A Short Account of the History of Mathematics · W. W. Rouse Ball
The subject is discussed at length by Cantor, chaps. xxxii--xxxv; by Hankel, pp. 172--293; by A. von Kremer in Kulturgeschichte des Orientes unter den Chalifen, Vienna, 1877; and by H. Suter in his "Die Mathematiker und Astronomen der Araber und ihre Werke," Zeitschrift für Mathematik und Physik, Abhandlungen zur Geschichte der Mathematik, Leipzig, vol. xlv, 1900. See also Matériaux pour servir à l'histoire comparée des sciences mathématiques chez les Grecs et les Orientaux, by L. A. Sédillot, Paris, 1845--9; and the following articles by Fr. Woepcke, Sur l'introduction de l'arithmétique Indienne en Occident, Rome, 1859; Sur l'histoire des sciences mathématiques chez les Orientaux, Paris, 1860; and Mémoire sur la propagation des chiffres Indiens, Paris, 1863.
The story of Arab mathematics is known to us in its general outlines, but we are as yet unable to speak with certainty on many of its details. It is, however, quite clear that while part of the early knowledge of the Arabs was derived from Greek sources, part was obtained from Hindoo works; and that it was on those foundations that Arab science was built. I will begin by considering in turn the extent of mathematical knowledge derived from these sources.
According to their traditions, in themselves very probable, the scientific knowledge of the Arabs was at first derived from the Greek doctors who attended the caliphs at Bagdad. It is said that when the Arab conquerors settled in towns they became subject to diseases which had been unknown to them in their life in the desert. The study of medicine was then confined mainly to Greeks and Jews, and many of these, encouraged by the caliphs, settled at Bagdad, Damascus, and other cities; their knowledge of all branches of learning was far more extensive and accurate than that of the Arabs, and the teaching of the young, as has often happened in similar cases, fell into their hands. The introduction of European science was rendered the more easy as various small Greek schools existed in the countries subject to the Arabs: there had for many years been one at Edessa among the Nestorian Christians, and there were others at Antioch, Emesa, and even at Damascus, which had preserved the traditions and some of the results of Greek learning.
The Arabs soon remarked that the Greeks rested their medical science on the works of Hippocrates, Aristotle, and Galen; and these books were translated into Arabic by order of the caliph Haroun Al Raschid about the year 800. The translation excited so much interest that his successor Al Mamun (813--833) sent a commission to Constantinople to obtain copies of as many scientific works as was possible, while an embassy for a similar purpose was also sent to India. At the same time a large staff of Syrian clerks was engaged, whose duty it was to translate the works so obtained into Arabic and Syriac. To disarm fanaticism these clerks were at first termed the caliph's doctors, but in 851 they were formed into a college, and their most celebrated member, Honein ibn Ishak, was made its first president by the caliph Mutawakkil (847--861). Honein and his son Ishak ibn Honein revised the translations before they were finally issued. Neither of them knew much mathematics, and several blunders were made in the works issued on that subject, but another member of the college, Tabit ibn Korra, shortly published fresh editions which thereafter became the standard texts.
In this way before the end of the ninth century the Arabs obtained translations of the works of Euclid, Archimedes, Apollonius, Ptolemy, and others; and in some cases these editions are the only copies of the books now extant. It is curious, as indicating how completely Diophantus had dropped out of notice, that as far as we know the Arabs got no manuscript of his great work till 150 years later, by which time they were already acquainted with the idea of algebraic notation and processes.
The Arabs had considerable commerce with India, and a knowledge of one or both of the two great original Hindoo works on algebra had been thus obtained in the caliphate of Al Mansur (754--775), though it was not until fifty or sixty years later that they attracted much attention. The algebra and arithmetic of the Arabs were largely founded on these treatises, and I therefore devote this section to the consideration of Hindoo mathematics.
The Hindoos, like the Chinese, have pretended that they are the most ancient people on the face of the earth, and that to them all sciences owe their creation. But it is probable that these pretensions have no foundation; and in fact no science or useful art (except a rather fantastic architecture and sculpture) can be definitely traced back to the inhabitants of the Indian peninsula prior to the Aryan invasion. This invasion seems to have taken place at some time in the latter half of the fifth century or in the sixth century, when a tribe of Aryans entered India by the north-west frontier, and established themselves as rulers over a large part of the country. Their descendants, wherever they have kept their blood pure, may still be recognised by their superiority over the races they originally conquered; but as is the case with the modern Europeans, they found the climate trying and gradually degenerated. For the first two or three centuries they, however, retained their intellectual vigour, and produced one or two writers of great ability.
