Now, the rule of operation for the procedure of inversion, in two verses:
48. In finding a quantity [i.e., an operand] when [the result] is given, make a divisor [into] a multiplier, a multiplier a divisor, a square a square-root, a square-root a square, a negative [quantity] a positive [one], [and] a positive [quantity] a negative [one].
49. But if it was increased or decreased by its own part, the denominator, increased or decreased by its [own] numerator, is the [corrected] denominator, and the numerator is unchanged. Then the rest [of the procedure] in inversion is as stated [above].
Here is an example:
50. Tell [me], quick-eyed girl, if you know the correct procedure for inversion, the number which, multiplied by three, added to three-fourths of the result, divided by seven, diminished by one-third of the result, multiplied by itself, decreased by fifty-two, having its square-root taken, increased by eight, and divided by ten, produces two.
Statement: The multiplier is 3; [the quantity] added, 3/4; the divisor, 7; [the quantity] subtracted, 1/3; the square; [the quantity] subtracted, 52; the square-root; [the quantity] added, 8; the divisor, 10; the given [result], 2. By means of the stated procedure, the resulting quantity is 28.
That is the rule for inversion.
Now, the rule of operation for methods of assumption [of some arbitrary quantity], with reduction of given [quantities] and remainders and simplification of fractional differences, in one verse:
51. As in the statement of the example, any desired number is multiplied, divided, or decreased and increased by fractions. The given quantity, multiplied by the desired [number] and divided by that [result], is the [required] quantity; [that] is called the operation with an assumed [quantity].
Example:
52. What quantity, multiplied by five, diminished by its own one-third, divided by ten, increased by one-third, one-half, and one-fourth of the [original] quantity, is seventy minus two?
Statement: The multiplier is 5; its own part is negative,
| 0 |
| 1/3 |
The quantity assumed here is 3. [It is] multiplied by five (15), decreased by its own third part (10), [and] divided by ten (1). [It is] added to the third, half, and quarter (3/3, 3/2, 3/4) of the quantity assumed here (3); the result is 17/4. The given [answer], 68, multiplied by the assumed [number], is divided by that. The resulting quantity is 48.
So however the [unknown] quantity in an example [when] multiplied or divided by something, or decreased or increased by a part of the quantity, [becomes] the given [answer], that is the way that [some] imagined given quantity [is transformed] via the operation in the [above-]mentioned explanation. Then divide the given [answer], multiplied by the assumed [quantity], by whatever [result] is obtained. The result [of that] is the [desired] quantity.
Now, an example [involving] reduction of the given [answer]:
53. One-third, one-fifth, and one-sixth of a quantity of spotless lotuses [were] offered to Siva, Visnu, and the Sun, and one-fourth to Parvati. The remaining six lotuses [were] offered at the feet of the teacher. Quickly say the number of all the lotuses.
Statement: 1/3, 1/5, 1/6, 1/4. The given [answer] is 6.
Taking the assumed quantity here [to be] unity (1) [and proceeding] as [explained] previously, the resulting quantity is 120.
Now, an example involving the reduction of the remainder.
54. A traveler on a pilgrimage gave one-half [of his money] at Prayaga, two-ninths of the rest at Kasi, one-fourth of the remainder in toll fees, and six-tenths of the remainder at Gaya. Sixty-three niskas [were] left over, [and he] returned with that to his own home. Tell [me] the [initial] amount of his money, if [the method of] reduction of remainders is clear [to you].
Statement: 1/1, 1/2, 2/9, 1/4, 6/10. The given [answer] is 63. Taking the assumed quantity here [to be] unity (1), subtracting the numerator from its denominator, multiplying the denominators together, and proceeding as explained, the resulting remainder is 7/60. Dividing this into the given [answer] 63, multiplied by the assumed [quantity], the resulting amount of money is 540. This may also be done by the rule for inversion.
55. Out of a swarm of bees, one-fifth went to a kadamba-flower, one-third to a plantain-flower, and three times the difference of those, O doe-eyed one, to a kutaja-flower. One remaining bee, tempted at the same time by the scent of a jasmine and a pandanus, hovered and wandered in the air; tell me, beloved, the number of bees.
Statement: 1/5, 1/3, 2/5. The given [answer] is 1. The resulting number of the swarm of bees is 15. And in other [examples, the procedure is] the same.
