Speaker 1: Chapter sixteen, Part one of a short account of the
Speaker 1: history of mathematics. This is a LibriVox recording. All LibriVox
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Speaker 1: mit dot edu forward slash p kJ. A short account
Speaker 1: of the history of mathematics by W. W. Rowseball. Chapter sixteen,
Speaker 1: The Life and Works of Newton. The mathematicians considered in
Speaker 1: the last chapter commenced the creation of those processes which
Speaker 1: distinguish modern mathematics. The extraordinary abilities of Newton enabled him,
Speaker 1: within a few years to perfect the more elementary of
Speaker 1: these processes, and to distinctly advance every branch of mathematical
Speaker 1: science then studied, as well as to create some new subjects.
Speaker 1: Newton was the contemporary and friend of Wallace, Huygens, and
Speaker 1: others of those mentioned in the last chapter. But though
Speaker 1: most of his mathematical work was done in between the
Speaker 1: years sixteen sixty five and sixteen eighty six, the Bulkovit
Speaker 1: was not printed at any rate in book form till
Speaker 1: some years later. I propose to discuss the works of
Speaker 1: Newton somewhat more fully than those of other mathematicians, partly
Speaker 1: because of the intrinsic importance of his discoveries, and partly
Speaker 1: because this book is mainly intended for English readers, and
Speaker 1: the development of mathematics in Great Britain was for a
Speaker 1: century entirely in the hands of the Newtonian school. Isaac
Speaker 1: Newton was born in Lincolnshire, near Grantham, on December twenty fifth,
Speaker 1: sixteen forty two, and died at Kensington, London, on March twentieth,
Speaker 1: seventeen twenty seven. He was educated at Trinity CID College, Cambridge,
Speaker 1: and lived there from sixteen sixty one till sixteen ninety six,
Speaker 1: during which time he produced the bulk of his work
Speaker 1: in mathematics. In sixteen ninety six he was appointed to
Speaker 1: a valuable government office and moved to London, where he
Speaker 1: resided till his death. His father, who had died shortly
Speaker 1: before Newton was born, was a Yeoman farmer, and it
Speaker 1: was intended that Newton should carry on the paternal farm.
Speaker 1: He was sent to school at Grantham, where his learning
Speaker 1: and mechanical proficiency excited some attention, and as one instance
Speaker 1: of his ingenuity. I may mention that he constructed a
Speaker 1: clock worked by water, which kept very fair time. In
Speaker 1: sixteen fifty six he returned home to learn the business
Speaker 1: of a farmer under the guidance of an old family servant. Newton, however,
Speaker 1: spent most of his time studying problems, making experiments, or
Speaker 1: devising mechanical models. His mother, noticing this, sensibly resolved to
Speaker 1: find some more congenial occupation for him, and his uncle,
Speaker 1: having been himself educated at Trinity College, Cambridge, recommended that
Speaker 1: he should be sent there. In sixteen sixty one, Newton
Speaker 1: accordingly entered as a subsidiar at Trinity College, where for
Speaker 1: the first time he found himself among surroundings which were
Speaker 1: likely to develop his powers. He seemed somewhat to have
Speaker 1: but little interest for the general society or for any
Speaker 1: pursuit save science and mathematics, and he complained to his
Speaker 1: friends that he found the other undergraduates disorderly. Luckily, he
Speaker 1: kept a diary, and we can thus form a fair
Speaker 1: idea of the course of education of the most advanced
Speaker 1: students at at English university at that time. He had
Speaker 1: not read any mathematics before coming into residence, but was
Speaker 1: acquainted with Sanderson's logic, which was then frequently read as
Speaker 1: preliminary to mathematics. At the beginning of his first October term,
Speaker 1: he happened to stroll down to Southridge Fair and there
Speaker 1: picked up a book on astrology, but could not understand
Speaker 1: it on account of the geometry and trigonometry. He therefore
Speaker 1: bought a Euclid and was surprised to find how obvious
Speaker 1: the preposition seemed. He thereupon read Aughtred's Clavis and Descartes's geometry,
Speaker 1: and the latter of which he managed to master by himself,
Speaker 1: though it's some difficulty. The interest he felt in the
Speaker 1: subject led him to take up mathematics rather than chemistry
Speaker 1: as serious study. His subsequent mathematical reading as an undergraduate
Speaker 1: was found on Kepler's Optics, the works of Vieta van
Speaker 1: Schutten's Miscellanies and Dascartes Geometre three, and Wallace's Arithmetica Infinitorum.
