Regula Falsi. The rule of false position is found in all the Hindu works.1 Bhâskara II gives prominence to the method and calls it iṭṣa-karma ("rule of supposition"). He describes the method thus:
"Any number, assumed at pleasure, is treated as specified in the particular question, being multiplied and divided, increased or diminished by fractions (of itself); then the given quantity, being multiplied by the assumed number and divided by that (which has been found) yields the number sought. This is called the process of supposition."2
Śrȋdhara takes the assumed number to be one.3 Mahâvȋra gives a large variety of problems to which he applies the rule.4 Gaṇeśa in his commentary on the Lȋlâvatȋ remarks, "In this method, multiplication, division, and fractions only are employed." The following examples will illustrate the nature of the problems solved by the rule of supposition:
(1) Out of a heap of pure lotus flowers, a third, a fifth, a sixth were offered respectively to the gods Śiva, Viṣṇunited and Sûrya and a quarter was presented to Bhavânȋ. The remaining six were given to the venerable preceptor. Tell quickly the number of lotuses.5
(2) The third part of a necklace of pearls, broken in Page 231 an amorous struggle, fell to the ground; its fifth part rested on the couch; the sixth part was saved by wench; and the tenth part was taken by her lover; six pearls remained strung. Say, of how many pearls was the necklace composed?1
(3) One-twelfth part of a pillar, as multiplied by 1⁄30 part thereof, was to be found under water; 1⁄20 of the remainder, as multiplied by 3⁄16 thereof, was found buried in the mire below; and 20 hasta of the pillar were found in the air (above the water). O friend, give out the length of the pillar.2
(4) A number of parrots descended on a paddy field, beautiful with crops bent down through the weight of ripe corn. Being scared away by men, all of them suddenly flew off. One-half of them went to the east, one-sixth went to the south-east; the difference between those that went to the east and those that went to the south-east, diminished by half of itself and again diminished by half of this (resulting difference), went to the south. The difference between those that went to the south and those that went to the south-east diminished by two-fifths of itself went to the south-west; the difference between those that went to the south and those that went to the south-west, went to the west; the difference between those that went to the south-west and those that went to the west, together with three-sevenths of itself went to the north-west; the difference between those that went to north-west and those that went to the west together with seven-eighths of itself, went to the north; the sum of those that went to the north-west and those that went to the north, diminished by two-thirds of itself went to the north-west; and 280 parrots were found to Page 232 remain in the sky above. How many were the parrots in all?1
The Method of Inversion. the method of inversion called vilomagati ("working backwards") is found to have been commonly used in India from very early times. Thus Âryabhaṭa I says:
"In the method of inversion multipliers become divisors and divisors multipliers, addition becomes subtraction and subtraction becomes addition."2
Brahmagupta's description is more complete. He says:
"Beginning from the end, make the multiplier divisior, the divisor multiplier; (make) addition subtraction and subtraction addition; (make) square square-root, and square-root square; this gives the required quantity."3
The following examples will illustrate the nature of problems solved by the above method:
(1) What is the quantity which when divided by 7, (then) multiplied by 3, (then) squared, (then) increased by 5, (then) divided by 3⁄5, (then) halved, and then reduced to its square-root happens to be the number 5?4
(2) The residue of degrees of the sun less three, being divided by seven, and the square-root from the quotient extracted, and the root less eight multiplied by nine, and to the product one being added, the amount is Page 233 a hundred. When does this take place on a Wednesday?1
Problems on Mixture. The Hindu works on pâtȋgaṇita contain a chapter relating to problems on mixture (miśraka-vyavahâra). Miscellaneous problems on interest, problems on allegation, and various other types of problems, in which quantities are to be separated from their mixture, form the subject matter of miśraka-vyavahâra. A chapter "on mixture" (De' mescolo) is found in early Italian works on arithmetic, evidently under Hindu influence.2
Some of the problems of this chapter are determinate and some are indeterminate. A few relating to interest and allegation have already been given.3 The following are some others:
(1) In the interior of a forest, 3 heaps of pomegranates were divided (equally) among 7 travellers, leaving 1 fruit as remainder; 7 (of such heaps) were divided among 9, leaving a remainder of 3 (fruits), again 5 (of such heaps) were divided among 8, leaving 2 fruits as remainder. O mathematician, what is the numerical value of a heap?4
(2) On a certain man bringing mango fruits home, his elder son took one fruit first and then half of what remained. The younger son did similarly with what was left. He further took half of what was left thereafter; and the other took the other half. Find the number of fruits brought by the father?5
Page 234 (3) A certain lay follower of Jainism went to a Jina temple with four gate-ways, and having taken (with him) fragrant flowers offered them in worship with devotion (at each gate). The flowers in his hand were doubled, trebled, quadrupled and quintupled (respectively in order) as he arrived at the gates (one after another). The number of flowers offered by him was sixty1 at each gate. How many flowers were originally taken by him?
