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 were made by Abul Wefa,[20] who solved geometrically $\scriptstyle{x^4=a}$ and $\scriptstyle{x^4+ax^3=b}$.

The solution of cubic equations by intersecting conics was the greatest achievement of the Arabs in algebra. The foundation to this work had been laid by the Greeks, for it was Menæchmus who first constructed the roots of $$\scriptstyle{x^3-a=0}$$ or $\scriptstyle{x^3-2a^3=0}$. It was not his aim to find the number corresponding to x, but simply to determine the side x of a cube double another cube of side a. The Arabs, on the other hand, had another object in view: to find the roots of given numerical equations. In the Occident, the Arabic solutions of cubics remained unknown until quite recently. Descartes and Thomas Baker invented these constructions anew. The works of Al Hayyami, Al Karhi, Abul Gud, show how the Arabs departed further and further from the Indian methods, and placed themselves more immediately under Greek influences. In this way they barred the road of progress against themselves. The Greeks had advanced to a point where material progress became difficult with their methods; but the Hindoos furnished new ideas, many of which the Arabs now rejected.

With Al Karhi and Omar Al Hayyami, mathematics among the Arabs of the East reached flood-mark, and now it begins to ebb. Between 1100 and 1300 A.D. come the crusades with war and bloodshed, during which European Christians profited much by their contact with Arabian culture, then far superior to their own; but the Arabs got no science from the Christians in return. The crusaders were not the only adversaries of the Arabs. During the first half of the thirteenth century, they had to encounter the wild Mongolian hordes, and, in 1256, were conquered by them under the leadership of Hulagu. The caliphate at BagdadBaghdad [sic] now ceased to exist. At the close of the fourteenth century still another empire was formed by