*From `A Short Account of the History of Mathematics' (4th edition, 1908)
by W. W. Rouse Ball.*

The mathematicians considered in the last chapter commenced the creation of those
processes which distinguish modern mathematics. The extraordinary abilities of
Newton enabled him within a few years to perfect the more elementary of those
processes, and to distinctly advance every branch of mathematical science then
studied, as well as to create some new subjects. Newton was the contemporary
and friend of Wallis, Huygens, and others of those mentioned in the last
chapter, but though most of his mathematical work was done between the years
1665 and 1686, the bulk of it was not printed - at any rate in book-form - till
some years later.

I propose to discuss the works of Newton more fully than those of other
mathematicians, partly because of the intrinsic importance of his discoveries,
and partly because this book is mainly intended for English readers, and the
development of mathematics in Great Britain was for a century entirely in the
hands of the Newtonian school.

*Isaac Newton* was born in Lincolnshire, near Grantham, on December
25, 1642, and died at Kensington, London, on March 20, 1727. He was educated at
Trinity College, Cambridge, and lived there from 1661 till 1696, during which
time he produced the bulk of his work in mathematics; in 1696 he was appointed
to a valuable Government office, and moved to London, where he resided till his
death.

His father, who had died shortly before Newton was born, was a yeoman
farmer, and it was intended that Newton should carry on the paternal farm. He
was sent to school at Grantham, where his learning and mechanical proficiency
excited some attention. In 1656 he returned home to learn the business of a
farmer, but spent most of his time solving problems, making experiments, or
devising mechanical models; his mother noticing this, sensibly resolved to find
some more congenial occupation for him, and his uncle, having been himself
educated at Trinity College, Cambridge, recommended that he should be sent
there.

In 1661 Newton accordingly entered as a student at Cambridge, where for the
first time he found himself among surroundings which were likely to develop his
powers. He seems, however, to have had but little interest for general society
or for any pursuits save science and mathematics. Luckily he kept a diary, and
we can thus form a fair idea of the course of education of the most advanced students
at an English university at that time. He had not read any mathematics before
coming into residence, but was acquainted with Sanderson's Logic,
which was then frequently read as preliminary to mathematics. At the beginning
of his first October term he happened to stroll down to Stourbridge Fair, and
there picked up a book on astrology, but could not understand it on account of
the geometry and trigonometry. He therefore bought a Euclid, and was surprised
to find how obvious the propositions seemed. He thereupon read Oughtred's Clavis
and Descartes's Géométrie, the latter of which he managed to
master by himself, though with some difficulty. The interest he felt in the
subject led him to take up mathematics rather than chemistry as a serious
study. His subsequent mathematical reading as an undergraduate was founded on
Kepler's Optics, the works of Vieta, van Schooten's Miscellanies,
Descartes's Géométrie, and Wallis's Arithmetica Infinitorum:
he also attended Barrow's lectures. At a later time, on reading Euclid more
carefully, he formed a high opinion of it as an instrument of education, and he
used to express his regret that he had not applied himself to geometry before
proceeding to algebraic analysis.

There is a manuscript of his, dated May 28, 1665, written in the same year
as that in which he took is B.A. degree, which is the earliest documentary
proof of his invention of fluxions. It was about the same time that he
discovered the binomial theorem.

On account of the plague the College was sent down during parts of the year
1665 and 1666, and for several months at this time Newton lived at home. This
period was crowded with brilliant discoveries. He thought out the fundamental
principles of his theory of gravitation, namely, that every particle of matter
attracts every other particle, and he suspected that the attraction varied as
the product of their masses and inversely as the square of the distance between
them. He also worked out the fluxional calculus tolerably completely: this in a
manuscript dated November 13, 1665, he used fluxions to find the tangent and
the radius of curvature at any point on a curve, and in October 1666 he applied
them to several problems in the theory of equations. Newton communicated these
results to his friends and pupils from and after 1669, but they were not
published in print till many years later. It was also whilst staying at home at
this time that he devised some instruments for grinding lenses to particular
forms other than spherical, and perhaps he decomposed solar light into
different colours.

Leaving out details and taking round numbers only, his reasoning at this
time on the theory of gravitation seems to have been as follows. He suspected
that the force which retained the moon in its orbit about the earth was the
same as terrestial gravity, and to verify this hypothesis he proceeded thus. He
knew that, if a stone were allowed to fall near the surface of the earth, the
attraction of the earth (that is, the weight of the stone) caused it to move
through 16 feet in one second. The moon's orbit relative to the earth is nearly
a circle; and as a rough approximation, taking it to be so, he knew the
distance of the moon, and therefore the length of its path; he also knew that
time the moon took to go once round it, namely, a month.

Hence he could easily find its velocity at any point such as
*M*. He could therefore find the distance *MT* through which it would
move in the next second if it were not pulled by the earth's attraction. At the
end of that second it was however at *M'*, and therefore the earth *E*
must have pulled it through the distance *TM'* in one second (assuming the
direction of the earth's pull to be constant). Now he and several physicists of
the time had conjectured from Kepler's third law that the attraction of the
earth on a body would be found to decrease as the body was removed farther away
from the earth inversely as the square of the distance from the centre of the
earth; if this were the actual law, and if gravity were the sole force which
retained the moon in its orbit, then *TM'* should be to 16 feet inversely
as the square of the distance of the moon from the centre of the earth to the
square of the radius of the earth. In 1679, when he repeated the investigation,
*TM'* was found to have the value which was required by the hypothesis,
and the verification was complete; but in 1666 his estimate of the distance of
the moon was inaccurate, and when he made the calculation he found that *TM'*
was about one-eighth less than it ought to have been on his hypothesis.

This discrepancy does not seem to have shaken his faith in the belief that
gravity extended as far as the moon and varied inversely as the square of the
distance; but from Whiston's notes of a conversation with Newton, it would seem
that Newton inferred that some other force - probably Descartes's vortices -
acted on the moon as well as gravity. This statement is confirmed by
Pemberton's account of the investigation. It seems, moreover, that Newton
already believed firmly in the principle of universal gravitation, that is,
that every particle of matter attracts every other particle, and suspected that
the attraction varied as the product of their masses and inversely as the
square of the distance between them; but it is certain that he did not then
know what the attraction of a spherical mass on any external point would be,
and did not think it likely that a particle would be attracted by the earth as
if the latter were concentrated into a single particle at its centre.

