Speaker 1: Chapter fifteen of A Short account of the History of mathematics.
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Speaker 1: p kJ. A Short account of the history of mathematics
Speaker 1: by W. W. Rowse Ball, Chapter fifteen, History of Mathematics
Speaker 1: from Descartes to Huygens circa sixteen thirty five to sixteen
Speaker 1: seventy five. I propose in this chapter to consider the
Speaker 1: history of mathematics during the forty years in the middle
Speaker 1: of the seventeenth century. I regard Descartes, Cavalieri, Pascal, Wallace,
Speaker 1: Fermat and Huygens as the leading mathematicians of this time.
Speaker 1: I shall treat them in that order, and I shall
Speaker 1: conclude with a brief list of the more eminent remaining
Speaker 1: mathematicians of the same date. I have already stated that
Speaker 1: the mathematicians of this period, and the remark applies more
Speaker 1: particularly to Descartes, Pascal, and Fermont, were largely influenced by
Speaker 1: the teaching of Kepler and de Rogues. And I would
Speaker 1: repeat again that I regard these latter and Galileo as
Speaker 1: forming a connecting link between the writers of the Renaissance
Speaker 1: and those of modern times. I should also add that
Speaker 1: the mathematicians considered in this chapter were contemporaries, and although
Speaker 1: I have tried to place them roughly in such an
Speaker 1: order that their chief works shall come in chronological arrangement,
Speaker 1: it is essential to remember that they were in relation
Speaker 1: with one another, and in general, were acquainted with one
Speaker 1: another's researches as soon as they were published Descartes. Subject
Speaker 1: to the above remarks, we may consider Descartes as the
Speaker 1: first of the modern school of mathematics. Renee Descout was
Speaker 1: born near Tours on March thirty first, fifteen ninety six,
Speaker 1: and died at Stockholm on February eleventh, sixteen fifty. He
Speaker 1: was thus a contemporary of Galileo and des Ragues. His father,
Speaker 1: who as the name implies, was of a good family,
Speaker 1: was accustomed to spend half the year at Rennes when
Speaker 1: the local parliament, in which he held a commission as
Speaker 1: a councilor, was in session, and the rest of the
Speaker 1: time on his family estate at Lake Harte at La Haye. Renee,
Speaker 1: the second of a family of two sons and one daughter,
Speaker 1: was sent at the age of eight years to the
Speaker 1: Jesuit school at La Fleche, and of the admirable discipline
Speaker 1: and education there given he speaks most highly. On account
Speaker 1: of his delicate health, he was permitted to lie in
Speaker 1: bed till late in the morning. This was a custom
Speaker 1: which he had always followed, and when he visited Pascal
Speaker 1: in sixteen forty seven, he told him that the only
Speaker 1: way to do good work in mathematics and to preserve
Speaker 1: his health was never to allow anyone to make him
Speaker 1: get up in the morning before he felt inclined to
Speaker 1: do so, an opinion which I chronicle for the benefit
Speaker 1: of any schoolboy into whose hands this work may fall.
Speaker 1: On leaving school in sixteen twelve, Descartes went to Paris
Speaker 1: to be introduced to the world of fashion. Here, through
Speaker 1: the medium of the Jesuits, he made the acquaintance of
Speaker 1: Medorges and renewed his schoolboy friendship with father Mersenne, and
Speaker 1: together with them he devoted the two years of sixteen
Speaker 1: fifteen and sixteen sixteen to the study of mathematics. At
Speaker 1: the time, a man of position usually entered either the
Speaker 1: army or the church. Des Cartes chose the former profession,
Speaker 1: and in sixteen seventeen joined the army of Prince Marie
Speaker 1: of Orange. Then at Breda. Walking through the streets, he
Speaker 1: saw a placard in Dutch, which excited his curiosity, and,
Speaker 1: stopping the first passer, asked him to translate it into
Speaker 1: either French or Latin. The stranger, who happened to be
Speaker 1: Isaac Beekman, the head of the Dutch college at Dort,
Speaker 1: offered to do so if Descartes would answer it, the
Speaker 1: placard being in fact a challenge to all the world
Speaker 1: to solve a geometrical problem there given. Descartes worked it
Speaker 1: out within a few hours, and a warm friendship between
Speaker 1: him and Beakman was the result. This unexpected test of
Speaker 1: his mathematical attainments made the uncongenial life of the army
Speaker 1: distasteful to him, but under family influence and tradition, he
Speaker 1: remained a soldier and was persuaded at the commencement of
Speaker 1: the Thirty Years War to volunteer under the Court at
Speaker 1: Boucquoi in the Army of Bavaria. He continued sank youed
Speaker 1: all this time to occupy his leisure with mathematical studies,
Speaker 1: and was accustomed to date the first ideas of his
Speaker 1: new philosophy and of his analytical geometry from three dreams
Speaker 1: he experienced on the night of November tenth, sixteen nineteen,
Speaker 1: at Newburgh, when campaigning on the Danube. He regarded this
Speaker 1: as the critical day of his life, and one which
Speaker 1: determined his whole future. He resigned his commission in the
Speaker 1: spring of sixteen twenty one and spent the next five
Speaker 1: years in travel, during most of which time he continued
Speaker 1: to study pure mathematics. In sixteen twenty six we find
Speaker 1: him settled at Paris, a little well built figure, modest
Speaker 1: clad in green taffety, and only wearing a sword and
Speaker 1: feather in token of his quality as a gentleman. During
Speaker 1: the first two years there he interested himself in general
Speaker 1: society and spent his leisure in the construction of optical instruments.
