# Indian Mathematics/Appendices—

APPENDIX I.
Extracts from Texts.
The Śulvasūtras.
1. In the following we shall treat of the different manners of building the agni. 2. We shall explain how to measure out the circuit of the area required for them.
45. The cord stretched across a square produces an area of twice the size.
46. Take the measure for the breadth, the diagonal of its square for the length: the diagonal of that oblong is the side of a square the area of which is three times the area of the square.
48. The diagonal of an oblong produces by itself both the areas which the two sides of the oblong produce separately.
49. This is seen in those oblongs whose sides are three and four, twelve and five, fifteen and eight, seven and twenty-four, twelve and thirty-five, fifteen and thirty-six.
51. If you wish to deduct one square from another square cut off a piece from the larger square by making a mark on the ground with the side of the smaller square which you wish to deduct. Draw one of the sides across the oblong so that it touches the other side. Where it touches there cut off. By this line which has been cut off the small square is deducted from the large one.
58. If you wish to turn a square into a circle draw half of the cord stretched in the diagonal from the centre towards the prāchī line. Describe the circle together with the third part of that piece of the cord which stands over.
Āryabhata's gaņita—(Circa. A.D. 520).
6. The area of a triangle is the product of the perpendicular common to the two halves and half the base. Half the product of this and the height is the solid with six edges.
10. Add four to one hundred, multiply by eight and add again sixty-two thousand. The result is the approximate value of the circumference when the diameter is twenty thousand.
13. The circle is produced by a rotation; the triangle and the quadrilateral are determined by their hypotenuses; the horizontal by water and the vertical by the plumb line.
29. The sum of a certain number of terms diminished by each term in succession added to the whole and divided by the number of terms less one gives the value of the whole.
Brahmagupta—(Born A.D. 598).
1. He who distinctly knows addition and the rest of the twenty operations and the eight processes including measurement by shadows is a mathematician.
14. The principal multiplied by its time and divided by the interest, and the quotient being multiplied by the factor less one is the time. The sum of principal and interest divided by unity added to the interest on unity is the principal.
17. The number of terms less one multiplied by the common difference and added to the first term is the amount of the last. Half the sum of the last and first terms is the mean amount, and this multiplied by the number of terms is the sum of the whole.
21. The product of half the sides and opposite sides is the rough area of a triangle or quadrilateral. Half the sum of the sides set down four times and each lessened by the sides being multiplied together—the square-root of the product is the exact area.
40. The diameter and the square of the radius respectively multiplied by three are the practical circumference and area. The square-roots extracted from ten times the square of the same are the exact values.
62. The integer multiplied by the sexagesimal parts of its fraction and divided by thirty is the square of the minutes and is to be added to the square of the whole degrees.
101. These questions are stated merely for gratification. The proficient may devise a thousand others or may solve by the rules taught problems set by others.
102. As the sun obscures the stars so does the expert eclipse the glory of other astronomers in an assembly of people by reciting algebraic problems, and still more by their solution.
Mahāvīra's Gaņita-Sāra-Sangraha—(Circa. A.D. 850).
i. 13–14. The number, the diameter and the circumference of islands, oceans and mountains; the extensive dimensions of the rows of habitations and halls belonging to the inhabitants of the world, of the interspace, of the world of light, of the world of the gods and to the dwellers in hell, and miscellaneous measurements of all sorts—all these are made out by means of computation.
vi. 147. Divide by their rate prices. Diminish by the least among them and then multiply by the least the mixed price of all the things and subtract from the given number of things. Now split up (this) into as many (as there are left) and then divide. These separated from the total price give the price of the dearest article of purchase. [This is a solution of example 36 below.]
vi. 169. It has to be known that the products of gold as multiplied by their colours when divided by the mixed gold gives rise to the resulting colour (varņa). [See examples 24 and 25 below.]
vii. 2. Area has been taken to be of two kinds by Jina in accordance with the result—namely, that which is for practical purposes and that which is minutely accurate.
vii, 233. Thus ends the section of devilishly difficult problems.
Śrīdhara's Triśatikā—(Circa A.D. 1030).
1. Of a series of numbers beginning with unity and increasing by one, the sum is equal to half the product of the number of terms and the number of terms together with unity.
32. In exchange of commodities the prices being transposed apply the previous rule (of three). With reference to the sale of living beings the price is inversely proportional to their age.
