A History of Mathematics/Middle Ages/The Arabs

by Florian Cajori

THE ARABS.

 

After the flight of Mohammed from Mecca to Medina in 622 A.D., an obscure people of Semitic race began to play an important part in the drama of history. Before the lapse of ten years, the scattered tribes of the Arabian peninsula were fused by the furnace blast of religious enthusiasm into a powerful nation. With sword in hand the united Arabs subdued Syria and Mesopotamia. Distant Persia and the lands beyond, even unto India, were added to the dominions of the Saracens. They conquered Northern Africa, and nearly the whole Spanish peninsula, but were finally checked from further progress in Western Europe by the firm hand of Charles Martel (732 A.D.). The Moslem dominion extended now from India to Spain; but a war of succession to the caliphate ensued, and in 755 the Mohammedan empire was divided,—one caliph reigning at Bagdad, the other at Cordova in Spain. Astounding as was the grand march of conquest by the Arabs, still more so was the ease with which they put aside their former nomadic life, adopted a higher civilisation, and assumed the sovereignty over cultivated peoples. Arabic was made the written language throughout the conquered lands. With the rule of the Abbasides in the East began a new period in the history of learning. The capital, Bagdad, situated on the Euphrates, lay half-way between two old centres of scientific thought,—India in the East, and Greece in the West. The Arabs were destined to be the custodians of the torch of Greek and Indian science, to keep it ablaze daring the period of confusion and chaos in the Occident, and afterwards to pass it over to the Europeans. Thus science passed from Aryan to Semitic races, and then back again to the Aryan. The Mohammedans have added but little to the knowledge in mathematics which they received. They now and then explored a small region to which the path had been previously pointed out, but they were quite incapable of discovering new fields. Even the more elevated regions in which the Hellenes and Hindoos delighted to wander—namely, the Greek conic sections and the Indian indeterminate analysis—were seldom entered upon by the Arabs. They were less of a speculative, and more of a practical turn of mind.

The Abbasides at Bagdad encouraged the introduction of the sciences by inviting able specialists to their court, irrespective of nationality or religious belief. Medicine and astronomy were their favourite sciences. Thus Haroun-al-Raschid, the most distinguished Saracen ruler, drew Indian physicians to Bagdad. In the year 772 there came to the court of Caliph Almansur a Hindoo astronomer with astronomical tables which were ordered to be translated into Arabic. These tables, known by the Arabs as the Sindhind, and probably taken from the Brahma-sphuta-siddhanta of Brahmagupta, stood in great authority. They contained the important Hindoo table of sines.

Doubtless at this time, and along with these astronomical tables, the Hindoo numerals, with the zero and the principle of position, were introduced among the Saracens. Before the time of Mohammed the Arabs had no numerals. Numbers were written out in words. Later, the numerous computations connected with the financial administration over the conquered lands made a short symbolism indispensable. In some localities, the numerals of the more civilised conquered nations were used for a time. Thus in Syria, the Greek notation was retained; in Egypt, the Coptic. In some cases, the numeral adjectives may have been abbreviated in writing. The Diwani-numerals, found in an Arabic-Persian dictionary, are supposed to be such abbreviations. Gradually it became the practice to employ the 28 Arabic letters of the alphabet for numerals, in analogy to the Greek system. This notation was in turn superseded by the Hindoo notation, which quite early was adopted by merchants, and also by writers on arithmetic. Its superiority was so universally recognised, that it had no rival, except in astronomy, where the alphabetic notation continued to be used. Here the alphabetic notation offered no great disadvantage, since in the sexagesimal arithmetic, taken from the Almagest, numbers of generally only one or two places had to be written.^[7]