(4) The first man has 16 azure-blue gems, the second has 10 emeralds, and the third has 8 diamonds. Each among them gives to each of the others 2 gems of the kind owned by himself; and then all three men come to be possessed of equal wealth. What are the prices of those azure-blue gems, emeralds and diamonds?2
(5) In what time will four fountains, being let loose together, fill a cistern, which they would severally fill in a day, in half a day, in a quarter and in a fifth part of a day?3
Problems involving Solution of Quadratic Equations. The solution of the quadratic equation has been known in India from the time of Âryabhaṭa I (499). Problems on interest requiring solution of the quadratic equation have already been mentioned. Mahâvȋra and Bhâskara II give many other problems. Mahâv&3523;ra divides those problems into two classes: (i) those that involve square-roots (mûla) and (ii) those Page 235 that involve square (varga) of the unknown. The first type gives a single positive answer, while the second type has two answers corresponding to two roots of the quadratic. Bhâskara II deals with the first type of problems only in his pâṭȋgaṇita, the Lȋlâvatȋ. The second type of problems, involving the square of the unknown has been treated by him in his Bȋjagaṇita (algebra). The following examples will illustrate the nature and scope of such problems:
(1) One-fourth of a herd of camels was seen in the forest; twice the square-root -from that had gone to mountain-slopes; and three times five camels were found to remain on the bank of a river. What was the numerical measure of that herd of camels?1
(2) Five and one-fourth times the square-root (of a herd) of elephants are sporting on a mountain slope; five-ninths of the remainder sport on the top of the mountain; five times the square-root of the remainder sport in a forest of lotuses; and there are six elephants then (left) on the bank of a river. How many are the elephants?2
(3) In a garden beautified by groves of various kinds of trees, in a place free from all living animals, many Page 236 ascetics were seated. Of them the number equivalent to the square-root of the whole collection were practising yoga at the foot of a tree. One-tenth of the remainder, the square-root (of what remained after this), 1⁄9 (of what remained after this), then the square-root (of what remained after this), 1⁄8 (of what remained after this), the square-root (of what remained after this), 1⁄7 (of what remained after this), the square-root (of what remained after this), 1⁄6 (of what remained after this), the square-root (of what remained after this), 1⁄5 (of what remained after this), the square-root (of what remained after this)—these parts consisted of those who were learned in the teaching of literature, in religious law, in logic, and in politics, as also of those who were versed in controversy, prosody, astronomy, magic, rhetoric and grammar, as well as of those who possessed an intelligent knowledge of the twelve varieties of the aṅga-śâstra; and at last 12 ascetics were seen (to remain without being included among those mentioned before). O excellent ascetic, of what numerical value was this collection of ascetics?1
(4) A single bee (out of a swarm of bees) was seen in the sky; 1⁄5 of the remainder (of the swarm) and 1⁄4 of the remainder (left thereafter) and again 1⁄3 of the remainder (left thereafter) and a number of bees equal to the square-root of the numerical value of the swarm, were seen in lotuses; and two bees were on a mango tree. How many were there?2
(5) Four times the square-root of half the number of a collection of boars went to a forest wherein tigers Page 237 were at play; 8 times the square-root of 1⁄10 of the remainder went to a mountain; and 9 times the square-root of 1⁄2 of the (next) remainder went to the bank of a river; and boars equivalent in (numerical) measure to 56 were seen to remain in the forest. Give the numerical measure of all those boars.1
