Speaker 1: Chapter sixteen, Part two of A Short Account of the
Speaker 1: History of Mathematics. This is a LibriVox recording. All LibriVox
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Speaker 1: MI I T dot E d U forward slash p
Speaker 1: K J. A Short Account of the History of Mathematics
Speaker 1: by W. W. Rowseball. Chapter sixteen, The Life and Works
Speaker 1: of Newton, Part two, instigated by Halley. Newton now returned
Speaker 1: to the problem of gravitation, and before the autumn of
Speaker 1: sixteen eighty four he had worked out the substance of
Speaker 1: propositions one through nineteen twenty one thirty thirty two through
Speaker 1: thirty five in the first book of the Principia. These,
Speaker 1: together with the notes on the laws of motion and
Speaker 1: various lemmas, were read for his lectures in the Michailmas
Speaker 1: Term sixteen eighty four. In November Halley received Newton's promised communication,
Speaker 1: which probably consisted of the substance of propositions one eleven
Speaker 1: and either seventeen or corollaries one of thirteen, and thereupon
Speaker 1: he again went to Cambridge, where he saw a curious
Speaker 1: treatise Desmo Too, drawn up since August. Most likely this
Speaker 1: contained Newton's manuscript notes of the lectures above alluded to.
Speaker 1: These notes are now in the University Library and are
Speaker 1: headed de motu cooporum. Halley begged that the results might
Speaker 1: be published, and finally secured a promise that they should
Speaker 1: be sent to the Royal Society. They were accordingly communicated
Speaker 1: to the Society not later than February sixteen eighty five,
Speaker 1: in the paper desmo Too, which contained the substance of
Speaker 1: the following propositions in the Principia Book I propositions one, four, six, seven, ten, eleven, fifteen,
Speaker 1: seventeen thirty two and Book two Propositions two, three and four.
Speaker 1: It seems also to have been due to the influence
Speaker 1: in tact of Halley at this visit in November sixteen
Speaker 1: eighty four, that Newton undertook to attack the whole problem
Speaker 1: of gravitation and practically pledged himself to publish his results.
Speaker 1: As yet, Newton had not determined the attraction of a
Speaker 1: spherical body on an external point, nor had he calculated
Speaker 1: the details of the planetary motions. Even if the members
Speaker 1: of the Solar system could be regarded as points. The
Speaker 1: first problem was solved in sixteen eighty five, probably either
Speaker 1: in January or February. No sooner, to quote from doctor
Speaker 1: Galicier's address on the Bison tenorary of the publication of
Speaker 1: the Principia, had Newton proved the superb theorem. And we
Speaker 1: know from his own words that he had no expectation
Speaker 1: of so beautiful a result till it emerged from his
Speaker 1: mathematical investigation that all the mechanism of the universe at
Speaker 1: once lay spread before him. When he discovered the theorems
Speaker 1: that from the first three sections of Book one, when
Speaker 1: he gave them in his lectures of sixteen eighty four,
Speaker 1: he was unaware that the Sun and Earth exerted their
Speaker 1: attractions as if they were but points. How different must
Speaker 1: these propositions have seen to Newton's eyes when he realized
Speaker 1: that these results, which he had believed to be only
Speaker 1: approximately true when applied to the Solar system, were really exact. Hitherto,
Speaker 1: they had been only true in so far as he
Speaker 1: could regard the Sun as a point compared to the
Speaker 1: distance of the planets, or the Earth as a point
Speaker 1: compared to the distance of the Moon, a distance amounting
Speaker 1: to all about sixty times the Earth's radius. But now
Speaker 1: they were mathematically true, excepting only for the slight deviation
Speaker 1: from a perfectly spherical form of the Sun, Earth, and planets.
Speaker 1: We can imagine the effect of this sudden transition from
Speaker 1: approximation to exactitude in stimulating Newton's mind to still greater efforts.
Speaker 1: It was now in his power to apply the mathematical
Speaker 1: analysis with absolute precision to the actual problems of astronomy.