As regards the form of the so-called Arabic numerals, the statement of the Arabic writer Albiruni (died 1039), who spent many years in India, is of interest. He says that the shape of the numerals, as also of the letters in India, differed in different localities, and that the Arabs selected from the various forms the most suitable. An Arabian astronomer says there was among people much difference in the use of symbols, especially of those for 5, 6, 7, and 8. The symbols used by the Arabs can be traced back to the tenth century. We find material differences between those used by the Saracens in the East and those used in the West. But most surprising is the fact that the symbols of both the East and of the West Arabs deviate so extraordinarily from the Hindoo Devanagari numerals (= divine numerals) of to-day, and that they resemble much more closely the apices of the Roman writer Boethius. This strange similarity on the one hand, and dissimilarity on the other, is difficult to explain. The most plausible theory is the one of Woepcke: (1) that about the second century after Christ, before the zero had been invented, the Indian numerals were brought to Alexandria, whence they spread to Rome and also to West Africa; (2) that in the eighth century, after the notation in India had been already much modified and perfected by the invention of the zero, the Arabs at Bagdad got it from the Hindoos; (3) that the Arabs of the West borrowed the Columbus-egg, the zero, from those in the East, but retained the old forms of the nine numerals, if for no other reason, simply to be contrary to their political enemies of the East; (4) that the old forms were remembered by the West-Arabs to be of Indian origin, and were hence called Gubar-numerals (= dust-numerals, in memory of the Brahmin practice of reckoning on tablets strewn with dust or sand; (5) that, since the eighth century, the numerals in India underwent further changes, and assumed the greatly modified forms of the modern Devanagari-numerals.^[3] This is rather a bold theory, but, whether true or not, it explains better than any other yet propounded, the relations between the apices, the Gubar, the East-Arabic, and Devanagari numerals.

It has been mentioned that in 772 the Indian Siddhanta was brought to Bagdad and there translated into Arabic. There is no evidence that any intercourse existed between Arabic and Indian astronomers either before or after this time, excepting the travels of Albiruni. But we should be very slow to deny the probability that more extended communications actually did take place.

Better informed are we regarding the way in which Greek science, in successive waves, dashed upon and penetrated Arabic soil. In Syria the sciences, especially philosophy and medicine, were cultivated by Greek Christians. Celebrated were the schools at Antioch and Emesa, and, first of all, the flourishing Nestorian school at Edessa. From Syria, Greek physicians and scholars were called to Bagdad. Translations of works from the Greek began to be made. A large number of Greek manuscripts were secured by Caliph Al Mamun (813–833) from the emperor in Constantinople and were turned over to Syria. The successors of Al Mamun continued the work so auspiciously begun, until, at the beginning of the tenth century, the more important philosophic, medical, mathematical, and astronomical works of the Greeks could all be read in the Arabic tongue. The translations of mathematical works must have been very deficient at first, as it was evidently difficult to secure translators who were masters of both the Greek and Arabic and at the same time proficient in mathematics. The translations had to be revised again and again before they were satisfactory. The first Greek authors made to speak in Arabic were Euclid and Ptolemæus. This was accomplished during the reign of the famous Haroun-al-Raschid. A revised translation of Euclid's Elements was ordered by Al Mamun. As this revision still contained numerous errors, a new translation was made, either by the learned Honein ben Ishak, or by his son, Ishak ben Honein. To the thirteen books of the Elements were added the fourteenth, written by Hypsicles, and the fifteenth by Damascius. But it remained for Tabit ben Korra to bring forth an Arabic Euclid satisfying every need. Still greater difficulty was experienced in securing an intelligible translation of the Almagest. Among other important translations into Arabic were the works of Apollonius, Archimedes, Heron, and Diophantus. Thus we see that in the course of one century the Arabs gained access to the vast treasures of Greek science. Having been little accustomed to abstract thought, we need not marvel if, during the ninth century, all their energy was exhausted merely in appropriating the foreign material. No attempts were made at original work in mathematics until the next century.

In astronomy, on the other hand, great activity in original research existed as early as the ninth century. The religious observances demanded by Mohammedanism presented to astronomers several practical problems. The Moslem dominions being of such enormous extent, it remained in some localities for the astronomer to determine which way the "Believer" must turn during prayer that he may be facing Mecca. The prayers and ablutions had to take place at definite hours during the day and night. This led to more accurate determinations of time. To fix the exact date for the Mohammedan feasts it became necessary to observe more closely the motions of the moon. In addition to all this, the old Oriental superstition that extraordinary occurrences in the heavens in some mysterious way affect the progress of human affairs added increased interest to the prediction of eclipses.^[7]

For these reasons considerable progress was made. Astronomical tables and instruments were perfected, observatories erected, and a connected series of observations instituted. This intense love for astronomy and astrology continued during the whole Arabic scientific period. As in India, so here, we hardly ever find a man exclusively devoted to pure mathematics. Most of the so-called mathematicians were first of all astronomers.

