Speaker 1: Chapter eighteen, Part one of a short account of the
Speaker 1: history of mathematics. This is a LibriVox recording. All LibriVox
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Speaker 1: by Paul King PJK dot scripts dot mit dot edu
Speaker 1: forward slash p kJ. A short account of the history
Speaker 1: of mathematics by W. W. Rousbaul. Chapter eighteen Lagrange, Laplace
Speaker 1: and their Contemporaries circa seventeen forty to eighteen thirty. The
Speaker 1: last chapter contains the history of two separate schools, the
Speaker 1: Continental and the British. In the early years of the
Speaker 1: eighteenth century, the English school appeared vigorous and fruitful, but
Speaker 1: decadence rapidly set in and after the death of McLaurin
Speaker 1: and Sennas Ubson. No British mathematician appeared who is at
Speaker 1: all comparable to the Continental mathematicians of the latter half
Speaker 1: of the eighteenth century. This fact is partly explicable by
Speaker 1: the isolation of the school, partly by its tendency to
Speaker 1: rely too exclusively on geometrical and fleuctional methods. Some attention was, however,
Speaker 1: given to practical science, But except for a few remarks
Speaker 1: on English physicists, I do not think it necessary to
Speaker 1: discuss English mathematicians further until about eighteen twenty, when analytical
Speaker 1: methods again come into vogue on the continent under the
Speaker 1: influence of John Bernoulli, the calculus had become an instrument
Speaker 1: of great analytical power, expressed in an admirable notation, and
Speaker 1: for practical applications it is impossible to overestimate the value
Speaker 1: of a good notation. The subject of mechanics remained, however,
Speaker 1: in much the condition in which Newton had left it,
Speaker 1: until de la Maire, in putting Newton's results into the
Speaker 1: language of the differential calculus, did something to extend it.
Speaker 1: Universal gravitation, as enunciated in the Principia, was accepted as
Speaker 1: an established fact, but the geometrical methods adopted in proving
Speaker 1: it were difficult to follow or use, or to use
Speaker 1: in analogous problems. McLaurin, Simpson and Clairot may be regarded
Speaker 1: as the last mathematicians of distinction who employed them. Lastly,
Speaker 1: the Newtonian theory of light was generally received as correct.
Speaker 1: The leading mathematicians of the era of which we are
Speaker 1: now entering are Euler, Lagrange, Laplace, and la Genre. Briefly,
Speaker 1: we may say that Euler extended, summed up, and completed
Speaker 1: the work of his predecessors, while Lagrange, with almost unrivaled skill,
Speaker 1: developed the infinitesimal calculus and theoretical mechanics into the form
Speaker 1: in which we now know them. At the same time,
Speaker 1: Laplace made some additions to the infinitesimal calculus and applied
Speaker 1: that calculus to the theory of universal gravitation. He also
Speaker 1: created a calculus of probabilities. Le Gendre invented spherical harmonic
Speaker 1: analysis and elliptic integrals, and added to the theory of numbers.
Speaker 1: The works of these writers are still standard authorities, and
Speaker 1: are hardly yet the subject matter of history. I shall
Speaker 1: therefore content myself with a mere sketch of their discoveries,
Speaker 1: referring anyone who wishes to know more to the works themselves. Lagrange, Laplace,
Speaker 1: and le Gendre created a French school of mathematics, of
Speaker 1: which the younger members are divided into two groups. Include
Speaker 1: Poissant and Fourier began to apply the mathematical analysis to physics,
Speaker 1: and the other including Monte Carnot and Poncelette created modern geometry.
Speaker 1: Strictly speaking, some of the great mathematicians of recent times,
Speaker 1: such as Gauss and Abel, were contemporaries of the mathematicians
Speaker 1: last named, But except for this remark, I think it
Speaker 1: convenient to defer any consideration of them to the next chapter,
Speaker 1: the development of analysis and mathematics. Euler Leonard Euler was
Speaker 1: born in Bail on April fifteenth, seventeen o seven and
Speaker 1: died at Saint Petersburg on September seventh, seventeen eighty three.
