A History of Mathematics/Middle Ages/The Hindoos

by Florian Cajori

THE HINDOOS.

The first people who distinguished themselves in mathematical research, after the time of the ancient Greeks, belonged, like them, to the Aryan race. It was, however, not a European, but an Asiatic nation, and had its seat in far-off India.

Unlike the Greek, Indian society was fixed into castes. The only castes enjoying the privilege and leisure for advanced study and thinking were the Brahmins, whose prime business was religion and philosophy, and the Kshatriyas, who attended to war and government.

Of the development of Hindoo mathematics we know but little. A few manuscripts bear testimony that the Indians had climbed to a lofty height, but their path of ascent is no longer traceable. It would seem that Greek mathematics grew up under more favourable conditions than the Hindoo, for in Greece it attained an independent existence, and was studied for its own sake, while Hindoo mathematics always remained merely a servant to astronomy. Furthermore, in Greece mathematics was a science of the people, free to be cultivated by all who had a liking for it; in India, as in Egypt, it was in the hands chiefly of the priests. Again, the Indians were in the habit of putting into verse all mathematical results they obtained, and of clothing them in obscure and mystic language, which, though well adapted to aid the memory of him who already understood the subject, was often unintelligible to the uninitiated. Although the great Hindoo mathematicians doubtless reasoned out most or all of their discoveries, yet they were not in the habit of preserving the proofs, so that the naked theorems and processes of operation are all that have come down to our time. Very different in these respects were the Greeks. Obscurity of language was generally avoided, and proofs belonged to the stock of knowledge quite as much as the theorems themselves. Very striking was the difference in the bent of mind of the Hindoo and Greek; for, while the Greek mind was pre-eminently geometrical, the Indian was first of all arithmetical. The Hindoo dealt with number, the Greek with form. Numerical symbolism, the science of numbers, and algebra attained in India far greater perfection than they had previously reached in Greece. On the other hand, we believe that there was little or no geometry in India of which the source may not be traced back to Greece. Hindoo trigonometry might possibly be mentioned as an exception, but it rested on arithmetic more than on geometry.

An interesting but difficult task is the tracing of the relation between Hindoo and Greek mathematics. It is well known that more or less trade was carried on between Greece and India from early times. After Egypt had become a Roman province, a more lively commercial intercourse sprang up between Rome and India, by way of Alexandria. A priori, it does not seem improbable, that with the traffic of merchandise there should also be an interchange of ideas. That communications of thought from the Hindoos to the Alexandrians actually did take place, is evident from the fact that certain philosophic and theologic teachings of the Manicheans, Neo-Platonists, Gnostics, show unmistakable likeness to Indian tenets. Scientific facts passed also from Alexandria to India. This is shown plainly by the Greek origin of some of the technical terms used by the Hindoos. Hindoo astronomy was influenced by Greek astronomy. Most of the geometrical knowledge which they possessed is traceable to Alexandria, and to the writings of Heron in particular. In algebra there was, probably, a mutual giving and receiving. We suspect that Diophantus got the first glimpses of algebraic knowledge from India. On the other hand, evidences have been found of Greek algebra among the Brahmins. The earliest knowledge of algebra in India may possibly have been of Babylonian origin. When we consider that Hindoo scientists looked upon arithmetic and algebra merely as tools useful in astronomical research, there appears deep irony in the fact that these secondary branches were after all the only ones in which they won real distinction, while in their pet science of astronomy they displayed an inaptitude to observe, to collect facts, and to make inductive investigations.

