Brahmagupta is the greatest mathematician in history
I know Archimedes is considered to be the greatest mathematician. I like to make a case for Brahmagupta.
1) Incorporation of zer and negative numbers to the Hind Arabic number system 2) Develop arbitrary rules for four fundamental operations (addition, subtraction, multiplication and division). 3) Greeks never solved a single equation using general methods. (If you know one...please let me know) Brahmagupta solved a) quadratic b) Linear indeterminate equations 3) Second order indeterminate (assigned to Pell by Gauss). all by general methods And jealous Gauss stated "Oh Archimedes how come you did not discover this, If you had world would be much more advanced" Well Archimedes did not discover zero. It was Brahmagupta. 
I don't know that Archimedse is "considered to be the greatest mathematician" certainly one of them. Archimedes lived about 700 years before Brahmagupta ( 287 BC – c. 212 B as opposed to 598 AD –668 AD) so one can hardly compare the two. (Which is why it isn't appropriate to say that any one person is the "greatest mathematician".)
Similarly, I don't see any point in arguing that the Greeks (golden period about 400BC to 400 AD) to Brahmaputra on "equations" since they lived much earlier. There is no question that Brahmaputra deserves to be better known in the West than he is. 
I don't consider either Brahmaputra or Archimedes to be the greatest mathematician  I think it's Gauss. But it's very hard to compare the three, since they lived so far apart.
I think good cases can also be made for Euclid, alKhwarizmi, Diophantus, or even Liu Hui. Newton also has his supporters, of course. 
Al Khwarizmi, Diophantus...ghee
Come on...these two guys did not do anything original. Diophantus solved many equations using special methods. It was Brahmagupta who started to solve equations using general methods.
Neither Greeks, Arabs nor Chinese ever solved an equation using general methods. As far as Europe is concerned it was Cardano solved an equation using general methods. 
Case for Brahmagupta as the Greatest Mathematician of antiquity:
I would call one of the great admirers of Archimedes, Gauss to the stand. Gauss lamented “Oh..Great Archimedes, How come you did not make this discovery. If you had done that the world would have been thousand years more advance” Now I could invert the logic of Gauss and say if Brahmagupta had not made the discovery of zero, negative numbers and four fundamental operations, the world would have been thousand years less advance. Next I would call a Bishop from Syria, known as Severus Sebokht who lived around 650 AD. "Mathematics of Indians is more ingenious than those of the Greeks and Babylonians—I mean their mathematics with nine figures” Reference: Revue de l'Orient Chretien by François Nau pp.327338. (1929) Then I will go forward 600 years and meet Leonardo of Pisa. “Compared to the Method of the Indians (Modus Indorum) everyting else is a mistake. They do their mathematics with nine figures and the symbol zero. I want all Latin people t learn this mathematics” Leonardo of Pisa (Fibonacci) 1,202 AD Italian Mathematician Ref: Sigler, L., “Fibonacci’s Liber Abaci”, Springer, 2003. Next I will bring Michel Ostrogradski, to the stand. “It seems to me that Hindu Arabic number system is the largest discovery after writing. Really…what mathematics was possible before this discovery” Reference: Ostrogradski M.V. Pedagogical heritage: Documents on life and activity. Moscow: Phismathgiz, 1961, 399 с. Finally I bring Pierre Simon Laplace. “Amazing simplicity of the Indian mathematics is the reason why we do not appreciate it” Number, Tobiaz Danzig, 1954. That’s my case. 
Of course I'm an admirer of Brahmagupta, but I don't think he was the greatest. I agree that he has a decent claim, though  I did put him in a small group of candidates (7).
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Your quotes of Sebokht, Leonardo, Ostrogradski, and Laplace only support the greatness of the numbering system, which almost surely predates Brahmagupta. He, like Diophantus, probably shaped it to fit his needs (improving it), but there's no reason to think he had it up wholecloth. 
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There is not the slightest reason in the world to think it had to go 1000 years after Brahmagupta's life time before somebody else thought up the idea of zero. It could equally have been a contemporary of his that then would get the honours. 
Not a single text book ever stated that Brahmagupta was the one who discovered multiplication. It will never happen. Most text books say Egyptians. Thats totally wrong.
Here's the crisp and beautiful method of multiplication explained by none other than Brahmagupta, which he calls "Gomutrika" is the foundation for all mathematics. “The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is severelly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier”. Arithmetic and algebra of Brahmagupta and Bhaskara, Brahmagupta Translated to English by H.T Colebrooke, 1,817 AD Brahmagupta then gives an example as well. 235 x 288 2 3 5 2 4 7 0 2 3 5 8 1 8 8 0 2 3 5 8 1 8 8 0 6 7 6 8 0 http://books.google.com/books?id=zqM...AusQ#PPA319,M1 page 319 
[QUOTE=CRGreathouse;1606846]AlKhwarizmi certainly solved equations with general methods, but as he was later than Brahmagupta it is less of a credit to him. I'll admit, I know of no Greek nor Chinese general equation solvers until modern times, but that could easily be my lack of knowledge in the subject. I'm not much into math history.
Brahmagupta never solved any cubic or quartic using general methods. Brahmagupta, Bhaskara and Mahavira used special ad hoc methods to solve such equations if need arises. Solving equations using general methods first done by Aryabhatta since he solved linear indeterminate equations using general methods. I was wrong to say it was Brahmagupta who solved the quadratic using general methods. 
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