65. If the gnomon be divided by twice the sum of the gnomon and the shadow the fraction of the day elapsed or which remains will be obtained.
Bhāskara—(Born A.D. 1114).
L. 1. I propound this easy process of calculation, delightful by its elegance, perspicuous with concise terms, soft and correct and pleasing to the learned.
L. 139. A side is put. From that multiplied by twice some assumed number and divided by one less than the square of the assumed number a perpendicular is obtained. This being set aside is multiplied by the arbitrary number and the side as put is subtracted—the remainder will be the hypotenuse. Such a triangle is termed 'genuine.'
L. 189. Thus, with the same sides, may be many diagonals in the quadrilateral. Yet, though indeterminate, diagonals have been sought as determinate by Brahmagupta and others.
L. 213. The circumference less the arc being multiplied by the arc the product is termed 'first.' From the quarter of the square of the circumference multiplied by five subtract that first product. By the remainder divide the first product multiplied by four times the diameter. The quotient will be the chord.
V. 170. In the like suppositions, when the operation, owing to restriction, disappoints the answer must by the intelligent be discovered by the exercise of ingenuity. Accordingly it said: 'The conditions—a clear intellect, assumption of unknown quantities, equation, and the rule of three—are means of operation in analysis.'
V. 224. The rule of three terms is arithmetic; spotless understanding is algebra. What is there unknown to the intelligent? Therefore for the dull alone it is set forth.
V. 225. To augment wisdom and strengthen confidence, read, read, mathematician, this abridgement elegant in style, easily understood by youth, comprising the whole essence of calculation and containing the demonstration of its principles—full of excellence and free from defect.
APPENDIX II.
Examples.
1. One-half, one-sixth, and one-twelfth parts of a pole are immersed respectively under water, clay, and sand. Two hastas are visible. Find the height of the pole?
Answer—8 hastas. S. 23.
2. The quarter of a sixteenth of the fifth of three-quarters of two-thirds of half a dramma was given to a beggar by a person from whom he asked alms. Tell how many cowries the miser gave if thou be conversant in arithmetic with the reduction termed sub-division of fractions?
Answer—1 cowrie. L. 32.
(1,280 cowries=1 dramma).
3. Out of a swarm of bees one-fifth settled on a blossom of kadamba, one-third on a flower of sīlīndhrī, three times the difference of those numbers flew to a bloom of kutaja. One bee, which remained, hovered and flew about in the air, allured at the same moment by the pleasing fragrance of a jasmine and pandanus. Tell me, charming woman, the number of bees?
Answer—15. L. 54, V. 108.
4. The third part of a necklace of pearls broken in an amorous struggle fell on the ground. Its fifth part was seen resting on the couch, the sixth part was saved by the lady and the tenth part was taken up by her lover. Six pearls ​remained on the string. Say, of how many pearls the necklace was composed?
Answer—30. S. 26.
5. A powerful, unvanquished, excellent black snake which is 32 hastas in length enters into a hole $\scriptstyle {7{\frac {1}{2}}}$ añgulas in $\scriptstyle {\frac {5}{14}}$ of a day, and in the course of a quarter of a day its tail grows by $\scriptstyle {2{\frac {3}{4}}}$ añgulas. O ornament of arithmeticians, tell me by what time this same enters fully into the hole?
Answer—$\scriptstyle {76{\frac {4}{5}}}$ days. M. v, 31.
(24 añgulas=l hasta.)
6. A certain person travels at the rate of 9 yojanas a day and 100 yojanas have already been traversed. Now a messenger sent after goes at the rate of 13 yojanas a day. In how many days will he overtake the first person?
Answer—25. M. vi, 327.
7. A white-ant advances 8 yavas less one-fifth in a day and returns the twentieth part of an añgula in 3 days. In what space of time will one, whose progress is governed by these rates of advancing and receding proceed 100 yojanas?
Answer—98042553 days. C. 283.
(8 yavas=1 añgula, 768000 añgulas=1 yojana).
8. Twenty men have to carry a palanquin two yojanas and 720 dīnāras for their wages. Two men stop after going two krośas, after two more krośas three others give up, and after going half the remaining distance five men leave. What wages do they earn?
Answers—18, 57, 155, 490. M. vi, 231.