That is the method of the assumed [quantity].
Now, the rule of operation for combination, in half a verse:
56. The sum is [separately] diminished and increased by the difference and [in each case] halved; that is called combination.
Here is an example:
57. Tell me, dear child, the two quantities whose sum is a hundred and one, and whose difference is twenty-five, if you understand combination.
Statement: The sum is 101, the difference 25; the two resulting quantities are 38, 63.
Now, the rule of operation for combination involving squares, in half a verse:
58. The difference of the squares [of the quantities] divided by the difference of the quantities [themselves] is the sum; from that, the two quantities [can be found] as previously stated.
Example:
59. Tell [me] quickly, intelligent calculator, what are the quantities whose difference is eight, and the difference of whose squares is four hundred?
Statement: The difference of the quantities is 8, the difference of [their] squares 400. The resulting two quantities are 21, 29.
That is the procedure with dissimilar [givens, i.e., difference of the quantities and difference of the squares].
Now a certain operation [for producing] squares is explained:
60. The square of an assumed number, multiplied by eight and decreased by one, then halved and divided by the assumed number, is one quantity; its square, halved and added to one, is the other.
61. Or else, one divided by double an assumed number and added to that is the first quantity, and one is the other. These give pairs of quantities, the sum and difference of whose squares, when decreased by one, are squares.
Example:
62. Tell [me], my friend, the numbers whose squares, subtracted and [separately] added [to each other] and [then] diminished by one, [produce] square-roots [i.e., are perfect squares]; [a problem] with which those skilled in algebra, [who] have gone beyond the algebra [techniques] called "six-fold," torment the dull-witted girls. %% f.??
In the first rule, the assumed quantity is considered [to be] 1/2. Its square 1/4, multiplied by eight, is 2. This, decreased by one (1) and halved, is 1/2. It is divided by the assumed quantity 1/2; the resulting first quantity is 1.
Its square (1) is halved (1/2) and increased by 1 (3/2). That is the other quantity. So the two quantities are 1/1, 3/2. In the same way, with one as the assumed quantity, the two numbers are 7/2, 57/8; with two, 31/4, 993/32.
Now in the second rule, the assumed quantity is 1. Unity (1) is divided by that multiplied by 2: 1/2. Added to the assumed quantity, the resulting first quantity is 3/2; the second is unity, 1. Thus the two quantities are 3/2, 1/1.
In the same way, with two as the assumed quantity, [they are] 9/4, 1/1; with three, 19/6, 1/1; with one-third, 11/6, 1/1.
Or, [the following] rule:
63. The square of the square of an assumed number, and the cube of that number, each multiplied by eight and the first product increased by one, are such quantities, in the manifest [arithmetic] just as in the unmanifest [algebra].
The assumed quantity is 1/2. The square of its square, 1/16, multiplied by eight (1/2), is added to 1; the resulting first quantity is 3/2. Again, the assumed quantity is 1/2; its cube, 1/8, is multiplied by eight. The resulting second quantity is 1/1. Thus the two quantities are 3/2, 1/1.
Now with one as the assumed quantity, [they are] 9, 8; with two, 129, 64; with three, 649, 216. And [computation can be done] to an unlimited extent in this way in all procedures, by means of assumed quantities.
64. Algebra, [which is] equivalent to the rules of arithmetic, appears obscure, but it is not obscure to the intelligent; and it is [done] not in six ways, but in many. Concerning the [rule of] three quantities, [with all of] arithmetic and algebra, the wise [have] a clear idea [even] about the unknown. Therefore, it is explained for the sake of the slow[-witted].
Now, the rule of operation for the multiplier of the square-root, in two verses:
65. The sum or difference of a quantity and a multiple of its square-root is given and the square of half the multiplier is added to the given number. The square root of their sum, increased or decreased by half the multiplier and then squared, is the desired quantity.
66. If that quantity is decreased or increased by a fraction, divide the given number and the multiplier of the root by one minus or plus the fraction. Then with those [results] the quantity is found as stated earlier.
A quantity minus its own square-root multiplied by some number is given; the square-root of [an amount equal to] that plus the square of half the multiplier of [the quantity's] square-root, is added to half the multiplier (or decreased by it, if the given number was increased [instead of decreased] by its own square-root times the multiplier). The square of that [result] is the [original] quantity.