Speaker 1: Of the three fundamental principles applied in the principia, we
Speaker 1: may say that the idea that every particle attracts every
Speaker 1: other particle in the universe was formed at least as
Speaker 1: early as sixteen sixty six. The law of equable distribution
Speaker 1: of areas its consequences, and the fact that if the
Speaker 1: law of attraction were that of the inverse square of
Speaker 1: the orbit of a particle about a center of force
Speaker 1: would be a conic where proved in sixteen seventy nine.
Speaker 1: And lastly, the discovery that a sphere whose density at
Speaker 1: any point depends only on the distance from the center
Speaker 1: attracts an external point as if the whole mass were
Speaker 1: collected at its center were made in sixteen eighty five.
Speaker 1: It was this last discovery that enabled him to apply
Speaker 1: the first two principles to the phenomenon of bodies of
Speaker 1: finite size. The draft of the first Book of the
Speaker 1: Principia was finished before the summer of sixteen eighty five,
Speaker 1: but the corrections and editions took some time, and the
Speaker 1: book was not presented to the Royal Society until April
Speaker 1: twenty eighth, sixteen eighty six. This book is given up
Speaker 1: to the consideration of the motion of particles or bodies
Speaker 1: in free space, either in known orbits, or under the
Speaker 1: action of known forces, or under their mutual attraction. In it,
Speaker 1: Uton generalizes the law of attraction into a statement that
Speaker 1: every particle of matter in the universe attracts every other
Speaker 1: particle with a force which varies directly as the product
Speaker 1: of their masses and inversely as a square of the
Speaker 1: distance between them, and he thence deduces the law of
Speaker 1: attraction for spherical shells of constant density. The book is
Speaker 1: prefaced by an introduction on the science of dynamics. The
Speaker 1: second Book of the Principia was completed by the summer
Speaker 1: of sixteen eighty six. This book treats of motion in
Speaker 1: a resting medium, and of hydrostatics and hydrodynamics, with a
Speaker 1: special application to waves, tides, and acoustics. He concludes it
Speaker 1: by shewing that the Cartesian theory of vortices is inconsistent
Speaker 1: both with the known facts and with the laws of motion.
Speaker 1: The next nine or ten months were devoted to the
Speaker 1: third book, probably for this he had originally no materials
Speaker 1: ready in it. The theorems obtained in the first book
Speaker 1: are applied to the chief phenomenon of the Solar System.
Speaker 1: The masses and distances of the planets, and, wherever sufficient
Speaker 1: data existed, of their satellites are determined. In particular, the
Speaker 1: motion of the Moon, the various inequalities therein, and the
Speaker 1: theory of the tides are worked out in detail. He
Speaker 1: also investigates the theory of comets shoes that they belong
Speaker 1: to the Solar system, explains how from three observations the
Speaker 1: orbit can be determined, and illustrates his results by considering
Speaker 1: certain special commets. The third book, as we have it,
Speaker 1: is but little more than a sketch of what Newton
Speaker 1: had finally proposed to himself to accomplish His original scheme
Speaker 1: is among the Portsmouth papers and his notes shoe that
Speaker 1: he continued to work at it for some years after
Speaker 1: the publication of the first edition of the Principia. The
Speaker 1: most interesting of his memoranda are those in which, by
Speaker 1: means of fluxions, he has carried out his results beyond
Speaker 1: the point at which he was able to translate them
Speaker 1: into geometry footnote. I take this opportunity of saying that
Speaker 1: I hope shortly to publish a memoir on the history
Speaker 1: and compilation of the Principia. The following brief summary of
Speaker 1: the contents of the work will give the reader a
Speaker 1: general idea of its arrangement. The Principia is preceded by
Speaker 1: a preface in which Newton says that his object is
Speaker 1: to apply mathematics to the phenomena of nature. Among these phenomena,
Speaker 1: motion is one of the most important. Now, motion is
Speaker 1: the effective force, and though he does not know what
Speaker 1: is the nature or origin of force, still many of
Speaker 1: its effects can be measured, and it is these that
Speaker 1: form the subject matter of the work. The work begins, therefore, naturally,
Speaker 1: with an introduction on dynamics, the science of motion, This
Speaker 1: commences with the eight definitions of the various terms such
Speaker 1: as mass, momentum, and so on. Newton then lays down
Speaker 1: three laws of motion, which are incapable of exact proof,
Speaker 1: but are confirmed partly by direct experiments, partly by the
Speaker 1: agreement with observation of the deductions from them. From these
Speaker 1: he deduces six fundamental principles of mechanics and adds an
Speaker 1: appendix on the motion of falling bodies, projectiles, oscillations, impact,
Speaker 1: and the mutual attractions of two bodies. The most important
Speaker 1: deduction is that of the parallelogram of velocities, accelerations, and forces.
