S`ulba-sutra-s of Baudha-ayana, Apa-stamba, Katya-ayana, and Manava


Contents

Pars & Capitulum

Titulus

Paginae

0

Introduction

pp. 1-14

2.1

Baudha-ayana S`ulba-sutra

77-100

2.2

Apa-stamba S`ulba-sutra

101-119

2.3

Katya-ayana S`ulba-sutra

120-125

2.4

Manava S`ulba-sutra

126-143

3.1

Commentary on Baudha-ayana S`ulba-sutra

147-233

3.2

Commentary on Apa-stamba S`ulba-sutra

234-263

3.3

Commentary on Katya-ayana S`ulba-sutra

264-271

3.4

Commentary on Manava S`ulba-sutra

272-281


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0. Introduction


p. 2 Vaidik s`akha-s ('branches', i.e., sectarian recensions) of authors of S`ulba-sutra-s

s`akha of Veda

its author[s] of S`ulba-sutra[-s]

Taittiriya of Kr.s.n.a Yajus

Baudha-ayana, Apa-stamba, Vadhula, Hiran.ya-kes`in

Maitra-ayan.i of Kr.s.n.a Yajus

Manava, Varaha

Kat.haka-Kapis.t.hala of K. Y.

Lauga-aks.in

S`ukla Yajus

Katya-ayana

Saman

Mas`aka

{Maitra-ayan.i is parent of ("PMMT") PURN.a : cf. /-PaURN.a-/ (infra p. 170).}

"PMMT" = "Purna Maitrayani-putra : Master of Preaching". http://www.btrts.org.sg/ven-purna-maitrayani-putra


p. 5 the 2 main types of fire-altars, with their sub-types

types

their sub-types

nitya ('perpetual')

garha-patya, ahavaniya, and daks.in.a-agni

kamya ('optional')

s`yena-cit, kanka-cit, alaja-cit, kurma-cit, etc.


p. 6 [according to Taittiriya Samhita 5:4:11:1-3] objectives of the kamya fire-altars

__-cit

its desired objective

Chandas

"Desiring cattle"

S`yena

"Desiring heaven"

Kanka

"Desiring heaven"

Alaja

"Desiring support from heaven"

Dron.a

"Gaining food"

Paricayya

"Desiring a region"

S`mas`ana

"Attaining the place where the forefathers have gone (pitr.loka)"


p. 7 [according to Taittiriya Samhita 5:6:8:2-3] depths of bricks of fire-altars for conquering worlds by means of poietic metres

# (1000s of bricks)

depth

world

metre

1st (1000)

knee-deep

"this world"

gayatri

2nd (2000)

navel-deep

atmosphaire

tris.tubh

3rd (3000)

neck-deep

"yonder world"

jagati


p. 8 area of layers of bricks

according to __

area in square purus.a-s

S`ata-ptha Brahman.a 10:2:3:17-18

101+1/2

Baudha-ayana 5:1, 5:8; Apa-stamba 8:3

7+1/2


pp. 9-10 dimensions of right-triangle bricken fire-altars : base, altitude, hypotenuse

p.

#

base, altitude, hypotenuse

*

S`ulba Sutra reference

9

a.i

4^2 + 3^2 = 5^2

1

Baudha-ayana 1:13


a.ii

12^2 + 9^2 = 15^2

3

Katya-ayana 2:5


a.iii

20^2 + 15^2 = 25^2

5

Apa-stamba 5:3


a.iv

16^2 + 12^2 = 20^2

4

Apa-stamba 5:3


b.ii

36^2 + 15^2 = 39^2

3

Apa-stamba 5:4

10

b.ix

12^2 + 5^2 = 13^2

1

Baudha-ayana 1:13; Apa-stamba 5:4


b.x

96^2 + 40^2 = 104^2

8

Maitra-ayan.i 1:4 - 1:6


c

7^2 + 24^2 = 25^2

1

Baudha-ayana 1:13


d

8^2 + 15^2 = 17^2

1

Baudha-ayana 1:13; Apa-stamba 5:5


e

12^2 + 35^2 = 37^2

1

Baudha-ayana 1:13; Apa-stamba 5:5

{In 3 cases, the hypotenuse differeth by 1 from another side :- (a.i) 5 -- 4; (b.ix) 13 -- 12; (c) 25 -- 24.

