| Part I |
| Part II |
| Part III |
I
The Arabs are before all else the pupils of the Greeks; their science is a continuation of Greek science which it preserves, cultivates, and on a number of important points develops and perfects. One of the greatest of them, al-Biruni, said in considering all the conditions necessary for scientific research-early education, knowledge of languages, long life, the possession of means sufficient to enable one to make journeys and acquire books and instruments: ‘all these conditions are rarely found in a single individual, especially in our day. That is why we ought to confine ourselves to what the ancients have dealt with and endeavour to perfect what can be perfected. The middle way is in all things the most praiseworthy; and he who attempts too much ruins himself and his estate’.
Al-Biruni is here, however, obviously too modest; for with this limited ambition the Arabs have really achieved great things in science; they taught the use of ciphers, although they did not invent them, and thus became the founders of the arithmetic of everyday life; they made algebra an exact science and developed it considerably and laid the foundations of analytical geometry; they were indisputably the founders of plane and spherical trigonometry which, properly speaking, did not exist among the Greeks. In astronomy they made a number of valuable observations. They preserved for us in their translations a number of Greek works, the originals of which have been lost: three books of the Conics of Apollonius, the Spherics of Menelaus, the Mechanics of Hero of Alexandria, the Pneumatics of Philo of Byzantium, a short book on the balance attributed to Euclid and another to Archimedes on the clepsydra-for which services we cannot be too grateful to them. Another reason for our interest in Arab sciences is the influence it has had in the West. The Arabs kept alive the higher intellectual life and the study of science in a period when the Christian West was fighting desperately with barbarism. The zenith of their activity may be placed in the ninth and tenth centuries, but it was continued down to the fifteenth. From the twelfth century every one in the West who had any taste for science, some desire for light, turned to the East or to the Moorish West. At this period the works of the Arabs began to the translated as those of the Greeks had previously been by them. The Arabs thus formed a bond of union, a connecting link between ancient culture and modern civilization. When at the renaissance the spirit of man was once again filled with the zeal for knowledge and stimulated by the spark of genius, if it was able to set promptly to work, to produce and to invent, it was become the Arabs had preserved and perfected various branches of knowledge, kept the spirit of research alive and eager and maintained it pliant and ready for future discoveries.
Before going into details, one fact must be impressed upon the render; in the history of the sciences, the words ‘Arab’ and ‘Muslim’ must be taken in a very wide sense. The majority of the learned men who have flourished in the world of Islam and under the protection of Muslim sovereigns were not Arabs by birth, and several were not even Muslims. The centre of intellectual life, which in the later Hellenistic period was at Alexandria in Egypt, was transferred in the most flourishing period of Arab learning to a district which now seems very remote at backward in civilization, to eastern Persia (Khorasan) and beyond to the valley of the Oxus, to Khwarizm, Turkistan and Bactria. Al-Khwarizmi for example was a native of Khiva,, al-Farghani of Transoxania, Abu’l –Wafa’ and al-Battani were of Persian origin, as was al-Biruni; al-Kindi was of pure Arab stock. Farabi was a Turk by origin and avicenna (Ibn Sina) hailed for near Balkh. Al-Ghazali and Nasir al-Din came from Tus in the east of the Persia. Omar Khayyam, who wrote his Algebra in Arabic, enjoys in our day a great fame as a Persian poet. Several of these scholars wrote in both languages. Avicenna made a Persian version of the one of his books which is perhaps the most important for physics: ‘The Philosophy dedicated to ‘Ala’; and Nasir al-Din al-Tusi wrote in the same language a very fine treatise on ethics and a little manual of astronomy. As to Averroes (Ibn Rushd), Arzachel (al-Zarkali), and Alpetragius (al-Bitruji), they were Arabs of Spain.
As to religion, Hunain b. Ishak, his son Ishak, Kusta b. Luka and others who great work as translators were Christians. Thabit b. Kurra, the great geometer, and al-Battani, the illustrious astronomer (Albategnius) belong to the Sabians, a pagan sect which worshipped the stars and was devoted to scientific studies and long survived under Islam. Others like Masha’ allah were Jews; and it may be mentioned that at the period of the Renaissance the Jews contributed very much by their translations and their teaching to the spread of Arab learning in the Latin West.
