authority by Brahmegupta, who flourished in the early part of the seventh century. Brahmegupta is again frequently referred to by Bhascara, whose works were published in the middle of the twelfth century, and exhibit those forms of decimal notation which are now universally adopted by civilised nations. "The first Arabian," says Dr. Peacock,* "who wrote upon Algebra and the Indian mode of computation is stated, with the common consent of Arabic authors, to have been Mohammed Ben Musa, the Khuwarezmite, who flourished about the end of the ninth century; an author who is celebrated as having made known to his countrymen other parts of Hindu science, to which he is said to have been very partial. Before the end of the tenth century those figures, which are called Hindasi from their origin, were in general use throughout Arabia; among others is mentioned the celebrated Alkindi, who was nearly contemporary with Ben Musa, and who, among his numerous other works, wrote one on the Indian mode of computation (Hisabu 'lHindi). The same testimony is repeated in almost every subsequent author on Arithmetic or Algebra, and is completely confirmed by their writing their figures from left to right after the manner of the Hindoos, but which is directly contrary to the order of their own writing.t *Ib. 413. ORIENTAL FORMS OF ARABIC NUMERALS. 230856789 १२३४५६७ te १० १२३४५६१८७१० 0 $ 0 8 0 6 9 6 addo ۱۱۲۳۴۵۶۷۸۹۱ १२३४ ४५७८७% †Though the method of writing the Arabic numerals from left to right has been deemed conclusive of their Sanscrit origin, independently of other evidence, might it not be maintained that the Arabic numerals "The use of this notation became general among Arabic writers, not merely on Arithmetic and Algebra, but likewise on Astronomy, about the beginning of the tenth century. We find it in the works of the astronomer Ebn Younis, who died in the year 1008, and it is found likewise in all subsequent astronomical tables. It was, of course, communicated to all those countries where their language and science were known. In the eleventh century the Moors were not merely in possession of the southern provinces of Spain, but had established a flourishing kingdom, where the favourite sciences of their Eastern ancestors were cultivated with uncommon activity and success; and from that quarter, and from the Moors in Africa, they chiefly appear to have been communicated to the Spaniards and other Europeans. "The learned Abbé Andres considers that the earliest example of the use of these figures, which is to be found in Spain or in Europe, is in a translation of Ptolemy in the year 1136; fac-similes of the former of these figures are said to be given in the Paleografia Española of Terreros, who found them in all the mathematical MSS. subsequent to that period, but in no other books or documents-not even in accounts which were kept in the Castilian, which differed little from the Roman numerals;—the calendars which were chiefly constructed in Spain, both in that age, and were invented by a people accustomed to write like the Hebrews, Arabs, and many other Oriental nations, from right to left, and not as we do, from left to right, on the very ground that the simplest of the arithmetical signs, the units or digits, are placed on the right side, and the larger and more complicated on the left. Arithmetical science, like every other, must have commenced with the simpler form, and have proceeded to the more elaborate. Had a people, writing as we are accustomed to write, from left to right, introduced a new system of notation, would it not be natural that the highest amounts should be the farthest from, and not the nearest to, the starting point? We should have probably written the units first, and then the tens, and then the hundreds, and then the thousands, and so on-just in the contrary way in which the record would be made by a Persian or an Arab. Language easily accommodates itself to the symbols which represent it, and the use of one and twenty is quite as familiar to our ears as twenty-one; three score, as sixty; or half a hundred as fifty. * It has been held by some writers that Leonardo's writings are to be referred to the beginning of the fifteenth century; but the great weight of evidence gives him two centuries earlier. until the end of the fourteenth century, and were sent from thence to other parts of Europe, continued to be written in the old notation." ANCIENT EUROPEAN FORMS OF ARABIC NUMERALS. 