Speaker 1: He was a son of a Lutheran minister who had
Speaker 1: settled at Bail, and was educated in his native town
Speaker 1: under the direction of John Bernoulli, with whose sons Daniel
Speaker 1: and Nicholas he formed a lifelong friendship. When in seventeen
Speaker 1: twenty five the younger Bernoullis went to Russia on the
Speaker 1: invitation of the Empress, they procured a place there for Euler,
Speaker 1: which in seventeen thirty three he exchanged for the share
Speaker 1: of mathematics then vacated by Daniel Bernoulli. The severity of
Speaker 1: the climate affected his eyesight, and in seventeen thirty five
Speaker 1: he lost the use of one eye completely. In seventeen
Speaker 1: forty one, he moved to Berlin at the request or
Speaker 1: rather command, of Frederic the Great. Here he stayed till
Speaker 1: seventeen sixty six, when he returned to Russia and was
Speaker 1: succeeded at Berlin by Lagrange. Within two or three years
Speaker 1: of his going back to Saint Petersburg he became blind.
Speaker 1: But in spite of this, and although his house, together
Speaker 1: with many of his papers, were burnt in seventeen seventy one,
Speaker 1: he recast and improved most of his earlier works. He
Speaker 1: died of ape apoplexy in seventeen eighty three. He was
Speaker 1: married twice. I think we may sum up Oiler's work
Speaker 1: by saying that he created analysis and revised almost all
Speaker 1: the branches of pure mathematics which were then known, filling
Speaker 1: up the details, adding proofs, and arranging the whole in
Speaker 1: consistent form. Such work is very important, and it is
Speaker 1: fortunate for science when it falls into the hands as
Speaker 1: competent as those of Euler. Euler wrote an immense number
Speaker 1: of memoirs on all kinds of mathematical subjects. His chief work,
Speaker 1: in which many of the results of earlier memoirs are embodied,
Speaker 1: are as follows in the first place. He wrote in
Speaker 1: seventeen forty eight, as in his Introductio in analyisim infiniteorum,
Speaker 1: which was intended to serve as an introduction to pure
Speaker 1: analytical mechanics. This is divided into two parts. The first
Speaker 1: part of the Analysis infinitorum contains the bulk of the
Speaker 1: matter which is to be found in the modern textbooks
Speaker 1: on algebra, theory of equations, and trigonometry. In the algebra,
Speaker 1: he paid particular attention to the expansion of various functions
Speaker 1: in series and to the summation of given series, and
Speaker 1: pointed out explicitly that an infinite series cannot be safely
Speaker 1: employed unless it is convergent. In the trigonometry, much of
Speaker 1: which is founded on FC. Mayer's Arithmetic of signs, which
Speaker 1: has been published in seventeen twenty seven, Euler developed the
Speaker 1: idea of John Bernoulli that the subject was a branch
Speaker 1: of analysis and not a mere appendage of astronomy or geometry.
Speaker 1: He also introduced, contemporaneously with Simpson the current abbreviations for
Speaker 1: the trigonometrical functions. Ensued that the trigonometrical and exponential functions
Speaker 1: were connected by the relation cosine theta plus isigin theta
Speaker 1: equals e to the power of eye times scasta. Here
Speaker 1: too we meet the symbol E used to denote the
Speaker 1: base of the Nypurian logarithms, namely the incommensurable number two
Speaker 1: point seven one a two eight, and the symbol PI
Speaker 1: used to denote the incommensurable number three point one four
Speaker 1: one five nine. The use of a single symbol to
Speaker 1: denote the number two point seven one eight two eight
Speaker 1: seems to be due to Coats, who denoted it by M. Newton, was,
Speaker 1: as far as I know, the first to employ the
Speaker 1: literal exponential notation. An Euler using the form A to
Speaker 1: the z has taken A as the base of any
Speaker 1: system of logarithms. It is probable that the choice of
Speaker 1: E for a particular base was determined by its being
Speaker 1: the vowel consecutive to A. The use of a single
Speaker 1: symbol to denote the number three point one four one
Speaker 1: five nine appears to have been introduced by John Bernoulli,
Speaker 1: who represented it by C. Euler in seventeen thirty four,
Speaker 1: denoted it by P and in a letter of seventeen
Speaker 1: thirty six in which he enunciated the theorem that the
Speaker 1: sum of the squares of the reciprocals of the natural
Speaker 1: numbers is one over six times pie squared. He used
Speaker 1: the letter C, Christian Goldbach in seventeen forty two used
Speaker 1: the Greek letter PI, and after the publication of Euler's analysis,
Speaker 1: the symbol pie was generally employed. The numbers E and
Speaker 1: PI would enter into the mathematical analysis from whatever side
Speaker 1: the subject was approached. The latter represents, among other things,
Speaker 1: the ratio of the circumference of a circle to the diameter.
