MOST ANCIENT GEOMETRY BOOK IN THE WORLD
BAUDHAYANA SULBASUTRA
One of the greatest ancient India vedic geometry books. Vedas have their six vedangas. They are Siksha, nirukta, vyaakarana, chadassastra, kalapasaastra and jyothisha. Among these kalapasaastra has four branches gruhyasoothra , dharma sootra, sroutha soothra and pitrumedha sootra. The sroutra sootra deals with all the customs and rituals connected with yaagas and yagyaas. SULBASUTRAS are the part of kalpasutra, which is the vedic geometry for building fire altars and all the goemtrical patterns are mentioned insuoba sutas. There are four major sulba sutras which bhauddhayanam, aapasthambam, kaatyaayanam and maanavam . the oldest one is the Baudhaayana sulba sutram which might have composed in the present form during 1000 Bc or before.detailed description of each line is given here which given in the book of INDIAN NATIONAL SCIENCE ACADEMY, New Dehi -2



1.1 The various constructions of sacrificial fires are now given.
1.2 We shall explain the methods of measuring areas of their (different) figures
(drawn on the ground.
1.3 Now, the measure of an Angela is 14 anus (grain of Panicum milliaceum); according to others, (it is ) 34 tilas (sesamum indicum) placed broad side on. One small pada is 10 angulas; one pradesa 12 angulas; one pytha and one uttarayuga 13 angulas each; one (big) pada 15 angulas. One isa measures 188 angulas; one aksa 104 angulas; one yuga 86 angulas; one janu 32 angulas; one samya and onebahu 36 angulas each. One prakrama equals 2 padas (30 angulas); one aratni 2 pradesas (24 angulas). But there are also instances of pada, yuga, prakrama, aratni and samya having different measures when these (words) are used as units of measurement, 5 aratnis (120 angulas) make one purusa; one uyama also has the same measure (5 aratnis); and 4 aratnis (96 angulas) make one uyayama.

1.4 Having desired (to construct) a square, one is to take a cord of length equal to the (side of the ) given square, make ties at both ends and mark it at its middle. The (east-west) line (equal to the cord) is drawn and a pole is fixed at its middle. The two ties (of the cord) are fixed in it (pole) and a circle is drawn with the mark (in the middle of the cord). Two poles are fixed at both ends of the diameter (east-west line). With one ties fastened to the eastern (pole), a circle is drawn with the other. A similar (circle) about the western (pole). The second diameter is obtained from the points of intersection of these two (circles); two poles are fixed at two ends of the diameter (thus obtained). With two ties fastened to the eastern (pole) a circle is drawn with the mark. The same (is to be done) with respect to the southern, the western and the northern (pole). The end points of intersection of these (four circles) produce the (required) square.
1.5 Now another (method). Ties are made at both ends of a cord twice the measure and a mark is given at the middle. This (halving of the cord) is for the east-west line (that is, the side of the required square). In the other half (cord) at a point shorter by one-fourth, a mark is given; this is the nyancana (mark). (Then) a mark is given at the middle (of the same half cord) for purposes of (fixing ) the corners (of the square). With the two ties fastened to the two ends of the east-west line (prsthya), the cord is to be stretched towards the south by the nyancana (mark); the middle mark (of the half cord) determines the western and the eastern corners (of the square).
1.6
When (the construction of) a rectangle is desired, two poles are fixed on the ground at a distance equal to the desired length. (This makes the east-west line). Two poles one on each side of each of the (two above mentioned) poles are fixed at equal distances (along the east-west line). A cord equal in length to the breadth (of the rectangle) is taken, its two ends are tied and a mark is given at the middle. With the two ties fastened to the two end poles (on either side of the pole) in the east, the cord is stretched to the south by the mark; at the mark (where it touches the ground) a sign is given. Both the ties are now fastened to the middle (pole at the east end of the praci), the cord is stretched towards the south by the mark over the sign (previously obtained) and a pole is fixed at the mark. This is the south-east corner. In this way are explained the north-east and the two western corners (of the rectangle).
1.7 When the eastern side is desired to be of shorter measure, a mark is given at half (the tiryanmant).

1.8Now another (method). Tiles are made at both ends of a cord of length equal to the measure increased by its half (so that the whole length of the cord is divided into three parts of half the measure each). In the third (extended) part on the western side a mark is given at a point shorter by one-sixth (of the third part); this is the nyancana. Another mark is made at the desired point for fixing the corners. With the two ties fastened to the two ends of the east-west line (prsthya), the cord is stretched towards the south by the nyancana, and the western and eastern corners (of the square) are fixed by the desired mark.
1.9The diagonal of a square produces double the area (of the square).
1.10 The breadth (of a rectangle) being the side of a given square (pramana) and the length the side of a square twice as large (dvikarani), the diagonal equals the side of a square thrice as large (trkarani).
1.11Thereby is explained the side of a square one-third the area of given square (trtiyakarani). It is the side of a square one-third the area of the square (explained in the preceding rule, that is, of the square on the tykarani).
1.12The areas (of the squares) produced separately by the length and the breadth of a rectangle together equal the area (of the square) produced by the diagonal.
1.13This is observed in rectangle having side 3 and 4, 12 and 5, 15 and 8, and 4, 12 and 35, 15 and 36.
2.1
If it is desired to combine two squares of different measure, a (rectangular) part is cut off from the larger (square) with the side of the smaller; the diagonal of the cut-off (rectangular) part is the side of the combined square. (Alternatively: If it is desired to combine two squares of different measures, a rectangle is formed with the side of the smaller (square) (as breadth) and that of the larger (as length); the diagonal of the rectangle (thus formed) is the side of the combined square).
2.2
If it is desired to remove a square from another, a (rectangular) part is cut off from the larger (square) with the side of the smaller one to be removed; the (longer) side of the cut-off (rectangular) part is placed across so as to touch the opposite side; by this contact (the side) is cut off. With the cut-off (part) the difference (of the two squares) is obtained.
2.3
A square intended to be transformed into a rectangle is cut off by its diagonal. One portion is divided into two (equal) parts which are placed on the two sides (of the other portion) so as to fit (them exactly).
2.4
Or else, if a square is to be transformed (into a rectangle), (a segment) of it is to be cut off by the side (of the rectangle); what is left out (of the square) is added to the other side. (Like Asl. 3.1 the rule is defective and does not lead to proper geometrical operation).
2.5
If it is desired to transform a rectangle into a square, its breadth is taken as the side of a square (and this square on the breadth is cut off from the rectangle). The remainder (of the rectangle) is divided into two equal parts and placed on two sides (one part on each). The empty space (in the corner) is filled up with a (square) piece. The removal of it (of the square piece from the square thus formed to get the required square) has been stated.
2.6
If it is desired to reduce one side of a square (that is, to make an isosceles trapezium) the reduced side is to be taken as the breadth (of a rectangular portion to be cut off from the square); the remaining part (of the square) is divided by the diagonal and (one half), after being inverted, is placed on the other side.
2.7
If it is desired to transform a square into (an isosceles) triangle, the square whose area is to be so transformed is doubled and a pole fixed at the middle of its east side; two cords with their ties fastened to it (the pole) are stretched to south-western and north-western corners (of the square); portions lying outside the cords are cut off.
2.8
If it is desired to transform a square into a double (isosceles) triangle (that is, rhombus), a rectangle twice as large as the square to be so transformed is made; a pole is fixed at the middle of its east side; two cords with their ties fastened to it (the pole) are stretched to the middle points of the southern and northern side (of the rectangle); portions lying outside the cords are cut off; thereby the (isosceles) triangle on the other side is explained.
2.9
If it is desired to transform a square into a circle, (a cord of length) half the diagonal (of the square) is stretched from the center to the east (a part of it lying outside the eastern side of the square); with one-third (of the part lying outside) added to the remainder (of the half diagonal), the (required) circle is drawn.
2.10
To transform a circle into a square, the diameter is divided into eight parts; one (such) part after being divided into twentynine parts is reduced by twentieth of them and further by the sixth (of the part left) less the eighth (of the sixth part).
2.11
Alternatively, divide (the diameter) into fifteen parts and reduce it by two of them; this gives the approximate side of the square (desired).
2.12
The measure is to be increased by its third and this (third) again by its own fourth less the thirty-fourth part (of that fourth); this is (the value of) the diagonal of a square (whose side is the measure).
3.1
Now, the placement of the ahavaniya from the garhapalya in the rrangement for the laying of sacrificial fires (will be discussed). According to tradition, the Brahmana has to place this fire (ahavaniya) (at a distance of) 8 prakramas, the prince 11 prakramas and the merchant 12 prakramas (from the garhapatya towards east).
3.2
Three squares of side one-third the distance (between the ahvaniya and the garhapatya) are made so as to be in contact with each other (along the east-west line); the garhapatya (fire) lies at the north-west and the daksinagni (anavaharyapa cana) at the south-east corner of the western square; the north-east corner of the eastern square marks the place of the ahavantya.
3.3
Alternatively, the distance between the garhapatya and the ahavaniya is divided into five or six (equal) parts, a sixth or a seventh part is added, the whole (of the cord measuring the original distance plus the added part) is divided into three (equal) parts, and a mark is given at the end of the second part from the eastern extremity. (With two ties) fastened to (poles at) the two ends of (the distance between) the garhapatya and the ahavaniya, the cord is stretched to the south by the mark and a pole fixed at (the spot reached by) the mark. This is the place of the daksinagni.
3.4
Or else, the measure (between the ahavaniya and the garhapatya) is increased by its fifty, the whole of it is divided into five parts, and a mark is given at the end of the second part from the western extremity. With two ties fastened to (poles at) the two ends of the east-west line (representing the distance between the two fires), the cord is stretched to the south by the mark and a pole fixed at (the spot reached by) the mark. This is the place of the daksinagni.
3.5
The utkara is explained by doing the opposite (that is, by reversing the cord and stretching it to the north).
3.6
To the west of the ahavaniya, as per tradition, is the altar for the new and full moon sacrifice (darsapaurnamasa), measuring 96 angulas (yajamanamatril) (in the east-west direction).
3.7
This (measure) less its third (64 angulas) forms the western side (of the altar) and half the measure (48 angulas) the eastern side; after making in this way a rectangle shorter on the side, poles are fixed at the (four) corners.
3.8
A tie is given at each end of a cord twice as long as the side (of the above altar) and a mark at the middle. With two ties fastened to (poles at the two ends of) the southern side, the cord is stretched to the south by the mark and a pole fixed at (the spot reached by) the mark. Fixing the two ties at this (pole), the southern side is circumscribed (with an are of a circle from the end to end) by the mark. Thereby the northern side is explained. The eastern side is circumscribed in the same way by a cord double its length, and likewise the western side.
3.9
The tradition has it that the altar for animal sacrifice (pasubandha) has 10 padas on its western side, 12 padas as its east-west line (praci) and 8 padas on its eastern side; how it is to be measured out has been explained. According to some, the alter is measured with the measures of a chariot (that is, with aksa (104 angulas) for the western side, isa (188 angulas) for the praci, and yuga (86 angulas) for the easterm side). According to others, the sides are 10 padas each.
3.10
According to tradition, the uttara vedi is four-cornered and measured (on each side) by a samya (36 angulas); in the absence of any particular direction, it is a square.
3.11
According to tradition, the paitrki vedi (used for performing pitr-rites) is formed with the third part. The third paart of the mahavedi is runed into a square; the side which produces on-third of this square makes that (of the paitrki vedi). Therefore, it is on-ninth of the area (of the mahavedi). According to others, its side measures 96 angulas (yajamanamatri) and it is four-cornered, the corners being pointed to the four cardinal direction.
3.12
For performing the sautramani sacrifice, the alter is advised to have an area one-third of the mahavedi. If a third part of the mahavedi is turned into a square its side will be 18 padas. It can also have, if one so desires, a shape in which one side is shorter and the other larger.
4.1
The sacrificial chamber (pragvamsa) is 16 prakramas long by 12 prakramas broad, or else 12 prakramas long by 10 prakramas broad.
4.2
(A length of) 12 prakramas is left in the middle between the sacrificial fires.
4.3
According to tradition, the mahavedi measures 30 padas or prakramas on its western side, 36 (padas or prakramas) along the east-west line and 24 (padas or prakramas) on its eastern side; how it is to be measured out has been explained. The mahavedi is 6 prakramas from the ahavantya (fire towards east).

