Speaker 1: Chapter sixteen, Part three of A Short Account of the
Speaker 1: History of Mathematics. This is a LibriVox recording. All LibriVox
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Speaker 1: by Paul King p J K dot scripts dot mit
Speaker 1: dot ed U forward slash p k J. A Short
Speaker 1: Account of the History of Mathematics by W. W. Rowseball,
Speaker 1: Chapter sixteen, The Life and Works of Newton, Part three.
Speaker 1: In seventeen o four, Newton published his Optics, which contains
Speaker 1: the results of papers already mentioned the first edition of
Speaker 1: this book were appended to two minor works which have
Speaker 1: no special connection with Optics, one being on cubic curves,
Speaker 1: the other on the quadrature of curves and u on fluxions.
Speaker 1: Both of them were old manuscripts with which his friends
Speaker 1: and pupils were familiar, but they were here published ubi
Speaker 1: et orbi for the first time. The first of these
Speaker 1: appendices is entitled Umeratio linierm terti rorubdinis. The object seems
Speaker 1: to be to illustrate the use of analytical geometry, and
Speaker 1: as the application to conics was well known, Newton selected
Speaker 1: the theory of cubics. He begins with some general theorems
Speaker 1: and classifies curves according as to whether their equations are
Speaker 1: algebraical or transcendental, the former being cut by a straight
Speaker 1: line in a number of points real or imaginary equal
Speaker 1: to the degree of the curve, the latter being cut
Speaker 1: by a straight line and an infinite number of points.
Speaker 1: Newton then choose that many of the most important properties
Speaker 1: of conics have their analogs in the theory of cubics,
Speaker 1: and he discusses the theory of asymptotes and curve linear diameters.
Speaker 1: After these general theorems, he commences his detailed examination of
Speaker 1: cubics by pointing out that a cubic must have at
Speaker 1: least one real point at infinity. If the asymptote or
Speaker 1: tangent at this point be a finite distance, it may
Speaker 1: be taken for the axis of y. This asymptote will
Speaker 1: cut the curve in three points altogether, of which at
Speaker 1: least two are an infinity. If the third point be
Speaker 1: a finite distance, then by one of his general theorems
Speaker 1: on asymptotes, the equation can be written in the form
Speaker 1: of x y squared plus h y equals ax cubed
Speaker 1: plus b x squared plus c x plus d, where
Speaker 1: the axes of X and Y are the asymptotes of
Speaker 1: the hyperbola, which is the locus of the middle points
Speaker 1: of all chords drawn parallel to the axis of While
Speaker 1: if on the third point in which this asymptote cuts
Speaker 1: the curve be also at infinity, the equation can be
Speaker 1: written in the form x y equals ax cube plus
Speaker 1: b x squared plus c x plus d. Next, he
Speaker 1: takes the case where the tangent at the real point
Speaker 1: at infinity is not at a finite distance. A line
Speaker 1: parallel to the direction in which the curve goes to
Speaker 1: infinity may be taken as the axis of Y. Any
Speaker 1: such line will cut the curve in three points altogether,
Speaker 1: of which one is by hypothesis at infinity and one
Speaker 1: is necessarily at a finite distance. He then choose that
Speaker 1: if the remaining point in which this line cuts the
Speaker 1: curve be at a finite distance, the equation can be
Speaker 1: written in the form hy squared equals ax cube plus
Speaker 1: b x squared plus c x plus d, while if
Speaker 1: it be at an infinite distance, the equation can be
Speaker 1: written in the form why equals axqube plus bx squared
Speaker 1: plus cx plus d. Any cubic is therefore reducible to
Speaker 1: one of the four characteristic forms. Each of these forms
Speaker 1: is then discussed in detail, and the possibility of the
Speaker 1: existence of double points, isolated ovals, et cetera is worked out.
