CHAPTER XIII

BEGINNINGS OF MODERN MATHEMATICAL SCIENCE

. All the sciences which have for their end investigations concerning order and measure, are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, accordingly, there ought to exist a general science which should explain all that can be known about order and measure, considered independently of any application to a particular subject, and that, indeed, this science has its own proper name, consecrated by long usage, to wit, mathematics. And a proof that it far surpasses in facility and importance the sciences which depend upon it is that it embraces at once all the objects to which these are devoted and a great many others besides. . Descartes.

As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection. -Lagrange.

The application of algebra has far more than any of his metaphysical speculations, immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences. - Mill.

The idea of coördinates which forms the indispensable scheme for making all processes visible, with its many-sided and stimulating applications in all branches of daily life, whether medicine, physical geography, political economy, statistics, insurance, the technical sciences the first beginnings of the calculus in their historical evolution, the development of the ideas of function and limit in connection with the elementary theory of curves, these are things without which in the present day not the slightest comprehension of the phenomena of nature can be attained, of which, however, the knowledge enables us as by magic to gain an insight with which in depth and range, but above all in certainty, scarcely any other can be compared. Voss.

How many celebrate the names of Newton and Leibnitz! How few have a real appreciation of that which these men have created of permanent value! Here lie the roots of our present-day knowledge, here the true continuation of the strivings of antique wisdom.

The invention of the differential calculus marks a crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a high level. -Whitehead.

MATHEMATICAL PHILOSOPHY. ANALYTIC GEOMETRY. DESCARTES.-The invention of analytic geometry by Descartes in 1637 and the almost contemporary introduction of integral calculus as the method of "indivisibles" may be regarded as the real beginning of modern mathematical science. Thanks to these fruitful ideas the science has during the three centuries that have since elapsed made extraordinary progress both in its own internal development and in its application throughout the range of the physical sciences. Descartes was born in Touraine in 1596, and after the education appropriate for a youth of family and some years of fashionable life in Paris, entered the army, then in Holland. His military career continued till 1621 with incidental opportunity for his favorite speculations in mathematics and philosophy. Some of his most fruitful ideas dated from dreams and his best thinking was habitually done before rising.

It is impossible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery Columbus when he first saw the Western shore, Franklin when the electric spark came from the string of his kite, Galileo when he first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and invented the method of coördinate geometry. Whitehead.

In order to devote himself more completely to his favorite studies he settled in Holland in 1629, devoting the next four years to writing a treatise, entitled Le Monde, upon the universe. In 1637 he published his great Discourse on the Method of Good Reasoning and of Seeking Truth in Science.1 This begins:

1 Discours de la Méthode pour bien conduire sa raison et chercher la vérité dans les sciences.

If this discourse seems too long to be read all at once, it can be divided into six parts. In the first will be found various considerations concerning the sciences; in the second, the chief rules of the method which the author has sought; in the third, some of those of ethics which he has deduced by this method; in the fourth, the reasons by which he proves the existence of God and of the human soul, which are the foundations of his metaphysics; in the fifth, the order of questions of physics which he has sought, and particularly the explanation of the movement of the heart and of some other difficulties which belong to medicine; also the difference which exists between our soul and that of the beasts; and in the last, what things he believes necessary in order to go farther in the investigation of nature than has been done, and what reasons have made him write.

Good sense is the most widely distributed commodity in the world, for every one thinks himself so well supplied with it that even those who are hardest to satisfy in every other respect are not accustomed to desire more of it than they have. In this it is not probable that all men are mistaken, but rather this testifies that the power of good judgment and of discriminating between the true and the false, which is properly what one calls good-sense or reason, is naturally equal in all men; and thus that the diversity of our opinions is not due to the fact that some are more reasonable than others, but only that we conduct our thought along different channels, and do not consider the same things. For it is not enough to have a good mind, but the principal thing is to apply it well. The greatest souls are capable of the greatest vices as well as of the greatest virtues: and those who only progress very slowly can advance much more, if they follow always the straight road than do those who run, departing from it.