Speaker 1: But these pursuits were merely the relaxations of one who
Speaker 1: failed to find in philosophy that theory of the universe,
Speaker 1: which he was convinced finally awaited him. In sixteen twenty eight,
Speaker 1: Cardinal Debray, the founder of the Oratorians, met Descartes, and
Speaker 1: was so much impressed by his conversation that he urged
Speaker 1: on him the duty of devoting his life to the
Speaker 1: examination of truth. Descartes agreed, and, the better to secure
Speaker 1: himself from interruption, moved to Holland, then, at the height
Speaker 1: of its power. There for twenty years he lived, giving
Speaker 1: up all his time to philosophy and mathematics. Science, he says,
Speaker 1: may be compared to a tree. Metaphysics is the root,
Speaker 1: physics is the trunk, and the three chief branches are mechanics, medicine,
Speaker 1: and morals, these forming the three applications of our knowledge,
Speaker 1: namely to the external world, to the human body, and
Speaker 1: to the conduct of life. And with these subjects alone
Speaker 1: his writings are concerned. He spent the first four years
Speaker 1: sixteen twenty nine to sixteen thirty three of his stay
Speaker 1: in Holland in writing Lemonde, which embodies an attempt to
Speaker 1: give a physical theory of the universe, but finding that
Speaker 1: its publication was likely to bring on him the hostility
Speaker 1: of the church, and having no desire to pose as
Speaker 1: a martyr, he abandoned it. The complete manuscript was published
Speaker 1: in sixteen sixty four. He then devoted himself to composing
Speaker 1: a treatise on universal science. This was published at Leyden
Speaker 1: in sixteen thirty seven, titled The Discourse on Method, and
Speaker 1: was accompanied with three appendices entitled Optics, Meteorology, and Geometry.
Speaker 1: It is from the last of these that the invention
Speaker 1: of analytical geometry dates. In sixteen forty one he published
Speaker 1: to work called Meditations, in which he explained at some
Speaker 1: length his views of philosophy as sketched out in the Discourse.
Speaker 1: In sixteen forty four he issued the Principia philosophy as
Speaker 1: a greater part of which was devoted to physical science,
Speaker 1: especially the laws of motion and the theory of vortices.
Speaker 1: In sixteen forty seven he received a pension from the
Speaker 1: French court in honor of his discoveries. He went to
Speaker 1: Sweden on the invitation of the Queen in sixteen forty nine,
Speaker 1: and died a few months later of inflammation of the lungs.
Speaker 1: In appearance, Descartes was a small man, with a large head,
Speaker 1: projecting brow, prominent nose, and black hair coming down to
Speaker 1: his eyebrows. His voice was feeble. Considering the range of
Speaker 1: his studies, he was by no means widely read. And
Speaker 1: he despised both learning and art unless something tangible could
Speaker 1: be extracted. Therefrom physician. He was cold and selfish. He
Speaker 1: never married and left no descendants, though he had one
Speaker 1: illegitimate daughter who died young. As to his philosophical theories,
Speaker 1: it would be sufficient to say that he discussed the
Speaker 1: same problems which have been debated for the last two
Speaker 1: thousand years. It is hardly necessary to say that the
Speaker 1: problems themselves are of great interest. But from the nature
Speaker 1: of the case, no solution ever offered is capable either
Speaker 1: of proof or disproof. And whenever a philosopher like Descartes
Speaker 1: believes that he has at last finally settled a question,
Speaker 1: it has been easy for his successors to point out
Speaker 1: the fallacy and his assumptions. All that can be effected
Speaker 1: is to make one explanation somewhat more probable than another.