Speaker 1: The final result is that in all, there are seventy
Speaker 1: eight possible forms which a cubic may take. Of these,
Speaker 1: Newton enumerated only seventy two. Four of the remainder were
Speaker 1: mentioned by Sterling in seventeen seventeen, and one by Nicole
Speaker 1: in seventeen thirty one, and one by Nicholas Bernoulli at
Speaker 1: about the same time. In the course of the work,
Speaker 1: Newton states the remarkable theorem that just as the shadow
Speaker 1: of a circle cast by a luminous point on a
Speaker 1: plane gives rise to all the conics, so the shadows
Speaker 1: of the curves represented by the equation y squared equals
Speaker 1: ax cube plus bx squared plus c x plus d
Speaker 1: give rise to all the cubics. This remained an unsolved
Speaker 1: puzzle until seventeen thirty one, when Nicole and Clairot gave
Speaker 1: demonstrations of it. A better proof is that given by
Speaker 1: Murdoch in seventeen forty, which depends on the classification of
Speaker 1: these curves into five species, according to as to whether
Speaker 1: their points of intersection with the axis of X are
Speaker 1: real and unequal, real and two of them equal, two
Speaker 1: cases real and all equal, or finally two imaginary and
Speaker 1: one real. In this tract, Newton also discusses double points
Speaker 1: in the plane and at infinity, and the description of
Speaker 1: curves satisfying given conditions, and the graphical solution of pro
Speaker 1: problems by the use of curves. The second appendix to
Speaker 1: the Optics is entitled des Quadratura corvorum. Most of it
Speaker 1: had been communicated to Barrow in sixteen sixty eight or
Speaker 1: sixteen sixty nine, and probably was familiar to Newton's pupils
Speaker 1: and friends from that time onwards. It consists of two parts.
Speaker 1: The bulk of the first part is a statement of
Speaker 1: Newton's method of affecting the quadrature and the rectification of
Speaker 1: curves by means of infinite series. It is noticeable as
Speaker 1: containing the earliest use in print of literal indices, and
Speaker 1: also the first printed statement of the binomial theorem, but
Speaker 1: these are introduced only Incidentally, the main object is to
Speaker 1: give rules for developing a function of X in a
Speaker 1: series of ascending powers of X, so as to enable
Speaker 1: mathematicians to effect the quadrature of any curve in which
Speaker 1: the ordinate y can be expressed as an explicit algebraical
Speaker 1: function of the ebscissa X. Wallace had shewn how this
Speaker 1: quadrature could be found when y was given as a
Speaker 1: sum of a number of multiples of powers of X,
Speaker 1: and Newton's rules of expansion here established rendered possible the
Speaker 1: similar quadrature of any curve whose ordinate can be expressed
Speaker 1: as a sum of an infinite number of such terms.
Speaker 1: In this way, he affects the quadrature of the curves
Speaker 1: y equals a square divided by the quantity b plus x,
Speaker 1: Y equals the square root of a squared plus or
Speaker 1: minus x squared, or y equals the square root of
Speaker 1: x minus x squared, or y equals the square root
Speaker 1: of one minus a x squared divided by the square
Speaker 1: root of one minus b x squared. But the results
Speaker 1: are of course expressed as infinite series. He then proceeds
Speaker 1: to curves whose ordinate is given as an implicit function
Speaker 1: of the obsessor, and he gives a method by which
Speaker 1: y can be expressed as an infinite series and ascending
Speaker 1: powers of x. But the application of the rule to
Speaker 1: any curve demands, in general such complicated numerical calculations as
Speaker 1: to render it of little value. He concludes this part
Speaker 1: by shewing that the rectification of a curve can be
Speaker 1: effected in a somewhat similar way. His process is equivalent
Speaker 1: to finding the integral with regard to x of the
Speaker 1: square root of one plus y squared in the form
Speaker 1: of an infinite series. I should add that Newton indicates
Speaker 1: the importance of determining whether the series are convergent, an
Speaker 1: observation far in advance of his time, But he knew
Speaker 1: of no general test for the purpose, and in fact
Speaker 1: it was not until Gaussenkoschi took up the question that
Speaker 1: the necessity of such limitations were commonly recognized. The part
Speaker 1: of the appendix which I have just described is practically
Speaker 1: the same as Newton's manuscript Diannealisi periqueisiones numero to menora minfinitas,
Speaker 1: which was subsequently printed in seventeen eleven. It is said
Speaker 1: that this was originally intended to form an appendix to
Speaker 1: Kinkhousen's algebra. The substance of it was communicated to Brow
Speaker 1: and by him to Collins and letters of July thirty
Speaker 1: first and August twelfth, sixteen sixty nine, and a summary
Speaker 1: of part of it was included in a letter of
Speaker 1: October twenty fourth, sixteen seventy six, sent to Leibniz. It
Speaker 1: should be read in connection with Newton's methodis differentialis published