These scholars, so very different in origin, have however several features in common. Their object was to simplify and make lucid. Without having sufficient genius to make generalization or any great synthesis, they are very good arrangers. They arrange logically. They classify and enumerate, and this simple gift of orderliness and lucidity is almost sufficient to explain the progress which they made. Their manner is didactivc; they appear to address themselves, not, like the Greeks, to some particular amateur or to some Maecenas interest in learning for itself alone, but rather to all intelligent students. Their books remind one of good secondary or university text-books. The Arabs were traders, travelers, and lawyers; they had the positive mind; their science therefore had a practical object; arithmetic had to serve the needs of commerce, and the division of estates; astronomy the requirements of travelers and those who cross the deserts, or of religion which has to know the hours of prayer, the azimuth of Mecca and the moment of the first appearance of the moon of Ramadan.
The Arab is always practical and never becomes lost in reveries. The Arabic languages moreover is dry, precise, and recalls somewhat the style of Voltaire in French. It is most suitable for an exact and precise science than for eloquence and poetic flights. It has the further advantage of lending itself readily to the formation of technical terms. The Arab scholars did not write in verse like the Hindus who composed their algebras in slokas; they did not propound historical problems like the Greeks; they had no taste for enormous numbers and vast periods of time. We do not find among them any kalpa, yoga, or ‘days of Brahma’ as among the Hindus, nor names for very high numerals. They are more positive than the Greeks themselves who were interest in very large numbers, as we see from the Arenarius and the cattle-problem of Archimedes and the ‘great year’ of Aristarchus of Samos.
We have no books of the time of the Umayyads; the documented history of Arab learning only begins with the ‘Abbasids.[1] Under the second caliph of that dynasty, al-Mansur, the centre of the Muslim Empire was transferred from the Byzantine to the Persian part of the Empire: al-Mansur founded Baghdad in 145 (762). He had at his court a number of learned men, engineers, and astronomers. The plans of the town were drawn up under the direction of the celebrated minister Khalid b. Barmark,m by the astromer Naubakht, a Persian, and by Masha’ allah, a Jew. In 154 (770) an astromer, Ya‘qubal-Fazari, presented at the court of al-Mansur a learned Hindu named Manka who introduced the Sindhind (Siddhanta), a treatise on astronomy according to Hindu methods. This work was translated by al-Fazari the young, but the translation is now lost. Al-Fazari was the first Muslim to construct an astrolabe. He wrote on the use of the armillary sphere and prepared tables according to the years of the Arabs. The translations from the Greek begin in the same period: Abu Yakya b. Batrik translated – in addition to medical works – the Quadripartitum of Ptolemy. Masha’allah (d. 815) is a scholar of importance; he wrote on astrology, on the astrolabe, and on meterology; his book on prices, de mercibus, is the oldest scientific work that we possess in Arabic. Johannes de Luna Hispalensis translated several of his works into Latin in the Middle Ages. ‘Omar b. al-Farrukhan (d. in 200-815), a friend of the vizier Yahya the Barmecide, was one of the engineers and architects of Baghdad; he translated some works from the Persian and annotated the Quadripartitum of Ptolemy.
This movement, begun under al-Mansur, developed still more under his grandson al-Ma’mun. A price with a fine intellect, a scholar, philosopher, and theologian, al-Ma’mun caused the works of the ancients to be sought out and established an office for translating them. Euclid as well as the Almagest was now translated into Arabic by al-Hajjaj b. Yusuf whose activity begins in the reign of Harun al-Rashid. His translation comprises the first six books of Euclid. Al-Ma’mun had a degree of the meridian measured in the plain of Sinjar by a method different from that of the Greeks: A number of observers setting out from the same point walked, some to the north, the others to the south, until they had seen the pole star rise and sink one degree. They then measured the distance traversed and took the mean of the results. They did not, however, actually keep to this mean but adopted the larger of the two values, 56 2/3 miles corresponding for the great circle to 47.325 kilometers, a result rather too large. At the same time observations were taken at Baghdad near the Shammasiya gate; its erection is attributed to Sind b. ‘Ali, a Jewish convert to Islam. From these observations the tables called ‘Tested Tables’ or ‘Tables of al-Ma’mun’ were prepared according to method of the sindhind. Al-Farghani (Al-Fraganus) is one of the astronomers of this time who were known to the medieval west. He belonged to Farghana in Transoxania. His Compendium of astronomy, a work much esteemed, was translated into Latin by Gerard of Cremona and by Johannes Hispalensis. Regiomontaus at the Renaissances studied it and the greate Melanchthon published an edition based on the work of Regiomontanus at Nuremberg in 1537.
