic Formulae been employed
when not requisite, either for the evolution of truth, or even its
apter illustration ? To each method belongs, undoubtedly, an
appropriate use. Newton, himself the inventor of Fluxions,
censured the handling of Geometrical subjects by Algebraical
calculations ; and the maturest opinions which he expressed were
additionally in favour of the Geometrical Method. His prefer
ence, so strongly marked, is not to be reckoned a mere matter oi
taste ; and his authority should bear with preponderating weight
upon the decision of every instructor in adopting what may be
deemed the best plan to insure the completes! mental develop
ment. Geometry, the vigorous product of remote time ; blended
with the earliest aspirations of Science and the earliest applica
tions of Art ; as well in the measures of music as in the move
ment of spheres ; as wholly in the structure of the atom as in that
of the world; directing MOTION and shaping APPEARANCE; in a
wonl, *t the moulding of the created all, is, in comprehensive
THE AMERICAN EDITION. Vll
view, the outward form of that Inner Harmony of which and in
which all things are. Plainly, therefore, this noble study has
other and infinitely higher uses than to increase the power of ab
straction. A more general and thorough cultivation of it should
oe strenuously insisted on. Passing from the pages of Euclid or
Legendre, might not the student be led, at the suitable time, to
those of the PRINCIPIA wherein Geometry may be found in varied
use from the familiar to the sublime ? The profoundest and the
happiest results, it is believed, would attend upon this enlargement
of our Educational System.
Let the PRINCIPIA, then, be gladly welcomed into every Hall
where a TRUE TEACHER presides. And they who are guided to
the diligent study of this incomparable work, who become
strengthened by its reason, assured by its evidence, and enlight
ened by its truths, and who rise into loving communion with the
great and pure spirit of its author, will go forth from the scenes
of their pupilage, and take their places in the world as strongminded,
right-hearted men such men as the Theory of our
Government contemplates and its practical operation absolutely
demands.
LIFE OF
SIE ISAAC NEWTON.
Nec fas est proprius mortal? attingere Divos. HALLEY.
FROM the thick darkness of the middle ages man s struggling
spirit emerged as in new birth ; breaking out of the iron control
of that period ; growing strong and confident in the tug and din
of succeeding conflict and revolution, it bounded forwards and
upwards with resistless vigour to the investigation of physical and
moral truth ; ascending height after height ; sweeping afar over
the earth, penetrating afar up into the heavens ; increasing in en
deavour, enlarging in endowment ; every where boldly, earnestly
out-stretching, till, in the AUTHOR of the PRINCIPIA, one arose,
who, grasping the master-key of the universe and treading its
celestial paths, opened up to the human intellect the stupendous
realities of the material world, and, in the unrolling of its harmo
nies, gave to the human heart a new song to the goodness, wis
dom, and majesty of the all-creating, all-sustaining, all-perfect
God.
Sir Isaac Newton, in whom the rising intellect seemed to attain,
as it were, to its culminating point, was born on the 25th of De
cember, O. S. 1642 Christmas day at Woolsthorpe, in the
parish of Colsterworth, in Lincolnshire. His father, John New
ton, died at the age of thirty-six, and only a few months after his
marriage to Harriet Ayscough, daughter of James Ayscough, oi
Rutlandshire. Mrs. Newton, probably wrought upon by the
early loss of her husband, gave premature birth to her only and
posthumous child, of which, too, from its extreme diminutiveness,
she appeared likely to be soon bereft. Happily, it was otherwise
decreed ! The tiny infant, on whose little lips the breath of life
10 LIFE OF SIR ISAAC NEWTON.
so doubtingly hovered, lived ;
lived to a vigorous maturity, to a
hale old age ; lived to become the boast of his country, the won
der of his time, and the "ornament of his srjecies."
Beyond the grandfather, Robert Newton, the descent of Sir
Isaac cannot with certainty be traced. Two traditions were held
in the family : one, that they were of Scotch extraction ; the
other, that they came originally from Newton, in Lancashire,
dwelling, for a time, however, at Westby, county of Lincoln, be
fore the removal to and purchase of Woolsthorpe about a hundred
years before this memorable birth.
The widow Newton was left with the simple means of a com
fortable subsistence. The Woolsthorpe estate together with
small one which she possessed at Sewstern, in Leicestershire, yield
ed her an income of some eighty pounds ; and upon this limited sum,
she had to rely chiefly for the support of herself, and the educa
tion of her child. She continued his nurture for three years,
when, marrying again, she confided the tender charge to the care
of her own mother.
