54 - Mathematics and Astronomy 

Sanandana said: 

1. I shall now set out the auxiliary (of the Veda) called Jyotisha which had been enunciated by Brahma in days of yore and through the mere knowledge of which men can attain the fulfilment of their ordained duties. 

2. O Brahmana! the science of Jyotisha, which has been expounded in four lakhs of verses, falls into three sections, devoted, respectively, to mathematics and astronomy (Ganita), horoscopy (Jataka) and natural astrology (Sanhita). 

3-4a. Topics of Mathematics and Astronomy. In the Ganita section have been set out the arithmetical operations (vv. 12b-59); computation of the mean and true positions of Planets (vv. 60-127); ‘the questions’ (on time, place and direction)1 (vv. 128-53a); lunar and solar eclipses, (vv. 153b-65a); the diagrammatic representation thereof,2 (gnomonic) shadow, (vv. 165b-67a); elevation of the lunar horns (vv. 167b-69), (planetary) conjunction (vv.170-73) and the (vyati-) patas3 (vv.174-87). 

1. Hindu texts on astronomy call this section as Triprashna-adhyaya, since it deals with ‘problems’ concerning the ‘three’ topics, viz., time, place and direction. 

2. Actually, however, this topic is not treated in the Purana. 

3. The two vyatipatas, being− malignant aspects of Sun and Moon, called Lala and Vaidhruta, occur at the moments when the sum of the longitudes of the Sun and the Moon is equal to 180° and 360°, respectively. These moments are considered to be highly inauspicious (vi-ati-pata). 

4b. Topics of Horoscopy. In the Jataka section are treated-the signs (rashi) (of the zodiac), their divisions (and properties); Nature of the planets (and their properties; (Mani fold inferior) births, 

5. Conception; Birth; Early death; Longevity; Order of divisions (and subdivisions in one's life): Vocations; Eight emplacements, (being those of the 7 planets and the lagna, in the horoscopic chart); ‘Royal’ planetary combinations; ‘Atmospheric’ planetary combinations; 

6. ‘Lunar’ planetary combinations; ‘Ascetic’ planetary combinations; Effect of the planets occupying the different signs; Effect of planets aspecting one another; Effect of the planets being in the several houses’; Effect of mutual association (of planets); Miscellaneous matters (relating to the association of planets): 

7. Malefic combination (of planets; Female horoscopy; Death; Reconstruction of lost horoscopes; and Effects of decanates. 

8. Topics of Natural astrology. The contents of the Samhita section are: Effects of the motion of the planets (across the different signs) (vv. 1-108); Abda Lakshana (Characteristics of the year) (vv. 109-33a); Tithi (Lunar day) (vv.133b-56a); Vara (Weekday) (vv. 156b-67a); Nakshatra (Asterism) (vv. 167b-211a); Yoga4 (vv. 211b-19a); Tithyartha or karana5 (vv. 219b-23). 

4. For the constitution and list of Yogas, see below, note under verse 211b. See also ch. 54, Verse 124 and footnote. 

5. For the nature of karana and the list thereof, see below, note under verse 219b. See also ch. 54, verses 126-27 and footnote. 

9. (Auspicious moment Muhurta) (vu.224-2a); Upagraha (Secondary atmospheric phenomena) (vv. 230-50a); (Sankranti (Sun’s transit into a sign) (vv. 250b-70); Gochara (Current motion of the planets) (vv. 271-82); Chandra-Tarabala (Astrological strength of the Moon and of the Asterisms) (vv. 283-89); Sarvalagna (Rising of the signs) (vv. 290-312a); Artava (First menstruation) (vv. 312b-17). 

10. Adhana (Conception) (vv.318-19); Pumsavana (Rite for the birth of a male child) (vv. 320-25); Jata-Nama-karma (Birth rites and Naming Ceremony) (vv. 326-30a); Annabhukti (Annaprasana, first feeding) (vv. 330b-34); Chaula (Tonsure) (vv. 335-43a); Ankurarpana (Auspicious sowing) (vv. 343b- 47); Maunjibandhana (and Upanayana, Tying the girdle and commencing studies) (w.348-79), Kshurikabandhana, (Girding the sword) (vv. 380-91a). 

11. Samavartana (Return from studies) (vv. 391b-94a); Vivaha (Marriage) (vv. 394b-523); Pratishtha (Installation of deities in temples) (vv. 524-39a); Sadma (Building of human residences) (vv.539b-619); Yatra (Travel for pilgrimage, war etc.) (vr.620-712): Praveshana (Return home) (vv. 713-20); Sadyovrushti (Immediate rain) (vv. 721-39a); Kurma-vibhaga (Division of the globe) (vv.739b-45); and Utpata (Portentuous phenomena (vv. 646-56a). I shall be setting out all these briefly.

MATHEMATICS

12b-14a. Notational places. Eka (one), dasha (ten), shata (hundred), sahasra (thousand), ayuta (ten thousand), laksha (lakh), prayuta (ten lakhs/million), koti (crore), arbuda, abja, kharva, nikharva, mahapadma, shanku, jaladhi, antya, madhya, parardha-these are the names (of the notational places), each succeeding one being ten times (the preceding). 

14b. additionand subtraction. Addition and subtraction (of numbers) can be done either in a forward or in a backward manner.6 

6. The idea is that the operation could start from the unit digits i.e., from the right end of the numbers, as is the general practice, or from the highest digits of the numbers at the left end. 

15a. Multiplication. In multiplication, the multiplicand (gunya) is multiplied up to its last digit (by the multiplier) (and the products added together).7

7. For the detailed working, see Lilāvati of Bhaskarāchārya, (edn. Hoshiarpur, 1975). So also, for the arithmetical operations set out below. 

15b. Division. O sage! In division that is the quotient (phala) which when multiplied by the divisor is completely subtractable (from the dividend). 

16a. Square. The product of a number multiplied by itself is called varga (square); the learned call it also (by the term) kruti. 

16b-18a. Square root. (Mark off the odd digits of the number whose square root is required.) Having subtracted the (greatest possible) square from the last odd place, keep that ‘root’ apart. Double that ‘root’ and divide the remainder and place the quotient alongside the previously obtained ‘root’. Subtract, O Brahmana, the square of that and again divide as before by the (newly formed) ‘root’. O great sage, by repeating as above (till all the digits are completed) the square root is obtained. 

18b-21a. Cube and cube root. The product of the multiplication thrice of the same number is called its ghana (cube). The method to derive the cube root (pada) isas follows: The first place (unit's digit of the number whose cube root is required) is termed ‘odd’; the next two digits (i.e., the tens and hundreds) are termed even.’ (Mark off the digits of the number into groups of three digits each in this manner, each group having one ‘odd’ and two ‘even-s’). Subtract from the last (group having an) odd place the (greatest possible) cube; that is the cube root (mula) (of that group). Divide (the next) even place with thrice the square of (the previous) cube root and place the quotient alongside the previous cube root. Square the new quotient and multiply it by three and by the last cube root and subtract it (from the next even place). Subtract, the cube (of the new cube root-digit) from the next odd place. Repeat the process and the cube root of the number is obtained. 

21b. Fractions. Two fractions are reduced to a common denominator when their numerators and denominators are multiplied by the denominator of one by that of the other. 

22. Fractions of fractions. O sage! Enquirers into science should understand that in fractions of fractions (bhaga-prabhaga), the products of their numerators and of their denominators give the correct figures of the numerator and denominator of the resultant fraction). 

23-24a. associated and dissociated fractions. associated fractions (bhaganubandha) and dissociated fractions (bhaga pavaha) are those in which a number is increased or decreased (by a fraction of its own). Here, multiply the whole number by the denominator (talasthahara), ascertain whether the fraction is positive or negative and accordingly add to or subtract from it the numerator. 

