ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

264 BRAHMAGUPTA AS AN ALGEBRAIST
Thus the two sides of the rational scalene triangle are 15
and 13. The base is .
½ ( 12²/6 - 6 ) + ½ ( 12²/8 - 8 ) = 9 + 5 = 14
The altitude is m = 12; area is equal to (base × altitude) / 2
= (14 × 12) / 2 = and the segments are √(13² - 12²) = 5 and
√(15² - 12²) = 9. Thus they are all integers.
Rational Isosceles Trapeziums
Brahmagupta has given us a method of obtaining such
isosceles trapeziums whose sides, diagonals, altitude, segments
and area are all rational numbers. His rule is as follows :
The diagonals of the rec-
tangle (generated) are the flank
sides of an isosceles trapezium;
the square of its side is divided
by an optional number and
then lessened by that optional
number and divided by two;
(the result) increased by the
upright is the base and lessened
by it is the face.¹
Fig. 21
Here in the figure, we have the isosceles trapezium ABCD
of which C D is the base and A B is known as the fase. Accord-
ing to Brahmagupta's rule, we have ( p being the optional
number).
CD = ½ ( (4m²n² - p) / p ) + ( m² - n² ) (base)
AB = ½ [ 4m²n²/p - p ] - [ m² - n² ] (face)
DH = ( m² - n² ) (upright)

1. आयतकर्णौ बाहू भुजकृतिरिष्टेन भाजितेष्टोना ।
   द्विहृता कोट्यधिका भूर्मुखमूना दिसमचतुरस्रे ॥ —BrSpSi. XII. 36

RATIONAL TRAPEZIUMS 265
AD = BC = m²+n² (the sides of the trapezium)
HC = base-upright = ½ [ 4m²n²/p - p ] (segment)
AC = BD = [ 4m²n²/p + p ] (diagonal)
AH = 2 mn (altitude)
ABCD = mn [ 4m²n²/p - p ] (area)
By chosing the values of m n and p suitably, the values
of all the dimensions of the isosceles trapezium can be made
integral. Pṛthūdaka Svāmī starts with the rectangle (5, 12, 13)
and suitably takes p as 6; then he calculates out the dimensions
of the trapezium : flank sides (AD and BC) = 13, base =14,
and base = 4, altitude (AH) = 12, segments of base (DH and
HC) = 5, and 9, diagonals (AC and BD) = 15, area ABCD =
108. All these values are integers.
In this example, the rectangle chosen is (5, 12, 13) which is
AA' DH, where AD = m² + n² = 13
and DH = m² - n² = 5
whence by adding the two we have
2m² = 18
This gives the value of m = 3, and hence n = 2. Pṛthū-
daka Svāmī has taken the value of p = 6 by choice. Putting
these values of m, n and p, the values for the dimensions of the
isosceles trapezium follow from the expressions given by Brahma-
gupta.
CD = ½ ( (4.3².2² / 6) - 6 ) + ( 3²-2² ) = 9+5 = 14 (base)
Face = 9-5=4
Sides AD = BC = 3²+2² = 13
and so on for the other dimensions.
Rational Trapeziums With
Three Equal Sides
This problem is very much the same as one of the rational
isosceles tpapezium with the only difference that in this case
one of the parallel sides is also equal to the slant sides. We

266 BRAHMAGUPTA AS AN ALGEBRAIST
have the following solution of this problem from Brahma-
gupta :
The square of the diagonal (of a generated rectangle)
gives three equal sides; the fourth (is obtained) by
subtracting the square of the upright from thrice the
square of the side (of that rectangle). If greater, it
is the base; if less, it is the face.¹
As before, the rectangle generated from m, n is given by
(m²–n², 2mn, m²+n²), that is these are the three sides of
the right triangle, which correspond to the two sides and the
diagonal of the rectangle generated by them. Let us suppose,
we have a trapezium ABCD whose sides AB, BC and AD are
equal, then
AB = BC = AD = (m²+n²)²
CD = 3(2mn)² – (m²–n²)² = 14 m²n² – m⁴ – n⁴
or CD = 3(m²–n²)² – (2mn)² = 3m⁴+3n⁴ – 10 m²n².
Pṛthudaka Svāmī has taken an illustration, where m=2,
n=1 and he deduces two rational trapeziums with three equal
sides (25, 25, 25, 39) and (25, 25, 25, 11).
The segment (CH), altitude (AH), diagonals (AC, BD)
and area of this trapezium are also rational, and given by :
CH (segment) = 6 m²n²—m⁴ – n⁴
AH (altitude) = 4 mn (m²—n²)
AC = BD (diagonals) = 4 mn (m²+n²)
ABCD (area) = 32m³n³ (m²—n²).
Rational Inscribed Quadrilaterals
We find in the Brāhmasphuṭasiddhānta a remarkable
proposition formulated by Brahmagupta :
To find all quadrilaterals which will be inscribable
within circles and whose sides, diagonals, perpendi-
culars, segments (of sides and diagonals by perpendi-
culars from vertices as also of diagonals by their
intersection), areas, and also the diameters of the