Speaker 1: But it is a mere accident that it is taken
Speaker 1: for its definition. De Morgan, in The Budget of Paradoxes,
Speaker 1: tells an anecdote which illustrates how little the use usual
Speaker 1: definition suggests its real use. He was explaining to an actuary,
Speaker 1: what was the chance that at the end of a
Speaker 1: given time a certain proportion of some group of people
Speaker 1: would be alive? And quoted the actuarial formula involving pie, which,
Speaker 1: in answer to a question he explained, stood for the
Speaker 1: ratio the circumference of a circle to its diameter. His acquaintance,
Speaker 1: who so far listened to the explanation with interest interrupted
Speaker 1: of men, explained, my dear friend that must be a delusion.
Speaker 1: What can a circle have to do with the number
Speaker 1: of people alive at the end of a given time.
Speaker 1: The second part of the analysis infinitetorum is on analytical geometry.
Speaker 1: Euler commenced this part by dividing curves into algebraical and
Speaker 1: transcendental and established a variety of propositions which are true
Speaker 1: for all algebraical curves. He then applied to the general
Speaker 1: equations of the second degree into in two dimensions, shewed
Speaker 1: that it represents the various conic sections and deduced most
Speaker 1: of their properties from the general equation. He also considered
Speaker 1: the classification of cubic, cortic and other algebraical curves. He
Speaker 1: next discussed the question as to what surfaces are represented
Speaker 1: by the general equation of the second degree in three dimensions,
Speaker 1: and how they may be discriminated one from the other.
Speaker 1: Some of these surfaces had not been previously investigated. In
Speaker 1: the course of this analysis, he laid down the rules
Speaker 1: for the transformation of coordinates in space. Here also we
Speaker 1: find the earliest attempt to bring the curvature of services
Speaker 1: within the domain of mathematics, and the first complete discussion
Speaker 1: of tortuous curves. The Analysis Infinitorum was followed in seventeen
Speaker 1: fifty five by the Instituciones Calculi Differentialis, to which it
Speaker 1: was intended as an introduction. This was the first textbook
Speaker 1: on the differential calculus which has any claim to be
Speaker 1: regarded as complete, and it may be said that most
Speaker 1: modern treatises on the subject are based on it. At
Speaker 1: the same time, it should be added that the exposition
Speaker 1: of the principles of the subject is often prolux and obscure,
Speaker 1: and sometimes not altogether accurate. This series of works was
Speaker 1: completed by the publication in three volumes in seventeen sixty
Speaker 1: eight to seventeen seventy of the Institutiononi's Calculi Integralis, in
Speaker 1: which the results of several of Euler's earlier memoirs on
Speaker 1: the same subject and on differential equations are included. This,
Speaker 1: like the similar Treatise on the differential Calculus, summed up
Speaker 1: what was then known on the subject, but many of
Speaker 1: the theorems were recast and the proofs improved. The beta
Speaker 1: and gamma functions were invented by Euler and discussed here,
Speaker 1: but only as illis sstrations of the methods of reduction
Speaker 1: and integration. His treatment of elliptical integrals as superficial. It
Speaker 1: was due to a theorem given by John Landon, a
Speaker 1: writer who was suggestive rather than powerful, in the Philosophical
Speaker 1: Transactions for seventeen fifty five, connecting the arcs of a
Speaker 1: hyperbola and an ellipse. Euler's works that form this trilogy
Speaker 1: have gone through numerous subsequent editions. The classic problems on
Speaker 1: isoperimetrical curves, the brachistochrone in a resulting medium, and the theory
Speaker 1: of geodesics, all of which had been suggested by his master,
Speaker 1: John Bernoulli, had engaged Euler's attention at an early date,
Speaker 1: and in solving them he was led to the calculus
Speaker 1: of variations. The general idea of this was laid down
Speaker 1: in his Corvera maxima minieve proprietate Gaudentium in ventio nova Akfasilis,
Speaker 1: published in seventeen forty four, but the complete development of
Speaker 1: the new calculus was first affected by Lagrange in seventeen
Speaker 1: fifty nine. The method used by Lagrange as described in
Speaker 1: Euler's Integral Calculus and is the same as that given
Speaker 1: in most modern textbooks on the subject. In seventeen seventy,
Speaker 1: Euler published the aneleisiic Zur Algebra in two volumes. The
Speaker 1: first volume treats of determinate Algebra. This contains one of
Speaker 1: the earliest attempts to place the fundamental process on a
Speaker 1: scientific basis. The same subject had attracted d. Lambert's attention.
Speaker 1: This work also includes the proof of the binomial theorem
Speaker 1: for an unrestricted index, which is still known by Euler's name.
Speaker 1: The proof is founded on the principle of the permanence
Speaker 1: of equivalent forms, but Euler made no attempt to investigate
Speaker 1: the convergency of the series that he should have omitted.