We shall now proceed to enumerate the names of the leading Hindoo mathematicians, and then to review briefly Indian mathematics. We shall consider the science only in its complete state, for our data are not sufficient to trace the history of the development of methods. Of the great Indian mathematicians, or rather, astronomers,—for India had no mathematicians proper,—Aryabhatta is the earliest. He was born 476 A.D., at Pataliputra, on the upper Ganges. His celebrity rests on a work entitled Aryabhattiyam, of which the third chapter is devoted to mathematics. About one hundred years later, mathematics in India reached the highest mark. At that time flourished Brahmagupta (born 598). In 628 he wrote his Brahma-sphuta-siddhanta ("The Revised System of Brahma"), of which the twelfth and eighteenth chapters belong to mathematics. To the fourth or fifth century belongs an anonymous astronomical work, called Surya-siddhanta ("Knowledge from the Sun"), which by native authorities was ranked second only to the Brahma-siddhanta, but is of interest to us merely as furnishing evidence that Greek science influenced Indian science even before the time of Aryabhatta. The following centuries produced only two names of importance; namely, Cridhara, who wrote a Ganita-sara ("Quintessence of Calculation"), and Padmanabha, the author of an algebra. The science seems to have made but little progress it this time; for a work entitled Siddhantaciromani ("Diadem of an Astronomical System"), written by Bhaskara Acarya in 1150, stands little higher than that of Brahmagupta, written over 500 years earlier. The two most important mathematical chapters in this work are the Lilavati (="the beautiful," i.e. the noble science) and Viga-ganita (="root-extraction"), devoted to arithmetic and algebra. From now on, the Hindoos in the Brahmin schools seemed to content themselves with studying the masterpieces of their predecessors. Scientific intelligence decreases continually, and in modern times a very deficient Arabic work of the sixteenth century has been held in great authority.^[7]

The mathematical chapters of the Brahma-siddhanta and Siddhantaciromani were translated into English by H. T. Colebrooke, London, 1817. The Surya-siddhanta was translated by E. Burgess, and annotated by W. D. Whitney, New Haven, Conn., 1860.

The grandest achievement of the Hindoos and the one which, of all mathematical inventions, has contributed most to the general progress of intelligence, is the invention of the principle of position in writing numbers. Generally we speak of our notation as the "Arabic" notation, but it should be called the "Hindoo" notation, for the Arabs borrowed it from the Hindoos. That the invention of this notation was not so easy as we might suppose at first thought, may be inferred from the fact that, of other nations, not even the keen-minded Greeks possessed one like it. We inquire, who invented this ideal symbolism, and when? But we know neither the inventor nor the time of invention. That our system of notation is of Indian origin is the only point of which we are certain. From the evolution of ideas in general we may safely infer that our notation did not spring into existence a completely armed Minerva from the head of Jupiter. The nine figures for writing the units are supposed to have been introduced earliest, and the sign of zero and the principle of position to be of later origin. This view receives support from the fact that on the island of Ceylon a notation resembling the Hindoo, but without the zero has been preserved. We know that Buddhism and Indian culture were transplanted to Ceylon about the third century after Christ, and that this culture remained stationary there, while it made progress on the continent. It seems highly probable, then, that the numerals of Ceylon are the old, imperfect numerals of India. In Ceylon, nine figures were used for the units, nine others for the tens, one for 100, and also one for 1000. These 20 characters enabled them to write all the numbers up to 9999. Thus, 8725 would have been written with six signs, representing the following numbers: 8, 1000, 7, 100, 20, 5. These Singhalesian signs, like the old Hindoo numerals, are supposed originally to have been the initial letters of the corresponding numeral adjectives. There is a marked resemblance between the notation of Ceylon and the one used by Aryabhatta in the first chapter of his work, and there only. Although the zero and the principle of position were unknown to the scholars of Ceylon, they were probably known to Aryabhatta; for, in the second chapter, he gives directions for extracting the square and cube roots, which seem to indicate a knowledge of them. It would appear that the zero and the accompanying principle of position were introduced about the time of Aryabhatta. These are the inventions which give the Hindoo system its great superiority, its admirable perfection.

