Speaker 1: Chapter fifteen, Part two of a short account of the
Speaker 1: history of mathematics. This is a LibriVox recording. All LibriVox
Speaker 1: recordings are in the public domain. For more information or
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Speaker 1: by Paul King p J K Dot scripts dot MI
Speaker 1: I T dot e edu forward slash p K J. Pascal.
Speaker 1: Among the contemporaries of Descartes, none displayed greater natural genius
Speaker 1: than Pascal, But his reputation rests more on what he
Speaker 1: might have done than on what he actually affected, as
Speaker 1: during a considerable part of his life he deemed it
Speaker 1: his duty to devote his whole time to religious exercises.
Speaker 1: Blaise Pascal was born at Clermont on June nineteenth, sixteen
Speaker 1: twenty three, and died at Paris on August nineteenth, sixteen
Speaker 1: sixty two. His father, a local jucture at Claremont and
Speaker 1: himself of some scientific reputation, moved to Paris in sixteen
Speaker 1: thirty one, partly to prosecute his own scientific studies, partly
Speaker 1: to carry on the education of his only son, who
Speaker 1: had already displayed exceptional ability. Pascal was kept at home
Speaker 1: in order to assure his not being overworked, and with
Speaker 1: the same object, it was directed that his education should
Speaker 1: be at first confined to the study of languages, and
Speaker 1: should not include any mathematics. This naturally excited the boy's curiosity,
Speaker 1: and one day, being then twelve years old, he asked
Speaker 1: in what geometry consisted. His tutor replied that it was
Speaker 1: the science of constructing exact figures and of determining the
Speaker 1: proportions between their different parts. Pascal, stimulated no doubt by
Speaker 1: the injunction against reading, it, gave up his place to
Speaker 1: the new study, and in a few weeks had discovered
Speaker 1: for himself many properties of figures, and in particular the
Speaker 1: proposition that the sum of the angles of a triangle
Speaker 1: is equal to two right angles. I have read somewhere,
Speaker 1: but I cannot lay my hand on the authority that
Speaker 1: his proof merely consisted in turning the angular points of
Speaker 1: a triangular piece of paper over so as to meet
Speaker 1: in the center of the inscribed circle. A similar demonstration
Speaker 1: can be got by turning the angular points over so
Speaker 1: as to meet at the foot of the perpendicular drawn
Speaker 1: from the biggest angle to the opposite side. His father,
Speaker 1: struck by this display of ability gave him a copy
Speaker 1: of Euclid's Elements, a book which Pascal read with avidity
Speaker 1: and soon mastered. At the age of fourteen, he was
Speaker 1: admitted to the weekly meetings of Robervelle, Mersenne Midorge and
Speaker 1: other French geometricians, from which the French Academy he ultimately sprung,
Speaker 1: being created by ordinance of Louis the fourteenth on December
Speaker 1: twenty second, sixteen sixty six. At sixteen, Pascal wrote an
Speaker 1: essay on conic sections, and in sixteen forty one, at
Speaker 1: the age of eighteen, he constructed the first arithmetical machine,
Speaker 1: an instrument which eight years later he further improved and patented.
Speaker 1: His correspondence with Fermat About this time Shews that he
Speaker 1: was then turning his attention to analytical geometry and physics.
Speaker 1: He repeated Torcelli's experiments by which the pressure of the
Speaker 1: atmosphere could be estimated as a weight, and he confirmed
Speaker 1: his theory of the cause of barometrical variations by obtaining
Speaker 1: at the same instant readings at different altitudes on the
Speaker 1: hill of Puis di Domme. In sixteen fifty, when in
Speaker 1: the midst of these researches, Pascal suddenly abandoned his favorite
Speaker 1: pursuits to study religion, or, as he says in his
Speaker 1: Ponte Say, to contemplate the greatness and the misery of man.
Speaker 1: And about the same time he persuaded the younger of
Speaker 1: his two sisters to enter the port Royal Society. In
Speaker 1: sixteen fifty three, he had to administer his father's estate.
Speaker 1: He now took up his old life again and made
Speaker 1: several experiments on pressure exerted by gases and liquids. It
Speaker 1: was also about this period that he invented the arithmetical triangle,
Speaker 1: and together with Framat, created the calculus of probabilities. He
Speaker 1: was meditating marriage when an accident again turned the current
Speaker 1: of his thoughts to a religious life. He was driving
Speaker 1: a foreign hand on November twenty third, sixteen fifty four,
Speaker 1: when the horses ran away. The two leaders dashed over
Speaker 1: the parapret of the bridge at Nui, and Pascal was
Speaker 1: only saved by the traces. Breaking. Always somewhat of a mystic,
Speaker 1: he considered this a special summon to abandon the world.
Speaker 1: He wrote an account of the accident on a small
Speaker 1: piece of parchment, which for the rest of his life
Speaker 1: he wore next to his heart to perpetually remind him
Speaker 1: of his covenant, and shortly moved to Port Royal, where
Speaker 1: he continued to live until his death in sixteen sixty two.
Speaker 1: Always delicate, he had injured his health by his incessant study.
Speaker 1: From the age of seventeen or eighteen, he suffered from insomnia,
Speaker 1: an acute dyspepsia, and at the time of his death
Speaker 1: was completely worn out. His famous provincial letters directed against
Speaker 1: the Jesuits and his Ponset were written towards the close
Speaker 1: of his life and are the first example of that
Speaker 1: finished form which is characteristic of the best French literature.