Speaker 1: The first book of the Principia is on the motion
Speaker 1: of bodies in free space, and is divided into fourteen sections.
Speaker 1: The first section consists of eleven preliminary lemmas treated by
Speaker 1: the method of prime and ultimate ratios and not by
Speaker 1: that of indivisibles. The second section commences by shewing that
Speaker 1: if a body such as a planet, revolve in an
Speaker 1: orbit subject to a force tending to affix points such
Speaker 1: as the Sun, the areas swept out by radii drawn
Speaker 1: from the body to the point are in one plane
Speaker 1: and are proportional to the times of describing them, and conversely,
Speaker 1: if the areas be proportional to the times, the force
Speaker 1: acting on the body must be directed to the point.
Speaker 1: Newton then chows how if the orbit be known and
Speaker 1: the center of force be given, the law of force
Speaker 1: can be determined, and he finds the law for various curves.
Speaker 1: In the third section, he applies these propositions to a
Speaker 1: body which describes a conic section about a focus, and
Speaker 1: proves that the force must vary inversely as a square
Speaker 1: of the distance, and that Kepler's third law would necessarily
Speaker 1: be true of such a system. He proves that if
Speaker 1: a body were projected in any way and subject to
Speaker 1: a central force which varied according to this law, then
Speaker 1: it must move in a conic section having the center
Speaker 1: of force in a focus. He concludes proposition seventeen corollaries
Speaker 1: three and four with a suggestion as to how the
Speaker 1: effects of distributing forces should be calculated. This was first
Speaker 1: done by the brilliant investigations of Laplace and Lagrange, and
Speaker 1: Laplace says that Lagrange's paper in the Berlin Memoirs for
Speaker 1: seventeen eighty six, on which the modern treatment of the
Speaker 1: subject is founded, was suggested by these remarks of Newton.
Speaker 1: The fourth and fifth sections are devoted to the geometry
Speaker 1: of conic sections, especially to the construction of conics, which
Speaker 1: satisfy five conditions. In section four. One of the conditions
Speaker 1: is that the focus is given. This includes the problem
Speaker 1: of finding the path of a comet from three observations,
Speaker 1: with which Newton says he found the most difficult problem
Speaker 1: of any which he had to solve curiously enough. He
Speaker 1: gave a second solution of the problem in Book three,
Speaker 1: proposition forty one, in which he recommended as more simple,
Speaker 1: but which is inapplicable in practice. The sixth section is
Speaker 1: devoted to determining what at any given time is the
Speaker 1: velocity and what is the position of a body, which
Speaker 1: is describing a given conic about a center of attraction
Speaker 1: in a focus, together with various converse problems. To effect this,
Speaker 1: Newton had to find the area of a sector of
Speaker 1: a conic. This is easily done for the parabola. He
Speaker 1: then endeavors to shew that the exact quadrature of any
Speaker 1: closed oval curve having no infinite branches, such as the ellipse,
Speaker 1: is impossible. The proof is not correct, as it stands
Speaker 1: since the result is not true for ovals of the
Speaker 1: form why to the power of two M equals the
Speaker 1: quantity A to the power of t two n minus
Speaker 1: x to the power of two n all multiplied by
Speaker 1: two to the power of two m times n to
Speaker 1: the power of two m multiplied by x to the
Speaker 1: power of two M times two n minus one, where
Speaker 1: M and n are positive integers. Newton seems himself to
Speaker 1: have felt some doubt about inserting it, though he believed
Speaker 1: the result to be true, an exact quadrature being impossible.