The first notable author of mathematical books was Mohammed ben Musa Hovarezmi, who lived during the reign of Caliph Al Mamun (814—833). He was engaged by the caliph in making extracts from the Sindhind, in revising the tablets of Ptolemæus, in taking observations at Bagdad and Damascus, and in measuring a degree of the earth's meridian. Important to us is his work on algebra and arithmetic. The portion on arithmetic is not extant in the original, and it was not till 1857 that a Latin translation of it was found. It begins thus: "Spoken has Algoritmi. Let us give deserved praise to God, our leader and defender." Here the name of the author, Hovarezmi, has passed into Algoritmi, from which comes our modern word algorithm, signifying the art of computing in any particular way. The arithmetic of Hovarezmi, being based on the principle of position and the Hindoo method of calculation, "excels," says an Arabic writer, "all others in brevity and easiness, and exhibits the Hindoo intellect and sagacity in the grandest inventions." This book was followed by a large number of arithmetics by later authors, which differed from the earlier ones chiefly in the greater variety of methods. Arabian arithmetics generally contained the four operations with integers and fractions, modelled after the Indian processes. They explained the operation of casting out the 9's, which was sometimes called the "Hindoo proof." They contained also the regula falsa and the regula duorum falsorum, by which algebraical examples could be solved without algebra. Both these methods were known to the Indians. The regula falsa or falsa positio was the assigning of an assumed value to the unknown quantity, which value, if wrong, was corrected by some process like the "rule of three." Diophantus used a method almost identical with this. The regula duorum falsorum was as follows:^[7] To solve an equation $\scriptstyle {f(x)=V}$, assume, for the moment, two values for x; namely, $\scriptstyle {x=a}$ and $\scriptstyle {x=b}$. Then form $\scriptstyle {f(a)=A}$ and $\scriptstyle {f(b)=B}$, and determine the errors $\scriptstyle {V-A=E_{a}}$ and $\scriptstyle {V-B=E_{b}}$; then the required $\scriptstyle {x={{bE_{a}-aE_{b}} \over {E_{a}-E_{b}}}}$ is generally a close approximation, but is absolutely accurate whenever $\scriptstyle {f(x)}$ is a linear function of x.

We now return to Hovarezmi, and consider the other part of his work,—the algebra. This is the first book known to contain this word itself as title. Really the title consists of two words, aldshebr walmukabala, the nearest English translation of which is "restoration" and "reduction." By "restoration" was meant the transposing of negative terms to the other side of the equation; by "reduction," the uniting of similar terms. Thus, $\scriptstyle {x^{2}-2x=5x+6}$ passes by aldshebr into $\scriptstyle {x^{2}=5x+2x+6}$; and this, by walmukabala, into $\scriptstyle {x^{2}=7x+6}$. The work on algebra, like the arithmetic, by the same author, contains nothing original. It explains the elementary operations and the solutions of linear and quadratic equations. From whom did the author borrow his knowledge of algebra? That it came entirely from Indian sources is impossible, for the Hindoos had no such rules like the "restoration" and "reduction." They were, for instance, never in the habit of making all terms in an equation positive, as is done by the process of "restoration." Diophantus gives two rules which resemble somewhat those of our Arabic author, but the probability that the Arab got all his algebra from Diophantus is lessened by the considerations that he recognised both roots of a quadratic, while Diophantus noticed only one; and that the Greek algebraist, unlike the Arab, habitually rejected irrational solutions. It would seem, therefore, that the algebra of Hovarezmi was neither purely Indian nor purely Greek, bat was a hybrid of the two, with the Greek element predominating.

The algebra of Hovarezmi contains also a few meagre fragments on geometry. He gives the theorem of the right triangle, but proves it after Hindoo fashion and only for the simplest case, when the right triangle is isosceles. He then calculates the areas of the triangle, parallelogram, and circle. For $\scriptstyle {\pi }$ he uses the value $\scriptstyle {3{\frac {1}{7}}}$, and also the two Indian, $\scriptstyle {\pi ={\sqrt {10}}}$ and $\scriptstyle {\pi ={\frac {62882}{20000}}}$. Strange to say, the last value was afterwards forgotten by the Arabs, and replaced by others less accurate. This bit of geometry doubtless came from India. Later Arabic writers got their geometry almost entirely from Greece.

