roots are equal to squares and numbers" (+cx=bx2+a, in modern notation), is constructed by means of a circle and the asymptotes of an equilateral hyperbola. The Risala Hisab is a short treatise on Arithmetic and Geometry written in the Persian language by Kazi Zadeh al Rumi, surnamed Ali Kushchi. He was one of the celebrated astronomers employed by Ulugh Begh to construct the astronomical tables which bear the name of that prince, who had caused an observatory to be erected in Samarcand for that purpose. Ulugh Begh was slain by his rebellious son, A.H. 853, or A.D. 1474. Baha Eddin was born at Balbec A.H. 953, and died A.H. 1031, or A.D. 1652. He was the author of a work on Algebra, entitled "Kholasat al Hisab," which was considered by learned Mohammedans as the most scientific treatise on the subject. This work was translated into the Persian language about sixty years after the death of the author. Mr. Strachey has given, in the twelfth volume of the Asiatic Researches, an analysis of the contents of this work, from which the abstract in the note will supply a more complete conception of the knowledge of Algebra among the Arabians. 1.The Arabic text of the Kholasat al Hisab, with a Persian Commentary by Roshan Ali, was printed at Calcutta in the year 1812. "The work is stated by the author, Baha Eddin, in his preface, to consist of an introduction, ten books, and a conclusion. The introduction contains the definitions of arithmetic, number, &c., and the author ascribes to the Indian sages the invention of the nine figures to express the numbers from one to nine. Book I. comprises the arithmetic of integers, with the rules of addition, duplation, subtraction, halving, multiplication, division, and the extraction of the square root; and the method of proving the operations by casting out the nines is described under each of these rules. Book II. contains the arithmetic of fractions. Book III. contains the rule for finding an unknown number by four proportionals. Book IV. gives a rule for finding an unknown number by assuming a known number, once or twice, and comparing the errors. Book V. gives a method of finding an unknown number by reversing all the steps of the process described in the question. Book VI. treats of mensuration in three chapters. Chapter 1 treats of the mensuration of lines and areas of rectilinear figures. Chapter 2 treats of curvilinear surfaces. Chapter 3 is on the mensuration of solids. Book VII., on practical geometry. Chapter 1. On levelling for the purpose of making canals. In this are described the plummet level and the water level, on the same principle as the spirit level. Chapter 2. On the mensuration of heights. Chapter 3. On the measuring of the breadth of rivers and the depth of wells. Book VIII., on finding unknown quantities by algebra. Chapter 1. Introduction. Chapter 2. On the six rules of algebra. The author remarks: "To find unknown quantities by algebra depends on acuteness and sagacity, an attentive consideration of The Lilavati of Bhascara was translated by the celebrated Fyzee into the Persian language in the year A.D. 1587. In the preface the the terms of the question, and a successful application of the invention to such things as may serve to bring out what is required. Call the unknown quantity shai, thing, and proceed with it according to the terms of the question, as has been said, till the operation ends with an equation. Let that side where there are negative quantities be made perfect, and let the negative quantity be added to the other side; this is called Jebr, restoration; let those things which are of the same kind, and equal on both sides, be thrown away; this is Mokabalah, opposition. Book IX. contains the following twelve rules respecting the properties of numbers : 1. To find the sum of the products of a number multiplied into itself and into all numbers below it; add one to the number, and multiply the sum by the square of the number; half the product is the number required. 2. To add the odd numbers in their regular order; add one to the last number, and take the square of half the sum. 3. To add even numbers from two upwards; multiply half the last even number by a number greater by one than that half. 4. To add the squares of the numbers in order; add one to twice the last number, and multiply a third of the sum by the sum of the numbers. 5. To find the sum of the cubes in succession; take the square of the sum of the numbers. 6. To find the product of the roots of two numbers; multiply one by the other, and the root of the product is the answer. 7. To divide the root of one number by that of another; divide one by the other, the root of the quotient is the answer. 8. To find a perfect number-that is, a number which is equal to the sum of its aliquot parts (Euclid, Book VII., def. 22). The rule is that delivered by Euclid, Book IX., prop. 36. 9. To find a square in a given ratio to its root; divide the first number of the ratio by the second; the square of the quotient is the square required. 10. If any number is multiplied and divided by another, the product multiplied by the quotient is the square of the first number. 