T

Al Kuhi, the second astronomer at the observatory of the emir at Bagdad, was a close student of Archimedes and Apollonius. He solved the problem, to construct a segment of a sphere equal in volume to a given segment and having a curved surface equal in area to that of another given segment. He, Al Sagani, and Al Biruni made a study of the trisection of angles. Abul Gud, an able geometer, solved the problem by the intersection of a parabola with an equilateral hyperbola.

this, but we are Creditable work

The Arabs had already discovered the theorem that the sum of two cubes can never be a cube. Abu Mohammed Al Hogendi of Chorassan thought he had proved told that the demonstration was defective. in theory of numbers and algebra was done by Fahri des Al Karhi, who lived at the beginning of the eleventh century. His treatise on algebra is the greatest algebraic work of the Arabs. In it he appears as a disciple of Diophantus. He was the first to operate with higher roots and to solve equations of the form x2+ax" = b. For the solution of quadratic equations he gives both arithmetical and geometric proofs. He was the first Arabic author to give and prove the theorems on the summation of the series: :

1+2*+3+..+n=(1+2+..+n),

13+23+33 + ··· + n3 = (1 + 2 + ··· + n)2.

...

...

Al Karhi also busied himself with indeterminate analysis. He showed skill in handling the methods of Diophantus, but added nothing whatever to the stock of knowledge already on hand. As a subject for original research, indeterminate analysis was too subtle for even the most gifted of Arabian minds. Rather surprising is the fact that Al Karhi's algebra shows no traces whatever of Hindoo indeterminate analysis.

But most astonishing it is, that an arithmetic by the same author completely excludes the Hindoo numerals. It is constructed wholly after Greek pattern. Abul Wefa also, in the second half of the tenth century, wrote an arithmetic in which Hindoo numerals find no place. This practice is the very opposite to that of other Arabian authors. The question, why the Hindoo numerals were ignored by so eminent authors, is certainly a puzzle. Cantor suggests that at one time there may have been rival schools, of which one followed almost exclusively Greek mathematics, the other Indian.

The Arabs were familiar with geometric solutions of quadratic equations. Attempts were now made to solve cubic equations geometrically. They were led to such solutions by the study of questions like the Archimedean problem, demanding the section of a sphere by a plane so that the two segments shall be in a prescribed ratio. The first to state this problem in form of a cubic equation was Al Mahani of Bagdad, while Abu Gafar Al Hazin was the first Arab to solve the equation by conic sections. Solutions were given also by Al Kuhi, Al Hasan ben Al Haitam, and others.20 Another difficult problem, to determine the side of a regular heptagon, required the construction of the side from the equation 203 2x+1=0. It was attempted by many and at last solved by Abul Gud.

x2

The one who did most to elevate to a method the solution of algebraic equations by intersecting conics, was Omar al Hayyami of Chorassan, about 1079 A.D. He divides cubics into two classes, the trinomial and quadrinomial, and each class into families and species. Each species is treated separately but according to a general plan. He believed that cubics could not be solved by calculation, nor bi-quadratics by geometry. He rejected negative roots and often failed to discover all the positive ones. Attempts at bi-quadratic equations

20

were made by Abul Wefa, who solved geometrically x = a and xax = 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 - a = 0 or x3-2 a3 = 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 Bagdad now ceased to exist. At the close of the fourteenth century still another empire was formed by Timur

or Tamerlane, the Tartar. During such sweeping turmoil, it is not surprising that science declined. Indeed, it is a marvel that it existed at all. During the supremacy of Hulagu, lived Nasir Eddin (1201–1274), a man of broad culture and an able astronomer. He persuaded Hulagu to build him and his associates a large observatory at Maraga. Treatises on algebra, geometry, arithmetic, and a translation of Euclid's Elements, were prepared by him. Even at the court of Tamerlane in Samarkand, the sciences were by no means neglected. A group of astronomers was drawn to this court. Ulug Beg (1393-1449), a grandson of Tamerlane, was himself an astronomer. Most prominent at this time was Al Kaschi, the author of an arithmetic. Thus, during intervals of peace, science continued to be cultivated in the East for several centuries. The last Oriental writer was Beha Eddin (15471622). His Essence of Arithmetic stands on about the same level as the work of Mohammed ben Musa Hovarezmi, written nearly 800 years before.

"Wonderful is the expansive power of Oriental peoples, with which upon the wings of the wind they conquer half the world, but more wonderful the energy with which, in less than two generations, they raise themselves from the lowest stages of cultivation to scientific efforts." During all these centuries, astronomy and mathematics in the Orient greatly excel these sciences in the Occident.

Thus far we have spoken only of the Arabs in the East. Between the Arabs of the East and of the West, which were under separate governments, there generally existed considerable political animosity. In consequence of this, and of the enormous distance between the two great centres of learning, Bagdad and Cordova, there was less scientific intercourse among them than might be expected to exist between peoples having the same religion and written language. Thus the

course of science in Spain was quite independent of that in Persia. While wending our way westward to Cordova, we must stop in Egypt long enough to observe that there, too, scientific activity was rekindled. Not Alexandria, but Cairo with its library and observatory, was now the home of learning. Foremost among her scientists ranked Ben Junus (died 1008), a contemporary of Abul Wefa. He solved some difficult problems in spherical trigonometry. Another Egyptian astronomer was Ibn Al Haitam (died 1038), who wrote on geometric loci. Travelling westward, we meet in Morocco Abul Hasan Ali, whose treatise on astronomical instruments' discloses a thorough knowledge of the Conics of Apollonius. Arriving finally in Spain at the capital, Cordova, we are struck by the magnificent splendour of her architecture. At this renowned seat of learning, schools and libraries were founded during the tenth century.

Little is known of the progress of mathematics in Spain. The earliest name that has come down to us is Al Madshriti (died 1007), the author of a mystic paper on 'amicable numbers.' His pupils founded schools at Cordova, Dania, and Granada. But the only great astronomer among the Saracens in Spain is Gabir ben Aflah of Sevilla, frequently called Geber. He lived in the second half of the eleventh century. It was formerly believed that he was the inventor of algebra, and that the word algebra came from 'Gabir' or 'Geber.' He ranks among the most eminent astronomers of this time, but, like so many of his contemporaries, his writings contain a great deal of mysticism. His chief work is an astronomy in nine books, of which the first is devoted to trigonometry. In his treatment of spherical trigonometry, he exercises great independence of thought. He makes war against the time-honoured procedure adopted by Ptolemy of applying "the rule of six quantities," and gives a new way of his own, based on the 'rule of four