Great genius is seldom marked by precocious development ;
and young Isaac, sent, at the usual age, to two day schools at
Skillington and Stoke, exhibited no unusual traits of character.
In his twelfth year, he was placed at the public school at Grantham,
and boarded at the house of Mr. Clark, an apothecary.
But even in this excellent seminary, his mental acquisitions con
tinued for a while unpromising enough : study apparentlv had no
charms for him ; he was very inattentive, and ranked low in the
school. One day, however, the boy immediately above our seem
ingly dull student gave him a severe kick in the stomach ; Isaac,
deeply affected, but with no outburst of passion, betook himself,
with quiet, incessant toil, to his books ; he quickly passed above
the offending classmate ; yet there he stopped not ; the strong
spirit was, for once and forever, awakened, and, yielding to itb
noble impulse, he speedily took up his position at the head of all.
His peculiar character began now rapidly to unfold itself.
Close application grew to be habitual. Observation alternated
with reflection. " A sober, silent, thinking lad," yet, the wisest
and the kindliest, the indisputable leader of his fellows. Gener
LIFE OF SIR ISA VC NEWTON. 11
osity, modesty, and a love of truth distinguished him then as ever
afterwards. He did not often join his classmates in play ; but he
would contrive for them various amusements of a scientific kind.
Paper kites he introduced ; carefully determining their best form
and proportions, and the position and number of points whereby
to attach the string. He also invented paper lanterns ; these
served ordinarily to guide the way to school in winter mornings,
but occasionally for quite another purpose ; they were attached to
the tails of kites in a dark night, to the dismay of the country people
dreading portentous comets, and to the immeasureable delight ol
his companions. To him, however, young as he was, life seemed
to have become an earnest thing. When not occupied with his
studies, his mind would be engrossed with mechanical contrivances ;
now imitating, now inventing. He became singularly skilful in the
use of his little saws, hatchets, hammers, and other tools. A
windmill was erected near Grantham ; during the operations ol
the workmen, he was frequently present ;
in a short time, he had
completed a perfect working model of it, which elicited general
admiration. Not content, however, with this exact imitation, he
conceived the idea of employing, in the place of sails, animal power ,
and, adapting the construction of his mill accordingly, he enclosed
in it a mouse, called the miller, and which by acting on a sort ot
treadvvheel, gave motion to the machine. He invented, too, a
mechanical carriage having four wheels, and put in motion with
a handle worked by the person sitting inside. The measurement
of time early drew his attention. He hrst constructed a water
clock, in proportions somewhat like an old-fashioned house clock.
The index of the dial plate was turned by a piece of wood acted
upon by dropping water. This instrument, though long used by
himself, and by Mr. Clark s family, did not satisfy his inquiring
mind. His thoughts rose to the sun ; and, by careful and oft-re
peated observations of the solar movements, he subsequently
formed many dials. One of these, named Isaac s dial, was the
accurate result of years labour, and was frequently referred to
for the hour of the day by the country people.
May we not discern in these continual efforts the diligent re
search^ the patient meditation, the aspiring glance, and the energy
12 LIFE OF SIR ISAAC NEWTON.
of discovery the stirring elements of that wondrous spirit,
which, clear, calm, and great, moved, in after years, through
deep onward through deep of Nature s mysteries, unlocking her
strongholds, dispelling darkness, educing order everywhere si
lently conquering.
Newton had an early and decided taste for drawing. Pictures,
taken sometimes from copies, but often from life, and drawn,
coloured and framed by himself, ornamented his apartment. He
was skilled also, in poetical composition,
" excelled in making
verses
;" some of these were borne in remembrance and repeated,
seventy years afterward, by Mrs. Vincent, for whom, in early
youth, as Miss Storey, he formed an ardent attachment. She
was the sister of a physician resident near Woolsthorpe ; but
Newton s intimate acquaintance with her began at Grantham.
where they were both numbered among the inmates of the same
house. Two or three years younger than himself, of great per
sonal beauty, and unusual talent, her society afforded him the
greatest pleasure ; and their youthful friendship, it is believed,
gradually rose to a higher passion ; but inadequacy of fortune
prevented their union. Miss Storey was afterwards twice mar
ried ; Newton, never; his esteem for her continued unabated
during life, accompanied by numerous acts of attention and
kindness.