11. The difference of two squares is equal to the product of the sum and difference of the roots. 12. If two numbers are divided by each other, and the quotients multiplied together, the result is always one. Book X. contains nine examples. "Conclusion. There are many questions in this science which learned men have to this time in vain attempted to solve; and they have stated some of these questions in their writings, to prove that this science contains difficulties, to silence those who pretend they find nothing in it above their ability, to warn arithmeticians against undertaking to answer every question that may be proposed, and to excite men of genius to attempt their solution. Of these I have selected seven : 1. To divide ten into two parts, such that when each part is added to its square root, and the sums are multiplied together, the product is equal to a supposed number. 2. What square number is that which being increased or diminished by ten, the sum and remainder are both square numbers? 3. A person said he owed Zaid ten all but the square root of what he owed translator remarks:-"By order of the Emperor Akbar, Fyzee1 translates into Persian from the Indian language the book Lilavati, so famous for the rare and wonderful arts of calculation and mensuration. He [Fyzee] begs leave to mention that the compiler of this book was Bhascara Acharya, whose birthplace, and the abode of his ancestors, was the city of Biddur in the country of the Deccan." The Vija Ganita was translated into Persian by Ata Alla Rasheedee, probably at Agra or Delhi. The Persian translator has prefixed the following introduction to his work : "By the grace of God, in the year 1044 Hejira [A.D. 1665], being the eighth year of the king's reign, I, Ata Alla Rasheedee, son of Ahmed Nadir, have translated into the Persian language, from Indian, the book of Indian Algebra, called Beej Gunnit [Vija Ganita], which was written by Bhasker Acharij [Bhascara Acharya], the author of the Leelawuttee [Lilavati]. In the science of calculation it is a discoverer of wonderful truths and nice subtleties, and it contains useful and important problems which are not mentioned in the Leelawuttee, nor in any Arabic or Persian book. I have dedicated the work to Shah Jehan, and I have arranged it according to the original in an introduction and five books." This Persian version of the Vija Ganita was translated into English by Edward Strachey, of the East India Company's Bengal Civil Service. Mr. Strachey remarks that the Persian does not afford a correct idea of the original Sanscrit, as a translation should do; and describes it as an undistinguished mixture of text and commentary. He gives a literal translation of extracts from the Persian, with re Amer, and that he owed Amer five all but the square root of what he owed Zaid. 4. To divide a cube number into two cube numbers. 5. To divide ten into two parts, such that if each is divided by the other, and the two quotients are added together, the sum is equal to one of the parts. 6. There are three square numbers in continued geometrical proportion, such that the sum of the three is a square number. 7. There is a square, such that when it is increased and diminished by its root and two, the sum and the difference are squares. 66 Know, reader, that in this treatise I have collected in a small space the most beautiful and best rules of this science, more than were ever collected before in one book. Do not underrate the value of this bride; hide her from the view of those who are unworthy of her, and let her go to the house of him only who aspires to wed her." 1 Faizee's Persian translation of the Lilavati was printed in 8vo at Calcutta in 1827. 2 In the Fitzwilliam Museum at Cambridge is preserved a splendid model in ivory of the palace tomb which Shah Jehan had erected to the memory of his wife. 3"Bija Ganita, or the Algebra of the Hindus," by Edward Strachey, of the East India Company's Bengal Civil Service. London, 1813. (Reviewed in the 'Edinburgh Review" for July, 1813.) 46 marks of his own. He has translated almost all the rules, some of the examples entirely, and others partly, including whatever he thought deserving attention, as giving a distinct idea of the work. The reference to Euclid' in the Persian can only be regarded as an interpolation by a later hand, as no references to Euclid have been discovered in Sanscrit copies of the Vija Ganita. Mr. Strachey has employed in his work the symbols and methods of modern algebra, and adds the following judicious remark of the Persian translator in reference to the teacher and learner. "In calculation, correctness is the chief point. A wise and considerate person will easily remove the veil from the object; but where the help of acuteness is wanting, a very clear explication is necessary.' 