A. A.
Krishnaswami Ayyangar (AAK), my father,
was born on 1 December, 1892. He got his M.A. in Mathematics at the age
of 18
and subsequently started teaching Mathematics at his own alma mater,

The subject matter of interest, presented in
this CD,
concerns only his works on the history of Indian Mathematics. This work
involved not only translations from the Sanskrit texts but also
transforming
some of them in modern notation and language. The originality of AAK’s
work was
brought home to Rajagopal on a casual conversation with Professor R.
Sridharan
of the Tata Institute of Fundamental Research, *Science in
the West and India*, edited by B. V. Subbarayappa and N. Mukunda,
(1995,
Himalaya Publishing House, Bombay). In this he points out that AAK‘s
article on
the Chakravala method (included in this CD) shows how it differs from
the
method of continued fractions. He recounted that this point was missed
by A.
Weil, who thought the Chakravala method was only an “experimental fact”
to the
Indians and attributed general proofs to Fermat and Lagrange much
later! On a
different occasion, Professor Subhash Kak of

Without the keen interest evinced by Professor
Kak and
his drive to get Indian contributions to the World of Astronomy,
Mathematics,
Music, and everything else, this collection would have suffered the
quiet death
of so many things Indian! Two new publications may be mentioned: **“ Indian
Mathematics: Redressing the balance”** by Ian G. Pearce (2002, http://www.history.mcs,st-andrews.ac.uk/history/Projects/Pearce/index.html)
and

It was not easy to obtain the relevant articles from the various journals particularly because AAK’s publications date back to early 1920’s and onwards. This collection of AAK’s articles on Indian Mathematics and Astronomy, in this CD, is a partial culmination of a much larger search for all his writings. In this form, this CD is our family’s tribute to our father’s memory. It is gratifying to find these works to be of interest to people engaged in writing about the World History of Mathematics and Astronomy to day. I hope this collection is a modest contribution to this endeavor.

Rajagopal’s many conversations with Professor
Kak on
Indian contributions to various human endeavors over the millennia
started all
this. Professor M. S. Rama Rao of

A.
K. Srinivasan

C-7
Hiranya

R
A Puram

Chennai
- 600 028,

email:
attipat22@yahoo.com

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Titles in
the ABSTRACTS and CONTENTS are linked
to the articles.

Move the mouse over anywhere on a particular title and click the left button.

This will
open the corresponding article.

**ABSTRACTS**

**THE TRAINING OF INDIAN ALMANAC
MAKERS
AND THEIR OBSERVATIONAL EQUIPMENT **

Mathematically,
the
five-fold specification of the *Panchanga (vara, thithi, nakshatra,
yoga,
& karana)* suggests that we are living in a five dimensional
world. We
require five coordinates to specify our place-time, a blended entity of
four
dimensions according to Einstein. A reliable panchanga gives these five
coordinates as accurately as possible for the orthodox persons
following
religious traditions. We have, now, a conflicting set of astronomical
systems
handed down from ancient times and followed by different sections of
people
with a blind and uncritical devotion to the letter of the texts. Those
who have
studied European astronomy do not have any conception of Hindu
astronomy. The Indian
astronomers following the Sanskrit texts have not an inkling of
European
astronomy. To bring the two together, Universities should teach Hindu
and
European Astronomy while in the Sanskrit Colleges European and Hindu
Astronomy
should be taught.

**THE EARLIEST
SOLUTION OF THE BIQUADTRATIC**

*Current
Science, Vol.7, No.4, Oct. 1938*

The
principle in
Bhaskara’s solution of the biquadratic was hidden away in a numerical
example.
Bhaskara’s is the earliest attempt at the solution of the biquadratic recognizing only the real positive roots.
This article generalizes his principle of his method of solving a
biquadratic.

**BHASKARA AND SAMCLISHTA KUTTAKA**

Saradakanta
Ganguly
draws attention to a rule regarding the solution of the general case of
Simultaneous Indeterminate Equations of the first degree, found in two
palm
leaf manuscript copies of Bhaskara’s *Lilavati**.
* The rule for the solution of the
general case of simultaneous indeterminate equations mentioned in the
manuscripts is known as *samclishta
bahusamanya kuttaka sutram **.*For brevity this is called*
samclishta sutra. *This note
proposes to reopen the question of genuineness of Bhaskara’s
authorship. It
examines the rule in the light of other Indian solutions of the same
problem
and also in the light of Indian mathematical tradition. Earliest Indian
attempt
of indeterminate analysis takes the form of solving simultaneous
congruences.