24b. addition and subtraction of fractions. O sage! (Addition and subtraction of fractions are affected by adding together or subtracting one from the other their numerators (after reducing the fractions) to a common denominator. 

25a. Denominator of an integer. When a denominator is not attached to a number, take l as its denominator. 

25b-26. Multiplication and division of fractions. The product of the multiplication of two fractions is obtained by dividing the product of the numerators by the product of the denominators. In the division of two fractions, the numerator and denominator of the divisor are inverted and the process of multiplication applied. 

27. Square etc. of fractions. For the square, cube, square root and cube root of fractions, calculate the respective squares etc. of the numerator and of the denominator. These for zero are always zero. 

28-29. Inverse operations. When the result (of certain operations) is ‘given’ (drushya) and the original number is to be found (rashiprasiddhaye), calculate taking the denominator as the numerator and the numerator as the denominator, the square as the square root and the square root as the square, minus as plus and plus as minus. However, in such cases of inverse operation where a part of an item) has been added to or subtracted from it, the denominator to which the numerator has been added to or subtracted from should be taken as the denominator, and the denominator as such should be taken as the numerator; the rest of the calculations are to be done) as before. 

30. Operations with assumed numbers (Ishtakarma). When an intended number (uddishta-rashi) has been multiplied, divided, has a part of it taken away from it or added to it and the result is drushta ‘known’), the intended number can be found by multiplying that result by an assumed number and dividing by the resultant (obtained by subjecting the assumed number to all the said operations). 

31. Operations with sums and differences. The difference (of two numbers) when added to or subtracted from the sum (of those two numbers) and (the result) divided by two, will give the two numbers.8 (This operation is called) Sankrama. The difference of the squares (of two numbers) will give their sum.9 (And from the said sum and difference), the individual numbers (can be found as stated above). 

8. Written in modern convention, this would be

9.

32-34. Methods to get perfect squares based on any assumed number (vargakarma). Multiply the square of an assumed number by 8 (gaja), subtract 1, halve it and divide by the assumed number. This would be one (number). Square this number and add 1. This would be another such number10 

10. 1. Verses 32-34 give three methods to derive, on the basis of any assumed number, sets of two numbers which when squared and added together or one subtracted from the other, and reduced by 1, give perfect squares. 

Method I. The numbers are: 

Sum of their squares minus 1= -2+=

Diff. of their squares minus 1=+6+4--=(8x2-2+/8x2)2

Method II. The numbers are: (x+1/2x) and 1.

Sum of their squares minus 1= (x+ 1/2x)2+ 12- 1 = (x+ 1/2x)2

Diff, of their squares minus 1= (x+1/2x)2-12-1= (x+1/2x)2

 Method III. The numbers are: (8x4+1) and 8x3

Sum of their Squares minus 1= (8x4 + 1)2+ (8x3)2—1= (8x4+ 4x2)

Diff, of their Squares minus 1= (8x4 + 1)2-(8x3)2-1=(8x4-4x2)2

Thus, all the results obtained are perfect squares. Or, 1 divided by twice the assumed number and the assumed number added is one (number). The other number is 1. The sum or difference of these two numbers reduced by one would be the squares of the two (desired) numbers. 

Or, multiply the square of the square and the cube of the assumed number and add to the first. The two desired numbers are obtained. These are according to the methods to be adopted in arithmetic (vyakta-ganita) and algebra (avyakta ganita). 

35-36. Operation involving the addition of a quantity to squares. When a resultant (drushta) is obtained by adding to or subtracting from a square its root multiplied by a multiplicand, that resultant should be added to the square of half the multiplicand and the root of the result calculated. To this result half the multiplicand is added (when gunaghanamula had been subtracted) and subtracted (when the gunaghanamula had been added) and the result squared. (This would give the required number). (This is called) Guna- (Karma). 

When, however, the resultant (drushya) is less or more by a part of the (required) number, the resultant and the root are to be appropriately reduced or increased and using them the required number is calculated as before. 

37. Rule of three (In a Trairashika) the Pramana (argument) and Iccha (requisition) are to be of the same denomination and (are to be placed) at the beginning and at the end. Phala (fruit) (which would be of a different denomination) would be placed in the middle. Phala multiplied by Iccha and divided by Pramana (adya) will yield the Iccha-phala. In the Inverse (rule of three), (the method is) reversed. 

38. Rule of five etc. In Pancharashika etc. the pramana side and the phala side are properly set down. By dividing the product of the larger number (of quantities) by the product of the smaller number (of quantities) the result is obtained. 

39a. Capital and interest. Capital is obtained through calculation with an assumed number (ishtakarma-vidhi, see verse 30, above). That subtracted from the composite amount (of capital and interest) will give the interest (kalantara). 

39b-40a. Rate multiplied by time, and the interest multiplied by the invested time are to be calculated and kept independently. Each divided by their sum and multiplied by the composite amount, would yield results which are respectively the capital and the interest. 

40b-41a. (In the matter of the loan of different amounts for different periods, the individual income is identical), while arranging the sides (as in Pancharashika etc., see above, verse 38), if the product of the smaller number of quantities and the months be greater than the product of the larger number of quantities, the interest divided by the total number of months, would give the rate. 

41b. (In the case of investments by different people making up the capital), the investment by each (kshepa) multiplied by the total income (mishra) and divided by the total investment (kshepayoga), would give the (proportionate) income (phala). 

42a. Time for filling a tank. (When different pipes individually take different times to fill a tank, in order to find the time required to fill the tank (if all the pipes are opened together), divide the denominators by the numerators of the individual fractions of time taken by each to fill (the tank), find the sum of the new fractions obtained) and (with that sum) divide 1. The time to fill (the tank) is obtained. 

42b-43. Geometrical progression (Gunottarashredhi). The sum of the series in geometrical progression (manam gunottare) (is found thus): when the number of terms (gaccha) is odd, reduce it by 1 and multiply; when it is even, halve it and square. (Continue the process) till the number ends. Then, beginning from the last term (gacchanta) perform backwards (vyasta) the operations of multiplication (guna) and squaring (varga) (in continuation). From the result subtract 1, divide the remainder by ‘gunaka-l’ and multiply by the first member of the series (prag-ghana).11 

11. Here, the first two lines give the method to calculate the last term of the series and the third line, the sum of the series. Thus, in a geometrical series, if m is the first member (mukha, prag), g is the common ratio (gunaka), r is the number of terms (gaccha) and s the sum of the series (dhana, sarvadhana) 

The last term =mx gr 

Sum of the series =m(gr-1)/(g-1) 

For detailed working, see Lilávati, op. cit., 125 (pp. 256-64). 

44-45a. Triangles and quadrilaterals. In the case of plane figures (kshetra), like triangles and quadrilaterals (tri-chaturshraka), the following apply; The hypotenuse (karna) is the square root of the sum of the squares of the base (bhuja) and altitude (koti). The base would be the square root of the difference between the squares of the hypotenuse and the altitude. and, the altitude would be the square root of the difference between the squares of the hypotenuse and the base.12 

12. Thus, in a right-angled triangle, if a is the altitude, b the base, and h the hypotenuse, a2+b2=h2. 

45b-46a. (The rational sides of a right-angled triangle, as calculated from any two numbers would be: (1) The square of the difference of two (natural) numbers added to twice the product of the numbers, which is equal to the sums of their squares; (ii) the product of the sums and differences of the two numbers, which is equal to the difference of their squares, and (iii) twice the product of the numbers.13 

13. The three sides would thus be i. (a - b)2+ 2ab =a2+ b2;

ii. (a+b) (a - b) = a2-b2; iii. 2ab, for (a2 – b2)2 (+2ab)2 = (a2+ b2)2. 

