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Joseph Louis Lagrange was born in Turin, Italy in 1736. Although his father wanted him to be a lawyer, Lagrange was attracted to mathematics and astronomy after reading a memoir by the astronomer Halley. At age 16, he began to study mathematics on his own and by age 19 was appointed to a professorship at the Royal Artillery School in Turin. The follwing year, Lagrange sent Euler a better solution he had discovered for deriving the central equation in the calculus of variations. These solutions and Lagrange's applications of them to celestial mechanics were so monumental that by age 25, he was regarded by many of his contemporaries as the greatest living mathematician.
In 1776, on the recommendation of Euler, he was chosen to succeed Euler as the director of the Berlin Academy. During his stay in Berlin, Lagrange distinguished himself not only in celestial mechanics, but also in algebraic equations and the theory of numbers. After twenty years in Berlin,he moved to Paris at the invitation of Louis XVI. Napoleon was a great admirer of Lagrange and showered him with honors--count, senator, and Legion of Honor. The years Lagrange spent in Paris were devoted primarily to didactic treatises summarizing his mathematical conceptions. One of Lagrange's most famous works is a memoir, Mecanique Analytique, in which he reduced the theory of mechanics to a few general formulas from which all other necessary equations could be derived.
In spite of his fame, Lagrange was always a shy and modest man. On his death in 1813, he was buried with honor in the Pantheon.
Lagrange made major contributions to many branches of mathematics. Some of the most important ones are on calculus of variations, solution of polynomial equations and power series and functions.
As for the calculus of variations, it started with the well-known letter that he sent to Euler in 1755 [Lettres Inedites de Joseph-Louis Lagrange a Leonard Euler, p.5-8] discussing a better way to solve his equation,known today as Euler's equation. In 1760, he started dealing with volumes and surface areas. He defined the volume and the surface area by and respectively, where the equation of the surface is given by z=f(x,y) and dz=Pdx+Qdy. He did not give enough explanations then but he noted that the double integral signs indicate that the two integrations must be performed successively. It was later in 1811, in the second edition of his Mecanique Analytique, that he introduced the general notion of a surface integral. He noted that if the tangent plane at dS,the element of surface, makes an angle with the xy-plane,then using simple trigonometry,we can write dxdy as . So if A is a function of three variables, then ,the second integral being taken over a region in the surface, the first over the projection of that region in the plane. Similarly, if is the angle the tangent plane makes with the xz-plane and that with the yz-plane, then and . Lagrange noted that , and could also be considered as the angles that a normal to the surface element makes with the x-, y-, and z-axes, respectively.
As for the solution of polynomial equations, Lagrange, in his Reflexions sur la Theorie Algebriques des Equations of 1770, tried to solve algebraically polynomial equations of degree five and higher starting with the procedure used by Cardano. He tried to generalize by considering permutations of the roots. However, he was unsuccessful with that and he was thus forced to abondon his quest. Nevertheless, his work did form the foundation on which all nineteenth-century work on the algebraic solutions of equations was based, [Katz, p.621], especially that of Cauchy who was able later to take the theory of permutations to a deeper level.
Finally, Lagrange's commitment to the necessity of an algebraic foundation for the calculus led him to the major accomplishment of the Fonctions Analytiques, in which he studied functions by means of their power series expansions. He believed that every function could be expanded into a power series, where for Lagrange, a function was defined as follows:One names a function of one or several quantities any mathematical expression in which the quantities enter in any manner whatever, connected or not with other quantities which one regards as having given and constant values, whereas the quantities of the function may take any possible values. [Fonctions Analytiques in Oeuvres de Lagrange Vol.9, p.15].
One of the basic results that followed in the Fonctions Analytiques is part of what is Known today as the fundamental theorem of calculus. This is how Lagrange put the theorem in his own words:
If is positive from x=a to x=b, for b>a, then f(b)-f(a) is positive. [Fonctions Analytiques in Oeuvres de Lagrange vol.9 p.78].
He started proving it by considering the formula f(x+i)=f(x)+iP, where P is a function of x and i and is defined as follows:
. As Lagrange said,since when i=0, , means that P(x,i) must be positive from i=0 up to a certain value of i which can be taken as small as desired. [Oeuvres de Lagrange vol.9 p.79]. We see here that Lagrange's definition of a derivative is very similar to ours. So now, since we know that is always positive, we can choose a positive small value for i such that f(x+i)-f(x) is positive. Then we divide the interval (which we would write [a,b]) into n+1 parts, each of length i, so that . By what we said before, we find that f(a+i)-f(a) is positive, f(a+2i)-f(a+i) is positive, ......f(a+(n+1)i)-f(a+ni) is positive, by setting successively x=a,a+i,....,a+ni, as long as the derivatives are positive for k=0,1,...n. So we see that the sum [f(a+(n+1)i)-f(a+ni)]+......+[f(a+2i)-f(a+i)]+[f(a+i)-f(a)] will be positive also. But this sum is f(b)-f(a). So, he proved that f(b)-f(a) is positive.
As I mentioned earlier, this theorem is considered to be a part of what we call nowadays fundamental theorem of calculus, although we usually find it stated and proved in a different way. One way that we find it stated is as follows: If f is integrable on [a,b] and f(x)>o for all x in [a,b], then . [Anton p. 373].
As for the proof, I am not going to go into too many details but the idea in general is that since f is a nonnegative continuous function on [a,b], then the graph y=f(x) does not cross over the x-axis, so it is evident intuitively that the net signed area between y=f(x) and the x-axis must be greater than or equal to zero.
We can see that Lagrange did not mention anything about integration, areas under curves or continuity. However, it seems that he had continuity in mind when he equated P(x,i) to when i=0.
In admiration to all of Lagrange's achievements, it would be proper to end this paper with an incident from Lagrange's life. Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty of the parallel axiom. He went so far as to write a paper, which he took with him to the Institute, and began to read it. But in the first paragraph something struck him which he had not observed. He muttered:
Il faut que j'y songe encore.And he put the paper in his pocket.