His four cardinal precepts were:

Never to receive anything for true which he did not recognize to be evidently so; that is, to avoid carefully precipitancy and prejudgment. Second, to divide each of the difficulties which he should examine into as many pieces as possible. Third, to conduct his thoughts in order, beginning with the simplest objects. The last, to make everywhere enumerations so complete and reviews so general that he should be assured of omitting nothing.

Three appendices dealt with optics, meteors, and geometry, the last containing the beginnings of analytic geometry. The relation of his philosophy to mathematics may be indicated in the following passages.

Considering that, among all those who up to this time made discoveries in the sciences, it was the mathematicians alone who had been able to arrive at demonstrations that is to say, at proofs cer

tain and evident - I did not doubt that I should begin with the same truths that they have investigated, although I had looked for no other advantage from them than to accustom my mind to nourish itself upon truths and not to be satisfied with false reasons.

When . . . I asked myself why was it then that the earliest philosophers would admit to the study of wisdom only those who had studied mathematics, as if this science was the easiest of all and the one most necessary for preparing and disciplining the mind to comprehend the more advanced, I suspected that they had knowledge of a mathematical science different from that of our time.

...

I believe I find some traces of these true mathematics in Pappus and Diophantus, who, although they were not of extreme antiquity, lived nevertheless in times long preceding ours. But I willingly believe that these writers themselves, by a culpable ruse, suppressed the knowledge of them; like some artisans who conceal their secret, they feared, perhaps, that the ease and simplicity of their method, if become popular, would diminish its importance, and they preferred to make themselves admired by leaving to us, as the product of their art, certain barren truths deduced with subtlety, rather than to teach us that art itself, the knowledge of which would end our admiration.

Those long chains of reasoning, quite simple and easy, which geometers are wont to employ in the accomplishment of their most difficult demonstrations, led me to think that everything which might fall under the cognizance of the human mind might be connected together in a similar manner, and that, provided only that one should take care not to receive anything as true which was not so, and if one were always careful to preserve the order necessary for deducing one truth from another, there would be none so remote at which he might not at last arrive, nor so concealed which he might not discover.

Descartes had attempted the solution of a historic geometrical problem propounded by Pappus. From a point P perpendiculars are dropped on m given straight lines and also on n other given lines. The product of the m perpendiculars is in a constant ratio to the product of the n; it is required to determine the locus · of P. Pappus had stated without proof that for m = n = 2 the locus is a conic section, Descartes showed this algebraically, — Newton afterwards conquering the difficulty by unaided geometry. Descartes distinguished geometrical curves for which x and y may be regarded as changing at commensurable rates, or as we should say, curves for which the slope is an algebraic function of the coördinates, from curves which do not satisfy this condition. These he called "mechanical," and did not discuss further. For the accepted definition of a tangent as a line between which and the curve no other line can be drawn, he introduced the modern notion of limiting position of a secant. In connection with this he considered a circle meeting the given curve in two consecutive points, a perpendicular to the radius of the circle being a common tangent to the circle and the given curve. The circle was not however that of curvature, but had its centre on an axis of symmetry of the given curve. He recognized the possibility of extending his methods to space of three dimensions, but did not work out the details. His geometry contained also a discussion of the algebra then known, and gave currency to certain important innovations, in particular the systematic use of a, b, and c, for known, x, y, and z, for unknown quantities; the introduction of exponents; the collection of all terms of an equation in one member; the free use of negative quantities; the use of undetermined coefficients in solving equations; and his rule of signs for studying the number of positive or negative roots of equations. He even fancied that he had found a method for solving an equation of any degree.

It is important to distinguish just what Descartes contributed to mathematics in his analytic geometry. Neither the combination of algebra with geometry nor the use of coördinates was new. From the time of Euclid quadratic equations had been solved geometrically, while latitude and longitude involving a