can be found. Another mode of trying for solutions is a combination of the preceding with the cuttaca (pulveriser).” These calculations were used in astronomy.

=

Doubtless this "cyclic method" constitutes the greatest invention in the theory of numbers before the time of Lagrange. The perversity of fate has willed it, that the equation yax2+1 should now be called Pell's problem, while in recognition of Brahmin scholarship it ought to be called the "Hindoo problem." It is a problem that has exercised the highest faculties of some of our greatest modern analysts. By them the work of the Hindoos was done over again; for, unfortunately, the Arabs transmitted to Europe only a small part of Indian algebra and the original Hindoo manuscripts, which we now possess, were unknown in the Occident.

Hindoo geometry is far inferior to the Greek. In it are found no definitions, no postulates, no axioms, no logical chain of reasoning or rigid form of demonstration, as with Euclid. Each theorem stands by itself as an independent truth. Like the early Egyptian, it is empirical. Thus, in the proof of the theorem of the right triangle, Bhaskara draws the right triangle four times in the square of the hypotenuse, so that in the middle

there remains a square whose side equals the difference between the two sides of the right

triangle. Arranging this square and the four triangles in a different way, they are seen, together, to make up the sum

of the square of the two sides. "Behold!" says Bhaskara,

without adding another word of explanation. Bretschneider conjectures that the Pythagorean proof was substantially the same as this. In another place, Bhaskara gives a second demonstration of this theorem by drawing from the vertex of

the right angle a perpendicular to the hypotenuse, and comparing the two triangles thus obtained with the given triangle to which they are similar. This proof was unknown in Europe till Wallis rediscovered it. The Brahmins never inquired into the properties of figures. They considered only metrical relations applicable in practical life. In the Greek sense, the Brahmins never had a science of geometry. Of interest is the formula given by Brahmagupta for the area of a triangle in terms of its sides. In the great work attributed to Heron the Elder this formula is first found. Whether the Indians themselves invented it, or whether they borrowed it from Heron, is a disputed question. Several theorems are given by Brahmagupta on quadrilaterals which are true only of those which can be inscribed on a circle a limitation which he omits to state. Among these is the proposition of Ptolemæus, that the product of the diagonals is equal to the sum of the products of the opposite sides. The Hindoos were familiar with the calculation of the areas of circles and their segments, of the length of chords and perimeters of regular inscribed polygons. An old Indian tradition makes = 3, also = √10; but Aryabhatta gives the value 1416. Bhaskara gives two values, the accurate,' 3927, and the 'inaccurate,' Archimedean value, 22. A commentator on Lilavati says that these values were calculated by beginning with a regular inscribed hexagon, and applying repeatedly the formula

10000

AD: √√2 √4 - AB, wherein AB is the side of the given

=

polygon, and AD that of one with double the number of sides. In this way were obtained the perimeters of the inscribed polygons of 12, 24, 48, 96, 192, 384 sides. Taking the radius=100, the perimeter of the last one gives the value which Aryabhatta used for π.

Greater taste than for geometry was shown by the Hindoos for trigonometry. Like the Babylonians and Greeks, they

divided the circle into quadrants, each quadrant into 90 degrees and 5400 minutes. The whole circle was therefore made up of 21,600 equal parts. From Bhaskara's 'accurate' value for it was found that the radius contained 3438 of these circular parts. This last step was not Grecian. The Greeks might have had scruples about taking a part of a curve as the measure of a straight line. Each quadrant was divided into 24 equal parts, so that each part embraced 225 units of the whole circumference, and corresponds to 3 degrees. Notable is the fact that the Indians never reckoned, like the Greeks, with the whole chord of double the arc, but always with the sine (joa) and versed sine. Their mode of calculating tables was theoretically very simple. The sine of 90° was equal to the radius, or 3438; the sine of 30° was evidently half that, or 1719. Applying the formula sina + cos2a = r2, they obtained sin 45° = 2431. Substituting for cos a