Speaker 1: I have read somewhere that philosophy has always been chiefly
Speaker 1: engaged with the interrelations of God, nature, and man. The
Speaker 1: earliest philosophers were Greeks, who occupied themselves mainly with the
Speaker 1: relations between God and Nature, and dealt with man separately.
Speaker 1: The Christian Church was so absorbed in the relation of
Speaker 1: God to man as to entirely neglect nature. Finally, modern
Speaker 1: philosophers concern themselves chiefly with the relations between man and nature.
Speaker 1: Whether this is a correct historical generalization of the views
Speaker 1: which have been successfully prevalent, I do not care to
Speaker 1: discuss here. But the statement as to the scope of
Speaker 1: modern philosophy marks the limitations of Descartes's writings, and they
Speaker 1: may be taken as the commencement of the modern school.
Speaker 1: Descartes's chief contributions to mathematics were his analytical geometry and
Speaker 1: his theory of vortices, and it is on his researches
Speaker 1: in connection with the former of these subjects that his
Speaker 1: reputation rests. Analytical geometry does not consist merely as a sometimes,
Speaker 1: as loosely said in the application of algebra to geometry,
Speaker 1: that had been done by Archimytes and many others, and
Speaker 1: had become the usual method of procedure in the works
Speaker 1: of the mathematicians of the sixteenth century. The great advance
Speaker 1: made by Descartes was that he saw that a point
Speaker 1: in a plane could be completely determined if its distances,
Speaker 1: say x and y form two fixed lines drawn at
Speaker 1: right angles in the plane were given with the convention
Speaker 1: familiar to us as to the interpretation of positive and
Speaker 1: negative values, and that though an equation f of xy
Speaker 1: equals zero was indeterminate and could be satisfied by an
Speaker 1: infinite number of values of x and y, yet these
Speaker 1: values of x and y determine the coordinates of a
Speaker 1: number of points which form a curve of which the
Speaker 1: equation f of xy equals zero expresses some geometrical property,
Speaker 1: that is, a property true of the curve at every point.
Speaker 1: Descartes assured that a point in space could be similarly
Speaker 1: determined by three coordinates, but he confined his attention to
Speaker 1: plain curves. It was at once seen by Descartes and
Speaker 1: his successors that in order to investigate the properties of
Speaker 1: a curve, it was sufficient to select any characteristic geometrical
Speaker 1: property as a definition, and to express it by means
Speaker 1: of an equation between the current coordinates of any point
Speaker 1: in the curve. That is, to translate the definition into
Speaker 1: a language of analytical geometry. The equation so obtained contains
Speaker 1: implicitly every property of the curve, and any particular property
Speaker 1: can be deduced from it by ordinary algebra without troubling
Speaker 1: about the geometry of the figure. The points in which
Speaker 1: the two curves intersect can be determined by finding the
Speaker 1: roots common to their two equations. I need not go
Speaker 1: further into details, for nearly everyone to whom the above
Speaker 1: is intelligible will have read analytical Geometry and be able
Speaker 1: to appreciate the value of its invention. Descartes Geometry is
Speaker 1: divided into three books. The first two of these treat
Speaker 1: of analytical geometry, and the third includes an analysis of
Speaker 1: the algebra then current. It is somewhat difficult to follow
Speaker 1: the reasoning, but the obscurity was intentional and due to
Speaker 1: the jealousy of Descartes. Translated, I omitted nothing says he
Speaker 1: except design. I had expected that people who have any
Speaker 1: math skill would not fail to say that. I had
Speaker 1: not said anything that has not already been said about
Speaker 1: the coordinate system. But I have hopefully made this topic
Speaker 1: more intelligible. The first book commences with an explanation of
Speaker 1: the principles of analytic geometry, and contains a discussion of
Speaker 1: a certain problem which has been propounded by Paswers in
Speaker 1: the seventh book of his Synagogue, and of which some
Speaker 1: particular cases have been considered by Euclid and Apollonius. The
Speaker 1: general theorem has baffled previous geometricians, and it was in
Speaker 1: the attempt to solve it that Descartes was led to
Speaker 1: the invention of analytical geometry. The full nunciation of the
Speaker 1: problem is rather involved, but the most important case is
Speaker 1: to find the locus of a point such that the
Speaker 1: product of the perpendiculars on m given straight lines shall
Speaker 1: be in a constant ratio to the product of the
Speaker 1: perpendiculars on n given straight lines. The ancients had solved
Speaker 1: this geometrically for the case em equals on n equals one,
Speaker 1: and the case m equals one, n equals two. Pappus
Speaker 1: had further stated that if m equals n equals to,
Speaker 1: the locus was a conic but he gave no proof.
