Speaker 1: Chapter fourteen of A Short Account of the History of Mathematics.
Speaker 1: This is a LibriVox recording. All LibriVox recordings are in
Speaker 1: the public domain. For more information or to volunteer, please
Speaker 1: visit LibriVox dot org. This is a recording by Paul
Speaker 1: King PJK dot scripts dot mit dot edu forward slash
Speaker 1: p kJ. A Short Account of the History of Mathematics,
Speaker 1: Chapter fourteen, Features of Modern Mathematics. The division between this
Speaker 1: period and that treated in the last six chapters is
Speaker 1: by no means so well defined as that which separates
Speaker 1: the history of Greek mathematics from the mathematics of the
Speaker 1: Middle Ages. The methods of analysis used in the seventeenth
Speaker 1: century and the kind of problems attacked changed but gradually,
Speaker 1: and the mathematicians at the beginning of this period were
Speaker 1: in immediate relations with those at the end of the
Speaker 1: last considered. For this reason, some writers have divided the
Speaker 1: history of mathematics into two parts, only treating the schoolmen
Speaker 1: as the lineal successors of the Greek mathematicians, and dating
Speaker 1: the creation of modern mathematics from the introduction of the
Speaker 1: Arab textbooks into Europe. The division I have given is
Speaker 1: I think more convenient for the introduction of analytical geometry
Speaker 1: and of the calculus completely revolutionize the development of the subject,
Speaker 1: and it therefore seems preferable to take their invention as
Speaker 1: marking the commencement of modern mathematics. The time that has
Speaker 1: elapsed since these methods were invented has been a period
Speaker 1: of incessant intellectual activity in all departments of knowledge, and
Speaker 1: the progress made in mathematics has been immense. The greatly
Speaker 1: extended range of knowledge and the rapid intercommunication of ideas
Speaker 1: due to printing increase the difficultficulties of a historian, while
Speaker 1: the mass of materials which has to be mastered, the
Speaker 1: absence of perspective, and even the echoes of old controversies
Speaker 1: combine to make it very difficult to give a clear
Speaker 1: and just account of the development of the subject. As, however,
Speaker 1: the leading facts are generally known, and the works published
Speaker 1: during this time are accessible to any student, I may
Speaker 1: deal more concisely with the lives and writings of modern
Speaker 1: mathematicians than with those of their predecessors, and confine myself
Speaker 1: more strictly than before to those who have materially affected
Speaker 1: the progress of the subject. Roughly speaking, we may say
Speaker 1: that five distinct stages in the history of this period
Speaker 1: can be discerned. First of all, there is the invention
Speaker 1: of analytical geometry by Descartes in sixteen thirty seven, and
Speaker 1: almost at the same time the introduction of the method
Speaker 1: of indivision, by the use of which areas volumes and
Speaker 1: the positions of centers of mass can be determined by summation,
Speaker 1: in a matter analogous to that affected nowadays by the
Speaker 1: aid of the integral calculus. The method of indivisibles was
Speaker 1: soon superseded by the integral calculus. Analytical geometry, however, maintains
Speaker 1: its position as part of the necessary training of every mathematician,
Speaker 1: and is incomparably more potent than the geometry of the
Speaker 1: ancients for all purposes of research. The latter is still,
Speaker 1: no doubt an admirable intellectual training, and it frequently affords
Speaker 1: an elegant demonstration of some proposition, the truth of which
Speaker 1: is already known. But it requires a special procedure for
Speaker 1: every particular problem attacked. The former, on the other hand,
Speaker 1: lays down a few simple rules by which property can
Speaker 1: be at once proved or disproved. In the second place,
Speaker 1: we have the invention of the flexitional or differential calculus
Speaker 1: about sixteen sixty six, and possibly an independent invention of
Speaker 1: it in sixteen seventy four. Wherever a quantity changes according
Speaker 1: to some continuous law, and most things in nature do
Speaker 1: so change, the differential calculus enables us to measure its
Speaker 1: rate of increase or decrease, and from its rate of
Speaker 1: increase or decrease, the integral calculus enables us to find
Speaker 1: the original quantity. Formerly, every separate function of X, such
Speaker 1: as one plus x, quantity to the power n, or
Speaker 1: the logarithm of one plus x, the sine of x,
Speaker 1: or the arctan of x, et cetera, could be expanded
Speaker 1: in ascending powers of X only by means of such
Speaker 1: special procedure as was suitable for that particular problem. But
Speaker 1: by the aid of the calculus, the expansion expansion of
Speaker 1: any function of X in ascending powers of X is
Speaker 1: in general reducible to one rule which covers all cases alike.
Speaker 1: So again, the theory of maxima and minima, the determination
Speaker 1: of the lengths of curves and the areas enclosed by them,
Speaker 1: the determination of surfaces of volumes and the centers of mass,
Speaker 1: and many other problems are each reducible to a single rule.
