Legendre also gave an effective way of extending his law to cases when p and q are not prime. The work of Bolyai and of Lobachevsky was poorly received. This he did, giving for the first time a rigorous foundation to all the elementary calculus of his day. The fundamental theorem of calculus asserts that which can be read as saying that the integral of the derivative of some function in an interval is equal to the difference in the values of the function at the endpoints of the interval.
Eventually Lagrange won, and the vision of mathematics that was presented to the world was that of an autonomous subject that was also applicable to a broad range of phenomena by virtue of its great generality, a view that has persisted to the present day. The honour of being the first to proclaim the existence of a new geometry belongs to two others, who did so in the late s: Since its invention it had been generally agreed that the calculus gave correct answers, but no one had been able to give a satisfactory explanation of why this was so. What kept it heading in the right direction was its rules of inference. For this, traditional ways of applying the calculus to the study of curves could be made to suffice. Because the similarities in the work of these two men far exceed the differences, it is convenient to describe their work together. All this work set the scene for the emergence of Carl Friedrich Gauss , whose Disquisitiones Arithmeticae not only consummated what had gone before but also directed number theorists in new and deeper directions. By imposing a measure on the space of all possible outcomes, the Russian mathematician Andrey Kolmogorov was the first to put probability theory on a rigorous mathematical footing. Rather, mathematics worked because its elementary terms were meaningless. However, no proof was ever discovered among his notebooks. An important property shared by some groups but not all is commutativity: Above all, he insisted that one should prove that solutions do indeed exist; it is not a priori obvious that every ordinary differential equation has solutions. The most influential worker in this direction was the Norwegian Sophus Lie. Projective geometry The French Revolution provoked a radical rethinking of education in France, and mathematics was given a prominent role. In Lagrange had analyzed all the successful methods he knew for second-, third-, and fourth-degree equations in an attempt to see why they worked and how they could be generalized. Mathematics in the 19th century Most of the powerful abstract mathematical theories in use today originated in the 19th century, so any historical account of the period should be supplemented by reference to detailed treatments of these topics. Mathematicians could now ask why they had believed Euclidean geometry to be the only one when, in fact, many different geometries existed. In Riemann published several papers applying his very general methods for the study of complex functions to various parts of mathematics. From his standpoint every conic section is equivalent to a circle , so his treatise contained a unified treatment of the theory of conic sections. In Germany Richard Dedekind patiently created a new approach, in which each new number called an ideal was defined by means of a suitable set of algebraic integers in such a way that it was the common divisor of the set of algebraic integers used to define it. He dispensed with the restriction to orthogonal projections and decided to investigate what properties figures have in common with their shadows. In his Erlanger Programm Klein proposed that Euclidean and non-Euclidean geometry be regarded as special cases of projective geometry. Differential equations Another field that developed considerably in the 19th century was the theory of differential equations. The work of the English mathematician George Boole and the American Charles Sanders Peirce had contributed to the development of a symbolism adequate to explore all elementary logical deductions. His work suggested that there were profound connections between the original question and other branches of number theory, a fact that he perceived to be of signal importance for the subject. Unlike real numbers, which can be located by a single signed positive or negative number along a number line, complex numbers require a plane with two axes, one axis for the real number component and one axis for the imaginary component. Elliptic integrals were intensively studied for many years by the French mathematician Adrien-Marie Legendre , who was able to calculate tables of values for such expressions as functions of their upper endpoint, x.
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