Speaker 1: He proceeds to give three ways, two arithmetical and one geometrical,
Speaker 1: of approximating to the sectorial area of an ellipse as
Speaker 1: closely as is desired. The seventh section is given up
Speaker 1: to the discussion of motion in a straight line under
Speaker 1: a force which varies inversely as a square of the
Speaker 1: distance and its comparison with the motion, and a conic
Speaker 1: under the same force. He concludes by giving a general
Speaker 1: solution for all the problems considered in this section. For
Speaker 1: X any law of force. He here determines geometrically, what
Speaker 1: is equivalent of finding the integral of X divided by
Speaker 1: the square root of the quantity ax minus x squared.
Speaker 1: The eighth section contains a general solutions for any orbit
Speaker 1: described under any central force of some of the problems
Speaker 1: previously considered. In proposition forty he states that the kinetic
Speaker 1: energy acquired by a body and moving from one point
Speaker 1: to another is equal to the total work done by
Speaker 1: the force between those two points. In the ninth section,
Speaker 1: he discusses the case or the orbit is in motion
Speaker 1: in its own plane round the center of force, and
Speaker 1: treats in detail of the motion of the apse line
Speaker 1: and the forces by which a given motion would be produced.
Speaker 1: Newton applied this reasoning proposition forty five corollary IWO to
Speaker 1: the case of the moon, but the resulting motion of
Speaker 1: the apses only came out about one half of the
Speaker 1: actual amount. The approximation was, in fact not carried to
Speaker 1: a sufficiently high order. Newton was aware of the discrepancy,
Speaker 1: and as he explained the similar difficulty in the case
Speaker 1: of the nodes, it had been long suspected Goldfrays Lunar Theory,
Speaker 1: second Edition, Article sixty eight that this scolium of the
Speaker 1: first edition to book three, Proposition thirty five meant that
Speaker 1: he had found the explanation. Nowhere in the principia does he, however,
Speaker 1: give any hint as to how this was affected, and
Speaker 1: that the true explanation of a difference which had long
Speaker 1: formed an obstacle to the universal acceptance of the Newtonian system,
Speaker 1: was first given by claireaut in seventeen fifty two. The
Speaker 1: Portsmouth Papers contained Newton's original work, and shew that he
Speaker 1: had obtained the true value by carrying the approximation to
Speaker 1: a sufficiently high order. It also seemed clear from these
Speaker 1: papers that Newton gave the corollary to Book one proposition
Speaker 1: forty five as a mere illustration to the motion of
Speaker 1: the apses and orbits, which are nearly circular, and did
Speaker 1: not mean it to apply to the moon. But by
Speaker 1: an inadvertence in the second and third editions, a reference
Speaker 1: to it as an authenticity for a result connected with
Speaker 1: the moon was added, which would naturally deceive any reader.
Speaker 1: Newton left most of the revision to the second edition
Speaker 1: to Cotes, and it is probable that the mistake is
Speaker 1: due to a blunder of the editor. Other questions connected
Speaker 1: with lunar and planetary irregularities are also discussed in this proposition,
Speaker 1: but the extreme conclusions of Newton misled all the early
Speaker 1: commentators and even Laplace in his se stem de Monde,
Speaker 1: published in seventeen ninety six, speaks of Newton as having
Speaker 1: only roughly sketched out this part of the subject, leaving
Speaker 1: it to be completed when the calculus should be further perfected.
Speaker 1: But in the last volume of his Mechanique Cerest, published
Speaker 1: in eighteen twenty five, he says that on more careful reading,
Speaker 1: he has no hesitation in regarding it as among the
Speaker 1: most profound parts of the work. The tenth section is
Speaker 1: devoted to the consideration of motion of bodies along given surfaces,
Speaker 1: but not in planes passing through the center of force,
Speaker 1: with special reference to the vibration of pendulums and the
Speaker 1: determination of the accelerating effect of gravity. In connection with
Speaker 1: the latter problem, Newton investigates the chief geometrical properties of cycloids, epicycloids,
Speaker 1: and hypocycloids. In the eleventh section are considered the problems
Speaker 1: connected with motions in orbits where the center of forces
Speaker 1: disturbed or the moving body is disturbed by other forces.