46b. Circumference of a circle. O sage, the diameter of a circle multiplied by 22 (akruti) and divided by 7 (adri) will give the circumference of a circle.14 

14. Thus the circumference of a circle is 22/7d.

47-48. Relation of Sine, Reversed sine and Diameter. Experts in trigonometry say: Reversed sine (shara) is given by multiplying together the sum and difference of the sine (jya) and the diameter (vyasa), finding the square root thereof and halving the result. The sine (jya) is given by (diameter-minus-reversed sine) multiplied by the reversed sine, finding its root and doubling it. And, the diameter is obtained by squaring half sine, dividing it by the reversed sine and adding to the result the reversed sine. 15 

15. Thus: shara= 1/2 (vyasa–(vydsa + jya) (vyåsa-jya) 

jya= 2 (Vyasa-shara) x shara

Vyāsa= (1/2 jya) + Shara

49-50a. Sine of an arc. Let the (circumference minus are) multiplied by the circumference be called ‘First’ (präg). One fourth the square of the circumference is multiplied by 5 and the ‘First’ subtracted therefrom. With the result divide (Diameter ‘First’). O brahmani! the result obtained would be the sine (jya) (of the relevant arc). 16 

16. If a is the arc, c circumference, d diameter and s sine; 

‘First’ = (c-a) a 

Sine = 4 d x a(c-a)/5/4c2-a(c-a)

50b-51a. Arc from sine. One fourth the sine is multiplied by the square of the circumference and divided by four times the diameter to which the sine is added. The result is subtracted from one fourth the square of the circumference and the square root of the same is calculated). Thus, when subtracted from half the circumference, gives the arc. 17 

17. Arc = c/2-c2/4-54c44/4d+s

51b-52a. Measure of corn heaped up in a cone. 

When coarse, medium and fine grain are heaped, the heights (vedha) (of the cones so formed) would, respectively, be one-ninth (anka) one-tenth (āšha) and one-eleventh (isha) of the respective circumferences. The measure of the grain in cubic cubits (ghana-kara) would be given by the square of one-sixth (anga) multiplied by the height.18 

18. Different heights for different grains are suggested for the reason that when made into a conical heap, coarse grain would pile up higher while fine grain would slip down and so pile up only to a lesser height. 

52b-53a. Measure of water in a tank. The length of the (stretch of) water multiplied by its breadth and height in inches and divided by 3100 (khakhendurāma) will give the volume of water in drona measures.19 

19. Drona is one of the bigger measures of volume used in ancient and mediaeval India. 

54b-55a. Measure of rubble. The height, breadth and length, in inches, O brahmana, of a heap of rubble, multiplied together and divided by 1150 (kha-aksha-isha) would give its volume in drona measures. 

55b-56a. Measure of metal. In the case of metal pieces, the length, breadth and height, in inches, multiplied together and divided by 585 (băņa-ibha-mårgana) should be declared to be the volume of the heap in drona measures. 

56b-57. Gnomon and the lighted lamp. O sage! the gnomon20 multiplied by the distance between the lamp and the gnomon and divided by the height of the lamp less the gnomon, gives the gnomonic shadow.21 

20. The gnomon in general use in ancient India was a strong, straight, cylindrical rod, made of metal or of wood, 12 inches in height, and pointed at the tip. In use, it was fixed firmly on a hard, level surface at the centre of a graduated circle. 

21. In the figure, LA=Height of the lamp 

GB=Gonomon 

CS=Gnomonic shadow 

By the application of the rules of

similar triangles, shadow, 

CS = GC x BC/LA 

Again, the gnomon multiplied by the distance between the gnomon and the lamp and divided by the gnomonic shadow, gives the height of the lamp.22 

22. Height of the lamp, 

BL=GC x BC /CS

58. The height of the lamp-minus-gnomon multiplied by the gnomonic shadow and divided by the gnomon will give the distance between the lamp and the gnomon.23

23. Lamp to gnomon, 

BC=LA x CS /GC

58b-59a. Two gnomons and the lighted lamp. The distance between the shadow tips multiplied by the shadow and divided by the difference between the shadows gives the base (bhūmi of the relevant shadow).24 

24. If LB, the lamp, GC and G’C’, the two 

gnomons, C’S’ and C’S’, the two 

shadows BS and BS’, the two 

bases, 

Base BS=SS’ x CS/C’S-CS

BS’=SS’ x C’S’/(C’S’-CS)

59. The base multiplied by the gnomon, and divided by the shadow gives the height of the lamp by the rule of three. 25

25. Lamp LB=BS x GC/CS=BS’ x G’C’/C’S’

ASTRONOMY

Mean Planets (Madhya-graha)

60. (Operational) mathematics has been set out concisely, as above. Now, shall be set out in brief, the computation of) the mean planets etc. according to (astronomical) mathematics. 

61-62a. The aeon (yuga). O brahmana! ‘The measure of the (great) aeon (maha-yuga, chaturyuga) is said to be 43,20,000 (khachatuṣhka-rada-arnava) (divine) years.26 Four tenths of it is said to be the kruta-yuga; three tenths form the Treta-yuga; two tenths the Dvapara-yuga; and one tenth the Kali-yuga. 

26. A ‘divine’ year (divya-varsha) is equal to 360 ‘mortal’ years or siderial years of current astronomical parlance. 

62b. Seventy-one yugas plus one Kruta-yuga period form (the time duration of) one Manu.27 

27. Manu is one of the mythical progenitors of man and the time mentioned here is called a manvantara. 14 manvantaras constitute a longer Puranic duration of time mentioned in the next verse, which is termed Kalpa. 

63. O foremost among brahmanas! fourteen Manus occur during the day-time of God Brahma. That much period, again, O foremost among brahmanas! is said to be His night, 

64. O Narada! the years that have gone by from the beginning of creation by God Brahma might be consolidated and the computation of planets could be commenced from that beginning. Alternatively, the computation could be done from the beginning of any desired yuga.

65. Revolutions of the planets. The number of eastward revolutions (bhagana) in a yuga (i.e., maha-yuga) or (chaturyuga), of the Sun, Mercury (Budha) and Venus (Shukra) and of the shighroccha28 of Mars, Saturn and Jupiter is 43,20,000. 

28. By Shighroccha is meant the farthest point from the centre of the earth in the orbit of the planet, or the higher apsis of the epicycle related to the equation of conjunction. 

66. The number of revolutions of the Moon is 5,77,53,336; that of Mars is 22,96,832. 

67. The number of revolutions of the shighroccha of Mercury (Budha) is 1,79,37,060. The revolutions of Jupiter are 3,64,220. 

68. The number of revolutions of the shighroccha of Venus is 70,22,376. The revolutions of Saturn are 1,46,568. 

69. The number of revolutions of the Moon’s mandoccha (apogee) is 4,88,203 and the retrograde revolutions of the Moon’s ascending node (pata) are 2,32,238. 

70. Terrestrial and Lunar days.29 The time from sunrise to sunrise is a terrestrial civil day (bhūmi-savana-vāsara). The number of terrestrial days in a (maha-yuga) is 1,57,79,17,828. The number of lunar days (tithi-s) in the yuga is 1,60,30,00,080. 

29. A lunar day (rithi) is equal to one-thirtieth of a lunar month (chandra-masa) which is the interval between two conjunctions (or oppositions) of the Moon, being the period of the moon’s synodical revolution, and is reckoned either from new moon to new moon or from full moon to full moon. 

71. Additive months30 and Subtractive days31 (In a maha-yuga) there are 15,93,336 additive months (adhimäsa) and 2,50,82,252 subtractive days (tithi-kşhaya). 

30. A lunar month, extending over 29.5306, days, being shorter than a solar month of 30.4380 days, the number of lunar months in about three solar years would be one more than that of the solar months. This extra lunar month is called adhimasa (additive or intercalary month). 