(4 krośas=1 yojana).
9. It is well known that the horses belonging to the sun's chariot are seven. Four horses drag it along being ​harnessed to the yoke. They have to do a journey of 70 yojanas. How many times are they unyoked and how many times yoked in four?
Answer—Every 10 yojanas, and each horse travels 40 yojanas. M. VI, 158.
10. If a female slave sixteen years of age brings thirty- two, what will one twenty cost?
Answer—$\scriptstyle {25{\frac {3}{5}}}$. L. 76.
11. Three hundred gold coins form the price of 9 damsels of 10 years. What is the price of 36 damsels of 16 years?
Answer—750. M. V, 40.
12. The price of a hundred bricks, of which the length, thickness and breadth respectively are 16, 8 and 10, is settled at six dīnāras, we have received 100,000 of other bricks a quarter less in every dimension. Say, what we ought to pay?
Answer—$\scriptstyle {2531{\frac {1}{4}}}$. C. 285.
13. Two elephants, which are ten in length, nine in breadth, thirty-six in girth and seven in height, consume one droņa of grain. How much will be the rations of ten other elephants which are a quarter more in height and other dimensions?
Answer—12 droņas, 3 prasthas, $\scriptstyle {1{\frac {1}{2}}}$ kudavas. C. 285.
(64 kudavas=16 prasthas=1 droņa).
14. One bestows alms on holy men in the third part of a day, another gives the same in half a day and a third distributes three in five days. In what time, keeping to these rates, will they have given a hundred?
Answer—$\scriptstyle {17{\frac {6}{7}}}$. C. 282.
15. Say, mathematician, what are the apportioned shares of three traders whose original capitals were respectively 51, 68 and 85, which have been raised by commerce conducted by them in joint stock to the aggregate amount of 300?
Answer—75, 100, 125. L. 93.
16. One purchases seven for two and sells six for three. Eighteen is the profit. What is the capital?
Answer—$\scriptstyle {18\div ({\frac {3/6}{2/7}}-1)=24}$. Bakhshāli Ms. 54.
17. If a pala of best camphor may be had for two nishkas, and a pala of sandal wood for the eighth part of a dramma and half a pala of aloe wood also for the eighth of a dramma, good merchant, give me the value of one nishka in the proportions of 1, 16 and 8; for I wish to prepare a perfume?
Answer—Prices—Drammas $\scriptstyle {14{\frac {2}{9}},~{\frac {8}{9}},~{\frac {8}{9}}+{\frac {4}{9}},~{\frac {64}{9}},~{\frac {32}{9}}}$ L. 98.
(16 drammas=1 nishka).
18. If three and a half mānas of rice may be had for one dramma and eight of beans for the same price, take these thirteen kākinīs, merchant, and give me quickly two parts of rice and one of beans: for we must make a hasty meal and depart, since my companion will proceed onwards?
Answer—$\scriptstyle {\frac {7}{12}}$ and $\scriptstyle {\frac {7}{24}}$. L. 97, V. 115.
(64 kākinīs=1 dramma).
19. If the interest on 200 for a month be 6 drammas, in what time will the same sum lent be tripled?
Answer—$\scriptstyle {66{\frac {2}{3}}}$ months. C. 287.
20. If the principal sum with interest at the rate of five on the hundred by the month amount in a year to one thousand, tell the principal and interest respectively?
Answer—625, 375. L. 89.
21. In accordance with the rate of five per cent, (per mensem) two months is the time for each instalment; and paying the instalments of 8 (on each occasion) a man became free in 60 months. What is the capital?
Answer—60. M. vi, 64.
22. Five hundred drammas were a loan at a rate of interest not known. The interest of that money for four months was lent to another person at the same rate and it accumulated in ten months to 78. Tell the rate of interest on the principal?
Answer—60. C. 288.
23. Subtracting from a sum lent at five in the hundred the square of the interest, the remainder was lent at ten in the hundred. The time of both loans was alike and the amount of interest equal?
Answer—Principal 8. V. 109.
24. There is 1 part of 1 varņa, 1 part of 2 varņas, 1 part of 3 varņas, 2 parts of 4 varņas, 4 parts of 5 varņas, 7 parts of 14 varņas, and 8 parts of 15 varņas. Throwing these into the fire make them all into one and then what is the varņa of the mixed gold?