Speaker 1: Daycartes also failed to prove this by pure geometry, but
Speaker 1: he shewed that the curve was represented by an equation
Speaker 1: of the second degree, that is, it was a conic section. Subsequently,
Speaker 1: Newton gave an elegant solution of the problem by pure geometry.
Speaker 1: In the second book, Descartes divides curves into two classes,
Speaker 1: namely geometrical and mechanical curves. He defines geometrical curves as
Speaker 1: those which can be generated by the intersection of two lines,
Speaker 1: each moving parallel to one coordinate axis, with commensurable velocities,
Speaker 1: by which he meant that dy by dx was an
Speaker 1: algebraical function, as for example is the case in the
Speaker 1: ellipse and the cissoid. He calls the curve mechanical when
Speaker 1: the ratio of the velocities of these lines is incommensurable,
Speaker 1: by which he meant that dy by dx was a
Speaker 1: transcendental function, as for example is the case in the
Speaker 1: cycloid and the quadratris. Descartes confined his discussion to algebraical
Speaker 1: curves das not treat of the theory of mechanical curves.
Speaker 1: The classification into algebraical and transcendental curves now usual is
Speaker 1: due to Newton. Daycart also paid particular attention to the
Speaker 1: theory of the tangents to curves, as perhaps might be
Speaker 1: inferred from his system of classification just alluded to. Then,
Speaker 1: the current definition of a tangent at a point was
Speaker 1: a straight line through the point such that between it
Speaker 1: and the curve no other straight line could be drawn,
Speaker 1: i e. The straight line of closest contact. Daycartes proposed
Speaker 1: to substitute for this that the tangent was the limiting
Speaker 1: position of the secin formatt and at a later date
Speaker 1: McLaurin and Lagrange adopted this definition. Barrow, followed by Newton
Speaker 1: and Leebanitz, considered the curve as the limit of an
Speaker 1: inscribed polygon when the sides become indefinitely small, and stated
Speaker 1: that the side of the polygon, when produced, became in
Speaker 1: the limit a tengent to the curve. Roberval, on the
Speaker 1: other hand, defined a tangent at a point as the
Speaker 1: direction of the motion at that instant of a point
Speaker 1: which was describing the curve. The results are the same
Speaker 1: whichever definition is selected, but the controversy as to which
Speaker 1: definition was the correct one was nonetheless lively. Descartes illustrated
Speaker 1: his theory by giving the general rule for drawing tangents
Speaker 1: and normals to a roulette. The method used by Descartes
Speaker 1: for finding the tangent or normal at any point of
Speaker 1: a given curve was substantially as follows. He determined the
Speaker 1: center and radius of a circle which should cut the
Speaker 1: curve into consecutive points. There, the tangent to the circle
Speaker 1: at that point will be the required tangent to the curve.
Speaker 1: In modern textbooks, it is usual to express the condition
Speaker 1: that two of the points in which a straight line
Speaker 1: such as y equals mx plas c cuts the curve
Speaker 1: shall coincide with the given point. This enables us to
Speaker 1: determine M and C, and thus the equation of the
Speaker 1: tangent there is determined. Descartes, however, did not venture to
Speaker 1: do this. But selecting a circle as the simplest curve
Speaker 1: and one to which we know how to draw a tangent,
Speaker 1: he so fixed his circle as to make it touch
Speaker 1: the given curve at the point in question, and thus
Speaker 1: reduce the problem to drawing a tangent to a circle.