31. A lunar day (tithi) being smaller than a terrestrial day (bhüdina), if the two begin simultaneously at a sunrise, the lunar day will end earlier. This difference will increase day by day and when it is equal to one day (24 hrs.), it constitutes one tithikshaya or subtractive day. 

72. Solar months32 and lunar months.33 There are (in a maha-yuga) 5,18,40,000 solar months (ravi-mäsa), and the number of lunar months is 5,34,33,336. 

32. The solar month (saura-masa), being one-twelfth of a solar year (saura-varsha) or siderial year (nakshatra-varsha), is the time required by the Sun to pass through one rashi (sign or 30°) of the Zodiac and is equal to 30.4380 terrestrial days. 

33. A lunar month (chandra-masa) is equal to 30 lunar days (tithis) and is equal to 29.5306 terrestrial days. 

73-74. Mandocchas or Apogees of the planets.34 The number of eastward revolutions of the Sun’s apogee (Surya-mandoccha) in a Kalpa period is 387; that of Mars 204; that of Mercury 368, that of Jupiter 900; that of Venus 535; and that of Saturn 39. 

34. Mandoccha is equivalent to the higher Apsis. The mandocchas of the Sun and the Moon are the same as their Apogees (points on the orbit farthest from the planet), while the mandocchas of the other planets are equivalent to their Aphelions (points on the orbit farthest from the Sun). 

74b-76a. Patas or Nodes of the planets,35 as Now, to the retrograde (váma) revolutions of the nodes (of the planets) in a Kalpa: Of Mars it is 214, of Mercury 488, of Jupiter 174, of Venus 903 and of Saturn 662. 

35. Pata (node) is the point at which the orbit of a planet intersects the ecliptic. 

76b-79. Ahargana or number of days from the epoch.36 The (solar) years, called bhagana that have elapsed in the current yuga are converted into (solar) months (by multiplying them by 12) and added to the lunar) months Madhu, (Chaitra) - Shukla (bright fortnight). etc, which have elapsed (in the current year), and the result is written down separately (in two places). It is then multiplied by the number of additive months in a yuga) and divided by the number of solar months (in a yuga). The quotient got (which will be the elapsed additive months) is added to the result (in months) got before and converted into days (by multiplying by 30). The number of days elapsed (in the current month) is added to it and the result written down in two places. (In one place) it is multiplied by the subtractive days (tithikshaya) (in the Yuga) and divided by the number of lunar days in the Yuga). The quotient obtained would be the elapsed subtractive days. These (elapsed subtractive days) are subtracted from the result (kept as above, in the second place). The result would be the number of elapsed terrestrial days (from the commencement of the yuga) to the previous midnight at Lankā.37 

36. The ahargana (lit. ‘total of days’) or bhüdina at any time is the number of (terrestrial) days that have elapsed upto the previous midnight from the time of the commencement of the epoch, which latter is generally taken as the current Yuga. 

37. Lanka is one of the cardinal hypothetical cities on the Earth’s equator where the meridian of the Indian city of Ujjain cuts it. In Indian astronomy, days are reckoned from midnight (or sunrise, according to different systems) at the meridian of Lanka. For a discussion on the position of Lanka, see Aryabhatiya, Ed. with Tr., (Delhi 1976), pp. 123-25. See also below, verses 86b-87a. 

79b-80a. Lords of the day etc. The Lords of the current day, month and year are reckoned, as counted from the Sun. Thus, (the ahargana) divided by 7 and (the remainder) counted from Sunday will give the name of the Lord of the day. (Again, the ahargana) is divided by the number of days in a month and that in a year (viz., 30 and 360); the quotients are then multiplied, respectively, by 2 and 3, and the products increased by 1. The results are divided by 7 and the remainders counted from the Sun will give the Lords of the present month and year, respectively. 

81b-82a. Mean planets. The number of revolutions of a planet (in a Mahāyuga) multiplied by the (currently) elapsed terrestrial days and divided by the number of terrestrial days (in the yuga) will give the elapsed revolutions of the planet (in signs, degrees etc.). 

82b-83a. Mean apogees and nodes. In the same manner, can be computed the mean positions of the apogees with direct motion, mentioned before. The nodes, too, (should be computed) similarly, but the results have to be subtracted from the circle (chakra, 180° or 12 signs) because of their retrograde motion). 

83b-84a. Measurements of the Earth. The diameter of the Earth is 1600 yojanas.38 The square there is to be multiplied by 10, and the square root of the product will give the circumference of the Earth. 

38. 1. A Yojana, according to this, will be about 4.95 miles or 7.56 km. 

Thus, if r is the radius of the Earth, Circumference=2r2x10= 2rx10= Here 10 is taken as as per the formula, cir. = 2r. This works out of 5059.556 yojanas.

84b. The Earth’s circumference multiplied by the sine colatitude (lambajya) (of a given place) and divided by the radius (trijiva)39 so is the exact circumference of the Earth at that place. 

39. Trijivā, called also trijya, tribhajyā, triguna, trirāshijyā, is sine 3 rashis (or 90) and is equal to the radius of the circle.

85-86a. Deshäntara correction to due terrestrial longitude. The Deshāntara (i.e., the distance of the place, in yojanas, along the said local circumference, from Zero or Lanka-Ujjain meridian) is multiplied by the daily motion of the planet (in minutes) and divided by the local circumference of the Earth. The quotient, which would be in minutes (kala), should be subtracted from the mean planet (at Lanka, vide verse 81b-82a, above) if the place is east of the meridian, and added if it is to the west of the meridian. The result would be the mean position of the planet at the given place. 

86b-87a. The central meridian. On the central meridian, which extends from the capital of the demons (Lanka) to the divine mountain (Meru), are the cities of) Avantika (Ujjain), Rohitaka and the one near the ‘Tank’ (Kurukshetra). 

87b-88a. Beginning of the weekday (Varapravrutti). A weekday (at a place) commences (at midnight at that place, which would be) midnight (at Lanka meridian, vide verse 79) to which the deshántara-nädis (time-difference due to terrestrial longitude) arc added (if the place is) to the east and subtracted from (if the place is) to the west (of the meridian). 

88b-89a. Mean-position of a planet at any time. The desired time in nadi-s (after the local midnight as calculated above) multiplied by the mean daily motion of the planet and divided by 60 gives a result in terms of minutes. This, when added (to the mean position at midnight) if the time taken after midnight and subtracted from it before midnight, will give the position (of the planet) at the desired time. 

89b-91a. Vikshepa or Celestial latitude of the Moon and the planets.40 The Moon is deflected by its node towards north and south from the limit of its declination, the maximum deviation being 1/80 of a circle (i.e., 4°30’). Jupiter (is similarly deflected) by twice one-ninth (i.e., 2/9) thereof (i.e., of the deflection of the Moon (i. e., 2/9 of 4°30’=1°), Mars thrice (i.e., 3/9 of 4°30 = 1° 30’) and Mercury, Venus and Saturn arc deflected four times (i.e., 4/9 of 4°30’= 20). 

40. Vikshepa (celestial latitude) is the deviation of the planets from the plane of the ecliptic.

TRUE PLANETS (SPHUȚA-GRAHA)

91b-93a. Primary Sines. The eighth part of the minutes contained in a sign (räshi) is the first sine (jyardha). 41 That divided by itself, the quotient subtracted from the sine and the remainder added to the sine will give the second sine. In the same manner, divide, successively, the sines found by the first sine, subtract (the sum of) the quotients from the divisor and add the remainder to the previous sine. The result will be the next sine. Thus the 24 sines are to be calculated successively.42 

41. Thus: The cycle=360°, räshi or sign = 1/12 x 360°= 30° 1/8 sign = 3°45’ or 225’. This is the first of the 24 signs contained in a quarter (90° or 3 signes). 