Answer—$\scriptstyle {10{\frac {1}{2}}}$. M. vi, 170.
[The term varņa corresponds to 'carat' or measure of 'purity of gold.'
25. Gold 1, 2, 3, 4 suvarnas, and losses 1, 2, 3, 4 māshakas.
The average loss is $\scriptstyle {{\frac {1.1+2.2+3.3+4.4}{1+2+3+4}}=3.}$ Bakhshāli Ms. 27.
26. Of two arithmetical progressions with equal sums and the same number of terms the first terms are 2 and 3, the increments 3 and 2 respectively and the sum 15. Find the number of terms?
Answer—3. Bakhshāli Ms. 18.
27. A merchant pays octroi on certain goods at three different places. At the first he gives $\scriptstyle {\frac {1}{3}}$ of the goods, at the second $\scriptstyle {\frac {1}{4}}$, and at the third $\scriptstyle {\frac {1}{5}}$. The total duty paid is 24. What was the original amount of the goods?
Answer—40. Bakhshāli Ms. 25.
28. One says: "Give me a hundred and I shall be twice as rich as you, friend!" The other replies: "If you deliver ten to me I shall be six times as rich as you. Tell me what was the amount of their respective capitals?
Answer—40 and 170. V. 106, 156.
29. A gives a certain amount, B gives twice as much as A, C gives 3 times as much as B, D gives 4 times as much as C and the total is 132.
Answer—A gives 4, etc. Bakhshāli Ms. 54.
30. Four jewellers possessing respectively 8 rubies, 10 sapphires, 100 pearls and 5 diamonds, presented each from his own stock one apiece to the rest in token of regard and gratification at meeting; and thus they became owners of stock of precisely the same value. Tell me, friend, what were the prices of their gems respectively?
Answer—24, 16, 1, 96 [These are relative values only]. L. 100.
31. The quantity of rubies without flaw, sapphires, and pearls belonging to one person, is five, eight and seven ​respectively. The number of like gems belonging to another is seven, nine and six. One has ninety, the other sixty-two rupees. They are equally rich. Tell me quickly, then, intelligent friend, who art conversant with algebra, the prices, of each sort of gem?
Answer—14, 1, 1, etc. V. 105 & 156.
[Bhāskara 'assumes' relative values.]
32. The horses belonging to these four persons respectively are five, three, six and eight; the camels belonging to them are two, seven, four and one; their mules are eight, two, one and three; and the oxen owned by them are seven, one, two and one. All are equally rich. Tell me severally, friend, the rates of the prices of horses and the rest?
Answer— 85, 76, 31, 4, etc. V. 157.
33. Say quickly, friend, in what portion of a day will four fountains, being let loose together fill a cistern, which, if opened one by one, would fill it in one day, half a day, the third and the sixth parts respectively?
Answer—$\scriptstyle {\frac {1}{12}}$. L. 95.
34. The son of Prithā, angered in combat, shot a quiver of arrows to slay Karna. With half his arrows he parried those of his antagonist; with four times the square-root of the quiverful he killed his horse; with six arrows he slew Salya; with three he demolished the umbrella, standard and bow; and with one he cut off the head of the foe. How many were the arrows which Arjuna let fly?
Answer—100. L. 67, V. 133.
35. For 3 paņas 5 palas of ginger are obtained, for 4 paņas 11 palas of long pepper and for 8 paņas 1 pala of ​pepper. By means of the purchase money of 60 paņas quickly obtain 68 palas? M. vi, 150.
Answer—Ginger 20, long pepper 44, pepper 4.
36. Five doves are to be had for three drammas; seven cranes for five; nine geese for seven and three peacocks for nine. Bring a hundred of these birds for a hundred drammas for the prince's gratification? V. 158–9; M. vi, 152.
Answer—Prices 3, 40, 21, 36.
Birds 5, 56, 27, 12.
(This class of problem was treated fully by Abú Kāmil-el-Misri (c. 900 A.D.). See H. Suter: Das Buch der Selten-heit, etc. Bibliotheca Mathematica 11 (1910–11), pp. 100–120.
37. In a certain lake swarming with red geese the tip of a bud of a lotus was seen half a hasta above the surface of the water. Forced by the wind it gradually advanced and was submerged at a distance of two hastas. Calculate quickly, O mathematician, the depth of the water?
