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

258 BRAHMAGUPTA AS AN ALGEBRAIST
this type was a problem, write Datta and Singh, which could
not be solved completely nor satisfactorily by Brahmagupta. In
fact, Brahmagupta had to depend on trial. Success in this direc-
tion was, however, remarkably attained by Bhāskara II. He evol-
ved a simple and elegant method which assisted in deriving an
auxiliary equation having the required interpolator ±1, ±2, or ±
4, simultaneously with its two integral roots, from another auxi-
liary equation empirically formed with any simple integral value
of the interpolator, positive or negative. This method has been
technically known as Cakravāla or the cyclic method. This is
so called because it proceeds as in a circle, the same set of opera-
tions being applied again and again in a continuous round. For
the details of this method, our reader is requested to consult
the Algebra of Bhāskara II and the narrative on this method as
given by Datta and Singh under the title "Cyclic Method" in
their History of Hindu Mathematics: Algebra, 1962 Edition,
pp. 161-72.
Solution of Indeterminate
Quadratic Equation
It is remarkable to see that Brahmagupta was the first
algebraist in the history of mathematics to find a general solu-
tion of the indeterminate quadratic equation
Nx²±c=y²
in positive integers. We have the following verse in the
Brāhmasphuṭasiddhānta in this connection:
From two roots (of a Square-nature or varga-prakṛti)
with any given additive or subtractive, by making
(combination) with the roots for the additive unity
other first and second roots (of the equation having)
the given additive or subtractive (can be found).¹
Let us take the following two equations:
a₁k=an+b; and b₁k=bn+Na
From them we get : by eliminating n
a₁b−ab₁=1

1. रूप प्रक्षेपपदे पृथगिष्टक्षेपशोध्यमूलाभ्याम् ।
   कृत्वाऽऽन्त्याथपदे ये प्रक्षेपे तेने ॥ —BrSpSi. XVIII. 66

SOLUTION OF INDETERMINATE 259
Hence b₁ = (a₁b - 1) / a = a whole number.
Now n² - N = ((a₁k - b)² - Na²) / a²
= (a₁²k² - 2bka₁ + k) / a²
= k(a₁²k - 2ba₁ + 1) / a²
Therefore (k / a²)(a₁²k - 2ba₁ + 1) is a whole number.
Since a, k have no common factor, it follows that
(a₁²k - 2ba₁ + 1) / a² = (n² - N) / k = k₁ = an integer.
Also k₁ = (n² - N) / k = (a₁²k - 2ba₁ + 1) / a²
= (a₁²(b² - Na²) - 2ba₁ + 1) / a²
= ((a₁b - 1) / a)² - Na₁².
Thus having known a single solution in positive integers
of the equation Nx² ± c = y², says, Brahmagupta, an infinite
number of other integral solutions can be obtained by making
use of the integral solutions of Nx² + 1 = y². If (p, q) be a
solution of the former equation found empirically and if (α, β)
be an integral solution of the latter, then by the principle of
Composition
x = pβ ± qα; y = qβ ± Npα
will be a solution of the former. Repeating the operations, we
can easily deduce as many solutions as we like.
FORM Mn²x² ± c = y² :
In this connection, Brahmagupta says :
If the remainder is that divided by a square, the first
root is that divided by its root¹.
This seems to mean that if we have the equation
Mn²x² ± c = y² (i)
such that the multiplier (i.e. the coefficient of x²) is divisible

1. वर्गेच्छिन्ने गुणके प्रथमं तन्मूल भाजितं भवति । —BrSpSi. XVIII.70

260 BRAHMAGUPTA AS AN ALGEBRAIST
by n², then we are justified in saying that if we put nx=u, the
equation (i) becomes Mu²±c=y² (ii),
and clearly the first root of (i) is equal to the first root of (ii)
divided by n. The corresponding second root will be the same
for both the equations.
FORM a²x²±c=y² :
We find Brahmagupta giving the following rule in this
connection : This is a solution of a particular form of a
varga-prakṛti or Square-nature.
If the multiplier be a square, the interpolator divided
by an optional number and then increased and
decreased by it, is halved. The former (of these
results) is the second root; and the other divided by
the square-root of the multiplier is the first root.¹
Thus the solutions of the equation
a²x²±c=y²
are :
x = 1/(2a) (±c/m - m)
y = 1/2 (±c/m + m)
where m is an arbitrary number.
Bhāskara II and Nārāyaṇa have also given the same
solutions as proposed by Brahmagupta.
Rational Geometrical Figures
In the days of the Taittirīya Saṁhitā and the Śatapatha
Brāhmaṇa, Indian mathematicians got familiarity with the
solution of such equations
x²+y²=z²
and the results were arrived geometrically on the basis of the
law of rectangle as propounded by Baudhāyana in the Śulba Sūtras
and which goes by his name. The reader is referred to the Chapter
on Baudhāyana, the first Geometer in the author's "Founders
of Sciences in Ancient India". Baudhāyana (c. 800 B.C.) gave a