Arithmetic and algebra also flourished alongside of astronomy. This was the period of the celebrated al-Khwarizmi, (i.e., the native of Khawarizm) [d. between 220 (835) and 230 (844)] whose name, corrupted by the Latin writers of the West, gave us, it is believed, the term algorism (sometimes written algorithm). Besides an important treatise on astronomy, he also wrote a book on the Indian (Hindi) method of calculation and another on algebra. The first was translated into Latin by Adelard of Bath, the two others by Gerard of Cremona; the treatise on astronomy and that on arithmetic are known only from these Latin translations.
The Algebra of al-Khawarizmi is lucid and well arranged. After dealing with equations of the second degree, the author discusses algebraic multiplication and division; he then treats of problems relating to the measurement of surfaces and deals with others relating to the division of estates or to various legal questions; these latter, which are generally equations of the first degree, although very complicated to look at, are all propounded in the form of numerical examples. The method of approaching the equation of the second degree is important. The author, following Diophantus, distinguishes six cases, one of which, however, is given only for the sake of completeness, for it is identical with the simplest case of the first degree, bx=c. The six cases are: squares equal to roots, ax2=bx; squares equal to numbers, ax2 + bx = c, squares and numbers equal to roots ax2 + C = b; roots are numbers equal to squares, bx + c = ax2 . We see from this list that the science of this time had not yet completely grasped the manipulation of signs since the different positions of the terms on the opposite sides of the equations seem to it to require separate solutions. The Arabs give the name mukabala (opposition, comparison) to this opposition of the two sides of an equation. This word is usually associated by them with the word jabr which means ‘restitution’. Jabr (al-jabr, algebra) is adding something to a given quantity or multiplying it so that it becomes exactly equal to another. This term seems to have originally meant the two simplest operations of algebra, a + x = b and ax=b; its application was later extended and it came to mean the whole subject. It is also found contrasted to hatt, ‘descent’, which means to diminish a number by subtraction or division so that it becomes equal to a given quantity:
a-x = b; a =b.
x
Al-Khwarizmi, having thus enumerated the six possible cases, gives the rules for their solution and in letters of the alphabet, for at this period algebraic notations were not yet invented. He then proves the rules. The demonstration is geometrical; the Arabs indeed were primarily geometers; they did not then conceive an algebra existing by itself and not based on geometry. This demonstration, repeated several times with the variations demanded by the differences of the cases, is rather a pretty one: here is an abbreviated example.
To solve the equation: a square and 10 roots are equal to 39 dirhams. Let us imagine a square the side of which is unknown; this is the one of which we desire to know the root: Let AB be
2.5 2.5
G
3
C
A
B
K
T
This square: If we multiply its side by a number, the product is this number of roots which we add to the square. Here we have to add ten roots: Let us therefore take a quarter of 10 or 2.5 and make on each side of the square the 4 parallelograms CGKT; the value of the square and these rectangles will be 39. But the small squares which are in the angles are each 2.5X2.5 or 6.2, that is 25 in all. The whole large square therefore amounts to 39+25 or 64. Its side is therefor 8 and if we deduct twice the side of the small squares at the angles, i.e. twice 2.5 or 5, there remains 3, which is the root of the square sought.
The question has been asked what difference there was in this case between the Hindu and Arab methods. According to M. Rodet, the Hindus were more analytical than the Arabs, less pure geometers; they had in addition the idea of the double sign; they transfer more easily a term from one side of an equation to the other; method with them is thus beginning to generalize. It must, however, be recognized that as regards exposition, their language, pompous and encumbered by its verse form, has not the clearness, exactness, and scientific simplicity of that of the Arabs.
There is a case in al-Khwarizmi where idea of the double sign seems just to emerge. It is the fifth: ax2+c= bx; ‘ in this case’, he says, ‘addition and subtraction may be equally well employed’. The theory of equations of the second degree remained down to the sixteenth century exactly as we find it in the Arab algebraist. In the eighteenth century Leonardo Fibonacci of Pisa, an algebraist of considerable importance, says that he owed a great deal to the Arabs. He traveled in Egypt, Syria, Greece, and Sicily, and learned the Arab method there, recognized it to be ‘superior to the method of Pythagoras’ and composed a Liber Abaci in fifteen chapters, the last of which deals with algebraic calculation. Leonardo of Pisa enumerates the six cases of the quadratic equation just as al-Khwarizmi gives them. The idea of negative and imaginary roots is not clearly defined till Cardan in his Ars Magna (1545).