Answer—$\scriptstyle {\frac {15}{4}}$. L. 153; V. 125.
38. If a bamboo measuring thirty-two hastas and standing upon level ground be broken in one place by the force of the wind and the tip of it meet the ground at sixteen hastas, say, mathematician, at how many hastas from the root it is broken?
Answer—12. L. 148.
39. A snake's hole is at the foot of a pillar 9 hastas high and a peacock is perched on the summit. Seeing a snake, at a distance of thrice the pillar, gliding towards his hole, he pounces obliquely on him. Say quickly at how many hastas from the snake's hole do they meet, both proceeding an equal distance?
Answer—12. L. 150.
40. From a tree a hundred hastas high, a monkey descended and went to a pond two hundred hastas distant, while another monkey, jumping a certain height off the tree, proceeded quickly diagonally to the same spot. If the space travelled by them be equal, tell me quickly, learned man, the height of the leap, if thou hast diligently studied calculation?
Answer—50. L. 155; V. 126.
41. The man who travels to the east moves at the rate of 2 yojanas, and the other man who travels northward moves at the rate of 3 yojanas. The latter having journeyed for 5 days turns to move along the hypotenuse. In how many days will he meet the other man?
Answer—13. M. vii, 211.
42. The shadow of a gnomon 12 añgulas high is in one place 15 añgulas. The gnomon being moved 22 añgulas further its shadow is 18. The difference between the tips of the shadows is 25 and the difference between the lengths of the shadows is 3. Find the height of the light?
Answer—100. C. 318; Ar. 16; L. 245.
43. The shadow of a gnomon 12 añgulas high being lessened by a third part of the hypotenuse became 14 añgulas. Tell, quickly, mathematician, that shadow?
Answer—$\scriptstyle {22{\frac {1}{2}}}$. V. 141.
44. Tell the perpendicular drawn from the intersection of strings mutually stretched from the roots to summits of two bamboos fifteen and ten hastas high standing upon ground of unknown extent?
Answer—6. L. 160.
45. Of a quadrilateral figure whose base is the square of four and the face two hastas and altitude twelve, the flanks thirteen and fifteen, what is the area?
Answer—108. S. 77.
46. In the figure of the form of a young moon the middle length is sixteen and the middle breadth is three hastas. By splitting it up into two triangles tell me, quickly, its area?
Answer—24. S. 83.
47. The sides of a quadrilateral with unequal sides are $\scriptstyle {13\times 15}$, $\scriptstyle {13\times 20}$ and the top side is the cube of 5 and the bottom side is 300. What are all the values here beginning with that of the diagonals?
Answers—315, 280, 48, 252, 132, 168, 224, 189, 44100. M. vii, 59.
If $\scriptstyle {A^{2}+B^{2}=C^{2}}$, and $\scriptstyle {a^{2}+b^{2}=c^{2}}$ then the quadrilateral Ac, Bc, aC, bC is cyclic and the diagonals are $\scriptstyle {Ab+aB}$ and $\scriptstyle {Aa+Bb}$, the area is $\scriptstyle {{\frac {1}{2}}(ABc^{2}+abC^{2})}$, &c. In the present case $\scriptstyle {A=15}$, $\scriptstyle {B=20}$, $\scriptstyle {C=25}$; $\scriptstyle {a=5}$, $\scriptstyle {b=12}$, $\scriptstyle {c=13}$. The diagonals are 315, 280; the area 44100. For full details see the Līlāvatī, § 193.
48. O friend, who knowest the secret of calculation, construct a derived figure with the aid of 3 and 5 as elements, and then think out and mention quickly the numbers measuring the perpendicular side, the other side and the hypotenuse?
Answer—16, 30, 34. M. vii, 94.
That is construct a triangle of the form $\scriptstyle {2mn}$, $\scriptstyle {m^{2}-n^{2}}$, $\scriptstyle {m^{2}+n^{2}}$, where m=5, n= 3.
49. In the case of a longish quadrilateral figure the perpendicular side is 55, the base is 48 and then the diagonal is 73. What are the elements here?
Answer—3, 8. M. vii, 121.
50. Intelligent friend, if thou knowest well the spotless Līlāvatī, say what is the area of a circle the diameter of which is measured by seven, and the surface of a globe or area like a net upon a ball, the diameter being seven, and the solid content within the same sphere?