4.4
The sadas (shed) lies 1 prakrama from there (east of the western edge of the mahavedi) and is 10 prakramas wide (in the east-west direction) and 27 aratnis, according to another opinion, 18 aratnis long in the south-north direction.
4.5
The havirdhana (shed for the soma-vehicles) lies 4 prakramas (to the east) from there; it is a square of 10 or 12 prakaramas; how it (such a square) is to be measured out has been explained.
4.6
The uttara vedi is measured out at a distance of half a prakrama to the west of the pole of the yupavata (sacrificial post fixed in pit). According to sima sacrifice, the uttara vedi measures 10 padas; how it is to be measured has been explained.
4.7
The catvala (pit in the ground) measures 36 angulas, or it may have any undefined measure.
4.8
The uparavas (holes over which the soma is ground) are each 1 pradesa long, the distance between two of them being 1 pradesa. A square of side equalling 1 aratni is made, poles are fixed at the (four) corners, and a circle of radius equal to half pradesa is drawn (with each pole at the corner as center).
4.9
Situated at a distance of 2 prakramas from the eastern half of the sadas (shed), the dhisnya (fires) are each 2 pradesas in diameter and separated from each other by the same distance (of 2 pradesas).
4.10
The side of the (covered) place for (kindling) the agnidhra (sacrificial fire) is 5 aratnis.
4.11
Thereby the marjaliya (covered place for cleansing sacrificial vessels) is explained; its door is made on the northern side.
4.12
The pits for sacrificial posts are (placed) at intervals of 1 aksa (104 angulas) and there are eleven of them as per tradition. The twentyfourth part of the sum of 10 aksas, 11 padas and 8 angulas is the prakrama. With this the altar is to be measured.
4.13
For the asvamedha (horse sacrifice), the twentyfourth part of the sum of 20 aksas, 21 padas and 8 angulas is the prakrama. With this the altar is to be measured.
4.14
For the making of 11 pits along the eastern side, a strip of breadth half a pada is cut off from the eastern half of the mahavedi and placed east of it in the east-west direction. In this (operation) 8 angulas are not taken into account, and there is no mutual connection.

4.15
The pits for the sacrificial posts are 1 pada (each) in diameter; the circumference of the base of the pits is 3 padas.
5.1
The area of the fire-altar made for the first time is 71/2 square purusa; that for the second time 81/2 (square purusa); that for the third time 91/2 (square purusa). Thus it begins with the 7-fold fire-alter (71/2 square purusa) and ends with the 101-fold.
5.2 Thereafter, to continue further, the 101-fold (fire-altar) is to be repeated (that is, after reaching101-fold, no further increase is to be made). Otherwise, the sacrificial rite is to be performed without a fire-altar.
5.3
The asvamedha (sacrifice) is an excpetion. If the asvamedha (requiring a firealtar of 21 ½ square purusa) is performed without (the required agni) being reached, one fold is added to get the next higher fold (that is, 22 ½ sq. purusa agni); no other procedure is allowed.
5.4
If (the required fire-altar is) surpassed, the fire-altar following the one surpassed is to be constructed.
5.5
But how is one fold to be added?
5.6
The excess (to be added) to be original form (of the fire-altar) should be divided into 15 parts and two parts be added to each fold (of 1 sq. purusa; after 13 parts are in this way added to 7 folds of 7 sq. purusa, the remaining part is added to ½ sq. purusa). The (new) fire-altar is to be laid with such (increadred) 7 ½ folds.
5.7
The height (of the fire-altar), according to some teaching, should be increased by the twenty-fourth part of the fifth of a janu (32 angulas).
5.8
Some construct the fire altar from one fold (1 ½ sq. purusa) upwards (upto 6 ½ sq. purusa) without wings and tail
5.9
This is not justified because it contradicts earlier and later precepts
5.10
In this connection some Brahmanas maintain that among the fire-altars the making of the falcon-shaped one is the first sacrificial ceremony
5.11
Other Brahmanas maintain that after having constructed alarger fire-altar a smaller one should not be laid
5.12
Our Brahmana teaches as dollows: he is winged for the unwinged cannot fly; the two wings are longer (than 1 purusa in each case) by 1 aratni, and this makes the birds strong in their wings; the two wings and the tail measure 1 vyama (eah).

5.13
A falcon without wings and tail does not exist; so the fire-altar which is not seven-fold has neither wings nor tail; moreover, the construction of one-fold fire-altar after the seven-fold has been laid is inadmissible; for all this the seven-fold is the fire-altar to be made for the first time.
5.14
The clefts are to be avoided; the meetings of edges (between bricks) in the upper and lower layer constitute these clefts, as per teachings. Such clefts, however, do not exist either in the peripheries or the two sides of a corner of the fire-altar.
5.15
One thousand bricks are to be used when (the fire-altar is) constructed for the first time.
5.16
This number is to be completed in the fifth layer.
5.17
Where two hundred bricks are desired (for each layer), pancacoda and nakasat (bricks) are to be counted together as one (that is, one of each type together to be considered as one brick).
6.1
The fire-altar indeed possesses the characteristics of an animal. The southern bones of an animal are on its southern (right)side; likewise its northern bones lie on its northern (left) side and vice versa. That (part) which is below (on the western side) is the same as what is above (on the eastern side). In the same manner bricks of different forms are to be placed (in the fire-altar).
6.2
(Bricks marked with) lines turned to the right are placed on the southern side, those with lines turned to the left on the northern side, those with straight lines on the east and the west side, and those with three lines in the middle (of the fire-altar). The placing (of the bricks) in the middle (along the east-west line of the fire-altar) is to be understood in the same way as the backbone of the animal which does not lie more on one side than the other (but passes strictly along the middle of its body).
6.3
On this the Brahmana has it that Prajapati indeed is Atharvan and Agni is Dadhyan, son of Atharvan, and the bricks are his bones.
6.4
(In a fire-altar) where exterior limbs (such as head, wings, tail, feet etc). are to be fitted (to the body, that is, the atman of the altar), the middle of the side of the limb (concerned) is to be joined to the middle of that side of the body with which it (the limb) is to be in contact.
6.5
According to tradition, it (the fire-altar) is to be laid (with its head) towards the east.
6.6
The number of bricks is not to be completed with those which are not made of clay or which are not bricks.

6.7
As per teachings of this Brahmana, one fire-altar is laid with bricks, another with animals.
6.8
For the fire-altar has the characteristics of an animal: the yoni of an animal is of different forms; before laying the bricks, the sacrificial formulas from the Yajas text are recited.
6.9
Things occupying space are to be placed in holds (in the ground).
6.10
By (diagrams in the form of) circle, bull, woman, signs made on the bricks are to be understood.
6.11
If the number of sacrificial formulas (recited) falls short of the number of bricks, the difference is to be made good by (the sacrificial formulas called) lokamprnas because their number is unspecified.
6.12
Types of bricks previously used are to be placed here.
6.13
(There are) five lokamprnas (in every fire-altar).
6.14
If the number of sacrificial formulas exceeds (the number of bricks), anointed pebbles are to be placed in the interstices (between the bricks).
6.15
By the statements ‘he places (the bricks) to the east’, he places (the bricks) to the west’ are meant the placement of types of bricks in a straight line (towards the specified direction).
6.16
By the statements ‘he places (the bricks) to the east’, he places (the bricks) to the west’ are meant the directions faced by the constructor (of the fire-altar).
6.17
(Bricks) in the east are to be placed oppositely towards west and those in the west oppositely towards east; such is the rule of restriction.
6.18
This sort (of arrangement) is suitable for a square (fire-altar with four corners).
7.1
One should not use (for the laying of the fire-altar) a broken brick, a brick which is cleft, a blackened brick (due to over or under heating), a damaged brick and a brick with scratching marks. In the layer where a brick full of natural holes (svayamatrnna) is used it is not be covered (by a brick).
7.2
The height of the brick is to be made a fifth of the janu (that is, 62/5 angula); that of the nakasal and the pancacoda half of that measure (that is, 31/5 angula).

7.3
What is lost by drying and burning is to be made good by loose earth because of the flexibility of its quantity.
7.4
According to tradition, the garhapatya fire has the measure of 1 vyayama.
7.5
It (garhapatya fire) is a square by one tradition and a circle by another.
7.6
The (garhapatya fire in the form of) square is to be divided into 7 parts (lengthwise) and then into 3 parts transversely. In the second layer, bricks are to be placed towards north (that is, the division in the first layer as aforesaid is to be reversed).
7.7
To place square bricks (instead of rectangular ones as indicated above), (square) bricks of sides one-sixth, one-fourth and one-third of 1 vyayama are made. Of them, 9 bricks of the first type and 12 of the second are placed in the first layer; 5 of the third type and 16 of the first are placed in the second layer.
7.8
Within the (garhapatya fire in the form of) circle a square of the maximum size possible is drawn and divided into 9 parts (squares). The segments of the circle (between the circumference and the square) are divided into 3 parts each. The second layer is placed in such a way that the corners (of the square within the circle) lie at the centres of the segments (of the first layer).
7.9
The dhisnya fires are one-layered in the form of a square or a circle.
7.10
Of these (dhinya fires), the agnidhriya is divided into 9 parts, and in one part a stone is to be placed (instead of a brick).
7.11
The dhisnya fire of the hotr priest is divided into 9 parts and the 3 parts on the eastern side are divided into 2 parts each.
7.12
The others (dhisnya fires) are each divided into 9 parts and then two parts, one in the center and the other in the east, are combined.
7.13
Now the marjdltya fire is divided into 3 parts and then the eastern and the western parts (taken together) are divided into 5 parts.
7.14
The bricks are to be made by mixing with ashes from the caldron.
7.15
This applies to those whose consecration lasts a year and not to those undergoing it for a smaller number of nights.
7.16
Thus (the laying of ) the fire-altar is completed with the sacrificial formulas (recited by the priest).

7.17
After (a fire-altar made of) three thousand (bricks), a fire-altar to be piled with metres (of the sacred hymns) is to be laid. This is because of a difference in the wish. This (fire-altar) is falcon-shaped, as it is natural (for all such fire-altars).
8.1
Now he who desires heaven is to construct a fire-altar in the form of a falcon.
8.2
It is of two different forms; one has its body in the form of a square and the other in the form of a falcon.
8.3
This is the tradition of the both the Brahmanas
8.4
Five (bricks) are placed in the southern corner and five in the northern. ‘(Let there be) the strength of the goat’,– with these (words) he places (the bricks) in the southern corner; ‘(Let there be) the strength of the bull’,–with these he places in the northern corner; ‘(Let there be) the strength of the tiger’,–with these he places in the northern wing; and ‘(Let there be) the strength of the man’,–with these he places in the middle. (This is one Brahmana).
8.5
The other Brahmana is: The fire-altar is that which is constructed in the likeness of the birds, that is, after the shadow cast by them while flying.
8.6
‘To distinguish between the divine and the uman (purposes), the fire-altar is to be constructed with square bricks’,–thus teaches the Matrauani Brahmana.
8.7
For (constructing) this (five-altar), (square) bricks (of sides) the fourth, the fifth, the sixth and the tenth part (of a purusa, that is 30, 24, 20 and 12 angulas respectively) are made.
8.8
Now the (area of the) fire-altar is measured not
8.9
Two holes are made on a bamboo rod at a distance equal to the height of a man with uplifted arms; a third hole is made at the middle. What (measurement) is done elsewhere with the cord is done here with the bamboo rod.
8.10
The body (of the fire-altar) is a square of 4 purusa; its (southern) wing is a square of 1 purusa made longer on the southern side by 1 aratni and its northern wing is explained in the same way; its tail is a square of 1 purusa lengthened on the western side by 1 pradesa. Thus, with the addition of (two) aratris and (one) pradesa, the seven-fold (fire-altar of 7 ½ sq. purusa) is accomplished.
8.11
In the placement (of bricks) at distance of one-third purusa (40 angulas) to the north from the end of the (southern) wind, 4 bricks of side equal to one fifth (of a purusa) and 2 quarter bricks (one-fourth of the area of a pancami brick, that is, 12 sq. angula) (are placed). Thereafter 8 bricks of side equal to one-fourth (of a purusa) (are placed). The remaining space of the (southern) wing is covered with bricks of side equal to one-sixth (of a purusa). Thereby the northern wing is explained.
8.12
Bricks of side equal to one-fourth (of a purusa) are to be placed on the eastern and the western side of the tail, and quarter bricks on its southern and northern side. The remaining space of the fire-altar is to be covered with bricks of side equal to one-fifth (of a purusa).
8.13
This is one layer of 200 bricks.
8.14
In the other layer, at a distance of half a vyayama (48 angulas) to the north from the (southern) end of the wing, 3 rows of 3 bricks each of side equal to one-sixth (of a purusa) alternating with 2 rows of 2 bricks each of side equal to 2 padas (30 angulas) are placed. The same (is done) for the northern (wing).
8.15
In the south-western corner (of the body), 9 bricks of side equal to one-sixth (of a purusa) are arranged in the form of a square; the same (is done) for the north-western corner.
8.16
9 bricks of side equal to one-sixth (of a purusa) alternating with 2 bricks of side equal to 2 padas are to be placed from the south-eastern corner (of the body) to the north-eastern.
8.17
The rest of the fire-altar is to be covered with bricks of side equal to one-fifth (of a purusa).
8.18
This is (another) layer of 200 bricks. (With these two types) alternating with each other, as many layers as desired are to be constructed.
9.1
Now another type (of square syenacit).
9.2
(For this are required bricks of side equal to) one-fifth (pancami) of a purusa; (those) with one side longer by half (adhyardha) of one-fifth (of a purusa); (these equal to) a half of its sixe (ardhya of the pancami); and (those equal to) a quarter of its size (padya of the pancami).