In 2 cases, the hypotenuse differeth by 2 from another side :- (d) 17 -- 15; (e) 37 -- 35.}

{Perhaps 3 & 4 & 5 (but not 2) are selected as multipliers (for a.ii, a.iv, and a.iii) because 3 & 4 & 5 appear in a.i.}

{Perhaps 8 is selected as multipliers (for b.x) because 8 = 13 - 5 (#s which appear in b.ix); and 3 (for b.ii) because 3 = 13 - 5*2 (where 2 is the cube root of 8).}


pp. 10-11 antiquity of the sum-of-squares theorem for right triangles

p. 10

"The tradition of attributing the theorem to Pythagoras is due to Cicero (c. 50 B.C.), Diogenes Laertius (second century A.D.), Athenaeus (c. 300 A.D.), Heron (third

p. 11

century A.D.), and Proclus (c. A.D. 460), and therefore started about five centuries after the death of Pythagoras. Junge ... emphasized uncertainties in the statements of Plutarch and Proclus. {It may be added that Cicero is known for inditing (De Natura Deorum 'On the Nature of the Gods') some very speculative conjectures (which he most likely extracted from some earlier authors of various speculative conjectures) alleging multiple deities of the same name, based on no more evidence than that those deities have mulitple myths about them; so he is likely to have done similarly to produce this conjecture. Praesumably, the authors were so very disreputable that he dared not mention their names.} ...

As to the relation 4^2 + 3^2 = 5^2 ... of rational triangle ..., very ancient {Middle-Kingdom} Egyptian knowledge is attested by the Kahun papyrus of the twelfth dynasty (c. 2000 B.C.) [fn. a : "Sen, Chapter on 'Mathematics', ... 148-149".]

As to the antiquity of the Pythagorean theorem in China, it is stated ... in the arithmetical classic Chou Pei Suan Ching (third or fourth century B.C.) ... .

The old Babylonians {Akkadians} of the second millennium B.C. left records on their cuneiform tablets of similar squared relationships ... . Neugebauer [pp. 28-42] ... derived ... the theorem ... from such Babylonian cuneiform tablets."

Marcus Tullius Cicero (transl. by Francis Brooks) : On the Nature of the Gods. http://oll.libertyfund.org/titles/539

M. Tullius Cicero : The Nature Of The Gods. http://thriceholy.net/Texts/Cicero.html

C. D. Yonge (transl.) : Cicero’s Tusculan Disputations; also, Treatises On

The Nature Of The Gods, and OnThe Commonwealth. NY : Harper & Brothers Publ, 1877. http://www.gutenberg.org/files/14988/14988-h/14988-h.htm pp. 209-355.

Sen = S. N. Sen, in :- D. M. Bose, S. N. Sen, & B. V. Subbarayappa : A Concise History of Science in India. New Delhi, 1971.

Neugebauer = Otto Neugebauer : The Exact Sciences in Antiquity. Copenhagen, 1952.


p. 12 a way of mentioning impropre moieties

/ardha/ ('1/2') before a # is indicatory of subtraction of 1/2 from it.

{Cf. Roman numeral with a smaller numeral before it (/I/ before /V/ or /X/, /X/ before /L/ or /C/, etc.) indicatory of subtraction of such. Also cf. numbers named in the Yoruba language by similar subtractions.}


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Baudha-ayana 1.


p. 147 (according to Baudha-ayana 1:3) units of measurement of distance

one __

aequaleth __

angula

14 an.u-s = 34 tila-s

(small) pada

10 angula-s

prades`a

12 angula-s

pada

15 angula-s

is.a

188 angula-s

aks.a

104 angula-s

yuga

86 angula-s

janu

32 angula-s

s`amya

36 angula-s

bahu

36 angula-s

prakrama

2 pada-s

aratni

2 prades`a-s = 24 angula-s

purus.a

5 aratni-s

vyayama

4 aratni-s

angula ('finger[-breadth]') is approximately 3/4 inch

{/Aratni/ is the distance "from the elbow to the tip of the little finger" (S-ED). The word for elbow is (in /Latin) ULNa/ : cognate with proto-Germanic */ULNa-/ > Old Norse /ULLr/, name of the god who remained amongst birds which his goddess-wife (Skadi) considered too noisy [she said, "Sleep I could not on ocean's couch for the wailing cry of the gull : from the wide sea faring, that bird awoke me" (EPE, p. 273)] : cf. arrival by Purn.a (cf. /-PaURN.a-/ infra p. 170) at "Matulagiri, where the incessant cries of birds disturbed him" (Samyukta Agama, chapter 35 http://www.btrts.org.sg/ven-purna-maitrayani-putra ).}

S-ED = Monier Monier-Williams : Sanskrit-English Dictionary. 1899.

EPE = Olive Bray (transl.) : The Elder or Poetic Edda, Commonly Known as Saemund's Edda. Viking Club, London, 1908. http://heathengods.com/library/The%20Elder%20or%20Poetic%20Edda%20Part%20I%20-%20O%20Bray.pdf


pp. 147-8 differences from Baudha-ayana in other s`ulba-sutra-s, concerning measures of distance

p. 147

"The unit pada has been made equal to 12 angulas by Katyayana (... 5.9).


The term vitasti has been used in place of prades`a by [Katyayana and Manava]."