1. कर्णकृतित्रिसम भुजास्त्रयश्चतुर्थो विशोष्य कोटि कृतिम् ।
   बाहुकृतेस्त्रिगुणाया यद्यधिको भूर्मुखं हीनः ॥ —BrSpSi. XII. 37

RATIONAL INSCRIBED QUADRILATERALS 267
circumscribed circles will be expressible in integers.
Such quadrilaterals we shall call as Brahmagupta
Quadrilaterals.
The solution of this formidable problem has been given by
Brahmagupta as follows :
The upright and bases of two right-angled triangles
being reciprocally multiplied by the diagonals of the
other will give the sides of a quadrilateral of unequal
sides : ( of these ) the greatest is the base, the least
is the face, and the other two sides are the two
flanks.¹
Taking Brahmagupta's integral solution, the sides of the
two right triangles of reference are given by :
(1) m²-n², 2 mn, m²+n²;
(ii) p²-q², 2 pq, p²+q²;
where m, n, p, q are integers. Then the sides of the Brah-
magupta Quadrilateral are.
(m²-n²)(p²+q²), (p²-q²)(m²+n²),
2mn(p²+q²), 2pq(m²+n²) (Arrangement A)
Pṛthūdaka Svāmī has-
illustrated the rational inscribed
quadrilateral by taking an
example of the right angle
triangles.
(i) (3,4,5) (m²-n²=3,
m²+n²=5, whence
m=2, n=1)
(ii) (5,12,13)(p²-q²=5,
p²+q²=13, whence
p=3, q=2)
Fig. 22
Substituting these values in the above equations, we get the
sides of the quardilateral as ( 39, 25, 52 and 60).²

1. जात्यद्वय कोटिभुजाः परकर्णगुणा भुजाश्चतुर्विषमे ।
   अधिको भूर्मुखंहीनो बाहुद्वितयं भुजावन्यौ ॥ —BrSpSi. XII. 38
   2 The diagonals of this quadrilateral are given by Bhāskara II as 56
   (=3.12+4.5) and 63 (=4.12+3.5). (Cont. on page 268)

270 BRAHMAGUPTA AS AN ALGEBRAIST
Put in other words, this means that one has to solve the
following equations :
(i) 5x—25 = y²
(ii) 10x—100 = y²
(iii) 83x—7635 = y²
Pṛthūdaka Svāmī, the commentator on the Brāhmasphuṭa-
siddhānta proceeds to solve these equations as follows :
(1.1) Suppose y = 10; then x = 125. Or put y = 5; then
x = 10.
(2.1) Suppose y = 10; then x = 20.
(3.1) Assume y = 1; then x = 92.
He then remarks that by virtue of the multiplicity of
suppositions there will be an infinitude of solutions in every
case, But no method has been given either by Brahmagupta or
his commentator to obtain the general solution.
Double Equations of the First Degree
Perhaps we have the earliest reference of the simultaneous
indeterminate quadratic equations of the type
x ± a = u²
x ± b = v²
in the Bhakaśālī Manuscript (Folio 59, recto).
Brahmagupta gives the solution of such simultaneous inde-
terminate quadratic equations of a general case as follows :
The difference of the two numbers by the addition
or subtration of which another number becomes a sq-
uare, is divided by an optional number and then incre-
ased or decreased by it. The square of half the result
diminished or increased by the greater or smaller (of
the given number) is the number (required).¹
Expressed in the language of algebra, shall have :
= ½ { ½ ( (a - b)/m ± m ) }² ∓ a

1. याभ्यां कृतिरधिको नस्तदन्तरं हृत युतो न मिष्टेन ।
   तद्दल कृतिरधिकोऽधिकयो रविको न यो राशिः ॥ —BrSpSi. XVIII. 74