Speaker 1: I should note, in passing that he only applied this
Speaker 1: method to curves which are symmetrical about an axis, and
Speaker 1: he took the center of the circle on the axis.
Speaker 1: Much of the reasoning in these two books is not
Speaker 1: easy to follow, but a Latin translation of them with
Speaker 1: explanatory notes was prepared by F. De Bone, and an
Speaker 1: addition of this with commentary by F. Van Schutten, was
Speaker 1: issued in sixteen fifty nine and had a wide circulation.
Speaker 1: The third book of Geometrie contains an analysis of the
Speaker 1: algebra then current, and it has affected the language of
Speaker 1: the subject. By fixing the custom of employing the letters
Speaker 1: at the beginning of the alphabet to denote known quantities
Speaker 1: and those at the end of the alphabet to denote
Speaker 1: unknown quantities. Descartes further introduced the system of indices now
Speaker 1: in use. But I would here remind the reader that
Speaker 1: the suggestion had been made by previous writers, though it
Speaker 1: had not been generally adopted, but very likely it was
Speaker 1: the original on the part of Descartes. I think also
Speaker 1: that Descartes was the first to realize that his letters
Speaker 1: might represent any quantities positive or negative, and that it
Speaker 1: was sufficient to prove a proposition for one general case
Speaker 1: compared to the old procedure. In this book, he made
Speaker 1: use of the rule for determining a limit to the
Speaker 1: number of pausesative and negative roots of an algebraical equation,
Speaker 1: which is still known by his name, and introduce the
Speaker 1: method of indeterminate coefficients for the solution of equations. He
Speaker 1: believed that he had given a method by which algebraical
Speaker 1: equations of any order could be solved, but in this
Speaker 1: he was mistaken. He made use of the method of
Speaker 1: indeterminate coefficients. Of the other two appendices to the discourse
Speaker 1: was one devoted to optics. The chief interest of this
Speaker 1: consists in the statement given of the law of refraction.
Speaker 1: This appears to have been taken from Snell's work, but
Speaker 1: not only is there no acknowledgment of the source from
Speaker 1: which it was obtained, but it is enunciated in such
Speaker 1: a way so as to lead a careless reader to
Speaker 1: suppose that it is due to the researches of Daycartes.
Speaker 1: Daiscartes would seem to have repeated Snell's experiments when in
Speaker 1: Paris in sixteen twenty six or sixteen twenty seven, and
Speaker 1: it is possible that he subsequently forgot how much he
Speaker 1: owed to the earlier investigations of Snell. A large part
Speaker 1: of the optics is devoted to determining the best shape
Speaker 1: for the lenses of a telescope, but the mechanical difficulties
Speaker 1: in grinding a surface of glass to a required form
Speaker 1: are so great as to render these investigations of little
Speaker 1: practical use. Daycartes seemed to have been doubtful whether to
Speaker 1: regard the rays of light as proceeding from the eye,
Speaker 1: and so to speak, touching the object as the Greeks
Speaker 1: had done, or as proceeding from the object and so
Speaker 1: affecting the eye. But since he considered the velocity of
Speaker 1: light to be infinite, he did not deem the point
Speaker 1: particularly important. The other appendix on meteors contains an explanation
Speaker 1: of numerous atmospheric phenomenon, including the rainbow. Daycart was unacquainted
Speaker 1: with the unequal refrangibility of rays of light of different colors,
Speaker 1: and the explanation of the latter is necessarily incomplete. Descartes's
Speaker 1: physical theory of the universe, embodying most of the results
Speaker 1: contained in his earlier and unpublished Le Monde, was given
Speaker 1: in his Principia sixteen forty four, and rests on a
Speaker 1: metaphysical basis. He commences with a discussion on motion, and
Speaker 1: then lays down ten laws of nature, of which the
Speaker 1: first two are almost identical with the first two laws
Speaker 1: of motion as given by Newton. The remaining eight laws
Speaker 1: are inaccurate. He next proceeds to discuss the nature of matter,
Speaker 1: which he regards as uniform in kind, though there are
Speaker 1: three forms of it. He assumes that the matter of
Speaker 1: the universe must be in motion and that the motion
Speaker 1: must result in a number of vortices. He states that
Speaker 1: the Sun is the center of an immense whirlpool of