"Manava (... 4.2 - 4.4) has ...

p. 148

1 angula = 6 yavas; and 1 prades`a = 10 vitastis."


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Baudha-ayana 2.


pp. 168-9 irrationality of certain square roots

p. 168

"Baudhayana, Apastamba and Katyayana gave the value of [the square root of] 2 ... with an additional term vis`es.a (approximate). ... According to Karavindasvami ..., the root s`is. when prefixed by vi denotes in all cases a 'correction in excess'. Datta [(2), pp. 198-202] has discussed the matter in detail ... . ...

p. 169

Hence vis`es.a refers to a small quantity, which is ... in excess ..., and cannot be accurately determined."

Datta (2) = B. Datta : The Science of S`ulba. Calcutta Univ, 1932.


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Baudha-ayana 3.


p. 169 prac-vams`a vedi

"the sacrificial ground ..., the pracina-vams`a or pragvams`a, as called by Baudhayana, Katyayana, and others, is erected at the western end, lying along ... the east-west line ... . The name is derived from the use of horizontal beams (vamsa) supported by four corner posts, on which {diagonally horizontal} corner beams are fastened ... . In this hall, ... the members of the family and friends ... can assemble."


pp. 169-70 within the prac-vams`a vedi : directional locations of the sacred fire

p.

fire

its direction

169

garhapatya ["is a perpetual sacred fire from which the other sacrificial fires are lighted." (p. 170)]

west


ahavaniya ["used for cooking oblations." (p. 170)]

east


daks.in.a-agni ["also called by the name anv[-]aharya-pacana, is used for cooking food." (p. 170)]

southwest


utkara ["is a rubbish pit dug out from the ground" (p. 170)]

northeast

170

dars`a-paurn.a-masika "for the full-moon sacrifice"

{/MAS-/ is Aztec for 'deer'. The deer-head is concealed in the dark of the moon (according to >al-Beruni).}

centre


p. 170 measurements of the saumikin vedi

"East of the pragvams`a and separated by a narrow space, the mahavedi or the saumiki vedi (the soma altar) is placed symmetrically about the east-west ... line.

This is a trapezium measuring

30 units on the western side,

24 on the eastern side and

36 units east-west along the spine [of symmetry]."


p. 170 (each measuring from its centre,) "The setting up of the mahavedi is described in the S`atapatha Brahman.a [3:5:1:3 - 3:5:1:6] as follows ... :"

sequence

__ paces (steps)

to the __

This is the __ of the altar for Soma.

1st

3

east

"intermediate"

2nd

15

south

"right hip"

3rd

15

north

"left hip"

4th

36

east

"fore-part"

5th

12

south

"right shoulder"

6th

12

north

"left shoulder"


pp. 170, 172 sadas-tent and 8 dhis.n.ya-hearths

p. 170

"the sadas tent is erected [within and] near the western base of the mahavedi. The tent is rectangular in area, the longer side lying south-north symmetrically about the east-west line, and is provided with doors on the western and eastern side. This tent is reserved for the priests to sit

p. 172

... : "And because all the gods {Vis`we-devah.} sat (sad) in it


therefore it is called sadas --

{But cf. the name of /SADyAS-jata/ (Sadyojata), one of the 5 Brahman-s (as worshipped in Bali etc.),}


and so do these these Brahman.as of every family {sarva-kula} now sit therein." [S`atapatha Brahman.a 3:5:3:5] ... In the middle of it,


a post of udumbara wood (Ficus glomerata) is fixed to the ground."

{/UDUMbara/ may be cognate with /ADUMnos/, name of a night-watchboy "frequently identified with" (WhWhCM, p. 26, s.v. "Adymnus") Phae:thon, whose name (< */BHaSaDhan/) may be cognate with /BSo^Dyah/, father of a setter-up of house-beams, of bolts, and of doors (Nh.emyah 3:6).}

WhWhCM = Adrian Room : NTC's Classical Dictionary. Contemporary Publ Gp, 1990. (reprinted as Who's Who in Classical Mythology. Gramercy Bks, 1997) http://library.atgti.az/categories/history/who%20is%20who%20in%20mythology.pdf

Nh.emyah 3:6 http://bibleapps.com/b/besodeiah.htm


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Baudha-ayana 4.


p. 177 shapes & measurements of altars

name

shape

measurement

agni-idhriya

square

side = 5 aratni-s

catvala

square

side = 36 angula-s

dhis.n.a-s

circles

diametre = 2 prades`a-s

havir-dhana

square

side = 10 or 12 prakrama-s

maha-vedi

isosceles trapezium

face = 24 pada-s; base = 30 pada-s; altitude = 36 pada-s (The units may be instead in prakrama-s.)