Arya-Bhata. The earliest of these, of whom we have definite information, is Arya-Bhata,
, edited by H. Kern, was published at Leyden in 1874; there is also an article on it by the same editor in the Journal of the Asiatic Society, London, 1863, vol. xx, pp. 371--387; a French translation by L. Rodet of that part which deals with algebra and trigonometry is given in the Journal Asiatique, 1879, Paris, series 7, vol. xiii, pp. 393--434. who was born at Patna in the year 476. He is frequently quoted by Brahmagupta, and in the opinion of many commentators he created algebraic analysis, though it has been suggested that he may have seen Diophantus's Arithmetic. The chief work of Arya-Bhata with which we are acquainted is his Aryabhathiya, which consists of mnemonic verses embodying the enunciations of various rules and propositions. There are no proofs, and the language is so obscure and concise that it long defied all efforts to translate it.
The book is divided into four parts: of these three are devoted to astronomy and the elements of spherical trigonometry; the remaining part contains the enunciations of thirty-three rules in arithmetic, algebra, and plane trigonometry. It is probable that Arya-Bhata regarded himself as an astronomer, and studied mathematics only so far as it was useful to him in his astronomy.
In algebra Arya-Bhata gives the sum of the first, second, and third powers of the first n natural numbers; the general solution of a quadratic equation; and the solution in integers of certain indeterminate equations of the first degree. His solutions of numerical equations have been supposed to imply that he was acquainted with the decimal system of enumeration.
In trigonometry he gives a table of natural sines of the angles in the first quadrant, proceeding by multiples of 33/4° , defining a sine as the semi-chord of double the angle. Assuming that for the angle 33/4° the sine is equal to the circular measure, he takes for its value 225, i.e. the number of minutes in the angle. He then enunciates a rule which is nearly unintelligible, but probably is the equivalent of the statement (n + 1) - n = n - (n - 1) - n , where alpha stands for 33/4^circ ; and working with this formula he constructs a table of sines, and finally finds the value of sin 90^circ to be 3438 . This result is correct if we take 3.1416 as the value of pi , and it is interesting to note that this is the number which in another place he gives for pi . The correct trigonometrical formula is (n + 1) - n = n - (n - 1) - 4 n ^2 12. Arya-Bhata, therefore, took 4sin^2 1/2alpha as equal to cosec alpha , that is, he supposed that 2 sin alpha = 1 + sin 2alpha : using the approximate values of sin alpha and sin 2alpha given in his table, this reduces to 2(225) = 1 + 449 , and hence to that degree of approximation his formula is correct. A considerable proportion of the geometrical propositions which he gives is wrong.
Brahmagupta. The next Hindoo writer of note is Brahmagupta, who is said to have been born in 598, and probably was alive about 660. He wrote a work in verse entitled Brahma-Sphuta-Siddhanta, that is, the Siddhanta, or system of Brahma in astronomy. In this, two chapters are devoted to arithmetic, algebra, and geometry.
The arithmetic is entirely rhetorical. Most of the problems are worked out by the rule of three, and a large proportion of them are on the subject of interest.
In his algebra, which is also rhetorical, he works out the fundamental propositions connected with an arithmetical progression, and solves a quadratic equation (but gives only the positive value to the radical). As an illustration of the problems given I may quote the following, which was reproduced in slightly different forms by various subsequent writers, but I replace the numbers by letters. "Two apes lived at the top of a cliff of height h , whose base was distant mh from a neighbouring village. One descended the cliff and walked to the village, the other flew up a height x and then flew in a straight line to the village. The distance traversed by each was the same. Find x ." Brahmagupta gave the correct answer, namely x = mh/(m+2) . In the question as enunciated originally h = 100 , m = 2 .