9.3
In the placement (of bricks), half bricks (half of pancami) turned towards north are placed on the eastern and the western side of the (southern)wing; the same (is done) for the northern (wing).
9.4
4 bricks longer by half and turned towards north (are placed) on each of the southern and northern side of the tail; 4 half bricks turned towards north (are placed) on the western side of the tail; and 2 quarter bricks on either side of them (that is, in two corners of the tail’s west end). I half brick turned towards east (is to be placed) at each of the two places where the tail is joined with the hind part (of the body) (that is, at two corners of the eastern side of the tail).
9.5
The rest of the fire-altar is to be covered with pancami (bricks).
9.6
This is one layer of 200 bricks.
9.7
In the other layer, 4 quarter bricks (are placed) on each in the 4 corners of the body; 2 half bricks on two sides of them (in each corner); 5 half bricks on the eastern front (of the body).
9.8
At the end of each wing, 3 bricks longer by half (are placed) oriented towards north and 1 half brick (is placed) in each of the interstices between them (adhyardha bricks).
9.9
The rest of the fire-altar is to be covered with pancami (bricks).
9.10
This is (another) layer of 200 bricks. (with these two layers) alternating with each other, as many layers as desired are to be constructed.
10.1
Now (is described the construction of a fire-altar in the form of a falcon) with curved wings and extended tail.
10.2
Bricks for this (fire-altar) are made with side equal to one-fourth (caturthi) of a purusa; (then those equal to) a half of its size (ardhya of caturthi) and a quarter of its size (padya of caturthi). The cutting (of the caturthi brick to obtain its half and quarter) is always to be done diagonally in the absence of any advice to the contrary.
10.3
(Then one should make) quarter bricks (with the same area as that of a caturthi-padya) bounded by four sides (measuring) ½ pada, 1 pada, 1 ½ pada and /2 PADA. Two of them touching other along their long sides (1 ½ pada) are made into (another) half brick (called hamsamukhi, swan beaked, with the same area as that of a caturthi-ardhya).
10.4
The fire-altar is now measured out. The body is 2 purusas (240 angulas) in length by 10 padas (150 angulas) in breadth. From its south-eastern corner towards north a mark is given at a distance of 1 ½ prakrama (45 angulas); the same (is done) towards west. By stretching a cord over these (two marks), the (south-eastern) corner is to be cut off. Thereby is explained the cutting off of other (three) corners. This makes the body (atman).
10.5
The head is of 5 ½ padas (82 ½ angulas) in length by ½ purusa (60 angulas) in breadth. The two eastern corners of it are cutt off with 1 prakrama (30 angulas).
10.6
The tail measures 6 padas (90 angulas) in the east-west and 2 purusas (240 angulas) in the south-north direction. The two eastern corners of it are cut off with 3 prakrmas (90 angulas) each.
10.7
The (southern) wing is of 12 padas (180 ngulas) in length (along north-south) and 10 padas (150 angulas) in breadth (along east-west). A pole is fixed at a distance of 6 padas (90 angulas) to the east from the middle (of its western side) and at each of the two south-western corners (of the rectangular wing). With a cord (stretching between these three poles, a triangular area) is to be enclosed. The (triangular) area enclosed by the cord is to be cut off and placed in the eastern side (of the wing) (with its vertex) pointing towards east. This is the bending (of the wing). Thereby the bending of the northern wing is explained.
10.8
At the end of each wing, 5 squares of side equal to 1 prakrama (30 angulas) are laid (in a row) so as to be in contact with each other; all of them are intersected diagonally in the downward direction (by joining the northeast corner to the south-east); and a half portion is removed (from each square).
10.9
Thus, with the addition of (two) aratnis and (one) pradela, the seven-fold (fire-altar of 7 ½ sq. purusa) is accomplished.
10.10
In the placement (of bricks), 1 caturthi is to be placed in the head of its junction (with the body) and 1 hamsamukhi (swan-beaked) to the east of it. 2 quarters bricks are placed on two sides (of the hamsamukhi), 3 four-sided quarter bricks below them on each side (of the head), the quarter bricks in the remaining space (of the head).
10.11
Alternatively, 1 hamsamukhi brick is to be placed at the (eastern) extremity of the head and 1 caturthi below it, to be flanked by 1 quarter brick on either side, 3 four-sided quarter bricks are placed to the west (of these two quarter bricks) and on each side (of the head) and quarter in the remaining space (of the head).

10.12
5 quarter bricks mutually joined with one another are to be placed to the west of the head (on the eastern end of the body) and the same to the east of the tail (on the western end of the body). Half bricks as well as quarter bricks are to be placed in the truncated parts.
10.13
The rest of the fire-altar is to be covered with caturthi bricks. The number (of 200 bricks) is to be completed with quarter bricks and half bricks.
10.14
This is one layer of 200 bricks
10.15
In the other layer, 4 hamsamukhi bricks are to be joined with 4 quarter bricks so as to form a rectangle; this is placed breadth-wise in the space of the svayamatrnna
10.16
At the junction of the tail (with the body), 2 hamsamukhi bricks, (with their vertices) turned towards west and their ½ pada sides lying with the body, are to be placed and below them and on both sides 3 quarter bricks (with their vertices) turned towards east.
10.17
At the western end of the tail 15 quarter bricks mutually joined with one another are to be placed
10.18
In the plumages of the wing 2 quarter bricks alternating with 1 half brick are to be placed (from the west) to the east.
10.19
In the truncated areas at the joints (between the body and the wing, the bendings of the wing etc.,) half bricks and quarter bricks are to be placed.
10.20
The rest of the fire-altar is to be covered with caturthi bricks. The number (of 200 bricks) is to be completed with quarter bricks and half bricks.
11.1 Now another type (of falcon-shaped fire-altar with curved wings and extended tail.
11.2
(In this case) the seven-fold (fire-altar) with (two) aratnis and (one) pradesa is accomplished with 187 ½ (square bricks) of side equal to one-fifth of a purusa (pancami).
11.3
The body can accommodate 52 of such (pancami bricks), the head 3 ½, the tail 15, southern wing 58 ½ and the northern wing the same (number of bricks as the southern).
11.4
The corners (of the rectangular body) are cut off with ½ vyayama (48 angulas); the tail is inclined; the bending of the two wings is done with 3

11.5
The different types of bricks (required for this fire-altar) are as follows.
11.6
Bricks of side equal to one-fifth of a purusa (pancami, 24 ang,X 24 ang.); bricks of which one side is longer by half (adhyardha) (36 ang. X 24 ang.); bricks of which one side is longer by a quarter (sapada, 30 ang. X 24 ang.); bricks which are quarter in size of those with side equal to one-fifth (of a purusa) (pancami-padya); bricks which are half on size (of the above, e.g., pancami ardhya); likewise, of bricks with side longer by half (that is, half and two types of quarter bricks made out of adhyardhas); triangular bricks made by joining two eighth parts, one from each of them (the eighth part of a pancami to be joined with the eighth part of an adhy;ardha, called ubhayi); and bricks of which one eighth the size of those with side equal to one-fifth (of a purusa). These are the ten (different types).
11.7
The pancami bricks and their halves are to be placed in the body and the same in the tail.
11.8
The adhyardha bricks and their halves (are to be placed ) in the two wings.
11.9
In the head (are to be placed) such bricks as are possible (as can be accommodated).
11.10
In the other layer, 1 ubhayi brick (formed by combining the eighth part of a pancami with the eighth part of an adhyardha) is to be placed at the easterm end of the (line of) junction of each wing (with the body); 1 ubhayi brick each at the western end (of the same line of junction); and 2 ubhayi bricks are to be placed on each side of the head.
11.11
At the western end of the tail, adhyardha bricks (with the longer side) turned towards east, and at the two sides (western corners), bricks of size one-fourth and one-eighth (of a pancami) (are to be placed).
11.12
Adhyardha bricks and parts thereof (are to be placed) in the two wings.
11.13
The remaining space (of the fire-altar) is to be filled with bricks such that these fit, the required number (of 200 bricks in the layer) is attained and the properties (of the fire-altar) are satisfied.
12.1
The body and the tail of the kite-shaped fire-altar (kankacit) are explained in the same manner(as those of the iyenacit just described).