Al-Khwarizmi’s other work, De Numero indico, raises the often-discussed question of the origin of the numerals. What the Arab scholar calls Indian (Hindi) calculation is counting with the numerals which we call Arabic in contrast to counting with the letters of the alphabet which was then usual in the East. It is evident from this qualification of Hindi that the Arabs did not claim to have invented the numerals; but we must not be too quick to conclude from this that they are really of Indian origin; for, as I have observed, the word Hindi is easily confused in the Arabic script with hindasi which means what relates to geometry or the art of the engineer; in various cases in which the word Hindi is used, the meaning of hindasi fits better; thus there is in astronomy a graduated circle which is called Hindi which ought perhaps to be translated ‘mathematical circle’. The numerals thus called might therefore be simply ‘the mathematical characters’. On the other hand the Persians call the numerals ‘figures of end’, which means in their language: ‘characters of the little or of small quantities’. As to the forms of the numerals, Woepcke wished to derive them from the initials of the names of the numerals in Sanskrit. But, apart from the fact that the connexion of the forms is not at all obvious, it may be objected that, in the arithmetical systems in which letters are used, it is not as a rule the initials of the numerals that are used, but the letters of the alphabet in their order; this was the case among the Greeks and among the Arabs themselves. The learned Arab al-Biruni in the tenth century says that the numerals came from ‘the most beautiful form the Indian figures’. He does not, however, say exactly what this form is nor in what part of India it was in use. It appears on the contrary that the numerals have a simpler and handier form among the Arabs than anywhere else; this must be their original form. The first five numerals are formed from 1,2,3,4,5 strokes ligatured; the next four seem to be formed by very simple conventions. The zero is a little circle or dot. It is very likely that the Arabs obtained these signs, like so much of their science, from the tradition of the neo-Platonic schools.
We know that in the numeral system the zero is of capital importance, for it is the zero that enables us to keep the figures in the series of powers of ten, units, tens, hundreds, &c. In the case where one of these powers is not represented. If we did not have a zero, we should have to use a table with columns, columns of units, tens, hundreds, &c., to keep each figure in its place. This table is what is known as the abacus. We find that the zero was known to the Arabs at least 250 years before it was known in the West. The abacus is first found in Rome with Boethius in the fifth century, but its use did not spread then. It reappears with Gerbert in the tenth century. Gerbert had travelled in Spain and studied the sciences of the Moors. He spread the use of the abacus but was not acquainted with the zero. It was not till the twelfth century that Christian arithmeticians began to write treatises on counting with the numerals, without columns, completed by the zero. This method was called algorithm. Now among the Arabs the numerals appear from the first with the zero. The author of the ‘Keys of the Sciences’, Mafatih al-‘Ulum, writing in the tenth century in a period when the use of the numerals had not yet become general, says that if a power of ten is not represented, a little circle is used ‘to keep the row’. This little circle is called sifr, ‘empty’. Some reckoners put a little bar called tarkin, form Nabataean rikan which also means ‘empty’, ‘nil’.
It may be noted that the Latin word cifra has a double meaning: it is sometimes zero, sometimes the ciphers themselves. In the sense of zero, it is evidently the Arabic sifr (with s) which means something written, a book or character. These words algebra, cipher, algorithm survive as witnesses of the part played by the Arabs in the foundation and diffusion of the science of calculation.
Under al-Ma’mun’s successors, especially Mu‘tadid, who was quite an important caliph, where flourished a number of scholars who threw a vivid light on Arab learning and most of whom were known to medieval scholars. Geometrical studies progressed and conic sections began to attract attention. There brothers, known as Banu Musa, distinguished themselves in this period; they were sons of a certain Shakir who, says a biographer, had been a brigand in his youth and harassed the roads of Khorasan; he became an intimate of al-Ma’mun and one of the most esteemed scholars of his time. We owe a number of works to these three brothers, one of which, on the measurement of plane and spherical surfaces, was translated into Latin by Gerard of Cremona under the title Liber Trium Fratrum. They wrote a treatise on mechanics which is preserved in the Vatican. This work does not deal mainly with the principles of mechanics nor with simple machines like that of Hero, which was translated into Arabic about the same time by Kusta b. Luka; it resembles the Pneumatics of Hero and Philo; in it are described automata and various apparatus constructed with great ingenuity. Another Arabic treatise of the same kind but later in date is that of Badi‘ul-Zaman al-Jazari, a copy of which with very fine miniatures is preserved in Constantinople. The Arabs were very skilful in the construction of clepsydras, water-clocks with automata; it will be remembered that Harun al-Rashid sent one as a present to Charlemagne.
Abu Ma‘shar of Balkh in Khorasan, who died at the age of a hundred in 272 (886), was an astronomer and astrologer of great renown. Four of his works, including the book De Conjunctionibut et annorum revolutionibus, were translated into Latin by Johannes Hispalensis and Adelard of Bath.