Speaker 1: He also attended Barrow's lectures at a later time. On
Speaker 1: reading Euclid more carefully, he formed a high opinion of
Speaker 1: it as an instrument of education, and he used to
Speaker 1: express his regret that he had not applied himself to
Speaker 1: geometry before proceeding to algebraic analysis. There is a manuscript
Speaker 1: of his dated May twenty eighth, sixteen sixty five, written
Speaker 1: in the same year as that which he took his
Speaker 1: BA degree, which is the earliest documentary proof of his
Speaker 1: invention of fluxions. It was about the same time that
Speaker 1: he discovered the binomial theorem. On account of the plague,
Speaker 1: the college was shut down in the summer of sixteen
Speaker 1: sixty five, and for a large part of the next
Speaker 1: year and a half Newton lived at home. This period
Speaker 1: was crowded with brilliant discoveries. He thought out the fundamental
Speaker 1: principles of his theory of gret gravitation, namely that every
Speaker 1: particle of matter attracts every other particle, and he suspected
Speaker 1: that the attraction varied as the product of their masses,
Speaker 1: and inversely as the square of the distance between them.
Speaker 1: He also worked out the fluctional calculus tolerably completely. Thus,
Speaker 1: in a manuscript dated November thirteenth, sixteen sixty five, he
Speaker 1: used fluxions to find the tangent and the radius of
Speaker 1: curvature at any point on a curve, and in October
Speaker 1: sixteen sixty six, he applied them to several problems in
Speaker 1: the theory of equations. Newton communicated these results to his
Speaker 1: friends and pupils from and after sixteen sixty nine, but
Speaker 1: they were not published in print till many years later.
Speaker 1: It was also while staying at home at this time
Speaker 1: that he devised some instruments for grinding lenses to particular
Speaker 1: forms other than spherical, and perhaps he decomposed solar light
Speaker 1: into different colors, leaving out details and taking round numbers only.
Speaker 1: His reasoning at this time on the theory of gravitation
Speaker 1: seems to have been as follows. He suspected that the
Speaker 1: force which retained the Moon in its orbit about the
Speaker 1: Earth was the same as terrestrial gravity, and to verify
Speaker 1: this hypothesis he proceeded thus. He knew that if a
Speaker 1: stone were allowed to fall near the surface of the Earth,
Speaker 1: the attraction of the Earth, that is, the weight of
Speaker 1: the stone, caused it to move through sixteen feet in
Speaker 1: one second. The Moon's orbit relative to the Earth is
Speaker 1: nearly a circle, and as a rough approximation, taking it
Speaker 1: to be so, he knew that the distance of the Moon,
Speaker 1: and therefore the length of its path. He also knew
Speaker 1: that time the Moon took to go once round it,
Speaker 1: namely a month. Hence he could easily find its velocity
Speaker 1: at any point such as m. He could therefore find
Speaker 1: the distance Mt through which he would move in the
Speaker 1: next second if it were not pulled by the Earth's attraction.
Speaker 1: At the end of that second, it was, however, at
Speaker 1: M prime, and therefore the Earth must have pulled it
Speaker 1: through the distance TM prime in one second, assuming the
Speaker 1: direction of the Earth pull to be constant. Now, he
Speaker 1: and several physicists of the time had conjectured from Kepler's
Speaker 1: third law that the attraction of the Earth on a
Speaker 1: body would be found to decrease as the body was
Speaker 1: removed further away from the Earth, in a proportion inversely
Speaker 1: as the square of the distance from the center of
Speaker 1: the Earth. If this were the actual law, and gravity
Speaker 1: were the sole force which retained the Moon in its orbit,
Speaker 1: then t M prime should be to sixteen feet in
Speaker 1: a proportion which inversely is the square of the distance
Speaker 1: of the Moon from the center of the Earth to
Speaker 1: the square the radius of the Earth. In sixteen seventy nine,
Speaker 1: when he repeated the investigation, t M prime was found
Speaker 1: to have the value which was required by the hypothesis,
Speaker 1: and the verification was complete. But in sixteen sixty six
Speaker 1: his estimate of the distance of the Moon was inaccurate,
Speaker 1: and when he made the calculation he found that t
Speaker 1: m prime was about one eighth less than it ought
Speaker 1: to have been on his hypothesis. This discrepancy does not
Speaker 1: seem to have shaken his faith in the belief that
Speaker 1: gravity extended to the Moon and varied inversely as the
Speaker 1: square of the distance. But from Wiston's notes of a
Speaker 1: conversation with Newton, it would seem that Newton inferred that
Speaker 1: some other force, probably Descartes's vortices, acted on the Moon
Speaker 1: as well as gravity. This statement is confirmed by Pemberton's
Speaker 1: account of the investigation. It seems, moreover that Newton all
Speaker 1: already believed firmly in the principle of universal gravitation, that is,
Speaker 1: every particle of matter attracts every other particle, and suspected
Speaker 1: that the attraction varied as the product of their masses
Speaker 1: and inversely as the square of the distance between them.