DOUBLE FIRST DEGREE EQUATIONS 271
or x = { ½ ( (a - b) / m ∓ m ) }² ∓ b
where m is an arbitrary number.
Datta and Singh has given the rationale of this method as
follows :
u² = x ± a; v² = x ± b,
From them, we have u² - v² = ± a ∓ b
Therefore u - v = m
and u + v = (± a ∓ b) / m ,
where m is arbitrary. Hence
u = ½ ( (± a ∓ b) / m + m ) = ± ½ ( (a - b) / m ± m )
Sincc it is obviously immaterial whether u is taken as posi-
tive or negative, we have
u = ½ ( (a - b) / m ± m )
Similarly v = ½ ( (a - b) / m ∓ m )
Therefore x = { ½ ( (a - b) / m ± m ) }² ∓ a,
or x = { ½ ( (a - b) / m ∓ m ) }² ∓ b,
where m is an arbitrary number.
Now we shall take up another particular case, for which
Brahmagupta has given a rule :
The sum of the two numbers tne addition and sub-
traction of which make another number (severally)
a square, is divided by an optional number and then
diminished by that optional number. The square
of half the remainder increased by the subtractive
number is the number (required)¹.
In the algebraic notations, we shall express it as follows :

1. यैरूनो यैश्च युतो रूपैर्वर्गस्तद्द्वयमिष्ट हृतम् ।
   इष्टोनं तद्दल कृतिरूनाऽभ्यधिका भवति राशिः ॥ —BrSpSi. XVIII. 71

274 BRAHMAGUPTA AS AN ALGEBRAIST
y = 1/a ((ad + bc)/m + b)
if b > c and m > (ad + bc)/m. If these conditions be reversed then x
and y will have their values interchanged.
Datta and Singh have given the following rationale of
these solutions :
axy = bx + cy + d,
or a²xy − abx − acy = ad,
or (ax − c) (ay − b) = ad + bc.
Suppose ax − c = m, a rational number;
then ay − b = (ad + bc)/m.
Therefore
x = 1/a (m + c)
y = 1/a ((ad + bc)/m + b)
Or, we may put ay − b = m;
in that case, we shall have ax − c = (ad + bc)/m;
whence x = 1/a ((ad + bc)/m + c),
y = 1/a (m + b).
Brahmagupta’s own rule.
Whilst the rule given above is ascribed to an unknown
author, Brahmagupta’s own rule for the solution of a quadratic
indeterminate equation involving a factum is as follows :
With the exception of an optional unknown, assume
arbitrary values for the rest of the unknowns, the
product of which forms the factum. The sum of the
product of these (assumed values) and the (respective)
coefficients of the unknowns will be absolute quantities.
The continued products of the assumed values and of
the coefficient of the factum will be the coefficient of
the optionally (left out) unknown. Thus the solution

BRAHMAGUPTA'S OWN RULE 275
is effected without forming an equation of the factum
Why then was it done so ?¹
Datta and Singh think that the reference in the latter por-
tion of this rule is to the method of the unknown author :
“Kiṁ kṛtaṁ tadataḥ” ? The principle underlying Brahma-
gupta's method is to reduce, like the Greek Diophantus (c.275
A.D.), the given indeterminate equation to a simple determi-
nate one by assuming arbitrary values for all the unknowns ex-
cept one. So undoubtedly it is inferior to the earlier method.
We now take an illustrative example from Brahmagupta :
On subtracting from the product of signs and degrees of
the Sun, three and four times (respectively) those
quantities, ninety is obtained. Determining the Sun
within a year (one can pass as a proficient) mathema-
tician.
If we presume x to denote the signs and y the degrees of
the Sun, then the equation would be :
xy - 3x - 4y = 90
Pṛthūdaka Svāmī solves it in two ways :
(i) Let us assume the arbitrary number to be 17. then
x = 1/1 ( (90.1 + 3.4) / 17 + 4 ) = 10
y = 1/1 (17 + 3) = 20
(ii) Let us assume arbitrarily y = 20. On substituting
this value of y in the above equation, we get
20x - 3x = 170
whence x = 10.