Answer—Area $\scriptstyle {38{\frac {2423}{5000}}}$; surface $\scriptstyle {153{\frac {1173}{1250}}}$; volume $\scriptstyle {179{\frac {1487}{2500}}}$ L. 204.
51. In a circle whose diameter is ten, what is the circumference? If thou knowest, calculate, and tell me also the area?
Answer—$\scriptstyle {\sqrt {1,000}}$, $\scriptstyle {\sqrt {6250}}$. S. 85.
52. The measure of Rāhu is 52, that of the moon 25, the part devoured 7.
Answer—The arrow of Rāhu is 2, that of the moon 5. C. 311.
This is an eclipse problem and means that circles of diameters 52 and 25 intersect so that the portion of the line joining the two centres common to the two circles is 7. The common chord cuts this into segments of 5 and 2.
53. The combined sum of the measure of the circumference, the diameter and the area is 1116. Tell me what the circumference is, what the calculated area, and what the diameter?
Answer—108, 972, 36. M. vii, 32.
The rule given is circumference$\scriptstyle {={\sqrt {12({\text{combined sum}}+64)}}-{\sqrt {64}}}$ which assumes that $\scriptstyle {\pi =3}$.
54. The circumferential arrows are 18 in number. How many are the arrows in the quiver?
Answer—37. M. 289.
The rule given is $\scriptstyle {n={\frac {(c+3)^{2}+3}{12}}}$ where c is the number of arrows in the outside layer.
55. Tell me, if thou knowest, the content of a spherical piece of stone whose diameter is a hasta and a half?
Answer—$\scriptstyle {1{\frac {25}{32}}}$. S. 93.
The rule given is $\scriptstyle {v=d^{3}({\frac {1}{2}}+{\frac {1}{2.18}})}$.
56. A sacrificial altar is built of bricks 6 añgulas high, half a hasta broad and one hasta long. It is 6 hastas long, 3 hastas broad and half a hasta high. Tell me rightly, wise man, what its volume is and how many bricks it contains.
Answer—9, 72. S. 96.
24 añgulas=1 hasta.
57. If thou knowest, tell me quickly the measure of a mound of grain whose circumference is 36 and height 4 hastas?
Answer—144. S. 102.
The rule used assumes that $\scriptstyle {\pi =3}$.
58. In the case of a figure having the outline of a bow, the string measure is 12, and the arrow measure is 6. The measure of the bow is not known. Find it, O friend.
Answer—$\scriptstyle {\sqrt {360}}$. M.. vii, 75.
59. In the case of a figure having the outline of a bow the string is 24 in measure, and its arrow is taken to be 4 in measure. What is the minutely accurate value of the area?
Answer—$\scriptstyle {\sqrt {5760}}$.M. vii, 72.
The rule used is $\scriptstyle {S={\frac {ac{\sqrt {10}}}{4}}}$.
60. Multiplier consisting of surds two, three and eight: multiplicand the surd three with the rational number five. Tell quickly the product?
Answer—$\scriptstyle {{\sqrt {9}}+{\sqrt {450}}+{\sqrt {75}}+{\sqrt {54}}}$. V. 32.
61. What is the number which multiplied by five and having the third part of the product subtracted, and the remainder divided by ten; and one-third, one half and a quarter of the original quantity added gives two less than seventy?
Answer—48. L. 51.
The solution may be summarised this: $\scriptstyle {f(x)=68}$, $\scriptstyle {f(3)=17/4}$ therefore $\scriptstyle {x={\frac {3\times 68}{17/4}}=48}$.
62. The eighth part of a troop of monkeys squared was skipping in a grove and delighted with their sport. Twelve remaining were seen on the hill amused with chattering to each other. How many were there in all?
Answer—48 or 16. V. 139.
63. The fifth part of the troop less three, squared, had gone to a cave and one monkey was in sight, having climbed on a branch. Say how many there were?
Answer—50 or 5. V. 140.
"But," Bhāskara says, "the second is not to be taken for it is incongruous. People do not approve a negative absolute number."
64. Say quickly what the number is which added to five times itself divided by thirteen becomes thirty?
Answer—$\scriptstyle {\frac {65}{3}}$. V. 168.
65. A certain unknown quantity is divided by another. The quotient added to the divisor and the dividend is fifty-three. What is the divisor?