Speaker 1: Until calculus of variations was invented by lug in seventeen
Speaker 1: fifty five, it was impossible to do more than sketch
Speaker 1: out the principles on which the problem should be solved,
Speaker 1: and laplace in his mechanique. Cerest was the first to
Speaker 1: work out most of the questions in any detail. Newton
Speaker 1: commences by considering the disturbance produced by the mutual action
Speaker 1: of two bodies revolving around one another. He then proceeds
Speaker 1: to consider the problem of three or more bodies which
Speaker 1: mutually attract one another. He first solves the question completely
Speaker 1: if the force of attraction varies directly as the distance.
Speaker 1: He next takes the case of the three bodies moving
Speaker 1: under their mutual attractions as in nature. This problem has
Speaker 1: not been yet solved generally, but in Newton's day it
Speaker 1: was beyond any analysis of which he had the command.
Speaker 1: He contrived, however, to work out roughly the chief effects
Speaker 1: of the disturbing action of the Sun on the motion
Speaker 1: of the moon proposition sixty six. Position was singled out
Speaker 1: by Lagrange as the most striking single illustration of the
Speaker 1: genius of Newton. To this proposition, twenty two corollaries are appended,
Speaker 1: in which it is applied to determine the motion in longitude.
Speaker 1: In latitude, the annual equation, the motion of the apse
Speaker 1: line and of the nodes, the evection, the change and
Speaker 1: inclination of the plane of the lunar orbit, the precession
Speaker 1: of the equinoxes, and the theory of the tides. The
Speaker 1: greater part of the third book consists of the numerical
Speaker 1: application of these principles to the case of the moon
Speaker 1: in the Earth. Lastly, Newton showed how from the motion
Speaker 1: of the nodes of the interior, constitution of the body
Speaker 1: could be roughly determined. Up to this point, Newton had
Speaker 1: generally treated the bodies with which he dealt as if
Speaker 1: they were particles. He now proceeds in section twelve to
Speaker 1: consider the attraction of spherical masses, which are either of
Speaker 1: uniform density or whose density at an point is a
Speaker 1: single valued function of the distance of the point from
Speaker 1: the center of the sphere. These are worked out for
Speaker 1: any law of attraction. In section thirteen he gives some
Speaker 1: general theorems on the theory of attractions and on some
Speaker 1: propositions dealing with the attractions of solids of revolution. But
Speaker 1: these problems are almost insoluble without the aid of the
Speaker 1: infinitesimal calculus and the Newtonian account of them is incomplete.
Speaker 1: The fourteenth section contains a statement of some theories and
Speaker 1: experiments in physical optics, and a solution by geometry of
Speaker 1: some problems in geometrical optics, particularly on the form of
Speaker 1: aplanatic refracting surfaces of revolution. The second book of the
Speaker 1: Principia is concerned with hydromechanics, and especially with motion in
Speaker 1: a resisting medium. These questions are not worked out so
Speaker 1: completely as those treated in the first book, and though
Speaker 1: this book provided the basis on which much of the
Speaker 1: subsequent work of Danielle Bernoulli, clairout De, Lambert, Euler and
Speaker 1: Laplace were erected, it is not of the same epoch
Speaker 1: making character as the first book. This book is divided
Speaker 1: into nine sections. The motion of bodies in a medium
Speaker 1: where the resistance varies directly as the velocity is considered
Speaker 1: in the first section. The motions where resistance varies as
Speaker 1: the square of the velocity is discussed in the second section.
Speaker 1: The motion where the resistance can be expressed as a
Speaker 1: sum of two terms, one of which varies as the velocity,
Speaker 1: and the others the square of the velocity, is dealt
Speaker 1: with in the third section. The second section contains proposition
Speaker 1: twenty five, a construction for the shape of the solid
Speaker 1: of least resistance. No proof is given, and it had
Speaker 1: been long somewhat of a mystery to know how Newton
Speaker 1: had contrived to solve the problem without the use of
Speaker 1: calculus of variations. Newton's demonstrations, there are two of them,
Speaker 1: have been recently discovered in the Portsmouth collection. The fourth
Speaker 1: section is devoted to a spiral motion in a resisting medium,
Speaker 1: the fifth to the theory of hydrostatics and elastic fluids,
Speaker 1: the sixth to the motion of pendulums in a resisting medium,
Speaker 1: the seventh to hydrodynamics, especially to the motion of projectiles
Speaker 1: in air and other fluids. The eighth to the theory
Speaker 1: of waves, including the principles from which the chief effects
Speaker 1: of the wave hypotheses in light and sound are calculated,
Speaker 1: and in particular the velocity of sound is determined. In
Speaker 1: a ninth section, Newton discusses the Cartesian theory of vortices.