Speaker 1: But it is certain that he did not then know
Speaker 1: what the attraction of a spherical mass on any external
Speaker 1: point would be, and he did not think it likely
Speaker 1: that a particle would be attracted by the Earth, as
Speaker 1: if the latter were concentrated into a single particle at
Speaker 1: its center. On his return to Cambridge in sixteen sixty seven,
Speaker 1: Newton was elected to a fellowship at his college and
Speaker 1: permanently took up his residence there. In the early part
Speaker 1: of sixteen sixty nine, or perhaps in sixteen sixty eight,
Speaker 1: he revised Barrow's lectures for him. The end of Lecture
Speaker 1: fourteen is known to have been written by Newton, but
Speaker 1: how much of the rest is due to his suggestions
Speaker 1: cannot now be determined. As soon as this was finished,
Speaker 1: he was asked by Barrow and Collins to edit and
Speaker 1: add notes to a translation of Kinghusen's Algebra, which he
Speaker 1: consented to do, but on condition that his name should
Speaker 1: not appear in the matter. In sixteen seventy he also
Speaker 1: began a systematic exposition of his analysis by infinite series,
Speaker 1: the object of which was to express the ordinate of
Speaker 1: a curve in an infinite algebraical series, every term of
Speaker 1: which could be integrated by Wallace's rule. His results on
Speaker 1: the subject have been communicated to Barrow, Collins, and others.
Speaker 1: In sixteen sixty nine, this was never finished. The fragment
Speaker 1: was published in seventeen eleven, but the substance of it
Speaker 1: had been printed as an appendix to the Optics in
Speaker 1: seventeen o four. These works were only the first of
Speaker 1: Newton's leisure, most of the time during these two years
Speaker 1: being given up to optical researches. In October sixteen sixty nine,
Speaker 1: Barrow resigned the Lucasian Chair in favor of Newton. During
Speaker 1: his tenure of the professorship, it was Newton's practice to
Speaker 1: lecture publicly once a week, from half an hour to
Speaker 1: an hour at a time, in one term of each year,
Speaker 1: probably dictating his lectures as rapidly as they could be
Speaker 1: taken down, and in the week following the lecture, to
Speaker 1: devote four hours to appointments, which he gave to students
Speaker 1: who wished to come to his rooms to discuss the
Speaker 1: results of the previous lecture. He never repeated a course,
Speaker 1: which usually consisted of nine or ten lectures, and generally
Speaker 1: the lectures of one course began from the point at
Speaker 1: which the preceding course had ended. The manuscripts of his
Speaker 1: lectures for seventeen of the first eighteen years of his
Speaker 1: tenure are extant. When first appointed, Newton chose optics for
Speaker 1: the subject of his lectures and researches, and before the
Speaker 1: end of sixteen sixty nine he had worked out the
Speaker 1: details of his discovery on the decomposition of a ray
Speaker 1: of white light into rays of different colors by means
Speaker 1: of a prism. The complete explanation of the theory of
Speaker 1: the rainbow followed from this discovery. These discoveries formed the
Speaker 1: subject matter of the lectures which he delivered as Lucasian
Speaker 1: Professor in the years sixteen sixty nine, sixteen seventy and
Speaker 1: sixteen seventy one. The chief new results were embodied in
Speaker 1: a paper communicated to the Royal Society in February sixteen
Speaker 1: seventy two, and subsequently published in the Philosophical Transactions. The
Speaker 1: manuscript of his original lectures was printed in seventeen twenty
Speaker 1: nine under the title Leccioni's Optiquet. This work is divided
Speaker 1: into two books, the first of which contains four sections,
Speaker 1: and the s second five. The first section of the
Speaker 1: first book deals with the decomposition of solar light by
Speaker 1: a prism in consequence of the unequal refragibility of the
Speaker 1: rays that compose it, and a description of his experiments
Speaker 1: is added. The second section contains an account of the
Speaker 1: method which Newton invented for the determining of the coefficients
Speaker 1: of refraction of different bodies. This is done by making
Speaker 1: a ray pass through a prism of the material so
Speaker 1: that the deviation is a minimum, and he proves that
Speaker 1: if the angle of the prism be i and the
Speaker 1: deviation of the ray be delta, then the refractive index
Speaker 1: will be the sign of one half ee plus delta
Speaker 1: multiplied by the cosec and half i. The third section
Speaker 1: is on refractions at plane surfaces. Here he sews that
Speaker 1: if a ray pass through a prism with minimum debas,
Speaker 1: the angle of incidents is equal to the angle of emergence.