Brahmagupta finds solutions in integers of several indeterminate equations of the first degree, using the same method as that now practised. He states one indeterminate equation of the second degree, namely, nx^2 + 1 = y^2 , and gives as its solution x = 2t/(t^2 - n) and y = (t^2 + n)/(t^2 - n) . To obtain this general form he proved that, if one solution either of that or of certain allied equations could be guessed, the general solution could be written down; but he did not explain how one solution could be obtained. Curiously enough this equation was sent by Fermat as a challenge to Wallis and Lord Brouncker in the seventeenth century, and the latter found the same solutions as Brahmagupta had previously done. Brahmagupta also stated that the equation y^2 = nx^2 - 1 could not be satisfied by integral values of x and y unless n could be expressed as the sum of the squares of two integers. It is perhaps worth noticing that the early algebraists, whether Greeks, Hindoos, Arabs, or Italians, drew no distinction between the problems which led to determinate and those which led to indeterminate equations. It was only after the introduction of syncopated algebra that attempts were made to give general solutions of equations, and the difficulty of giving such solutions of indeterminate equations other than those of the first degree has led to their practical exclusion from elementary algebra.
In geometry Brahmagupta proved the Pythagorean property of a right-angled triangle (Euc. i, 47). He gave expressions for the area of a triangle and of a quadrilateral inscribable in a circle in terms of their sides; and shewed that the area of a circle was equal to that of a rectangle whose sides were the radius and semiperimeter. He was less successful in his attempt to rectify a circle, and his result is equivalent to taking sqrt 10 for the value of pi . He also determined the surface and volume of a pyramid and cone; problems over which Arya-Bhata had blundered badly. The next part of his geometry is almost unintelligible, but it seems to be an attempt to find expressions for several magnitudes connected with a quadrilateral inscribed in a circle in terms of its sides: much of this is wrong.
It must not be supposed that in the original work all the propositions which deal with any one subject are collected together, and it is only for convenience that I have tried to arrange them in that way. It is impossible to say whether the whole of Brahmagupta's results given above are original. He knew of Arya-Bhata's work, for he reproduces the table of sines there given; it is likely also that some progress in mathematics had been made by Arya-Bhata's immediate successors, and that Brahmagupta was acquainted with their works; but there seems no reason to doubt that the bulk of Brahmagupta's algebra and arithmetic is original, although perhaps influenced by Diophantus's writings: the origin of the geometry is more doubtful, probably some of it is derived from Hero's works, and maybe some represents indigenous Hindoo work.
Bhaskara. To make this account of Hindoo mathematics complete I may depart from the chronological arrangement and say that the only remaining Indian mathematician of exceptional eminence of whose works we know anything was Bhaskara, who was born in 1114. He is said to have been the lineal successor of Brahmagupta as head of an astronomical observatory at Ujein. He wrote an astronomy, of which four chapters have been translated. Of these one termed Lilavati is on arithmetic; a second termed Bija Ganita is on algebra; the third and fourth are on astronomy and the sphere; some of the other chapters also involve mathematics. This work was, I believe, known to the Arabs almost as soon as it was written, and influenced their subsequent writings, though they failed to utilize or extend most of the discoveries contained in it. The results thus became indirectly known in the West before the end of the twelfth century, but the text itself was not introduced into Europe till within recent times.
The treatise is in verse, but there are explanatory notes in prose. It is not clear whether it is original or whether it is merely an exposition of the results then known in India; but in any case it is most probable that Bhaskara was acquainted with the Arab works which had been written in the tenth and eleventh centuries, and with the results of Greek mathematics as transmitted through Arabian sources. The algebra is syncopated and almost symbolic, which marks a great advance over that of Brahmagupta and of the Arabs. The geometry is also superior to that of Brahmagupta, but apparently this is due to the knowledge of various Greek works obtained through the Arabs.
The first book or Lilavati commences with a salutation to the god of wisdom. The general arrangement of the work may be gathered from the following table of contents. Systems of weights and measures. Next decimal numeration, briefly described. Then the eight operations of arithmetic, namely, addition, subtraction, multiplication, division, square, cube, square-root, and cube-root. Reduction of fractions to a common denominator, fractions of fractions, mixed numbers, and the eight rules applied to fractions. The "rules of cipher," namely, a+/-0 = a, 0^2 = 0, sqrt(0) = 0, a÷0 = infinity . The solution of some simple equations which are treated as questions of arithmetic. The rule of false assumption. Simultaneous equations of the first degree with applications. Solution of a few quadratic equations. Rule of three and compound rule of three, with various cases. Interest, discount, and partnership. Time of filling a cistern by several fountains. Barter. Arithmetical progressions, and sums of squares and cubes. Geometrical progressions. Problems on triangles and quadrilaterals. Approximate value of pi . Some trigonometrical formulae. Contents of solids. Indeterminate equations of the first degree. Lastly, the book ends with a few questions on combinations.