Speaker 1: this matter, in which the planets float and are swept
Speaker 1: round like straws in a whirlpool of water. Each planet
Speaker 1: is supposed to be the center of a secondary whirlpool
Speaker 1: by which its satellites are carried. These secondary whirlpools are
Speaker 1: supposed to produce variations of density in the surrounding medium
Speaker 1: which constitute the primary whirlpool, and so cause the planets
Speaker 1: to move on in ellipses and not circles. All these
Speaker 1: assumptions are arbitrary and unsupported by any investigation. It is
Speaker 1: not difficult to prove that on his hypothesis, the Sun
Speaker 1: would be in the center of these ellipses, and not
Speaker 1: at a focus, as Kepler had shewn was the case,
Speaker 1: and that the weight of a body at every place
Speaker 1: on the surface of the Earth except the equator, would
Speaker 1: act in a direction which was not vertical. But it
Speaker 1: will be sufficient here to say that Newton, in the
Speaker 1: second book of his Principia in sixteen eighty seven, considered
Speaker 1: the theory in detail, enchewed that its consequences are not
Speaker 1: only inconsistent with each of Kepler's laws and with the
Speaker 1: fundamental laws of mechanics, but are all also at variance
Speaker 1: with the ten laws of nature assumed by Descartes. Still,
Speaker 1: in spite of its crudeness and its inherent defects, the
Speaker 1: theory of vortices marks a fresh era in astronomy, for
Speaker 1: it was an attempt to explain the phenomenon of the
Speaker 1: whole universe by the same mechanical laws which experiments shoes
Speaker 1: to be true here on Earth. Cavalieri, almost contemporaneously with
Speaker 1: the publication in sixteen thirty seven of Descartes's geometry, the
Speaker 1: principles of the integral calculus, so far as they are
Speaker 1: concerned with summation, were being worked out in Italy. This
Speaker 1: was affected by what was called the principle of indivisibles,
Speaker 1: and was the invention of Cavalieri. It was applied to
Speaker 1: numerous problems connected with the quadrature of curves and surfaces,
Speaker 1: the determination of volumes, and the positions of centers of mass,
Speaker 1: to the complete exclusion of the tedious method of exhaustions
Speaker 1: used by the Greeks. In principle, the methods are the same,
Speaker 1: but the notation of indivisibles is more concise and convenient.
Speaker 1: It was in its turn superseded at the beginning of
Speaker 1: the eighteenth century by the integral calculus, but its use
Speaker 1: will be familiar to all mathematicians who have read any
Speaker 1: commentary on the first section of the first book of
Speaker 1: Newton's Principia, in the application of lemmas two and three
Speaker 1: to the determination of areas, volumes, and so on. Bonaventura
Speaker 1: Calvieri was born at Milan in fifteen ninety eight and
Speaker 1: died at Bologna in November twenty seventh, sixteen forty seven.
Speaker 1: He became a Jesuit at an early age, and on
Speaker 1: the recommendation of the order, he was in sixteen twenty
Speaker 1: nine made professor of mathematics at Bologna, and he continued
Speaker 1: to occupy the chair there until his death. I have
Speaker 1: already mentioned Cavalieri's name for the part of the book
Speaker 1: that he took in introducing the use of logarithms into
Speaker 1: to Italy. He was one of the most influential mathematicians
Speaker 1: of his time, but his subsequent reputation rests mainly on
Speaker 1: his invention of the principle of indivisibles. The principle of
Speaker 1: indivisibles has been used by Kepler in sixteen o four
Speaker 1: and sixteen fifteen in somewhat crude form. It was first
Speaker 1: stated by Cavalieri in sixteen twenty nine, but he did
Speaker 1: not publish his results till sixteen thirty five. In his
Speaker 1: early annunciation of the principle in sixteen thirty five, Cavalieri
Speaker 1: asserted that a line was made up of an infinite
Speaker 1: number of points, each without magnitude, and a surface of
Speaker 1: an infinite number of lines, each without breadth, and a
Speaker 1: volume of infinite number of surfaces, each without thickness. To
Speaker 1: meet the objections of Gouldinis and others, the statement was
Speaker 1: recast and in its final form used by the mathematicians
Speaker 1: of the seventeenth century. It was published in calim Calviieri's
Speaker 1: six Geometrical Exercises in sixteen forty seven, the third of
Speaker 1: which is devoted to a defense of the theory. These
Speaker 1: exercises contain the first rigid demonstration of the properties of pappus.