On his return to Cambridge in 1667 Newton was elected to a fellowship at his
college, and permanently took up his residence there. In the early part of
1669, or perhaps in 1668, he revised Barrow's lectures for him. The end of the
fourteenth lecture is known to have been written by Newton, but how much of the
rest is due to his suggestions cannot now be determined. As soon as this was
finished he was asked by Barrow and Collins to edit and add notes to a
translation of Kinckhuysen's Algebra; he consented to do this, but
on condition that his name should not appear in the matter. In 1670 he also
began a systematic exposition of his analysis by infinite series, the object of
which was to express the ordinate of a curve in an infinite algebraical series
every term of which can be integrated by Wallis's rule; his results on this
subject had been communicated to Barrow, Collins, and others in 1669. This was
never finished: the fragment was published in 1711, but the substance of it had
been printed as an appendix to the Optics in 1704. These works
were only the fruit of Newton's leisure, most of his time during these two
years being given up to optical researches.

In October 1669, Barrow resigned the Lucasian chair in favour of Newton.
During his tenure of the professorship, it was Newton's practice to lecture
publicly once a week, for from half-an-hour to an hour at a time, in one term
of each year, probably dictating his lectures as rapidly as they could be taken
down; and in the week following the lecture to devote four hours to
appointments which he gave to students who wished to come to his rooms to
discuss the results of the previous lecture. He never repeated a course, which
usually consisted of nine or ten lectures, and generally the lectures of one
course began from the point at which the preceding course had ended. The
manuscripts of his lectures for seventeen out of the first eighteen years of
his tenure are extant.

When first appointed Newton chose optics for the subject of his lectures and
researches, and before the end of 1669 he had worked out the details of his
discovery of the decomposition of a ray of white light into rays of different
colours by means of a prism. The complete explanation of the theory of the
rainbow followed from this discovery. These discoveries formed the
subject-matter of the lectures which he delivered as Lucasian professor in the
years 1669, 1670 and 1671. The chief new results were embodied in a paper
communicated to the Royal Society in February, 1672, and subsequently published
in the Philosophical Transactions. The manuscript of his original
lectures was printed in 1729 under the title Lectiones Opticae.
This work is divided into two books, the first of which contains four sections
and the second five. The first section of the first book deals with the
decomposition of solar light by a prism in consequence of the unequal
refrangibility of the rays that compose it, and a description of his experiments
is added. The second section contains an account of the method which Newton
invented for determining the coefficients of refraction of different bodies.
This is done by making a ray pass through a prism of the material so that the
deviation is a minimum; and he proves that, if the angle of the prism be *i*
and the deviation of the ray be ,
the refractive index will be sin ½ (*i* + )
cosec ½ *i*. The
third section is on refractions at plane surfaces; he here shews that if a ray
pass through a prism with minimum deviation, the angle of incidence is equal to
the angle of emergence; most of this section is devoted to geometrical
solutions of different problems. The fourth section contains a discussion of
refractions at curved surfaces. The second book treats of his theory of colours
and of the rainbow.

By a curious chapter of accidents Newton failed to correct the chromatic
aberration of two colours by means of a couple of prisms. He therefore
abandoned the hope of making a refracting telescope which should be achromatic,
and instead designed a reflecting telescope, probably on the modal of a small
one which he had made in 1668. The form he used is that still known by his
name; the idea of it was naturally suggested by Gregory's telescope. In 1672 he
invented a reflecting microscope, and some years later he invented the sextant
which was rediscovered by J. Hadley in 1731.

His professorial lectures from 1673 to 1683 were on algebra and the theory
of equations, and are described below; but much of his time during these years
was occupied with other investigations, and I may remark that throughout his
life Newton must have devoted at least as much attention to chemistry and
theology as to mathematics, though his conclusions are not of sufficient
interest to require mention here. His theory of colours and his deductions from
his optical experiments were at first attacked with considerable vehemence. The
correspondence which this entailed on Newton occupied nearly all his leisure in
the years 1672 to 1675, and proved extremely distasteful to him. Writing on
December 9, 1675, he says, ``I was so persecuted with discussions arising out
of my theory of light, that I blamed my own imprudence for parting with so
substantial a blessing as my quiet to run after a shadow.'' Again, on November
18, 1676, he observes, ``I see I have made myself a slave to philosophy; but if
I get rid of Mr. Linus's business, I will resolutely bid adieu to it eternally,
excepting what I do for my private satisfaction, or leave to come out after me;
for I see a man must either resolve to put out nothing new, or to become a
slave to defend it.'' The unreasonable dislike to have his conclusions doubted
or to be involved in any correspondence about them was a prominent trait in
Newton's character.

Newton was deeply interested in the question as to how the effects of light
were really produced, and by the end of 1675 he had worked out the corpuscular
or emission theory, and had shewn how it would account for all the various
phenomena of geometrical optics, such as reflexion, refraction, colours,
diffraction, etc. To do this, however, he was obliged to add a somewhat
artificial rider, that his corpuscules had alternating fits of easy reflexion
and easy refraction communicated to them by an ether which filled space. The
theory is now known to be untenable, but it should be noted that Newton
enunciated it as a hypothesis from which certain results would follow: it would
seem that he believed that wave theory to be intrinsically more probable, but
it was the difficulty of explaining diffraction on that theory that led him to
suggest another hypothesis.

Newton's corpuscular theory was expounded in memoirs communicated to the
Royal Society in December 1675, which are substantially reproduced in his Optics,
published in 1704. In the latter work he dealt in detail with his theory of
fits of easy reflexion and transmission, and the colours of thin plates, to
which he added an explanation of the colours of thick plates [bk. II, part 4]
and observations on the inflexion of light [bk. III].

Two letters written by Newton in the year 1676 are sufficiently interesting
to justify an allusion to them. Leibnitz, who had been in London in 1673, had
communicated some results to the Royal Society which he had supposed to be new,
but which it was pointed out to him had been previously proved by Mouton. This
led to a correspondence with Oldenburg, the secretary of the Society. In 1674
Leibnitz wrote saying that he possessed ``general analytical methods depending
on infinite series.'' Oldenburg, in reply, told him that Newton and Gregory had
used such series in their work. In answer to a request for information, Newton
wrote on June 13, 1676, giving a brief account of his method, but adding the
expansions of a binomial (that is, the binomial theorem) and of ;
from the latter of which he deduced that of sin *x*: this seems to be the
earliest known instance of a reversion of series. He also inserted an
expression for the rectification of an elliptic arc in an infinite series.

Leibnitz wrote on August 27 asking for fuller details; and Newton in a long
but interesting replay, dated October 34, 1676, and sent through Oldenburg,
gives an account of the way in which he had been led to some of his results.

