Speaker 1: Chapter fifteen, 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: A short account of the history of mathematics by W. W. Rowseball.
Speaker 1: History of Mathematics from Descartes to Huygens circa sixteen thirty
Speaker 1: five to sixteen seventy five, Part three. Huygens Christian Huygens
Speaker 1: was born at the Hague on April fourteenth, sixteen twenty nine,
Speaker 1: and died in the same town on June eighth, sixteen
Speaker 1: ninety five. He generally wrote his name is Hougens h
Speaker 1: u g E N S, but I follow the usual
Speaker 1: cust In spelling it as Huygens. It is also sometimes
Speaker 1: written as Huyghens. His life was uneventful and is a
Speaker 1: mere record of the dates of his various works. In
Speaker 1: sixteen fifty one he published an essay in which he
Speaker 1: shewed the fallacy in a system of quadratures. Proposed by
Speaker 1: Gregois de Saint Veisant, who was well versed in the
Speaker 1: geometry of the Greeks, but had not grasped the essential
Speaker 1: points in the more modern methods. This essay was followed
Speaker 1: by tracts on the quadrature of the conics and the
Speaker 1: approximate rectification of the circle. In sixteen fifty four, his
Speaker 1: attention was directed to the improvement of the telescope. In
Speaker 1: conjunction with his brother, he devised a new and better
Speaker 1: way of grinding and polishing lenses. As a result of
Speaker 1: these improvements, he was able during the following two years
Speaker 1: sixteen fifty five and sixteen fifty six to resolve numerous
Speaker 1: astronomical questions, as, for example, the nature of Saturn's appendage.
Speaker 1: His astronomical observations required some exact means of measuring time,
Speaker 1: and he was thus led in sixteen fifty six to
Speaker 1: invent the pendulum clock, as described in his tract on
Speaker 1: Holurogium sixteen fifty eight. The time pieces previously in use
Speaker 1: had been balanced clocks. In the year sixteen fifty seven,
Speaker 1: Huygens wrote a small work on the calculus of probabilities,
Speaker 1: founded on the correspondence of Pascal and Format. He spent
Speaker 1: a couple of years in England. About this time. His
Speaker 1: reputation was now so great that in sixteen sixty five
Speaker 1: Louis the fourteenth offered him a pension if he would
Speaker 1: live in Paris, which accordingly then became his place of residence.
Speaker 1: In sixteen sixty eight he sent to the Royal Society
Speaker 1: of London in answer to a problem. They had proposed
Speaker 1: a memoir in which, simultaneously with Wallace and Wren, he
Speaker 1: proved by experiment that the momentum in a certain direction
Speaker 1: before the collision of two bodies is equal to the
Speaker 1: momentum in that direction after the collision. This was one
Speaker 1: of the points in mechanics on which Descartes had been mistaken.
Speaker 1: The most important of Huygens's work was in his Horologium Oscilatorium,
Speaker 1: published in Paris in sixteen seventy three. The first chapter
Speaker 1: is devoted to the pendulum clocks. The second chapter contains
Speaker 1: a complete account of the descent of heavy bodies under
Speaker 1: their own weights in a vacuum, either vertically down or
Speaker 1: on smooth curves. The second chapter contains a complete account
Speaker 1: of the us the descent of heavy bodies under their
Speaker 1: own weights in a vacuum, either vertically down or on
Speaker 1: smooth curves. Amongst other propositions, he shows that the cycloid
Speaker 1: is tautochronous. In the third chapter, he defines evolutes and involutes,
Speaker 1: proves some of their more elementary properties, and illustrates his
Speaker 1: methods by finding the evolutes of the cycloid and the parabola.
Speaker 1: These are the earliest instances in which the envelope of
Speaker 1: a moving line is determined. In the fourth chapter, he
Speaker 1: solves the problem of the compound pendulum and chows that
Speaker 1: the centers of oscillation and suspension are interchangeable. In the
Speaker 1: fifth and last chapter, he discusses again the theory of clocks,
Speaker 1: points out that if the bob of the pendulum were
Speaker 1: made by means of cycloidal checks to oscillate in a cycloid,
Speaker 1: the oscillations would be isochronous, and finishes by shewing that
Speaker 1: the centrifia force on a body which moves in a
Speaker 1: circle of radius R with uniform velocity V varies directly
Speaker 1: as V squared and inversely as R. This work contains
Speaker 1: the first attempt to apply dynamics to bodies of finite size,
Speaker 1: and not merely to particles. In sixteen seventy five, Huygens
Speaker 1: proposed to regulate the motion of watches by use of
Speaker 1: the balanced spring, in the theory of which he had
Speaker 1: been perhaps anticipated in a somewhat ambiguous and incomplete statement
Speaker 1: made by Hook in sixteen fifty eight. Watches or portable
Speaker 1: clocks had been invented early in the sixteenth century, and
Speaker 1: by the end of that century were not very uncommon,
Speaker 1: but they were clumsy and unreliable, being driven by a
Speaker 1: main spring and regulated by a conical pulley and verge escapement. Moreover,
Speaker 1: until sixteen eighty seven, they had only one hand. The
Speaker 1: first watch whose motion was regulated by a balanced spring
Speaker 1: was made at Paris under Huygen's directions and presented by
Speaker 1: him to Louis the fourteenth The increasing intolerance of the
Speaker 1: Catholics led to his return to Holland in sixteen eighty one,
Speaker 1: and after the revocation of the Edict of Nantes, he
Speaker 1: refused to hold any further communication with France. He now
Speaker 1: devoted himself to the construction of lenses of enormous focal length.