An example of the given diminished by its square-root in that way:
67. O maiden, one pair of geese playing in the water saw seven-halves of the square-root [of the total] of the flock, growing tired of playing, go to the shore. Tell me the number of the flock of geese.
Statement: the multiplier of the root is 7/2, the given number 2. From that given number 2, increased by the square of half the multiplier 49/16 [to give] 81/16, the square-root is 9/4. Added to half the multiplier, 7/4, [it is] 4. Squared, it produces the number of the flock of geese, 16.
Now an example of the given increased by its square-root in that way:
68. Learned one, what is the number which, added to nine times its square-root, is twelve hundred plus forty?
Statement: the multiplier of the square-root is 9, the given number 1240. By the stated method, the resulting quantity is 961.
Example:
69. Out of a flock of geese, ten times the square-root [of the total] went to the Manasa lake when a cloud approached, one-eighth went to a forest filled with hibiscus, and three couples were seen playing in the water. Tell me, maiden, the number of the flock.
Statement: the multiplier of the root is 10, the fraction 1/8, the given number 6. "If that quantity is decreased or increased by a fraction": here, after dividing the given number and the multiplier of the root by one (1) minus the fraction (7/8), the given number becomes 48/7, the multiplier of the root 80/7. With these, proceeding by the first rule, the resulting number of the flock is 144.
Example:
70. Enraged in the battle, Partha [Arjuna] shot a host of arrows to kill Karna. With half the arrows he turned aside the host of arrows of that [opponent]; with four times the square-root of the total, he killed his horses; with six arrows, he killed [his charioteer] Salya; then with three arrows he destroyed the umbrella, the banner, and the bow [of his enemy], and with one, he cut off his head. How many arrows did Arjuna shoot? [This is from an episode in the Mahabharata.]
Statement: the multiplier of the root is 4, the fraction 1/2, the given number 10. "If that quantity is decreased or increased by a fraction": with that first rule, the resulting number of arrows is 100.
71. The square-root of one-half of a swarm of bees and eight-ninths of the swarm went to a jasmine shrub, and one female is buzzing to one remaining male buzzing trapped within a lotus, to which its fragrance attracted him at night. Say, beloved, the number of bees.
Here eight-ninths of the quantity, and the square-root of its half, are negative with respect to the quantity. The given number is two, [and] it is negative. And the given number, halved, is [the source of] half the desired quantity.
Thus the statement: the multiplier of the root is .1/2, the fraction .8/9, the given number 1. Here, as [explained] above, the result is half the quantity, 36. This times two is the number of the swarm of bees, 72.
An example where the given [answer] is added to a root and a fraction:
72. Find quickly the quantity that, added to eighteen [times] its own square-root and to one-third of the quantity, [gives] the result twelve hundred, if you [have] skill in arithmetic.
Statement: The multiplier of the root is 18; the fraction, 1/3; the given [answer] 1200. Then divide the given [answer], multiplied by the root, by one plus the fraction (4/3); [proceeding] as [stated] above, the resulting quantity is 576.
Now, the rule of operation for the [rule] of three quantities, in one verse:
73. The [given] amount and the desired [amount], [being of] the same type, are [written] in the first and last [positions, respectively]. The result of that [given amount], [being of] a different type, is [put] in the middle. That, multiplied by the desired [amount] and divided by [the given amount in] the first [position], is the result of the desired [amount]. In the inverse [rule of three quantities], the procedure is reversed.
Example:
74. If two and a half palas of saffron are obtained for three-fourths of a niska, tell me at once, best of merchants, how much of that [can be bought] with nine niskas.
Statement:
| 3/7 | 5/2 | 9/1 |
And also:
75. If a hundred plus four niskas are obtained for sixty-three palas of good-quality camphor, then think [and] tell [me], friend, what [can be obtained] with twelve and a quarter palas?
Statement:
| 63 | 104 | 49/4 |
And also:
76. If one and one-eighth kharikas of rice can be obtained for two drammas, say at once what [amount] of that [can be bought] for seventy panas.