Thabit b. Kurra of Harran in Mesopotamia is often regarded as the greatest Arab geometer. It was he who did science the service of translating into Arabic seven of the eight books of the conic sections of Apollonius and thus preserving three which are now lost in the original text. The Banu Musa were associated with him in this work: they presented him a pension of 500 dinars a month. Thabit knew Greek and Syriac and made translations from these languages into Arabic. He improved the translation of Euclid’s Elements by Ishak b. Hunain and that of the Almagest by the same. He wrote a number of short treatises or memoirs on astronomy and geometry, elucidating numerous passages in ancient works, inventing new propositions, annotating and facilitating study. Almost all the scientific subjects that could be studied in his day seen to be touched upon in his works. There are references to memoirs by him on the postulates and axioms of Euclid, on the transversal figure (translated into Latin by Gerard of Cremona), on method in geometry, on mechanics, on irrationals, conceived in the manner of Euclid and Plato, and an introduction to Euclid, a work much esteemed. His work on the shadows of the gnomon, i.e. on the sundial, is the earliest that we know on this subject. His treatise on the balance, Liber Carastonis sive de Statera, was translated into Latin by Gerard of Cremona. Arabic literature contains several treatises on the balance, one of which, that of al-Khazini, is of particular interest. The idea of equilibrium and of gravity is highly developed in it; specific gravities are also discussed.
Thabit made astronomical observations in Baghdad, notably to determine the altitude of the sun and the length of the solar year. He recorded his observations in a book. Belonging to the pagan sect of the Sabians and at heart deeply attached to paganism, this scholar is one of the most eminent representatives in the Middle Ages of the traditions of classical culture.
In the next generation there stands out one of the most illustrious scholar of the East, perhaps the one whom the Latin scholars of the Middle Ages and Renaissance most admired and eulogized, al-Battani (Albategnius). He made his astronomical observations from 264 to 306 (877-918). He wrote a large treatise and compiled astronomical tables,[4] which show in many respects an advance on the work of al-Khwarizmi and diverge still farther from Indian methods. Calculations or observations like those relating to the first appearance of the new moon, to the inclination of ecliptic, to the length of the tropic and sidereal year, to lunar anomalies, to eclipses, to parallaxes, are more complicated and more accurate in al-Battani than in al-Khwarizmi. But his greatest claim to fame is undoubtedly that if he did not discover he at least popularized the first notions of trigonometrical ratios as we use them to-day. Ptolemy used chords, for the calculation of which he had only one main theorem, a very clumsy one. Al-Battani substituted the sine for the chord. He used tangent and cotangent and he we acquainted with two or three fundamental relations in trigonometry. The sine is called in Arabic jaib, which means ‘a bay or curve’ (in Latin sinus), and this is evidently the origin of the word sine. The cotangent to the Arab astronomer is the ‘horizontal shadow’ of the gnomon, and tangent is the ‘vertical shadow’. They did not yet reckon directly according to the arcs of the circle; but the gnomon itself is divided into 12 parts. Habash, contemporary of al-Battani, divides it into 60 parts. Hence we get tables of cotangents in parts of the gnomon, based on the equation cot a =
12. The altitudeof the sun is deter mined, starting
from the cotangent, by the formula
sin (90-a) = cot a . 60
√ (122 +cot2 a)
The formulae
Sin a = tan a , cos a = 1
(1+tn2 a (1+tan 2 a)
are explained in al-Battani. This brings us very far beyond the point reached by the Greeks and really opens the era of modern science.
Later by some sixty years than Albategnius, another astronomer of great renown, Abu’l-Wafa’, continued his work. Several modern scholars have thought they could see in the Almagest of this author the discovery of the third lunar inequality, which we call the ‘variation’, the first two being already known to the Greeks. A long discussion developed in the Academie des Sciences of Paris in which the most eminent scholars took part, Biot, Arago, Le Verrier, Joseph Bertrand, which lasted from 1836 till1817. In the end it was not proved that the variation was really known to Abu’l-Wafa’. The Arab astronomers do not distinguish the first two lunar inequalities quite as we do; they break them up differently and this is what gave rise to some uncertainty.