Speaker 1: The only mathematical work that he produced after retiring to
Speaker 1: Port Royal was the Essay of the Cycloid in sixteen
Speaker 1: fifty eight. He was suffering from leaplessness and a toothache
Speaker 1: when the idea occurred to him, and to his surprise,
Speaker 1: his teeth immediately ceased to ache. Regarding this as a
Speaker 1: divine intimation to proceed with the problem, he worked incessantly
Speaker 1: for eight days at it and completed a tolerably full
Speaker 1: account of the geometry of the cycloid. I now proceed
Speaker 1: to consider his mathematical works in rather greater detail. His
Speaker 1: early essay of the geometry of conics, written in sixteen
Speaker 1: thirty nine but not published until seventeen seventy nine, seems
Speaker 1: to have been founded on the teachings of Derog. Two
Speaker 1: of the results are important as well as interesting. The
Speaker 1: first of these is the theorem known now as Pascal's theorem,
Speaker 1: namely that if a hexagon is inscribed in a conic,
Speaker 1: the points of the intersection of the opposite sides will
Speaker 1: lie in a straight line. The second, which is really
Speaker 1: due to Derague, is that if a quadrilateral be inscribed
Speaker 1: in a and a straight line be drawn cutting the
Speaker 1: sides taken in order in the points abcn, d and
Speaker 1: the conic in P and Q, then PA times pc
Speaker 1: over PD times pd equals Qa times QC over QB
Speaker 1: times QD. Pascal's arithmetical triangle, known today as Pascal's triangle,
Speaker 1: was written in sixteen fifty three but not printed until
Speaker 1: sixteen sixty five. The triangle is constructed as in the
Speaker 1: annexed figure. By the way, the annexed figure shows Pascal's
Speaker 1: triangle as being kind of on its side by today's standards,
Speaker 1: A one at the apex, followed by a one to one,
Speaker 1: and then in the third row one two to one,
Speaker 1: and in the fourth row one three three one, and
Speaker 1: in the fifth row one four six four one. So
Speaker 1: they will refer to these lines as diagonal lines. The
Speaker 1: whole horizontal lines, the way they're depicted are one one one, one, one,
Speaker 1: one two, three, four, five, one three six, ten, fifteen,
Speaker 1: one four ten, twenty thirty five, and one five fifteen
Speaker 1: thirty five, seventy. And they have that written in actual
Speaker 1: rows and columns. So, taking the latter description, each horizontal
Speaker 1: line being formed from the one above it by making
Speaker 1: every number in it equal to the sum of those
Speaker 1: above it and to the left of it in the
Speaker 1: row immediately above. For example, in the fourth line, twenty
Speaker 1: is equal to one plus three plus six plus ten.
Speaker 1: Then Pascal's arithmetical triangle to any required order is got
Speaker 1: by drawing a diagonal downwards from the right to left.
Speaker 1: Is in the figure. The numbers are what are now
Speaker 1: called the figure it numbers. Those in the first line
Speaker 1: are called the numbers of the first order. Those in
Speaker 1: the second line one, two, three, four, five, and so
Speaker 1: on are called natural none numbers or numbers of the
Speaker 1: second order, those in the third line numbers of the
Speaker 1: third order, and so on. It is easily shown that
Speaker 1: the m number in the n th row is M
Speaker 1: plus n minus two factorial divided by m minus one
Speaker 1: factorial times n minus one factorial. The numbers in any
Speaker 1: diagonal give the coefficients of the expansion of a binomial.
Speaker 1: For example, the figures in the fifth diagonal, namely one, four, six,
Speaker 1: four one, are the coefficients in the expansion of a
Speaker 1: plus b raised to the power four. Pascal use the
Speaker 1: triangle partly for this purpose, and partly to find the
Speaker 1: numbers of combinations of m things taken n at a time,
Speaker 1: which you stated correctly to be n plus one times
Speaker 1: n plus two times n plus three dot dot dot
Speaker 1: times m divided by m minus n factorial. Perhaps as
Speaker 1: a mathematic Pascal is best known in connection with his
Speaker 1: correspondence with Fermont in sixteen fifteen four, in which he
Speaker 1: laid down the principles of the theory of probabilities. This
Speaker 1: correspondence arose from a problem proposed by a gamester, the
Speaker 1: Chevalier de Maire, to Pascal, who communicated it to Fermat.
Speaker 1: The problem was this, two players of equal skill want
Speaker 1: to leave the table before finishing the game. Their scores
Speaker 1: and the number of points which constitute the game being given,
Speaker 1: it is desired to find in what proportion should they
Speaker 1: divide the stakes. Pascal and Fermat agreed on the answer,
Speaker 1: but gave different proofs. The following is a translation of
Speaker 1: Pascal's solution. That of Fermat is given later. The following
Speaker 1: is my method for determining the share of each player. When,
Speaker 1: for example, two players play a game of three points
Speaker 1: and each player has staked thirty two piece fo suppose
Speaker 1: that the first player had gained two points and then
Speaker 1: the second player one point. They now have to play
Speaker 1: for a point on this condition that if the first
Speaker 1: player gain, he takes all the money which is at stake,
Speaker 1: namely sixty four pietols, while if the second player gain,
Speaker 1: each player has two points, so that they are on
Speaker 1: terms of equality, and if they leave off playing, each
Speaker 1: ought to take thirty two piecetoles. Thus, if the first
Speaker 1: player gain, then sixty four pistoles belonged to him, and
Speaker 1: if he loses, then thirty two piecetoles belong to him.
Speaker 1: If therefore, the players do not wish to play this game,
Speaker 1: but to separate without playing it. The first player would
Speaker 1: say to the second I am certain of thirty two
Speaker 1: piece stoles, even if I lose this game, and for
Speaker 1: the other thirty two piece stoles, perhaps I shall have them,
Speaker 1: and perhaps you will have them. The chances are equal.
Speaker 1: Let us divide these thirty two pistoles equally and give
Speaker 1: me also the thirty two pistoles of which I am certain.