Speaker 1: in seventeen thirty six. Some additional theorems are there given,
Speaker 1: and he discusses his method of interpolation, which which had
Speaker 1: been briefly described in the letter of October twenty fourth,
Speaker 1: sixteen seventy six. The principle is this, if y equals
Speaker 1: fi x be a function of x, and if when
Speaker 1: x is successively put equal to a one, a, two,
Speaker 1: and so on, the values of y be known, and b,
Speaker 1: b one, b two, and so on, then a parabola
Speaker 1: whose equation is y equals p plus q x plus
Speaker 1: r x squared plus so on can be drawn through
Speaker 1: the points A one, b one, a two, b two,
Speaker 1: and so on, And the ordinate of this parabola may
Speaker 1: be taken as an approximation to the ordinate of the
Speaker 1: curve the degree of the parabola will, of course be
Speaker 1: one less than the number of the given points. Newton
Speaker 1: points out that in this way the areas of any
Speaker 1: curves can be approximately determined. The second part of this
Speaker 1: appendix to the optics, contained in a description of Newton's method,
Speaker 1: is fluction. This is best considered in connection with Newton's
Speaker 1: manuscript on the same subject, which was published by John
Speaker 1: Coulson in seventeen thirty six, and of which it is
Speaker 1: a summary. The fluctional calculus is one form of the
Speaker 1: infinitesimal calculus expressed in a certain notation, just as the
Speaker 1: differential calculus is another aspect of the same calculus expressed
Speaker 1: in a different notation. Newton assumed that all geometrical magnitudes
Speaker 1: must be conceived as generated by continuous motion. Thus a
Speaker 1: line may be considered as generated by the motion of
Speaker 1: a point, a surface by that of a line, a
Speaker 1: solid by that of a surface, a plane angle by
Speaker 1: the rotation of a line, and so on. The quantity
Speaker 1: thus generated was defined by him as the fluent or
Speaker 1: flowing quantity. The velocity of the moving magnitude was defined
Speaker 1: as the fluxion of the f fluent. This seems to
Speaker 1: be the earliest definite recognition of the idea of a
Speaker 1: continuous function, though it had been foreshadowed in some of
Speaker 1: Napier's papers. The following is a summary of Newton's treatment
Speaker 1: of fluctions. There are two kinds of problems. The object
Speaker 1: of the first is to find the fluxion of a
Speaker 1: given quantity, or more generally, the relation of the fluence
Speaker 1: being given, to find the relation of their fluctions. This
Speaker 1: is equivalent to differentiation. The object of the second, or
Speaker 1: inverse method of fluctions is from the fluxion or some
Speaker 1: relations involving it, to determine the fluent or more generally,
Speaker 1: an equation being proposed exhibiting the relation of the fluxions
Speaker 1: of quantities to find the relations of those quantities or
Speaker 1: fluence to one another. This is equivalent either to integration,
Speaker 1: which Newton termed the method of quadrature, or to the
Speaker 1: solution of a differential equation, which was called by Newton
Speaker 1: the inverse method of tangents. The methods for solving these
Speaker 1: problems are discussed at considerable length. Newton then went on
Speaker 1: to apply these results to questions connected with the maxima
Speaker 1: and minima of quantities. The method of drawing tangents to
Speaker 1: curves and the curvature of curves, namely the determination of
Speaker 1: the center of curvature, the radius of curvature, and the
Speaker 1: rate at which the radius of curvature increases. He next
Speaker 1: considered the quadrature of curves and the rectification of curves.
Speaker 1: In finding the maximum and minimum of functions of one variable,
Speaker 1: we regard the change of sign of the difference between
Speaker 1: two consecutive values of the function as the true criterion.
Speaker 1: But his argument is that when a quantity increasing has
Speaker 1: attained its maximum, it can have no further increment, or
Speaker 1: when decreasing, it has attained its minimum and can have
Speaker 1: no further decrement. Consequently, the fluction must be equal to nothing.
Speaker 1: It has been remarked that neither Newton nor Lobbyingits produced
Speaker 1: a calculus that is a classified collection of rules, and
Speaker 1: that the problems they discussed were treated from first principles.
Speaker 1: That no doubt is the usual sequence in the history
Speaker 1: of such discoveries, though the fact is frequently forgotten by
Speaker 1: subsequent writers. In this case, I think the statement, so
Speaker 1: far as Newton's treatment of the differential or fluctional part
Speaker 1: of the calculus is concerned, is incorrect, as the foregoing
Speaker 1: account sufficiently shoes. If a flowing quantity or fluent were
Speaker 1: represented by x, Newton denoted its fluxion by x dot,
Speaker 1: and the fluxions of x dot or second flunction of
Speaker 1: x by x dot dot, and so on. Similarly, the
Speaker 1: fluent of x was denoted by x written in a box,
Speaker 1: or sometimes by x prime or x inside square brackets.
