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<v Speaker 1>Chapter fourteen of A Short Account of the History of Mathematics.

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<v Speaker 1>This is a LibriVox recording. All LibriVox recordings are in

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<v Speaker 1>the public domain. For more information or to volunteer, please

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<v Speaker 1>visit LibriVox dot org. This is a recording by Paul

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<v Speaker 1>King PJK dot scripts dot mit dot edu forward slash

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<v Speaker 1>p kJ. A Short Account of the History of Mathematics,

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<v Speaker 1>Chapter fourteen, Features of Modern Mathematics. The division between this

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<v Speaker 1>period and that treated in the last six chapters is

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<v Speaker 1>by no means so well defined as that which separates

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<v Speaker 1>the history of Greek mathematics from the mathematics of the

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<v Speaker 1>Middle Ages. The methods of analysis used in the seventeenth

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<v Speaker 1>century and the kind of problems attacked changed but gradually,

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<v Speaker 1>and the mathematicians at the beginning of this period were

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<v Speaker 1>in immediate relations with those at the end of the

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<v Speaker 1>last considered. For this reason, some writers have divided the

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<v Speaker 1>history of mathematics into two parts, only treating the schoolmen

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<v Speaker 1>as the lineal successors of the Greek mathematicians, and dating

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<v Speaker 1>the creation of modern mathematics from the introduction of the

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<v Speaker 1>Arab textbooks into Europe. The division I have given is

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<v Speaker 1>I think more convenient for the introduction of analytical geometry

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<v Speaker 1>and of the calculus completely revolutionize the development of the subject,

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<v Speaker 1>and it therefore seems preferable to take their invention as

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<v Speaker 1>marking the commencement of modern mathematics. The time that has

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<v Speaker 1>elapsed since these methods were invented has been a period

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<v Speaker 1>of incessant intellectual activity in all departments of knowledge, and

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<v Speaker 1>the progress made in mathematics has been immense. The greatly

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<v Speaker 1>extended range of knowledge and the rapid intercommunication of ideas

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<v Speaker 1>due to printing increase the difficultficulties of a historian, while

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<v Speaker 1>the mass of materials which has to be mastered, the

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<v Speaker 1>absence of perspective, and even the echoes of old controversies

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<v Speaker 1>combine to make it very difficult to give a clear

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<v Speaker 1>and just account of the development of the subject. As, however,

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<v Speaker 1>the leading facts are generally known, and the works published

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<v Speaker 1>during this time are accessible to any student, I may

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<v Speaker 1>deal more concisely with the lives and writings of modern

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<v Speaker 1>mathematicians than with those of their predecessors, and confine myself

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<v Speaker 1>more strictly than before to those who have materially affected

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<v Speaker 1>the progress of the subject. Roughly speaking, we may say

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<v Speaker 1>that five distinct stages in the history of this period

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<v Speaker 1>can be discerned. First of all, there is the invention

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<v Speaker 1>of analytical geometry by Descartes in sixteen thirty seven, and

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<v Speaker 1>almost at the same time the introduction of the method

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<v Speaker 1>of indivision, by the use of which areas volumes and

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<v Speaker 1>the positions of centers of mass can be determined by summation,

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<v Speaker 1>in a matter analogous to that affected nowadays by the

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<v Speaker 1>aid of the integral calculus. The method of indivisibles was

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<v Speaker 1>soon superseded by the integral calculus. Analytical geometry, however, maintains

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<v Speaker 1>its position as part of the necessary training of every mathematician,

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<v Speaker 1>and is incomparably more potent than the geometry of the

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<v Speaker 1>ancients for all purposes of research. The latter is still,

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<v Speaker 1>no doubt an admirable intellectual training, and it frequently affords

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<v Speaker 1>an elegant demonstration of some proposition, the truth of which

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<v Speaker 1>is already known. But it requires a special procedure for

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<v Speaker 1>every particular problem attacked. The former, on the other hand,

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<v Speaker 1>lays down a few simple rules by which property can

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<v Speaker 1>be at once proved or disproved. In the second place,

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<v Speaker 1>we have the invention of the flexitional or differential calculus

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<v Speaker 1>about sixteen sixty six, and possibly an independent invention of

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<v Speaker 1>it in sixteen seventy four. Wherever a quantity changes according

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<v Speaker 1>to some continuous law, and most things in nature do

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<v Speaker 1>so change, the differential calculus enables us to measure its

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<v Speaker 1>rate of increase or decrease, and from its rate of

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<v Speaker 1>increase or decrease, the integral calculus enables us to find

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<v Speaker 1>the original quantity. Formerly, every separate function of X, such

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<v Speaker 1>as one plus x, quantity to the power n, or

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<v Speaker 1>the logarithm of one plus x, the sine of x,

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<v Speaker 1>or the arctan of x, et cetera, could be expanded

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<v Speaker 1>in ascending powers of X only by means of such

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<v Speaker 1>special procedure as was suitable for that particular problem. But

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<v Speaker 1>by the aid of the calculus, the expansion expansion of

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<v Speaker 1>any function of X in ascending powers of X is

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<v Speaker 1>in general reducible to one rule which covers all cases alike.

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<v Speaker 1>So again, the theory of maxima and minima, the determination

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<v Speaker 1>of the lengths of curves and the areas enclosed by them,

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<v Speaker 1>the determination of surfaces of volumes and the centers of mass,

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<v Speaker 1>and many other problems are each reducible to a single rule.