Speaker 1: Thus the first player will have forty eight pistoles in
Speaker 1: the second sixteen pistoles. Next, suppose that the first player
Speaker 1: has gained two points in the second player none, and
Speaker 1: they are about to play for a point. The condition
Speaker 1: is that if the first player gain this point, he
Speaker 1: secures the game and takes sixty four pistols, And if
Speaker 1: the second player gain this point, then the players will
Speaker 1: be in the situation already examined, in which the first
Speaker 1: player is entitled the forty eight pistoles and the second
Speaker 1: to sixteen pistos. Thus, if they do not wish to play,
Speaker 1: the first player would say to the second if I
Speaker 1: gain this point, I gain sixty four pistos. If I
Speaker 1: lose it, I am titled to forty eight pistos. Give
Speaker 1: me then the forty eight pistoles of which I am certain,
Speaker 1: and divide the other sixteen. Since our chances of gaining
Speaker 1: the point are equal. Thus the first player will have
Speaker 1: fifty six pistoles and the second player eight pistos. Finally,
Speaker 1: suppose that the first player had gained one point and
Speaker 1: the second player none. If they proceed to play for
Speaker 1: a point, the condition is that if the first player
Speaker 1: gain it, the players will be in the situation first examined,
Speaker 1: in which the first player is entitled to fifty six pistos.
Speaker 1: If the first player lose the point, each player has
Speaker 1: then a point, and each is entitled to thirty two pistoles.
Speaker 1: Give me the thirty two pistoles of which I am certain,
Speaker 1: and divide the remainder of the fifty six pistoles equally,
Speaker 1: that is, divide the twenty four pistoles equally. Thus the
Speaker 1: first player will have the sum of thirty two and
Speaker 1: twelve pistoles, that is, forty four pistoles, and consequently the
Speaker 1: second will have twenty pistoles. Pascal proceeds next to consider
Speaker 1: the similar problem when the game is won by whomever
Speaker 1: first obtains M plus en points, and one player has
Speaker 1: m while the other has end points. The answer is
Speaker 1: obtained by using the arithmetical triangle or Pascal's triangle. The
Speaker 1: general solution, in which the skill of the players is unequal,
Speaker 1: is given in many modern textbooks on algebra, and agrees
Speaker 1: with Pascal's result, though of course the notation of the
Speaker 1: latter is different and less convenient. Pascal made a most
Speaker 1: illegitimate use of the new theory in the seventh chapter
Speaker 1: of his Ponce. He practically puts his argument that, as
Speaker 1: the value of eternal happiness must be infinite, then even
Speaker 1: if the probability of a religious life ensuring eternal happiness
Speaker 1: be very small, still the expectation, which is measured by
Speaker 1: the product of the two must be of sufficient MEAs
Speaker 1: magnitude to make it worth while to be religious. The argument,
Speaker 1: if worth anything, would apply equally to any religion which
Speaker 1: has promised eternal happiness to those who accepted its doctrines.
Speaker 1: If any conclusion may be drawn from the statement, it
Speaker 1: is the undesirability of applying mathematics to questions of morality,
Speaker 1: of which some of the data are necessarily outside the
Speaker 1: range of an exact science. It is only fair to
Speaker 1: add that no one had more contempt than Pascal for
Speaker 1: those who changed their opinions according to the prospect of
Speaker 1: material benefit, and this isolated passage is at variance with
Speaker 1: the spirit of his writings. The last mathematical work of
Speaker 1: Pascal was that on the Cycloid of sixteen fifty eight.
Speaker 1: The cycloid is the curve traced out by the point
Speaker 1: on the circumference of a circular hoop which rolls along
Speaker 1: a straight line. Galileo in sixteen thirty had been the
Speaker 1: first to call attention to this curve, and had suggested
Speaker 1: that the arches of bridges should be built in the
Speaker 1: form of it. It is a graceful curve, but the
Speaker 1: only bridge with a cycloidal arch of which I have
Speaker 1: heard is the one built by Essex in the grounds
Speaker 1: of Trinity College at Cambridge. Four years later, in sixteen
Speaker 1: thirty four, Roberval found the area of the cycloid. Descartes
Speaker 1: thought little of this solution and defied him to find
Speaker 1: its tangents, the same challenge also being sent to Fermont,
Speaker 1: who at once solved the problem. Several questions connected with
Speaker 1: the curve and with the surface and volume generated by
Speaker 1: its revolution, about its axis base or the tangent at
Speaker 1: its vertex, were then proposed by various mathematicians. These and
Speaker 1: some analogous questions, as well as the positions of the
Speaker 1: centers of mass of the solids there formed, were solved
Speaker 1: by Pascal in sixteen fifty eight, and the results were
Speaker 1: issued as a challenge to the world. Wallace succeeded in
Speaker 1: solving all the questions except those connected with the center
Speaker 1: of mass. Pascal's own solutions were affected by the method
Speaker 1: of indivisibles, and are similar to those which a modern
Speaker 1: mathematician would give. By the aid of the integral calculus.
Speaker 1: He obtained by summation what are equivalent to the following integrals,
Speaker 1: the integral of sign phi de phi, also the integral
Speaker 1: of sign squared phi de phi, and also the integral
Speaker 1: of Phi times sign phi d phi, one limit being
Speaker 1: either zero or half Pi. He also investigated the geometry
Speaker 1: of the Archimedean spiral. These researches, according to de Lambert,
Speaker 1: form a connecting link between the geometry of archimytes and
Speaker 1: the infinitesimal calculus of Newton Wallace. John Wallace was born
Speaker 1: at Ashford on November twenty second, sixteen sixteen, and died
Speaker 1: ah Oxford on October twenty eighth seventeen o three, when
Speaker 1: fifteen years old, he happened to see a book on
Speaker 1: arithmetic in the hands of his brother. Struck with curiosity
Speaker 1: at the odd signs and symbols in it, he borrowed
Speaker 1: the book and in a fortnight mastered the subject. It
Speaker 1: was intended that he should be a doctor, and he
Speaker 1: was sent to Emmanuel College, Cambridge. While there he kept
Speaker 1: an act on the doctrine of the circulation of the blood.
Speaker 1: This is said to have been the first occasion in
Speaker 1: Europe on which this theory was publicly maintained in disputation.