In 1656, Newton s mother was again left a widowr
, and took
up her abode once more at Woolsthorpe. He was now fifteen
years of age, and had made great progress in his studies
; but she,
desirous of his help, and from motives of economy, recalled him
from school. Business occupations, however, and the manage
ment of the farm, proved utterly distasteful to him. When sent to
Grantham Market on Saturdays, he would betake himself to his
former lodgings in the apothecary s garret, where some of Mr.
Clark s old books employed his thoughts till the aged and trust
worthy servant had executed the family commissions and announced
the necessity of return : or, at other times, our young philosopher
would seat himself under a hedge, by the wayside, and continue
his studies till the same faithful personage proceeding alone to
the town and completing the day s business stopped as he re
LIFE OF SIR ISAAC NEWTON, 13
turned. The more immediate affairs of the farm received no
better attention. In fact, his passion for study grew daily more
absorbing, and his dislike for every other occupation more in
tense. His mother, therefore, wisely resolved to give him all the
advantages which an education could confer. He was sent back
to Grantham school, where he remained for some months in busy
preparation for his academical studies. At the recommendation
of one of his uncles, who had himself studied at Trinity College,
Cambridge, Newton proceeded thither, and was duly admitted.
on the 5th day of June 1660, in the eighteenth year of his age.
The eager student had now entered upon a new and wider
field
; and we find him devoting himself to the pursuit of know
ledge with amazing ardour and perseverance. Among other sub
jects, his attention was soon drawn to that of Judicial Astrology
He exposed the folly of this pseudo-science by erecting a figure
with the aid of one or two of the problems of Euclid ; and thus
began his study of the Mathematics. His researches into this
science were prosecuted with unparallelled vigour and success.
Regarding the propositions contained in Euclid as self-evident
truths, he passed rapidly over this ancient system a step which
he afterward much regretted and mastered, without further pre
paratory study, the Analytical Geometry of Descartes. Wallis s
Arithmetic of Infinites, Saunderson s Logic, and the Optics of
Kepler, he also studied with great care ; writing upon them
many comments ; and, in these notes on Wallis s work was un
doubtedly the germ of his fluxionary calculus. His progress was
so great that he found himself more profoundly versed than his tutor
in many branches of learning. Yet his acquisitions were not
gotten with the rapidity of intuition
; but they were thoroughly
made and firmly secured. Quickness of apprehension, or Intel
lectual nimbleness did not belong to him. He saw too far : his,
insight was too deep. He dwelt fully, cautiously upon the least
subject ; while to the consideration of the greatest, he brought a
massive strength joined with a matchless clearness, that, regard
less of the merely trivial or unimportant, bore with unerring sa
gacity upon the prominences of the subject, and, grappling with
its difficulties, rarely failed to surmount them.
14 LIFE OF SIR ISAAC NEWTON
His early and fast friend, Dr. Barrow in compass of inven
tion only inferior to Newton who had been elected Professor
of Greek in the University, in 1660, was made Lucasian Profes
sor of Mathematics in 1663, and soon afterward delivered his
Optical Lectures : the manuscripts of these were revised by New
ton, and several oversights corrected, and many important sug
gestions made by him ; but they were not published till 1669.
In the year 1665, he received the degree of Bachelor of Arts ;
and, in 1666, he entered upon those brilliant and imposing dis
coveries which have conferred inappreciable benefits upon science,
and immortality upon his own name.
Newton, himself, states that he was in possession of his Method
of Fluxions, " in the year 1666, or before." Infinite quantities
had long been a subject of profound investigation ; among the
ancients by Archimedes, and Pappus of Alexandria ; among the
moderns by Kepler, Cavaleri, Roberval, Fermat and Wallis.
With consummate ability Dr. Wallis had improved upon the lahours
of his predecessors : with a higher power, Newton moved
forwards from where Wallis stopped. Our author first invented
his celebrated BINOMIAL THEOREM. And then, applying this
Theorem to the rectification of curves, and to the determination
of the surfaces and contents of solids, and the position of their
centres of gravity, he discovered the general principle of deducing
the areas of curves from the ordinate, by considering the area as
a nascent quantity, increasing by continual fluxion in the propor
tion of the length of the ordinate, and supposing the abscissa
to increase uniformly in proportion to the time. Regarding lines
as generated by the motion of points, surfaces by the motion of
lines, and solids by the motion of surfaces, and considering that
the ordinates, abscissae, &c., of curves thus formed, vary accord
ing to a regular law depending on the equation of the curve,
he deduced from this equation the velocities with which these
quantities are generated, and obtained by the rules of infinite
series, the ultimate value required. To the velocities with which