Speaker 1: Of these three of focal length one hundred twenty three
Speaker 1: feet one hundred eighty feet and two hundred ten feet
Speaker 1: was subsequently given by him to the Royal Society in London,
Speaker 1: in whose possession they still remain. It was about this
Speaker 1: time that he discovered the achromatic eyepiece for a telescope,
Speaker 1: which is known by his name. In sixteen eighty nine
Speaker 1: he came from Holland to England in order to make
Speaker 1: the acquaintance of Newton, whose Principia had been published in
Speaker 1: sixteen eighty seven, the extraordinary merits of which Huygens had
Speaker 1: at once recognized. On his return. In sixteen ninety Huygens
Speaker 1: published as Treatise on Light, in which the undulatory theory
Speaker 1: was expounded and explained. Most of this had been written
Speaker 1: as early as sixteen seventy eight. The general idea of
Speaker 1: the theory had been suggested by Robert Hook in sixteen
Speaker 1: sixty four, but he had not investigated its consequences in
Speaker 1: any detail. This publication falls outside the years considered in
Speaker 1: this chapter, but here it may be briefly said that,
Speaker 1: according to the wave or undulatory theory, space is filled
Speaker 1: with an extremely rare ether, and light is caused by
Speaker 1: a series of waves or vibrations in this ether, which
Speaker 1: are set in motion by the pulsations of the luminous body.
Speaker 1: From this hypothesis, Huygens deduced that the laws of reflection
Speaker 1: and refraction explained the phenomenon a double refraction, and gave
Speaker 1: a construction for the extraordinary ray in biaxial crystals. While
Speaker 1: he found by experiment the chief phenomenon of polarization, the
Speaker 1: immense reputation and unrivaled powers of Newton led to the
Speaker 1: disbelief in the theory, which he rejected, and to the
Speaker 1: general adoption of Newton's emission theory. But it should be
Speaker 1: noted that Huygen's explanation of some phenomenon, such as the
Speaker 1: colors of thin plates, was inconsistent with the results of experiments,
Speaker 1: Nor was it until Young and Wallaston at the beginning
Speaker 1: of this century revived the undulatory theory and modified some
Speaker 1: of its details, and Fresnel elaborated their views that its
Speaker 1: acceptance could be fully justified. Besides these works, Huygens took
Speaker 1: part in most of the contrary verses and challenges which
Speaker 1: then played so large a part in the mathematical world,
Speaker 1: and wrote several minor tracts in one of these, he
Speaker 1: investigated the form and properties of the caternary. In another,
Speaker 1: he stated in general terms the rule for finding maxima
Speaker 1: and minima, of which format had made use, and shewed
Speaker 1: that the subtangent of an algebraical curve fx y equal
Speaker 1: zero was equal to y f y by fx, where
Speaker 1: f y is the derivative of fx y, regarded as
Speaker 1: a function of y. In some posthumous works issued at
Speaker 1: Leyden in seventeen o three, he further shewed how from
Speaker 1: the focal lengths of the component lenses the magnifying power
Speaker 1: of a telescope could be determined, and explains some of
Speaker 1: the phenomenon connected with halos and parhelia. I should add
Speaker 1: that almost all of its demonstrations, like those of Newton,
Speaker 1: are rigidly geometrical, and he would seem to have made
Speaker 1: no no use of the differential or fluctional calculus, though
Speaker 1: he admitted the validity of the methods used therein. Thus,
Speaker 1: even when first written, his works were expressed in an
Speaker 1: archaic language and perhaps received less attention than their intrinsic
Speaker 1: merits deserved. I have now traced the development of mathematics
Speaker 1: for a period which we may take roughly as dating
Speaker 1: from sixteen thirty five to sixteen seventy five, under the
Speaker 1: influence of Descartes, Cavalieri, Pascal Wallace, Fermat and Huygens. The
Speaker 1: life of Newton partly overlaps this period. His works and
Speaker 1: influence are considered in the next chapter. I may dismiss
Speaker 1: the remaining mathematicians of this time, whom I dare to
Speaker 1: mention with comparatively slight notice. The following is an alphabetical
Speaker 1: list of the more remarkable among them. The dates given
Speaker 1: to those of the birth and death of the mathematician,
Speaker 1: to who whose name they are appended. Bauchet fifteen eighty
Speaker 1: one to sixteen thirty eight, Barrow sixteen thirty to sixteen
Speaker 1: seventy seven, Brokner sixteen twenty to sixteen eighty four, Collins
Speaker 1: sixteen twenty five to sixteen eighty three, Coursier sixteen oh
Speaker 1: four to sixteen ninety two, Des Bonnes sixteen oh one
Speaker 1: to sixteen fifty two, De la Lubert sixteen hundred to
Speaker 1: sixteen sixty four, Frenikre sixteen o five to sixteen seventy,
Speaker 1: Gregory sixteen thirty eight to sixteen seventy five, Hook sixteen
Speaker 1: thirty five to seventeen oh three, Hood sixteen thirty three
Speaker 1: to seventeen oh four, kink Housens sixteen thirty to sixteen
Speaker 1: seventy nine, Mercadix sixteen twenty to sixteen eighty seven, Marsenne
Speaker 1: fifteen eighty eight to sixteen forty eight, Midorge fifteen eighty
Speaker 1: five to sixteen forty seven, Pell sixteen ten to sixteen
Speaker 1: eighty five, Richie sixteen nineteen to sixteen ninety two, Roberval
Speaker 1: sixteen oh two to sixteen seventy five, Rohmer sixteen forty
Speaker 1: four to seventeen ten, Saint Vincent fifteen eighty four to
Speaker 1: sixteen sixty seven, Sleuz sixteen twenty two to sixteen eighty five,
Speaker 1: Torricelli sixteen oh eight to sixteen forty seven, Chernhausen sixteen
Speaker 1: thirty one to seventeen oh eight, Van Schutten died in
Speaker 1: sixteen sixty one, and Renn sixteen thirty two to seventeen
Speaker 1: twenty three. In the following notes, I have arranged the
Speaker 1: above mentioned mathematicians so that, as far as possible, their
Speaker 1: chief contributions shall come in chronological order. Bachet Claude Gaspar
Speaker 1: Baluchet de mess Iraq was born in Bourg in fifteen
Speaker 1: eighty one and died in sixteen thirty eight. He wrote
Speaker 1: the problem pleisante sixteen twelve, second and enlarged editions sixteen
Speaker 1: twenty four, which contains an interesting collection of arithmetical tricks
Speaker 1: and questions, many of which are quoted in chapter one
Speaker 1: of My Mathematical Recreations and Problems. Also lesali Monte Arithmetique,
Speaker 1: which exists in manuscript, and a translation of arithmetic by Diafontis.