12.2
5(pancami) bricks are to be accommodated in the head whose shape has been explained.
12.3
57(pancami) bricks are to be accommodated in the southern wing and the same in the northern.
12.4
The bending of the two wings is done with 1 vyaydma plus 1 pradela (that is, with 108 angulas), 6 plumages (at each end of the two wings) are to be formed with 6- pancami half bricks. An area equivalent to) 1 ½ pancami is left.
12.5
With this (area left out), two feet each measuring 1 aratni (24 angulas) long by 1 pradesa (12 angulas) broad are made on the western end of the tail at a distance of 1 aratni from each other; at each side of the western end (of each foot) 2 bricks of size one-eighth (of the pancami) (are placed).
12.6
Thus, with the addition of (two) aratnis and (one) pradesa, the seven-fold (fire-altar of 7 ½ sq. purusa) is accomplished.
12.7
The different types of bricks (required for this fire-altar) are as follows: bricks of side equal to one-fifth (of a purusa) and parts thereof (half, quarter and one-eighth of pancami bricks); quarter bricks (having the area of a quarter pancami) bounded by four sides (measuring) 1 ½ pradesa (6 angulas). 1 ½ pradesa (18 angulas), 1 pradesa (12 angulas) and XX/2 pradesa (12 X X/2 angulas); adhyardha bricks (having the area of 1 ½ pancami) bounded by four sides (measuring) ½ vyayama (48 angulas), 1 aratni (24 angulas), 1 aratni (24 angulas) and /XX2 aratni (24 /XX2 angulas). These make six (types).
12.8
Of them, four-sided quarter bricks together with the one-eighths are placed in two feet, and the measuring space is to be filled with bricks such that these fit, the required number (of 200 bricks in the layer) is attained and the properties (of the fire-altar) are satisfied.
13.1
The body, the head and the tail of the fire-altar in the form of an alaja bird are explained in the same manner (as those of the kankacit) with the two feet withdrawn.
13.2
63 (pancami) bricks are to be accommodated in the southern wing and the same in the northern.
13.3
The bending of the two wings is done with 1 purusa (120 angulas).
13.4
From ( a pole fixed at a distance of ) 1 aratni (24 angulas) towards east from the western bend a, a cord is streched along the (line of)intersection of the westernmost plumage, and (the part lying west of the cord) cut off.
13.5
In this was (an area equivalent to) 5 ½ pancami (bricks) stands removed.
13.6
1 quarter b rick is placed at each western bend (to fill up the triangular void caused by the aforesaid removal). Out of the brick types the four-sides quarter b ricks and the one-eights are to be taken away and the remaining space (of the fire-altar) is to be filled with (remaining types of) bricks su;ch that these fit, the required number (of 200 bricks in the layer) is attained and the properties (of the fire-altar) are satisfied).
14.1
A fire-altar in the form of an isosceles triangle (prauga) is to be constructed as follows.
14.2
An isosceles triangle equal in area to the (seven-fold) fire-altar with (two) aratnis and (one) pradesa (that is, 7 ½ sq. purusa) is laid. Bricks (called brhati) of length equal to one-twelfth of its western side and breadth equal to half (of the length) are to be made; then bricks which are half and quarter (of the brhatils).
14.3
Of them, two bhalf bricks with their hypotenuses turned outwards are to be placed in the apex and half bricks on both sides.
14.4
The rest of the fire-altar is to be covered with brhati and the number (of 200 bricks) is to be completed with half bricks.
14.5
In the other layer, 47 quarter bricks mutually joined with one another are to be placed on the western side (of the triangle).
14.6
1 sulapadya (short-based quarter brick) in the apex (is to be placed).
14.7
4 quarter bricks, – 2 wide-based (dtrghapadya) and 2 of the other type (e.g.short-based, sulapadya), are to be placed in the space of the suayamatrnna, and half bricks on the two sides.
14.8
The rest of the fire-altar is to be covered with brhali bricks (with length) turned towards east, and the number (of 200 bricks) is to be completed with half bricks.
15.1
A fire-altar in the form of a rhombus (made of two isosceles triangles, (ubhayata prauga) is to be constructed as follows:
15.2
A rhombus equal in area to the (seven-fold) fire-altar with (two) aratnis and (one) pradesa (that is, 7 ½ sq. purusa) is laid. As in the case of fire-altar in the form of isosceles triangle, bricks and their variations are to be made with the ninth part of the breadth (of the rectangle used for the construction of the rhombus).
15.3
The placement (of bricks in the first layer) is the same as before (as that of the isosceles triangle).
15.4
In the second layer, 2 (short-based) quarter bricks are to be placed in the apices and 2 (wide-based) quarter bricks at the meeting places (of the two isosceles triangles).
15.5
4 quarter bricks, -2 wide-based (dilrgha-padya) and 2 of the other type (iulapadya) are to be placed in the space of the suayamatrnna and half bricks on the two sides.
15.6
The rest of the fire-altar is to be covered with brhati bricks (with length) tu;rned towards east, and the number (of 200 bricks) is to be completed with half bricks.
16.1
According to tradition, a fire-altar in the form of a chariot wheel is to be construced.
16.2
The chariot wheels are indeed of two types, e.g. those with spokes and those (formed) by the joining of circular segments (to a central square piece). In the absence of any distinction between the two, both are taken into consideration and described.
16.3
Now, the (area of the) fire-altar is measured out. A circle of area equal to that of the (seven-fold) fire-altar with (two) aratnis and (one) pradesa is made, the largest possible square is inscribed in it, and bricks are made with the twelfth part of its side.
16.4
6 of these (bricks) are placed in each circular segment and the remaining space (of the segment) is divided into 8 parts.
16.5
The other layer is to be so oriented that the corners (of the square) lie in the centres of the segments (of the first layer).
16.6
Now the other type.
16.7
Square bricks of area equal to the fifteenth part of half a (square) purusa are made for purposes of measurement.
16.8
With 225 of them (of such bricks) is accomplished the seven-fold (filre-altar) with (two) aratnils and (one) pradesa.
16.9
To these (225 bricks) another 64 (bricks of the same kind) are added and with them (289 bricks) a square is made (as follows). (At first) a square is made with a side containing 16 bricks (in which 256 bricks are used up), leaving a balance of 33 bricks. These (33 bricks) are placed on all sides (actually on two adjoining sides, so as to obtain a square of side containing 17 bricks).
16.10
16 (bricks) at the center constitute the nave; 64 (bricks thereafter) constitute the spokes and 64 the empty spaces (between spokes); the remaining (145 bricks) from the felly.
16.11
(The square shaped) nave at its borders is transformed into a circle (by the method previously decribed). The outer and the inner (squares) enclosing the felly are transformed into (two) circles. After dividing the space between the felly and the nave into 32 equal parts, the alternate ones are removed. In this way, the added area (equivalent of 64 bricks) stands withdrawn.
16.12
After dividing the felly into 64 equal parts and drwing (radial) lines (through these divisions), a circle is drawn through the middle (of the felly), making the number (of parts in the felly) equal to 128.
16.13
The spokes are each divided into 4 parts; the nave is divided into 8 parts.
16.14
This is the first layer (of 200 parts or bricks).
16.15
In the other layer, a circle is to be drawn within the nave at a distance equal to one-fourth (of the radius) from its inner edge. The same (is to be done) within the felly (at a distance equal to one-fourth of the felly’s breadth) from the inner circumference.
16.16
After dividing the inner edge of the felly (that is, the circle drawn within it) into 64 equal parts, (radial) lines are to be drawn (so as to divide the felly into 64 parts)..
16.17
(The space in each of) the spokes is divided into 5 parts from circle (in the nave) to circle (in the felly).
16.18
The space in each interstice of the felly is divided into 2 parts, and there is 1 part in each interstice of the nave.
16.19
The remaining space of the nave is to be divided into 8 parts.
16.20
These are the 16 types (of bricks required) in (the construction of) the fire-altar in the form of a chariot wheel with spokes.
17.1 According to tradition, a fire-altar in the form of a trough is to be constructed.

17.2
The troughs are indeed of two types, e.g. the square-shaped and the circular. In the absence of any distinction between the two, both are taken into consideration.
17.3
Now the (area of the) fire-altar is measured out. The body is a square of side equal to 2 2/3 purusas.
17.4
Its handle lies at the western side (of the body) and is ½ purusa and 10 angulas (that is, 70 angulas) long towards east and 2/3 purusa (80 angulas) broad towards north.
17.5
Thus, with the addition of (two) aratnis and (one) pradesa, the seven-fold (fire-altar of 7 ½ sq. purusas) is accomplished.
17.6
The different types of bricks (required for this fire-altar) are as follows: bricks of side equal to 1/6 purusa (sasthi); bricks of side longer on one side by half (adhyardha), half bricks (of the sasthi) transversely cut; and bricks of side equal to ¼ purusa (caturthi).
17.7
Of these, 6 sasthi bricks are placed on each of the two parts of the western side (of the body) between the handle and the corner, the rest of the fire-altar is to be covered with brhati (that is, adhyardha) bricks, and the number (of 200 bricks) is to be complted with half bricks.
17.8
In the other layer, 1 adhyardha is to be placed in the south-eastern corner and the same on the north-eastern.
17.9
Sasthi bricks are to be placed on the easterm front (between the 2 adyardhas).
17.10
Bricks of side equal to ¼ purusa (caturthi) are to be placed on the southern and the northern side (of the body).
17.11
2 caturthi bricks are to be placed on each corner of the east side of the handle, 2 adyardha bricks turned towards north-south below them on each side, and 2 sasthi bricks below them in the middle along east.
17.12
The rest of the fire-altar is to be covered with brhati (adhyardha) bricks turned towards east and the number (of 200 bricks) is to be completed with half bricks.
18.1
Now the other type (of dronacti in the form of a circle).
18.2
120 (square) bricks, each 1/16 of a (square) purusa (sodasi, side = ¼ pu. Or 30 ang.) give the area of the seven-fold (fire-altar of 7 ½ sq. purusa) with (two) aratnis and (one) pradesa.

18.3
One of them is taken away, and (the area equivalent to) the remaining (119 bricks) is transformed into a circle.
18.4
This (kind of transformation into circle) has been explained in the case of the fire-altar in the form of a chariot wheel of the first type.
18.5
The sodasi (bricks which is taken away) is placed in the middle of the east side (of the square equivalent to the area of 119 bricks) and with it the circle (of the same area as that of the square) is to be drawn.
18.6
The western part (of the sodasi brick) cut off (by the circle) is placed on its eastern side.
18.7
The (four) circular segments (obtained by drawing the maximum possible square within the above circle and after placing 6 bricks of side equal to 1/12 of the side of the inscribed square on the base of the segment, as in the case of the chariot wheel) are each divided into 7 parts.
18.8
Bricks in the middle of the segments are each 1 prakrama (30 angulas) wide.
18.9
The number (of 200 bricks) is to be completed by bricks half of the square bricks (made with 1/12 of the side of the inscribed square).
18.10
In the other layer, the brick in the middle of the (eastern) segment is placed in the lip (-shaped handle) and the space below it is divided into 2 equal parts.
18.11
This is the fire-altar in the form of a circular trough involving (the use of) nine types (of bricks).
18.12
The construction of the samuhya and the paricayya (fire-altars) is explained in the same way as that of the fire-altar in the form of the chariot wheel as already discussed.
18.13
Pits are dug out in the (four) cardinal directions of (the space to be occupied by) the samuhya, and the earth is collected from them and plaed on the samuhya (instead of the bricks).
18.14
The positioning of bricks in the paricayya is different (from that of the chariot wheel); these are placed all around in (concentric) circles turning towards right.
19.1
According to tradition, a fire-altar in the form of a pyre (smasanacit) is to be constructed.

19.2
The entire (area of the) fire-altar is divided into 15 square-parts. How to do this has been stated.
19.3
A rectangle is made with its length equal to thrice (the side of) the square-part and breadth equal to half (the side of the square-part). Lines are drawn from the middle of its eastern side to the two western corners and the two outer parts are removed (so as to obtain an isosceles triangle). This (isosceles triangle) is divided into 10 parts.
19.4
The entire fire-altar is composed of 20 of them (of such isosceles triangles).
19.5
In the other layer, one of the (five) isosceles triangles (into which the entire fire-altar can be divided) is to be vertically bisected. Each half is divided into 6 parts. These two (half isosceles triangles each divided into 6 parts) are to be placed on either side (the southern and the northern side of th fire-altar).
19.6
Bricks of length equal to one-third (the side of the square part) and breadth equal to one-fourth (the side of the square part) are to be made, and then half of such bricks by transverse bisection.
19.7
After placing them (the half bricks) on the two ends (the eastern and the western), the rest of the fire-altar is to be covered with the brhati bricks turned towards east, and the number (of 200 bricks) is to be completed with half bricks.
19.8
The height of the fire-altar is to be increased by one-fifth (of the janu, that is, 62/5 angula).
19.9
The whole of it (the height including the added 5th part) is divided into three parts and bricks are made with the fourth or the ninth or the fourteenth part of the two of these three parts (according as the fire-altar is intended to have 5,10 or 15 layers).
19.10
With these bricks, 4 or 9 or 14 layers are made, the remaining layer (of thickness equal to one-third of the height) is diagonally cut in the downward direction and half of it removed.
19.11
Its division is exact. Larger and smaller bricks are taken according as these fit.
20.1
According to tradition, a fire-altar in the form of a tortoise (kurmacit) is to be constructed by one desiring to win the world of the supreme Spirit (Brahmaloka).

20.2
The tortoises are indeed of two types, e.g. those with twisted limbs and those with rounded ones. In the absence of any distinction between the two, both are taken into consideration and described.
20.3
The fire-altar is measured out as follows. The body is a square of side euqal to 10 prakramas (300 anguals); its corners are cut off with 2 prakramas (60 angulas) on each side.
20.4
4 squares each of side equal to 1 prakrama (30 angulas) are made in (the middle of) the eastern front and 2 of them lying at either extreme are cut off by their diagonals. The same is done on the southern, western and northern front. This is the body.
20.5
The head is 5 padas (75 angulas) long by ½ purusa (60 angulas) broad, of which the two eastern corners are to be cut off with 1 prakrama (30 angulas) on each side.
20.6
The feet are to be raised where the corners (of the body) have been cut off. The foot (in the south-eastern) is 2XX padas (30XX2 angulas) broad by twice that measure (that is, 60XX2 angulas) long and its eastern coner is cut off by 2XX2 padas. Thereby is explained the cutting off of the other (three) feet. Of the two feet at the western (corners of the body), their western corners are to be cut off.
20.7
Thus, with the addition of (two) aratnis and (one) pradesa, the seven-fold (fire-altar of 7 ½ sq. purusa) is accomplished.
20.8
(Square) bricks of side equal to one-fourth of a purusa, and their halves and quarters (obtained by dividing the first type diagonally) are to be made for this (fire-altar).
20.9
(Then one should make) adhyardha quarter (that is, one-fourth of acturthis longer on one side by half) bricks bounded by four sides (measuring) 1 prakrama, 1 pada, 1 pada and XX2 pada.
20.10
Two of them touching each other along their long sides (1 prakrama) are to be made into another (type of) brick (hamsamukhi).
20.11
(Then one should make) another (type of ) square brick of side equal to half of 2X2 pada (that is, X2 or 15X2 angulas).
20.12
In the placement (of bricks), a square brick (of side equal to 15X2 angulas) is to be placed at the top of the head, followed by two hamsamukhi (bricks) below it.
20.13
5 square bricks and 2 quarter bricks (pancami-padya) are to be placed in each foot.

20.14
Half bricks are to be placed wherever a corner has been cut off.
20.15
The rest of the fire-altar is to be covered with caturthi bricks. The number (of 200 bricks) is to be completed with half bricks.
20.16
In the other layer, I hamsamukhi at the top of the head and 1 quarter brick on either side of it are to be placed.
20.17
To the west of these on each side (of the head) 2 (four-sided) adhyardha quarter bricks oppositely oriented are to be placed.
20.18
To the west of these on both sides are to be placed 2 quarter (caturthi-padya) bricks in alignment with the intersection.
20.19
2 caturthi (here called dvipadas or squares of side equal to 2 padas or 30 angulas) and 3 half bricks are to be placed in each foot.
20.20
Half bricks and quarter bricks are to be placed wherever a corner has been cut off.
20.21
The rest of the fire-altar is to be covered with caturthi bricks. The number (of 200 bricks) is to be completed with half bricks.
21.1 Now the other type (of kurmacit with rounded limbs).
21.2 120 (square) bricks of side equal to 1/16 purusa (sodasi) give the area of the seven-fold (fire-altar of 7 ½ sq, purusa) with (two) aratnis and (one) pradesa).
21.3 Of them 5 sodasi bricks are taken away, and (the area equivalent to) the remaining (115 bricks) is transformed into a circle. This (kind of transformation into a circle) has been explained in the case of the fire-altar in the form of a trough of the second type.
21.4
With 5 sodasi bricks, (four) feet in (four) intermediate directions and the head in the east are to be raised. How to round off these bricks (for fitting with the circular body) has been explained.
21.5
The (four) circular segments (obtained by drawing the maximum possible square within the above circle and after placing 6 bricks of side equal to 1/12 of the side of the inscribed square on the base of the segment as in the case of the trough) are each divided into 7 parts. Bricks in the middle of the segments are each 1 prakrama (30 angulas) wide.
21.6
The resulting bricks in excess (of 200) are adjusted by square bricks longer by half on one side (adhyardha).