Answer—5, 8. M. vi, 274.
66. What number is that which multiplied by nought and added to half itself and multiplied by three and divided by nought amounts to the given number sixty-three?
Answer—14. This assumes that $\scriptstyle {{\frac {0}{0}}=1}$. L. 46.
67. What four numbers are such that their product is equal to twenty times their sum, say, learned mathematician who art conversant with the topic of the product of unknown quantities?
Answer—5, 4, 2, 11. V. 210.
Bhāskara puts arbitrary values for three of the quantities and gets 11 for the fourth.
68. If you are conversant with operations of algebra tell the number of which the fourth power less double the sum of the square and of two hundred times the simple number is ten thousand less one?
Answer—11. V. 138.
This may be expressed by $\scriptstyle {x^{4}-2(x^{2}+200x)=9999}$. It is the only case in which the fourth power occurs.
69. The square of the sum of two numbers added to the cube of their sum is equal to twice the sum of their cubes?
Answer—1, 20; 5, 76, etc. V. 178.
70. Tell me, if you know, two numbers such that the sum of them multiplied respectively by four and three may when added to two be equal to the product of the same numbers?
Answer—5, 10 and 11, 6. V. 209, 212.
71. Sav quickly, mathematician, what is the multiplier by which two hundred and twenty-one being multiplied and sixty-five added to the product, the sum divided by a hundred and ninety-five becomes cleared?
Answer—5, 20, 35 &c. L. 253; V. 65.
72. What number divided by six has a remainder of five, divided by five has a remainder of four, by four a remainder of three and by three one of two?
Answer—59. Br. xviii, 7; V. 160.
73. What square multiplied by eight and having one added to the product will be a square? V. 82.
Here $\scriptstyle {8u^{2}+1=t^{2}}$ and u=6, 35, etc. t=17, 99, etc.
74. Making the square of the residue of signs and minutes on Wednesday multiplied by ninety-two and eighty-three respectively with one added to the product an exact square; who does this in a year is a mathematician. Br. xviii, 67.
$\scriptstyle {(1)\ 92u^{2}+1=t^{2}\quad (2)\ 83u^{2}+1=t^{2}}$.
Answer—$\scriptstyle {(1)\ u=120,\ t=1151.\quad (2)\ u=9,\ t=82}$.
75. What is the square which multiplied by sixty-seven and one being added to the product will yield a square-root; and what is that which multiplied by sixty-one with one added to the product will do so likewise? Declare it, friend, if the method of the 'rule of the square' be thoroughly spread, like a creeper, over thy mind? V. 87.
$\scriptstyle {(1)\ 67u^{2}+1=t^{2}.\quad (2)\ 61u+1=t^{2}.}$
Answers—(1) u=5967, t=48842. (2) u=226,153,980, t=1,766,319,049.
76. Tell me quickly, mathematician, two numbers such that the cube-root of half the sum of their product and the smaller number, and the square-root of the sum of their ​squares, and those extracted from the sum and difference increased by two, and that extracted from the difference of their squares added to eight, being all five added together may yield a square-root—excepting, however, six and eight? V. 190.
$\scriptstyle {{\sqrt[{3}]{(xy+y)^{2}}}+{\sqrt {x^{2}+y^{2}}}+{\sqrt {x+y+2}}+{\sqrt {x-y+2}}+{\sqrt {^{x2}-y^{2}+8}}=t^{2}}$
Answers—x=8; 1677/4; 15128, etc.; y=6, 41; 246, etc.
CHRONOLOGY.
Circ.
Pythagoras‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥B.C. 530
Euclid‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 290
Archimedes‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 250
The S'ulvasūtras‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥"?
The Nine Sections‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 150
Hipparchus‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 130
Nicomachus‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥A.D. 100
Ptolemy‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 150
Sassanian period begins‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 229
The Sea Island Arithmetic‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 250
Gupta period begins‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 320
Diophantus‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 360
Hypatia‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 410
Bœthius‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 470 b.
Āryabhata‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 476 b.
Damascius‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 480 b.
Athenian schools closed‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 530
Chang ch'iu-chien‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 550
Varāha Mihira‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 587 d.
Brahmagupta‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 598 b.
