Joseph Louis LaGrange biography sites:

http://www-history.mcs.st-andrews.ac.uk/Biographies/Lagrange.html

Sophie Germain biography sites:

http://www-history.mcs.st-andrews.ac.uk/Biographies/Germain.html

http://www.agnesscott.edu/Lriddle/WOMEN/germain.htm

Quotes attributed to LaGrange:

"The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure."   Preface to Mécanique Analytique.

[said about the chemist Lavoisier:]  "It took the mob only a moment to remove his head; a century will not suffice to reproduce it." Quoted in H Eves, An introduction to the history of mathematics (Philadelphia 1983).

"I do not know."

"As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection."

Regarding Sophie Germain:

"Sophie Germain proved to the world that even a woman can accomplish something in the most rigorous and abstract of sciences."  Carl Frederic Gauss quoted in D MacHale, Comic Sections (Dublin 1993)

"All things considered, she was probably the most profoundly intellectual woman that France has ever produced. And yet, strange as it may seem, when the state official came to make out her death certificate, he designated her as a "rentière-annuitant" (a single woman with no profession)—not as a "mathématicienne." Nor is this all. When the Eiffel Tower was erected, there was inscribed on this lofty structure the names of seventy-two savants. But one will not find in this list the name of that daughter of genius, whose researches contributed so much toward establishing the theory of the elasticity of metals—Sophie Germain. Was she excluded from this list for the same reason she was ineligible for membership in the French Academy—because she was a woman? If such, indeed, was the case, more is the shame for those who were responsible for such ingratitude toward one who had deserved so well of science, and who by her achievements had won an enviable place in the hall of fame."    H. J. Mozans, 1913

Joseph LaGrange is known for contributions to the theories of determinants, continued fractions, and many other fields. He developed partial differential equations far beyond those of d'Alembert, developed the calculus of variations far beyond that of the Bernoullis, and developed terminology and notation (e.g. the use of f'(x) and f''(x) for a function's 1st and 2nd derivatives). He proved a fundamental Theorem of Group Theory. He was a master of number theory, proving difficult and historic theorems including Wilson's theorem (p divides (p-1)! + 1 when p is prime); Lagrange's Four-Square Theorem (every positive integer is the sum of four squares); and that n·x² + 1 = y² has solutions for every positive non-square integer).  Lagrange's many contributions to physics include understanding of vibrations (he found an error in Newton's work and published the definitive treatise on sound), celestial mechanics (including an explanation of why the Moon keeps the same face pointed towards the Earth), the principle of least action (which Hamilton compared to poetry), and the discovery of the Lagrangian points (e.g., in Jupiter's orbit). Lagrange's textbooks were noted for clarity and inspired most of the 19th century mathematicians on this list.

Sophie Germain is known for her work on Fermat's Last Theorem, and a theorem which has become known as Germain's Theorem. This was to remain the most important result related to Fermat's Last Theorem from 1738 until the contributions of Kummer in 1840.

NOVA's  "Math's Hidden Woman" http://www.pbs.org/wgbh/nova/proof/germain.html

Sophie Germain Prime: http://mathworld.wolfram.com/SophieGermainPrime.html