Speaker 1: Bachet was the earliest writer who discussed a solution of
Speaker 1: indetermined equations by means of continued fractions. Midorge Claude Mid'orge,
Speaker 1: born in Paris in fifteen eight eighty five and died
Speaker 1: in sixteen forty seven, belonged to a distinguished family of
Speaker 1: the Robe, and was himself a councilor at Chatellers, then
Speaker 1: treasurer to the local parliament at Amiens. He published some
Speaker 1: works on optics, of which one issued in sixteen thirty
Speaker 1: one as extant, and in sixteen forty one a treatise
Speaker 1: and conic sections. He also left a manuscript containing solutions
Speaker 1: of over one thousand geometrical problems, many of which are
Speaker 1: said to be ingenious. The annunciations were published by M.
Speaker 1: Charles Henry in eighteen eighty two. Mersennes Marin Merceinne born
Speaker 1: in fifteen eighty eight and died in Paris in sixteen
Speaker 1: forty eight was a Franciscan friar who made it his
Speaker 1: business to be acquainted and correspond with the French mathematicians
Speaker 1: of that date and many of their foreign contemporaries. In
Speaker 1: sixteen thirty four he published a translation of Galileo's Mechanics.
Speaker 1: In Si sixteen forty four he issued is Kojitata Physico Mathematica,
Speaker 1: by which he is best known, containing an account of
Speaker 1: some experiments in physics. He also wrote a synopsis of mathematics,
Speaker 1: which was printed in sixteen sixty four. The preface to
Speaker 1: the Cogitata contains a statement, probably due to fermat that
Speaker 1: in order that two P minus one may be prime,
Speaker 1: the only values of P not greater than two hundred
Speaker 1: and fifty seven which are possible are one, two, three, five, seven, thirteen, seventeen,
Speaker 1: nineteen thirty one, sixty seven, one hundred twenty seven, and
Speaker 1: two hundred fifty seven, to which list her Silhoff has shewn,
Speaker 1: we must add the number sixty one. With this addition,
Speaker 1: the statement has been verified for all except twenty three
Speaker 1: values of P, namely sixty one, seventy one, eighty nine,
Speaker 1: one hundred and one, one hundred and three, one hundred
Speaker 1: and seven, one hundred and nine, one hundred twenty seven,
Speaker 1: one hundred thirty seven, one hundred thirty nine, one hundred
Speaker 1: forty nine, one hundred fifty seven, one hundred sixty three,
Speaker 1: one hundred sixty seven, one hundred seventy three, one hundred
Speaker 1: eighty one, one ninety three, one ninety seven, one ninety nine,
Speaker 1: two twenty seven, two twenty nine, two forty one, and
Speaker 1: two fifty seven. Of these values, Mersenne asserted that P
Speaker 1: equals sixty seven, P equals one twenty seven, and P
Speaker 1: equals two fifty seven make two to the exponent P
Speaker 1: minus one a prime, and that the other values make
Speaker 1: two P minus one a composite number. It is most
Speaker 1: likely that these results are particular cases of some general
Speaker 1: theorem on the subject, which remains to be discovered. The
Speaker 1: number two to the power sixty one minus one contains
Speaker 1: nineteen digits and is the highest number at present known
Speaker 1: to be prime. Its value in digits is two three, o,
Speaker 1: five eight, four, three, zero, zero nine two one three
Speaker 1: six nine, three nine five one. The theory of perfect
Speaker 1: numbers depends directly on that of Mersenne's numbers. It is
Speaker 1: probable that all perfect numbers are included in the formula
Speaker 1: two to the power of P minus one, as opposed
Speaker 1: to negative one plus two to the P. Where negative
Speaker 1: one plus two to the P is prime. You could
Speaker 1: prove that any number of this form is perfect. Euler
Speaker 1: showed that the formula includes all even perfect numbers, and
Speaker 1: there is reason to believe though a rigid demonstration is wanting,
Speaker 1: then a odd number cannot be perfect. If we assume
Speaker 1: that the last of these statements is true, then every
Speaker 1: perfect number is of the above form. Thus, if P
Speaker 1: equals two, three, five, seven, thirteen, seventeen, nineteen, thirty one,
Speaker 1: sixty one, then my ver sends rule. The corresponding values
Speaker 1: of negative one plus two to the P are prime.