Speaker 1: His interests, however, centered on mathematics. He was elected to
Speaker 1: a fellowship at Queen's College, Cambridge and subsequently took orders,
Speaker 1: but on the whole adhered to the Puritan Party, whom
Speaker 1: he rendered great assistance in deciphering the Royalist despatches. He however,
Speaker 1: joined the Moderate Presbyterians in signing the Remonstrance against the
Speaker 1: Execution of Charles the First, by which he incurred the
Speaker 1: lasting hostility of the Independence. In spite of their opposition,
Speaker 1: he was appointed in sixteen forty nine to the civilian
Speaker 1: chair of Geometry at Oxford, where he lived until his
Speaker 1: death on October twenty eighth, seventeen o three. Besides his
Speaker 1: mathematical works, he wrote on theology, logic, and philosophy, and
Speaker 1: was the first to devise a system for teaching deaf mutes.
Speaker 1: I confine myself to a few notes on his more
Speaker 1: important mathematical writings. They are notable partly for the introduction
Speaker 1: of the use of infinite series as an ordinary part
Speaker 1: of analysis, and partly for the fact that they revealed,
Speaker 1: explained to all students the principles of those new methods
Speaker 1: which distinguish modern from classical mathematics. The most important of
Speaker 1: Wallace's works was his Arithmetica Infinitorum, which was published in
Speaker 1: sixteen fifty six. In this treatise, the methods of analysis
Speaker 1: of Descartes and Cavalieri were systematized and greatly extended, but
Speaker 1: their logical exposition is open to criticism. It at once
Speaker 1: became the standard book on the subject, and is constantly
Speaker 1: referred to by subsequent writers. It is prefaced by a
Speaker 1: short tract on conic sections, which was subsequently expanded into
Speaker 1: a separate treatise. He commences by proving the law of
Speaker 1: indices chews that X to the zero x to the
Speaker 1: minus one, x to the minus two, and so on
Speaker 1: represent one one over x, one over x squared, and
Speaker 1: so on. That x to the power of one half
Speaker 1: represents the square root of X, that x to the
Speaker 1: power of two thirds represents the qb root of x squared,
Speaker 1: and that generally that x to the exponent minus n
Speaker 1: represents the reciprocal of x to the power n, and
Speaker 1: that x to the p over q represents the q
Speaker 1: th root of x to the p. Leaving the numerous
Speaker 1: algebraical applications of this discovery, he next proceeds defined by
Speaker 1: the method of indivisibles, the area inclosed between the curve
Speaker 1: y equals x to the m the axis of x
Speaker 1: and any ordinate x equals h, and he proves that
Speaker 1: the ratio of this area is equal to that of
Speaker 1: the parallelogram on the same base and of the same altitude,
Speaker 1: and is equal to the ratio one over M plus one.
Speaker 1: He apparently assumed that the same result would be true
Speaker 1: also for the curve y equals a, x rays to
Speaker 1: the power m, where A is any constant an n
Speaker 1: m is any number positive or negative. But he only
Speaker 1: discusses the case of which the parabola of m equals two,
Speaker 1: and that of the hyperbola in which m equals minus one.
Speaker 1: In the latter case, his interpretation of the result is incorrect.
Speaker 1: He then chows that similar results might be written down
Speaker 1: for any curve of the form Y equals the sum
Speaker 1: of a x rays to the power m, and hence
Speaker 1: that if the ordinate Y on a curve can be
Speaker 1: expanded in powers of the obsisa x, its quadrature can
Speaker 1: be determined. Thus, he said that if the equation of
Speaker 1: a curve where y equals x to the zero plus
Speaker 1: x to the one plus x squared, and so on,
Speaker 1: its area would be x plus one half x squared
Speaker 1: plus one third x cubed, and so on. He then
Speaker 1: applies this to the quadrature of the curves. Y equals
Speaker 1: quantity x minus x squared to the power zero. Y
Speaker 1: equals the quantity x minus x squared to the power one.
Speaker 1: Y equals x minus x squared quantity to the power two,
Speaker 1: Y equals the quantity x minus x square to the
Speaker 1: power three, and so on. Taken between the limits x
Speaker 1: equals zero and x equals one, and shows that the
Speaker 1: areas are respectively one one over six, one over thirty
Speaker 1: one over one, forty, and so on. He next considers
Speaker 1: the curve of the form y equals x to the
Speaker 1: power one over M, and establishes the theorem that the
Speaker 1: area bounded by the curve the axis of x and
Speaker 1: the ordinate x equals one is to the area of
Speaker 1: the rectangle on the same base and the same altitude
Speaker 1: as M over M plus one. This is equivalent of
Speaker 1: finding the value of the integral from zero to one
Speaker 1: of x rays to the power one over md x.
Speaker 1: He illustrates this by the parabola in which M equals two.