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<v Speaker 1>The theories of differential equations of the calculus, of variations

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<v Speaker 1>of finite differences, et cetera, are the developments of the

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<v Speaker 1>ideas of the calculus. These two subjects, analytical geometry and

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<v Speaker 1>the calculus, became the chief instruments of further progress in mathematics.

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<v Speaker 1>In both of them, a sort of machine was constructed

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<v Speaker 1>to solve a problem. It was only necessary to put

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<v Speaker 1>in the particular function dealt with, or the equation of

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<v Speaker 1>the particular curve or surface considered, and on performing certain

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<v Speaker 1>simple operations, the result came out. The validity of the

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<v Speaker 1>process was proved once and for all, and it was

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<v Speaker 1>no longer requisite to invent some special method for every

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<v Speaker 1>separate function, curve, or surface. In the third place, Huygens

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<v Speaker 1>laid the foundation of a satisfactory treatment of dynamics, and

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<v Speaker 1>Newton reduced it to an exact science. The latter mathematician

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<v Speaker 1>proceeded to apply the new analytical methods not only to

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<v Speaker 1>numerous problems in the mechanics of solids and fluids on

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<v Speaker 1>the Earth, but to the Solar system. The whole of mechanics,

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<v Speaker 1>terrestrial and celestial was thus brought within the domain of mathematics.

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<v Speaker 1>There is no doubt that Newton used the calculus to

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<v Speaker 1>obtain many of his results, but he seems to have

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<v Speaker 1>thought that if his demonstrations were established by the aid

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<v Speaker 1>of a new science, which was at that time generally unknown,

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<v Speaker 1>his critics, who would not understand the fluctional calculus, would

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<v Speaker 1>fail to realize the truth and importance of his discoveries.

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<v Speaker 1>He therefore determined to give geometrical proofs of all of

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<v Speaker 1>his results. He accordingly cast the Principia into a geometrical form,

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<v Speaker 1>and thus presented it to the world in a language

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<v Speaker 1>which all men could then understand. The theory of mechanics

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<v Speaker 1>was extended and was systematized into its modern form by

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<v Speaker 1>Laplace and Lagrange toward the end of the eighteenth century.

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<v Speaker 1>In the fourth place, we may say that during this

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<v Speaker 1>period the chief branches of physics have been brought within

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<v Speaker 1>the scope of mathematics. This extension of the domain of

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<v Speaker 1>mathematics was commenced by Huygens on Newton when they probounded

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<v Speaker 1>their theories of light, but it was not until the

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<v Speaker 1>beginning of this century that sufficiently accurate observations were made

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<v Speaker 1>in the most physical subjects to enable mathematical reasoning to

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<v Speaker 1>be applied to them. From the results of the observations

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<v Speaker 1>and experiments which have been since published, numerous and far

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<v Speaker 1>reaching conclusions have been obtained by the use of mathematics.

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<v Speaker 1>But we now want some more simple hypotheses from which

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<v Speaker 1>we can deduce those laws which at present from our

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<v Speaker 1>starting point. If to take one example, we could say

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<v Speaker 1>in what electricity consisted, we might get some simple laws

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<v Speaker 1>or hypotheses from which, by the aid of mathematics, all

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<v Speaker 1>the observed phenomenon could be then deduced in the same

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<v Speaker 1>way as Newton deduced all the results of physical astronomy

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<v Speaker 1>from the law of gravitation. All lines of research seem, moreover,

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<v Speaker 1>to indicate that there is an intimate connection between the

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<v Speaker 1>different branches of physics e g. Between light, heat, electricity,

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<v Speaker 1>and magnetism. The ultimate explained The nation of this and

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<v Speaker 1>of the leading facts in physics seems to demand a

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<v Speaker 1>study of molecular physics. A knowledge of molecular physics, in

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<v Speaker 1>its turn, seems to require some theory as to the

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<v Speaker 1>constitution of matter. It would further appear that the key

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<v Speaker 1>to the constitution of matter is to be found in

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<v Speaker 1>chemistry or chemical physics. So the matter stands at present.

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<v Speaker 1>Helmholds in Germany, and Maxwell and Lord Kelvin Sir William

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<v Speaker 1>Thompson in Great Britain have done a great deal in

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<v Speaker 1>applying mathematics to physics. But the connection between the different

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<v Speaker 1>branches of physics and the fundamental laws of those branches,

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<v Speaker 1>if there be any simple ones, are riddles which are

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<v Speaker 1>not yet solved. This history does not pretend to treat

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<v Speaker 1>of problems which are now the subject of investigation. And

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<v Speaker 1>though mathematical physics forms a large part of modern mathematics,

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<v Speaker 1>I shall not discuss it in any detail. Fifthly, this

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<v Speaker 1>period has seen an immense extension of here mathematics. Much

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<v Speaker 1>of this is the creation of comparatively recent times, and

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<v Speaker 1>I regard the details of it as outside the limits

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<v Speaker 1>of this book, though in chapter nineteen I have allowed

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<v Speaker 1>myself to mention some of the subjects discussed. The most

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<v Speaker 1>striking features of this extension are the developments of higher geometry,

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<v Speaker 1>of higher arithmetic or the theory of numbers, of higher algebra,

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<v Speaker 1>including the theory of forms and of the theory of equations,

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<v Speaker 1>and also the discussion of functions of double and multiple periodicity,

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<v Speaker 1>and notably the creation of a theory of functions. End

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<v Speaker 1>of section twenty. Recording by Paul King PJK dot scripts

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<v Speaker 1>dot mit dot edu. Forward slash p kJ