marjaliya

square

side = 5 aratni-s

prac-vams`a

rectangle

length = 16 prakrama-s, breadth = 12 prakrama-s; or else, length = 12 prakrama-s, breadth = 10 prakrama-s

sadas

rectangle

length = 27 aratni-s, breadth = 10 prakrama-s; or else length = 18 aratni-s, breadth = 10 prakrama-s


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Baudha-ayana 5-7.


p. 181 layers of bricks constituting a fire-altar

"In the usual five-layer construction, the height {stature} of the altar is 32 ang., height {stature} of each layer being 32/5 ang. (Bs`l. 5.7 and 7.2). According to Karavindasvami, it {the thickness/stature of each layer} is 6 ang. ... . ... the 3rd and 5th layer are the replica of the first and the 4th layer that of the 2nd. In the placement of bricks in different layers clefts between two layers are avoided. An altar f daily fire like the garhapatya has 21 bricks in each layer, while a kamya fire-altar is constructed with 1000 bricks, each layer containing 200 bricks. ...

Pan~ca[-]cod.a and nakasat bricks have half the thickness of ordinary bricks ... . They are usually on {in} the fifth layer placed at the top."


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Baudha-ayana 10.


p. 190 S`ata-patha Brahman.a 10:2:1:1

"Prajapati was desirous of going up to the world of heaven; ... . He saw this bird-like body, the fire-altar, and constructed it. ... By contracting and expanding (the wings) he did fly up : whence even to this day can only fly up when they contract their wings and spread their feathers"."


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Baudha-ayana 11. S`yena-cit


p. 197 "types of bricks ... for covering the fire-altar"

#

description

name

dimensions (ang.)

B1

square, 1/5 pu

pan~cami

24 * 24

B2

rectangular longer than pan~cami by half

adhyardha-pan~cami

24 * 36

B3

rectangular longer than pan~cami by a quarter

pan~cami-sapada

24 * 30

B4

triangular, half of one-fifth

pan~cami-ardha

24, 24

B5

triangular, quarter of one-fifth

pan~cami-padya

24, 12

B6

triangular, half of adhyardha

adhyardha-ardha

36, 24


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Baudha-ayana 12. Kanka-cit


p. 204 "types of bricks ... for covering the fire-altar"

description

name

dimensions (ang.)

triangular, one-eighth of one-fifth

as.t.ami

12, 12

4-sided, of area 1/4 pan~cami

caturas`ra-padya

6, 12, 18

4-sided, of area 1+1/2 pan~cami

caturas`ra-adhyardha

24, 24, 48


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Baudha-ayana 20. Kurma-cit


p. 228 "Fire-altars in the form of a tortoise [kurma, a swimming turtle]". {Cf. 'Great Tortoise' /Maha-kas`yapa/, who understood when silently the Buddha displayed a flower ("FS") : much as "a flower by way of invitation" (to visit the district Sunaparanta) was sent to the Buddha by Purn.a (PN"P").}

"FS" = "Flower Sermon". http://www.tamqui.com/buddhaworld/Flower_Sermon

PN"P" = Pali Names "Pun.n.a". http://www.palikanon.com/english/pali_names/pu/punna.htm ).}

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Apa-stamba 8 & 9 & 10. Caturas`ra-s`yena-cit


p. 242 bricks

name

description

dimensions (ang.)

adhyardha

rectangular

36 * 24

ardhya

rectangular half of pan~cami

24 * 12

prades`a

square quarter of one-fifth

12 * 12


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Apa-stamba 11.


p. 244 bricks

#

description (fraction of purus.a)

name

dimensions (ang.)

B1

one fourth

an.uka

30 * 30

B2

one-fifth

aratni

24 * 24

B3

one-sixth

urvasthi

20 * 20

B4

quarter of one-fourth

an.ukapada

15 * 15


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Katya-ayana.


pp. 266-7 terminology of geometry

p.

term

its meaning

266

tiryanmani {tiryag-}

'shorter side' of a rectangle


s`at.akamukha

'isosceles triangle' (litterally, 'cart-mouth')

267

pras`vamani

'longer side' of a rectangle


aks.n.aya

'diagonal' of a rectangle


bhumi

'area'


ks.etra

'figure' in plane geometry


ks.etrajn~ana-m

'knowledge of plane-figures'


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Manava 7 & 8.


p. 276 "various bricks such as vis`vajyoti, r.tavya, svayamatr., apasya {cf. name of Apasyanta}, pran.abhr.t, vais`vadevi {Vaidik deity-category}, vayavya {cf. name of "Vayaviya Samhita" of S`iva Upa-puran.a}, chanda, viraja {cf. name of Vaidik goddess Viraj}, vikarn.i have been used."


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S. N. Sen & A. K. Bag : The S`ulbasutras of Baudhayana, Apastamba, Katyayana and Manava. Indian National Science Academy, New Delhi, 1983.