Speaker 1: He states, but does not prove, that the corresponding result
Speaker 1: of the curve of the form y equals x rays
Speaker 1: to the p over q. This work contains also one
Speaker 1: of the earliest investigations of the formation and properties of
Speaker 1: continued fractions, a discussion that was suggested by Bruckner's use
Speaker 1: of these fractions. Wallace shewed considerable ingenuity in reducing the
Speaker 1: equations of curves to the forms given above, but as
Speaker 1: he was unacquainted with the binomial theorem, he could not
Speaker 1: affect the quadrature of the circle whose equation is y
Speaker 1: equals quantity x minus x square to the half. Since
Speaker 1: he was unable to expand this in powers of x,
Speaker 1: he laid down. However, the principle of interpolation. Thus, as
Speaker 1: the ordinate of the circle y equals x minus x
Speaker 1: squared raised to the half, is a geometrical mean between
Speaker 1: the ordinate of the curve y equals quantity x minus
Speaker 1: x squared rays to the zero, and y equals quantity
Speaker 1: x minus x squared rays to the one. It might
Speaker 1: be supposed that, as an approximation the area of the semicircle,
Speaker 1: the integral from zero to one of quantity x minus
Speaker 1: x squared dx, which is one eighth Pi, might be
Speaker 1: taken as the geometrical mean between the values of the
Speaker 1: integral from zero to one of quantity x minus x
Speaker 1: squared rays to the zero dx, and the integral of
Speaker 1: zero to one of quantity x minus x squared rays
Speaker 1: to the one dx. That is, the values one and
Speaker 1: one sixth. This is equivalent to taking four multiplied by
Speaker 1: the root of two thirds or three point two six
Speaker 1: approximately as the value of pi. But Wallace argued, we
Speaker 1: have in fact a series one one sixth one over thirty,
Speaker 1: one over one forty and so on, and therefore the
Speaker 1: term interpolated between one and one sixth ought to be
Speaker 1: so chosen as to obey the law of the series. This,
Speaker 1: by an elaborate method which I need not describe in detail,
Speaker 1: leads to a value for the interpolated term, which is
Speaker 1: equivalent to taking pie equals two times two over one
Speaker 1: times two thirds times four thirds times four fifths times six,
Speaker 1: fifth times six over seven times eight over seven times
Speaker 1: eight over nine, and so on. The subsequent mathematicians of
Speaker 1: the seventeenth century constantly used interpolation to obtain results which
Speaker 1: we should attempt to obtain by direct analysis. A few
Speaker 1: years later, in sixteen fifty nine, Wallace published a tract
Speaker 1: containing the solution of the problems on the cycloid which
Speaker 1: had been proposed by Pascal. In this he incidentally explained
Speaker 1: how the principles laid down in his Arithmetica Infinitorum could
Speaker 1: be used for the rectification of algebraic curves, and gave
Speaker 1: a solution of the problem to rectify the semi cubic
Speaker 1: parabola ex cubed equals ayz squared, which had been discovered
Speaker 1: in sixteen fifty seven by his pupil, William Neel. This
Speaker 1: was the first case in which the length of a
Speaker 1: curved line was determined by mathematics, and since all attempts
Speaker 1: to rectify the ellipse in the hyperbolea had been necessarily ineffectual,
Speaker 1: it had been previously supposed that no curves could be rectified,
Speaker 1: as indeed Descartes had definitely asserted to be the case.
Speaker 1: The cycloid was a second curve rectified, and this was
Speaker 1: done by Wren in sixteen fifty eight. Early in sixteen
Speaker 1: fifty eight, a similar discovery, independent at that of Nile,
Speaker 1: was made by Van Harat, and this was published by
Speaker 1: Van Schutten in his edition of Descartes metric in sixteen
Speaker 1: fifty nine. Van Heret's method is as follows. He supposes
Speaker 1: the curve to be referred to rectangular axes. If this
Speaker 1: be so, and if x y be the coordinates of
Speaker 1: any point on it and end the length of the normal,
Speaker 1: and if another point whose coordinates are x new to
Speaker 1: be taken such that new to h equals the ratio
Speaker 1: N to y, where h is a constant, then if
Speaker 1: ds be the element of the length of the required curve,
Speaker 1: then we have by similar triangles the ratio d s
Speaker 1: to dx equals nd to y. Therefore hds equals new dx. Hence,
Speaker 1: if the area of the locus of the point x
Speaker 1: new can be found, the first curve can be rectified.
Speaker 1: In this way. Van Herat affected the rectification of the
Speaker 1: curve yqubed equals a X squared, and added that the
Speaker 1: rectification of the parabola y squared equals as is impossible
Speaker 1: since it requires the quadrature of the hyperbola. The solutions
Speaker 1: given by Nile and Wallace are somewhat similar to that
Speaker 1: given by Van Herret, but no general rules enunciated, and
Speaker 1: the analysis is clumsy. A third method was suggested by
Speaker 1: fermat in sixteen sixty, but it is both inelegant and laborious.
Speaker 1: In sixteen sixty five, Wallace published the first systematic Treatise
Speaker 1: on Analytical conic sections. I have already mentioned that the
Speaker 1: geometrie of Descartes is both difficult and obscure, and to
Speaker 1: many of his contemporaries, to whom the method was new,
Speaker 1: it must have been incomprehensible. Wallace made the method intelligible
Speaker 1: to all mathematicians. This is the earliest book in which
Speaker 1: these curves are considered and defined as curves of the
Speaker 1: second degree and not as sections of a cone on
Speaker 1: a circular base. The theory of the col collision of
Speaker 1: bodies was propounded by the Royal Society in sixteen sixty
Speaker 1: eight for the consideration of mathematicians. Wallace, Wren, and Huygens
Speaker 1: sent correct and similar solutions, all depending on what is
Speaker 1: now called the conservation of momentum. But while Ren and
Speaker 1: Huygens confine their theory to perfectly elastic bodies, Wallace considered
Speaker 1: also imperfectly elastic bodies. This was followed in sixteen sixty
Speaker 1: nine by a work on statics centers of gravity, and
Speaker 1: in sixteen seventy by one on dynamics. These provide a
Speaker 1: convenient synopsis of what was then known on the subject.