42. The second sine, being the sine for the second section (khanda) 225’ +225’=450’, would be: 225/225=1; 450-1=449’. 

The third sine, being the sine for the third section (Khanda), 225’+ 225’+ 225=675’, would be: Present quotient 449/225=2; the sum of the quotients 2+1=3; the sum reduced from the first sine 225=225 - 3 = 222. The result added /to the second sine = 449+222=671, the third sine. 

The process is repeated. The several sines calculated thus are given below: 

No.   Khanda in minutes Sine  No.   Khanda in Minutes Sine

0                0                         000          

1                225                     225   13              2925                  2585

2                440                    449  14              3150                   2728

3                675                     671   15              3375                   2859

4                900                    890  16              3600                  2978

5                1125                   1105 17              3825                  3084

6                1350                   1315 18              4050                  3177

7                1575                   1520 19              4275                  3256

8                1800                  1719 20             4500                  3321

9                2025                  1910 21              4725                  3372

10              2250                  2093 22             4950                  3409

11               2475                  2267   23            5175                   3431

12              2700                  2431 24             5400                  3438

93b-94a. Kranti or declination. The sine of Maximum declination (Parama-apakramajya) is 1397.43 When any sine is multiplied by this and divided by trijivă (sine 90°, i.e., 3438), the arc of the result would be the declination of the planet required).

43. In Hindu astronomy, the maximum declination (obliquity of the ecliptic) is taken as 24’; and 1397 represents sine 24o. 

94b-95a. Sines re. planetary positions. When (the longitude of) the planet is subtracted from that of its mandoccha (higher apsis of the equation of the centre) or from its shighroccha (higher apsis of the equation of conjunction), the remainder is its kendra anamoly); the pada (quadrant) (of the kendra is noted, and from that) its base-sine (bhuja-jya) and perpendicular-sine (koti-jyā) are found.44 

44. While the mandocchas of the Sun and the Moon are the same as as their apogees, that of the other planets are equal to their aphelions. For details on the concept of and computations with mandocchas and Shighrocchas in Hindu astronomy, see E. Burgess, Translation of the Süryasiddhanta, ed. by P. Gangooly, Calcutta, 1935, pp. 53-56.

95b-96a. In an odd (vishama) quadrant the base-sine is reckoned from the part gone by (gata) and the perpendicular sine from the part yet to be covered (gamya). In an even quadrant (sama), the base-sine is reckoned from the part yet to be covered and the perpendicular-sine from the part gone by. 

96b-98a. Derivation of sines of arcs. (To derive the sine of any arc, e.g., the kendra-minus-planet, convert the arc) to minutes and divide by 225 (tattva-lochana); the result would be the number of the preceding tabular sine (jya-pindaka). Multiply the remainder in minutes) by the difference of the preceding and following tabular sines and divide by 225 (tattva lochana). The quotient obtained is added to the preceding tabular sine; the result would give the sine (of the are taken). 

The same procedure is to be adopted also for versed sines (utkramajya). 

100b-101.45 Derivation of arcs from sines. Subtract from the given sine the next less tabular sine; multiply the remainder by 225 tattvāśhvi) and divide by the difference between the next lower and next higher tabular sines. Add the quotient to the product of the serial number of the next less sine and 225. The result would be the arc (of the sine taken). 

45. The Venkateshvara Press edn. of Narada Purana used for this translation, repeats, here, haplographically, two verses, thus erratically increasing serial number of the verses by two. While the repeated verses arc dropped in the translation the increased serial number is retained so that there might not be any discrepancy in the number of the verses in the text edition and the present translation.

101b-103a. Manda-paridhi (Epicycle of the apsis or the equation of the centre). The number of degrees of the Sun’s mandaparidhi is 14, and that of the Moon 32, at the end of the even quadrants; at the end of the odd quadrants, they are 20 minutes less in each case). 

(In the case of the other planets, they are), at the end of even quadrants, 75 for Mars, 30 (for Mercury), 33 (for Jupiter), 12 (for Venus) and 19 (for Saturn). At the end of the odd quadrants, (they arc) 72 for Mars, 28 (for Mercury), 32 for Jupiter, 11 (for Venus) and 48 (for Saturn). 

103b-105a. Shighraparidhi or Epicycles of the equation of conjunction. The shighraparidhis at the end of the even quadrants are 235 for Mars, 133 (for Mercury), 70 (for Jupiter), 202 (for Venus) and 39 (for Saturn). At the end of the odd quadrants, they are 232 for Mars, 132 (for Mercury), 72 (for Jupiter) 260 (for Venus) and 40 (for Saturn). 

105b-106a. Sphutaparidhi or corrected epicycle. The base sine (bhujajyā) should be multiplied by the difference of the epicycles at the odd and even quadrants and divided by the Radius (trijya) and the result, (which would be in minutes), should be applied to the even epicycle (Yugma-vrutta): (these minutes are) additive if the even epicycle is less than the odd epicycle and subtractive otherwise. The corrected (sphuta) epicycle (is thus obtained). 

106b-107a. Mandaphala or Equation of the centre. The base-sine (bhujajya) and perpendicular sine (kotijya) should be multiplied by the corrected epicycle and divided by the number of degrees in a circle (360). (The result would be the corresponding bhujaphala and kotiphala, respectively, in minutes). The arc corresponding to the base-sine (bhujajya) would be the equation of the centre (mandaphala) in minutes etc. 

107b-108. Shighraphala or Equation of conjunction. When the kendra is in the half-orbit beginning with capricorn (makaradi), the result from the perpendicular-sine (koti-phala) of the distance from the conjunction (kendra) is to be added to Radius (trijivā) and subtracted when in that beginning with Cancer (karkyādi). The square of this sum or difference is added to the result from the base-sine (bhujaphala). The square root of their sum is called chalakarna (variable hypotenuse). 

110b-111.46 The result from the base-sine (bhujaphala) is multiplied by radius and divided by the variable hypotenuse (chalakarna). The arc corresponding to the quotient is in minutes and will be the equation of conjunction (shaighrya-phala). This (the Shaighryaphala) is to be employed in the first and fourth process of correction for Mars and other planets. 

46. The Venkateshvara Press edition of Narada Purana used for this translation repeats here haplographically three earlier lines, increasing the serial numbering of the verses correspondingly. Here, the changed verse number is adopted according to the edition; the repeated lines are not translated.

112. Computation of True planets. For the Sun and the Moon, mandakarņa alone is required. That for Mars etc. is now stated: (First) that for conjunction, then that for the apsis, again that for apsis and for conjunction--the four in succession (half the corrections being applied of the first two and the entire correction of the last two). 

113. When the kendra is in the half or bit beginning with Aries (Ajādi) the equation is additive (dhana) for all planets, both in the correction for conjunction and for the apsis; they are all subtractive in the half orbit beginning with Libra (Tuladi). 

114. Bhujantara correction for the equation of time. The daily motion (bhukti) of a planet multiplied by the sun’s result from the base-sine and divided by the number of minutes in a circle (bhachakra). The result, which would be in minutes, is applied to the True planet got (verses 112-13 above) in the same direction as the equation applied to the Sun. 

115-16. Mean daily motion. The equation of a planet’s daily motion is to be calculated like that for the Mean planet in the process for the apsis. The daily motion is multiplied by the difference of the tabular sines corresponding to the base-sine (doriyantara) of anamoly and then divided by 225 (tattvanetra). The result is multiplied by the corresponding epicycle of the apsis (mandaparidhi) and divided by the number of degrees in a circle (bhagana); (the result) is additive when in the half-orbit beginning with cancer, and subtractive when in the half-orbit beginning with Capricorn. 