But Abu’l-Wafa’s services to trigonometry are indisputable. With him trigonometry becomes still more explicit and acquires the formula for the addition of the angles
Sin (a+b) = sin a cos b + sin b cos a
R
This formula, discovered at this time did not, however, become known to the Latin world and Copernicus seems to have been unaware of it. Rhaeticus, the pupil and editor of Copernicus, rediscovered it very laboriously in his Opus palatinum de triangulis, after having given another formula much more complicated than Abu’l-Wafa’s.[5] This is not the end of the services rendered by Abu’l-Wafa’ to science. A geometer of great ingenuity, he dealt with a number of problems and studied the quadrature of the parabola and the volume of the paraboloid; in algebra he translated Diophantus.
During these two centuries in which the final from was given to these discoveries which are now at the foundations of all our modern civilization, a number of powerful minds were dealing with other problems relating to the philosophy of the sciences, and to physical and natural sciences. Without reaching final solutions, they trained the mind, elaborated ideas and prepared the way for future discoveries. Al-Kindi (d. 260 = 873), the first of the great scholastics, wrote on meteorology and optics. His treatise on the rains and winds and his improved version of the Optics of Euclid were translated into Latin. He also endeavoured to ascertain the laws that govern the fall of a body, a question with which the Arabs were not often concerned. Farabi, ‘the second master’ after Aristotle, an eminent neo-Platonist with a profound knowledge of ancient philosophy, wrote a remarkable treatise on music, an art in which he excelled. In this treatise we find the germ of the idea of logarithms. We know how music is related to mathematics. As early as the time of Pythagoras the necessity for expressing by sections of chords the first musical intervals, the octave, the quarter, and the fifth, had given a stimulus to the study of fractions. All the musical theory of the Arabs is expressed in terms of fractions. It contains logarithms in posse because the addition of the intervals, fourths, tones, semitones, quarter-tones, &c., corresponds to the multiplication of the lengths of chords which define them and the subtraction of the intervals corresponds to the division of these terms; the notes on a stringed instrument are connected by a logarithmic law. Avicenna and Algazel discuss the question of infinite quantities, sometimes in connexion with religion and sometimes in connexion with physics. Is a past infinite series possible? Is there on a straight line a first point where it is met by another straight line inclined towards it?-and questions connected with atomism-a square being regularly divided into atoms, how can the diagonal contain more atoms than the side? How in a line of atoms can an atom remain indivisible when it is in contact on either side with two different atoms? Can movement, heat, and light be conceived in terms of atoms? These are problems of he same kind as the sophisms of Zeno of Elea. They represent the gropings of the human mind before the discovery of the differential calculus. Al-Biruni, a scholar of exceptional erudition, a very keen critic, compiled a learned work on the chronology of the different nations. He travelled for a considerable period in India, tells us about the arithmetic of the Hindus, notes peculiarities connected with the game of chess, and deals with several questions of mathematical geography (projection, azimuths). He also did something to advance trigonometry.
We now come to a scholar who needs no introduction to our readers, for very few authors enjoy such fame; namely the celebrated Omar Khayyam (‘Umar b. Ibrahim al-Khayyami), poet and mathematician (d.517=1123). His skill as a geometer is equal to his literary erudition and reveals real logical power and penetration. His Algebra[6] is a book of the first rank and one which represents a much more advanced state of this science than that we see among the Greeks. Omar also marks a considerable advance on al-Khwarizmi: in the first place as regards the degree of the equations; the greater part of his book is actually devoted to cubic equations, while al-Khwarizmi only dealt with quadratics. Then as regards the discussion of the problems, possible and impossible solutions, and the limits of these solutions, it marks an enormous advance on the Greeks. Omar, however, is still under the influence of Diophantus, inasmuch as he endeavours to solve equations into whole numbers; he has therefore not completely freed himself from indeterminate algebra. He classifies equations of the third degree into 27 classes which are again divided into 4 categories, the last two of which consist of trinomial equations and of quadrinomial equations (with four terms). The fourth category contains the three classes: x3 +bx2 = cx+d; x3 +cx = bx2 +d; x3 +d = bx2 +cx. The gives us an idea of the difficulty of the problems conceived. The method used to deal with these problems is geometrical analysis, a kind of analytical geometry as it was conceived before Descartes at a period when the systems of co-ordinates and mathematical notations were not yet established. The last class, for example, is solved with the help of two hyperbolas, constructed according to the data of the problem, and according as b, the coefficient of the squares represented by a certain line, is equal to or less than the height of a parallelepiped constructed according to the isolated number and that of the roots, the conics intersect or do not intersect. ‘There are different cases of this variety,’ says Omar, ‘some of them impossible. It has been solved by means of the properties of the two hyperbolas.’ Such a method demanded a profound knowledge of the works of Apollonius and great skill in their application. Omar defends his originality as regards the Greeks and several Arab students who had preceded him. ‘In this study’, he says at the beginning of his treatise, ‘we meet with propositions depending on certain very difficult kinds of preliminary theorems, in the solution of which most of those who have tried have failed. No work of he ancients dealing with them has come down to us.’ This method of solving equations of the third degree is again found almost identically in the Geometrie of Descartes. As to the purely algebraical solution of the equation of the third degree, it did not see the light till the Renaissance when we find it in the writings of Scipione del Ferro, of Tartaglia, and of Cardan, but still an obscure and uncertain form which gave rise to much disputation.