21.7
In the other layer, the feet are divided in the same manner as the head (in the first layer) and the head is divided like the feet (as in the first layer).
21.8
Alternating with each other as many layers as desired are to be constructed.
21.9
A little loose earth is to be placed on the edge of the tortoise (shaped fire-alter) and a large quantity in the middle of it. In the (fire-altar in the form of) trough, the opposite is the case.
21.10
According to some other (teachers) the fire-altars beginning with the one-fold should be (in the form of) issoceles triangles.
21.11
According to some other (teachers), (these should be in the form of) squares. Bricks are to be made with 1/12 of the side of the square and half and quarter bricks thereof.
21.12
The increase (in the area) of the fire-altar for the asvamedha (sacrifice) takes place by the purusa and not by (two aratnis and (one) pradesa.
21.13
This (asvamedha fire-altar) is original and three times as large. Being three times as large, this fire-altar is twentone-fold, and such is the tradition contained in both Brahmanas.

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DISCOVERY AND USE OF ZERO Gaayathre shadsankhyaamardhe apaneethe dvayanke avasishtasthrayastheshu roopamapaneeya dvayankaadha: soonyam sthaapyam In gayatri chandas, one pada has six letters. When this number is made half, it becomes three (i.e the pada can be divided into two). Remove one from three and make it half to get one. Remove one from it, thus gets the zero (Soonya). PINGALACHARYA IN CHANDA SASTRA 200 B.C.



CALCULATIONS WITH ZERO Vikaaramaayaanthi dhanarunakhaani na soonya samyoga viyogathasthu soonyaaddhi suddham swamrunam kshayam swam vadhaadinaa kham khaharam vibhakthaa: Nothing happens (to the number) when a positive or negative number is added with 0. When +ve and -ve numbers are subtracted from 0, the +ve number becomes negative and -ve number becomes +ve. When multiplied with 0, the values of both +ve and -ve numbers become 0, when divided by 0, it becomes infinity (khahara). SRIPATI IN SIDDHANTHA SEKHARA 1039 AD Yathaa ekarekhaa sathasthaane satham dasasthane dasaiam chaikasthaane yathaa cha ekathvepi sthree mathaa cha uchyathe duhithaa svasaa cha ithi In the unit place the digit has the same value, in 10th place, 10 times the value and in 100th place 100 times the value, is given. VYASA BHASHAYA TO YOGA SUTRA 650 AD

DISCOVERY OF PLACE VALUES – II
Yathaachaikaapi rekha sthaananyathvena nivisamaanaika dasa satha sahasraadi sabda prathyaya bhedhamanubhavathi
One and the same numerical sign when occupying different places is conceived as measuring 1, 10, 100, 1000 etc.
SANKARACHARYA VEDANTA SUTRA BHASHAYA

KNOWLEDGE ON INFINITY
Asmin vikara khahare na raasaavapi praveshteshvapi ni: srutheshu bahushvapi syaallaya srushtikaalenanthe chyuthe bhoothaganeshu yaddhath
Nothing happens to the (huge number) infinity, when any number enters (added) or leaves (subtrated) the infinity. During pralaya many things get dissolved in Mahavishnu and after pralaya, during srushti all those things get out of him. This happens without affecting the lord himself. Like that, whatever number is added to infinity or whatever is subtracted from it, the infinity remains unchanged.
BRAHMAGUPTHA IN BRAHMASPHUTA SIDDHANTA 600 AD
BHAKARACHARYA II – BEEJAGANITA 1148 AD

USE OF AVERAGE VALUES
Ganayithva visthaaram bahushusthaneshu thadyuthirbhaayyaa sthaanakamithyaa samamithirevam dairgye cha vedhe cha
(For length, breadth and depth) the measurements should be taken at many places and the sum should be divided by the number of times (places) the measurement is taken. BHASKARACHARYA II IN LILAVATI 1150 AD

USE OF FRACTIONS
Drammaardha thrilavadvayasya sumathe paadathrayam yadbhaveth that panchaamsaka shoda saamsa charana: sampraarthithenaa- rthinaa datto yenavaraatakaa: kathi kadaryenarpithastena me broohithvam yadi vetsi vatsaganitha jaathim prabhagaabhidhaam
One man has given to a beggar fraction of 1 dramma (a unit of money). That fraction is one fourth of the one sixth of one fifth of the three fourth of the two third of the half of a dramma. Then tell how much kowdi (a unit fraction of the amount dramma) was given to the beggar?
BHASKARACHARYA I – ARYABHATEEYA BHASHAYA 628 AD

USE OF RATIO AND PROPORTION
Ashtow daanthaa sthryo damyaa ithi gaava: prakeerthi thaa: ekaagrasya sahasrasya kathi daanthaa: katheetharai:
(Out of 11 cattle) Eight are tamed and 3 are to be tamed and (how many are) to be tamed) if the number of cows is 1001?
BHASKARACHARYA I – ARYABHATEEYA BHASHAYA 628 AD

PERMUTATIONS AND COMBINATION – I
Katukathiktha kashaayaamla lavana madhurai: sakhe rasai: shadbhi: vidadhaathi soopakaaro vyanchanamaachakshva kathibhedam
Friend, a cook prepared varieties of food with 6 savours: pungent, bitter, astringent, acid, saline and sweet. Say what is the possible number of varieties of food that can be made with these savours.
SRIDHARACHARYA IN PATIGANITA 990 AD

PERMUTATIONS AND COMBINATION – II
Paasankusaahi damarooka kapaala soolai: khadvangasakthi sara chaapayuthairbhavanthi anyonya hastha kalithai: kathi moorthibhedaa: sambho haririva gadaari saroja sankachakrai:
Pasa, ankusa, serpant, damaru, kapala, soola, khatvanga, sakti, chapa, sara with these (ten) items how many permutations and combinations are possible for Lord Siva. Similarly with the four items, sanku, chakra, gadha and padma holding in the hands, how many combinations are possible for Lord Vishnu?
BHASKARACHARYA II IN LILAVATI 1114 AD

PARTNERSHIP AND SHARES
Samavaayakaasthu vanija: panchaikaikottharaadhi mooladhanaa: laabha: sahasra sankhyo vada kasmai thathra kim deyam
Five partners collaborate in a business. The capital invested by them are (in the ratio) one and the same number increasing successively by one (i.e 1,2,3,4, & 5) respectively. Profit that accrued amounts to 1000. Say what should be given to whom.
BHASKARACHARYA I – IN ARYABHATEEYA BHASHYA 628 AD

LOANS AND INTERESTS
Kutumbaarthamasakthena gruheetham vyaadhithena vaa upaplava nimittham cha vidyaathaapalkrutham thath kanyaavaivahikam chaiva prethakaaryeshu yathkrutham ethath sarvam pradaathavyam kutumbena krutham prabho
Loans are taken for meeting the expenditure connected with economic problems due to family burden, health problems, treatment, education, expenditure during accident, marriage of daughter, for performing rituals connected with the demise of the family members, etc.
VISHNUSMRUTHI 100 BC

INTEREST CALCULATION
Maasena sathasya phalam panchaiko bhavyakerdhamaya vruttho lekhakapaado varshe panchaadika navasatheemisram
The rate of interest being 5% per month, the commission of surety 1% per month, fee for accountant ½% and charges of the scribe 1/4% per month, certain sum amounts to 905 a year. Find the capital, the interest and the shares of the surety?
SRIDHARACHARYA IN PATIGANITA 990 AD

RULES OF CHARGING INTEREST
Atha utthamarna: adhamarnaadyathaa datthamartham gruhneeyaath dvikam thrikam chathushkam panchakam cha satham prathimaasam
The loans can be given and taken between borrower and lender. Generally charged interest rates are 2, 3, 4, or 5% per month.
Sa paadapanaa dharmyaa maasavruddhi: panassathasya panchapanaa vyaavaharikee
Reasonable (dharmic) rate of interest is 1.25% per month (i.e 15% per annum) on the transactions with common man for non commercial purposes. But for commercial purposes (for making profit out of it) interest rate can be 5% per month.
VISHNU SMRUTHI 100 B.C

RULES OF BODIES IN MOTION
Bhakthe vilomavivare gathiyogenaanulomavivare dvow gathyantharena labdow dviyogakaalaavatheethaishyow
Whenever two bodies are travelling in the opposite directions, the distance between them is to be divided by the sum of their speeds. If they move in the same direction, the distance is to be divided by the difference of their speeds. This gives the time required for meeting of the bodies or the time elapsed after meeting of the moving bodies.
ARYABHATA I – ARYABHATEEYA 499 AD
Ekow naa yojananyashtow yaathyanyo yojanadvayam yojanaanaan satham panthaa: sangama: kva gamaagame One man travels at 8 yojana speed per day. Another travels at 2 yojana per day, starting simultaneously from the same place. After reaching the destination, the first man comes back. If the length of the track is 100 yojana. Say where is the meeting place of the two? (One going forward and the other traveller returning).
SREEDHARACHARYA PATIGANITHA 990 AD

PROGRESSION OF THE TYPE
12 + 22 + 32 + 42 + ….
Sapthaanaam ashtaanaam saptadasaanaam chathurbhu jaaschithaya: ekavidyaanaam vaachyam padastharaasthaa hi vargaakhyaa:
There are (three pyramidal) piles on square bases having 7, 8 and 17 layers which are also squares. Say the number of units there in.
BHASKARACHARYA I – ARYABHATEEYA BHASHYA 628 AD
PROGRESSION OF THE TYPE 13 + 23 + 33 + 43 +
Chathurasraghanaschithaya: panchachathurnavastharaa vinirdesyaa: ekaavaghatithaasthaa: samachathura sreshtakaa: kramasa:
There are three pyramidal piles having 5, 4 and 9 cuboidal layers. They are cuboidal bricks (of unit dimension) with one brick in the topmost layer. Find the number of bricks used in them.
BHASKARACHARYA I – ARYABHATEEYA BHASHYA 628 AD
PROGRESSION OF THE TYPE
n + n2 + n3 + n 4
Sankalithakruthighanaanaam sankalithasamaasamaanaam me kathaya shannaam sakhe padaanaam ganayithvaa yadivijaanaasi Friend, if you know, then say after calculation (i) the sum of successive sum of 6 natural numbers (ii) the sum of the squares of the first 6 natural numbers and (iii) the sum of the cubes of first 6 natural numbers.
SREEDHARACHARYA – IN PATIGANITHA 900 AD

FIRST DEGREE INDETERMINATE EQUATION
Mudgaanaam kudavaa: saptha labhyanthe navabhi: pane: panena kudavasyaardham thandulaanaamavaapyathe thatha: panathrayam saardham gruheethvaaasu vaningmama thandulaanaam prayacchaamsa mudgaanaam cha dvisangunam
7 kudavas (unit of measurement) of mudga are obtained for 9 panas and ½ kudava of rice is obtained for one pana. Then O! merchant take 3½ panas and quickly give me one part of rice and two parts of mudga.
SREEDHARACHARYA – IN PATIGANITHA 900 AD

FIRST ORDER EQUATION – I
Ye nirjaraa dinadinaardha thrutheeya shashtai: sampoorayanthi pruthak pruthakeva mukthaa: vaapeem yadaa yugapadeva sakhe vimukthaasthe kenavaasaralavena thadaa vadaasu
By opening 4 inlets separately, one pond gets filled respectively within 1, ½, 1/3, and 1/6 days. If all the four inlets are opened together, how much time (in fraction of the day) is required to fill the pond ?
BHASKARACHARYA II – IN LILVATI 1114 AD
FIRST ORDER EQUATION – II
Nava gulikaa saptha (cha) roopakasamaasthrayaanaam (thu) gulikaanaam thrayodasaanaam cha roopakaanaam thadaa kim gulikaa moolyam If 9 gulika and 7 rupaka are equal to 3 gulika and 13 rupaka, what is the price of one gulika? (the answer can be determined through the same method followed above)
SREEDHARACHARYA PATIGANITHA 990 AD