1 2 3 0 9 6 1 8 9 10 1 2 3 4 4 6 4 8 910 2 3 4 9 6 7 8 9 7 1 2 3 4 5 6 7 8 9 19° The Arabs indeed do not claim the honour of having invented, the decimal system of numeration, but attribute it to the Hindoos, whose name in Arabic* it bears; but its progress is not very clearly traceable in Europe. It is certain that in the beginning of the sixteenth century Roman figures were used by merchants and accountants. They lingered longer in England than in any other part of the European world, having found an asylum in the dark and dull regions of the Exchequer. According to the Hindus, numeration is of divine origin; the invention of nine figures (anca), with the device of places to make them suffice for all numbers, being ascribed to the beneficent Creator of the universe, in Bhascara's "Vásaná," and its glossary, and in Crishna's "Commentary on the Vija-ganita." Here nine figures are specified; the place where none belongs to it being shown by a blank, Súnja, which, to obviate mistake, is denoted by a dot, or small circle. From the right, where the first and lowest number is placed, towards the left hand, increasing regularly in decuple proportion namely, unit, ten, hundred, thousand, myriad, hundred thousands, million, ten millions, hundred millions, thousand millions, ten thousand millions, hundred thousand millions, billion, ten billions, hundred billions, thousand billions, ten thousand billions, hundred thousand billions. A passage of the "Véda,” which is cited by Súrya-dása, exhibits the decimal notation thus:- "Be these milch-kine before me, one, ten, a hundred, a thousand, ten thousand, a hundred thousand, a million. . Be these milch-kine my guides in this world."† Ganésa observes, that numeration has been carried to a greater number of places by Sríd'hara and others; but adds, that the names are omitted on account of the numerous contradictions, and the little utility of those designations. The text of the Ganita-Sara, or abridgement of Sríd'hara, does not correspond with this reference, for it exhibits the same eighteen places, and no more.‡ The subject of numbers is approached with reverence by the great Hindoo writers. The Brahmagapta is introduced by this invocation:- "Salutation to Ganesa, resplendent as a blue and spotless lotus, and delighting in the tremulous motion of the dark * Hindi. + "Colebrooke's Hindoo Algebra," p. 4. serpent, which is perpetually twining within his throat." And the volume is ushered in by this flowery announcement :-" Having bowed to the deity, whose head is like an elephant's,* whose feet are adored by gods; who, when called to mind, relieves his votaries from embarrassment, and bestows happiness on his worshippers; I propound this easy process of computation,† delightful by its elegance,‡ perspicuous, with words concise, soft, and correct, and pleasing to the learned." § While on the subject of Hindoo numerals, it may be amusing to see a specimen or two of the delicate and winning forms in which arithmetical questions are propounded to the student for solution:-" Beautiful and dear Lilávati, whose eyes are like a fawn's! tell me what are the numbers resulting from one hundred and thirty-five, taken into twelve? If thou be skilled in multiplication, by whole, or by parts, whether by subdivision of form, or separation of digits, tell me, auspicious woman, what is the quotient of the product divided by the same multiplier ?" Colebrooke, p. 6. Again," Pretty girl, with tremulous eyes, if thou knowest the correct method of inversion, tell me what is the number, which multiplied by three, and added to the three-quarters of the product, and divided by seven, and reduced by subtraction of a third part of the quotient, and then multiplied into itself, and having fifty-two subtracted from the product, and the square root of the remainder extracted, and eight added, and the sum divided by ten, yields two?" To a question so complicated, it is hardly fair to keep back the solution. Statement:-Multiplier, 3; additive, &; divisor, 7; decrease, ; square—; subtractive, 52; square root—; additive, 8 ; divisor 10; given number, two. Answer:-Proceeding as directed, the result is 28, the number sought. Colebrooke, pp. 21-2. All the operations are inverted. The known number 2, multiplied by the divisor 10, converted into a multiplicator, makes 20; from which the additive 8 being subtracted, leaves 12; the square whereof (extraction of the root being directed) is 144; and adding * Gánésa, represented with an elephant's head and human body. † Arithmetic. Páti ganita; Páti, Paripàti, or Vyaeta ganita. Lilavati-delightful,- -an allusion to the title of the book. § Colebrooke's Translation of "Brahmagapta and Bhascara," 4to, 1817. p. 1. |