In this letter Newton begins by saying that altogether he had used three
methods for expansion in series. His first was arrived at from the study of the
method of interpolation by which Wallis had found expressions for the area of a
circle and a hyperbola. Thus, by considering the series of expressions ,
,
,...,
he deduced by interpolations the law which connects the successive coefficients
in the expansions of ,
,...;
and then by analogy obtained the expression for the general term in the
expansion of a binomial, that is, the binomial theorem. He says that he
proceeded to test this by forming the square of the expansion of ,
which reduced to 1 - *x*²; and he proceeded in a similar
way with other expansions. He next tested the theorem in the case of
by extracting the square root of 1 - *x*², *more
arithmetico*. He also used the series to determine the areas of the circle
and the hyperbola in infinite series, and he found that the results were the
same as those he had arrived at by other means.

Having established this result, he then discarded the method of
interpolation in series, and employed his binomial theorem to express (when
possible) the ordinate of a curve in an infinite series in ascending powers of
the abscissa, and thus by Wallis's method he obtained expressions in infinite
series for the areas and arcs of curves in the manner described in the appendix
to his Optics and in his De Analysi per Equationes Numero
Terminorum Infinitas. He states that he had employed this second method
before the plague in 1665-66, and goes on to say that he was then obliged to
leave Cambridge, and subsequently (presumably on his return to Cambridge) he
ceased to pursue these ideas, as he found that Nicholas Mercator had employed
some of them in his Logarithmo-technica, published in 1668; and he
supposed that the remainder had been or would be found out before he himself
was likely to publish his discoveries.

Newton next explains that he had also a third method, of which (he says) he
had about 1669 sent an account to Barrow and Collins, illustrated by
applications to areas, rectification, cubature, etc. This was the method of
fluxions; but Newton gives no description of it here, though he adds some
illustrations of its use. The first illustration is on the quadrature of the
curve represented by the equation

which he says can be effected as a sum of (*m* + 1)/*n*
terms if (*m* + 1)/*n* be a positive integer, and which he thinks
cannot otherwise be effected except by an infinite series. [This is not so, the
integration is possible if *p* + (*m* + 1)/*n* be an integer.]
He also gives a list of other forms which are immediately integrable, of which
the chief are

,

,

,

,

;

where *m* is a positive integer and *n* is any
number whatever. Lastly, he points out that the area of any curve can be easily
determined approximately by the method of interpolation described below in
discussing his Methodus Differentialis.

At the end of his letter Newton alludes to the solution of the ``inverse
problem of tangents,'' a subject on which Leibnitz had asked for information.
He gives formulae for reversing any series, but says that besides these
formulae he has two methods for solving such questions, which for the present
he will not describe except by an anagram which, being read, is as follows,
``Una methodus consistit in extractione fluentis quantitatis ex aequatione
simul involvente fluxionem ejus: altera tantum in assumptione seriei pro
quantitate qualibet incognita ex qua caetera commode derivari possunt, et in
collatione terminorum homologorum aequationis resultantis, as eruendos terminos
assumptae seriei.''

He implies in this letter that he is worried by the questions he is asked
and the controversies raised about every new matter which he produces, which
shew his rashness in publishing ``quod umbram captando eatenus perdideram
quietem meam, rem prorsus substantialem.''

Leibnitz, in his answer, dated June 21, 1677, explains his method of drawing
tangents to curves, which he says proceeds ``not by fluxions of lines, but by
the differences of numbers''; and he introduces his notation of *dx* and *dy*
for the infinitesimal differences between the co-ordinates of two consecutive
points on a curve. He also gives a solution of the problem to find a curve
whose subtangent is constant, which shews that he could integrate.

In 1679 Hooke, at the request of the Royal Society, wrote to Newton
expressing a hope that he would make further communications to the Society, and
informing him of various facts then recently discovered. Newton replied saying
that he had abandoned the study of philosophy, but he added that the earth's
diurnal motion might be proved by the experiment of observing the deviation
from the perpendicular of a stone dropped from a height to the ground - an
experiment which was subsequently made by the Society and succeeded. Hooke in
his letter mentioned Picard's geodetical researches; in these Picard used a
value of the radius of the earth which is substantially correct. This led
Newton to repeat, with Picard's data, his calculations of 1666 on the lunar
orbit, and he thus verified his supposition that gravity extended as far as the
moon and varied inversely as the square of the distance. He then proceeded to
consider the general theory of motion of a particle under a centripetal force,
that is, one directed to a fixed point, and showed that the vector would sweep
over equal areas in equal times. He also proved that, if a particle describe an
ellipse under a centripetal force to a focus, the law must be that of the
inverse square of the distance from the focus, and conversely, that the orbit
of a particle projected under the influence of such a force would be a conic
(or, it may be, he thought only an ellipse). Obeying his rule to publish nothing
that could land hum in a scientific controversy these results were locked up in
his notebooks, and it was only a specific question addressed to him five years
later that led to their publication.

The Universal Arithmetic, which is on algebra, theory of
equations, and miscellaneous problems, contains the substance of Newton's
lectures during the years 1673 to 1683. His manuscript of it is still extant;
Whiston extracted a somewhat reluctant permission from Newton to print it, and
it was published in 1707. Amongst several new theorems on various points in
algebra and the theory of equations Newton here enunciates the following
important results. He explains that the equation whose roots are the solution
of a given problem will have as many roots as there are different possible
cases; and he considers how it happens that the equation to which a problem
leads may contain roots which do not satisfy the original question. He extends
Descartes's rule of signs to give limits to the number of imaginary roots. He uses
the principle of continuity to explain how two real and unequal roots may
become imaginary in passing through equality, and illustrates this by
geometrical considerations; thence he shews that imaginary roots must occur in
pairs. Newton also here gives rules to find a superior limit to the positive
roots of a numerical equation, and to determine the approximate values of the
numerical roots. He further enunciates the theorem known by his name for
finding the sum of the *n*th powers of the roots of an equation, and laid
the foundation of the theory of symmetrical functions of the roots of an
equation.

The most interesting theorem contained in the work is his attempt to find a
rule (analogous to that of Descartes for real roots) by which the number of
imaginary roots of an equation can be determined. He knew that the result which
he obtained was not universally true, but he gave no proof and did not explain
what were the exceptions to the rule. His theorem is as follows. Suppose the
equation to be of the *n*th degree arranged in descending powers of *x*
(the coefficient of
being positive), and suppose the *n* + 1 fractions

to be formed and written below the corresponding terms of
the equation, then, if the square of any term when multiplied by the
corresponding fraction be greater than the product of the terms on each side of
it, put a plus sign above it: otherwise put a minus sign above it, and put a
plus sign above the first and last terms. Now consider any two consecutive
terms in the original equation, and the two symbols written above them. Then we
may have any one of the four following cases: ()
the terms of the same sign and the symbols of the same sign; ()
the terms of the same sign and the symbols of opposite signs; ()
the terms of opposite signs and the symbols of the same sign; ()
the terms of opposite signs and the symbols of opposite signs. Then it has been
shewn that the number of negative roots will not exceed the number of cases (),
and the number of positive roots will not exceed the number of cases ();
and therefore the number of imaginary roots is not less than the number of
cases ()
and ().
In other words the number of changes of signs in the row of symbols written
above the equation is an inferior limit to the number of imaginary roots.
Newton, however, asserted that ``you may almost know how many roots are
impossible'' by counting the changes of sign in the series of symbols formed as
above. That is to say, he thought that in general the actual number of
positive, negative and imaginary roots could be got by the rule and not merely
superior or inferior limits to these numbers. But though he knew that the rule
was not universal he could not find (or at any rate did not state) what were
the exceptions to it: this problem was subsequently discussed by Campbell,
Maclaurin, Euler, and other writers; at last in 1865 Sylvester succeeded in
proving the general result.