1. भावितके यद्घातो विनष्टवर्णेन तत्प्रमाणानि ।
   कृतेष्टानि तदाहत वर्णैक्यं भवति रूपाणि ॥
   वर्ण प्रमाणभावित घातो भर्वातेष्ट वर्ण संख्यैवन् ।
   सिध्यति विनाऽपि भावित-समकरणात् किं कृतं तदतः ॥ —BrSpSi. XVIII. 62-63
2. भानो राश्यंशवधात् त्रिचतुर्गुणितान् विशोध्य राश्यंशान् ।
   नवतिं दृष्ट्वा सूर्यं कुर्वन्नावत्सराद् गणकः ॥ —BrSpSi. XVIII. 61.
   — : o : —
   Reference
   H.T. Colebrooke : Algebra with Arithmetic and Mensuration
   from the Sanscrit of Brahmagupta and Bhas-
   cara, London, 1817.
   B. Datta and A.N. Singh : History of Hindu Mathematics Pt. I
   and II, 1962.

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CHAPTER X
Arabic and Indian
Divisions of the
Zodiac
I It has long been a debated question whether the Indian
and Arabian divisions of the zodiac had a common origin. Sir
William Jones thought that they had not; but Colebrooke holds
a contrary view. The coincidence, in the two systems of division
is so exact that he thinks, it could not be due to chance. Coleb-
rooke has discussed this point in details in one of his Papers entit-
led "On the Indian and Arabian divisions of the zodiac", Asiatic
Researches Vol. ix.. p. 323-376, reproduced in the Miscellaneous
Essays, Vol. II., p. 321-373, 1872.

1. Aśvinī, now the first nakṣatra, but anciently the last
   but one, probably obtained its present situation at the head of
   the asterisms, when the beginning of the zodiac was referred to
   the first degree of Meṣa (the Ram). As measuring a portion of
   the zodiac, it occupies the first 13°20' of Meṣa; and its beginning
   follows immediately after the principal star in the last nakṣatra
   Revatī, reckoned by some exactly, by others nearly, opposite
   to the very conspicuous one, which forms the fourteenth asterism.
   As a constellation, Aśvinī comprises three stars (Aries α, β, γ)
   figured as a horse-head; and the principal, which is also the
   northern one, is stated by all ancient authorities, in 10°N and
   8°E. from the beginning of the Meṣa.
   According to Arabs, the first manzil or lunar mansion is
   entitled Sheraṭan (by Persians, Sheratain), and comprises two
   stars of the third magnitude on the head of Aries. in lat. 6°36'
   and 7°51' N and long. 26°13' and 27°7'. With the addition of a

278 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC
third, also in the head of the Ram, the asterism is denominated
Āshrait. The bright star of the second or third magnitude
which is out of the figure of the Ram, according to Ulugh Beg,
but on the nose according to Hipparchus, cited by this author
from Ptolemy, is determined Nātih : It is placed in lat. 9°30'N
and long. 1 0° 43', and is apparently the same with the principal
star of the Indian asterism; for Muhammad of Tizin, in his table
of declination and right ascension, expressly terms it the first star
of the Sheratain.
2. Bharaṇī, the second asterism, comprises three stars
(35, 39, 41 Aries) figured by the yoni or pudendum muliebre and
the principal and southern star of this nakṣatra is placed in 12°N.
On the Arabian system, the second manzil, entitled Butain is
placed by Ulugh Beg in lat. 1°12' and 3°12', and this cannot
possibly be reconciled with the Indian constellation. But
Muhammad of Tizin assigns to the bright star of Butain a decli-
nation of 23°N exceeding by nearly 2° the declination allotted by
him to Nātih or his first star in Sheratain. This agrees with the
difference between the principal stars of Aśvinī and Bharaṇī; and
it may be inferred, that some among the Mohammadan astrono-
mers have concurred with the Hindus in referring the second
constellation to stars that form Musca.
3. Kṛttikā, now the third, formerly the first, nakṣatra
consists of six stars figured as knife or razor, and the principal
and southern star is placed in 4½ or 5°N and in 65 sixths of
degrees (or 10°50′) from its own commencement (cf. the Sūryasid-
dhānta), or 37° 28′ to 38° from the beginning of the Meṣa (the
Siddhānta-Śiromaṇi or the Graha Lāghava) respectively. This
longitude of the circle of declination corresponds nearly with
that of the bright star in the Pleiades, which is 40° of longitude
distant from the principal star of Revatī.
The stars indicated by Ulugh Beg for Thurayyā, also corres-
pond exactly with the Pleiades.
4. Rohiṇī, is the fourth nakṣatra, the Arabic name for the
fourth mansion is Debarān (or with the article Aldebarān).
It corresponds to the bright star called the Bull's eye, and
which is unquestionably the same with the principal
and eastern star of Rohiṇī, placed in 4½° or 5°S and 49½°E
by the Hindu writers on Astronomy. This nakṣatra is