Here, in order [to use] the operation of reduction on the amount, the statement of the two drammas converted to panas is
| 32 | 9/8 | 70 |
Now the rule of operation in the inverse [rule of] three quantities:
77. [Sometimes] decrease in the result occurs when there is increase in the desired [quantity], or [vice versa, i.e.] increase when decrease. So the inverse [rule of] three quantities should be known by those who understand calculation.
When there is decrease in the result when there is increase in the desired [quantity], or increase in the result in the case of decrease [in the desired quantity], then the inverse [rule of] three quantities [is used]. That is as follows:
78. In the case of the cost of living beings [according to their] age, and the weight and alloy of gold, and the subdivision of amounts, the [rule of] three quantities should be inverted.
An example involving the price with respect to the age of a living being:
79. If a woman [slave] sixteen years old is bought for [a price of] 32, what is [the price of one] twenty years old? [If] an ox after two years of labor is bought for four niskas, then what is [the price of one] after six years of labor?
Statement:
| 16 | 32 | 20 |
| 25 |
| 3/4 |
Statement of the second [problem]:
| 2 | 4 | 6 |
| 1 |
| 1/3 |
An example involving the weight of impure gold:
80. If a gadyanaka of gold with an alloy of ten is obtained for one niska, then say how much [is the price of the same amount] with an alloy of fifteen.
Statement:
| 10 | 1 | 15 |
Example involving subdivision of amounts:
81. If a heap of grain measured out with a measure of seven adhakas produced a hundred [of those] measures, then what [will it be] with a measure of five adhakas?
Statement:
| 7 | 100 | 5 |
That is the inverse [rule of] three quantities.
Now, the rule of operation in the [rule of] five or more quantities, in one verse:
82. In the case of five, seven, nine, etc., quantities, reverse the result and the divisors. When the product of the larger [set] of terms is divided by the product of the smaller [set] of terms, [that quotient] is the result.
Here is an example:
83. If the interest on one hundred for a month is five, tell [me] what is [the interest] on sixteen when a year has passed? Also, tell [me], mathematician, the time from the principal and the interest, and the amount of the principal when the time and the result are known.
Statement:
| 1 | 12 |
| 100 | 16 |
| 5 |
| 9 |
| 3/5 |
Now, in order to know the time, the statement is
| 1 | |
| 100 | 16 |
| 5 | 48/5 |
In order [to find] the amount of the principal, the statement is
| 1 | 12 |
| 100 | |
| 5 | 48/5 |
84. If the interest on one hundred for a month and one-third is five and one-fifth, say what is the interest on sixty-two and one-half for three months and one-fifth?
Statement:
| 4/3 | 16/5 |
| 100 | 125/2 |
| 26/5 |
The resulting interest is
| 7 |
| 4/5 |
Now, an example of the [rule of] seven quantities:
85. If eight colored pieces of cloth of best-quality silk, three karas in width and eight karas in length, are obtained for one hundred, then say quickly, merchant, what another such piece, three and a half karas long and half a hasta wide, will cost, if you understand business.
Statement:
| 3 | 1/2 |
| 8 | 7/2 |
| 8 | 1 |
| 100 |
Now, an example of the [rule of] nine quantities:
86. If thirty benches, twelve angulas thick, the square of four angulas in width, and fourteen karas long, cost one hundred, then tell me, friend, what will be the cost [of] fourteen benches which are four [units] less in width, length, and thickness?
Statement:
| 12 | 8 |
| 16 | 12 |
| 14 | 10 |
| 30 | 14 |
| 100 |
| 16 |
Now, an example of the [rule of] eleven quantities:
87. If the cost of renting carts to carry the benches of the size first described for a distance of one gavyuti is eight drammas, tell [me] what is the cost of renting carts to carry the other benches, described subsequently [as] four [units] less [in each dimension], for six gavyutis?
Statement:
| 12 | 8 |
| 16 | 12 |
| 14 | 10 |
| 30 | 14 |
| 1 | 6 |
| 8 |
Now, the rule of operation in barter, in half a verse:
88. The same [process of] transposition is also [done] in barter, [and] there the two prices always [are exchanged] too.
Example:
89. If three hundred mangoes can be bought in this market for a dramma, and thirty ripe pomegranates for a pana, say quickly, friend, how many pomegranates are obtained by barter for ten mangoes?
Statement:
| 16 | 1 |
| 300 | 30 |
| 10 |