Speaker 1: The infinitely small part by which a fluence such as
Speaker 1: x increased in a small interval of time measured by
Speaker 1: omicron was called the moment of the fluent, and its
Speaker 1: value was shown to be x dot omicron. Newton adds
Speaker 1: the important remark that thus we may, in any problem
Speaker 1: neglect the terms multiplied by the second and higher powers
Speaker 1: of omicron, and we can always find an equation between
Speaker 1: the coordinates of x y of a point on a
Speaker 1: curve and their fluxions x dot y dot. It is
Speaker 1: an application of this principle which constitutes one of the
Speaker 1: chief values of calculus. For if we desire to find
Speaker 1: the effect produced by several causes of a system, then
Speaker 1: when and if we can find the effect produced by
Speaker 1: each cause when acting alone in a very small time,
Speaker 1: the total effect produced in that time will be equal
Speaker 1: to the sum of the separate effects. I should here
Speaker 1: note the fact that Vince and other English writers in
Speaker 1: the eighteenth century used x dot to denote the increment
Speaker 1: of x, and not the velocity with which it increased.
Speaker 1: That is, x dot in their writing stands for what
Speaker 1: Newton would have expressed by x dot omicron, and what
Speaker 1: Leibnitz would have written as dx. I need not discuss
Speaker 1: in detail the manner in which Newton treated the problem
Speaker 1: above mentioned. I will only add that, in spite of
Speaker 1: the form of his definition, the introduction into geometry of
Speaker 1: the idea of time was evaded by supposing that some
Speaker 1: quantity e g. The absissa of a point on a
Speaker 1: curve increased equably, and the required results then depend on
Speaker 1: the rate at which other quantities e g. The ordinate
Speaker 1: or the radius of curvature increased relatively to the one
Speaker 1: so chosen. The fluence so chosen is what we now
Speaker 1: call the independent variable. Its fluction was termed the principal fluction,
Speaker 1: and of course if it were denoted by x, then
Speaker 1: x dot was constant, and consequently x dot dot equalled zero.
Speaker 1: There is no question that Newton used the method of
Speaker 1: Fluctions in sixteen sixty six, and it is practically certain
Speaker 1: that accounts of it were communicated in manuscript to friends
Speaker 1: and pupils from and after sixteen sixty nine. The manuscript
Speaker 1: from which most of the above summary has been taken
Speaker 1: is believed to have been written between sixteen seventy one
Speaker 1: and sixteen seventy seven, and to have been in circulation
Speaker 1: at Cambridge from that time onward. It was unfortunate that
Speaker 1: it was not published at once. Strangers at a distance
Speaker 1: naturally judged of the method by the letter to Wallace
Speaker 1: in sixteen ninety two or by the Tractatus did Quadratura crverum,
Speaker 1: and were not aware that it had been so completely
Speaker 1: developed at an earlier date. This was the cause of
Speaker 1: numerous misunderstandings. At the same time, it must be added
Speaker 1: that all mathematical analysis was leading up to the idea
Speaker 1: and methods of the infinitesimal calculus. Foreshadowings of the principles
Speaker 1: and even of the language of that calculus can be
Speaker 1: found in the writings of Napier, Kepler, Cavalieri, Pascal Fermat, Wallace,
Speaker 1: and Berrow. It was Newton's good luck to come at
Speaker 1: a time when everything was ripe for the discovery, and
Speaker 1: his ability enabled him to construct almost at once a
Speaker 1: complete calculus. The notation of the fluctional calculuses, for most
Speaker 1: purposes less convenient than that of the differential calculus. The
Speaker 1: latter was invented by Libaneates in sixteen seventy five and
Speaker 1: published in sixteen eighty four, some nine years before the
Speaker 1: earliest printed account of Newton's method of fluxis But the
Speaker 1: question whether the general idea of the calculus expressed in
Speaker 1: that notation was obtained by Libyaneths from Newton, or whether
Speaker 1: it was invented independently, gave rise to a long and
Speaker 1: bitter controversy. The leading facts are given in the next chapter.