117-18. Subtract the daily motion of the planet corrected for the apsis from the daily motion of its conjunction (shighra). Multiply the remainder by the difference between the last hypotenuse (antya-karna) and the radius and divide by the variable hypotenuse (chalakarna), verses 107b-108). The result is to be added to the daily motion when the hypotenuse is greater than the radius and subtractive when it is less; (in the latter case, if the result is greater (than the daily motion) subtract the latter from it; the remainder will be the retrograde (vakra) daily motion of the planet). 

119-120a. Retrogression of planets. Mars and other planets would commence to be in retrograde motion (vakri) when the degrees of their kendra in the fourth process (verse 112) are, respectively, (Mars) 164, (Mercury) 144, (Jupiter) 130, (Venus) 163 and (Saturn) 115. They cease to be retrograde from when the degrees (of their kendras) are equal to the above said numbers subtracted from the degrees in) a circle. 

120b-121a. Length of day and night. Sine declination (krantijya) multiplied by the equinoctial shadow (vishuvadbha)47 and divided by 12 is the Earth-sine (kshitijyā).48 This multiplied by Radius (trijya) and divided by the ‘day-radius’ (dina-vyāsa) 49 (gives the sine of the ascensional difference, chara). The correspondic arc (in minutes) would be the ascensional difference in pranas50 (charåsavah). 

47. Vișhuvad-bha or Equinoctial shadow at a place is the shadow of the Sun cast by a gnomon of 12 digits at Midday on the day of the Vernal equipox (March 21) or Autumnal equinox (Sept.23) at that place.

48. Kșhitijya (Earth-sine) is the sine of the arc of the diurnal circle intercepted between the horizon and the six o’clock line. 

49. Dina-vyāsa-dala (day-radius), called also dyu-jya (‘day-sine’) is the radius of the diurnal circle, in contrast to trijya which is the radius of the ‘great circle’ or the ‘tabular radius’. 

50. Prana or Asu (‘respiration’) is the period of time required for one respiration and is /equal to 4 seconds.

121b-122. The said are added to and subtracted from the fourth part of the day and night, separately, will give the duration of half day and half night respectively, when the declination is north (udak-kránti). The reverse would be the case when the declination is south (yāmya-krānti). Double these (half days and nights) would give (the lengths of the day and night, respectively. 

123. Position of a planet in an asterism, 51 (Since 27 nakshatra-s or bham-s or asterisms make up the full ecliptic of 360°), the extent in it of one asterism (bha-bhoga) is 800 minutes (or 13° 20’). (And, since the Moon gains in longitude over the Sun one circle or 360° in 30 lunar days or tithis), the extent of a tithi (tithi-bhoga) is 720 minutes. The asterisms crossed by a planet is got by dividing the longitude of the True planet by 800. (The remainder divided by the daily motion of the planet will give the days etc. (traversed by the planet in the next asterism). 

51. The ecliptic is divided into 27 asterisms or lunar mansions of equal extent of 13°20’ or 800’ each, called Ashvini, Bharani etc. The nakshatra forms one of the members of the five-member, almanac (Panchanga) of the Hindus, the other four members being, Vara (weekday), tithi (lunar day), Yoga and Karana, for which see below.

124. Yoga at a given time.52 The sum of the true longitudes of the Sun and the Moon at the required time, reduced to minutes, if divided by 800, will give the number of Yogas which have elapsed. The portion gone (gata) and to go (gamya) in the current Yoga multiplied by 60 and divided by the sum of the daily motion of the Sun and the Moon will give the corresponding nädikas thereof. 

52. Yoga, one of the members of the five-membered Hindu almanac, is used only for astrological purposes. It is a period of time of variable length during which the joint motion of the longitudes of the Sun and the Moon amounts to 13°20’ or 800’, being the extent of a lunar mansion. They are 27 in number, and are mentioned in Hindu almanacs for each day. They have individual names, as follows: 

1. Vishkambha, 2. Priti; 3. Ayuşhmán; 4. Saubhagya; 5. Shobhana, 6. Atiganda, 7. Sukarma, 8. Dhruti; 9. Shula; 1p, Ganda; ll. Vruddhi; 12. Dhruva; 13. Vyäghåta; 14. Harshana; 15. Vajra, 16. Siddhi, 17. Vyatipata, 18. Variyas; 19. Parigha, 20. Shiva, 21. Siddha; 22. Sadhya; 23. Shubha; 24. Shukla, 25. Brahman, 26. Indra, and 27. Vaidhruti.

125. Tithi at a desired time. Subtract the longitude of the Sun in minutes from the longitude of the Moon in minutes and divide by the extent of a tithi (tithibhoga, 720’); the result will be the tithis elapsed. The nådis gone or to go in the current tithi at the desired time are derived by multiplying the remainder by 60 and dividing by the difference between the daily motion of the Sun and Moon at the desired time. 

126-127. The Karana at the desired time.53 The tithis elapsed after the first half of the first tithi of the bright fortnight are multiplied by two and divided by seven (naga). The remainder, counted as Bava, Bälava, Kaulaka, Taitila, Gara, Vanij and Vishți, would give the elapsed karanas. The karanas from the latter half of the fourteenth tithi of the dark fortnight (to the first half of the first tithi, of the bright fortnght) are Shakuni, Naga, Chatushpät and Kimstughna. 

53. The karana is also an entity made use of only in astrology and forma a member of the five-membered (panchanga), Hindu almanac. Each Karana extends over half a tithi. The four dhruva (‘fixed’) karanas viz., Shakuni, Naga, Chatushpät and Kimstugna occupy the four half-tithis as stated above, and the other seven karanas repeat eight time through the next 56 half-tithis, when the cycle of 60 karanas in a lunar month is completed. The cycle is then repeated.

ON DIRECTION, PLACE AND TIME

128. Setting the gnomon. On a stone slab, levelled with water, or on hard level plaster, describe a circle with any radius measured in gnonomonic digits. 

129-131. At its centre fix the gnomon, of twelve digits of the measure of the gnomonic digits used above. Mark the two points where the (gnomonic) shadow meets the circumference of the circle, before and after noon; these two points are to be called the west and east points, (respectively). Midway between them, draw, (using a pair of compasses), by means of a fish-figure (timi), a north-south line. Also, Midway between the north and south directions, draw, by means of a fish figure, an east-west line. In the same manner, by means of fish-figures (matsya), draw the intermediate directions between the four cardinal directions. 

132. Draw a square circumscribing the circle), along with the (eight) lines emanating from the centre. Any given shadow is reckoned by the digits of its base-sine (bhujā-sūtra) projected on the square. 

133. Prime vertical etc. The east-west line is called the prime vertical (sama-mandala); it is termed also the equatorial horizon (six o’clock circle, unmandala) and equinoctial circle (celestial equator, vishuvanmandala). 

134. Agra (Amplitude). (In the circle) draw another east-west line through the extremity of the equinoctial shadow (vishuvadbha); the interval between any given shadow and the line of the equinoctial shadow is termed the amplitude (agrā). 

135. The square root of the sum of the squares of the gnomon and of the shadow is the hypotenuse (karna). If, from the square of the latter, the square of the gnomon be subtracted, the square roo of the remainder is the shadow; the gnomon is found by the converse process. 

136. Precession of the equinoxes (Ayanachalana), at any desired time. In a Yuga, the circle of asterisms librates eastward thirty score (i.e., 600) times. This (number) multiplied by the terrestrial days elapsed (at the desired time) and divided by the number of days in a Yuga gives the elapsed liberations, (signs, degrees) etc. 

137. Derive the sine (of the ayana-sphuta thus got), multiply it by three and divide by ten. The result will be the amount of the precession of the equinoxes (at the desired time). The longitude of the planets should be corrected (by adding to or subtracting from the longitudes) the said precession. It is from this corrected longitude of the planets) that their declination (krānti), gnomonic shadow (chaya), ascensional difference (chara) etc. are to be calculated.

