By Omar Al-khayam
Omar Al-Khayyam's recognized ebook on algebra and equations is taken into account to be possibly his most vital contribution to arithmetic - the cream of his paintings. students of arithmetic regard the ebook as a vital merchandise in either their own libraries and institutional collections. This ebook offers with the answer of quadratic and cubic equations. Al-Khayyam solved all attainable circumstances of such equations by utilizing geometrical methods, occasionally regarding conic sections comparable to parabolas and hyperbolas. The proofs offered are distinctive and extremely deep. the writer made due acknowledgment and referral to the paintings and contributions of others who got here earlier than him. a latest reader could be astonished by means of the top of the range of the paintings, linguistically in addition to mathematically. Historians of technological know-how, academics of arithmetic and mathematicians themselves will locate the publication either attention-grabbing and informative.
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Extra resources for An Essay by the Uniquely Wise 'Abel Fath Omar Bin Al-khayam on Algebra And Equations: Algebra Wa Al-Muqabala (Great Books of Islamic Civilization)
So the location of z is known because the locations of both the circle and the (conic) section are known. From the point z we draw two perpendiculars, zt and za, to hk and ha respectively. The rectangle zh equals the rectangle bk. Removing the common rectangle hl, we are left with two equal rectangles, zb and lk. So the ratio of zl to lc is the same as the ratio of hb to bl because hb equals tl; further, their squares are also proportional. But the ratio of the square of zl to the square of lc is the same as the ratio of dl to lc – for the circle – so the ratio of the square of hb to the square of bl is the same as the ratio of dl to lc.
Construct a parallelepiped whose base is the square bh and (whose volume) equals the given number. Let its height ab be perpendicular to bh. We let bc be along ab and is equal to the assumed number of squares. Then we complete ah and we extend hb along hm regardless of the magnitude. On the given hm we construct the surface hn that equals ah. So the location of the point n is known. At n, we construct the hyperbola ntk that does not meet hm and hs, and so it is of known location. We construct another hyperbola, lct, with vertex at c, and axis along bc, and each of its sides, the perpendicular and the oblique, equals ac.
At h, we construct the hyperbola hm, that does not meet zd or dy. So the location of hm is known. We construct another hyperbola amt, with vertex at the point a and axis ab, and each of its sides, the perpendicular and the oblique, equals ac. This hyperbola must intersect the other section. Let the two sections intersect at m, so the location of m is known. From m we draw two perpendiculars mk and ml whose location and magnitude are known. The rectangle md equals the rectangle hd which in turn equals the rectangle ad, and dk is common.
An Essay by the Uniquely Wise 'Abel Fath Omar Bin Al-khayam on Algebra And Equations: Algebra Wa Al-Muqabala (Great Books of Islamic Civilization) by Omar Al-khayam