Speaker 1: This essential step is the more curious, as he had
Speaker 1: himself recognized the necessity of concus littering the convergency of
Speaker 1: the infinite series. The second volume treats of the indeterminate
Speaker 1: or Diophantine algebra. This contains a solution of some of
Speaker 1: the problems proposed by Fermont, which had hitherto remained unsolved.
Speaker 1: A French translation of the algebra with numerous and valuable
Speaker 1: editions by Lagrange was brought out in seventeen ninety four,
Speaker 1: and a treatise on arithmetic by Euler was appended to it.
Speaker 1: These four works comprise most of what Euler produced in
Speaker 1: pure mathematics. He also wrote numerous memoirs on nearly all
Speaker 1: subjects of applied mathematics and mathematical physics, then studied the
Speaker 1: chief results in them are as follows. In the mechanics
Speaker 1: of a grid system, he determined the general equation of
Speaker 1: a motion of a body about a fixed point, which
Speaker 1: are ordinarily written of the form A multiplied by di
Speaker 1: omega i by d t minus in brackets b minus
Speaker 1: c closed bracket times omega two times omega three equals L.
Speaker 1: And he gave the general equation of the motion of
Speaker 1: a free body, which are usually presented in the form
Speaker 1: d y d t times mu minus m v theta
Speaker 1: three plus m w theta two equals x, and dh
Speaker 1: one prime by d t equals h two prime theta
Speaker 1: three plus h three prime theta two equals L. He
Speaker 1: also defended and elaborated the theory of least action, which
Speaker 1: had been propounded by Marpetuis in seventeen fifty one. In
Speaker 1: his Sa de cosmal g In hydrodynamics, Euler established the
Speaker 1: general equations of motion which are commonly expressed in the
Speaker 1: form one over row d row by d x equals
Speaker 1: x minus d u by d s minus u times
Speaker 1: du by dx minus v times du by d y
Speaker 1: minus w times du by d z. At the time
Speaker 1: of his death he was engaged in writing a treatise
Speaker 1: on hydromechanics, in which the treatment of the subject would
Speaker 1: have been completely recast. His most important works on astronomy
Speaker 1: are as Theorea Mortrum Planetarum a comatarum, published in seventeen
Speaker 1: forty four, is Theoria Motus Lunarius, published in seventeen fifty three,
Speaker 1: and is Theoria Motum lune published in seventeen seventy two.
Speaker 1: In these he attacked the problem of three bodies. He
Speaker 1: supposed the body considered e g. The Moon, to carry
Speaker 1: three rectangular axes with it in its motion, the axes
Speaker 1: moving parallel to themselves, and to these axes all the
Speaker 1: motions were referred. This method is not convenient, but it
Speaker 1: was for from Euler's results that Meyer constructed the lunar tables,
Speaker 1: from which his widow in seventeen seventy received five thousand pounds,
Speaker 1: being the prize offered by the English Parliament, and in
Speaker 1: recognition of Euler's services, A sum of three hundred pounds
Speaker 1: was voted as an honorarium to him. Euler was much
Speaker 1: interested in optics. In seventeen forty six he discussed the
Speaker 1: relative merits of the omission and undulatory theories of light. He,
Speaker 1: on the whole, preferred the latter. In seventeen seventy to
Speaker 1: seventeen seventy one he published his optical researches in three
Speaker 1: volumes under the title Dioptrica. He also wrote an elementary
Speaker 1: work on physics and Fundamental Principles of mathematical philosophy. This
Speaker 1: originated from an invitation he received when he first went
Speaker 1: to Berlin to give the Lessons on Physics to the
Speaker 1: Princess of an Halt Desso. These lectures were published in
Speaker 1: seventeen sixty eight to seventy teen seventy two in three
Speaker 1: volumes under the title Lettres ser carcurg suget de physique,
Speaker 1: and for half a century remained a standard treatise on
Speaker 1: the subject. Of course, Euler's magnificent works were not the
Speaker 1: only textbooks containing original matter produced at this time. Amongst
Speaker 1: numerous writers, I would be I would specially single out
Speaker 1: Daniel Bernoulli, Simpson, Lambert Bizou, Tremblay, and abrogast As having
Speaker 1: influenced the development of mathematics. To the first two mentioned,
Speaker 1: I have already alluded in the last chapter, end of
Speaker 1: section thirty. Recording by Paul King, Oakville, Ontario, p j
Speaker 1: K dot scripts dot mit dot edu. Forward slash p
Speaker 1: kJ