Speaker 1: He begins by shewing that if there were no internal friction,
Speaker 1: the motion would be impossible. He must therefore assume some
Speaker 1: law of friction, and as a working hypothesis he supposes
Speaker 1: that the resistance arising from want of lubricity in the
Speaker 1: parts of a fluid is catius peribus proportional to the
Speaker 1: velocity of which the parts of the fluid are separated
Speaker 1: from each other. This hypothesis, as he himself remarks, is
Speaker 1: probably not altogether correct, but he thinks that it will
Speaker 1: give a general idea of the motion. He next proves
Speaker 1: that on this hypothesis that the motion would be unstable.
Speaker 1: He must therefore suppose that some constraining force prevents this catastrophe,
Speaker 1: and he then chows that in that case, Kepler's third
Speaker 1: law could not be true. Lastly, he chows, by independent
Speaker 1: reasoning that the hypothesis must lead to results which are
Speaker 1: inconsistent with Kepler's other two laws, and that both the
Speaker 1: vortices and the motion of the planets would necessarily be unstable.
Speaker 1: Several continental mathematicians made attempts to modify the Cartesian hypothesis
Speaker 1: so as to avoid these conclusions, but they could never
Speaker 1: explain one phenomenon without introducing fresh difficulties. It may be
Speaker 1: taken that by seventeen fifty the Cartesian theory was finally abandoned.
Speaker 1: The third book is headed on the system of the world,
Speaker 1: and is concerned chiefly with the application of the results
Speaker 1: of the first book to the Solar System. It is
Speaker 1: introduced by certain rules of philosophizing and as a list
Speaker 1: of certain data obtained from astronomical observations. The rules are one,
Speaker 1: we may only assume as the possible causes of phenomenon,
Speaker 1: such causes as are sufficient to explain them, and are
Speaker 1: also verai kausi, a vera causa being one which is
Speaker 1: capable of detection and such that its connection with the
Speaker 1: phenomenon can be ultimate shewn by independent evidence. Two effects
Speaker 1: of a similar kind must have similar causes. Three, whatever
Speaker 1: properties of bodies are found by experience to be invariable
Speaker 1: should be assumed to be so in places where direct
Speaker 1: experiments cannot be made. Newton commences by illustrating the universality
Speaker 1: of the law of gravitation and sketches out the principles
Speaker 1: which lead him to think that the Solar system is
Speaker 1: necessarily stable. He determines that the mass of the Moon,
Speaker 1: the masses of the planets, their distances from the Sun,
Speaker 1: and their figures. In the first edition, he estimated proposition
Speaker 1: thirty seven that the ratio of the mass of the
Speaker 1: Moon to that of the Earth was approximately that of
Speaker 1: one to twenty six. In the second and third editions
Speaker 1: this was altered to a ratio which is nearly that
Speaker 1: of one in forty. But except for the mass of
Speaker 1: the moon, he approximates to the results now known with
Speaker 1: astonishing closeness. He finds the disturbing force exerted by the
Speaker 1: Sun and the Moon, and considers the five chief irregularities
Speaker 1: in the orbit of the Moon. He next discusses the
Speaker 1: solar and lunar tides, determines the precession of the equinoxes,
Speaker 1: and finally shows how the elements of a comet can
Speaker 1: be determined by these observations, and applies his results to
Speaker 1: certain comets. Before this time it had been commonly believed
Speaker 1: that the comets had nothing to do with the Solar system,
Speaker 1: though in sixteen eighty one d'orfel had shewn that the
Speaker 1: path of the Great Comet of sixteen eighty was a
Speaker 1: parabola having the Sun at its focus. Lastly, the precipia
Speaker 1: is concluded by a general scolium containing the reflection on
Speaker 1: the constitution of the universe and on the eternal, the
Speaker 1: infinite and perfect being by whom it is governed. The
Speaker 1: chief alterations in the second edition, published in seventeen thirteen
Speaker 1: were the substitutions of simpler proofs for some of the
Speaker 1: propositions in the second edition of the first book, a