Speaker 1: Most of this section is devoted to geometrical solutions of
Speaker 1: different problems. The fourth section contains a discussion of refractions
Speaker 1: at curved surfaces. The second book treats of the theory
Speaker 1: of colors of the rainbow. By a curious chapter of accidents,
Speaker 1: Newton failed to correct the chromatic aberration of two colors
Speaker 1: by means of a couple of prisms. He therefore abandoned
Speaker 1: the hope of making a refracting telescope which should be achromatic,
Speaker 1: and instead designed a reflecting telescope, probably on the model
Speaker 1: of a small one which he had made in sixteen
Speaker 1: sixty eight. The form he used is that still known
Speaker 1: by his name. The idea of it was naturally suggested
Speaker 1: by Gregory's telescope. In sixteen seventy two he invented a
Speaker 1: reflecting microscope, and some years later he invented the sixtent,
Speaker 1: which was rediscovered by Hadley in seventeen thirty one. His
Speaker 1: professorial lectures from sixteen seventy three to sixteen eighty three
Speaker 1: were on algebra and the theory of equations, and are described below.
Speaker 1: But much of his time during these years was occupied
Speaker 1: with other investigations, and I may remark that throughout his
Speaker 1: life Newton must have devoted at least as much attention
Speaker 1: to the chemistry and theology as to mathematics, though his
Speaker 1: conclusions are not of sufficient interest to require mention here.
Speaker 1: His theory of colors and his deductions from his optical
Speaker 1: experiments were attacked with considerable vehemence by parodies of France,
Speaker 1: Linus and Lucas at lieges Hooke in England and Huygens
Speaker 1: in Paris, but his opponents were finally refuted. The correspondence
Speaker 1: which this entailed on Newton occupied nearly all his life
Speaker 1: pleasure in the years sixteen seventy two to sixteen seventy five,
Speaker 1: and proved extremely distasteful to him. Writing on December ninth,
Speaker 1: sixteen seventy five, he says, I was so persecuted with
Speaker 1: discussions arising out of my theory of light that I
Speaker 1: blame my own imprudence for parting with so substantial a
Speaker 1: blessing as my quiet to run after a shadow again.
Speaker 1: After November eighteenth, sixteen seventy six, he observes, I see
Speaker 1: I have made myself a slave to philosophy. But if
Speaker 1: I get rid of mister Linus's business, I will resolutely
Speaker 1: bid adieu to it eternally, accepting what I do for
Speaker 1: my private satisfaction, or leave to come out after me.
Speaker 1: For I see a man must either resolve to put
Speaker 1: out nothing new or to become a slave to defend it.
Speaker 1: The unreasonable dislike to have his conclusions doubted or to
Speaker 1: be involved in any correspondence about was a prominent trait
Speaker 1: in Newton's character. He next set himself to examine the
Speaker 1: problem as to how light was really produced, and by
Speaker 1: the end of sixteen seventy five he had worked out
Speaker 1: the corpuscular or emission theory, a theory to which he
Speaker 1: was perhaps led by his researches on the theories of attraction.
Speaker 1: Only three ways have been suggested in which light can
Speaker 1: be produced mechanically. Either the eye may be supposed to
Speaker 1: send out something which, so to speak, feels the object,
Speaker 1: as the Greeks believed, or the object perceived may send
Speaker 1: out something which hits or affects the eye, as assumed
Speaker 1: in the emission theory. Or there may be some medium
Speaker 1: between the eye and the object, and the object may
Speaker 1: cause some change in the form or nature of this
Speaker 1: intervening medium, and thus affect the eye, as Hook and
Speaker 1: Huygen's supposed in the wave or undulatory theory. It will
Speaker 1: be enough here to say that on either of the
Speaker 1: two latter theories, all the obvious phenomena of geometrical optics,
Speaker 1: such as reflection, refraction, et cetera, can be accounted for.