(6) The sum of two (quantities, which are respectively equivalent to the) square-root (of the numerical value) of a collection of swans and (the square-root of the same collection) as combined with 68, amounts to 34. How many swans there are in that collection?2
(7) Pârtha (Arjuna), irritated in a fight, shot a quiver of arrows to slay Karṇa. With half his arrows, he parried those of his antagonist, with four times the square-root of the quiver-full, he killed his horses; with three he demolished the umbrella, standard and bow; and with one he cut off the head of his foe. How many were the arrows, which Arjuna let fly?3
(8) The square-root of half the number of a swarm of bees is gone to a shrub of jasmin; and so are eight-ninths of the whole swarm; a female is buzzing to one remaining mail that is humming within a lotus, in which he is confined, having been allured to it by its Page 238 fragrance at night. Say, lovely woman, what is the number of bees.1
(9) One-twelfth part of a pillar, as multiplied by 1⁄30 part thereof, was found under water; 1⁄20 of the remainder, as multiplied by 3⁄16 thereof, was found buried in the mire, and 20 hasta of the pillar were found in the air. O friend, give the measure of the length of the pillar.2
(10) A number of elephants (equivalent to) 1⁄10 of the herd minus 2, as multiplied by the same (1⁄10 of the herd minus 2), is found playing in a forest of sallakȋ trees. The remaining elephants of the herd equal in number to the square of 6 are moving on a mountain. How many are the elephants?3
Page 230 1 The method originated in India and went to Europe through Arabia. There is a mediæval MS., published by Libri in his Histoire, I, 304 and possibly due to Rabbi ben Ezra in which the method is attributed to the Hindus. For further details and references, see Smith, History, II, p. 437, foot-note 1.
2 L, p. 10.
3 See the rule on stambhoddeśa, Triś, p. 13.
4 These problems occur in chapters iii and iv of the Gaṇita-sâra-saṁgraha.
5 L, p. 11. Cf. GSS, p. 48 (7).
Page 231 1 Triś, p. 14, cf. GSS, p. 49 (17–22) for a similar example.
2 GSS, p. 55(60). Cf. Triś, p. 13.
Page 232 1 GSS, pp. 48f (12–16).
2 Â, Gaṇitapâda, 28.
3 BrSpSi, p. 301. The method occurs also in GSS, p. 102 (286); MSi, p. 149; L, p. 9; etc.
4 GSS, p. 102 (287). Examples of this type are very common in Hindu arithmetic. They were also very common in Europe. Smith in his History, II, quotes two such problems from an American arithmetic of the 16th century.
Page 233 1 Colebrooke, cha, p. 333 (18).
2 Smith, History, II, p. 588, note 4.
3 See commercial problems, pp. 216ff; also problems on proportionate division (prakṣepa-karaṇa): Triś, p. 26; GSS, p. 75(79½); MSi, pp. 154–155.
4 GSS, p. 82 (128½). Such problems are given under the rule of vallikâ-kuṭṭȋkâra by Mahâvȋra.
5 GSS, p. 82 (131½).
Page 234 1 GSS, p. 79 (112½–113½). The printed text has pañca ("five"). According to it the answer is 43/12 which appears absurd. There are some other problems in the printed edition which give such absurd results. All those are, we presume, due to the defects of the mss. consulted by the editor. So here we have made the emendation ‘sixty.’
2 GSS, p. 87 (165–166).
3 BrSpSi, p. 177 (com.); L, p. 23.
Page 235 1 GSS, p. 51 (34). The problem belongs to the type of the mûla-jâti, and leads to an equation of the form x−(bx+c√x+a)=0. The method of solution is given in GSS, p. 50 (33).
2 GSS, p. 52 (46). The problem is one of the śeṣa-mûla variety. It gives the equation
Page 236 1 GSS, p. 52 (42–45). The problem is of same variety as the above one. The substitution will have to be made 6 times to reduce the resulting equation.
2 GSS, p. 53 (48). This problem is of the dviragra-śeṣa-mûla variety.
2 GSS, p. 56 (68). This problem is of the mûla-miŚra variety, wherein the sum of square-roots is involved. It gives an equation of the form √x + √x±d = m.
3 L, p. 16.
Page 238 1 L, p. 16.
2 GSS, p. 55 (60). The problem gives the equation
3 GSS, p. 55 (63).
All content from History of Hindu Mathematics