Next to be noticed are the three sons of Musa ben Sakir, who lived in Bagdad at the court of the Caliph Al Mamun. They wrote several works, of which we mention a geometry in which is also contained the well-known formula for the area of a triangle expressed in terms of its sides. We are told that one of the sons travelled to Greece, probably to collect astronomical and mathematical manuscripts, and that on his way back he made acquaintance with Tabit ben Korra. Recognising in him a talented and learned astronomer, Mohammed procured for him a place among the astronomers at the court in Bagdad. Tabit ben Korra (836-901) was born at Harran in Mesopotamia. He was proficient not only in astronomy and mathematics, but also in the Greek, Arabic, and Syrian languages. His translations of Apollonius, Archimedes, Euclid, Ptolemy, Theodosius, rank among the best. His dissertation on amicable numbers (of which each is the sum of the factors of the other) is the first known specimen of original work in mathematics on Arabic soil. It shows that he was familiar with the Pythagorean theory of numbers. Tabit invented the following rule for finding amicable numbers: If $\scriptstyle {p=3\cdot 2^{n}-1}$, $\scriptstyle {q=3\cdot 2^{n-1}-1}$, $\scriptstyle {r=9\cdot 2^{2n-1}-1}$ (n being a whole number) are three primes, then $\scriptstyle {a=2^{n}pq,~b=2^{n}r}$ are a pair of amicable numbers. Thus, if $\scriptstyle {n=2}$, then $\scriptstyle {p=11}$, $\scriptstyle {q=5}$, $\scriptstyle {r=71}$, and $\scriptstyle {a=220}$, $\scriptstyle {b=284}$. Tabit also trisected an angle.

Foremost among the astronomers of the ninth century ranked Al Battani, called Albategnius by the Latins. Battan in Syria was his birthplace. His observations were celebrated for great precision. His work, De scientia stellarum, was translated into Latin by Plato Tiburtinus, in the twelfth century. Out of this translation sprang the word 'sinus,' as the name of a trigonometric function. The Arabic word for "sine," dsckiba, was derived from the Sanscrit jiva, and resembled the Arabic word dschaib, meaning an indentation or gulf. Hence the Latin "sinus."^[3] Al Battani was a close student of Ptolemy, but did not follow him altogether. He took an important step for the better, when he introduced the Indian "sine" or half the chord, in place of the whole chord of Ptolemy. Another improvement on Greek trigonometry made by the Arabs points likewise to Indian influences. Propositions and operations which were treated by the Greeks geometrically are expressed by the Arabs algebraically. Thus, Al Battani at once gets from an equation $\scriptstyle {{{\sin \theta } \over {\cos \theta }}=D}$, the value of $\scriptstyle {\theta }$ by means of $\scriptstyle {\sin \theta ={D \over {\sqrt {1+D^{2}}}}}$,—a process unknown to the ancients. He knows, of course, all the formulas for spherical triangles given in the Almagest, but goes further, and adds an important one of his own for oblique-angled triangles; namely, $\scriptstyle {\cos a=\cos b\cos c+\sin b\sin c\cos A}$.

At the beginning of the tenth century political troubles arose in the East, and as a result the house of the Abbasides lost power. One province after another was taken, till, in 945, all possessions were wrested from them. Fortunately, the new rulers at Bagdad, the Persian Buyides, were as much interested in astronomy as their predecessors. The progress of the sciences was not only unchecked, but the conditions for it became even more favourable. The Emir Adud-ed-daula (978-983) gloried in having studied astronomy himself. His son Saraf-ed-daula erected an observatory in the garden of his palace, and called thither a whole group of scholars.^[7] Among them were Abul Wefa, Al Kuhi, Al Sagani.