There appear to have been several notations in use in different parts of India, which differed, not in principle, but merely in the forms of the signs employed. Of interest is also a symbolical system of position, in which the figures generally were not expressed by numerical adjectives, but by objects suggesting the particular numbers in question. Thus, for 1 were used the words moon, Brahma, Creator, or form; for 4, the words Veda, (because it is divided into four parts) or ocean, etc. The following example, taken from the Surya-siddhanta, illustrates the idea. The number 1,577,917,828 is expressed from right to left as follows: Vasu (a class of 8 gods) + two + eight + mountains (the 7 mountain-chains) + form + digits (the 9 digits) + seven + mountains + lunar days (half of which equal 15). The use of such notations made it possible to represent a number in several different ways. This greatly facilitated the framing of verses containing arithmetical rules or scientific constants, which could thus be more easily remembered.

At an early period the Hindoos exhibited great skill in calculating, even with large numbers. Thus, they tell us of an examination to which Buddha, the reformer of the Indian religion, had to submit, when a youth, in order to win the maiden he loved. In arithmetic, after having astonished his examiners by naming all the periods of numbers up to the 53d, he was asked whether he could determine the number of primary atoms which, when placed one against the other, would form a line one mile in length. Buddha found the required answer in this way: 7 primary atoms make a very minute grain of dust, 7 of these make a minute grain of dust, 7 of these a grain of dust whirled up by the wind, and so on. Thus he proceeded, step by step, until he finally reached the length of a mile. The multiplication of all the factors gave for the multitude of primary atoms in a mile a number consisting of 15 digits. This problem reminds one of the 'Sand-Counter' of Archimedes.

After the numerical symbolism had been perfected, figuring was made much easier. Many of the Indian modes of operation differ from ours. The Hindoos were generally inclined to follow the motion from left to right, as in writing. Thus, they added the left-hand columns first, and made the necessary corrections as they proceeded. For instance, they would have added 254 and 663 thus: $\scriptstyle {2+6=8}$, $\scriptstyle {5+6=11}$, which changes 8 into 9, $\scriptstyle {4+3=7}$. Hence the sum 917. In subtraction they had two methods. Thus in $\scriptstyle {821-348}$ they would say, $\scriptstyle {8{\text{ from }}11=3}$, $\scriptstyle {4{\text{ from }}11=7}$, $\scriptstyle {3{\text{ from }}7=4}$. Or they would say, $\scriptstyle {8{\text{ from }}11=3}$, $\scriptstyle {5{\text{ from }}12=7}$, $\scriptstyle {4{\text{ from }}8=4}$. In multiplication of a number by another of only one digit, say 569 by 5, they generally said, $\scriptstyle {5\cdot 5=25}$, $\scriptstyle {5\cdot 6=30}$, which changes 25 into 28, $\scriptstyle {5\cdot 9=45}$, hence the 0 must be increased by 4. The product is 2845. In the multiplication with each other of many-figured numbers, they first multiplied, in the manner just indicated, with the left-hand digit of the multiplier, which was written above the multiplicand, and placed the product above the multiplier. On multiplying with the next digit of the multiplier, the product was not placed in a new row, as with us, but the first product obtained was corrected, as the process continued, by erasing, whenever necessary, the old digits, and replacing them by new ones, until finally the whole product was obtained. We who possess the modern luxuries of pencil and paper, would not be likely to fall in love with this Hindoo method. But the Indians wrote "with a cane-pen upon a small blackboard with a white, thinly liquid paint which made marks that could be easily erased, or upon a white tablet, less than a foot square, strewn with red flour, on which they wrote the figures with a small stick, so that the figures appeared white on a red ground."^[7] Since the digits had to be quite large to be distinctly legible, and since the boards were small, it was desirable to have a method which would not require much space. Such a one was the above method of multiplication. Figures could be easily erased and replaced by others without sacrificing neatness.

	7		3		5
1
		7		3		5
2	1				1
		4		6		0
8	8		2		0

But the Hindoos had also other ways of multiplying, of which we mention the following: The tablet was divided into squares like a chess-board. Diagonals were also drawn, as seen in the figure. The multiplication of $\scriptstyle {12\times 735=8820}$ is exhibited in the adjoining diagram.^[3] The manuscripts extant give no information of how divisions were executed. The correctness of their additions, subtractions, and multiplications was tested "by excess of 9's." In writing fractions, the numerator was placed above the denominator, but no line was drawn between them.