Speaker 1: Within the present century, crucial experiments have been devised which
Speaker 1: give different results. According as one or the other theory
Speaker 1: is adopted. All these experiments agree with the results of
Speaker 1: the undulatory theory and differ from the results of the
Speaker 1: Newtonian theory. The latter is therefore untenable, but whether the
Speaker 1: former represents the whole truth and nothing but the truth
Speaker 1: is still an open question. Until, however, the theory of
Speaker 1: interference suggested by Young was worked out by Fresnel. The
Speaker 1: hypothesis of Huygens failed to account for all the facts
Speaker 1: and was open to more questions than that of Newton.
Speaker 1: It should be noted that Newton nowhere expresses an opinion
Speaker 1: that the corpuscular theory is true, but always treats it
Speaker 1: as a hypothesis from which, if true, certain results would follow.
Speaker 1: It would, moreover seem that he believed that the wave
Speaker 1: theory to be intrinsically more probable, and it was only
Speaker 1: the difficulty of explaining diffraction on that theory that led
Speaker 1: him to reject it. His remarks on the other physical
Speaker 1: subjects show a similar caution. Newton's corpuscular theory was expounded
Speaker 1: in memoirs communicated to the Royal Society in December sixteen
Speaker 1: seventy five, which are substantially reproduced in his Optics, published
Speaker 1: in seventeen o four. In the latter work he dealt
Speaker 1: in detail with his theory of fits of easy reflection
Speaker 1: and transmission, and the colors of thin plates, to which
Speaker 1: he added an explanation of the colors of thick plates
Speaker 1: and observations on the inflection of light. Two letters written
Speaker 1: by Newton in the year sixteen seventy six are sufficiently
Speaker 1: interesting to justify an allusion to them. Liibinets, who had
Speaker 1: been in London in sixteen seventy three, communicated some results
Speaker 1: to the Royal Society which he had supposed to be new,
Speaker 1: but which, it was pointed out to him, had been
Speaker 1: previously proved by Mouton. This led to a correspondence with oldenburg,
Speaker 1: the secretary of the Society, in sixteen seventy four. Leibniz
Speaker 1: wrote saying that he possessed general analytical methods depending on
Speaker 1: the infinite series. Oldenburgen reply told him that Newton and
Speaker 1: Gregory had used such series in their work. In answer
Speaker 1: to her request for information, Newton wrote on June thirteenth,
Speaker 1: sixteen seventy six, giving a brief account of his method,
Speaker 1: but adding the expansions of a binomial i e. The
Speaker 1: binomial theorem and of the arc sign of x, from
Speaker 1: the latter of which he deduced that of sign of x.
Speaker 1: This seems to be the earliest known incense of the
Speaker 1: reversion of a series. He also inserted an expression for
Speaker 1: the rectification of an elliptic arc in an infinite series.
Speaker 1: Leibnitz wrote on August twenty seventh asking for fuller details,
Speaker 1: and Newton, in a long but interesting reply, day on
Speaker 1: October twenty fourth, sixteen seventy six, and sent through Oldenburg,
Speaker 1: gives an account of the way in which he had
Speaker 1: been led to some of his results. In this letter,
Speaker 1: Newton begins by saying that altogether he had used three
Speaker 1: methods for expansion in a series. His first was arrived
Speaker 1: at from the study of the method of interpolation, by
Speaker 1: which Wallace had found expressions for the area of a
Speaker 1: circle and hyperbola. Thus, by considering the series of expressions
Speaker 1: one minus x squared quantity to the power zero over two,
Speaker 1: one minus x squad quantity to the powers two over two,
Speaker 1: and one minus x gred quantity to the four over two,
Speaker 1: and so on, he deduced by interpolations the law which
Speaker 1: connects the successive coefficial in the expansions of one minus
Speaker 1: x squared to the power of a half, one minus
Speaker 1: x square to the power of three halves, and so on,
Speaker 1: and then by analogy obtained the expression for the general
Speaker 1: term in the expansion of a binomial i e. The
Speaker 1: binomial theorem. He says that he proceeded to test this
Speaker 1: by forming the square of the expansion one minus x
Speaker 1: squared to the power of a half, which reduced to
Speaker 1: one minus x squared, and he proceeded in a similar
Speaker 1: way with other expansions. He next tested the theorem in
Speaker 1: the case of one minusx squared quantity to the power
Speaker 1: of one half by extracting the square root of one
Speaker 1: minus x squared mare arithmetico. He also used the series
Speaker 1: to determine the areas of the circle and the hyperbola
Speaker 1: in infinite series, and found that the results were the
Speaker 1: same as those he had arrived at by other means.