Speaker 1: The question is one of considerable difficulty, But I will
Speaker 1: hear only say that from what I have read of
Speaker 1: the voluminous literature on the question, I think on the
Speaker 1: whole it points to the fact that Lebanets obtained the
Speaker 1: idea of the differential calculus from a manuscript of Newton's
Speaker 1: which he saw in sixteen seventy five. I believe, however,
Speaker 1: that the prevalent opinion is that the inventions were independent.
Speaker 1: The remaining events of Newton's life require little or no comment.
Speaker 1: In seventeen o five he was knighted. From this time onward,
Speaker 1: he devoted much of his leisure to theology, and wrote
Speaker 1: at great length on the prophecies and predictions, subjects which
Speaker 1: had always been of interest to him. His Universal arithmetic
Speaker 1: was published by Wiston in seventeen o seven, and his
Speaker 1: Analysis by Infinite Series in seventeen eleven, but Newton had
Speaker 1: nothing to do with the preparation of either these or
Speaker 1: the press. His evidence before the House of Commons in
Speaker 1: seventeen fourteen on the determination of longitude at sea marks
Speaker 1: an important epoch in the history of navigation. The dispute
Speaker 1: with Leibnitz as to whether he had derived the ideas
Speaker 1: of the differential calculus from Newton or invented it independently,
Speaker 1: originated about seventeen o eight and occupied much of Newton's time,
Speaker 1: especially between the years seventeen o nine and seventeen sixteen.
Speaker 1: In seventeen o nine, Newton was persuaded to allow Cotes
Speaker 1: to prepare the long talked of second edition of the Principia.
Speaker 1: It was issued in March of seventeen thirteen. A third
Speaker 1: edition was published in seventeen twenty six, under the direction
Speaker 1: of Henry Pemberton. In seventeen twenty five, Newton's health began
Speaker 1: to fail. He died on March twentieth, seventeen twenty seven,
Speaker 1: and eight days later was buried with great state in
Speaker 1: Westminster Abbey. His chief works, taking them in their order
Speaker 1: of publication, are the Principia published in sixteen eighty seven,
Speaker 1: The Optics with Appendices on cubic curves, the Quadrature and
Speaker 1: Rectification of Curves by use of the Infinite Series, and
Speaker 1: the Method of Fluxions published in seventeen four, and the
Speaker 1: Universal Arithmetic published in seventeen o seven. The Analysis per
Speaker 1: series Fluxions et cetera published in seventeen eleven, Leccioni's Optique
Speaker 1: published in seventeen twenty nine, Method of Fluxions et cetera
Speaker 1: I E. Newton's Manuscript on Fluxions, translated by J. Colson
Speaker 1: and published in seventeen thirty six, and the methodis differentialis
Speaker 1: also published in seventeen thirty six. In appearance, Newton was short, and,
Speaker 1: towards the close of his life, rather stout, but well set,
Speaker 1: with a square lower jaw, brown eyes and a very
Speaker 1: broad forehead and a rather sharp features. His hair turned
Speaker 1: gray before he was thirty, and remained thick and white
Speaker 1: and silver till his death. As to his manners, he
Speaker 1: dressed slovenly, was rather languid, and was often so absorbed
Speaker 1: in his own thoughts as to be any but a
Speaker 1: lively companion. Many anecdotes of his extreme absence of mind
Speaker 1: when engaged in any investigation have been preserved. Thus Once,
Speaker 1: when riding home from Granthem, he dismounted to lead his
Speaker 1: horse up a steep hill. When he turned at the
Speaker 1: top to remount, he found that he had the bridle
Speaker 1: in his hand, while his horse had slipped it and
Speaker 1: gone away again. On the few occasions when he sacrifices
Speaker 1: time to entertain his friends, if he left him to
Speaker 1: get more wine or for any similar reason, he would
Speaker 1: as often as not be found after the lapse of
Speaker 1: some time working out a problem. Oblivious alike of his
Speaker 1: expectant guests and of his errand, he took no exercise,
Speaker 1: indulged in no amusements, and worked incessantly, often spending eighteen
Speaker 1: or nineteen hours out of the twenty four in writing.