Speaker 1: They are three, seven, thirty one, one hundred, twenty seven,
Speaker 1: eight thousand, one hundred ninety one, one hundred and thirty,
Speaker 1: one thousand and seventy one, five hundred and twenty four thousand,
Speaker 1: two hundred eighty seven two billion, one hundred forty seven million,
Speaker 1: four hundred eighty three thousand, six hundred forty seven, and finally,
Speaker 1: two quintillion, three hundred five quadrillion, eight hundred and forty
Speaker 1: three trillion, nine billion, two hundred and thirteen million, six
Speaker 1: hundred ninety three thousand, nine hundred fifty one, and the
Speaker 1: corresponding perfect numbers are six twenty eight, four hundred ninety six,
Speaker 1: eight thousand, one hundred and twenty eight, thirty three million,
Speaker 1: five hundred and fifty thousand, three hundred thirty six, eight billion,
Speaker 1: five hundred eighty nine million, eight hundred sixty nine thousand,
Speaker 1: fifty six, one hundred and thirty seven billion, four hundred
Speaker 1: and thirty eight million, six hundred ninety one thousand and
Speaker 1: three hundred twenty eight. Next is two quintillion, three hundred
Speaker 1: and five quadrillion, eight hundred and forty three trillion, eight billion,
Speaker 1: one hundred thirty nine million, nine hundred and fifty two
Speaker 1: thousand and one hundred twenty eight, and finally two undecillion,
Speaker 1: six hundred and fifty eight decillion, four hundred and fifty five,
Speaker 1: none million, nine hundred ninety one octillion, five hundred and
Speaker 1: sixty nine septillion, eight hundred and thirty one heccillion, seven
Speaker 1: hundred and forty four, quintillion, six hundred fifty four quadrillion,
Speaker 1: six hundred ninety two trillion, six hundred fifteen billion, nine
Speaker 1: hundred and fifty three million, eight hundred and forty two thousand,
Speaker 1: one hundred and seventy six. De Beaume Coone de Beaume
Speaker 1: was born in Blois in sixteen oh one and died
Speaker 1: in sixteen fifty two. Wrote explanatory notes on the obscure
Speaker 1: and difficult analytical geometry of Descartes. He also discussed as
Speaker 1: superior and inferior limits to the roots of an equation.
Speaker 1: This was not published till sixteen fifty nine. Roberval Jill
Speaker 1: Persigner de Roberval, born at Roberval in sixteen o two
Speaker 1: and died at Paris in sixteen seventy five, described himself
Speaker 1: from the place of his birth as de Roberval, a
Speaker 1: seigniorial title to which he had no right. He discussed
Speaker 1: the nature of the tangents to the curves, solved some
Speaker 1: of the easier questions connected with the cycloid, generalized Archimedes's
Speaker 1: theorems on the spiral and wrote on mechanics and the
Speaker 1: method of indivisibles, which he rendered more precise and logical.
Speaker 1: He was a professor in the University of Paris and
Speaker 1: in correspondence with nearly all of the leading mathematicians of
Speaker 1: his time. A complete edition of his works was included
Speaker 1: in the Old Memoir of the Academy of Sciences, published
Speaker 1: in sixteen ninety three. Van Schutten Franz Vanschuten, to whom
Speaker 1: we owe an edition of Vieta's works, succeeded his father,
Speaker 1: who had taught mathematics to Huygen's hood, and slews as
Speaker 1: a professor at Leyden in sixteen forty six. He brought
Speaker 1: out in sixteen fifty nine a Latin translation of Descartes geometry,
Speaker 1: and in sixteen fifty seven a collection of mathematical exercises,
Speaker 1: in which he recommended the use of coordinates in space
Speaker 1: of three dimensions. He died in sixteen sixty one. Saint
Speaker 1: Vincent Gregoire de Saint Vincent, a Jesuit born at Bruges
Speaker 1: in fifteen eighty four and died at Ghent in sixteen
Speaker 1: sixty seven, discovered the expansion of the logarithm of one
Speaker 1: plus x in ascending powers of X. Although a circle squarer,
Speaker 1: he is worthy of mention for the numerous theorems of
Speaker 1: interest which he discovered in his search after the impossible,
Speaker 1: and Montucla ingeniously remarks that no one ever squared the
Speaker 1: circle with so much ability, except for his principal object,
Speaker 1: with so much success. He wrote two books on the subject,
Speaker 1: one published in sixteen forty seven and the other in
Speaker 1: sixteen sixty eight, which covers some two or three thousand
Speaker 1: closely printed pages. The fallacy in the quadrature was pointed
Speaker 1: out by Huygens in the former work, he had used indivisibles.
Speaker 1: An earlier work entitled Theoremata Mathematica, published in sixteen twenty four,
Speaker 1: contains a clear account of the method of exhaustions, which
Speaker 1: is applied to several quadratures, notably that of the hyperbola.
Speaker 1: For further details of Saint Vincent's life and works, see
Speaker 1: La j Quaterie histois des science che le Bleges, Brussels,
Speaker 1: eighteen sixty six. Torricelli Evangelista Torricelli born at Fenza in
Speaker 1: October fifteenth, sixteen oh eight, died at Florence in sixteen
Speaker 1: forty seven, wrote on the quadrature of the cycloid and conix,
Speaker 1: the theory of the barometer, the value of gravity found
Speaker 1: by observing the motion of two weights connected by a
Speaker 1: string passing over a fixed pulley, the theory of projectiles,
Speaker 1: and the motion of fluids. His mathematical writings were published
Speaker 1: in sixteen forty four. Hood Johann Hood, a burgomaster of Amsterdam,
Speaker 1: was born there in sixteen thirty three and died in
Speaker 1: the same town in seventeen oh four. He wrote two
Speaker 1: tracts in sixteen fifty nine. One was on the reduction
Speaker 1: of equations which have equal roots, and the other he stated,
Speaker 1: what is equivalent to the proposition that if f of
Speaker 1: xy equals zero be the algebraical equation of a curve,
Speaker 1: then that subtangent is negative y of the partial derivative
Speaker 1: of x with respect to y, divided by the partial
Speaker 1: derivative of x with respect to x. But being ignorant
Speaker 1: of the notation of the calculus, his annunciation is involved.