This is the earliest known work which contains a systematic exposition of the decimal system of numeration. It is possible that Arya-Bhata was acquainted with it, and it is most likely that Brahmagupta was so, but in Bhaskara's arithmetic we meet with the Arabic or Indian numerals and a sign for zero as part of a well-recognised notation. It is impossible at present to definitely trace these numerals farther back than the eighth century, but there is no reason to doubt the assertion that they were in use at the beginning of the seventh century. Their origin is a difficult and disputed question. I mention below the view which on the whole seems most probable, and perhaps is now generally accepted, and I reproduce there some of the forms used in early times.
To sum the matter up briefly, it may be said that the Lilavati gives the rules now current for addition, subtraction, multiplication, and division, as well as for the more common processes in arithmetic; while the greater part of the work is taken up with the discussion of the rule of three, which is divided into direct and inverse, simple and compound, and is used to solve numerous questions chiefly on interest and exchange--the numerical questions being expressed in the decimal system of notation with which we are familiar.
Bhaskara was celebrated as an astrologer no less than as a mathematician. He learnt by this art that the event of his daughter Lilavati marrying would be fatal to himself. He therefore declined to allow her to leave his presence, but by way of consolation he not only called the first book of his work by her name, but propounded many of his problems in the form of questions addressed to her. For example, "Lovely and dear Lilavati, whose eyes are like a fawn's, tell me what are the numbers resulting from 135 multiplied by 12. If thou be skilled in multiplication, whether by whole or by parts, whether by division or by separation of digits, tell me, auspicious damsel, what is the quotient of the product when divided by the same multiplier."
I may add here that the problems in the Indian works give a great deal of interesting information about the social and economic condition of the country in which they were written. Thus Bhaskara discusses some questions on the price of slaves, and incidentally remarks that a female slave was generally supposed to be most valuable when 16 years old, and subsequently to decrease in value in inverse proportion to the age; for instance, if when 16 years old she were worth 32 nishkas, her value when 20 would be represented by (16×32)÷20 nishkas. It would appear that, as a rough average, a female slave of 16 was worth about 8 oxen which had worked for two years. The interest charged for money in India varied from 31/2 to 5 per cent per month. Amongst other data thus given will be found the prices of provisions and labour.
The chapter termed Bija Ganita commences with a sentence so ingeniously framed that it can be read as the enunciation of a religious, or a philosophical, or a mathematical truth. Bhaskara after alluding to his Lilavati, or arithmetic, states that he intends in this book to proceed to the general operations of analysis. The idea of the notation is as follows. Abbreviations and initials are used for symbols; subtraction is indicated by a dot placed above the coefficient of the quantity to be subtracted; addition by juxtaposition merely; but no symbols are used for multiplication, equality, or inequality, these being written at length. A product is denoted by the first syllable of the word subjoined to the factors, between which a dot is sometimes placed. In a quotient or fraction the divisor is written under the dividend without a line of separation. The two sides of an equation are written one under the other, confusion being prevented by the recital in words of all the steps which accompany the operation. Various symbols for the unknown quantity are used, but most of them are the initials of names of colours, and the word colour is often used as synonymous with unknown quantity; its Sanskrit equivalent also signifies a letter, and letters are sometimes used either from the alphabet or from the initial syllables of subjects of the problem. In one or two cases symbols are used for the given as well as for the unknown quantities. The initials of the words square and solid denote the second and third powers, and the initial syllable of square root marks a surd. Polynomials are arranged in powers, the absolute quantity being always placed last and distinguished by an initial syllable denoting known quantity. Most of the equations have numerical coefficients, and the coefficient is always written after the unknown quantity. Positive or negative terms are indiscriminately allowed to come first; and every power is repeated on both sides of an equation, with a zero for the coefficient when the term is absent. After explaining his notation, Bhaskara goes on to give the rules for addition, subtraction, multiplication, division, squaring, and extracting the square root of algebraical expressions; he then gives the rules of cipher as in the Lilavati; solves a few equations; and lastly concludes with some operations on surds. Many of the problems are given in a poetical setting with allusions to fair damsels and gallant warriors.
Fragments of other chapters, involving algebra, trigonometry, and geometrical applications, have been translated by Colebrooke. Amongst the trigonometrical formulae is one which is equivalent to the equation d(sin theta) = cos theta dtheta .