In August, 1684, Halley came to Cambridge in order to consult Newton about
the law of gravitation. Hooke, Huygens, Halley, and Wren had all conjectured
that the force of the attraction of the sun or earth on an external particle
varied inversely as the square of the distance. These writers seem
independently to have shewn that, if Kepler's conclusions were rigorously true,
as to which they were not quite certain, the law of attraction must be that of
the inverse square. Probably their argument was as follows. If *v* be the
velocity of a planet, *r* the radius of its orbit taken as a circle, and *T*
its periodic time, *v* = 2*r*/*T*.
But, if *f* be the acceleration to the centre of
the circle, we have *f* = 4²*r*/*T*²
Now, by Kepler's third law, *T*² varies
as *r*³; hence *f* varies inversely as *r*². They
could not, however, deduce from the law the orbits of the planets. Halley
explained that their investigations were stopped by their inability to solve
this problem, and asked Newton if he could find out what the orbit of a planet
would be if the law of attraction were that of the inverse square. Newton
immediately replied that it was an ellipse, and promised to send or write out
afresh the demonstration of it which he had found in 1679. This was sent in
November, 1684.

Instigated by Halley, Newton now returned to the problem of gravitation; and
before the autumn of 1684, he had worked out the substance of propositions
1--19, 21, 30, 32--35 in the first book of the Principia. These
together with notes on the laws of motion and various lemmas, were read for his
lectures in the Michaelmas Term, 1684.

In November Halley received Newton's promised communication, which probably
consisted of the substance of propositions 1, 11 and either proposition 17 or
the first corollary of proposition 13; thereupon Halley again went to
Cambridge, where he saw ``a curious treatise, De Motu, drawn up
since August.'' Most likely this contained Newton's manuscript notes of the
lectures above alluded to: these notes are now in the university library and
are headed ``*De Motu Corporum*.'' Halley begged that the results might
be published, and finally secured a promise that they should be sent to the
Royal Society: they were accordingly communicated to the Society not later than
February, 1685, in the paper De Motu, which contains the substance
of the following propositions in the Principia, book I, props. 1,
4, 6, 7, 10, 11, 15, 17, 32; book II, props. 2,3,4.

It seems also to have been due to the influence and tact of Halley at his
visit in November, 1684, that Newton undertook to attack the whole problem of
gravitation, and practically pledged himself to publish his results: these are
contained in the Principia. As yet Newton had not determined the
attraction of a spherical body on an external point, nor had he calculated the
details of the planetary motions even if the members of the solar system could
be regarded as points. The first problem was solved in 1685, probably either in
January or in February. ``No sooner,'' to quote from Dr. Glaisher's address on
the bicentenary of the publication of the Principia, ``had Newton
proved this superb theorem - and we know from his own words that he had no
expectation of so beautiful a result till it emerged from his mathematical
investigation - than all the mechanism of the universe at once lay spread
before him. When he discovered the theorems that form the first three sections
of book I, when he gave them in his lectures of 1684, he was unaware that the
sun and earth exerted their attractions as if they were but points. How
different must these propositions have seemed to Newton's eyes when he realised
that these results, which he had believed to be only approximately true when
applied to the solar system, were really exact! Hitherto they had been true
only in so far as he could regard the sun as a point compared to the distance
of the planets, or the earth as a point compared to the distance of the moon -
a distance amounting to only about sixty times the earth's radius - but now
they were mathematically true, excepting only for the slight deviation from a
perfectly spherical form of the sun, earth and planets. We can imagine the
effect of this sudden transition from approximation to exactitude in
stimulating Newton's mind to still greater efforts. It was now in his power to
apply mathematical analysis with absolute precision to the actual problems of
astronomy.''

Of the three fundamental principles applied in the Principia we
may say that the idea that every particle attracts every other other particle
in the universe was formed at least as early as 1666; the law of equable
description of areas, its consequences, and the fact that if the law of
attraction were that of the inverse square the orbit of a particle about a
centre of force would be a conic were proved in 1679; and, lastly, the
discovery that a sphere, whose density at any point depends only on the
distance from the centre, attracts an external point as if the whole mass were
collected at its centre was made in 1685. It was this last discovery that
enabled him to apply the first two principles to the phenomena of bodies of finite
size.

The draft of the first book of the Principia was finished
before the summer of 1685, but the corrections and additions took some time,
and the book was not presented to the Royal Society until April 28, 1686. This
book is given up to the consideration of the motion of particles or bodies in
free space either in known orbits, or under the action of known forces, or
under their mutual attraction; and in particular to indicating how the effects
of disturbing forces may be calculated. In it also Newton generalizes the law
of attraction into a statement that every particle of matter in the universe
attracts every other particle with a force which varies directly as the product
of their masses, and inversely as the square of the distance between them; and
he thence deduces the law of attraction for spherical shells of constant
density. The book is prefaced by an introduction on the science of dynamics,
which defines the limits of mathematical investigation. His object, he says, is
to apply mathematics to the phenomena of nature; among these phenomena motion
is one of the most important; now motion is the effect of force, and, though he
does not know what is the nature or origin of force, still many of its effects
can be measured; and it is these that form the subject-matter of the work.

The second book of the Principia was completed by the summer of
1686. This book treats of motion in a resisting medium, and of hydrostatics and
hydrodynamics, with special applications to waves, tides, and acoustics. He concludes
it by shewing that the Cartesian theory of vortices was inconsistent both with
the known facts and with the laws of motion.

The next nine or ten months were devoted to the third book. Probably for
this originally he had no materials ready. He commences by discussing when and
how far it is justifiable to construct hypotheses or theories to account for
known phenomena. He proceeds to apply the theorems obtained in the first book
to the chief phenomena of the solar system, and to determine the masses and
distances of the planets and (whenever sufficient data existed) of their
satellites. In particular the motion of the moon, the various inequalities
therein, and the theory of the tides are worked out in detail. He also
investigates the theory of comets, shews that they belong to the solar system,
explains how from three observations the orbit can be determined, and
illustrates his results by considering certain special comets. The third book
as we have it is but little more than a sketch of what Newton had finally
proposed to himself to accomplish; his original scheme is among the
``Portsmouth papers,'' and his notes shew that he continued to work at it for
some years after the publication of the first edition of the Principia:
the most interesting of his memoranda are those in which by means of fluxions
he has carried his results beyond the point at which he was able to translate
them into geometry.