Speaker 1: Frenich Bernard Franich de Bessi born in Paris circa sixteen
Speaker 1: o five and died in sixteen seventy wrote numerous papers
Speaker 1: on combinations and on the theory of numbers, and also on
Speaker 1: magic squares. It may be interesting to add that he
Speaker 1: challenged Huygens to solve the following system of equations in
Speaker 1: integers x squared plus y squared equals z squared, x
Speaker 1: squared equals U squared plus v squared, and x minus
Speaker 1: y equals U minus v. A solution was given by m.
Speaker 1: Pepin in eighteen eighty. Frenko's misscellaneous works, edited by de
Speaker 1: la Haire, was published in the Memoir de l Academie,
Speaker 1: volume five, sixteen ninety one. De la Lubert Antoine de
Speaker 1: la Lubert, a Jesuit born in Languedoc in sixteen hundred
Speaker 1: died at Toulouse in sixteen sixty four, is chiefly celebrated
Speaker 1: for an incorrect solution of Pascal's problems on the cycloid,
Speaker 1: which he gave in sixteen sixty, but he has a
Speaker 1: better claim to distinction in having been the first mathematician
Speaker 1: to study the properties of the helix. Kinkhusen Gerrard Kinkhusen
Speaker 1: born in Holland in sixteen thirty and died in sixteen
Speaker 1: seventy nine, wrote in sixteen sixty a textbook on analytical
Speaker 1: conics in sixteen sixty one an algebra, and in sixteen
Speaker 1: sixty nine formed a collection of geometrical problems solved by
Speaker 1: analytical geometry. Corsier Pierre Coursier, a Jesuit, born at Troy's
Speaker 1: in sixteen o four and died at Auxeire in sixteen
Speaker 1: ninety two, wrote on the curves of intersection of a
Speaker 1: sphere with a cylinder or cone, also on spherical polygons.
Speaker 1: The latter work was published in sixteen sixty three. Richie
Speaker 1: Michel Angrichi born in sixteen nineteen, made a cardinal in
Speaker 1: sixteen eighty one and died in Rome in sixteen ninety two,
Speaker 1: wrote in sixteen sixty six a treatise in which he
Speaker 1: solved by Greek geometry those problems on maxima and minima,
Speaker 1: and on tangents to curves which had been considered by
Speaker 1: Descartes Pascal informat. N mcketter Nicholas Macketter, sometimes known as Kaufmann,
Speaker 1: born in Holstein about sixteen twenty, resided most of his
Speaker 1: life in England. He went to France in sixteen eighty three,
Speaker 1: where he designed and constructed the fountains at Versailles, but
Speaker 1: when they were finished, Louis the fourteenth refused to make
Speaker 1: him the payment agreed on unless he would turn Catholic.
Speaker 1: He died of vexation and poverty in Paris in sixteen
Speaker 1: eighty seven. He wrote a treatise on logarithms entitled Logarithmotechnia,
Speaker 1: published in sixteen sixty eight, and discovered the series log
Speaker 1: of one plus x equals x minus x squared over
Speaker 1: two plus x squared over three minus x to the
Speaker 1: power four over four, and so on. He proved this
Speaker 1: by writing the equation of a hyperbola in the form
Speaker 1: of y equals one over one plus x, which would
Speaker 1: be equal to one minus x plus x squared minus
Speaker 1: xque plus and so on, to which Wallace's formula stated
Speaker 1: before could be applied. The same series had been independently
Speaker 1: discovered by Saint Vincent Barrow. Isaac Barrow was born in
Speaker 1: London in sixteen thirty and died at Cambridge in sixteen
Speaker 1: seventy seven. He went to school first at Charterhouse, where
Speaker 1: he was so troublesome that his father was heard to
Speaker 1: pray that if it pleased God to take any of
Speaker 1: his children, he could best spare Isaac, and subsequently to Felstedt.
Speaker 1: He completed his education at Trinity College, Cambridge, after taking
Speaker 1: his degree in sixteen forty eight. He was elected to
Speaker 1: a fellowship in sixteen forty nine. He then resided for
Speaker 1: a few years in college, but in sixteen fifty five
Speaker 1: he was driven out by the persecution of the Independence.
Speaker 1: He spent the next four years in the East of Europe,
Speaker 1: and after many adventures, returned to England in sixteen fifty nine.
Speaker 1: He was ordained the next year and appointed to the
Speaker 1: professorship of Greek at Cambridge. In sixty two he was
Speaker 1: made professor of Geometry at Gresham College, and in sixteen
Speaker 1: sixty three was selected as the first occupier of the
Speaker 1: Lucasian Chair at Cambridge. He resigned the latter to his
Speaker 1: pupil Newton in sixteen sixty nine, whose superior abilities he
Speaker 1: recognized and frankly acknowledged. For the remainder of his life
Speaker 1: he devoted himself to the study of divinity. He was
Speaker 1: appointed the Master of Trinity College in sixteen seventy two,
Speaker 1: and he held the post until his death. He is
Speaker 1: described as low in stature, lean and of a pale complexion,
Speaker 1: slovenly in dress, and an inveterate smoker. He was noted
Speaker 1: for his strength and courage, and once, when traveling in
Speaker 1: the East, he saved the ship by his own prowess
Speaker 1: from capture by pirates. A ready and caustic wit made
Speaker 1: him a favorite of Charles the second and induced the
Speaker 1: courtiers to respect even if they did not appreciate him.