Speaker 1: Having established this result, he then discarded the method of
Speaker 1: interpolation in series and employed his binomial theorem to express,
Speaker 1: when possible, the ordinate of a curve in an infinite
Speaker 1: series in ascending powers of the abscissa, and thus, by
Speaker 1: Wallace's method he obtained expressions in an infinite series for
Speaker 1: the areas and arcs of curves, in the manner described
Speaker 1: in the Appendix to his Optics and his di anuisi
Speaker 1: perre equessiones numero to menorum infinitorum. He states that he
Speaker 1: had employed his second method before the plague in sixteen
Speaker 1: sixty five sixty six, and goes on to say that
Speaker 1: he was then obliged to leave Cambridge, and subsequently i e.
Speaker 1: Presumably on his return to Cambridge, he ceased to pursue
Speaker 1: these ideas, as he found that Nicholas Mercat had employed
Speaker 1: some of them in his Logarithmimotechnica, published in sixteen sixty eight,
Speaker 1: and he supposed that the remainder had been or would
Speaker 1: be found out before he himself was likely to publish
Speaker 1: his discoveries. Newton next explains that he had also a
Speaker 1: third method, of which he says he had about sixteen
Speaker 1: sixty nine sent an account to Barrow and Collins, illustrated
Speaker 1: by applications to areas rectification, qubature, and et cetera. This
Speaker 1: was the method of fluxions, but Newton gives no description
Speaker 1: of it here, though he adds some illustrations of its use.
Speaker 1: The first illustration is on the quadrature of the curve
Speaker 1: represented by the equation y equals ax to the power
Speaker 1: m multiplied by the quantity b plus c x to
Speaker 1: the power n quantity to the power p, which he
Speaker 1: says can be affected as a sum of quantity m
Speaker 1: plus one divided by n terms if quantity m plus
Speaker 1: one divided by m be a positive integer, and which
Speaker 1: he thinks cannot otherwise be affected except by an infinite series.
Speaker 1: He also gives a list of other forms which are
Speaker 1: immediately integrable, of which g f are x is raised
Speaker 1: to the power of m times n minus one multiplied
Speaker 1: by open bracket a plus b x the power n
Speaker 1: plus c x to the power of two n close
Speaker 1: bracket raise the power of minus one. Next x to
Speaker 1: the power of open bracket m plus one half close
Speaker 1: bracket n minus one multiplied by open bracket a plus
Speaker 1: b x the power of n plus c x of
Speaker 1: the power of two n close bracket raised to the
Speaker 1: power of minus one. The next term x to the
Speaker 1: power of m times n minus one, multiplied by open
Speaker 1: bracket a plus b x to the power of n
Speaker 1: plus c x of the power of two en close
Speaker 1: bracket raise to the power of plus or minus a half.
Speaker 1: Next term x to the power of M times n
Speaker 1: minus one multiplied open bracket A plus b x to
Speaker 1: the power of n close bracket plus or minus a
Speaker 1: half open bracket c plus d x of the power
Speaker 1: of n close bracket, raise the power of minus one.
Speaker 1: Next term x to the power of m times n
Speaker 1: minus n minus one multiplied by open bracket A plus
Speaker 1: b x the power of n close bracket. Raise to
Speaker 1: the power of one half multiplied by open bracket c
Speaker 1: plus d x to the power of n close bracket.
Speaker 1: Raise to the power of negative one half. Where m
Speaker 1: is a positive integer and n is any number whatever. Lastly,
Speaker 1: he points out that the area of any curve can
Speaker 1: be easily determined approximately by the method of interpretation described below.
Speaker 1: In discussing his methodis differentialis. At the end of his letter,
Speaker 1: Newton alludes to the solution of the inverse pro album
Speaker 1: of tangents, a subject on which Leibniz had asked for information.