The demonstrations throughout the book are geometrical, but to readers of
ordinary ability are rendered unnecessarily difficult by the absence of
illustrations and explanations, and by the fact that no clue is given to the
method by which Newton arrived at his results. The reason why it was presented
in a geometrical form appears to have been that the infinitesimal calculus was
then unknown, and, had Newton used it to demonstrate results which were in
themselves opposed to the prevalent philosophy of the time, the controversy as
to the truth of his results would have been hampered by a dispute concerning
the validity of the methods used in proving them. He therefore cast the whole
reasoning into a geometrical shape which, if somewhat longer, can at any rate
be made intelligible to all mathematical students. So closely did he follow the
lines of Greek geometry that he constantly used graphical methods, and
represented forces, velocities, and other magnitudes in the Euclidean way by
straight lines (*ex. gr.* book I, lemma 10), and not by a certain number
of units. The latter and modern method had been introduced by Wallis, and must
have been familiar to Newton. The effect of his confining himself rigorously to
classical geometry is that the Principia is written in a lnaguage
which is archaic, even if not unfamiliar.

The adoption of geometrical methods in the Principia for
purposes of demonstration does not indicate a preference on Newton's part for
geometry over analysis as an instrument of research, for it is known now that
Newton used the fluxional calculus in the first instance in finding some of the
theorems, especially those towards the end of book I and in book II; and in
fact one of the most important uses of that calculus is stated in book II,
lemma 2. But it is only just to remark that, at the time of its publication and
for nearly a century afterwards, the differential and fluxional calculus were
not fully developed, and did not possess the same superiority over the method
he adopted which they do now; and it is a matter for astonishment that when
Newton did employ the calculus he was able to use it to so good an effect.

The printing of the work was slow, and it was not finally published till the
summer of 1687. The cost was borne by Halley, who also corrected the proofs,
and even put his own researches on one side to press the printing forward. The
conciseness, absence of illustrations, and synthetical character of the book
restricted the numbers of those who were able to appreciate its value; and
though nearly all competent critics admitted the validity of the conclusions,
some little time elapsed before it affected the current beliefs of educated
men. I should be inclined to say (but on this point opinions differ widely)
that within ten years of its publication it was generally accepted in Britain
as giving a correct account of the laws of the universe; it was similarly
accepted within about twenty years on the continent, except in France, where
the Cartesian hypothesis held its ground until Voltaire in 1738 took up the
advocacy of the Newtonian theory.

The manuscript of the Principia was finished by 1686. Newton
devoted the remainder of that year to his paper on physical optics, the greater
part of which is given up to the subject of diffraction.

In 1687 James II, having tried to force the university to admit as a master
of arts a Roman Catholic priest who refused to take the oaths of supremacy and
allegiance, Newton took a prominent part in resisting the illegal interference
of the king, and was one of the deputation sent to London to protect the rights
of the university. The active part taken by Newton in this affair led to his
being in 1689 elected member for the university. This parliament only lasted
thirteen months, and on its dissolution he gave up his seat. He was
subsequently returned in 1701, but he never took any prominent part in
politics.

On his coming back to Cambridge in 1690 he resumed his mathematical studies
and correspondence, but probably did not lecture. The two letters to Wallis, in
which he explained his method of fluxions and fluents, were written in 1692 and
published in 1693. Towards the close of 1692 and throughout the following two
years, Newton had a long illness, suffering from insomnia and general nervous
irritability. Perhaps he never quite regained his elasticity of mind, and,
though after his recovery he shewed the same power in solving any question
propounded to him, he ceased thenceforward to do original work on his own
initiative, and it was somewhat difficult to stir him to activity in new
subjects.

In 1694 Newton began to collect data connected with the irregularities of the
moon's motion with the view of revising the part of the Principia
which dealt with that subject. To render the observations more accurate, he
forwarded to Flamsteed a table of corrections for refraction which he had
previously made. This was not published till 1721, when Halley communicated it
to the Royal Society. The original calculations of Newton and the papers
connected with them are in the Portsmouth collection, and shew that Newton
obtained it by finding the path of a ray, by means of quadratures, in a manner
equivalent to the solution of a differential equation. As an illustration of
Newton's genius, I may mention that even as late as 1754 Euler failed to solve
the same problem. In 1782 Laplace gave a rule for constructing such a table,
and his results agree substantially with those of Newton.

I do not suppose that Newton would in any case have produced much more
original work after his illness; but his appointment in 1696 as warden, and his
promotion in 1699 to the mastership of the Mint, at a salary of £ 1500 a year,
brought his scientific investigations to an end, though it was only after this
that many of his previous investigations were published in the form of books.
In 1696 he moved to London, in 1701 he resigned the Lucasian chair, and in 1703
he was elected president of the Royal Society.

In 1704 Newton published his Optics, which contains the results
of the papers already mentioned. To the first edition of this book were
appended two minor works which have no special connection with optics; one
being on cubic curves, the other on the quadrature of curves and on fluxions.
Both of them were manuscripts with which his friends and pupils were familiar,
but they were here published *urbi et orbi* for the first time.

The first of these appendices is entitled Enumeratio Linearum Tertii
Ordinis; the object seems to be to illustrate the use of analytical
geometry, and as the application to conics was well known, Newton selected the
theory of cubics.

He begins with some general theorems, and classifies curves according as
their equations are algebraical or transcendental; the former being cut by a
straight line in a number of points (real or imaginary) equal to the degree of
the curve, the latter being cut by a straight line in an infinite number of
points. Newton then shews that many of the most important properties of conics
have their analogues in the theory of cubics, and he discusses the theory of
asymptotes and curvilinear diameters.