Speaker 1: In sixteen eighty five Wallace published in Algebra, preceded by
Speaker 1: historical account of the development of the subject, which contains
Speaker 1: a great deal of valuable information. The second edition, issued
Speaker 1: in sixteen ninety three, in forming the second volume of
Speaker 1: his opera, is considerably enlarged. This algebra is noteworthy as
Speaker 1: containing the first systematic use of formulae. A given magnitude
Speaker 1: is here represented by the numerical ratio, which it bears
Speaker 1: to the unit of the same kind of magnitude. Thus,
Speaker 1: when Wallace wants to compare two lengths, he regards each
Speaker 1: as containing so many units of length. This perhaps will
Speaker 1: be made clearer if I say that the relation between
Speaker 1: the space described in any time by a particle moving
Speaker 1: with uniform velocity would be denoted by Wallace by the
Speaker 1: formula S equals vt, where s is the number representing
Speaker 1: the ratio of the space described to the unit of length,
Speaker 1: while previous writers would have denoted the same relation by
Speaker 1: stating what is equivalent to the proportion S one over
Speaker 1: s two equals v one T one over v two
Speaker 1: T two c for example, Newton's Principia Bookie one, Section one,
Speaker 1: Lemma ten or eleven. It is curious to note that
Speaker 1: Wallace rejected as absurd the now usual idea of a
Speaker 1: negative number as being less than nothing, but accepted the
Speaker 1: view that it is something greater than infinity. The latter
Speaker 1: opinion may be right and consistent with the former, but
Speaker 1: it is hardly a more simple one. Vermat While Descartes
Speaker 1: was laying down the foundation of analytical geometry, the same
Speaker 1: subject was occupying the attention of another and hardly less
Speaker 1: distinguished Frenchman. This was Fermat Pierre de Fermat, who was
Speaker 1: born near Montabon in sixteen o one and died at
Speaker 1: Castree in January twelfth, sixteen sixty five, was the son
Speaker 1: of a leather merchant. He was educated at home. In
Speaker 1: sixteen thirty one he obtained the post of councilor for
Speaker 1: the local parliament at Toulouse, and he discharged the duties
Speaker 1: of the office with scrupulous accuracy and fidelity. There devoting
Speaker 1: most of his leisure to mathematics. He spent the remainder
Speaker 1: of his life, a life which, but for a somewhat
Speaker 1: acrimonious dispute with Descartes on the validity of analysis used
Speaker 1: by the latter, was unruffled by any event which calls
Speaker 1: for special notice. The dispute was due chiefly to the
Speaker 1: obscurity of Descartes, but the tact and courtesy of Vermat
Speaker 1: brought it to a friendly conclusion. Vermat was a good scholar,
Speaker 1: and amused himself by conjecturally restoring the work of Apollonius
Speaker 1: on plain loci. Except for a few isolated papers, Vermat
Speaker 1: published nothing in his lifetime, and gave no systematic exposition
Speaker 1: of his methods. Some of the most striking of his
Speaker 1: results are found after his death on loose sheets of
Speaker 1: paper written on the margins of works which he had
Speaker 1: read and annotated, and are unaccompanied by any proof. It
Speaker 1: is thus somewhat difficult to estimate the dates and originality
Speaker 1: of his work. After his death, his papers and correspondences
Speaker 1: were printed by his nephew at Toulouse in two volumes
Speaker 1: sixteen seventy and sixteen seventy nine. A summary of it
Speaker 1: with notes was published by Brazen and Toulouse in eighteen
Speaker 1: fifty three, and a reprint of it was issued at
Speaker 1: Berlin in eighteen sixty one. A new edition is now
Speaker 1: being issued by the French government, which will include some
Speaker 1: letters on his discoveries and methods in the theory of
Speaker 1: numbers recently found at Leyden by M. Charles Henry Fermatt
Speaker 1: was constitutionally modest and retiring, and does not seem to
Speaker 1: have intended his papers to be published. It is probable
Speaker 1: that he revised his notes as occasion required, and that
Speaker 1: his published works represent the final form of his retire searches,
Speaker 1: and therefore cannot be dated much earlier than sixteen sixty.
Speaker 1: I shall consider separately one his investigations into the theory
Speaker 1: of numbers, two his use in geometry of analysis and
Speaker 1: of infinitesimals, and three his method of treating questions of probability. One.
Speaker 1: The theory of numbers appears to have been the favorite
Speaker 1: study of Fermat. He prepared an addition of Diophantus, and
Speaker 1: the notes and comments thereon contain numerous theorems of considerable elegance.
Speaker 1: This forms the first of two volumes of his works.
Speaker 1: Most of the proofs of Fermat are lost, and it
Speaker 1: is possible that some of them were not rigorous, an
Speaker 1: induction by analogy and the intuition of genius sufficing to
Speaker 1: lead him to correct results. The following examples will illustrate
Speaker 1: these investigations. A. If p, b, prome, and A be
Speaker 1: a prime to P, then A to the power of
Speaker 1: P minus one subtract one is divisible by p. That is,
Speaker 1: when you subtract one from A to the power of
Speaker 1: P minus one, this is equivalent to zero modulus P.
Speaker 1: A proof of this, first given by Euler, is well known.
Speaker 1: A more general theorem, that of a to the power
Speaker 1: of five of n subtracting one from that result is
Speaker 1: equivalent to zero mod n. Where a is prime to
Speaker 1: n and five n is the number of integers less
Speaker 1: than n and prime to it. B A prime greater
Speaker 1: than two can be expressed as the difference of two
Speaker 1: square integers in one and only one way for MAT's
Speaker 1: proof is as follows. Let N be the prime, and
Speaker 1: suppose it equal to x squared minus y squared, that is,
Speaker 1: to the product x plus y times x minus y. Now,
Speaker 1: by hypothesis, the only integral factors of N are n
Speaker 1: a unity. Hence x plus y equals n n x
Speaker 1: minus y equals one. Solving these equations, we get x
Speaker 1: equals a half times n plus one, and y equals
Speaker 1: a half times n minus one. C He gave a
Speaker 1: proof of the statement by Diophantis quoted above that the
Speaker 1: sum of the squares of two integers cannot be of
Speaker 1: the form four n minus one, and that he added
Speaker 1: a corollary, which I take to mean that it is
Speaker 1: impossible that the product of a square and a prime
Speaker 1: of the form four n minus one, even if multiplied
Speaker 1: by a number prime to the latter, can be either
Speaker 1: a square or the sum of two squares. For example,
Speaker 1: forty four is a multiple of eleven, which is of
Speaker 1: the form four times three minus one by four, hence
Speaker 1: it cannot be expressed is a sum of two squares.