Speaker 1: He gives formulae for reversing any series, but says that
Speaker 1: besides these formulae, he has two methods for solving such questions,
Speaker 1: which for the present he will not describe except by
Speaker 1: an anagram, which, being read is as follows. Une methodis
Speaker 1: consisted in extraccioni fluentis countitatis ex equacioni simul irvalvente fluccionim
Speaker 1: ejus altera tantum in assumcioni serie pro quantitate qualibet incognicia
Speaker 1: ex quaquetera commodi derivari posunt et in collagioni terminorum homologorum
Speaker 1: equagiis ras altantis add rouendos terminus assumtai serie. He implies
Speaker 1: in this letter that he is worried by the questions
Speaker 1: he has asked and the controversial raised about every new
Speaker 1: matter which he produces, which shew his rationalness in publishing
Speaker 1: quad ombram captando a tennis padiderum quietumnem remprosa substantiarim. Leibnitz
Speaker 1: did not reply to this letter till June twenty first,
Speaker 1: sixteen seventy seven. In his answer, he explains his method
Speaker 1: of drawing tangents to curves, which he says proceeds not
Speaker 1: by fluxions of lines, but by the differences of numbers,
Speaker 1: and he introduces his notation of d x and d
Speaker 1: y for the infinitesimal differences between the coordinates of two
Speaker 1: consecutive points on a curve. He also gives a solution
Speaker 1: of the problem to find a curve whose subtangent is constant,
Speaker 1: which shews that he could integrate. In sixteen seventy nine, Hook,
Speaker 1: at the request of the Royal Society, wrote to Newton,
Speaker 1: expressing a hope that he would make further communications to
Speaker 1: the Society, and informing him of various facts then recently discovered.
Speaker 1: Newton replied, saying that he had abandoned the study of philosophy,
Speaker 1: but he added that the Earth's diurnal motion might be
Speaker 1: proved by the experiment of observing the deviation from the
Speaker 1: perpendicular of a stone dropped from a height to the ground,
Speaker 1: an experiment which was subsequently made by the Society and succeeded.
Speaker 1: Hook in his letter mentioned Picard's geodetical researches. In these
Speaker 1: Picard used a value of the radius of the Earth,
Speaker 1: which is substantially correct. This led Newton to repeat with
Speaker 1: Picard's data his calculations of sixteen sixty six on the
Speaker 1: lunar orbit, and he found the verification of his view
Speaker 1: was complete. He then proceeded to the General theory of
Speaker 1: motion under a centrepetal force, and demonstrated one the equitable
Speaker 1: description of areas two, that if an ellipse were described
Speaker 1: about a focus under a centripetal force, then the law
Speaker 1: was that of the inverse square of the distance three,
Speaker 1: and conversely that of the orbit of a particle projected
Speaker 1: under the influence of such a force. Was a conic
Speaker 1: or it may be he thought only an ellipse, obeying
Speaker 1: his rule to publish nothing which could lend him in
Speaker 1: scientific controversy. These results were locked up in his notebooks,
Speaker 1: and it was only a specific question addressed to him
Speaker 1: five years later that led to their publication. The Universal Arithmetic,
Speaker 1: which is on algebra, theory of equations, and miscellaneous problems,
Speaker 1: contains the substance of Newton's lectures during the years sixteen
Speaker 1: seventy three to sixteen eighty three. His manuscript of it
Speaker 1: is still extant. Wiston extracted somewhat reluctant permission from Newton
Speaker 1: to print it, and it was published in seventeen o seven.
Speaker 1: Amongst several new theorems on various points in algebra and
Speaker 1: the theory of equations, Newton here enunciated the following important results.
Speaker 1: He explained that the equations whose roots are the solution
Speaker 1: of a given problem will have as many roots as
Speaker 1: there are different possible cases, and he considered how it
Speaker 1: happened that the equation to which a problem led might
Speaker 1: contain roots which did not satisfy the original question. He
Speaker 1: extended Descartes's rule of signs to give limits to the
Speaker 1: number of imaginary roots. He used the principle of continuity
Speaker 1: to explain how two real and unequal roots might become
Speaker 1: imaginary in passing through equality, and illustrated this by geometrical considerations.