Speaker 1: more full and accurate investigation found on some fresh experiments
Speaker 1: made by Newton about the year sixteen ninety and the
Speaker 1: resistance of fluids in the seventh section of the second book,
Speaker 1: and the addition of a detailed examination of the causes
Speaker 1: of the precession of the equinoxes and the theory of
Speaker 1: comets in the third book. The chief alterations in the
Speaker 1: third edition, published in seventeen twenty six were in the
Speaker 1: Scholium of Inflections and the edition of a new Scolium
Speaker 1: on the Motion of the Moon's nodes, Book three, Proposition
Speaker 1: fifty three, end of footnote. The demonstrations throughout the work
Speaker 1: are geometrical, but to readers of ordinary ability are rendered
Speaker 1: unnecessarily difficult by the absence of illustrations and explanations, by
Speaker 1: the fact that no clue is given to the method
Speaker 1: by which Newton arrived at his results. The reason why
Speaker 1: it was presented in a geometrical form appears to have
Speaker 1: been that the infinitetesimal calculus was then unknown, and had
Speaker 1: Newton used it to determine the results, which were in
Speaker 1: themselves opposed to the prevalent philosophy of the time, the
Speaker 1: controversy as to the truth of his results would have
Speaker 1: been hampered by a dispute concerning the validity of the
Speaker 1: methods used in proving them. He therefore cast the whole
Speaker 1: reasoning into a geometrical shape, which, if somewhat longer, can
Speaker 1: at any rate be made intelligible to all mathematical students.
Speaker 1: So closely did he follow the lines of Greek geometry
Speaker 1: that he constantly used graphical methods and represented forces, velocities,
Speaker 1: and other magnitude in the Euclidean way by straight lines
Speaker 1: Book one lemma En, and not by certain number of units.
Speaker 1: The latter and modern method had been introduced by Wallace
Speaker 1: and must have been familiar to Newton. The effect of
Speaker 1: his confining himself rigorously to classical geometry is that the
Speaker 1: Principia is written in a language which is archaic, even
Speaker 1: if not unfamiliar. The adoption of geometrical methods in the
Speaker 1: Principia for purposes of demonstration does not indicate a preference
Speaker 1: on Newton's part for geometry over analysis as an instrument
Speaker 1: of research. For it is known now that Newton used
Speaker 1: the fluctional calculus in the first instance in finding some
Speaker 1: of the theorems, especially those towards the end of Book one
Speaker 1: and in Book two, And in fact one of the
Speaker 1: most important uses of that calculus is stated in Book
Speaker 1: two Lemma I. But it is only just to remark
Speaker 1: that at the time of its publication, and for nearly
Speaker 1: a century afterwards, the differential and fluctional calculus were not
Speaker 1: fully developed and did not possess the same superiority or
Speaker 1: over the method he adopted by which they do now.
Speaker 1: And it is a matter of astonishment that when Newton
Speaker 1: did employ the calculus, he was able to use it
Speaker 1: to so good an effect. The ability shewn in the
Speaker 1: translation in a few months of theorems so numerous and
Speaker 1: of so great complexity into the language of the geometry
Speaker 1: of our Chimytes and Apollonius is I suppose unparalleled in
Speaker 1: the history of mathematics. The printing of the work was slow,
Speaker 1: and it was not finally published till the summer of
Speaker 1: sixteen eighty seven. The whole cost was borne by Halle,
Speaker 1: who also corrected the proofs and even put his own
Speaker 1: researches on one side to press the printing forward. The conciseness,
Speaker 1: absence of illustrations, and synthetical character of the book restricted
Speaker 1: the numbers of those who were able to appreciate its value,
Speaker 1: And though nearly all competent critics admitted the validity of
Speaker 1: the conclusions, so little time elapse before it affected the
Speaker 1: current beliefs of educated men. I should be inclined to say,
Speaker 1: but on this point opinions differ widely that within ten
Speaker 1: years of its publication it was generally accepted in Britain
Speaker 1: as giving a correct account of the laws of the universe.