Speaker 1: He also stated that a number of the form a
Speaker 1: squared plus b squared, where a is prime to b,
Speaker 1: cannot be divided by a prime of the form four
Speaker 1: n minus one d. Every prime of the form four
Speaker 1: n plus one is expressible, and that in one way
Speaker 1: only as a sum of two squares. This problem was
Speaker 1: first solved by Euler, who shewed that the number of
Speaker 1: the form two to the power m times four n
Speaker 1: minus one can be always expressed as a sum of
Speaker 1: two squares. E If a, b c are integers such
Speaker 1: that A squared plus b squared equals c squared, then
Speaker 1: A times b cannot be a square. Lagrange gave a
Speaker 1: solution of this f the determination of a number x
Speaker 1: such an x squared n plus one may be a
Speaker 1: square Where n is a given integer which is not
Speaker 1: a square. G There is only one integral solution of
Speaker 1: the equation x squd plus two equals y cubed, and
Speaker 1: there are only two integral solutions of the equation x
Speaker 1: squard plus four equals y cubed. The required solution are
Speaker 1: evidently for the first equation x equals five, and for
Speaker 1: the second equation x equals two and x equals eleven.
Speaker 1: This question was issued as a challenge to the English
Speaker 1: mathematicians Wallace and Digby. H No integral values of x,
Speaker 1: y z can be found to satisfy the equation x
Speaker 1: to the n plus y to the n equals z
Speaker 1: to the n if n be an integer greater than two.
Speaker 1: This proposition, known as Fermat's last theorem, has acquired extraordinary
Speaker 1: celebrity from the fact that no general demonstration of it
Speaker 1: has ever been given, but there is no reason to
Speaker 1: doubt that it is true. Probably Fermat discovered its truth
Speaker 1: the case of an equals three, and then for the
Speaker 1: case of an equals four. His proof for the former
Speaker 1: of these cases is lost, but that for the latter
Speaker 1: is extant, and a similar proof for the case of
Speaker 1: an equals three was given by Euler. These proofs depend
Speaker 1: on shewing that of three integral values of x y
Speaker 1: z can be found which satisfy the equation, then it
Speaker 1: will be possible to find three and other similar integers
Speaker 1: which satisfy it. In this way, finally, we shew that
Speaker 1: the equation must be satisfied by three values, which obviously
Speaker 1: do not satisfy it. Thus no integral solution is possible.
Speaker 1: It would seem that this method is inapplicable to any
Speaker 1: cases except those of an equals three and an equals four.
Speaker 1: For Matt's discovery of the general theorem was made later.
Speaker 1: An easy demonstration can be given on the assumption that
Speaker 1: a number can be resolved into prime or complex factors
Speaker 1: in one and only one way. The assumption has been
Speaker 1: made by some writers, but it is not universally true.
Speaker 1: It is possible that Fermatt made some such supposition, though
Speaker 1: it is perhaps more likely that he discovered a rigorous demonstration.
Speaker 1: In eighteen twenty three, le Gendre obtained a proof for
Speaker 1: the case of an equals five. In eighteen thirty two,
Speaker 1: le Jeund Richle gave one for an equals fourteen, and
Speaker 1: in eighteen forty la Mae and Lebezgue gave proofs for
Speaker 1: an equals seven. The proposition appears to be true universally,
Speaker 1: and in eighteen forty nine Coumer, by means of ideal primes,
Speaker 1: proved it to be so for all numbers, except if
Speaker 1: any which satisfy three conditions. It is not certain whether
Speaker 1: any number can be found to satisfy these conditions, but
Speaker 1: there is no number less than one hundred which does so.
Speaker 1: The proof is complicated and difficult, and there can be
Speaker 1: no doubt is based on the considerations unknown to Fermat.
Speaker 1: I may add that to prove the truth of the
Speaker 1: proposition where n is greater than four, it is obviously
Speaker 1: sufficient to confine ourselves to cases where n is a prime,
Speaker 1: and the first step in Coumer's demonstration is to shew
Speaker 1: that in such cases one of the numbers x, y
Speaker 1: z must be divisible by n. A letter exists now
Speaker 1: in the University at Leyden, which gave an idea of
Speaker 1: Format's methods. The letter is undated, but it would appear
Speaker 1: that at the time Fermatt wrote it he had proved
Speaker 1: the proposition h above only for the case when n
Speaker 1: equals three two. I next proceed to mention Fermat's use
Speaker 1: in geometry of analysis and of infinitesimals. It would seem
Speaker 1: from his correspondence that he had thought out the principles
Speaker 1: of analytic geometry for himself before reading Descartes geometri and
Speaker 1: had realized that from the equation, or, as he calls it,
Speaker 1: the specific property of a curve, all its properties could
Speaker 1: be deduced. His extent papers on geometry deal, however, mainly
Speaker 1: with the application of infinitesimals, to the determination of the
Speaker 1: tangents to curves and to the quadrature of curves, and
Speaker 1: to the questions of maxima and minima. Probably these papers
Speaker 1: are a revision of his original manuscripts, which he destroyed
Speaker 1: and were written about sixteen sixty three. But there is
Speaker 1: no doubt that he was in possession of the general
Speaker 1: idea of his methods for finding maxima and minima as
Speaker 1: early as sixteen twenty eight and sixteen twenty nine. He
Speaker 1: obtained the sub tangent to the eclipse cycloid cissoid, conchoid
Speaker 1: and quadratrics by making the ordinance of the curve and
Speaker 1: a straight line the same for two points whose absissay
Speaker 1: were x and xus e. But there is nothing to
Speaker 1: indicate that he was aware that the process was general,
Speaker 1: and though in the course of his work he used
Speaker 1: the principle, it is probable that he never separated it,
Speaker 1: so to speak, from the symbols of the particular problem
Speaker 1: he was considering. The first definite statement of the method
Speaker 1: was due to Barrow and was published in sixteen sixty nine.