Speaker 1: Thence chewed that imaginary roots must occur in pairs. New
Speaker 1: Newton also here gave rules to find a superior limit
Speaker 1: to the positive roots of a numerical equation and to
Speaker 1: determine the approximate values of the numerical roots. He further
Speaker 1: enunciated the theorem known by his name for finding the
Speaker 1: sum of the n th powers of the roots of
Speaker 1: an equation, and laid the foundation of the theory of
Speaker 1: symmetrical functions of the roots of an equation. The most
Speaker 1: interesting theorem contained in the work is his attempt to
Speaker 1: find a rule analogous to that of descartes for real
Speaker 1: roots by which the number of imaginary roots of an
Speaker 1: equation can be determined. He knew that the result which
Speaker 1: he obtained was not universally true, but he gave no
Speaker 1: proof and did not explain what were the exceptions to
Speaker 1: the rule. His theorem was as follows. Suppose the equation
Speaker 1: to be of the nth degree, arranged in descending powers
Speaker 1: of X, the coefficient of X to the end being positive. Hence,
Speaker 1: oppose the n plus one fractions one n overix, n
Speaker 1: plus one multiplied by two over one, n minus one
Speaker 1: over n minus two, multiplied by three over two, and
Speaker 1: so on, and then the middle term of the series
Speaker 1: would be N minus p plus one, all divided by
Speaker 1: N minus p that fraction multiplied by p plus one
Speaker 1: over p, and so on, and the second last term
Speaker 1: being two over one multiplied by the fraction n over
Speaker 1: n minus one, and finally the final term one to
Speaker 1: be formed and written below the corresponding terms of the equation.
Speaker 1: Then if the square of any term, when multiplied by
Speaker 1: the corresponding fraction, be greater than the product of the
Speaker 1: terms on each side of it, put a plus sign
Speaker 1: above it. Otherwise, put a minus sign above it, and
Speaker 1: put a plus sign above the first and last terms.
Speaker 1: Now consider any two consecutive terms in the original equation
Speaker 1: and the two symbols written above them. Then we may
Speaker 1: have any one of the four following cases alpha the
Speaker 1: terms of the same sign and the symbols of the
Speaker 1: same sign, Beta, the terms of the same sign and
Speaker 1: the symbols of the opposite signs, Gamma, the terms of
Speaker 1: opposite signs and the symbols of the same sign. Delta
Speaker 1: the terms of the opposite signs and the symbols of
Speaker 1: the opposite signs. Then it has been shown that the
Speaker 1: number of negative roots will not exceed the number of
Speaker 1: cases alpha, and the number of positive roots will not
Speaker 1: exceed the number of cases gamma, and therefore the number
Speaker 1: of imaginary roots is not less than the number of
Speaker 1: cases beta and delta. In other words, the number of
Speaker 1: changes of signs in the row of symbols written above
Speaker 1: the equation is an inferior limit to the number of
Speaker 1: imaginary roots. Newton, however, asserted that you may also know
Speaker 1: how many roots are impossible by counting the changes of
Speaker 1: sign in the series of symbols formed as above. That
Speaker 1: is to say, he thought that in general, the actual
Speaker 1: number of positive, negative, and imaginary roots could be got
Speaker 1: only by the rule, and not merely superior or inferior
Speaker 1: to the limits of these numbers. But though he knew
Speaker 1: that the rule was not universal, he could not find
Speaker 1: what were the exceptions to it. This theorem was subsequently
Speaker 1: discussed by Kempbell, Maclaurin, Euler, and other writers. At last,
Speaker 1: in eighteen sixty five, Sylvester succeeded in proving the general result.
Speaker 1: In August sixteen eighty four, Halley came to Cambridge in
Speaker 1: order to consult Newton about the law of gravitation. Hook, Huygens, Halley,
Speaker 1: and Wren had all conjectured that the force of the
Speaker 1: attraction of the Sun or Earth on an external particle
Speaker 1: varied inversely as the square of the distance. These writers
Speaker 1: seemed to have independently shewn that if kept conclusion were
Speaker 1: rigorously true, as to which they were not quite certain,
Speaker 1: the law of attraction must be that of the inverse square,
Speaker 1: but they could not deduce from the law of the
Speaker 1: orbits of the planets. Halley explained that their investigations were
Speaker 1: stopped by their inability to solve this problem, and asked
Speaker 1: Newton if he could find out what the orbit of
Speaker 1: a planet would be if the law of attraction were
Speaker 1: that of an inverse square. Newton immediately replied that it
Speaker 1: was an ellipse, and promised to send or write out
Speaker 1: afresh the demonstration of it which he had found in
Speaker 1: sixteen seventy nine. This was sent in November of sixteen
Speaker 1: eighty four. End of Section twenty four, Chapter sixteen, Part one.
Speaker 1: Recording by Paul King PJK dot scripts dot mit dot edu.
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