Speaker 1: He wrote with a sustain and somewhat stately eloquence, and
Speaker 1: with his blameless life and scrupulous conscientiousness, he was an
Speaker 1: impressive personage of the time. His earliest work was a
Speaker 1: complete edition of the Elements of Euclid, which he issued
Speaker 1: in sixteen fifty five. He published an English translation in
Speaker 1: sixteen sixty and in sixteen fifty seven an edition of
Speaker 1: the data. His lectures delivered in sixteen sixty four, sixteen
Speaker 1: sixty five, and sixteen sixty six were published in sixteen
Speaker 1: eighty three under the title of Leccion's Mathematicae. These are
Speaker 1: mostly on the metaphysical basis for mathematical truths. His lectures
Speaker 1: for sixteen sixty seven were published in the same year
Speaker 1: and suggest the analysis by which Archimmittes was led to
Speaker 1: his chief results. In sixteen sixty nine he issued his
Speaker 1: Leccion's Opticate Geometricee. It is said in the preface that
Speaker 1: Newton revised and corrected these lectures, adding matter of his own,
Speaker 1: but it seems probable from Newton's remarks in the Fluctional Controversy,
Speaker 1: that the editions were confined to the parts which dealt
Speaker 1: with optics. This which is his most important work in mathematics,
Speaker 1: was republished with a few minor alterations in sixteen seventy four.
Speaker 1: In sixteen seventy five he published an edition with numerous
Speaker 1: comments on the first four books of the conics of
Speaker 1: Apollonius and of the extent works of Archemedes and Theodosius.
Speaker 1: In the Optical Lectures, many problems connected with the reflection
Speaker 1: and refraction of light are treated with great ingenuity. The
Speaker 1: geometrical focus of a point seen by reflection or refraction
Speaker 1: is defined, and it is explained that the image of
Speaker 1: an object is the locus of the geometrical foci of
Speaker 1: every point on it. Barrow also worked out a few
Speaker 1: of the easier propers parties of thin lenses, and considerably
Speaker 1: simplified the Cartesian explanation of the rainbow. The Geometrical Lectures
Speaker 1: contained some new ways of determining the areas of the
Speaker 1: tangents of curves. The most celebrated of these is the
Speaker 1: method given for the determination of tangent's two curves, and
Speaker 1: that is sufficiently important to require a detailed notice because
Speaker 1: it illustrates the way in which Barrow, Hood and Slews
Speaker 1: were working on the line suggested by Fermat towards the
Speaker 1: methods of the differential calculus. Fermat had observed that the
Speaker 1: tangent at point P on a curve was determined if
Speaker 1: one other point besides P on it were known. Hence,
Speaker 1: if the length of the sub tangent mt could be found,
Speaker 1: thus determining the point T, then the line t P
Speaker 1: would be the required tangent. Now, Barrow remarked that if
Speaker 1: the abscissa and ordinate at a point Q adjacent to
Speaker 1: P were drawn, he got a small triangle p q R,
Speaker 1: which he called the differential triangle because its sides p
Speaker 1: R and p q were the differences of the obscissa
Speaker 1: and ordinance of P and Q, so that t M
Speaker 1: to m P equals q R to r P. To
Speaker 1: find q R to r P, he supposed that x
Speaker 1: y were the coordinates of p and x minus e,
Speaker 1: Y minus a those of Q. Barrow actually used P
Speaker 1: for x and M for y, but I alter these
Speaker 1: to agree with the modern practice. Substituting the coordinates of
Speaker 1: Q in the equation of the curve, and neglecting the
Speaker 1: squares and higher powers of E, n, A as compared
Speaker 1: with their first powers, he obtained the ratio E to A.
Speaker 1: The ratio A over E was subsequently, in accordance with
Speaker 1: a suggestion made by Slews, termed the angular coefficient of
Speaker 1: the tangent at the point. Barrow applied the method to
Speaker 1: the curves. One x squared multiplied by the quantity x
Speaker 1: squared plus y squared equals are squared y squared, two
Speaker 1: xqube plus y cubed equals are cubed, N three x
Speaker 1: cube plus y cubed equals r x y called la Galande.
Speaker 1: Four y equals the quantity R minus x multiplied by
Speaker 1: the tangent of pi x divided by two R the quadratrix,
Speaker 1: and five y equals r times ten pi x over
Speaker 1: two R. It will be sufficient here if I take
Speaker 1: as an illustration the simpler case of the parabola y
Speaker 1: squared equals px. Using the notation given above, we have
Speaker 1: for the point p y squared equals px, and for
Speaker 1: the point q y minus a quantity squared equals p
Speaker 1: multiplied by the quantity x minus e. Subtracting, we get
Speaker 1: two A y minus a squared equals p e. But
Speaker 1: if A be an infinitesimal quantity, A squared must be
Speaker 1: infinitely smaller and therefore may be neglected. Hence, the ratio
Speaker 1: e to A equals the ratio two y t P. Therefore,
Speaker 1: t M t y equals e to a equals two
Speaker 1: y t p. That is, t m equals two y
Speaker 1: square divided by p, which is equal to two x.