EQUATIONS OF HIGHER ORDER- I
Vaanarakulathribhaga: svathryamsa samanvi1tha: sara: prayayow moolam cha pipaasathi dvow choothathale sthithow seshow
One third of a troop of monkey with one third of itself has gone to the tank; the square root of the whole troop is afflicted with thirst, and the remaining 2 monkeys are sitting under the mango tree. What is the total number of monkeys? 1/3 a + 1/9 a + a + 2 = a.
REEDHARACHARYA – PATIGANITHA 990 AD
EQUATIONS OF HIGHER ORDER- II
Bale maralakula mooladalaani saptha theere vilaasabhara manthara gaanyapasyam kurvancha keleekalaham kalahamsayugmam sesham jale vada maraalakula pramaanam
I saw that one half of 7 times of the square root of the total number of swans were slowly moving away in the river. Remaining 2 are playing in water. What is the number of total swans? (equation: 7/2 a+2=a)
BHASKARACHARYA – LILAVATI 1114 AD

PYTHAGORUS THEOREM
DISCOVERED BY BOUDHAYANA
Samachathurasrasyakshnayaa rajju dvishtavathim bhoomim karothi
The diagonal of a square produces double the area of the square. Deerghachathurasrasyakshnayaarajju: paarsvamaani thiryanmaani cha yatpruthakbhoothe kuruthasthadubhayam karoti
Areas produced separately by the length and breadth of rectangle together equal to the area of the (square) produced by the diagonal.
BOUDHAYANA BOUDHAYANA SULBASUTRA 700 BC

EXPLANATION OF BINOMIAL THEOREM
If a three syllablic Madhya Chanda based on guru and lakhu sounds were followed, then variation of guru and lakhu sound will be on the following pattern: 3 guru sound occur once, 2 guru and 1 lakhu occur thrice, 1 guru and 2 lakhu sounds occur thrice, 3 lakhu occur once. The equation can be derived easily. If guru is g and lakhu is 1 then,
(g+1)3 = g3+3g21+3g12+l3. This equation is the same as (x+y)3. Similarly for finding the pratishta Chanda, in the Chanda sastra of Pingalacharya, the following equation can be indirectly applied in this form: (g+1)4 which is expanded as g4+4g31+4g212+4g13+14 I.e 4 guru sound occur once, 3 guru and 1 lakhu occur four times, 2 guru and 2 lakhu occur four times, 1 guru and 3 lakhu occur four times and 4 lakhu occur once.
PINGALACHARYA – CHANDASASTRA 200 BC

GEOMETRY IN SULBASUTRA-II
Thaasaam trika chathushkayordvaadasikapanchikayo: panchadasikaashti kayo: saaptikachathurimsathikayo: dvaadasika panchathrimsathikayo: panchadasikashad- thrimsikayo: ithyethaasoopalabdhi:
Hypotenuse in rectangles having sides 3 and 4 (= 5), 12 and 5 (= 13), 15 and 8 (= 17), 7 and 24 (= 25), 12 and 35 (= 37) and 15 and 36 (= 39) (I.49).
BOUDHAYANA BOUDHAYANA SULBASUTRA 700 BC

ANGULAR DIMENSIONS
Angagunavedahuthaasaa: kalikaa vikalaa: samudrajaladhaya: svalpajalakhaashtasasi dhruthisasina: kalikaa: saraagnayo vikalaa: thrijyaakruthivarashta navathribhuvo visve jinaamsajyaa.
Thribhujasya phalasareeram samadalakoti bhujaardha samvarga:
The area of a triangle is the product of the prependicular and half the base.
ARYABHATTA I ARYABHATEEYA 499 AD
Karnasthrayodasa syaath panchadasaanyo mahee drisapthaiva vishamasthri bhujasya sakhe phalasankhyaa kaa bhavedasya
What is the area of a scalene traingle in which one lateral side is 13 units, other 15 unit and the base is 14 units.
Ashtaadasakocchrayovamso vaathena paathithomoolaath shadgathvaavasow pathithaasthribhujam kruthvaa kva bhaghna: syaath
A bamboo of beight 18 cubits fell by the wind, it falls at a distance of 6 cubits from the root, thus forming a right triangle, where is the break?
BHASKARA I COMMENTARY TO ARYABHATEEYA 628 AD

POLYGONAL
Thribdhyankaagninabha schandraisthri bhaanaa shtayugaashtabhi: vedaagni baanakhaaschaicha khakhaabhraa bhrarasai: kramaath baaneshu nakha baanai schadvidvi nandeshu saagarai: kuraamadasavedaischa vruthhavyaase samaahathe khakhakhaabhraarka sambhakthe labhyanthe kramasobhujaa: vrutthaantha sthraya poorvaanaam navaasraantham pruthak pruthak
For cyclic equilateral triangle, cyclic square, cyclic equilateral pentagon,…. to cyclic equilateral nonagon, (cyclic figures having 3 to 9 sides with equal side measurements) their sides can be calculated respectively when diameter is multiplied separetely with 103923 (triangle) 84854 (quadrilateral) 70534 (pentagon), 60000 (hexagon) 52055 (septagon) 45922 (octagon) and 41031 (nonagon) and divided by 120000, the value will be the measurements of the sides of cyclic equilateral triangles to cyclic equilateral nonagon. Bhaskaracharya has given the example: If 2000 is the diameter of circle, equilateral geometrical figures inscribed inside that circle will have sides as follows:
Geometrical figure Bhaskara’s value Modern value
Triangle 1732 + .05 1732.043
Square 1414 + .021 1414.211
Pentagon 1175 + .056 1175.5619
Hexagon 1000 + .00 999.996
Septagon 867 + .58 867.5799
Octagon 765 + .36 765.3636
Nonagon 683 + .85 683.85
BHASKARA II – LILAVATI 1114 AD

CIRCLE – VALUE OF Π
Chathuradhikam sathatmashtagunam dvaashashtisthathaa sahasraanaam ayuthadvya vishkambasyaasannoo vruthhaparinaaha:
When 100 increased by 4 multiplied by 8 and added to 62,000 gives an approximate value for the circumference of a circle having diameter 20,000 units.
ARYABHATA I ARYABHATEEYA 499 AD
Ashtadvaadasa shadkaa: vishkambhasthathvatho mayaa drushtaa: theshaam samavrutthaanaam parithiphalam me pruthak broohi
Diameter of 3 circles are correctly seen by me to be 8, 12 and 6 units respectively. Tell me separately the circumference and areas of the circles.
BHASKARACHARYA I – 628 AD

SOMAYAJI’S THEOREMS
Vyaasaath vanasangunithaath pruthagaaptam thryaadyayugvimoola ghanai: thrigunavygaase svamrunam kramasa: kruthvaapi paridhiraaneyu:
Multiply the diameter of a circle with 4 and keep it at different places and divide each with the odd numbers beginning from 3, 5, 7,… as their cubes subtracted by the same value. Repeat this and add/subtract alternatively the results to three times the diameter of the circle to get the circumference with the highest degree of accuracy. This theorem can be mathematically represented as follows:
Circumference = 3D+4D/(33-3)-4D/(53-5)+4D/73 7)-..
Vargairyujaam vaa dvigunairnirekair vargeekruthair varji thayugma vargai: vyaasam cha chadghnam vibhajeth phalam svam vyaase thrinighne paridhi sthadaasyaath
Six times the diameter is divided separetely by the square of twice the square of even integers 2,4,6…. minus one, diminished by the squares of even integers themselves. The sum of the resulting quotient by thrice the diameter is the circumference.
This can be mathematically written as follows:Circumference =
3D+6D([1/2x22-1]2-22) + ([1/2x42-1]2-42)+[(1/2x62-1)2-62])+….
PUTHUMANA SOMAYAJI – KARANAPADDHATI 1450 AD

AREA OF CIRCLE AND SPHERE
Vrutthakshethre paridhigunitha vyaasapaada: phalam thath kshunnam vedairupari paritha:kandukasyeva jaalam golasyaivam thadapi cha phalam prushtajam vyaasanighnam shadbhirbhaktham bhavathi niyatham golagarbhe ghanaakhyam
When circumference is multiplied with diameter and that result divided by 4, that will give the area of a circle. This when multiplied with 4 gives the surface area of the globe which is like surface of a ball. This when multiplied with diameter and divided by 6 gives the volume of the sphere of globe.
Mathematically it can be written as 2r x 2r/4 =r2
BHASKARACHARYA II – LILAVATI – 1114 A.D

NEWTON GAUSS (1670AD) BACKWARD
INTERPOLATION DISCOVERED BY VATESWARACHARYA
Dhanushaaptha bhuktha jeevaghaathe labdham saroopakam dalitham labdaghna vivarahatham cha samsodhya niyogya vikalajyaa
In modern mathematical form this interpolation formula can be written as f(x) = f(xi)+ (x-xi)1/h Df(xi-h) + (x-xi)1/h. (x-xi+h)1/h. D2f(xi-h)½.
VATESWARA VATESWARA SIDDHANTA 904 AD

ARC AND CHORD
Svalpachaapaacchaghanashashta bhaagatho vistaraardhakruthir- bhaktha varjitham sishtachaapamihasinjanee bhaveth thadyuth oalpaka gunoasakruthdhanu:
The chord of an arc of a circle is obtained from the result of the cube of the length of the arc divided by six times the cube of radius and subtracted from the arc. This can be mathematically presented as follows: Chord (R Sine ) = s – (s3 / 6r3). Here length of the arc s is in angular dimensions, r is the radius and  is the angle of the arc.
PUTHUMANA SOMAYAJI – KARANA PADDHATHI – 1450 AD
Paridhe: shadbhaagajyaa vishkambhaardhena saa thulyaa
The chord of one sixth of circumference is equal to the radius of that circle.
ARYABHATTA I – ARYABHATEEYA 499 AD

LENGTH OF ARC – CHORD
Vyaasaabdhighaathayuthamourvikayaa vibhaktho jeevangghri panchagunitha: paridhesthuvarga: labdhonithaath paridhivarga chathurtha bhaagaadaapte pade vruthidalaath pathithedhanu: syaath.
One fourth of five times the chord multiplied with square of circumference divided by four times the diameter added with the chord. This value is subtracted from one fourth of the square of circumference. Square root of this is taken and subtracted from half of the circumference to get the arc.
BHASKARA II – LILAVATI 1114 AD

ARC AND ARROW
Jyaavyaasayogaanthara ghaathamoolam vyaasasthadoono dalitha: sara: syaath vyaasaaccharonaacchara sangunaa cha moolam dvinighnam bhavatheeha geevaa yeevaardhavarge sarabhaktha yukthe vyaasapramaanam pravadanthi vrutthe
When the sum and differences of diameter and the chord are multiplied, and their square root is taken and if half of that is subtracted from the diameter, the arrow is obtained. The difference of diameter and the arrow multiplied with the arrow, twice the square root of that value gives the chord. The square of half the chord divided by arrow and added with arrow gives the diameter of the circle.
BHASKARA II – LILAVATI 1114 AD

NEWTON’S INFINITE GP CONVERGENT SERIES
DISCOVERED BY NILAKANTA SOMAYAJI
Evam yasthuthya ccheda paramabhaaga paramaparyayaa ananthaayaa api samyoga: thasya ananthaanaam api kalpyamaanasya yogasyaaddhyaavayavina: parasparama cchedaad ekonacchedaa mamsa saadhyam sarvathraapi samaanam eva…
Thus the sum of an infinite series, whose later terms (after the first) are got by diminishing the preceding or by the same divisor, is always equal to the first term divided by one less than the common mutual divisor.
NILKANTA ARYABHATEEYA BHASHAYA 1444
SINE, COSINE, RADIUS AND ARC
Anyonya kotihathayorabhimatha gunayosthrijeejavayaa hathayo: yogaviyogow syaathaamabhimathagunachaapa yogavivaragunow
The sum of the products of Sin A and Cos B and when angles are exchanged, Sin B and Cos A, gives the Sin of the sum of the angles. Similarly the difference of the above gives the value of the sin of angular difference. Sin (A+B) = Sin A Cos B+Cos A Sin B And Sin (A-B) = Sin A Cos B – Cos A Sin B.
Yadveshta chaapagunatha ccharavargayoga moolaardhamishta dhanurardhaguna: pradishta: jyaanaam nijathriguna vargaviseshamoolam kotisthadoona sahithow thrigunow svabhaanow
Square root, of the square of a chord (R sin ) diminished from squares of radius gives the koti (R cos ). This subtracted from radius gives the (small) arrow of arc. This added to radius is big arrow of the arc…..
PUTHUMANA SOMAYAJI – KARANA PADDHATI 1450

