MATHEMATICIANS

GALILEO

Galileo discovered the laws of inertia, falling bodies (including parabolic trajectories), and the pendulum. He even anticipated Einstein's special relativity. He was a great inventor: in addition to being first to conceive of the pendulum clock, he developed a new type of pump, and the best telescope, thermometer, hydrostatic balance, and cannon sector of his day. As a famous astronomer, he pointed out that Jupiter's Moons, which he discovered, provide a natural clock and allow a universal time to be determined by telescope anywhere on Earth. (This was of little use in ocean navigation since a ship's rocking prevents the required delicate observations.) Galileo is also renowned for early discoveries with microscope.
Galileo is often called the "Father of Modern Science" because of his emphasis on experimentation. He understood that results needed to be repeated and averaged (he used mean absolute difference as his curve-fitting criterion, two centuries before Gauss and Legendre introduced the mean squared-difference criterion). For his experimental methods and discoveries, his laws of motion, and for (eventually) helping to spread Copernicus' heliocentrism, Galileo is considered to be one of the most influential scientists ever; he ranks #12 on Hart's list of the Most Influential Persons in History. Despite his extreme importance to mathematical physics, Galileo doesn't usually appear on lists of greatest mathematicians. However Cavalieri and Torricelli were his students; to encourage them, he probably avoided competing with them. Galileo derived certain centroids using a rudimentary calculus before Cavalieri did, named (and may have been first to discover) the cycloid curve, and did other mathematical work. Moreover, Galileo may have been first to write about infinite equinumerosity (the "Hilbert's Hotel Paradox"). Galileo once wrote "Mathematics is the language in which God has written the universe.
EUCLID
Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that there are infinitely many prime numbers; he stated and proved the unique factorization theorem; and he devised Euclid's algorithm for computing gcd. He introduced the Mersenne primes and observed that (M2+M)/2 is always perfect (in the sense of Pythagoras) if M is Mersenne. (The converse, that any even perfect number has such a corresponding Mersenne prime, was tackled by Alhazen and proven by Euler.) He proved that there are only five "Platonic solids," as well as theorems of geometry far too numerous to summarize; among many with special historical interest is the proof that rigid-compass constructions can be implemented with collapsing-compass constructions. Among several books attributed to Euclid are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), a book on spherical geometry, a book on logic fallacies, and his comprehensive math textbook The Elements. Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem (a pair of triangles is coaxial if and only if it is copolar) was proved in one of these lost works; this is the fundamental theorem which initiated the study of projective geometry. Euclid ranks #14 on Michael Hart's famous list of the Most Influential Persons in History. The Elements introduced the notions of axiom and theorem; was used as a textbook for 2000 years; and in fact is still the basis for high school geometry, making Euclid the leading mathematics teacher of all time. Some think his best inspiration was recognizing that the Parallel Postulate must be an axiom rather than a theorem.
There are many famous quotations about Euclid and his books. Abraham Lincoln abandoned his law studies when he didn't know what "demonstrate" meant and "went home to my father's house [to read Euclid], and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."

ARISTOTLE
Aristotle is considered the greatest scientist of the ancient world, and the most influential philosopher and logician ever; he ranks #13 on Michael Hart's list of the Most Influential Persons in History. (His science was a standard curriculum for almost 2000 years, unfortunate since many of his ideas were quite mistaken.) His writings on definitions, axioms and proofs may have influenced Euclid. He was also the first mathematician to write on the subject of infinity. His writings include geometric theorems, some with proofs different from Euclid's or missing from Euclid altogether; one of these (which is seen only in Aristotle's work prior to Apollonius) is that a circle is the locus of points whose distances from two given points are in constant ratio. Even if, as is widely agreed, Aristotle's geometric theorems were not his own work, his status as the most influential logician and philosopher makes him a candidate for the List.

ALBERT EINSTEIN
Albert Einstein is generally agreed to be one of the two greatest physicists in all of history. The atomic theory achieved general acceptance only after Einstein's 1905 paper which showed that atoms' discreteness explained Brownian motion. Another famous 1905 paper introduced the famous equation E = mc2; yet Einstein published other papers that same year, two of which were more important and influential than either of the two just mentioned! No wonder that physicists speak of the Miracle Year without bothering to qualify it as Einstein's Miracle Year! (Altogether Einstein published at least 300 books or papers on physics.)
Although most famous for his Special and General Theories of Relativity, Einstein's discovery of the photon made him the key pioneer in quantum theory (and led to his only Nobel Prize). Many years later, he was first to call attention to the "spooky" nature of quantum entanglement.

Einstein certainly has the breadth, depth, and historical importance to qualify for this list; but his genius and significance were not in the field of pure mathematics. (He acknowledged his limitation, writing "I admire the elegance of your [Levi-Civita's] method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.") Einstein was a mathematician, however; he pioneered the application of tensor calculus to physics and invented the "Einstein summation notation." I've chosen to include him on this list because his extreme greatness overrides his focus away from math. Einstein ranks #10 on Michael Hart's famous list of the Most Influential Persons in History. His General Theory of Relativity has been called the most creative and original scientific theory ever. Einstein once wrote "... the creative principle resides in mathematics [; thus] I hold it true that pure thought can grasp reality, as the ancients dreamed."