I have departed from the chronological order in treating here of Bhaskara, but I thought it better to mention him at the same time as I was discussing his compatriots. It must be remembered, however, that he flourished subsequently to all the Arab mathematicians considered in the next section. The works with which the Arabs first became acquainted were those of Arya-Bhata and Brahmagupta, and perhaps of their successors Sridhara and Padmanabha; it is doubtful if they ever made much use of the great treatise of Bhaskara.
It is probable that the attention of the Arabs was called to the works of the first two of these writers by the fact that the Arabs adopted the Indian system of arithmetic, and were thus led to look at the mathematical text-books of the Hindoos. The Arabs had always had considerable commerce with India, and with the establishment of their empire the amount of trade naturally increased; at that time, about the year 700, they found the Hindoo merchants beginning to use the system of numeration with which we are familiar, and adopted it at once. This immediate acceptance of it was made the easier, as they had no works of science or literature in which another system was used, and it is doubtful whether they then possessed any but the most primitive system of notation for expressing numbers. The Arabs, like the Hindoos, seem also to have made little or no use of the abacus, and therefore must have found Greek and Roman methods of calculation extremely laborious. The earliest definite date assigned for the use in Arabia of the decimal system of numeration is 773. In that year some Indian astronomical tables were brought to Bagdad, and it is almost certain that in these Indian numerals (including a zero) were employed.
In the preceding sections of this chapter I have indicated the two sources from which the Arabs derived their knowledge of mathematics, and have sketched out roughly the amount of knowledge obtained from each. We may sum the matter up by saying that before the end of the eighth century the Arabs were in possession of a good numerical notation and of Brahmagupta's work on arithmetic and algebra; while before the end of the ninth century they were acquainted with the masterpieces of Greek mathematics in geometry, mechanics, and astronomy. I have now to explain what use they made of these materials.
Alkarismi. The first and in some respects the most illustrious of the Arabian mathematicians was Mohammed ibn Musa Abu Djefar Al-Khwarizmi. There is no common agreement as to which of these names is the one by which he is to be known: the last of them refers to the place where he was born, or in connection with which he was best known, and I am told that it is the one by which he would have been usually known among his contemporaries. I shall therefore refer to him by that name; and shall also generally adopt the corresponding titles to designate the other Arabian mathematicians. Until recently, this was almost always written in the corrupt form Alkarismi, and, though this way of spelling it is incorrect, it has been sanctioned by so many writers that I shall make use of it.
We know nothing of Alkarismi's life except that he was a native of Khorassan and librarian of the caliph Al Mamun; and that he accompanied a mission to Afghanistan, and possibly came back through India. On his return, about 830, he wrote an algebra, which is founded on that of Brahmagupta, but in which some of the proofs rest on the Greek method of representing numbers by lines. He also wrote a treatise on arithmetic: an anonymous tract termed Algoritmi De Numero Indorum, which is in the university library at Cambridge, is believed to be a Latin translation of this treatise. Besides these two works he compiled some astronomical tables, with explanatory remarks; these included results taken from both Ptolemy and Brahmagupta.
The algebra of Alkarismi holds a most important place in the history of mathematics, for we may say that the subsequent Arab and the early medieval works on algebra were founded on it, and also that through it the Arabic or Indian system of decimal numeration was introduced into the West. The work is termed Al-gebr we' l mukabala: al-gebr, from which the word algebra is derived, means the restoration, and refers to the fact that any the same magnitude may be added to or subtracted from both sides of an equation; al mukabala means the process of simplification, and is generally used in connection with the combination of like terms into a single term. The unknown quantity is termed either "the thing" or "the root" (that is, of a plant), and from the latter phrase our use of the word root as applied to the solution of an equation is derived. The square of the unknown is called "the power." All the known quantities are numbers.
The work is divided into five parts. In the first Alkarismi gives rules for the solution of quadratic equations, divided into five classes of the forms ax^2 = bx , ax^2 = c , ax^2 + bx= c , ax^2 + c= bx , and ax^2 = bx + c , where a, b, c are positive numbers, and in all the applications a=1 . He considers only real and positive roots, but he recognises the existence of two roots, which as far as we know was never done by the Greeks. It is somewhat curious that when both roots are positive he generally takes only that root which is derived from the negative value of the radical.