After these general theorems, he commences his detailed examination of
cubics by pointing out that a cubic must have at least one real point at
infinity. If the asymptote or tangent at this point be a finite distance, it
may be taken for the axis of *y*. This asymptote will cut the curve in
three points altogether, of which at least two are at infinity. If the third
point be at a finite distance, then (by one of his general theorems on
asymptotes) the equation can be written in the form

where the axes of *x* and *y* are the asymptotes
of the hyperbola which is the locus of the middle points of all chords drawn
parallel to the axis of *y*; while, if the third point in which this
asymptote cuts the curve be also at infinity, the equation can be written in
the form

Next he takes the case where the tangent at the real point
at infinity is not at a finite distance. A line parallel to that direction in
which the curve goes to infinity may be taken as the axis of *y*. Any such
line will cut the curve in three points altogether, of which one is by
hypothesis at infinity, and one is necessarily at a finite distance. He then
shews that if the remaining point in which this line cuts the curve be at a
finite distance, the equation can be written in the form

while if it be at an infinite distance, the equation can be
written in the form

Any cubic is therefore reducible to one of four
characteristic forms. Each of these forms is then discussed in detail, and the
possibility of the existence of double points, isolated ovals, etc., is worked
out. The final result is that in all there are seventy-eight possible forms
which a cubic may take. Of these Newton enumerated one seventy-two; four of the
remainder were mentioned by Stirling in 1717, one by Nicole in 1731, and one by
Nicholas Bernoulli about the same time.

In the course of the work Newton states the remarkable theorem that, just as
the shadow of a circle (cast by a luminous point on a plane) gives rise to all
the conics, so the shadows of the curves represented by the equation

give rise to all the cubics. This remained an unsolved
puzzle until 1731, when Nicole and Clairaut gave demonstrations of it; a better
proof is that given by Murdoch in 1740, which depends on the classification of
these curves into five species according as to whether their points of
intersection with the axis of *x* are real and unequal, real and two of
them are equal (two cases), real and all equal, or two imaginary and one real.

In this tract Newton also discusses double points in the plane and at
infinity, the description of curves satisfying given conditions, and the
graphical solution of problems by the use of curves.

The second appendix to the Optics is entitled De Quadratura
Curvarum. Most of it had been communicated to Barrow in 1668 or 1669,
and probably was familiar to Newton's pupils and friends from that time
onwards. It consists of two parts.

The bulk of the first part is a statement of Newton's method of effecting
the quadrature and rectification of curves by means of infinite series; it is
noticeable as containing the earliest use in print of literal indices, and also
the first printed statement of the binomial theorem, but these novelties are
introduced only incidentally. The main object is to give rules for developing a
function of *x* in a series in ascending powers of *x*, so as to
enable mathematicians to effect the quadrature of any curve in which the
ordinate *y* can be expressed as an explicit algebraical function of the
abscissa *x*. Wallis had shewn how this quadrature could be found when *y*
was given as a sum of a number of multiples of powers of *x*, and Newton's
rules of expansion here established rendered possible the similar quadrature of
any curve whose ordinate can be expressed as the sum of an infinite number of
such terms. In this way he effects the quadrature of the curves

,

,

,

,

but naturally the results are expressed as infinite series.
He then proceeds to curves whose ordinate is given as an implicit function of
the abscissa; and he gives a method by which *y* can be expressed as an
infinite series in ascending powers of *x*, but the application of the
rule to any curve demands in general such complicated numerical calculations as
to render it of little value. He concludes this part by shewing that the rectification
of a curve can be effected in a somewhat similar way. His process is equivalent
to finding the integral with regard to *x* of
in the form of an infinite series. I should add that Newton
indicates the importance of determining whether the series are convergent - an
observation far in advance of his time - but he knew of no general test for the
purpose; and in fact it was not until Gauss and Cauchy took up the question
that the necessity of such limitations was commonly recognized.

The part of the appendix which I have just described is practically the same
as Newton's manuscript De Analysi per Equationes Numero Terminorum
Infinitas, which wa subsequently printed in 1711. It is said that this
was originally intended to form an appendix to Kinckhuysen's Algebra,
which, as I have already said, he at one time intended to edit. The substance
of it was communicated to Barrow, and by him to Collins, in letters of July 31
and August 12, 1669; and a summary of it was included in the letter of October
24, 1676, sent to Leibnitz.

It should be read in connection with Newton's Methodus Differentialis,
also published in 1711. Some additional theorems are there given, and he
discusses his method of interpolation, which had been briefly described in the
letter of October 24, 1676. The principle is this. If *y* = be
a function of *x*, and if, when *x* is successively put equal to ,
,...,
the values of *y* be known and be ,
,...,
then a parabola whose equation is *y* = *p* + *qx* + *rx*²
+ ... can be drawn through the points ,
,...,
and the ordinate of this parabola may be taken as an approximation to the
ordinate of the curve. The degree of the parabola will of course be one less
than the number of given points. Newton points out that in this way the areas
of any curves can be approximately determined.

The second part of this appendix to the Optics contains a
description of Newton's method of fluxions. This is best considered in
connection with Newton's manuscript on the same subject which was published by
John Colson in 1736, and of which it is a summary.

The invention of the infinitesimal calculus was one of the great
intellectual achievements of the seventeenth century. This method of analysis,
expressed in the notation of fluxions and fluents, was used by Newton in or
before 1666, but no account of it was published until 1693, though its general
outline was known to his friends and pupils long anterior to that year, and no
complete exposition of his methods was given before 1736.

The idea of a fluxion or differential coefficient, as treated at this time,
is simple. When two quantities - e.g. the radius of a sphere and its volume -
are so related that a change in one causes a change in the other, the one is
said to be a function of the other. The ratio of the rates at which they change
is termed the differential coefficient or fluxion of the one with regard to the
other, and the process by which this ratio is determined is known as
differentiation. Knowing the differential coefficient and one set of
corresponding values of the two quantities, it is possible by summation to
determine the relation between them, as Cavalieri and others had shewn; but
often the process is difficult, if, however, we can reverse the process of
differentiation we can obtain this result directly. This process of reversal is
termed integration. It was at once seen that problems connected with the
quadrature of curves, and the determination of volumes (which were soluble by
summation, as had been shewn by the employment of indivisibles), were reducible
to integration. In mechanics also, by integration, velocities could be deduced from
known accelerations, and distances traversed from known velocities. In short,
wherever things change according to known laws, here was a possible method of
finding the relation between them. It is true that, when we try to express
observed phenomena in the language of the calculus, we usually obtain an
equation involving the variables, and their differential coefficients - and
possibly the solution may be beyond our powers. Even so, the method is often
fruitful, and its use marked a real advance in thought and power.

I proceed to describe somewhat fully Newton's methods as described by
Colson. Newton assumed that all geometrical magnitudes might be conceived as
generated by continuous motion; thus a line may be considered as generated by
the motion of a point, a surface by that of a line, a solid by that of a
surface, a plane angle by the rotation of a line, and so on. The quantity thus
generated was defined by him as the fluent or flowing quantity. The velocity of
the moving magnitude was defined as the fluxion of the fluent. This seems to be
the earliest definite recognition of the idea of a continuous function, though
it had been foreshadowed in some of Napier's papers.