Speaker 1: This is exactly the procedure of the differential calculus, except
Speaker 1: that we there have a rule by which we can
Speaker 1: get the ratio A over e or d y by
Speaker 1: d x directly without the labor of going through a
Speaker 1: calculation similar to the above for every separate case. Bruckner,
Speaker 1: William Viscount Bruckner, one of the founders of the Royal
Speaker 1: Society of London, born in sixteen twenty and died on
Speaker 1: April fifteenth, sixteen eighty four, was among the most brilliant
Speaker 1: mathematicians of his time, and was in intimate relations with Wallace,
Speaker 1: fermat and other leading mathematicians. I mentioned on page one
Speaker 1: fifty five his curious reproduction of Brahmagupta's solution of a
Speaker 1: certain indetermined equation. Bruckner proved that the area enclosed between
Speaker 1: the equilateral hyperbola x y equal one and the axis
Speaker 1: of X and the ordnance ex equals one and ex
Speaker 1: equals two is equal either two one over one times
Speaker 1: two plus one, over three times four plus one, over
Speaker 1: five times six plus, and so on, or to one
Speaker 1: minus one half plus one third minus a quarter and
Speaker 1: so on. He also worked out similar expressions for different
Speaker 1: areas bounded by the hyperbola and straight lines. He wrote
Speaker 1: on the rectification of the parabola and of the cycloid.
Speaker 1: It is noticeable that he used infinite series to express
Speaker 1: quantities whose values he could not otherwise determine. In answer
Speaker 1: to a request of Wallace to attempt the quadrature of
Speaker 1: the circle, he shewed that the ratio of the area
Speaker 1: of the circle to the area of the circumscribed square,
Speaker 1: that is, the ratio of Pie four, is equal to
Speaker 1: the ratio one over one plus one squared over two
Speaker 1: plus three squared over two plus five squared over two
Speaker 1: plus seven squared over two, and so on to one.
Speaker 1: Continued fractions had been introduced by Cataldi in his Treatise
Speaker 1: on Finding the Square Roots of Numbers, published at Bologna
Speaker 1: in sixteen thirteen, but he treated them as common fractions.
Speaker 1: Bruckner was the first writer who investigated or made any
Speaker 1: use use of their properties. For further details, see C.
Speaker 1: Hutton's Mathematical Dictionary. James Gregory James Gregory born at Drumoake,
Speaker 1: near Aberdeen in sixteen thirty eight and died in Edinburgh
Speaker 1: in October sixteen seventy five, was successively professor at Saint
Speaker 1: Andrew's and Edinburgh. In sixteen sixty he publishes Optica Promota,
Speaker 1: in which the reflecting telescope known by his name is described.
Speaker 1: In sixteen sixty seven he issued a z vera circuli
Speaker 1: at hyperbole quadratia, in which he showed the areas of
Speaker 1: the circle and hyperbola could be obtained in the form
Speaker 1: of infinite convergent series, and here I believe for the
Speaker 1: first time we find a distinction drawn between convergent and
Speaker 1: divergent series. This work contains a remarkable geometrical proposition to
Speaker 1: the effect that the ratio of the area of any
Speaker 1: arbitrary sector to that of the inscribed or circumscribe regular
Speaker 1: polygons is not expressible by a finite number of algebraical terms. Hence,
Speaker 1: he inferred that the quadrature was impossible. This was accepted
Speaker 1: by Montucla, but it is not conclusive, for it is
Speaker 1: conceivable that some particular sector might be squared in this
Speaker 1: particular sector might be the whole circle. This book contains
Speaker 1: also the earliest annunciations of the expansions in the series
Speaker 1: of sin x, cosin of x, inverse sin of x,
Speaker 1: and inverse cosin of X. It was reprinted in sixteen
Speaker 1: sixty eight with an appendix geometrae Pals, in which Gregory
Speaker 1: explained how the volumes of solids of revolution could be determined.
Speaker 1: In sixteen seventy one, or perhaps earlier, he established the
Speaker 1: theorem that theta equals ten theta minus one third ten
Speaker 1: cub theta minus one fifth tenth to the fifth theta,
Speaker 1: and so on, the result being true only if theta
Speaker 1: lie between negative one quarter pie and positive one quarter pie.
Speaker 1: This is the theorem on which the work of most
Speaker 1: of the subsequent calculation of approximations to the numerical value
Speaker 1: of pie has been based. For further detail see C.
Speaker 1: Hutton's mathematical dictionary. Reren Sir Christopher Wren was born in
Speaker 1: Noyle in sixteen thirty two and died in London in
Speaker 1: seventeen twenty three. Wren's reputation as a mathematician has been
Speaker 1: overshadowed by his fame as an architect, but he was
Speaker 1: Ciphilian Professor of Astronomy at Oxford from sixteen sixty one
Speaker 1: to sixteen seventy three, and for some time president of
Speaker 1: the Royal Society. Together with Wallace and Huygens, he investigated
Speaker 1: the laws of co of bodies. He also discovered the
Speaker 1: two systems of generating lines on the hyperboloid of one sheet,
Speaker 1: though it is probable that he confined his attention to
Speaker 1: the hyperboloid of revolution. Besides these, he communicated papers on
Speaker 1: the resistance of fluids and the motion of the pendulum.
Speaker 1: He was a friend of Newton, and like Huygen's Hook,
Speaker 1: Halley and others had made attempts to shew that the
Speaker 1: force under which the planets move varies inversely as a
Speaker 1: square of the distance from the Sun. Wallace, Bruckner, Wren,
Speaker 1: and Boyle, the last name being a chemist and physicist
Speaker 1: rather than a mathematician, where the leading philosophers who had
Speaker 1: founded the Royal Society of London. The society arose from
Speaker 1: the self styled Indivisible College in London in sixteen forty five.
