By Omar Al-khayam
Omar Al-Khayyam's well-known e-book on algebra and equations is taken into account to be possibly his most crucial contribution to arithmetic - the cream of his paintings. students of arithmetic regard the e-book as an important merchandise in either their own libraries and institutional collections. This e-book bargains with the answer of quadratic and cubic equations. Al-Khayyam solved all attainable instances of such equations by utilizing geometrical techniques, occasionally regarding conic sections comparable to parabolas and hyperbolas. The proofs offered are unique and intensely deep. the writer made due acknowledgment and referral to the paintings and contributions of others who got here ahead of him. a latest reader could be astonished via the prime quality of the paintings, linguistically in addition to mathematically. Historians of technology, lecturers of arithmetic and mathematicians themselves will locate the booklet either attention-grabbing and informative.
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Additional 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 p is known because the two sections are of known locations. From p we construct two perpendiculars pn and hpu, whose locations and magnitudes are known. The rectangle da equals the rectangle dp, so nh equals zh as we have shown many times. Their sides are equivalent and so are the squares of their sides. But the ratio of the square of ph to the square ha is the same as the ratio ch to ha. So the ratio of the square of bd to the square of bh is the same as the ratio ch to ha. The parallelepiped whose base is the square of bd and height ha equals (in volume) the parallelepiped whose base is the square of bh and height ch.
We make dm common. It follows that nl equals hl because their sides are equal and so are the squares of their sides. Hence, the ratio of the square of ab to the square of bl is the same as the ratio of the square of ml to the square of ld. But the ratio of the square of ml to the square of ld is the same as the ratio of cl to ld as we have shown repeatedly. Consequently, the ratio of the square of ab to the square of bl is the same as the ratio of cl to ld. Thus, the parallelepiped – whose height is ld and base equals the square of ab – equals the parallelepiped whose base is bl and height lc.
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. The surface (rectangle) hb equals (in area) the rectangle am; so their sides are equivalent and hence the squares of their sides are equivalent.
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