Speaker 1: It was similarly accepted within about twenty years on the continent,
Speaker 1: except in France, where the Cartesian hypothesis held its ground
Speaker 1: until Voltaire in seventeen thirty eight took up the advocacy
Speaker 1: of the Newtonian theory. The manuscript of the Principia was
Speaker 1: finished by sixteen eighty six. Newton devoted the remainder of
Speaker 1: that year to his paper on Physical Optics, the greater
Speaker 1: part of which is given up to the subject of diffraction.
Speaker 1: In sixteen eighty seven, James the Second having tried to
Speaker 1: force the university to admit as a Master of Arts,
Speaker 1: a Roman Catholic priest who refused to take oaths of
Speaker 1: supremacy and allegiance. Newton took a prominent part in resisting
Speaker 1: the illegal interference of the King, and was one of
Speaker 1: the deputation sent to London to protect the rights of
Speaker 1: the university. The act of partaken by Newton in this
Speaker 1: affair led to his being in sixteen eighty nine elected
Speaker 1: member for the University. This parliament only lasted thirteen months,
Speaker 1: and on its disillusion he gave up his seat. He
Speaker 1: was subsequently returned in seventeen oh one, but he never
Speaker 1: took any prominent part in politics. On his coming back
Speaker 1: to Cambridge in sixteen ninety he resumed his mathematical studies
Speaker 1: and correspondence. If he lectured at this time, which is doubtful,
Speaker 1: it was on the subject matter of the Principia. The
Speaker 1: two letters to Wallace which he explained his methods of
Speaker 1: Fluxion's influence, which were written in sixteen ninety two and
Speaker 1: published in sixteen ninety three. Towards the close of sixteen
Speaker 1: ninety two and throughout the two following year years Newton
Speaker 1: had a long illness, suffering from insomnia and general nervous irritability.
Speaker 1: Perhaps he never quite regained his elasticity of mind, and
Speaker 1: though after his recovery he chewed the same power in
Speaker 1: solving any question propounded to him, he ceased there thenceforward
Speaker 1: to do original work of his own initiative, and it
Speaker 1: was somewhat difficult to stir him to activity in new subjects.
Speaker 1: In sixteen ninety four, Newton began to collect data connected
Speaker 1: with the irregularities of the moon's motion, with the view
Speaker 1: of revising the part of the Principia which dealt with
Speaker 1: that subject to render the observations more accurate. He forwarded
Speaker 1: to Flamstead a table of corrections for refraction which he
Speaker 1: had previously made. This was not published till seventeen twenty one,
Speaker 1: when Halley communicated it to the Royal Society. The original
Speaker 1: calculations of Newton and the papers connected with it, or
Speaker 1: in the Portsmouth Collection, enschw that Newton attained it by
Speaker 1: finding the path of array by means of quadratures, in
Speaker 1: a manner equivalent to the solution of a differential equation.
Speaker 1: As an illustration of Newton's genius. I may mention that
Speaker 1: even as late as seventeen fifty four Euler failed to
Speaker 1: solve the same problem. In seventeen eighty two, Laplace gave
Speaker 1: a rule for constructing such a table, and his results
Speaker 1: agree substantially with those of Newton. I do not suppose
Speaker 1: that Newton would in any case have produced much more
Speaker 1: original work after his illness. But his appointment in sixteen
Speaker 1: ninety six as warden and his promotion in sixteen ninety
Speaker 1: nine to the mastership of the Mint at a salary
Speaker 1: of one thousand, five hundred pounds a year, brought his
Speaker 1: scientific investigations to an end, though it was only after
Speaker 1: this that many of his previous investigations were published in
Speaker 1: the form of books. In sixteen ninety six he moved
Speaker 1: to London. In seventeen o one he resigned the Lacasian chair,
Speaker 1: and in seventeen o three he was elected President of
Speaker 1: the Royal Society. End of section twenty five. Recording by
Speaker 1: Paul King, Oakville, Ontario, p J K dot scripts dot
Speaker 1: M I T dot E d U forward slash p
Speaker 1: K J