Speaker 1: Cavalieri's work on the subject were reissued with his later
Speaker 1: corrections in sixteen fifty three. The method of indivisibles is
Speaker 1: simply that any magnitude may be divided into an infinite
Speaker 1: number of small quantities, which can be made to bear
Speaker 1: any required ratios, for example, equality one to the other.
Speaker 1: The analysis given by Cavaliery is hardly worth quoting, except
Speaker 1: as being one of the first steps taken toward the
Speaker 1: formation of an infinitesimal calculus. One example will suffice. Suppose
Speaker 1: it be required to find the area of a right
Speaker 1: angled triangle. Let the base contain n points. In the
Speaker 1: other side n times a points, then the ordinance at
Speaker 1: the successive points of the base will contain A two A,
Speaker 1: and so on to n times a points. Therefore, the
Speaker 1: number of points in the figure is A plus two
Speaker 1: A plus dot dot dot plus na, the sum of
Speaker 1: which is half n squared a plus one half n
Speaker 1: times a. Since n is very large, we may neglect
Speaker 1: the one half Na is inconsiderable compared with the one
Speaker 1: half n squared A, and the area is one half
Speaker 1: na times n, that is, one half the altitude times
Speaker 1: the base. There is no difficulty in criticizing such a proof,
Speaker 1: But although the form in which it is presented is indefensible,
Speaker 1: the substance of it is correct. It would be misleading
Speaker 1: to give the above as the only specimen of the
Speaker 1: method of indivisibles, and I therefore quote another example taken
Speaker 1: from a later writer, which will fairly illustrate the US
Speaker 1: use of the method when modified and corrected by the
Speaker 1: method of limits. Let it be required to find the
Speaker 1: area bounded by the parabola apc the tangent at A
Speaker 1: and any diameter dc complete the parallelogram ab cd. Divide
Speaker 1: AD into n equal parts. Let AM contain r of them,
Speaker 1: and Mn be the r plus one part. Draw mp
Speaker 1: and n q parallel to a B, and draw pr
Speaker 1: parallel to a D. Then, when n becomes indefinitely large,
Speaker 1: the curve linear area apcd will be the limit of
Speaker 1: the sum of all parallelograms like PN. Now, the ratio
Speaker 1: of the area Pn to the area BD equals the
Speaker 1: ratio MP times mn is to DC times a D.
Speaker 1: But by properties of the parabola, Mp over d C
Speaker 1: equals AM squared over a D squared, which is also
Speaker 1: equal two are squared over n squared, and Mn over
Speaker 1: AD equals one over n. Hence MP times m N
Speaker 1: over dc times AD equals r squared over n cubed. Therefore,
Speaker 1: the area of PN over the area of bd equals
Speaker 1: are squared over n cubed. Therefore, ultimately the area of
Speaker 1: a pc d over area bd equals one squared plus
Speaker 1: two squared plus three squared plus dot dot dot plus
Speaker 1: n minus one squared all over n cubed, which is
Speaker 1: also equal to one sixth of n times n minus
Speaker 1: one times two n minus one all over n cubed,
Speaker 1: which in the limit one to three. It is perhaps
Speaker 1: worth noticing that Cavaliery and its successors always use the
Speaker 1: method to find the ratios of two areas, volumes, or
Speaker 1: magnitudes of the same kind of dimensions. That is, they
Speaker 1: never thought of an area as containing so many units
Speaker 1: of area. The idea of comparing a magnitude with the
Speaker 1: unit of the same kind seems to have been due
Speaker 1: to Wallace. It is evident that in its direct form
Speaker 1: the method is applicable to only a few curves. Cavalieri
Speaker 1: proved that if M be a positive integer, then the
Speaker 1: limit when n is infinite of one to the M
Speaker 1: plus two to the M plus three to the m
Speaker 1: dot dot dot plus n to the m all over
Speaker 1: end to the M plus one is one over M
Speaker 1: plus one, which is equivalent to saying that he found
Speaker 1: the integral to x of x to the m from
Speaker 1: x equals zero to x equals one. He also discussed
Speaker 1: the quadrature of the hyperbola. End of Chapter twenty one,
Speaker 1: Part one. Recording by Paul King H T T, P
Speaker 1: p j K dot scripts dot m I T dot
Speaker 1: ed U forward slash p kJ station