Newton's treatment of the subject is as follows. There are two kinds of
problems. The object of the first is to find the fluxion of a given quantity,
or more generally ``the relation of the fluents being given, to find the
relation of their fluxions.'' This is equivalent to differentiation. The object
of the second or inverse method of fluxions is from the fluxion or some
relations involving it to determine the fluent, or more generally ``an equation
being proposed exhibiting the relation of the fluxions of quantities, to find
the relations of those quantities, or fluents, to one another.'' This is
equivalent either to integration which Newton termed the method of quadrature,
or to the solution of a differential equation which was called by Newton the
inverse method of tangents. The methods for solving these problems are
discussed at considerable length.

Newton then went on to apply these results to questions connected with the
maxima and minima of quantities, the method of drawing tangents to curves, and
the curvature of curves (namely, the determination of the centre of curvature,
the radius of curvature, and the rate at which the radius of curvature
increases). He next considered the quadrature of curves and the rectification
of curves. In finding the maximum and minimum of functions of one variable we
regard the change of sign of the difference between two consecutive values of
the function as the true criterion; but his argument is that when a quantity
increasing has attained its maximum it can have no further increment, or when
decreasing it has attained its minimum it can have no further decrement;
consequently the fluxion must be equal to nothing.

It has been remarked that neither Newton nor Leibnitz produced a calculus,
that is, a classified collection of rules; and that the problems they discussed
were treated from first principles. That, no doubt, is the usual sequence in
the history of such discoveries, though the fact is frequently forgotten by
subsequent writers. In this case I think the statement, so far as Newton's
treatment of the differential or fluxional part of the calculus is concerned,
is incorrect, as the foregoing account sufficiently shews.

If a flowing quantity or fluent were represented by *x*, Newton denoted
its fluxion by ,
the fluxion of
or second fluxion of *x* by ,
and so on. Similarly the fluent of *x* was denoted by , or
sometimes by *x'* or [*x*]. The infinitely small part by which a
fluent such as *x* increased in a small interval of time measured by *o*
was called the moment of the fluent; and its value was shewn to be *o*.
Newton adds the important remark that thus we may in any problem neglect the
terms multiplied by the second and higher powers of *o*, and we may always
find an equation between the co-ordinates *x*, *y* of a point on a
curve and their fluxions ,
.
It is an application of this principle which constitutes one of the chief
values of the calculus; for if we desire to find the effect produced by several
causes on a system, then, if we can find the effect produced by each cause when
acting alone in a very small time, the total effect produced in that time will
be equal to the sum of the separate effects. I should here note the fact that
Vince and other English writers in the eighteenth century used
to denote the increment of *x* and not the velocity with
which it increased; that is
in their writings stands for what Newton would have expressed by *o*
and what Leibnitz would have written as *dx*.

I need not discuss in detail the manner in which Newton treated the problems
above mentioned. I will only add that, in spite of the form of his definition,
the introduction into geometry of the idea of time was evaded by supposing that
some quantity *ex. gr.* the abscissa of a point on a curve) increased
equably; and the required results then depend on the rate at which other
quantities (*ex. gr.* the ordinate or radius of curvature) increase
relatively to the one so chosen. The fluent so chosen is what we now call the
independent variable; its fluxion was termed the ``principal fluxion''; and, of
course, if it were denoted by *x*, then
was constant, and consequently
= 0.

There is no question that Newton used a method of fluxions in 1666, and it
is practically certain that accounts of it were communicated in manuscript to
friends and pupils from and after 1669. The manuscript, from which most of the
above summary has been taken, is believed to have been written between 1671 and
1677, and to have been in circulation at Cambridge from that time onwards,
though it is probable that parts were rewritten from time to time. It was
unfortunate that it was not published at once. Strangers at a distance
naturally judged of the method by the letter to Wallis in 1692, or by the Tractatus
de Quadrature Curvarum, and were not aware that it had been so
completely developed at an earlier date. This was the cause of numerous
misunderstandings. At the same time it must be added that all mathematical
analysis was leading up to the ideas and methods of the infinitesimal calculus.
Foreshadowings of the principles and even of the language of that calculus can
be found in the writings of Napier, Kepler, Cavalieri, Pascal, Fermat, Wallis,
and Barrow. It was Newton's good luck to come at a time when everything was ripe
for the discovery, and his ability enabled him to construct almost at once a
complete calculus.

The infinitesimal calculus can also be expressed in the notation of the
differential calculus: a notation which was invented by Leibnitz probably in
1675, certainly by 1677, and was published in 1684, some nine years before the
earliest printed account of Newton's method of fluxions. But the question
whether the general idea of the calculus expressed in that notation was
obtained by Leibnitz from Newton, or whether it was discovered independently,
gave rise to a long and bitter controversy. The leading facts are given in the
next chapter.

The remaining events of Newton's life require little or no comment. In 1705
he was knighted. From this time onwards he devoted much of his leisure to
theology, and wrote at great length on prophecies and predictions, subjects
which had always been of interest to him. His Universal Arithmetic
was published by Whiston in 1707, and his Analysis by Infinite Series
in 1711; but Newton had nothing to do with the preparation of either of these
for the press. His evidence before the House of Commons in 1714 on the
determination of longitude at sea marks an important epoch in the history of
navigation.

The dispute with Leibnitz as to whether he had derived the ideas of the
differential calculus from Newton or invented it independently originated about
1708, and occupied much of Newton's time, especially between the years 1709 and
1716.

In 1709 Newton was persuaded to allow Cotes to prepare the long-talked-of
second edition of the Principia; it was issued in March 1713. A
third edition was published in 1726 under the direction of Henry Pemberton. In
1725 Newton's health began to fail. He died on March 20, 1727, and eight days
later was buried in Westminster Abbey.

His chief works, taking them in their order of publication, are the Principia,
published in 1687; the Optics (with appendices on *cubic curves*,
the *quadrature and rectification of curves by the use of infinite series*,
and the *method of fluxions*), published in 1704; the Universal
Arithmetic, published in 1707; the Analysis per Series, Fluxiones,
etc., and the Methodus Differentialis, published in 1711; the Lectiones
Opticae, published in 1729; the Method of Fluxions, etc.
(that is Newton's manuscript on fluxions), translated by J. Colson
and published in 1736; and the Geometrica Analytica, printed in
1779 in the first volume of Horsley's edition of Newton's works.

In appearance Newton was short, and towards the close of his life rather
stout, but well set, with a square lower jaw, brown eyes, a broad forehead, and
rather sharp features. His hair turned grey before he was thirty, and remained
thick and white as silver till his death.