figured as a wheel cart, and comprises five stars, out of the seven
which the Greeks named the Hyades. The Arabs, however, like
the Hindus, reckon five stars only in the asterism. Sir William
Jones supposes them to be in the head and neck of the Bull; they
probably are α, ρ, γ, δ, ε Tauri, agreeably to Mons. Bally's
conjecture.
5. Mṛgaśirā, the fifth nakṣatra, represented by an antelope
head, contains three stars; the same which constitute the fifth
lunar mansion Hakāh; for the distance of 10°S assigned to the
northern star of this nakṣatra, will agree with no other but one
of the three in the head of Orion. The difference of longitude
(24° to 25½°) from Kṛttikā corresponds with sufficient exactness;
and so does the longitude of its circle of declination (62° to 63°)
from the end of Rewatī; since the true longitude of λ Orionis,
from the principal star in Revatī (ζ piscium) is 63½°.
6. Ārdrā, the sixth nakṣatra, consists of a single bright
star, described as a gem, and placed in 9°S (by some in 11°) and
at the distance of 4⅓ to 4° in longitude from the last asterism.
This indicates the star in the shoulder of Orion (a orionis). The
sixth lunar mansion is named by the Arabs as Hanāh; and
comprises two stars in the feet of the second twin, according to
Ulugh Beg, though others make it to be a shoulder. Mohammad
of Tizin allots five stars to this constellation; and the Kāmūs,
among various meanings of Hanāh, says, that it is a name for
five stars in the left arm of Orion; remarking also, that the
lunar mansion is named Tahāyī, comprising three stars called
Tahyāt. Obviously here the Indian and Arabian asterisms are
irreconcileable.
7. Punarvasū (used in a dual number) is the seventh
nakṣtra, and is represented by a house, or even a bow, and it
includes four stars, among which the principal and eastern one
is 30° or 32° from the fifth asterism; but has been placed by all
authorities in 6°N. This agrees with (β Geminorum) one of the
two stars in the heads of the Twins, which together constitute
the seventh lunar mansion ziraā, according to Mohammad of
Tus and Mohammad of Tizin and other Arabian authorities.
The seventh lunar mansion of Arabs is named ziraā ul ased
according to Jauhari and other cited by Hyde in his Commen-

250 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC
tary on Ulugh Beg, and that the Kāmūs makes this term to be
the name of eight stars in the form of a bow.
8. Puṣya, the eighth asterism, is described as an arrow, and
consists of three stars, the chief of which being also the middle
most, has no latitude, and is 12° to 13° distant from the seventh
asterism, being placed by Hindu astronomers in 106° of longitude.
This is evidently δ Cancri; and does not differ widely from the
eighth lunar mansion Nethrah, which according to Ulugh Beg and
others consists of two stars, including the nebula of Cancer.
The Indian constellation comprises two other stars besides δ
Cancri, which are perhaps γ and β of the same constellation.
9. Aśleṣā, the ninth asterism, contains five stars figured
as a potter’s wheel, and of which the principal or eastern one is
placed in 7°S, and according to different tables, 107°, 108° or 109°
E. This appears to be intended for the bright star in the south-
ern claw of Cancer (α Cancri), and cannot be reconciled with
the lunar mansion Tarf or Tarfah, which comprises two stars
near the lion’s (siṁha) eye, the northernmost being placed by
Mohammad of Tīzīn in 24° of N. declination.
10. Maghā, the tenth asterism, contains like the last, five
stars, but which are figured as a house. The principal of the
Southern one has no latitude; and according to all authorities,
has 129° longitude. This is evidently Regulus (α Leonis) : which
is exactly 129½° distant from the last star in Revatī. The tenth
lunar mansion of Arabians is Jebhah, which comprises three
(some say, four) stars, nearly in the longitude of the lion’s heart.
In this instance, therefore, the Indian and Arabian divisions of
the zodiac coincide. This nakastra consists of α, γ, η and ν
Leonis.
11. Pūrva-Phālgunī is the eleventh nakṣatra and is repre-
sented by a couch or bedstead; it consists of two stars deter-
mined by the place of the chief star (the northernmost, accord-
ing to the Sūrya-Siddhānta) in 12°N and 144°E., or according to
Brahmagupta, the Śiromaṇi and the Grahalāghava 147° or 148°
E. They are probably δ and θ Leonis. The Arabian name for
this lunar mansion is zubrah or Khertan.