1 2 It also appears that the peculiar notation of the Hindus is entirely wanting in the Persian, where the algebraic processes are always expressed in words at length. Most of the technical terms used in the Persian translation are Arabic. The product of two quantities in general is called their rectangle. It appears that the Persians are indebted for whatever mathematical knowledge they acquired to the Arabians, who had beforederived all their knowledge from the Hindus and the Greeks. The Persians have borrowed the word jebr and mokabalah, together with the greater part of their mathematical terminology, from the Arabs. The following extract from a short treatise on Algebra in 1 The reference to Euclid is contained in the solution of the question, “What right-angled triangle is that, the sum of whose three sides is 40, and the rectangleof the two sides about the right angle 120?" In the solution occurs the expression, "For the sum of the two sides is always greater than the hypotenuse, by the asses' proposition." In a note, the English translator adds, "Meaning by the asses' proposition, the 20th of the first book of Euclid, which, we are told, was ridiculed by the Epicureans as clear even to asses. This tradition of the East, however, differs from that of the West; since it has been handed down, that the 5th, and not the 20th proposition of the first Book of Euclid, bears that designation. A Cambridge witty wag of former days has supplied some small comfort for the halting and feebleminded who are too weak to proceed, in the following pithy couplet : "If this be rightly called the bridge of asses, He's not the ass who stops, but he who passes." Mr. Strachey has printed the following extract, a translation from a Persian treatise on arithmetic, in p. 184 of vol. xii. of the Asiatic Researches. "The Indian sages, wishing to express numbers conveniently, invented the nine figures. The first figure on the right hand they made stand for units, the second for tens, the third for hundreds, the fourth for thousands. Thus, after the third rank, the next following is units of thousands, the second tens of thousands, the third hundreds of thousands, and so on. Every figure, therefore, in the first rank, is the number of units it expresses, every figure in the second, the number of tens which the figure expresses, in the third, the number of hundreds, and so on. When in any rank a figure is wanting, write a cipher like a small circle o, to preserve the rank. Thus ten is written 10, a hundred 100, five thousand and twenty-five 5025." Persian verse, by Mohammed Nadjm-Eddin Khan, appended to the Calcutta edition of the Kholasat al Hisab, will serve as an illustration of this remark : : "Complete the side in which the expression illa (less, minus) occurs, and add as much to the other side, O learned man: this is in correct language called jebr. In making the equation mark this it may happen that some terms are cognate and equal on each side, without distinction; these you must on both sides remove, and this you call mokabalah." In the thirteenth volume of the "Asiatic Researches" is printed an Essay by J. Tytler, Esq., on the "Binomial Theorem as known to the Arabians." The essay contains an English translation, with the Arabic original, of a passage from the Ayoun-ul-Hisab, or Rules of Arithmetic, composed by Mohammed Baquier, in the reign of Shah Abbas I., about the year A.D. 1600. In the example given, the number 12 is employed, and the mode of solution appears to be the same as that given in the Lilavati, in which the number is 6, for finding the number of combinations of 6 things taken 1, 2, 3, &c., at a time. It may reasonably be doubted whether the Hinas or the Arabians had attained to the idea that the successive coeficients of an expanded binomial are respectively equal to the numbers of combinations of a number (equal to the exponent) of things taken 1, 2, 3, &c., at a time, when the exponent itself is a positive integer. During the middle ages the countries of Western Europe appear to have produced few students of eminence in the mathematical sciences. The scientific knowledge, such as it was in these times, was not derived directly either from Greek or Hindu sources. There were, however, individuals who, inspired with a thirst for knowledge, both in England and on the continent, travelled into foreign countries in search of knowledge. Notices of such persons are extant, as Adelard, a monk of Bath, who first brought the knowledge of Euclid's Elements into England by a translation from the Arabic; and John of Basingstoke, who introduced the knowledge of the Greek numerical characters. And others, such as Roger Bacon, a diligent student of the works of nature, whose attempts to promote the advancement of the sciences brought him under the suspicion of the clergy in those times of practising magic and of having dealings with the devil. The seventh century of the Christian era was almost barren of writers on science. The irruption and wars of the Saracens, and the increasing corruptions of the Church, mutually assisted in the destruction of all learning and science. The eighth century was nearly of the same character, with a few exceptions of great eminence. Isidorus Hispalensis flourished in the early part of the seventh century, and governed the Church of Seville as Archbishop for forty years. He was the most eminent scholar of his age. His writings were numerous; one of the most |