TAYLOR (1685 AD) SERIES OF SINE AND
COSINE DISCOVERED BY NILAKANTA
ista-dohkotidhanushoh svasamipasamirate jye dve saavayave nyasya kuryaad unaadhikam dhanuh dvighna talliptikaptaikasarasailasikhindavah nyasyacchedaaya cha mithastatsamskaaravidhitsaya anyasyam atha taam dvighnaam tathaa syam iti samskriti: santha te krtasamskare svagunau dhanusas tayo:
Placing the sine and cosine chords nearest to the arc, whose sine and cosine chords are required, get the arc difference to be subtracted or added. For making the correction, 13,751 should be divided by twice the arc difference in minutes and the quotient is to be placed as the divisor, divide the one (sine or cosine) by this divisor and add to or subtract from the other (cosine or sine) according as the arc difference is to be added or subtracted. Double this result and do as before. Add or subtract the result to or from the first sine or cosine to get the desired sine or cosine chords.
NILAKANTA – TANTRA SANGRAHA 1444 AD

NEWTON GAUSS (1670) INTERPOLATION FORMULA DISCOVERED BY GOVINDASWAMI
gacchad-yata-gunantharavapuryathaishya-disvasanaa cchedaabhyaasa-samuha-kaarmukakrti-praapthath tribhisthaadithah vedaihi sadbhir avaaptam antyagunaje rasyo: kramad antyabhe ganthavaahata-varthamaana-gunajaaccha paatham ekaadibhi:antyad utkramatah kramena vishamai: sankhyaviseshai: khsipedbhankthvaptam, yadi maurvikavidhir ayam makhyah kramad vartate sodhyam vyutkramathaa stathakrthaphlam…..
Mathematicaly this formula is summarised as follows: F(x+nh)=f(x)+nf(x)+½n(n-1)(f(x)-f(x-h) Multiply the difference of the last and the current sine differences by the square of the elemental arc and further mutiply by three. Now divide the result so obtained by four in the first rasi, or by six in the second rasi. The final result thus obtained should be added to the portion of the current sine difference (got by linear proportion). In the last rasi, multiply the linearly promotional part of the current sine differences by the remaining part of the elemental arc and divide by the elemental arc. Now, divide the result by the odd numbers according to the current sine difference, when counted from the end in the reverse order. Add the final result thus obtained to the portion of the current sine difference. These are the rules for computing true sine differences for sines. In the case of versed sines, apply the rules in the reverse order and the above corrections are to be subtracted from the respective differences.
GOVINDASWAMI – COMMENTARY FOR MAHABHASKAREEYA 800 AD

NEWTON’S (1660 AD) POWER SERIES
DISCOVERED BY SOMAYAJI
nihatya chapavargena chapam tatthathphalani cha haret samulayugvargaistrijyavargahatai: kramaat chapam phlani chadhodhonyasyoparyupari tyajet jivaptyai, sangraho syaiva vidvan-ityadina krtha: nihathya chapavargena rupam tattatphalani cha hared vimulayugvargaistrijyavargahatai: kramat kintu vyasadalenaiva dvighnenadyam vibhajyataam phalanyadhodha: kramaso nyasyoparyupari tyajet saraptyai, sangraho asyaiva stenastri-tyadinaa krta:
Multiply repeatedly the arc by its square and divide by the square of even numbers increased by that number and then multiplied by the square of radius. Place the arc and result one below the other and subtract each from what is above it. To derive the arc, which are collected, beginning with the expression Vidvan (katapayadi number). Multiply repeatedly, the unit measurement which is the radius, by the square of the arc and divide by the square of even numbers decreased by that number and then multiplied by the square of radius; the first is, however, to be divided by twice the radius. Place the results one below the other and subtract each from the one above it. That is the method to derive the saras, which are collected in the beginning with stena. (This equation is now known as Newton power series.)
PUTHUMANA SOMAYAJI – KARANAPADDHATI (1450 AD)

VOLUMES OF CONES
Samakhaatha phalathryamasai: soochikhathe phalam bhavathi
The one third of the volume of the uniform cylinder is the volume of the cone.
Pardhirbhitthilagrasya raasesthrimsathkara: kila anthakonasthithasyaapi thithithulyakara: sakhe bahishkona sthithasyaapi panchaghnanava sammitha: theshaa ma chakshva me kshipram ghanahasthaath pruthak pruthak
Friend, the food grains are kept at a circumference of 30 cubit in the floor, outside corner of the room, inside corner and side of the wall. Find out the volume of the grain if the height is 45 cubit.
BHASKARA II LILAVATI 1114 AD

LHUILER’S (1782 AD) FORMULA
DISCOVERED BY SOMAYAJI
Doshnamdvayordvayor ghaatayutaanaam tisraanaam vadhaat ekaikonetarattraikyam catushkavadhabhajitam Iabdha mulena yadvrttam vishkambhaardhena nirmitam sarvam caturbhujakshetram tasminneva tisthtahathe
The three sums of the product of sides, taken two at a time are to be multiplied together and divided by the product of the sums of the sides taken three at a time and diminished by the fourth. If a circle is drawn with the square root of this quantity as radius, the whole quadrilateral will be situated inside it.
PARAMESWARA COMMENTARY FOR LILAVATI (1360 AD)

GREGORY’S (1632 AD) SERIES
FOR INVERSE TANGENT
DISCOVERED BY MADHAVA CHARYA
istajya-trijyayorghathath kotyaptam prathamam phalam jyavargam gunakam kritva kotivargam cha haarakam pratha maadiphalebhyo atha neya phalakrtir muhu: eka-tryaady-ojasankhyabhirabhakteshveteshv anukramaat ojanam samyutesthyaktva
yugmayogam dhanur bhavet doh-kotyor alpameveha kalpaniyam iha smrtam labdhinam avasanam syanna thathaapi muhu: krte
Obtain the first result of multiplying the jya (R sine ) by the trijya (radius) and dividing the product by koti (R cos ). Multiply this result by the square of the jya and divide the square by the koti. Thus we obtain a second result a sequence of the further results by repeatedly multiply by the square of the jya and dividing by the square of the koti. Divide the terms of the sequence in order by the odd numbers 1,3,5,…; after this, add all the odd terms and subtract from them all the even terms (without disturbing the order of the terms). Thus is obtained the dhanus whose two elements are the given jya and koti. (Here the smaller of the two elements should be taken as the jya, since other wise the series obtained will be non finite) (use of Tangent)
MADHAVA YUKTI BHASHA? (1350 AD)

DE MOIVRE’S (1650 AD) APPROXIMATION DISCOVERED BY MADHAVA CHARYA
Asmat sukshmataroanyo vilikhyate kashcanapi samskara: ante samasankhyadalavarga saiko guna:, sa eva puna: yugagunito rupayuta: samasankhyadalahato bhaved haara: trisaradivisa mashankhyaharanat param etad eve va karyam
A correction for cirumference still more precise is being stated here. The multiplier is the square of half the even integer increased by unity. This multiplier multiplied by 4, then increased by unity and then multiplied by half the even integer is the divisor. This correction may be applied after the division by odd integers,3, 5, etc. i.e Circumference = 4D (1-1/3+1/5-1/7….. + ..-1/n(½(n+1)2+1((½(n+1)2 x 4 +1) (½(n+1))
MADHAVA KRIYA KRAMAKARI (1350 AD)

DE MOIVRE’S (1650 AD) APPROXIMATION
yatsankhyaaatra harane krte nivrtta hrtis tu jamitaya tasya urdhvagatasyas samasankhya taddalam guno ante syat tadvargai rupahato haaro vyasabdhighatata: pragvat tasyam aptam svamrne krte dhane sodhanan cha karaniyam sukhma: paridhi: sa syat bahukrtvo haranato atisukshmas cha
………. Let the process stop at a certain stage, giving rise to a finite sum, multiply four times the diameter by half the even integer subsequent to the last odd integer used as divisor and then divide by the square of the integer increased by unity. The result is the correction to be added to or subtracted from finite sum. The choice of addition or subtraction is depending on sign of the last term in the sum. The final result is the circumference determined more accurately than by taking a large number of terms:
MADHAVA YUKTIBHASHA? (1350 AD)

HORIZON Aaveshtamaanamatha thaani dalapravruthyaa yadvrutthamathra harijam kshithijam thadaahu: yasmin bhaveth samudayasthamayo akhilaanaam praachyaam kramaadaparadisyudu khecharaanaam
The great circle which goes round them, dividing each of them into two equal parts, is called harija or kshitija. This in modern astronomy is horizon. This is the circle on which rising and setting of stars and planets take place towards east and west respectively.
VATESWARA SIDDHANTA 880 AD

ASTRONOMICAL DEFINITIONS
Urdhvamadho apara poorvamihaadyam praahuridam samamandala manyath thadvadihotthara dakshinadikstham vrutthayugam vidisorapi thadvath
Vertical circle passing through the west and east cardinal points is the first circle: this is called the samamandala. (This circle is the prime vertical. Another similar vertical circle (called the yaamyottara-vrutta) which passes through the north and south cardinal points is called the meridian.
VATESWARA – VATESWARA SIDDHANTA 880 AD

TYCHO BRAHE REDUCTION OF ECLIPTIC
DISCOVERED BY ACHYUTA PISHAROTI
Patonasya vidhostu kotibhujayorjive mithastadayet antyakshepasarahatam vadhamamum vikshepakotyaharet labdham vyasadaloddhrtam himakare svarnam, vipate vidhau yugmaayugmapadopage; vidhurayam spashto bhagole bhavet
Multiply the tabular cosine and sine of the moon minus node and the product by the tabular versine of the maximum latitude of the moon. Divide this by the tabular cosine of the latitude at the particular moment and the quotient is to be divided again by the tabular radius. The result is to be added to or subtracted from the moon’s longitude, as the moon minus node is in an even or an odd quadrant, respectively. The true moon measured on the ecliptic is thus obtained.
ACHYUTA PISHAROTI SPHUTANIRNAYA

EQUATOR
Khasvasthikaad dakshinatho akshabhaagow paathaa (la) samjnachha thathottharena naadyankitham vaishuvatham thaduktham vruttham bhagolasya khagolamadhye
The sphere of the asterisms lie within the sphere of the sky. Great circle of the sphere of asterisms which lies towoards the south of the zenith by an amount equal to the degrees of local latitude and towards the north of nadir by the same amount and which is graduated with the division of nadis is the vishuvathvrutta. This circle is called the equator.
VATESWARA SIDDHANTA 880 AD

6 O’CLOCK CIRCLE Poorvaaparakshithija sangamayorgathamcha yaamyaadadha: palalavai:
kshithijaadvi lagnam soumyaadathopari samadruvamarga samstham unmandalam dinaniso: kshayavruddhikruthaath.
Passing through the two points of intersection of prime vertical and horizon, lying below the south cardinal point by the degrees of local latitude, fastened to the horizon, and lying above the north cardinal point, passing through the north celestial pole, is the Unmandala, the cause of decrease and increase of the day and night. (This in modern astronomy is known as the 6’o clock circle.)
VATESWARA SIDDHANTA 880 AD

CIRCLE OF DIURNAL MOTION
Harije parapoorva mandala dyujaavruthha visesha sinjinee udayaagraguno dyumandale bhoojyothavruttha kujaan tharaamsajeevaa:
R sine of the arc of the horizon lying between the prime vertical and the diurnal circle of the planet is the R sine of agra (now known as the rising point of the planet) and the R sine of the degrees of diurnal circle lying between six o’ clock circle and the horizon is bhoojya (bhujya) which is termed as Earthsine.
VATESWARA SIDDHANTA 880 AD

DAY RADIUS
Kraanthijyaa vargonaath thrijyaavargaath padam dyujeevaa syaath thrijyaakraanthi yaanthara samaasa ghaathasya moolam vaa
Day radius is equal to the square root of the difference obtained by subtracting the squares of R sine of the declination from the square of the radius or the square root of the product of the difference and the sum of the radius and the R sine of the declination.
VATESWARA SIDDHANTA 880 AD

ECLIPTIC
Naaddyaahvavrutthaajathulaadilagnam jinaamsakairadakshinatho mrugaadow soumye seetha mandiraadaav apakramaakhyam thadusanthi vruttham
Fastened to the so called nadivrutta or the equator at the points of Aries and Libra and lying 24 degrees of the south (of equator) at the first point of Capricon and 24 degrees to the north (of equator) at the first point of Cancer, there is a great circle called the apakrama vrutta (now known as the ecliptic)
VATESWARA SIDDHANTA 880 AD