√

2

its equal sin (90 — a), and making a = 60°, they obtained

√32 sin 60° = = 2978. With the sines of 90, 60, 45, and 30 2 as starting-points, they reckoned the sines of half the angles by the formula ver sin 2 a = 2 sin2 a, thus obtaining the sines of 22° 30', 11° 15', 7° 30', 3° 45'. They now figured out the sines of the complements of these angles, namely, the sines of 86° 15', 82° 30', 78° 45', 75°, 67° 30′; then they calculated the sines of half these angles; then of their complements; then, again, of half their complements; and so on. By this very simple process they got the sines of angles at intervals of 3° 45'. In this table they discovered the unique law that if a, b, c be three successive arcs such that a − b = b − c = 3° 45', sin b then sin a sin b = (sin b — sin c) This formula was

225

afterwards used whenever a re-calculation of tables had to

be made. No Indian trigonometrical treatise on the triangle

is extant. In astronomy they solved plane and spherical right triangles.18

It is remarkable to what extent Indian mathematics enters into the science of our time. Both the form and the spirit of the arithmetic and algebra of modern times are essentially Indian and not Grecian. Think of that most perfect of mathematical symbolisms - the Hindoo notation, think of the Indian arithmetical operations nearly as perfect as our own, think of their elegant algebraical methods, and then judgė whether the Brahmins on the banks of the Ganges are not entitled to some credit. Unfortunately, some of the most brilliant of Hindoo discoveries in indeterminate analysis reached Europe too late to exert the influence they would. have exerted, had they come two or three centuries earlier.

THE ARABS.

After the flight of Mohammed from Mecca to Medina in 622 A.D., an obscure people of Semitic race began to play an important part in the drama of history. Before the lapse of ten years, the scattered tribes of the Arabian peninsula were fused by the furnace blast of religious enthusiasm into a powerful nation. With sword in hand the united Arabs subdued Syria and Mesopotamia. Distant Persia and the lands beyond, even unto India, were added to the dominions of the Saracens. They conquered Northern Africa, and nearly the whole Spanish peninsula, but were finally checked from further progress in Western Europe by the firm hand of Charles Martel (732 A.D.). The Moslem dominion extended now from India to Spain; but a war of succession to the caliphate ensued, and in 755 the Mohammedan empire was divided, one caliph reigning at Bagdad, the other at Cordova

in Spain. Astounding as was the grand march of conquest by the Arabs, still more so was the ease with which they put aside their former nomadic life, adopted a higher civilisation, and assumed the sovereignty over cultivated peoples. Arabic was made the written language throughout the conquered lands. With the rule of the Abbasides in the East began a new period in the history of learning. The capital, Bagdad, situated on the Euphrates, lay half-way between two old centres of scientific thought, India in the East, and Greece in the West. The Arabs were destined to be the custodians of the torch of Greek and Indian science, to keep it ablaze during the period of confusion and chaos in the Occident, and afterwards to pass it over to the Europeans. Thus science. passed from Aryan to Semitic races, and then back again to the Aryan. The Mohammedans have added but little to the knowledge in mathematics which they received. They now and then explored a small region to which the path had been previously pointed out, but they were quite incapable of discovering new fields. Even the more elevated regions in which the Hellenes and Hindoos delighted to wander namely, the Greek conic sections and the Indian indeterminate analysis were seldom entered upon by the Arabs. were less of a speculative, and more of a practical turn of mind.

They

The Abbasides at Bagdad encouraged the introduction of the sciences by inviting able specialists to their court, irrespective of nationality or religious belief. Medicine and astronomy were their favourite sciences. Thus Haroun-alRaschid, the most distinguished Saracen ruler, drew Indian. physicians to Bagdad. In the year 772 there came to the court of Caliph Almansur a Hindoo astronomer with astronomical tables which were ordered to be translated into Arabic. These tables, known by the Arabs as the Sindhind, and