138. Latitude and co-latitude from shadow. The radius multiplied separately by the gnomon (12) and the equinoctial shadow (chaya) and divided by the equinoctial hypotenuse (vishuvatkarna)54 will give the cosine and sine, respectively, of the latitude. The arcs of these sines will be the co-latitude and latitude. These two will always be inclined to the south. 

54. Vishuvatkarna, or equinoctial hypotenuse, is the hypotenuse of the equinoctial shadow found by calculating the square root of the sum of the squares of the equinoctial -shadow and the gnomon.

139. Zenith distance of the Sun. The Zenith distance of the Sun (at any time) would be the sum of the latitude of the place and the Sun’s declination, when both are in the same direction; and, their difference when in opposite directions. From the Zenith distance, its sine and cosine are to be found. 

140. The shadow and its hypotenuse from Zenith distance. The sine (of the Zenith distance, found as above) and the Radius 55 multiplied by the length of the gnomon in digits (i.e., 12) and divided by the cosine (of the Zenith distance), give, respectively, the shadow of the gnomon and its hypotenuse at midday (ahardala). 

55. By ‘Radius’ (with capital ‘R’) is meant the radius of the great circle or the tabular radius. In Sanskrit it is called trijya, trijiva, tribhajya etc., meaning the sine of three sines or 90°.

141. Sun’s declination and longitude from latitude of a place and zenith distance. Find the difference between the degrees of the latitude (of the place) and those of the Sun’s zenith distance at noon when both are in the same direction and their sum, otherwise. The sun’s declination (apakrama) would be detained. 

141b-143a. Multiply sine declination by the Radius and divide by the maximum declination of the Sun (i.e., 1397 minutes). (Taking the quotient as the sine) find its arc. This arc will be the longitude of the Sun (in the first quarter of the ecliptic) beginning with Aries. When the Sun is in the second quarter) beginning with Cancer, subtract (the arc) from 6 signs (chakrārdha). When (in the third quarter beginning with Libra), add (the arc) to 6 signs (bhardha). When (in the fourth quarter) beginning with Capricorn subtract (the arc) from 12 signs (chakra); in each case, the result will be) the true longitude (sphuta) of the Sun at midday. 

143b. Mean Sun. Apply to this the equation of the apsis (mandaphala) repeatedly in the opposite sign and the Sun’s mean longitude will be got. 

144. Ahoratrasu of a planet. The diurnal motion (in minutes) of any planet (on any day) is to be multiplied by the number of pranas (asus or respirations, of time) contained in the rising periods of the sign (räshi) occupied by the planet and divided by 1800. The quotient added to the number of pranas in a circle is termed the day-night duration in pranas of the planet (on that day). 

145-146. Right ascensions of the Signs at Lanka. (Towards finding the right ascensions of the ends of the first three signs, Aries, Taurus and Gemini, find the declinations of the said ends), multiply the day-radius of three signs (tribhadyukarnárdha) and divide by their own respective day-radii (svähoraträrdha), in order, the sines of one, of two and of three signs. The quotients when converted into arc and subtracted, each from the one following, give, beginning with Aries, the times of rising (in pranas) (of the three signs) at Lanka. They are, respectively, 1670, 1795 and 1935. 

147a. Right ascensions at any place. The above, diminished each by its portion of ascensional difference (charakhanda), as calculated for a place, give the times of rising at that place. 

147b-148a. For the three signs beginning with Cancer, invert the times of rising at Lanka and add the portions of ascensional difference of the respective signs inverted. The above six, in inverse order, will be the times for the other six commencing with Libra. 

148b-149a. Udaya-Lagna (rising point of the ecliptic) at any time. The ascensional equivalents of the parts of the sign (occupied by the planet) which are gone (bhukta) and to come (bhogya) are to be calculated from the longitude of the Sun at the given time. They will be given by the number of degrees traversed and to be traversed, multiplied by the ascensional equivalent (udayāsavah) of the sign and divided by 30. 

149b-150a. From the desired time in nädikas (as reckoned from sunrise) reduced to pranas, subtract the equivalent in pranas of the part of the sign to come, and also the ascensional equivalents of the further signs, in succession. In the same manner, subtract the equivalents of the part which has gone by and of the signs which have to go, in inverse order. 

150b-151a. If there be a remainder, multiply by 30 and divide by the equivalent of the unsubtracted sign; subtract or add the quotient, appropriately, to the whole signs. The result would be the point of the ecliptic at the horizon at that time. 

1516-152a. Madhyalagna (Point of the ecliptic at the meridian). From the east or west hour-angle of the Sun in nadis (nata-nädi), calculate as above using the equivalents in right ascension (lankodayasavah) and apply the result as an additive or subtractive equation to the Sun’s longitude. The point of the ecliptic upon the meridian (Madhya-lagna) at that moment would result. 

152b-153-a. Time from Lagna. (In order to find the instant when a given point of the ecliptic would be upon the horizon), add together the ascensional equivalents, in pranas, of: (1) the part of the sign to be traversed by the given point (on the ecliptic) if it is less (than the longitude of the Sun), (2) of the part traversed if it is greater than the longitude of the Sun), and (3) of the intervening signs. The sum in pranas) will give the instant (for the given lagna).

SOLAR AND LUNAR ECLIPSES

153b-154a. Possibility of a lunar eclipse. (Compute the True Sun, Moon and Node at the syzygies). If sine (Sun-minus-Node) is less than 14 (Indra), there is a possibility of an eclipse. The said sine in minutes is to be multiplied by 11 (Shiva) and divided by 7 (shaila). The result is called Shara in digits (angulas), and would be directed towards the hemi sphere in which (Sun-minus-Node) lies. 

154b. Eclipser and the eclipsed. (In the solar eclipse, which occurs only at new moon), the Moon obscures the Sun; and in the lunar eclipse, (which occurs only at full moon), the shadow of the Earth obscures the Moon. 

154b-155a. Half the (angular) diameters of the eclipsed and the eclipser minus the shara is called channa (ka), (the eclipsed portion, reckoned in digits).56 Subtract, (if possible), the eclipsed body from the channa; the result would be kha-cchanna (empty space eclipse). 

56. It might be noted that if the shara is larger and cannot be subtracted from half the (angular) diameters of the eclipsed and the eclipser, there will be no channa (eclipsed portion) and so no eclipse.

155b-156b. Computation of the lunar eclipse. Half the angular diameters is to be added to the shara, the sum multiplied by ten and divided by channa. The square root (of the product) is found and a sixth of it is subtracted from it and divided by the angular diameter of the Moon (glau-vapuh). The result will give the half-duration of the eclipse (sthityardha) in ghatikā etc. 

156b-158a. The sthityardha (is placed at two places). Sine Sun-minus-Node in degrees is doubled and the number taken as palas. They are subtracted from or added to the sthityardha if Sun-minus-Node is less than 6 signs or 12 signs, respectively, and vice versa if otherwise. The results got would be the true sthityardha (as reckoned) from the commencement and completion of the eclipse. The eclipsed portion (in digits) multiplied by 20 (nakha) and divided by the angular diameter of the eclipsed body will give the results called Vi (m) shopaka-s. 

158b-159. Computation of the solar eclipse. (In the solar eclipse), the mid-eclipse occurs at the moment of conjunction. Calculate the tribhona-lagna (rising point of the ecliptic-minus- three signs) of the Sun for the moment of conjunction (parvänta) and keep it apart. (For the tribhona-lagna) calculate the parallax in latitude in degrees (natāmsha) by combining, appropriately, the declination (krānti) and latitude (aksha). Divide it by 22 and square the result. If the square is less than 2, halve it and add to the square; if the square is more than 2, subtract 2 from it, halve the result and add to the square. The result thus got is added to 12 and the sum is called ‘divisor (hara). 