DAY DIAMETER
Vishuvajyaa aayaa mardha varga vislesha moolamavalambaka: kranthithrijyaakruthyo rantharapadam dvigunam dinavyaasa
Square the sine of latitude and deduct from the square of the radius. Its square root is the sine of the co-latitude (its arc being the co-latitude). Square the sine of the declination deduct from the square of the radius and find its root. Twice the result is the day diameter.
PANCHASIDDHANTIKA 4-23 – VARAHA MIHIRA 505 AD

SETTING POINT OF ECLIPTIC
Praachyaam kuja apakrama vrutthasanga praaglagnamaahu (paritho asthalagnam) (lagnaadbhaveth) sa (pta) ma (raa) si (ra) stha thasyaa (stha) kaalo abhyudayosya bhooyath
Point of intersection of horoizon and the ecliptic in the eastern half of the celestial sphere is called praglagna. I.e. the rising point of ecliptic; the same in the western half is called astalagna, known as setting point of ecliptic.
VATESWARA SIDDHANTA 880 AD

RISING – SETTING LINE
Vyaasaardha vrutthe antharam ethayo: syaaccharaardha jeevaa parapoorvayosthath agraagrayoryad harijenibaddham soothram grahaanaam udayaastha samjnam
The arcual distance between the six o’clock circle and the horizon measure, along the R circle trijyavrutta known as great circle of the celestial sphere, supposed to be of radius 3438’ (minute of angle) is the charardhajya. It is called the R sine of the Ascensional difference. A thread tied to the extremities of the agra on the eastern and western halves of the horizon is called the udayaastasutra. (In moderen astronomy it is known as the rising – setting line of planets).
TESWARA SIDDHANTA 880 AD

DAY RADIUS AND EARTHSINE
Kraanti thribhaantharajyaa dyujyaa vaa charadalajeevayaa hruthaa thrijyaa kshithi jeevaghnaa svaahoraathraardhajeevaa vaa
Rsine of the difference between the three signs and the declination is also equal to the day radius. Day radius multiplied by earthsine and divided by the R sine of the Ascensional difference gives the day radius.
VATESWARA SIDDHANTA 3(4)-3) – 880 AD

SUN’S PRIME VERTICAL
Urdhvamadho aparapoorvamihaadyam praahuridam samamandala manyath thadvathihottharadakshina dikstham vrutthayugam vidisorapi thadvath.
Vertical circle passing through the west and east cardinal points is the first circle called samamandala or the prime vertical.
VATESWARA SIDDHANTA- GOLA. 3-1, 2 – 880 AD

PARALLAX-I
Thithernathasya kramasinjanee hathaa khamadhya lagnaprabhavena sankunaa kshamaashadangkaabhi saraankanethrahrud vilambane syaad ghatikaadi vaa phalam.
R sine of the hour angle at the amavasya multiplied by R sine of the altitude of the meridian ecliptic point and divided by 2954961 gives the parallax in ghatikas at mid eclipse (Sishyadhi vruddhi Tantra 6-8)
LALLACHARYA SISHYADHI VRUDDHI TANTRA
PARALLAX-II
Thriraasijeevaa valanajyakaa hruthaa sileemukhai rankulathaam vrajanthi thaa: dvisankunaa drushtigathi: saraachalairvibhaajithaa lambana naadikaa phalam
Radius and the valanajya when divided by 5, are converted into angulas. The R sine of driggati multiplied by 2 and divided by 75 gives ghatika of the parallax in longitude. (Sishyadhi vruddhi Tantra 13-11)
LALLACHARYA SISHYADHI VRUDDHI TANTRA 700 AD
PARALLAX-III
Nathakramajyaambara sankunighnaa syaallambanam thathvarase shuhrudvaa drukshepabhukthyanthara yoscha ghaatha: khabaanayugmaa kshihrutho nathi: syaath
R sine of the hour angle multiplied by Rsine of altitude of the merdian ecliptic point and divided by 5625 gives parallax in longitude. The Difference of true motions of the Sun and the moon multiplied by the Rsine of drikshepa and divided by 2250 gives the parallax in latitude. (Sishyadhi vruddhi Tantra 13-12)
LALLACHARYA SISHYADHI VRUDDHI TANTRA 700 AD

APOGEE, PERIGEE AND ORBIT OF EARTH
Svochhaath shadbhaagaadhyadhiko yadaa thadaa bhavathi svaneechastha: doorenochhaga urvyaa: karnavasaannochhago nikate
When a planet is at a distance of 6 signs from its apogee, it is said to be at the perigee or neecha. When a planet is at the apogee, it is farthest from the earth when at the perigee, it is nearest to the earth. This is so because of the length of the hypotenuse in each case (Sishyadhi vruddhi Tantra 14-10)
LALLACHARYA SISHYADHI VRUDDHI TANTRA 700 AD

VELOCITY OF PLANETS PER DAY
Sun 59’ 8” 10’’’ 13’’’’ (gopaajnayaa dinadhaama)
34” 51’’’ 36’’’’
(Chandikeso bharga snigdhosow)
Mars 31’ 26” 29’’’ 42’’’’ (Prabhurdharaachakra paala)
Mercury 245’ 32” 36’’’ 32’’’’
(Rageethumbururganeswara)
Jupiter 4’ 59” 7’’’ 2’’’’
(Prajnaasanoo dharmavaan)
: Venus 96’ 7” 37’’’ 51’’’’
(Kasi saambasanna chola:)
Saturn 2’ 0” 23’’’ 32’’’’
(Prabhalapraajno nara:)
The modern values of angular motions are Earth/Sun 59.14’, Mars 31.45’, Mercury 245.7’, Juptiter 4.99’ Venus 96.13’, and Saturn 2’.
PUTHUMANA SOMAYAJI KARANAPADHATI (1450 AD)

SHAPE OF EARTH
Gaganamarudaagni jalamrunmayo mahaabhootha gunayutha:khastha: kakshaabhiraavrutho ayam bhapan charaanthascha bhoogola
Spherical earth, made of ether, fire, air, water and clay (Panchabhoothas) and thus have all the properties of the five elements, surrounded by the orbits and extending upto the sphere of stars, remain in the space (Sishyadhi vruddhi Tantra 17-1)
LALLACHARYA SISHYADHI VRUDDHI TANTRA 700 AD
Praguna paridhe: sathaamsako ganithajnaa: kathayanthi drusyathe prathi bhaathi thadaa samaa mahee vishaye yanthra thathaiva gamyathe
Mathematicians say that one hundredth of the cirucumference of the earth appears to be plane. So, that portion of the earth appears to be plane to an observer (Sishyadhi vruddhi Tantra 20-35)
LALLACHARYA SISHYADHI VRUDDHI TANTRA 700 AD

ROTATION OF EARTH – I
Pranenaithi kalaam bhooryadi tharhi kutho vrajeth kamadhyaanam aavarthana murvyaa schenna pathanthi samucchrayaa: kasmath
If earth rotates at a speed of 1’ of an angle in 4 seconds, will not the things on the loft fall? Where does the earth go in this speed? (Brahmasphuta siddhanta 11-17).
BRAHMAGUPTA BRAHMASPHUTA SIDDHANTA 629 AD
FOUR QUADRANTS OF EARTH
Udayo yo lankaayaam soasthamayo: savithureva siddhapure madhyahno yavakotyaam romake vishaye ardharaathramsyaath
When it is Sunrise in Lanka, the same Sun sets in Siddhapura. (Gautimaala). It is noon in Yavakoti (Korea) and midnight in Romaka (Rome) (Aryabhateeyam 4-13).
ARYABHATA -I ARYABHATEEYA (499 AD)

GLOBE
Samavrutthaprushtamaanam sookshmam golam prasaadhya daarumayam sthagithaarka samaankitha kaala bhogarekaadvaye paridhov
Perfectly circular throughout and spherical, made of wood, marked with degrees and minutes, incorporated with lines both longitude and latitude at ends, is the golayantra. (Panchasiddhantika 14-23)
VARAHAMIHIRA PANCHASIDDHANTIKA (505 AD)
Kaashtamayam samavruthham samanthatha: samagurum laghum golam paaradathaila jalaistham bhramayeth svadhiyaa cha kaalasamam
Made of wood, fully circular, uniform, equally dense throughout and spherical shaped golayantra, which rotates at a fixed rate of time as the earth does by the help of mercury, oil and water, by the application of our intelligent calculation, is the golayantra-Globe.
……….
Nrushiyojanam, njilaa bhoovyaaso
8000 Nr units is equal to one yojana. The diameter of earth is 1050 yojana.
ARYABHATA-I ARYABHATEEYA (499 AD)

ROTATION OF EARTH – II
Ku ngi si bu nlru shru khru praak
Eastward rotations of the earth in one Yuga is 1582237500
Anuulomagathirnoustha: pasyathyachalam vilomagam yadvath achalaani bhaani thadvath samapaschimagaani lankaayaam
Just as a man in a boat moving forward sees the stationary objects as moving backward, so are the stationary stars and celestial bodies seen by the people at equator (Lanka) as moving exactly towards west.
Ku aavarthaaschaapi naakshathraa:
The rotation of the earth is the cause of days (Aryabhateeyam 3-5).
ARYABHATA-I ARYABHATEEYA (499 AD)
MERIDIAN
Lankaayaamekam sankukeelam prathishtaapya thenaikam soothraagram baddhvaa punarmerorupari thadagramanyath baddhvaa yathaayathaa drusyatha….. thadvath bhoomaavapi kaachidrekha lankaatha: kharapuratha….. merumasthakaanavagaahya sthithaa saa punarathra desaanthara vidhaayini syaath
Fix a pole in Lanka, tie thread on that, take the other end to the North pole, tie it there also, then one can see the line of the thread passing through Lanka, Kharapuri, Arctic point and so many other countries upto the top of Meru. This is international meridian line (Sankaranarayana on Laghubhaskareeya I-23)
SANKARANARAYANA I LAGHUBHASKAREEYA (950 AD)
GRAVITY
Aakrushti sakthischa mahee thayaa yath khastham guru svaabhimukham svasakthyaa aakrushyathe thathpathatheeva bhaathi same samanthaath kva pathathyayam khe:
This earth attracts whatever solid materials are in the space, by her own force of attraction towards her (earth). All those subjected to this attractional force fall, to the earth. Due to equal force of attraction among the celestial bodies, where can each among them fall? (Siddhanta siromani Bhuvanakosham 6)
BHASKARA II SIDDHANTA SIROMANY (1114 AD)
MERIDIAN AND TIME
Desaanthara ghatee kshunnah madhyaa bhukthir dyuchaarinaam shashtyaa bhaktham runam praachyaam rekhaayaa: paschime dhanam.
The time is calculated based on the meridian. Divide the time by 60… and the longitude is calculated. Towards the east subtract and towards the west add the number (Laghubhaskareeyam 1-31)
BHASKARA I LAGHUBHASKAREEYA (628 AD)
MERIDIAN AND TIME
Panchaasathaa thribhisthryamsaamyuthairyojanaischa naaddyekaa samapoorva paschimasthairnithyam sodhyaa cha deyaa cha
One nadi for every 53 1/3yojanas has to be deducted or added (to Ujjaini) by the people in places east and west, respectively of the Ujjaini meridian. (Panchasiddhantika 9-10)
VARAHAMIHIRA – PANCHASIDDHANTIKA (605 AD)
ECLIPSE-I
Kimartham asura: kaschidraahurnaama saimhikeyoarkam chandram cha grasatha ithi srooyathe sraapi pouraanika sruthireva! ka: punariha raahurithyuchyathe

What does it mean that Asura is responsible for the eclipse? Others say that a snake Rahu swallows the Sun and the Moon! Those are puranic stories! Then what is called the Rahu?
SANKARANARAYANA COMMENTRAY TO LAGHUBHASKAREEYA 950 AD
Cchadayathi sasi sooryam sasinam mahathee cha bhoocchaayaa
Moon covers (shadows) the Sun and the great shadow of the earth covers the moon (which causes the eclipse)
ARYABHATAI ARYABHATEEYA (499 AD)
ECLIPSE-II
Atha eva bhoocchayaa chandragrahanasya kaaranam
That is why it is said that the shadow of the earth is the cause for the lunar eclipse.
SANKARANARAYANA COMMENTRAY TO LAGHUBHASKAREEYA
Asuro yadi maayayaa yutho niyatho athigrastheethi they mantham ganithena katham sa labhyathe grahakrutha parva vinaa kathanchana
If you are of the opinion that an artifical demon is always the cause of an eclipse by swallowing, then how is it that an eclipse can be determined by means of calculations. Moreover why is then not an eclipse occur on a day other than the day of new or full moon (Sishyadhi vruddhi Tantra 20-22)
LALLACHARYA SISHYADHI VRUDDHI TANTRA 700 AD

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