We shall now proceed to the consideration of some arithmetical problems and the Indian modes of solution. A favourite method was that of inversion. With laconic brevity, Aryabhatta describes it thus: "Multiplication becomes division, division becomes multiplication; what was gain becomes loss, what loss, gain; inversion." Quite different from this quotation in style is the following problem from Aryabhatta, which illustrates the method:^[3] "Beautiful maiden with beaming eyes, tell me, as thou understandst the right method of inversion, which is the number which multiplied by 3, then increased by $\scriptstyle {\frac {3}{4}}$ of the product, divided by 7, diminished by $\scriptstyle {\frac {1}{3}}$ of the quotient, multiplied by itself, diminished by 52, the square root extracted, addition of 8, and division by 10, gives the number 2?" The process consists in beginning with 2 and working backwards. Thus, $\scriptstyle {(2\cdot 10-8)^{2}+52=196}$, $\scriptstyle {{\sqrt {196}}=14}$, and $\scriptstyle {14\cdot {\frac {3}{2}}\cdot 7\cdot {\frac {4}{7}}\div 3=28}$, the answer.

Here is another example taken from Lilavati, a chapter in Bhaskara's great work: "The square root of half the number of bees in a swarm has flown out upon a jessamine-bush, $\scriptstyle {\frac {8}{9}}$ of the whole swarm has remained behind; one female bee flies about a male that is buzzing within a lotus-flower into which he was allured in the night by its sweet odour, but is now imprisoned in it. Tell me the number of bees." Answer, 72. The pleasing poetic garb in which all arithmetical problems are clothed is due to the Indian practice of writing all school-books in verse, and especially to the fact that these problems, propounded as puzzles, were a favourite social amusement. Says Brahmagupta: "These problems are proposed simply for pleasure; the wise man can invent a thousand others, or he can solve the problems of others by the rules given here. As the sun eclipses the stars by his brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more if he solves them."

The Hindoos solved problems in interest, discount, partnership, alligation, summation of arithmetical and geometric series, devised rules for determining the numbers of combinations and permutations, and invented magic squares. It may here be added that chess, the profoundest of all games, had its origin in India.

The Hindoos made frequent use of the "rule of three," and also of the method of "falsa positio," which is almost identical with that of the "tentative assumption" of Diophantus. These and other rules were applied to a large number of problems.

Passing now to algebra, we shall first take up the symbols of operation. Addition was indicated simply by juxtaposition as in Diophantine algebra; subtraction, by placing a dot over the subtrahend; multiplication, by putting after the factors, bha, the abbreviation of the word bhavita, "the product"; division, by placing the divisor beneath the dividend; square-root, by writing ka, from the word karana (irrational), before the quantity. The unknown quantity was called by Brahmagupta yâvattâvat (quantum tantum). When several unknown quantities occurred, he gave, unlike Diophantus, to each a distinct name and symbol. The first unknown was designated by the general term "unknown quantity." The rest were distinguished by names of colours, as the black, blue, yellow, red, or green unknown. The initial syllable of each word constituted the symbol for the respective unknown quantity. Thus yâ meant x; kâ (from kâlaka = black) meant y; yâ kâ bha, "x times y"; $\scriptstyle {ka~15~ka~{\overset {.}{10}}}$, "$\scriptstyle {{\sqrt {15}}-{\sqrt {10}}}$."

The Indians were the first to recognise the existence of absolutely negative quantities. They brought out the difference between positive and negative quantities by attaching to the one the idea of 'possession,' to the other that of 'debts.' The conception also of opposite directions on a line, as an interpretation of + and — quantities, was not foreign to them. They advanced beyond Diophantus in observing that a quadratic has always two roots. Thus Bhaskara gives $\scriptstyle {x=50}$ and $\scriptstyle {x=-5}$ for the roots of $\scriptstyle {x^{2}-45x=250}$. "But," says he, "the second value is in this case not to be taken, for it is inadequate; people do not approve of negative roots." Commentators speak of this as if negative roots were seen, but not admitted.