Speaker 1: For Matt also obtained the areas of parabolas and hyperbolas
Speaker 1: of any order, and determine the center of mass of
Speaker 1: a few simple curves and of a paraboloid of revolution.
Speaker 1: As an example of his method of solving these equations,
Speaker 1: I will quote his solution of the problem to find
Speaker 1: the area but between the parabola yqube equals p x
Speaker 1: squared on the axis of X and the line x
Speaker 1: equals a. He says that if the several ordinates at
Speaker 1: the points for which x is equal to a A
Speaker 1: times the quantity one minus e, and A times the
Speaker 1: quantity one minus e to the power of two, and
Speaker 1: so on be drawn, the area will be split into
Speaker 1: a number of little rectangles whose areas are respectively a
Speaker 1: E times p A squared to the power of one
Speaker 1: third a e times one minus e, multiplied by the
Speaker 1: quantity p A squared times the quantity one minus e squared,
Speaker 1: all under a cube root. The sum of these is
Speaker 1: P to the one third A to the five thirds
Speaker 1: times e divided by the quantity d one minus one
Speaker 1: minus e to the five thirds, and by a subsidiary proposition,
Speaker 1: for of course he was not acquainted with the binomial theorem.
Speaker 1: He finds the limit of this when e vanishes to
Speaker 1: be three fifths p to the one third eight of
Speaker 1: the five thirds. The theorems last mentioned were published only
Speaker 1: after his death, and probably they were not written till
Speaker 1: he had read the works of Cavalieri and Wallace. Kepler
Speaker 1: had remarked that the values of a function immediately adjacent
Speaker 1: to and on either side of, a maximum or minimum
Speaker 1: value must be equal. For mad applied this principle to
Speaker 1: a few examples. Thus, to find the maximum value of
Speaker 1: x times the quantity A minus x, his method is
Speaker 1: essentially equivalent to taking a consecutive value of x, namely
Speaker 1: x minus e, where e is very small, and putting
Speaker 1: x times quantity a minus x equal two quantity x
Speaker 1: minus e multiplied by the quantity a minus x plus e.
Speaker 1: Simplifying and ultimately putting e equals zero, we get x
Speaker 1: equals half a. This value of x makes the given
Speaker 1: expression a maximum three. Fermat must share with Pascal the
Speaker 1: honor of having founded the theory of probabilities. I have
Speaker 1: already mentioned the problem proposed to Pascal, and which he
Speaker 1: communicated to Fermat, and have there given Pascal's solution. Format's
Speaker 1: solution depends on the theory of combinations, and will be
Speaker 1: sufficiently illustrated by the following example, the substance of which
Speaker 1: was taken from a letter dated August twenty fourth, sixteen
Speaker 1: fifty four, which occurs in the correspondence with Pascal. Fermat
Speaker 1: discusses the case of two players and supposes that the
Speaker 1: first one same two points to win, and the second
Speaker 1: three points. The game will be then certainly decided in
Speaker 1: the course of four trials. Take the letters A and
Speaker 1: B and write down all the combinations that can be
Speaker 1: formed of four letters. These combinations are the following sixteen.
Speaker 1: In all. Now let big A denote the player who
Speaker 1: wants two points, and big B denote the player who
Speaker 1: wants three points. Then in these sixteen combinations, every combination
Speaker 1: in which A occurs twice or oftener represents a case
Speaker 1: favorable to big A, and every combination in which little
Speaker 1: B occurs three times or oftener represents a case favorable
Speaker 1: to player big B. Thus, on counting them, it will
Speaker 1: be found that there will be eleven cases favorable to
Speaker 1: A and five cases favorable to B, and since these
Speaker 1: cases are all equally likely, a's chance of winning the
Speaker 1: game is to be's chances as eleven is to five.
Speaker 1: The only other problem on this subject, which as far
Speaker 1: as I know, attracted the attention of Framat, was also
Speaker 1: proposed to him by Pascal, and was as follows. A
Speaker 1: person undertakes to throw a six with a die in
Speaker 1: eight throws, supposing him to have made three throws without success,
Speaker 1: what proportion of the stake should he be allowed to
Speaker 1: take on condition of giving up his fourth throw. For
Speaker 1: Matt's reasoning is his follows. The chance of success is
Speaker 1: one in six, so that he should be allowed to
Speaker 1: take one sixth of the stake on condition of giving
Speaker 1: up his throw. But if we wish to estimate the
Speaker 1: value of the fourth throw before any throw is made,
Speaker 1: then the first throw is worth one six of the steak.
Speaker 1: The second is worth one sixth of what remains, and
Speaker 1: that is five thirty sixths of the steak. The third
Speaker 1: throw is worth one six of what now remains, That
Speaker 1: is twenty five over two hundred and sixteenths of the steak.
Speaker 1: The fourth throw is worth one sixth of what now remains,
Speaker 1: and that is one hundred and twenty five over one thousand,
Speaker 1: two hundred ninety sixths of the steak. For Matt does
Speaker 1: not seem to have carried the matter much further, But
Speaker 1: his correspondence with Pascal shoes that his views on the
Speaker 1: fundamental principles of the subject were accurate, those of Pascal
Speaker 1: were not altogether correct. For Maat's reputation is quite unique
Speaker 1: in the history of science. The problems on numbers which
Speaker 1: he had proposed long defied all efforts to solve them,
Speaker 1: and many of them yielded only to the skill of Euler.
Speaker 1: One still remains on solved. This extraordinary achievement has overshadowed
Speaker 1: his other work, but in fact it is all of
Speaker 1: the highest order of excellence, and we can only regret
Speaker 1: that he thought fit to write so little. End of
Speaker 1: Part two read by Paul King, p J K dot
Speaker 1: scripts dot MI I T dot ed U forward slash
Speaker 1: p K J