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It was at once seen that in order to investigate the properties of a curve it was sufficient to select, as a definition, any characteristic geometrical property, and to express it by means of an equation between the (current) co-ordinates of any point on the curve, that is, to translate the definition into the language of analytical geometry.
Descartes’ chief contributions to mathematics were his analytical geometry and his theory of vortices, and it is on his researches in connection with the former of these subjects that his mathematical reputation rests.
Analytic geometry does not consist merely (as is sometimes loosely said) in the application of algebra to geometry; that had been done by Archimedes and many others, and had become the usual method of procedure in the works of the mathematicians of the sixteenth century.
It is hardly necessary to say that the problems themselves are of importance and interest, but from the nature of the case no solution ever offered is capable either of rigid proof or of disproof; all that can be effected is to make one explanation more probable than another, and whenever a philosopher like Descartes believes that he has at last finally settled a question it has been possible for his successors to point out the fallacy in his assumptions.
I have read somewhere that philosophy has always been chiefly engaged with the inter-relations of God, Nature, and Man.
On account of his delicate health he was permitted to lie in bed until late in the mornings; this was a custom which he always followed, and when he visited Pascal in 1647 he told him that the only way to do good work in mathematics and to preserve his health was never to allow anyone to make him get up in the morning before he felt inclined to do so; an opinion which I chronicle for the benefit of any schoolboy into whose hands this work may fall.
On leaving school in 1612 Descartes went to Paris to be introduced to the world of fashion.Finally, modern philosophers concern themselves chiefly with the relations between Man and Nature.Whether this is a correct historical generalization of the views which have been successively prevalent I do not care to discuss here, but the statement as to the scope of modern philosophy marks the limitations of Descartes’ writings.His father, who, as the name implies, was of good family, was accustomed to spend half the year at Rennes when the local parliament, in which he held a commission as councilor, was in session, and the rest of the time on his family estate of Les Cartes at La Haye.René, the second of a family of two sons and one daughter, was sent at the age of eight years to the Jesuit School at La Flêche, and of the admirable discipline and education there given he speaks most highly.The great advance made by Descartes was that he saw that a point in a plane could be completely determined if its distances, say x and y, from two fixed lines drawn at right angles in the plane were given, with the convention familiar to us as to the interpretation of positive and negative values; and that though an equation f(x,y) = 0 was indeterminate and could be satisfied by an infinite number of values of x and y, yet these values of x and y determined the co-ordinates of a number of points which form a curve, of which the equation f(x,y) = 0 expresses some geometrical property, that is, a property true of the curve at every point on it.Descartes asserted that a point in space could be similarly determined by three co-ordinates, but he confined his attention to plane curves.In 1647 he received a pension from the French court in honor of his discoveries.He went to Sweden on the invitation of the Queen in 1649, and died a few months later of inflammation of the lungs. Considering the range of his studies he was by no means widely read, and he despised both learning and art unless something tangible could be extracted therefrom.In 1628 Cardinal de Berulle, the founder of the Oratorians, met Descartes, and was so much impressed by his conversation that he urged on him the duty of devoting his life to the examination of truth.Descartes agreed, and the better to secure himself from interruption moved to Holland, then at the height of his power.