Gebel. hen Aphla, who lived in the llth century, substi tuted, instead of the ancient method, three or four theo rems, which are the foundation of modern trigonometry. The Arabs also simplified trigonometrical calculation, by substituting the sines of arcs, instead of the chords of the double arcs ; and this was even one of their earliest in ventions, for it is found in the writings of Albatenius, who flourished about the year 880 of our era. The names of many Arabian geometers are. known; we shall, however, only mention Bagdadin, or Mahomet Al-Bagdadi, (of Bag dad,) the author of an elegant work on mensuration, which has been translated and published in 1570 ; and Alhazen, the celebrated author of a work on optics, which shows him to have been an excellent geometer. In general, the Arabian geometers had little invention, they were almost all compilers or commentators on the ancients.
Persia has also had its geometers. The most celebrated was Nassir-Eddin Al-Tussi ; he wrote a learned commen tary on Euclid, which was printed in 1590 at the press of the Medici. He also revised the conics of Apollonius, and wrote a commentary on the subject ; this was useful to Dr Halley, in restoring the fifth, sixth, and seventh books of that precious work. The geometer next in es teem was Maimon-Resehid he wrote a commentary on Euclid, and is said to have indulged in a singular whim : he had conceived such an affection for one of the proposi tions of the first book of the Elements, that he wore the diagram as an ornament embroidered on his sleeve. Ge ometry has, in modern times, been respected among the Persians, but they have not made any improvements in the science. The traveller Chardin has given some traits of the pedantry of their literati. " They have given," says he, a a name to every proposition of the Elements. They call the 47th proposition of the first book of Euclid the figure of the bride, probably because it is to become the mother of a numerous progeny of other theorems. The 48th proposition, again, they call the bride's sister ; and they, with reason, denominate geometry the difficult sci ence.
The Turks have not altogether neglected geometry. In the libraries of Constantinople, the greater number of the Greek mathematicians may be found translated into Arabic, and some into the Turkish language ; but it does not appear that they pay attention to any thing be yond what is contained in Euclid's Elements, and indeed they have never made one discovery in the sciences.
There are hardly any traces of geometry among the an cient Hebrews. Every one knows that when Solomon's temple was built, Hiram king of Tyre furnished architects and navigators, a proof that geometry must then have been very little known in Palestine. It was not until the se cond dispersion among the nations that they began to cul tivate the sciences. In the ninth century, the Jews, after the example of the Arabians, began to translate the Greek geometers into their language ; but they have discovered Ilothing whatever in geometry.
The researches of the learned have brought to light astronomical tables in India, which mn have been con structed by the principles of geometry ; but the period at which they have been formed has by no means been com pletely ascertained. Some arc of opinion, that they have been framed from observations made at a very remote pe riod, not less than three thousand years before the Chris tian xra ; and if this opinion be well founded, the science of geometry must have been cultivated in India to a con siderable extent, long before the period assigned to its origin in the West ; so that many of the elementary pro positions may have been brought from India to Greece. The Hindoos have a treatise called the Suryci Sidhanta, which professes to be a revelation from heaven, commu• nieated to Meya, a man of great sanctity, about four mil lion of years ago ; but setting aside this fabulous origin, it has been supposed to be of great antiquity, and to have been written at least two thousand years before the Christian era. Interwoven with many absurdities, this book con tains a rational system of trigonometry, which differs en tirely from that first known in Greece or Arabia : In fact, it is founded on a geometrical theorem, which was not known to the mathematicians of Europe before the time of Vieta, about two hundred years ago. And it employs the sines of arcs, a thing unknown to the Greeks, who used the chords of the double arcs. The invention of sines has been attributed to the Arabs, but it is possible that they may have received this improvement in trigonometry as well as the numeral characters, from India.
According to the natural progress of knowledge, the sciences of astronomy and geometry Enlist have been long cultivated, and carried to some degree of perfection, be fore a system of trigonometry would be formed; we may therefore infer, that geometry had an earlier origin in In dia than the Snub Sidhanta. It is, however, proper we should state, that the high antiquity both cf the Indian as tronomy and the Sztrya Sidhanta has been controverted ; but we cannot find room to enter on this point here. The antiquity of the Indian geometry has been asserted by Bailly in his Astronomic Indienne, and Professor Mayfair in his Remarks on the Astronomy of the Brahmins, Echo. Trans. vol. ii. and Observations on the Trigonometry of the Brahmins, Edin. Trans. vol. iv. with great eloquence and strength of reasoning (See our article ASTRONOMY.) On this side of the question, the Edinburgh Review, vol. x. p. 455, may also be consulted. And on the opposite side, La Place, Systeme du Minide, 2d edit. p. 239 ; Bentley On the Hindoo Systems of Astronomy, in the Asiatic Researches, vol. viii. ; Edinburgh Review, vol. xviii. p. 210 ; Leslie's Geometry, 2d edit. p. 456. Mr Leslie is of opinion, that the Hindoos derived their knowledge of mathematics from the West. In opposition to this, consult Strachey, in the Pre face to Bija Ganita ; and a review of the work in Edin burgh Review, vol. xxi. p. 364.