Omar’s Algebra marks a stage in the advance of this branch of mathematics. Following on the excellent edition which he published in 1851, the learned editor Woepcke collected several other problems which were popular with Arab mathematicians and also presuppose a knowledge of conics, like the problem of the two proportional means, the trisection of an angle, the construction of regular polygons and especially of the enneagon. Several solutions of the problem of the trisection of an angle were known to the Arabs. An able geometer, Sijzi, gives one which includes all; it depends on the intersection of a hyperbola and of a circle. The construction of the regular polygon with nine sides, given by Ibn al-Laith, depends on the intersection of a hyperbola and a parabola. A lemma not proved by Archimedes (de Sphaera et Cylindro, ii. 6-7) provoked research by Ibn al-Haitham and others. Al-Kuhi puts the problem in this form: to construct a segment of a sphere equal in volume to a segment of given sphere in surface to another segment of the given sphere. He solves it very cleverly with the help of two auxiliary cones and conics: an equilateral hyperbola and a parabola, and he then discusses the limits.
In arithmetic the Arabs made several discoveries: about magic squares[7] and ‘amicable’ numbers. The invention of the proof by ‘casting out the nines’ is attributed to them, and the process known as the ‘rule of the double false position’ (regula duorum falsorum) which we again find in our arithmeticians of the seventeenth and eighteenth centuries. One of them enunciates the famous theorem of Fermat: the sum of two cubes is never a cube in whole numbers, but no proof is given. Al-Karkhi (d. c.420=1029) gives by a very neat and simple geometrical process the sum of the third powers of the successive series 13+23+33…+n3 and later al-Kashi, physician and astronomer to Ulugh Beg in Samarkand, gives the summation of the fourth powers, which suggests on mean degree of talent.
In the eleventh and twelfth centuries Arab astronomy was in a flourishing condition in Spain: it was for long afterwards studied in the East and continued to retain the interest of scholars of medieval Europe. In Spain al-Zarkali (Arzachel) who lived c. 420-80 (1029-1087) is famous as an which he wrote a treatise out of which a whole literature developed. A Jew of Montpellier translated it into Latin; King Alfonso of Castille made two translations of it into Romance (Spanish), and Regiomontanus in the fifteenth century published a collection of problems on the ‘noble instrument of the safiha’. Copernicus quotes Arzachel along with Albategnius in his book De Revolutionibus orbium coelestium. Al-Bitruji (Alpetragius), a pupil of the philosopher Ibn Tufail (twelfth century), had original ideas on the movements of the planets. He left a book which was translated into Hebrew by Moses ben Tibbon, then into Latin in the sixteenth century by Kalonymos ben David. The Alfonsine Tables compiled in the thirteenth century by Alfonso X the wise are a development of Arab astronomy. The longitudes are referred to the meridian of Toledo.
These scholars had unfettered and inquiring minds; they do not hesitate to criticize Ptolemy, and with Averroes they declare themselves against the theory of the multiplicity and eccentricity of the spheres. They look for more simple and more ‘natural’ systems. Al-Biruni had already admitted that astronomical hypotheses were all relative, that one could equally well, as Aristarchus of Samos and Seleucus of Babylon had proposed two thousand years before Copernicus, and, at a period not so remote, several Hindus attribute the diurnal movement to the earth and make it turn on its own axis and around the sun, while ‘saving appearances’, that is to say, explaining and calculating all the movements of the stars. The spirit of Arabian research at this period was not hampered by any fixed views or dogmatism.