Another important generalisation, says Hankel, was this, that the Hindoos never confined their arithmetical operations to rational numbers. For instance, Bhaskara showed how, by the formula $\scriptstyle {{\sqrt {a+{\sqrt {b}}}}={\sqrt {a+{\sqrt {a^{2}-b}} \over 2}}+{\sqrt {a-{\sqrt {a^{2}-b}} \over 2}}}$ the square root of the sum of rational and irrational numbers could be found. The Hindoos never discerned the dividing line between numbers and magnitudes, set up by the Greeks, which, though the product of a scientific spirit, greatly retarded the progress of mathematics. They passed from magnitudes to numbers and from numbers to magnitudes without anticipating that gap which to a sharply discriminating mind exists between the continuous and discontinuous. Yet by doing so the Indians greatly aided the general progress of mathematics. "Indeed, if one understands by algebra the application of arithmetical operations to complex magnitudes of all sorts, whether rational or irrational numbers or space-magnitudes, then the learned Brahmins of Hindostan are the real inventors of algebra."^[7]

Let us now examine more closely the Indian algebra. In extracting the square and cube roots they used the formulas $\scriptstyle {(a+b)^{2}=a^{2}+2ab+b^{2}}$ and $\scriptstyle {(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}}$. In this connection Aryabhatta speaks of dividing a number into periods of two and three digits. From this we infer that the principle of position and the zero in the numeral notation were already known to him. In figuring with zeros, a statement of Bhaskara is interesting. A fraction whose denominator is zero, says he, admits of no alteration, though much be added or subtracted. Indeed, in the same way, no change takes place in the infinite and immutable Deity when worlds are destroyed or created, even though numerous orders of beings be taken up or brought forth. Though in this he apparently evinces clear mathematical notions, yet in other places he makes a complete failure in figuring with fractions of zero denominator.

In the Hindoo solutions of determinate equations, Cantor thinks he can see traces of Diophantine methods. Some technical terms betray their Greek origin. Even if it be true that the Indians borrowed from the Greeks, they deserve great credit for improving and generalising the solutions of linear and quadratic equations. Bhaskara advances far beyond the Greeks and even beyond Brahmagupta when he says that "the square of a positive, as also of a negative number, is positive; that the square root of a positive number is twofold, positive and negative. There is no square root of a negative number, for it is not a square." Of equations of higher degrees, the Indians succeeded in solving only some special cases in which both sides of the equation could be made perfect powers by the addition of certain terms to each.

Incomparably greater progress than in the solution of determinate equations was made by the Hindoos in the treatment of indeterminate equations. Indeterminate analysis was a subject to which the Hindoo mind showed a happy adaptation. We have seen that this very subject was a favourite with Diophantus, and that his ingenuity was almost inexhaustible in devising solutions for particular cases. But the glory of having invented general methods in this most subtle branch of mathematics belongs to the Indians. The Hindoo indeterminate analysis differs from the Greek not only in method, but also in aim. The object of the former was to find all possible integral solutions. Greek analysis, on the other hand, demanded not necessarily integral, but simply rational answers. Diophantus was content with a single solution; the Hindoos endeavoured to find all solutions possible. Aryabhatta gives solutions in integers to linear equations of the form $\scriptstyle {ax\pm by=c}$, where a, b, c are integers. The rule employed is called the pulveriser. For this, as for most other rules, the Indians give no proof. Their solution is essentially the same as the one of Euler. Euder's process of reducing $\scriptstyle {\frac {a}{b}}$ to a continued fraction amounts to the same as the Hindoo process of finding the greatest common divisor of a and b by division. This is frequently called the Diophantine method. Hankel protests against this name, on the ground that Diophantus not only never knew the method, but did not even aim at solutions purely integral.^[7] These equations probably grew out of problems in astronomy. They were applied, for instance, to determine the time when a certain constellation of the planets would occur in the heavens.