Speaker 1: In character, he was religious and conscientious, with an exceptionally
Speaker 1: high standard of morales, having, as Bishop BRUNEI said, the
Speaker 1: whitest soul he ever knew. Newton was always perfectly straightforward
Speaker 1: and honest, but in his controversies with Leibanez's Hook and others,
Speaker 1: though scrupulously just, he was not generous, and it would
Speaker 1: seem that he had frequently took offense at a chance
Speaker 1: of expression when none was intended. He modestly attributed his
Speaker 1: discoveries largely to the admirable work done by his predecessors,
Speaker 1: and once explained that if he had seen farther than
Speaker 1: other men, it was only because he stood on the
Speaker 1: shoulders of giants. He summed up his own estimate of
Speaker 1: his work in the sentence, I do not know what
Speaker 1: I may appear to the world, but to myself I
Speaker 1: seemed to have been only like a boy playing on
Speaker 1: the sea shore and diverting myself in now and then
Speaker 1: finding a smoother pebble or a prettier shell than ordinary,
Speaker 1: whilst the great ocean of truth lay all undiscovered before me.
Speaker 1: He was morbidly sensitive to being involved in any discussions.
Speaker 1: I believe that, with the exception of his papers on optics,
Speaker 1: every one of his works was published only under pressure
Speaker 1: from his friends and against his own wishes. There are
Speaker 1: several instances of his communicating papers and results on condition
Speaker 1: that his name should not be published. Thus, when in
Speaker 1: sixteen sixty nine he had, at Collier's requests, solved some
Speaker 1: problems on the harmonic series and on annuities which had
Speaker 1: previously baffled investigation, he only gave permission that his results
Speaker 1: should be published. So it be as he says, without
Speaker 1: my name to it. For I see not what there
Speaker 1: is desirable in public esteem. Were I able to acquire
Speaker 1: and maintain it, it would perhaps increase my acquaintance the
Speaker 1: thing which I chiefly studied, to decline in intellect. He
Speaker 1: has never been surpassed, and probably never been equalled. Of this.
Speaker 1: His extant works are the only proper test. Perhaps the
Speaker 1: most wonderful single illustration of his powers was the composition
Speaker 1: in seven months of the first book of the Principia.
Speaker 1: As specific illustrations of his ability, I may mention his
Speaker 1: solutions of the problems of Pappus, of John Bernoulli's challenge,
Speaker 1: and of the question of orthogonal trajectories. The problem of
Speaker 1: Pappus is to find the locus of a point such
Speaker 1: that the rectangle under its distances from two different straight
Speaker 1: lines shall be in a given ratio to the rectangle
Speaker 1: under its distances from two other given straight lines. Many
Speaker 1: geometricians from the time of Apollonius had tried to find
Speaker 1: a geometrical solution and had failed. But what had proved
Speaker 1: insuperabowl to his predecessors seemed to have presented life little
Speaker 1: difficulty to Newton, who gave an elegant demonstration as that
Speaker 1: the locust was a conic geometry, said Lagrange was when
Speaker 1: recommending the study of analysis to his pupils. Is a
Speaker 1: strong bow, but it is one which only a Newton
Speaker 1: could fully utilize. As another example, I may mention that
Speaker 1: in sixteen ninety six John Bernoulli challenged mathematicians one to
Speaker 1: determine the brachistochrone, and two to find a curve such
Speaker 1: that if any line drawn from a fixed point O
Speaker 1: cut it in P and Q, then O P to
Speaker 1: the power n plus q o q to the power
Speaker 1: n would be a constant leiben It solved the first
Speaker 1: of these questions after an interval of rather more than
Speaker 1: six months, and then suggested that they should be sent
Speaker 1: as a challenge to Newton and others. Newton received the
Speaker 1: problems on January twenty ninth, seen ninety seven, and the
Speaker 1: next day gave the complete solutions of both, at the
Speaker 1: same time generalizing the second question. An almost exactly similar
Speaker 1: case occurred in seventeen sixteen, when Newton was asked to
Speaker 1: find the orthogonal trajectory of a family of curves. In
Speaker 1: five hours, Newton solved the problem in the form in
Speaker 1: which it was propounded to him, and laid down in
Speaker 1: the principles for finding trajectories. It is almost impossible to
Speaker 1: describe the effect of Newton's writings without being suspected of exaggeration.
Speaker 1: But if the state of mathematical knowledge in sixteen sixty
Speaker 1: nine or at the death of pascalar fermat be compared
Speaker 1: with what was known in sixteen eighty seven, it will
Speaker 1: be see how immense was the advance. In fact, we
Speaker 1: may say that it took mathematicians half a century or
Speaker 1: more before they were able to assimilate the work with
Speaker 1: which Newton produced in those twenty years sure geometry. Newton
Speaker 1: did not establish any new methods, but no modern writer
Speaker 1: has shewn the same power in using those of classical geometry.
