His most important theoretical invention, however,may be his incorporating infinitesimals;he invented them to solve combinatorial games like Go,but they have pure mathematical significance as thelargest possible ordered field.
Weil made profound contributions to several areas of mathematics,especially algebraic geometry, which he showed to have deep connectionswith number theory.
If Wit so much from Ign'rance undergo,
Ah let not Learning too commence its Foe!
Of old, those met Rewards who cou'd excel,
And such were Prais'd who but endeavour'd well:
Tho' Triumphs were to Gen'rals only due,
Crowns were reserv'd to grace the Soldiers too.
Now, they who reached Parnassus' lofty Crown,
Employ their Pains to spurn some others down;
And while Self-Love each jealous Writer rules,
Contending Wits becomes the Sport of Fools:
But still the Worst with most Regret commend,
For each Ill Author is as bad a Friend.
To what base Ends, and by what abject Ways,
Are Mortals urg'd thro' Sacred Lust of praise!
Ah ne'er so dire a Thirst of Glory boast,
Nor in the Critick let the Man be lost!
Good-Nature and Good-Sense must ever join;
To err is Humane; to Forgive, Divine.
Al-Biruni (Alberuni) was an extremely outstanding scholar,far ahead of his time, sometimes shown withAlkindus and Alhazen as one of the greatest Islamic polymaths,and sometimes compared to Leonardo da Vinci.
LEARN then what MORALS Criticks ought to show,
For 'tis but half a Judge's Task, to Know.
'Tis not enough, Taste, Judgment, Learning, join;
In all you speak, let Truth and Candor shine:
That not alone what to your Sense is due,
All may allow; but seek your Friendship too.
Like many great mathematicians he emphasized the importance of intuition, writing"The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it."
Hausdorff had diverse interests: he composed musicand wrote poetry, studied astronomy, wrote on philosophy,but eventually focused on mathematics, where he did important work inseveral fields including set theory, measure theory,functional analysis, and both algebraic and point-set topology.
(He also showed that the rationals have the same cardinalityas the integers; and that the reals have the same cardinalityas the points of N-space and as the power-set of the integers.)Although there are infinitely many distinct transfinite numbers,Cantor conjectured that , the cardinality ofthe reals, was the second smallest transfinite number.
Poincaré also found time to becomea famous popular writer of philosophy, writing,"Mathematics is the art of giving the same name to different things;"and "A [worthy] mathematician experiences in his workthe same impression as an artist; hispleasure is as great and of the same nature;"and "If nature were not beautiful, it would not be worth knowing,and if nature were not worth knowing, life would not be worth living."With his fame, Poincaré helped the world recognize the importance of thenew physical theories of Einstein and Planck.
Jordan was a great "universal mathematician",making revolutionary advances in group theory, topology, andoperator theory;and also doing important work in differential equations, number theory,measure theory,matrix theory, combinatorics, algebra and especially Galois theory.
Dedekind was far ahead of his time, so Noether became famousas the creator of modern algebra; but she acknowledged her great predecessor,frequently saying "It is all already in Dedekind."Dedekind was concerned with rigor, writing"nothing capable of proof ought to be accepted without proof."Before him, the real numbers, continuity, and infinity alllacked rigorous definitions.
While Einstein, Newton, and Galileo may be the Top Three,Maxwell is a strong candidate for "fourth greatest scientist ever."Recalling Newton's comment about "standing on the shoulders" ofearlier greats, Einstein was asked whose shoulders stood on; he didn't name Newton: he said "Maxwell."Maxwell has been called the "Father of Modern Physics"; he ranks #24 onHart's list of the Most Influential Persons in History.
But then, in 1864 James Clerk Maxwell stunned the world bypublishing the equations of electricity and magnetism andshowing that light itself is linked to the electro-magnetic force.
Methods always seemed to be born in his mind in some mysterious way."Hermite's other famous students included Darboux, Borel, andHadamard who wrote of "how magnificent Hermite's teaching was,overflowing with enthusiasm for science, which seemed to come to lifein his voice and whose beauty he never failed to communicate to us,since he felt it so much himself to the very depth of his being."Although he and Abel had proved that the general quintic lackedalgebraic solutions, Hermite introduced an elliptic analog to thecircular trigonometric functions and used these to providea general solution for the quintic equation.