Passing by the subject of linear equations with more than two unknown quantities, we come to indeterminate quadratic equations. In the solution of $\scriptstyle {xy=ax+by+c}$, they applied the method re-invented later by Euler, of decomposing $\scriptstyle {(ab+c)}$ into the product of two integers $\scriptstyle {m\cdot n}$ and of placing $\scriptstyle {x=m+b}$ and $\scriptstyle {y=n+a}$.

Remarkable is the Hindoo solution of the quadratic equation $\scriptstyle {cy^{2}=ax^{2}+b}$. With great keenness of intellect they recognised in the special case $\scriptstyle {y^{2}=ax^{2}+1}$ a fundamental problem in indeterminate quadratics. They solved it by the cyclic method. "It consists," says De Morgan, "in a rule for finding an indefinite number of solutions of $\scriptstyle {y^{2}=ax^{2}+1}$ (a being an integer which is not a square), by means of one solution given or found, and of feeling for one solution by making a solution of $\scriptstyle {y^{2}=ax^{2}+b}$ give a solution of $\scriptstyle {y^{2}=ax^{2}+b^{2}}$. It amounts to the following theorem: If p and q be one set of values of x and y in $\scriptstyle {y^{2}=ax^{2}+b}$ and $\scriptstyle {p'}$ and $\scriptstyle {q'}$ the same or another set, then $\scriptstyle {qp+pq'}$ and $\scriptstyle {app'+qq'}$ are values of x and y in $\scriptstyle {y^{2}=ax^{2}+1}$. From this it is obvious that one solution of $\scriptstyle {y^{2}=ax^{2}+1}$ may be made to give any number, and that if, taking b at pleasure, $\scriptstyle {y^{2}=ax^{2}+b^{2}}$ can be solved so that x and y are divisible by b, then one preliminary solution of $\scriptstyle {y^{2}=ax^{2}+1}$ 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 $\scriptstyle {y^{2}=ax^{2}+1}$ 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

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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 $\scriptstyle {\pi =3}$, also $\scriptstyle {={\sqrt {10}}}$; but Aryabhatta gives the value $\scriptstyle {\frac {31416}{10000}}$. Bhaskara gives two values,—the 'accurate,' $\scriptstyle {\frac {3927}{1250}}$, and the 'inaccurate,' Archimedean value, $\scriptstyle {\frac {22}{7}}$. A commentator on Lilavati says that these values were calculated by beginning with a regular inscribed hexagon, and applying repeatedly the formula $\scriptstyle {AD={\sqrt {2-{\sqrt {4-{\overline {AB^{2}}}}}}}}$, 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 $\scriptstyle {\pi }$.

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 $\scriptstyle {\pi }$ 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 correspond to $\scriptstyle {3{\frac {3}{4}}}$ 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 $\scriptstyle {\sin ^{2}a+\cos ^{2}a=r^{2}}$, they obtained sin 45°$\scriptstyle {={\sqrt {\frac {r^{2}}{2}}}=2431}$. Substituting for cos a its equal $\scriptstyle {\sin(90-a)}$, and making $\scriptstyle {a=60}$°, they obtained sin 60°$\scriptstyle {={\frac {\sqrt {3r^{2}}}{2}}=2978}$. With the sines of 90, 60, 45, and 30 as starting-points, they reckoned the sines of half the angles by the formula $\scriptstyle {{\text{ver }}\sin 2a=2\sin ^{2}a}$, thus obtaining the sines of 22° 30′, 11° 16′, 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&prime'; 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 $\scriptstyle {a-b=b-c=}$3° 45′, then $\scriptstyle {\sin a-\sin b=(\sin b-\sin c)-{\frac {\sin b}{225}}}$. This formula was 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 judge 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.