An Essay by the Uniquely Wise 'Abel Fath Omar Bin Al-khayam by Omar Al-khayam

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By Omar Al-khayam

Omar Al-Khayyam's well-known 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 booklet as a vital merchandise in either their own libraries and institutional collections. This ebook bargains with the answer of quadratic and cubic equations. Al-Khayyam solved all attainable situations of such equations through the use of geometrical techniques, occasionally related to conic sections equivalent to parabolas and hyperbolas. The proofs offered are special 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 may be astonished via the prime quality of the paintings, linguistically in addition to mathematically. Historians of technological know-how, lecturers of arithmetic and mathematicians themselves will locate the booklet either fascinating and informative.

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If c falls on the circumference of the circle or outside it, then we extend cz and construct a rectangle, with one of its angles at c, in such a way that if we construct at the opposite angle to c a section as described before, it will meet the circle by intersection or tangency. This can be proved by a simple deduction that I have left as a mathematical exercise for the reader of my essay. Whoever could not prove the deduction will gain nothing from this essay, it being based on the previously mentioned three books.

The two sides, the perpendicular and the oblique, are equal. Hence, the square of ht equals the product of bt and tc, and so the ratio of bt to th is the same as the ratio of th to tc. But the square of hm (which is equal to bt) equals the product of bm and ba, as was shown by proposition yb of article a in the book of conic sections. So the ratio of ab to bt is the same as the ratio of bt to bm and is as the ratio of bm (which equals ht) to tc. Thus, the four lines are proportional. Hence, the ratio of the square of ab (the first) to the square of bt (the second) is the same as the ratio of bt (the second) to tc (the fourth).

One of my friends suggested that I should show the flaw in the proof of Abu-Aljood Mohammad Ben Al-Laith of the fifth type of the six triplicate (three terms) types that can be solved by conic sections which is: a cube plus a number equals squares. Abu-Aljood said: We set the line ab to equal the number of squares, and we cut from ab the segment bc to be a side of a cube that equals the assumed number. So the line bc is either equal to, greater than, or smaller than ca. He said: If they are equal, we complete the rectangle ch then construct a hyperbola at d that does not meet ab and bh.

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