160. Find the difference between tribhona-lagna and the Sun, reduce it by a tenth thereof, multiply by 14 (purandara) and divide by the ‘divisor’ (found in verse 159). The result would be the Lambana (parallax in longitude, in nādikas). If tribhona-lagna is more than 12 (arka), the lambana is positive and, if otherwise, negative. 

161. The lambana-nädikas are multiplied by 13 (viśhva); the result is to be reckoned in terms of minutes and added to or subtracted from Sun-minus-Node appropriately and (from its sine) its shara is determined (vide verse 154a). The lambana therefrom is multiplied by six, After appropriate addition and subtraction thereof, re. tribhona-lagna, the natāmshas are again calculated 

162-163a. A tenth part of these natamshas are subtracted from 18 and the result multiplied by the said tenth part. The product is subtracted from 18 minutes and divided by 6. The result will give parallax in longitude (nati) in the same direction of the (previous) natāmshas. Appropriate application of this subtractively or additively in accordance with opposite directions or the same direction of the two, will render the shara accurate. 

163b-165a. Using the said shara, the (exact) eclipsed portion (channa) and half duration of the eclipse are to be found (as before). The half-duration is multiplied by 6; the result, which would be in degrees, is kept at two places and the tribhona-lagna subtracted from or added to it, respectively, and the lambana calculated from the two, as before. The results being appropriately applied, the exact (first and second) half durations are obtained. These subtracted from and added to, as the case may be to the mid-eclipse (madhya-kala), will give the exact times of the commencement and the conclusion of the eclipse.

MISCELLANEOUS MATTERS

165b-166a. Heliacal visibility of the planets57. The kālāmsha (degrees of time) of the planets beginning with the Moon are: 12 (Moon), 17 (Mars), 13 (Mercury), 11 (Jupiter), 9 (Venus) and 15 (Saturn). The heliacal setting and rising of these planets occur, respectively, at times got by subtracting (the kālāmshas) from sunrise and by adding them to sunset. 

57. For a little time before sunrise and after sunset, the planets, near the Sun, would be invisible on account of the sun’s brilliance. The times after sunset and before sunrise they would be visible are indicated here.

166b-167a. Shadow of a planet. Consider the reflection of a planet (in water or in a mirror) and ascertain the ocular altitude (drugauchyam lambam). The distance between the foot of the gnomon and the spot of the reflection, multiplied by 12 (ravi) and divided by the ocular altitude, will give the measure of the shadow of the planet (in digits). 

167b-168. Elongation of the Moon’s horns. At sunset (on the desired day) the tithi and its divisions (nådikā, vinādikā etc.) gone and to go are accurately determined. The tithis are multiplied by 16 and the square of the tithis subtracted therefrom. The result is multiplied by the equinoctial shadow (akshabha) and divided by 15. The direction of the result is taken as north and corrected appropriately with the declination (in minutes) of the Sun and also by shara of the Moon in minutes applied reversely. The result is divided by twice the tithis. The valana (deflection) in digits towards the direction of the correction (i.e., the direction in which the Sun is with reference to the Moon) is got. 

169. Subtract from the tithi one-fifth of itself; (the measure of) the illuminated part of the Moon (phase, sita) is obtained. The born of the Moon will be elongated in the direction of valana. The measure in digits of the elongation shall be ascertained by means of a diagram. 

170-171a. Conjunction of the planets. The numbers 5, 6, 7, 9 and 5 (corresponding to the five planets Mars etc.) are multiplied by Trijyā-minus-the respective shighrakarna and (placed at two places. One set is divided, respectively by 21, 12, 6, 24 and 3 and the result is applied to other set, i.e., subtracted if the hypotenuse is larger than trijya and added otherwise. The results divided by 3 will be the angular diameters of the discs of the several planets beginning from Mars. 

171b-172a. Time of conjunction. (When it is necessary to determine when two planets will meet): If the two planets are both regular or retrograde, the difference in their longitudes is divided by the difference of their rates of motion; if one is regular and the other retrograde, (the difference of their longitudes) is divided by the sum of their rates of motion. The result in days etc. would give the time to pass for their conjunction. 

172b-173. Mutual non-obscuration of planets in conjunction. (The two planets in conjunction) should be corrected by their respective parallaxes in latitude (nati) according to their sharas, it being addition if in the same direction and subtraction otherwise. When the north-south difference of the two planets is less than half the sum of their diameters, there will be (visual) distinction between the two, (there being no obscuration). True (sphuta) results can, of course, be arrived by working with the parallax in longitude (lambana) etc. as in the case of the solar eclipse.

VYATIPĂTA AND VAIDHRTA: MALIGNANT SITUATIONS OF SUN AND MOON

174. Vaidhruta.58 When the Sun and the Moon are in the same ayana, i.e., on the same side of either solstice, if the sum (of their longitudes) is one circle (360), and both have equal declinations, that situation is called Vaidhruta. 

58. This section, called Patadhikara (Section on patas) in treatises, treats of two astronomical situations called Vyatipata and Vaidhruta which are highly malignant in character. They are computed and used for astrological purposes, viz., to avoid auspicious acts being held at these situations.

175. Vyatipata. When the Sun and the Moon are on the opposite sides of either solstice and have equal minutes of declination, that situation is called Vyatipata, the sum (of their longitudes) being a half-circle. 

176. Time when declinations are equal. When the longitudes of the Sun and the Moon corrected by the degrees of precession of the equinoxes as found by observation, is equal to 12 signs or 6 signs, respectively, compute their declinations. 

177. Then, if the declination of the Moon, which is in an odd quadrant, and had been corrected for its latitude (Vikshepa), is greater than the declination of the Sun, the situation of pata is already past. 

178. If less, it is still to come. In an even quadrant, the reverse is the case. If the Moon’s declination is subtractible from its latitude, the rules as to the quadrant are to be reversed. 

179. Multiply the sines of the two declinations separately by Radius (trijya) and divide by the sine of the maximum declination (parakrantijyā, viz., 1397) and find the arcs (of the two results). The difference of the arcs) or half that difference is to be added to the Moon’s longitude when the pata is yet to occur. 

180. It is to be subtracted from the Moon’s longitude when the pala is past. The said (difference) when multiplied by the Sun’s daily motion and divided by the Moon’s daily motion, gives a correction, in minutes, to be duly applied to the longitude of the Sun, (being subtracted from the Sun if the pata is past and added to it if the pata is yet to occur). 

181. A similar correction is to be applied, in the reverse order, to the Moon’s mode. This operation is repeated until the declinations of the Sun and the Moon) are equal. 

182. Mean time of the Pata. The pata occurs at the moment when the declinations are equal. (To find out whether a pata is past or yet to come): If the Moon’s longitude at the pata found by applying the correction to the Moon (vide verse 180) is less or greater than (the Moon’s longitude) at midnight (of that day), the pata is past or is yet to occur, respectively. 

183. True time of Pata. The minutes of the interval between the Moon’s longitude (at the Mean pata), as ascertained, and that at midnight, when multiplied by 60 and divided by the Moon’s daily motion, will give the time of the pata in nädikas. 

184. Half-duration of the Pata. Multiply half the sum of the diameters of the Sun and the Moon by 60 and divide by the difference of their daily motions. The result will be the half- duration, (sthityardha (of the pata), in nādika etc. 

185. Beginning, Middle and End of the Pala. The true time (of the pata (vide verse 183) is the middle (moment of the pata). That diminished by the half-duration (vide verse 184) is the moment of its commencement and that increased by the half-duration is the moment of its end. 

186. Consequence of the Pata. Compared to the beginning and the end, the middle moment of the pala is extremely harmful. It is like burning fire and is to be avoided in all (auspicious) rites. 

187. O brahmana! Thus, has been stated, in brief, matters relating to mathematics and astronomy (ganita). I shall, now, set out horoscopy (jataka) commencing with the statement of) the nomenclature of the räshi-s (signs).