Abul Wefa (940-998) was born at Buzshan in Chorassan, a region among the Persian mountains, which has brought forth many Arabic astronomers. He forms an important exception to the unprogressive spirit of Arabian scientists by his brilliant discovery of the variation of the moon, an inequality usually supposed to have been first discovered by Tycho Brahe.^[11] Abul Wefa translated Diophantus. He is one of the last Arabic translators and commentators of Greek authors. The fact that he esteemed the algebra of Mohammed ben Musa Hovarezmi worthy of his commentary indicates that thus far algebra had made little or no progress on Arabic soil. Abul Wefa invented a method for computing tables of sines which gives the sine of half a degree correct to nine decimal places. He did himself credit by introducing the tangent into trigonometry and by calculating a table of tangents. The first step toward this had been taken by Al Battani. Unfortunately, this innovation and the discovery of the moon's variation excited apparently no notice among his contemporaries and followers. "We can hardly help looking upon this circumstance as an evidence of a servility of intellect belonging to the Arabian period." A treatise by Abul Wefa on "geometric constructions" indicates that efforts were being made at that time to improve draughting. It contains a neat construction of the corners of the regular polyedrons on the circumscribed sphere. Here, for the first time, appears the condition which afterwards became very famous in the Occident, that the construction be effected with a single opening of the compass.

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.

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 this, but we are told that the demonstration was defective. Creditable work 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 $\scriptstyle {x^{2n}+ax^{n}=b}$. 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:—

${\begin{aligned}\scriptstyle {1^{2}+2^{2}+3^{2}+\cdots +n^{2}}&\scriptstyle {=(1+2+\cdots +n)^{{2n+1} \over 3}{\text{,}}}\\\scriptstyle {1^{3}+2^{3}+3^{3}+\cdots +n^{3}}&\scriptstyle {=(1+2+\cdots +n)^{2}{\text{.}}}\\\end{aligned}}$

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 $\scriptstyle {x^{3}-x^{2}-2x+1=0}$. It was attempted by many and at last solved by Abul Gud.

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 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 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 (1547-1622). 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 quantities.' This is: If $\scriptstyle {PP_{1}}$ and $\scriptstyle {QQ_{1}}$ be two arcs of great circles intersecting in $\scriptstyle {A}$ and if $\scriptstyle {PQ}$ and $\scriptstyle {P_{1}Q_{1}}$ be arcs of great circles drawn perpendicular to $\scriptstyle {QQ_{1}}$, then we have the proportion

$\scriptstyle {\sin AP:\sin PQ=\sin AP_{1}:\sin P_{1}Q_{1}}$.

From this he derives the formulas for spherical right triangles. To the four fundamental formulas already given by Ptolemy, he added a fifth, discovered by himself. If a, b, c, be the sides, and A, B, C, the angles of a spherical triangle, right-angled at A, then $\scriptstyle {\cos B=\cos b\sin C}$. This is frequently called "Geber's Theorem." Radical and bold as were his innovations in spherical trigonometry, in plane trigonometry he followed slavishly the old beaten path of the Greeks. Not even did he adopt the Indian 'sine' and 'cosine,' but still used the Greek 'chord of double the angle.' So painful was the departure from old ideas, even to an independent Arab! After the time of Gabir ben Aflah there was no mathematician among the Spanish Saracens of any reputation. In the year in which Columbus discovered America, the Moors lost their last foot-hold on Spanish soil.

We have witnessed a laudable intellectual activity among the Arabs. They had the good fortune to possess rulers who, by their munificence, furthered scientific research. At the courts of the caliphs, scientists were supplied with libraries and observatories. A large number of astronomical and mathematical works were written by Arabic authors. Yet we fail to find a single important principle in mathematics brought forth by the Arabic mind. Whatever discoveries they made, were in fields previously traversed by the Greeks or the Indians, and consisted of objects which the latter had overlooked in their rapid march. The Arabic mind did not possess that penetrative insight and invention by which mathematicians in Europe afterwards revolutionised the science. The Arabs were learned, but not original. Their chief service to science consists in this, that they adopted the learning of Greece and India, and kept what they received with scrupulous care. When the love for science began to grow in the Occident, they transmitted to the Europeans the valuable treasures of antiquity. Thus a Semitic race was, during the Dark Ages, the custodian of the Aryan intellectual possessions.