In the East in the troubled period of the Mongol invasions there flourished a great scholar with a fine synthetic brain, Nasir al-Din Tusi (d. 672=1274). He made observations at Maragha in Asia Minor in an observatory founded by the munificence of the Mongol Khans and drew up the astronomical tables called after the regal title of these conquerors ‘The Ilkhanian Tables’. The instruments at Maragha were much admired. The Arab astronomers devoted great attention to the perfecting of instruments. The most important was the armillary sphere, which was known to the ancients and represented in a general way the celestial sphere; it consisted of three rings corresponding to the meridian, the ecliptic, and colure of the solstices, and of two rings of observation. The Arabs completed and perfected the sphere of Ptolemy and of the Alexandrians. They added to it two rings giving the co-ordinates of the stars with respect to the horizon, then a ring for the observation of the altitudes. They endeavoured to make their instruments as large as possible in order to minimize error’ they then began to make special instruments, each being devoted to a special class of observations. In the observatory at Maragha there were instruments made of rings for special purposes: ecliptical, solsticial, and equatorial armillaries. The ecliptical, which was very much used, had five rings, the largest of which was some twelve feet across. It was graduated in degrees and minutes. When Alfonso of Castille wanted to construct an armillary sphere, which would be the finest and best that had yet been made, it was to the Arabs that he turned for information. At the Renaissance, Regiomontanus, in order to reconstruct the ecliptical of Ptolemy, used Arabic books and it was from them that he became acquainted with the alidade, the name of which is Arabic.
Nasir al-Din is equally important as a geometer. He edited most of the mathematical works of antiquity to the number of sixteen, which, with four books of the Muslim period, practically constituted the whole scientific knowledge of the period. Among the books added was one by Nasir al-Din himself, namely the Treatise on the Quadrilateral,[8] a work on spherical trigonometry of the first rank. In it he expands this subject in a most orderly and lucid fashion, at first according to the method of Menelaus and Ptolemy and then according to new methods the advantages of which he points out. The rule which he calls that of the ‘supplementary figure’, which dispenses with the use of Ptolemy’s theorem of the quadrilateral, is simply the statement that the sines of angles are proportional to those of sides,
sin A = sin b = sin c
sin a sin B sin C
To this rule he adds a ‘method of the tangent’ based on the relation sin b= tan c . Trigonometry, plane and spherical, is now well established tan C expression. In a short paragraph Nasir al-Din recalls his Arab predecessors who had a share in its invention.
Lastly, we must mention the astronomers of Samarkand, whose tables, prepared in 1437 for a prince of the family of Tamerlane under the title Tables of Ulugh Beg, were highly esteemed in the West and published in part in England in the eighteenth century.[9]
Such then in its broad outlines was the scientific work of the Arabs. It came to an end when that of the Western genius began, that is to say in the fifteenth century. It is sometimes asked what were the causes of this cessation of intellectual activity in the Muslim world. Whence came this torpor after a period of such prolific activity ? This however, is a question which raises very obscure problems of general psychology about which on one has yet put forward any very definite theory and, as I have none to propound myself, I do not think I ought to attempt to discuss it.
CRRA DE VAUX.
[1]. Much has been written, more particularly in recent years, about science among the Arabs; it is a subject which would require a long bibliography, a bibliography will be found in Volume II of my Les Penseurs de l’Islam, Paris, Geuthner, 1921. Specialk mention must, however,l be made here of the excellent work work done by Prof. E. Wiedemann of Erlangen who has gathered round him a body of pupils and collaborators. Cf. also H. Suter, Die Matbematiker and Astronomen der Araber und ibre Werke, Leipzig, 1900.
[2] Edited with English introduction and translation by F. Rosen, London, 1831.
[3] Latin text edited by Prince Boncompagni in the series Trattati d’ Aritmetica, Rome, 1857, No. 1
[4] Edited in Arabic and Latin by Nallino, 1903.
Al-Khwarizmi’s astronomical treatise has been edited in Latin by H. Suter from Adelard of Bath’s version (Copenhagen, 1914). At this period the Arab astronomers reckoned longitudes from the meridian of ‘Arin’. This name is a corruption: it is really Ujjain, a town in Central India which had then an observatory; at a much latter date in the eighteenth century Jay Singh reestablished the observatory there.
[5] We also find the secant in Abu’l-Wafa’ which he calls ‘the diameter of the shadow’; its introduction is usually credited to Copernicus.
[6] Edited and translated into French by F. Woepcke, Paris, 1857.
[7] Magic squares were used in talismans. I noticed recently in a treatise of the Arab occultist al-Buni (d.622=1225) a very ingenious general solution of the problem of magic squares. This solution enables one, when a square of side n is known, whether n be odd or even, to construct a square of side n +2.
[8] Edited with a French translation by Caratheodory Pasha, Constantinople, 1891.
[9] Edited by J. Greaves and T. Hyde, in Persian and Latin, London, 1650 and 1665. sedillot translated into French the prolegomena to these Tables (Paris, 1846).