He next gives geometrical proofs of these rules in a manner analogous to that of Euclid ii, 4. For example, to solve the equation x^2 + 10x = 39 , or any equation of the form x^2 + px = q , he gives two methods of which one is as follows. Let AB represent the value of x , and construct on it the square ABCD (see figure below). Produce DA to H and DC to F so that AH=CF=5 (or 1/2p ); and complete the figure as drawn below. Then the areas AC , HB , and BF represent the magnitudes x^2 , 5x , and 5x . Thus the left-hand side of the equation is represented by the sum of the areas AC , HB , and BF , that is, by the gnomon HCG . To both sides of the equation add the square KG , the area of which is 25 (or 1/4p^2 ), and we shall get a new square whose area is by hypothesis equal to 39+25, that is, to 64 (or q + 1/4p^2 ) and whose side therefore is 8. The side of this square DH , which is equal to 8, will exceed AH , which is equal to 5, by the value of the unknown required, which, therefore, is 3.
In the third part of the book Alkarismi considers the product of (x+/- a) and (x+/- b) . In the fourth part he states the rules for addition and subtraction of expressions which involve the unknown, its square, or its square root; gives rules for the calculation of square roots; and concludes with the theorems that asqrt(b) = sqrta^2b and sqrt(a)sqrt(b)=sqrt(ab) . In the fifth and last part he gives some problems, such, for example, as to find two numbers whose sum is 10 and the difference of whose squares is 40.
In all these early works there is no clear distinction between arithmetic and algebra, and we find the account and explanation of arithmetical processes mixed up with algebra and treated as part of it. It was from this book then that the Italians first obtained not only the ideas of algebra, but also of an arithmetic founded on the decimal system. This arithmetic was long known as algorism, or the art of Alkarismi, which served to distinguish it from the arithmetic of Boethius; this name remained in use till the eighteenth century.
Tabit ibn Korra. The work commenced by Alkarismi was carried on by Tabit ibn Korra, born at Harran in 836, and died in 901, who was one of the most brilliant and accomplished scholars produced by the Arabs. As I have already stated, he issued translations of the chief works of Euclid, Apollonius, Archimedes, and Ptolemy. He also wrote several original works, all of which are lost with the exception of a fragment on algebra, consisting of one chapter on cubic equations, which are solved by the aid of geometry in somewhat the same way as that given later.
Algebra continued to develop very rapidly, but it remained entirely rhetorical. The problems with which the Arabs were chiefly concerned were solution of equations, problems leading to equations, or properties of numbers. The two most prominent algebraists of a later date were Alkayami and Alkarki, both of whom flourished at the beginning of the eleventh century.
Alkayami. The first of these, Omar Alkayami, is noticeable for his geometrical treatment of cubic equations by which he obtained a root as the abscissa of a point of intersection of a conic and a circle. The equations he considers are of the following forms, where a and c stand for positive integers, (i) x^3 + b^2x = b^2c , whose root he says is the abscissa of a point of intersection of x^2 = by and y^2 = x(c - x) ; (ii) x^3 + ax^2 = c^3 , whose root he says is the abscissa of a point of intersection of xy = c^2 and y^2 = c(x + a) ; (iii) x^3± ax^2 + b^2x = b^2c , whose root he says is the abscissa of a point of intersection of y^2=(x+/- a)(c-x) and x(b+/- y) = bc . He gives one biquadratic, namely, (100-x^2)(10-x)^2 = 8100 , the root of which is determined by the point of intersection of (10 - x)y = 90 and x^2 + y^2 = 100 . It is sometimes said that he stated that it was impossible to solve the equation x^3 + y^3 = z^3 in positive integers, or in other words that the sum of two cubes can never be a cube; though whether he gave an accurate proof, or whether, as is more likely, the proposition (if enunciated at all) was the result of a wide induction, it is now impossible to say; but the fact that such a theorem is attributed to him will serve to illustrate the extraordinary progress the Arabs had made in algebra.
Alkarki. The other mathematician of this time (circ. 1000) whom I mentioned was Alkarki. He gave expressions for the sums of the first, second, and third powers of the first n natural numbers; solved various equations, including some of the forms ax^2p± bx^p± c=0 and discussed surds, shewing, for example, that sqrt(8)+sqrt(18)=sqrt(50) .