Fall of Alexandria‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 640
Gupta period ends‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 650
Sassanian period ends‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 652
Muhammad b. Mūsā‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 820 d.
Mahāvīra‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 850 ?
'Tabit b. Qorra‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 826 ?
el-Battānī‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 877 b.
el-Birūni‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 972 b.
Ibn Sīna‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 980 b.
S'rīdhara‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 991 b.
el-Karchi‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 1016
'Omar b. Ibrāhīm el-Chaijāmi‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 1046 b.
Bhāskhara‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥" 1114 b.
BIBLIOGRAPHY.
(For a more complete bibliography see that given in the Journal of the Asiatic Society of Bengal, VII, 10, 1911.)
First Period.
Thibaut, G.—On the S'ulvasūtras, J.A.S.B., XLIV, 1875.—The Baudhāyana S'ulvasūtra, The Pandit (Benares) 1875–6.—The Kātyāyana S'ulvasūtra, Ib., 1882.
Bürk, A.—Das Apastamba-S'ulba-Sūtra, Z.D.M.G., 55, 1901; 56, 1902.
Second Period.
Burgess, E. and Whitney, G.—The Sūrya Siddhānta, Jour. Am. Or. Soc., VI, 1855.
Bapu Deva Sastri and Wilkinson, L.—The Sūrya Siddhānta and the S'iddhānta S'iromani, Calcutta, 1861.
Thibaut, G. and Sudharkar Dvivedi.—The Pañcha-Siddhāntikā of Varaha Mihira, Benares, 1889.
Rodet, L.—Leŗons de Calcul d'Aryahhata, Paris, 1879.
Kaye, G. R.—Āryabhata, J.A.S.B., IV, 17, 1908.
Third Period.
Colebrooke, H. T.—Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhascara, London, 1817.
Rangacarya, M.—The Gaņita-Sāra-Saṅgraha of Mahā­vīracārya, Madras, 1908.
Ramanujacharia, N. and Kaye, G. R.—The Triśatikā of S'rīdharāchārya, Bib. Math., XIII, 3, 1913.
Notations.
Buhler, G.—Indische Palæographie, Strassburg, 1896.
Bayley, E. C.—On the Genealogy of Modern Numerals, London, 1882.
Woepcke, F.—Mémoire sur la propagation des Chiffres indieus, Jour. Asiatique, 1863.
Kaye, G. R.—Indian Arithmetical Notations, J. A. S. B., III, 7, 1907. The Use of the Abacus in Ancient India, J. A. S. B., IV, 32, 1908. Old Indian Numerical Systems, Indian Antiquary, 1911.
Fleet, J. F.—Äryabhata's system of expressing Numbers, J. R. A. S., 1911. The Use of the Abacus in India, J. R. A. S., 1911.
Smith, D. E. and Karpinski, L. C.—The Hindu Arabic Numerals, Boston, 1911.
Other Works.
Sachau, E. C.—Alberuni's India, London, 1910.
Thibaut, G.—Astronomie, Astrologie und Mathematik, Grundriss der Indo-Arischen Philologie, III, 9, 1899.
Hoernle, R.—The Bakhshālī Manuscript, Indian Antiquary, XVIII, 1888.
Kaye, G. R.—Notes on Hindu Mathematical Methods, Bib. Math., XI, 4, 1911.—Hindu Mathematical Methods, Indian Education—1910–1913. The Source of Hindu Mathematics, J. R. A. S., 1910. The Bakhshāli Manuscript, J. A. S. B., VII, 9, 1912.
Heath, T, L.—Diophantus of Alexandria, Cambridge, 1910.
Rosen, F.—The Algebra of Mohammed ben Musa, London, 1831.
Suter, H.—Die Mathematiker und Astronomen der Araber und Ihre Werke, Leipzig, 1900.
Yoshio Mikami.—The development of Mathematics in China and Japan, Leipzig, 1912.
The general works on the history of mathematics by Cantor, Gunther, Zeuthen, Tannery and v. Braunmuhl and the articles by Woepcke, Rodet, Vogt, Suter and Wiedemann should also be consulted.
XI———82.
* ↑ These numbers refer to Baudhāyana's edition as translated by Dr. Thibaut.
* ↑ L=the Līlāvatī, V=Vīja Gaņita, both by Bhāskara, M=Mahāvīra, S=Srīdhara, C=Chaturveda.