1. वर्गे गुणके क्षेपः केनचिदुद्धृतयुतोनितो दलितः ।
   प्रथमोऽन्यन्मूलमन्त्यो गुणकारपदोद्धृतः प्रथमः ॥ —BrSpSi. XVIII. 69

, is there a space or a matra?
Let's compare त in कोटिः or तद्दलं: त has a left loop.
In आयतचतुरस्रस्य: आ य त च तु र स्र स्य.
No, it's आयतचतुरस्रस्य. Wait, let me re-examine: आयताचतुरस्रस्य?
Wait, look at त: the right side of त goes down, then there is a च. It's आयतचतुरस्रस्य.
Wait, what about चेष्टस्येष्टाधिका?
Let's look at चे ष्टस्येष्टाधिका:
चे has a space after it?
Wait, look at the footnote line 1:
1. इष्टस्य भुजस्य कृतेर्भक्तो नेष्टेन तद्दलं कोटिः ।
Line 2:
आयतचतुरस्रस्य चे ष्टस्येष्टाधिका कर्णः ॥ —BrSpSi. XII. 35
Wait, look at चे ष्टस्येष्टाधिका:
There is चे, then a space, then ष्टस्येष्टाधिका.
Wait, could it be चेष्टस्येष्टाधिका with just a broken space? Yes, in De

262 BRAHMAGUPTA AS AN ALGEBRAIST
The sum of the squares of two unequal numbers is the
side; their product multiplied by two is the altitude,
and twice the difference of the squares of those two
unequal numbers is the base of an isosceles triangle.¹
Thus if m,n be two integers such that m is not equal to n,
the sides of all rational isosceles triangles with integral sides are
given by
m² + n², m² + n², 2(m² - n²)
and the altitude of the triangle is 2mn.
This method was also followed by Mahāvīra and other
Indian mathematicians. In fact, their solutions are based on
the juxtaposition of two rational right triangles, equal so that
they have a common leg. It is remarkably a powerful device, for
every rational triangle or quadrilateral may be formed by the
juxtaposition of two or four rational right triangles.
Isosceles Triangles with a Given Altitude
Here we have a rule given by Brahmagupta for finding out
all rational isosceles triangles possessing the same altitude :
The (given) altitude is the producer (karaṇī). Its
square divided by an optional number is increased and
diminished by that optional number. The smaller is the
base and half the greater is the side.²
Thus if m be any rational number then for a given definite
altitude a, the sides of the rational isosceles triangles are
½ (a²/m + m) each and the base is (a² - m²)/m . We shall illustrate it
by an example taken from the commentary of Pṛthūdaka Svāmī
The given altitude is 8; let us take any rational number m = 4
then the two equal sides of the isosceles are given by ½ ((8² + 4²)/4) =
10 each and the base is (8² - 4²)/4 = 12. Thus the three sides of the

1. कृति युतिर सदृशराश्योर्बाहुर्घातो द्विसंगुणो लम्बः ।
   कृत्यन्तरमसदृशयोर्द्विगुणं द्विसमत्रिभुज भूमिः ॥ —BrSpSi. XII. 33
2. करणी लम्बस्तत्कृतिरिष्टहतेष्टोन संयुताऽल्पा भूः ।
   अधिको द्विहृतो बाहुः संक्षेप्यो यद्वधो वर्गः ॥ —BrSpSi. XII. 37.

ISOSCELES TRIANGLES 263
rational isosceles triangle with altitude 8 are (10,10, 12).
Rational Scalene Triangles: Brahmagupta lays down the
following rule in the case of rational scalene triangle :
The square of an optional number is divided twice
by two arbitrary numbers; the moieties of the sums
of the quotients and (respective) optional numbers are
the sides of a scalene triangle; the sum of the moieties
of the differences is the base.¹
In other words, if m, p, q are any rational numbers, then
the sides of a rational scalene triangle are :
½ ( m²/p + p ), ½ ( m²/q + q ),
½ ( m²/p - p ) + ½ ( m²/q - q )
Here the altitude (m), area and segments of the base of
this triangle are all rational.
Thus putting m=12, p=6, and q=8 in Brahmagupta's gene-
ral equation, Pṛthūdaka Svāmī derives a scalene triangle with
sides (13,15) and (14) altitude (12), area (84 and the segments of
the base (5) which are all integral numbers.
½ ( m²/p + p ) = ½ ( 12²/6 + 6 ) = 15;
½ ( m²/q + q ) = ½ ( 12²/8 + 8 ) = 13
B' A C' A' B' C'
B H C H B C
Fig. 19 Fig. 20

1. इष्टद्वयेन भक्तो द्विधेष्ट वर्ग फलेष्टयोगार्धे ।
   विषमत्रिभुजस्य भुजाविष्टोनफलार्धयोगो भूः ॥
   —BrSpSi. XII, 34.

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