ARABIC AND INDIAN DIVISIONS OF THE ZODIAC 281
It may be mentioned here that Brahmagupta and Bhāskara
selected the southern for the principal star; while the Sūrya-
Siddhānta took the northern. Hence the latitude stated by
several Hindu authorities is the mean between both stars; and
the difference of longitude, compared to the preceding and subse-
quent asterisms, may be exactly reconciled upon this suppo-
sition.
12. Uttara-Phalgunī, which is the twelfth nakṣatra, con-
sists of two stars, and is figured as a bed or cot. These stars are
ascertained by the place of one of them (the northernmost) 13°N.
and 155° E. This indicates β Leonis; the same which singly
constitutes the Arabian Lunar mansion Serfah, though Moham-
mad of Tīzīn seems to hint that it consists of more than one
star.
13. Hasta, the thirteenth nakṣatra, has the name and figure
of a hand; and is suitably made to contain five stars. The
principal one towards the west, next to the north-western star,
is placed according to all authorities in 11° and 170° E. This
can only belong to the constellation Corvus ; and accordingly
five stars in that constellation (α, β, γ, δ and ε Corvi). The
thirteenth lunar mansion of Arabs is Awwā, which is also
described to contain five stars, situated under Virgo and so
disposed as to resemble the letter Alif. They are placed by
Ulugh Beg in the wing. Here obviously, there is nothing common
between the Hindu and Arabian specification of the asterism.
The agreement is only in the number of stars and in the longi-
tude.
14. Citrā, the fourteenth nakṣatra, is figured as pearl. It
is placed by the Sūrya-Siddhānta in 2° S and 180° E. and by
Brahmagupta, the Śiromaṇi and Grahalāghava in 1¾ or 2°S and
183°E. This agrees with the Virgin's spike (α Virginis). The
same star constitutes the fourteenth lunar mansion of the Arabs
named from it Simāc ul aāzi.
15. Svāti, the fifteenth nakṣatra, is represented by a coral
bead. The Sūrya-Siddhānta, Brahmagupta, the Śiromaṇi and
Grahalāghava, all concur in placing it at 37°N. They differ one
degree in longitude of its circle of declination, three of them

282 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC
making it 199° and the other 198°. The Indian asterism totally
disagrees with the lunar mansion Ghafr which is the fifteenth
Arabian mansion. and which consists of three stars in the Virgin's
(Kanyā) foot, according to Ulugh Beg. but in or near the
balance (Tulā), according to others.
16. Viśākhā, the sixteenth nakṣatra, consists of four stars
described as a festoon. All the authorities place the principal
and northernmost star in 1°, 1°20′ or 1°30′ S and in 212°, 212°5′
or 213° E. The latitude seems to indicate the bright star in the
Soutehrn Scale (α Librae), though the longitude disagrees (suggest-
ing possibly a remote star κ Librae). Colebrooke suggests the four
stars to be α ν ι Librae and γ Scorpii. The sixteenth lunar
mansion according to Arabs is Zubanah or Zubāniyah according
to Mohammad of Tizin, the bright star in the northern scale (β
Librae).
17. Anurādhā, the seventeenth nakṣatra, consists of four
stars and is described as a row of oblations in a right line. Its
chief or middlemost star is placed in 3°, or 2° or 1°45′ S and in 224°
or 224°5′E, thus placing it near the head of the Scorpion (Vṛścika)
(δ Scorpionis) and the asterism comprises β, δ, π, and ρ Scorpionis.
The seventeenth lunar mansion of Arabs is called Iklīl or Iklīlu'l-
jebhah, which is said to contain 4, 3, or 6 stars lying in a straight
line. Those assigned by Ulugh Beg for this mansion are β, δ, ν
and π Scorpionis. Thus here the Indian and Arabian astronomers
both concur exactly.
18. Jyeṣṭha, the eighteenth nakṣatra, comprises three stars
figured as a ring. The principal and middlemost star is placed
in 4°·3½° or 3° S and in 229°, 229°5′ or 230°E; this position indicates
Antares or the Scorpion's heart (α Scorpionis), which is also
the eighteenth lunar mansion, named Kalb or Kalbul'akrab.
The three stars of Indian asterism may be α, σ and τ Scorpionis.
19. Mūla, the nineteenth nakṣatra, is represented by a
lion's tail, and it contains eleven stars, of which the charac-
teristic one, the easternmost, is placed in 9°, 8½° or 8° S and in
241° or 242° E. This probably (not exactly) indicates ν Scor-
pionis. This agrees with the eighteenth lunar mansion of Arabs
known as Shaulah, consisting of two stars near the Scorpion's