Speaker 1: Most of its members moved to Oxford during the Civil
Speaker 1: War were Hook, who was then an assistant in Boyle's laboratory,
Speaker 1: joined in their meetings. The Society was formally constituted in
Speaker 1: London in sixteen sixty and was incorporated on fifteenth of
Speaker 1: July sixteen sixty two. Hook Robert Hook born at Freshwater
Speaker 1: on July eighteenth, sixteen thirty five and died in London
Speaker 1: on March third, seventeen o three. Was educated at Westminster
Speaker 1: and christ Church, Oxford, and in sixteen sixty five he
Speaker 1: became professor of geometry at Gresham College, a post which
Speaker 1: he occupied till his death. He is still known by
Speaker 1: the law which he discovered that the tension exerted by
Speaker 1: a stretched string is within certain limits proportional to the extension, or,
Speaker 1: as it is better stated, that the stress is proportional
Speaker 1: to the strain. He invented and discovered thus the conical pendulum,
Speaker 1: and was the first to state explicitly that the motions
Speaker 1: of the heavenly bodies were merely dynamical problems. He was
Speaker 1: as jealous as he was vain and irritable, and accused
Speaker 1: both Newton and Huygens of unfairly appropriating his results. Like Huygens,
Speaker 1: Wren and Halley, he made efforts to find the law
Speaker 1: of force under which the planets move about the Sun,
Speaker 1: and he believed the law to be that of the
Speaker 1: inverse square of the distance. He, like Huygens, discovered that
Speaker 1: the small oscillations of a coiled spiral spring were practically isochronous,
Speaker 1: and was thus led to recommend, possibly in sixteen fifty eight,
Speaker 1: the use of the balanced spring in watches. He had
Speaker 1: a watch of this kind made in London in sixteen
Speaker 1: seventy five. It was finished just three months later than
Speaker 1: the one made under the direction of Huygens in Paris.
Speaker 1: Collins John Collins born near Oxford on March fifth, sixteen
Speaker 1: twenty five, and died in London on November tenth, sixteen
Speaker 1: eighty three, was a man of great natural ability, but
Speaker 1: of slight education, being devoted to mathematics. He spent his
Speaker 1: spare time in correspondence with the leading mathematicians of the time,
Speaker 1: for whom he was always ready to do anything in
Speaker 1: his power, and he has been described not inaptly as
Speaker 1: the English mersenne to him, we are indebted for much
Speaker 1: information on the details of the discoveries of the period Pell.
Speaker 1: Another mathematician who devoted considerable part of his time to
Speaker 1: making known the discoveries of others and to correspondence with
Speaker 1: leading mathematicians was John Pell. Pell was born at Sussex
Speaker 1: on March first, sixteen ten, and died in London on
Speaker 1: December tenth, sixteen eighty five. He was educated at Trinity College, Cambridge.
Speaker 1: He occupied in succession the mathematical chairs at Amsterdam and Breda.
Speaker 1: He then entered the English Diplomatic Service, but finally settled
Speaker 1: in sixteen sixty one in London, where he spent the
Speaker 1: last twenty years of his life. His chief works were,
Speaker 1: in addition with considerable new Matter of the Algebra by
Speaker 1: Brenker en Rohneus London, sixteen sixty eight and A Table
Speaker 1: of Square Numbers London, sixteen seventy two, for further detail
Speaker 1: see My History of mathematics at Cambridge. Slews Grene Francois
Speaker 1: Wartre de Sleuse or Sleusius Canon of Lieges born on
Speaker 1: July seventh, sixteen twenty two and died on March nineteen,
Speaker 1: sixteen eighty five found for the subtangent of a curve
Speaker 1: fx wi by equals zero, an expression was which is
Speaker 1: the equivalent to negative y multiplied by the partial derivative
Speaker 1: delta f by delta y divided by the partial derivative
Speaker 1: delta f by dx. He wrote numerous tracts and in
Speaker 1: particularly discussed at some lengths spirals and points of inflection.
Speaker 1: Chernhausen Erinfried Walter von Chernhausen was born at Kislingswald on
Speaker 1: April tenth, sixteen thirty one, and died at Dresden on
Speaker 1: October eleventh, seventeen oh eight. In sixteen eighty two, he
Speaker 1: worked at the theory of caustics by reflection, or as
Speaker 1: they were usually called, catacoustics, and shewed that they were rectifiable.
Speaker 1: This was the second case in which the envelope of
Speaker 1: a moving line was determined. He constructed the burning mirrors
Speaker 1: of great power, the transformation by which he removed certain
Speaker 1: intermediate terms from a given algebraical equation, is well known.
Speaker 1: Romer Olof Romer was born at Arnus in September twenty fifth,
Speaker 1: sixteen forty four and died in Copenhagen on September nineteenth,
Speaker 1: seventeen ten, and was the first to measure the velocity
Speaker 1: of light. This was done in sixteen seventy five by
Speaker 1: means of the eclipses of Jupiter's satellites. He brought the
Speaker 1: transit and mural circle into common use, the altazimuth having
Speaker 1: been previously generally employed, and it was on his recommendation
Speaker 1: that astronomical observations of stars were subsequently made in general
Speaker 1: on the meridian. He was also the first to introduce
Speaker 1: micrometers and reading microscopes into an observatory. He also deduced
Speaker 1: from the properties of epicycloids the form of the teeth
Speaker 1: and the toothed wheels best fitted to secure a uniform motion.
Speaker 1: End of Part twenty three.