Speaker 1: The theories of differential equations of the calculus, of variations
Speaker 1: of finite differences, et cetera, are the developments of the
Speaker 1: ideas of the calculus. These two subjects, analytical geometry and
Speaker 1: the calculus, became the chief instruments of further progress in mathematics.
Speaker 1: In both of them, a sort of machine was constructed
Speaker 1: to solve a problem. It was only necessary to put
Speaker 1: in the particular function dealt with, or the equation of
Speaker 1: the particular curve or surface considered, and on performing certain
Speaker 1: simple operations, the result came out. The validity of the
Speaker 1: process was proved once and for all, and it was
Speaker 1: no longer requisite to invent some special method for every
Speaker 1: separate function, curve, or surface. In the third place, Huygens
Speaker 1: laid the foundation of a satisfactory treatment of dynamics, and
Speaker 1: Newton reduced it to an exact science. The latter mathematician
Speaker 1: proceeded to apply the new analytical methods not only to
Speaker 1: numerous problems in the mechanics of solids and fluids on
Speaker 1: the Earth, but to the Solar system. The whole of mechanics,
Speaker 1: terrestrial and celestial was thus brought within the domain of mathematics.
Speaker 1: There is no doubt that Newton used the calculus to
Speaker 1: obtain many of his results, but he seems to have
Speaker 1: thought that if his demonstrations were established by the aid
Speaker 1: of a new science, which was at that time generally unknown,
Speaker 1: his critics, who would not understand the fluctional calculus, would
Speaker 1: fail to realize the truth and importance of his discoveries.
Speaker 1: He therefore determined to give geometrical proofs of all of
Speaker 1: his results. He accordingly cast the Principia into a geometrical form,
Speaker 1: and thus presented it to the world in a language
Speaker 1: which all men could then understand. The theory of mechanics
Speaker 1: was extended and was systematized into its modern form by
Speaker 1: Laplace and Lagrange toward the end of the eighteenth century.
Speaker 1: In the fourth place, we may say that during this
Speaker 1: period the chief branches of physics have been brought within
Speaker 1: the scope of mathematics. This extension of the domain of
Speaker 1: mathematics was commenced by Huygens on Newton when they probounded
Speaker 1: their theories of light, but it was not until the
Speaker 1: beginning of this century that sufficiently accurate observations were made
Speaker 1: in the most physical subjects to enable mathematical reasoning to
Speaker 1: be applied to them. From the results of the observations
Speaker 1: and experiments which have been since published, numerous and far
Speaker 1: reaching conclusions have been obtained by the use of mathematics.
Speaker 1: But we now want some more simple hypotheses from which
Speaker 1: we can deduce those laws which at present from our
Speaker 1: starting point. If to take one example, we could say
Speaker 1: in what electricity consisted, we might get some simple laws
Speaker 1: or hypotheses from which, by the aid of mathematics, all
Speaker 1: the observed phenomenon could be then deduced in the same
Speaker 1: way as Newton deduced all the results of physical astronomy
Speaker 1: from the law of gravitation. All lines of research seem, moreover,
Speaker 1: to indicate that there is an intimate connection between the
Speaker 1: different branches of physics e g. Between light, heat, electricity,
Speaker 1: and magnetism. The ultimate explained The nation of this and
Speaker 1: of the leading facts in physics seems to demand a
Speaker 1: study of molecular physics. A knowledge of molecular physics, in
Speaker 1: its turn, seems to require some theory as to the
Speaker 1: constitution of matter. It would further appear that the key
Speaker 1: to the constitution of matter is to be found in
Speaker 1: chemistry or chemical physics. So the matter stands at present.
Speaker 1: Helmholds in Germany, and Maxwell and Lord Kelvin Sir William
Speaker 1: Thompson in Great Britain have done a great deal in
Speaker 1: applying mathematics to physics. But the connection between the different
Speaker 1: branches of physics and the fundamental laws of those branches,
Speaker 1: if there be any simple ones, are riddles which are
Speaker 1: not yet solved. This history does not pretend to treat
Speaker 1: of problems which are now the subject of investigation. And
Speaker 1: though mathematical physics forms a large part of modern mathematics,
Speaker 1: I shall not discuss it in any detail. Fifthly, this
Speaker 1: period has seen an immense extension of here mathematics. Much
Speaker 1: of this is the creation of comparatively recent times, and
Speaker 1: I regard the details of it as outside the limits
Speaker 1: of this book, though in chapter nineteen I have allowed
Speaker 1: myself to mention some of the subjects discussed. The most
Speaker 1: striking features of this extension are the developments of higher geometry,
Speaker 1: of higher arithmetic or the theory of numbers, of higher algebra,
Speaker 1: including the theory of forms and of the theory of equations,
Speaker 1: and also the discussion of functions of double and multiple periodicity,
Speaker 1: and notably the creation of a theory of functions. End
Speaker 1: of section twenty. Recording by Paul King PJK dot scripts
Speaker 1: dot mit dot edu. Forward slash p kJ