sting. The Hindu asterism probably includes all the stars in the
Scorpion’s tail (ε, μ, ζ, η, θ, ι, κ, λ, υ and ν Scorpionis).
20. Pūrva-Āṣāḍha, the twentieth nakṣatra, is figured as an
elephant’s tooth or as a couch, and it consists of two stars, of
which the most southern one is placed in 5½°, 5⅓° or 5° S and
254° or 255° E. This corresponds well with δ Sagittarii, and
which also corresponds with the twentieth lunar mansion of
Arabs called Nāaim. The Arabian mansion consists of four,
or according to some eight, stars. The Indian nakṣatra corres-
ponds to δ and ε Sagittarii.
21. Uttara-Āṣāḍha, the twenty-first nakṣatra, is represented
by a couch or by an elephant’s tooth. The principal or the
most northerly star is placed in 5° S and 260° or 261° E, agreeing
with a star in the body of Sagittarius (τ Sagittarii), and the
other star is perhaps the one marked ζ. The Arabian lunar
mansion corresponding to it is Baldah, consisting of six stars,
two, of which are placed by Mohammad of Tīzīn in declination
21° and 16°. One of these must be a star in the head of Sagit-
tarius. Some authors, on the contrary, describe the lunar
mansion as destitute of stars. Here the Arabs and Hindus do
not show reconciliation.
22. Abhijit, the twenty-second asterism, consists of three
stars figuring as a triangle or as a nut of floating Trapa (in
modern Indian astronomy, it does not occupy an equal portion
of the ecliptic with other nakṣatras). Its brightest star is very
remote from the zodiac, being in 60° or 62° N. The longitude
of its circle of declination is 265°, 266° 40′ or 268° according to
different authorities. The corresponding lunar mansion of
Arabs is Zābih, consisting of two stars (according to some, four)
in the horns of Capricorn. This totally disagrees with Indian
asterism.
23. Śravaṇa, the twenty-third nakṣatra, is represented by
three footsteps, and contains three stars of which the middlemost
is placed in 30° N (all authorities agree), and longitude 280°
(Sūrya-Siddhānta) or 278° (Brahmagupta and Śiromaṇi), or 275°
(Grahalāghava). The assigned latitude indicates the bright
star in the Eagle, whence the three may be inferred to be α, β

284 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC
and γ Aquilae. According to Arabs, the twenty-third lunar
mansion is Balā, which consists of two stars in the left hand of
Aquarius. Here again Arabian and Hindu divisions are at
variance.
24. Dhaniṣṭhā, the twenty-fourth nakṣatra, is represented
by a drum or tabor. It comprises four stars, the westernmost
of which is placed in 36° N and according to Brahmagupta,
Śiromaṇi and the Sūrya-Siddhānta in 290° E (Grahalāghava
gives 286°). This longitude of the circle of declination and the
distance of the star on it from the ecliptic indicate the Dolphin :
and the four stars are α, β, γ and δ Dolphini. The correspond-
ing lunar mansion of Arabs is Sāud, which comprises two stars
in Aquarius (β and ζ Aquarii). Here again the two divisions
disagree completely.
24. Śatabhiṣak, the twenty fifth nakṣatra, is a cluster of 100
stars figured by a circle. The principal or the brightest has no
latitude; or only a third, or at utmost half, a degree of south
latitude; and longitude 320°. This best corresponds with λ
Aquarii. According to Arabs, the twenty-fifth lunar mansion
is known as Akhbiyah which consists of three stars only, placed
in the wrist of the right hand of Aquarius. However, it appears
from Ulugh Beg's tables, as well as from Mohammad of Tīzīn's,
that four stars are assigned to this mansion. The Indian and
Arabian systems of division differ considerably but less widely
according to some.
26. Pūrva-Bhādrapada, the twenty-sixth nakṣatra, consists
of two stars represented by a couch or bed, or else by a double
headed figure, one of which is placed in 24° N and 325° or 326°
E. The only conspicuous star nearly in that position is the bright
star in Pegasus (α Pegasi) and the other may be the nearest
considerable star in the same constellation (ζ Pegasi). The
twenty-sixth Arabian lunar mansion is Mukaddim, consisting
of two brightest stars in Pegasus (α and β). Here the Indian
and Arabian divisions show concurrence.
27. Uttar-Bhādrapada, the twenty-seventh nakṣatra, consists
of two stars, figured as a twin or a person with double face, or
else as a couch. The position of the most northerly of the two