Speaker 1: In algebra and the theory of equations, he introduced a
Speaker 1: system of literal indices, established the binomial theorem, created no
Speaker 1: inconsiderable part of the theory of equations. One rule which
Speaker 1: he enunciated in this subject remained till a few years ago,
Speaker 1: an unsolved riddle which had overtaxed the resources of succeeding mathematicians.
Speaker 1: In analytical geometry, he introduced the modern classification of curves
Speaker 1: into the algebraical and transcendental, and established many of the
Speaker 1: fundamental properties of asymptotes, multiple points, and isolated loops, illustrated
Speaker 1: by a discussion of cubic curves. The fluxitional or infinitesimal
Speaker 1: calculus was invented by Newton in or before the year
Speaker 1: sixteen sixty seve and circulated in manuscripts amongst his friends
Speaker 1: in and after the year sixteen sixty nine, though no
Speaker 1: account of the method was printed till sixteen ninety three.
Speaker 1: The fact that the results are nowadays expressed in a
Speaker 1: different notation has led to Newton's investigations on this subject
Speaker 1: being somewhat overlooked. Newton further was the first to place
Speaker 1: dynamics on a satisfactory basis, and from dynamics he deduced
Speaker 1: the theory of statics. This was in the introduction to
Speaker 1: the Principia, published in sixteen eighty seven. The theory of attractions,
Speaker 1: the application of the principles of mechanics to the solar system,
Speaker 1: the creation of physical astronomy, and the establishment of the
Speaker 1: law of universal gravitation are wholly due to him, and
Speaker 1: were first published in the same work. The particular questions
Speaker 1: connected with the motion of the Earth and the Moon
Speaker 1: were worked out as fully as was then possible. The
Speaker 1: theory of hydrodynamics was created in the second book of
Speaker 1: the Principia, and he added considerably to the theory of hydrostatics,
Speaker 1: which may be said to have been first discussed by Pascal.
Speaker 1: The theory of the propagation of waves, and in particular
Speaker 1: the application to determine the velocity of sound, is due
Speaker 1: to Newton, and was published in sixteen eighty seven. In
Speaker 1: geometrical optics, he explained, amongst other things, the decomposition of
Speaker 1: light and the theory of the rainbow. He invented the
Speaker 1: reflecting telescope known by his name, and the sextant. In
Speaker 1: physical optics he suggested and elaborated the emission theory of light.
Speaker 1: The above list does not exhaust the subjects he investigated,
Speaker 1: but it will serve to illustrate how marked was his
Speaker 1: influence on the history of mathematics, on his writings, and
Speaker 1: on their effects. It will be enough to quote the
Speaker 1: remarks of two or three of those who were subsequently
Speaker 1: concerned with the subject matter of the Principia. Lagrange described
Speaker 1: the Principia as the greatest production of the human mind,
Speaker 1: and said that he felt dazed that such an illustration
Speaker 1: of what a man's intellect might be capable. In describing
Speaker 1: the effect of his own writings and those of la Place,
Speaker 1: it was a favorite remark of his that Newton was
Speaker 1: not only the greatest genius that had ever existed, but
Speaker 1: he was also the most fortunate. For as there is
Speaker 1: but one universe, it can happen but to one man
Speaker 1: in the world's history to be the interpreter of its laws. Laplace,
Speaker 1: who is in general very sparing of his praise, makes
Speaker 1: of Newton the one exception and the words in which
Speaker 1: he enumerates the causes which will always assure the Principias
Speaker 1: of pre eminence above all other productions of the human
Speaker 1: intellect have often been quoted. Not less remarkable is the
Speaker 1: homage rendered by Gauss. For other great mathematicians or philosophers,
Speaker 1: he used the epithets magnus or clarus or clarissimus. For
Speaker 1: Newton alone, he kept the prefix summus. Finally, Billott, who
Speaker 1: had made a special study of Newton's works, sums up
Speaker 1: his remarks by saying, come geometrie e comic experimental ternuton
Speaker 1: sanzegale paleri de sidu genre de genipe eller plo ere
Speaker 1: sons exemple end of section twenty six. Recording by Paul King, Oakville, Ontario,
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Speaker 1: e ed U forward slash p K J