As to his manners, he dressed slovenly, was rather languid, and was often so
absorbed in his own thoughts as to be anything but a lively companion. Many
anecdotes of his extreme absence of mind when engaged in any investigation have
been preserved. Thus once when riding home from Grantham he dismounted to lead
his horse up a steep hill; when he turned at the to remount, he found that he
had the bridle in his hand, while his horse had slipped it and gone away.
Again, on the few occasions when he sacrificed his time to entertain his
friends, if he left them to get more wine or for any similar reason, he would
as often as not be found after the lapse of some time working out a problem,
oblivious alike of his expectant guests and of his errand. He took no exercise,
indulged in no amusements, and worked incessantly, often spending eighteen or
nineteen hours out of the twenty-four in writing.

In character he was religious and conscientious, with an exceptionally high
standard of morality, having, as Bishop Burnet said, ``the whitest soul'' he
ever knew. Newton was always perfectly straightforward and honest; but in his
controversies with Leibnitz, Hooke and others, though scrupulously just, he was
not generous; and it would seem that he frequently took offence at a chance
expression when none was intended. He modestly attributed his discoveries
largely to the admirable work done by his predecessors; and once explained
that, if he had seen further than other men, it was only because he had stood
on the shoulders of giants. He summed up his own estimate of his work in the
sentence, ``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,
whilst the great ocean of truth lay all undiscovered before me.'' He was
morbidly sensitive to being involved in any discussions. I believe that, with
the exception of his papers on optics, every one of his works was published
only under pressure from his friends and against his own wishes. There are
several instances of his communicating papers and results on condition that his
name should not be published: thus when in 1669 he had, at Collins's request,
solved some problems on harmonic series and on annuities which had previously
baffled investigation, he only gave permission that his results should be
published ``so it be,'' as he says, ``without my name to it; for I see not what
there is desirable in public esteem, were I able to acquire and maintain it: it
would perhaps increase my acquaintance, the things which I chiefly study to
decline.''

Perhaps the most wonderful single illustration of his powers was the
composition in seven months of the first book of the Principia,
and the expression of the numerous and complex results in classical geometrical
form. As other illustrations of his ability I may mention his solutions of the
problems of Pappus, of John Bernoulli's challenge, and of the question of
orthogonal trajectories. The problem of Pappus, here alluded to, is to find the
locus of a point such the rectangle under its distances from two given straight
lines shall be in a given ratio to the rectangle under its distances from two
other given straight lines. Many geometricians from the time of Apollonius had
tried to find a geometrical solution and had failed, but what had proved
insuperable to his predecessors seems to have presented little difficulty to
Newton who gave an elegant demonstration that the locus was a conic. Geometry,
said Lagrange when recommending the study of analysis to his pupils, is a
strong bow, but it is one which only a Newton can fully utilize. As another
example I may mention that in 1696 John Bernoulli challenged mathematicians (i)
to determine the brachistochrone, and (ii) to find a curve such that if any
line drawn from a fixed point *O* cut it in *P* and *Q* then
would be constant. Leibnitz solved the first of these questions
after an interval of rather more than six months, and then suggested that they
be sent as a challenge to Newton and others. Newton received the problems on
Jan. 29, 1697, and the next day gave the complete solutions to both, at the
same time generalising the second question. An almost exactly similar case
occurred in 1716 when Newton was asked to find the orthogonal trajectory of a
family of curves. In five hours Newton solved the problem in the form in which
it was propounded to him, and laid down the principles for finding
trajectories.

It is almost impossible to describe the effect of Newton's writings without
being suspected of exaggeration. But, if the state of mathematical knowledge in
1669 or at the death of Pascal or Fermat be compared with what was known in
1700 it will be seen how immense was the advance. In fact we may say that it
took mathematicians half a century or more before they were able to assimilate
the work produced in those years.

In pure geometry Newton did not establish any new methods, but no modern
writer has shewn the same power in using those of classical geometry. In
algebra and the theory of equations he introduced the system of literal
indices, established the binomial theorem, and created no inconsiderable part
of the theory of equations: one rule which he enunciated in this subject
remained till a few years ago an unsolved riddle which had overtaxed the
resources of succeeding mathematicians. In analytical geometry, he introduced
the modern classification of curves into algebraical and transcendental; and
established many of the fundamental properties of asymptotes, multiple points,
and isolated loops, illustrated by a discussion of cubic curves. The fluxional
or infinitesimal calculus was invented by Newton in or before the year 1666,
and circulated in manuscript amongst his friends in and after the year 1669,
though no account of the method was printed till 1693. The fact that the
results are nowadays expressed in a different notation has led to Newton's
investigations on this subject being somewhat overlooked.

Newton, further, was the first to place dynamics on a satisfactory basis,
and from dynamics he deduced the theory of statics: this was in the
introduction to the Principia published in 1687. The theory of
attractions, the application of the principles of mechanics to the solar
system, the creation of physical astronomy, and the establishment of the law of
universal gravitation are due to him, and were first published in the same
work, but of the nature of gravity he confessed his ignorance, though he found
inconceivable the idea of action at a distance. The particular questions
connected with the motion of the earth and moon were worked out as fully as was
then possible. The theory of hydrodynamics was created in the second book of
the Principia, and he added considerably to the theory of
hydrostatics which may be said to have been first discussed in modern times by
Pascal. The theory of the propagation of waves, and in particular the
application to determine the velocity of sound, is due to Newton and was
published in 1687. In geometrical optics, he explained amongst other things the
decomposition of light and the theory of the rainbow; he invented the
reflecting telescope known by his name, and the sextant. In physical optics, he
suggested and elaborated the emission theory of light.

The above list does not exhaust the subjects he investigated, but it will
serve to illustrate how marked was his influence on the history of mathematics.
On his writings and on their effects, it will be enough to quote the remarks of
two or three of those who were subsequently concerned with the subject-matter
of the Principia. Lagrange described the Principia as
the greatest production of the human mind, and said he felt dazed at such an
illustration of what man's intellect might be capable. In describing the effect
of his own writings and those of Laplace it was a favourite remark of his that
Newton was not only the greatest genius that had ever existed, but he was also
the most fortunate, for as there is but one universe, it can happen but to one
man in the world's history to be the interpreter of its laws. Laplace, who is
in general very sparing of his praise, makes of Newton the one exception, and
the words in which he enumerates the causes which ``will always assure to the Principia
a pre-eminence above all the other productions of human genius'' have been
often quoted. Not less remarkable is the homage rendered by Gauss; for other
great mathematicians or philosophers he used the epithets magnus, or clarus, or
clarissimus: for Newton alone he kept the prefix summus. Finally Biot, who had
made a special study of Newton's works, sums up his remarks by saying, ``comme
géomètre et comme expérimentateur Newton est sans égal; par la réunion de ces
deux genres de génies à leur plus haut degré, il est sans exemple.''

This page is included in a collection of
mathematical biographies taken from A Short Account of the History of
Mathematics by W. W. Rouse Ball (4th Edition, 1908).

Transcribed by

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