ARABIC AND INDIAN DIVISIONS OF THE ZODIAC 285
is in 26° or 27°N and 337° E. which probably indicates the bright
star in the head of Andromeda, and the other star to be the
one in the extremity of the wing of Pegasus (γ Pegasi). This
exactly agrees with the twenty-seventh lunar mansion of Arabs
named as Muakkher. Ulugh Beg assigns those stars to it.
28. Revatī, the twenty-eighth nakṣatra, comprises thirty-
two stars figured as a tabor. The principal star is the southern-
most one, it has no latitude, and two of them assert no longitude,
but some make it ten minutes short of the origin of the ecliptic,
viz. 359° 50'. This clearly marks the star on the ecliptic in the
string of the Fishes (ζ Piscium). The ascertainment of this star is
important in regard to the adjustment of the Hindu sphere. The
Arabic name for this mansion is Risha, signifying a cord. But
the constellation as described by Jauhari and cited by Golius,
consists of a multitude of stars in the shape of a fish and termed
Betnu'lhūt; in the navel of which is the lunar mansion. Moham-
mad of Tīzīn alse makes this lunar mansion to be the same with
Betnu'lhūt, which appears, however, to be the bright star in the
girdle of Andromeda (β Andromedae) though others describe it
as the northern fish, extending, however, to the horns of Ram.
The lunar mansion and the Indian asterism, therefore, are not
reconcileable in this last instance.
I leave it to the readers to draw an inference as to the
concurrence of the divisions of zodiac in Indian and Arabian
systems. I would personally agree with Sir William Jones that
the agreements are by chance. Arabs derived the idea of
dividing zodiac in 27 or 28 mansions from Indians, or may have
got it from Greeks, and then they proceeded in their own way
for details. I do not agree with those scholars who sometimes
state that the Hindus took the hint of dividing the ecliptic from
Greeks. The Atharvaveda devotes a number of Suktas or
hymns on Nakṣatras, and I have shown elsewhere that inspired
by these hymns, Gārgya was the first Ṛṣi who detailed out
the nakṣatras. This happened much before Greeks developed
even their first notions of astronomy. While the concept of 27
nakṣatras is Vedic and most ancient and of purely Indian origin
the concept of 12 Rāśis (signs) or twelve constellations is proba-
bly inspired from Greeks. [The names Kanyā, (virgo), Tulā

286 ARABIC AND INDIAN DIVISIONS OF THE ZODIAC
(Libra),Vṛścika (Scorpio), Dhanu, (Sagittarius), Makara, (Capri-
corn), Kumbha (Aquarius), Mīna (pisces), Meṣa (Aries), Vṛṣa
(Taurus), Mithuna (Gemini), Karka (Cancer), and Siṁha (Leo)
were not used for Rāśis or signs in the Vedic times]. I shall con-
clude this description with a passage from Colebrooke :
The result of comparison shows, I hope satis-
factorily, that the Indian asterisms, which mark the
divisions of the ecliptic, generally consist of nearly the
same stars, which constitute the lunar mansions of
the Arabians : but in a few instances, they essentially
differ. The Hindus have likewise adopted the divi-
sion of the ecliptic and zodiac into twelve signs or
constellations, agreeing in figure and designation with
those of the Greeks; and differing merely in the place
of the constellations, which are carried on the Indian
sphere a few degrees further west than on the Grecian.
That the Hindus took the hint of this mode of
dividing the ecliptic from the Greeks, is not perha-
ps altogether improbable; but if such be the origin of it
they have not implicitly received the arrangement
suggested to them, but have reconciled and adapted
it to their own ancient distribution of the ecliptic into
twenty-seven parts.
In like manner, they may have either received or given
the hint of an armillary sphere as an instrument for
astronomical observation ; but certainly they have not
copied the instrument which was described by Ptolemy,
for the construction differs considerably.
Names, Shapes, and Number
of the Stars of the Nakṣatras
The Muhūrta-cintāmaṇi provides a list of shapes associated
to the nakṣatras (MuC. II. 59-60), In this list we are giving the
number of stars as indicated by Varāhamihira, Brahmagupta
and Lalla. The identification given here is as indicated by
E. Burgess, in his Translation of the Sūrya-Siddhānta 1935, p. 378.
(Calcutta). This table has been reproduced here from the
Mahābhāskarīya of Bhāskara I, edited by K.S. Shukla.

NAMES, SHAPES AND NUMBER OF THE STARS 287