Even where the methods of Arab algebra are quite general the applications are confined in all cases to numerical problems, and the algebra is so arithmetical that it is difficult to treat the subjects apart. From their books on arithmetic and from the observations scattered through various works on algebra, we may say that the methods used by the Arabs for the four fundamental processes were analogous to, though more cumbrous than, those now in use; but the problems to which the subject was applied were similar to those given in modern books, and were solved by similar methods, such as rule of three, &c. Some minor improvements in notation were introduced, such, for instance, as the introduction of a line to separate the numerator from the denominator of a fraction; and hence a line between two symbols came to be used as a symbol of division. Alhossein (980--1037) used a rule for testing the correctness of the results of addition and multiplication by "casting out the nines." Various forms of this rule have been given, but they all depend on the proposition that, if each number in the question be replaced by the remainder when it is divided by 9, and if these remainders be added or multiplied as directed in the question, then this result when divided by 9 will leave the same remainder as the answer whose correctness it is desired to test when divided by 9: if these remainders differ, there is an error. The selection of 9 as a divisor was due to the fact that the remainder when a number is divided by 9 can be obtained by adding the digits of the number and dividing the sum by 9.
I am not concerned with the views of Arab writers on astronomy or the value of their observations, but I may remark in passing that they accepted the theory as laid down by Hipparchus and Ptolemy, and did not materially alter or advance it. I may, however, add that Al Mamun caused the length of a degree of latitude to be measured, and he, as well as the two mathematicians to be next named, determined the obliquity of the ecliptic.
Albategni. Albuzjani. Like the Greeks, the Arabs rarely, if ever, employed trigonometry except in connection with astronomy; but in effect they used the trigonometrical ratios which are now current, and worked out the plane trigonometry of a single angle. They are also acquainted with the elements of spherical trigonometry. Albategni, born at Batan in Mesopotamia, in 877, and died at Bagdad in 929, was among the earliest of the many distinguished Arabian astronomers. He wrote the Science of the Stars, which is worthy of note from its containing a mention of the motion of the sun's apogee. In this work angles are determined by "the semi-chord of twice the angle," that is, by the sine of the angle (taking the radius vector as unity). It is doubtful whether he was acquainted with the previous introduction of sines by Arya-Bhata and Brahmagupta; Hipparchus and Ptolemy, it will be remembered, had used the chord. Albategni was also acquainted with the fundamental formula in spherical trigonometry, giving the side of a triangle in terms of the other sides and the angle included by them. Shortly after the death of Albategni, Albuzjani, who is also known as Abul-Wafa, born in 940, and died in 998, introduced certain trigonometrical functions, and constructed tables of tangents and cotangents. He was celebrated as a geometrician as well as an astronomer.
Alhazen. Abd-al-gehl. The Arabs were at first content to take the works of Euclid and Apollonius for their text-books in geometry without attempting to comment on them, but Alhazen, born at Bassora in 987 and died at Cairo in 1038, issued in 1036 a collection of problems something like the Data of Euclid. Besides commentaries on the definitions of Euclid and on the Almagest, Alhazen also wrote a work on optics, which includes the earliest scientific account of atmospheric refraction. It also contains some ingenious geometry, amongst other things, a geometrical solution of the problem to find at what point of a concave mirror a ray from a given point must be incident so as to be reflected to another given point. Another geometrician of a slightly later date was Abd-al-gehl (circ. 1100), who wrote on conic sections, and was also the author of three small geometrical tracts.
It was shortly after the last of the mathematicians mentioned above that Bhaskara, the third great Hindoo mathematician, flourished; there is every reason to believe that he was familiar with the works of the Arab school as described above, and also that his writings were at once known in Arabia.
The Arab schools continued to flourish until the fifteenth century. But they produced no other mathematician of any exceptional genius, nor was there any great advance on the methods indicated above, and it is unnecessary for me to crowd my pages with the names of a number of writers who did not materially affect the progress of the science in Europe.
From this rapid sketch it will be seen that the work of the Arabs (including therein writers who wrote in Arabia and lived under Eastern Mohammedan rule) in arithmetic, algebra, and trigonometry was of a high order of excellence. They appreciated geometry and the applications of geometry to astronomy, but they did not extend the bounds of the science. It may be also added that they made no special progress in statics, or optics, or hydrostatics; though there is abundant evidence that they had a thorough knowledge of practical hydraulics.
The general impression left is that the Arabs were quick to appreciate the work of others--notably of the Greek masters and of the Hindoo mathematicians--but, like the ancient Chinese and Egyptians, they did not systematically develop a subject to any considerable extent. Their schools may be taken to have lasted in all for about 650 years, and if the work produced be compared with that of Greek or modern European writers it is, as a whole, second-rate both in quantity and quality.
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End of part 12 of 28
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