Aryabhata
mentions
only two congruences but can be extended to more than two cases by
repeating
the process. This is done by Mahaviracharya. This paper gives an
example to
solve a set of congruences which is a repetition of a process with no
new
principle involved. Thus it does not justify an explicit enunciation by
Bhaskara. If he had done so, he has played a role of a commentator.
Therefore,
it is likely that a commentator has interpolated it in Bhaskara’s text.

Aryabhata
was the
first to hint the general problem of finding a number having given
residues
with respect to given modulii. Mahaviracharya has given a rule
different from
the samclishta rule. Bhaskara suggests a novel method (** Samclishta
Kuttaka)** different from

5 x ≡
7 (mod. 63)

30 x ≡
14 (mod. 63)

Bhaskara’s
method will give 35 x ≡ 49 (mod. 63) or 5 x ≡ 49
(mod. 9) as the representative congruence with x ≡ 5 as the least
value.
This does not satisfy the given congruences and have to choose the next
higher
value 14.

This
note gives new
proofs (by AAK) of Apollonius and Brahmagupta’s theorems.

More
than thousand
years before Euler, Indians were the first to give a general solution
to the
equation

x^{2}
– N y^{2} = 1 (1)

It was
based on the
principle of composition of quadratic forms.

Euler’s
theorem

If x =
a, y = b satisfies α x^{2} + p = y^{2},
(2) and

x = c, y = d satisfies α x^{2}
+ q = y^{2}, (3)** **then

x = bc
± ad, y = bd ± αac
satisfies α x^{2} + pq = y^{2
}(4)

has
been dealt by
Brahmagupta in his ‘Vargaprakriti’ and drawing important corollaries
from it.

Special identities are also given by him for deriving integral
solutions of Ny^{2}+1=x^{2}
and of Ny^{2}±4=x^{2}.

About
1150 A.D.,
Bhaskara gives integral solutions, designating it as ‘Chakravala’ after
the
ancients. Western writers attribute the cyclic method to Brahmagupta
but the
credit should go to Bhaskara. Bhaskara may have given credit to his
predecessors for the name ‘chakravala’, as applied to all iterative
operations
in mathematics.

Brahmagupta’s
formula for the composition of quadratic forms is also repeated by
Bhaskara.
Using this both of them derive the solutions of equation (1) from those
of

N y^{2}±
2 = x^{2 } or
N y^{2} ± 4 = x^{2}
(5)

Bhaskara
goes one
step further to show that the roots of the equations (1) and (5) can be
derived, by successive reduction, from the more general equation N y^{2}
± k = x^{2}, where k is an integer. The method of such
reduction is the
** chakravala method (Cyclic Method). **The first complete
proof of
Bhaskara’s Cyclic Method is sketched in this paper.

This
paper discusses
the following:

·
Bhaskara’s
Condition for Reduction

·
Reduced
Forms

·
Properties
of Bhaskara’s Forms

·
Bhaskara
and Fermat

**SOME GLIMPSES OF
ANCIENT HINDU
MATHEMATICS**

The
Mathematics
Student, Vol. I, 1933

The
early Hindus
developed mathematics as a limb of Vedas (Vedanta). The world owes some
of the
basic ideas such as place value system of notation in Arithmetic, the
generalizations of Algebra, the sine function of Trigonometry, and the
foundations of Indeterminate Analysis. The early texts contain merely
rules,
results, and a number of problems but rarely a full worked out
mathematical
argument.

·
Vedanga
Jyotisha is the oldest work bearing astronomy. The
style of the text is characterized by brevity, archaisms, and want of
connection between consecutive verses. The text gives rules to
calculate
relative lengths of day and night. It gives positions of the moon at
different
parts of the year and the time of the day. The early Hindus, evidently,
were well
acquainted with arithmetical manipulations including fractions.

·
Sulva
Sutra (about 200 A.D.) deal with the construction of
sacrificial altars. It gives construction of squares, rectangles,
triangles and
parallelograms with given specifications. It also contains the first
arithmetical solution of indeterminate equation x^{2} + y^{2}
=
z^{2} and rational approximations to √2, (√п)/2.

·
Bhakshali
Manuscript, a work on Arithmetic, was discovered
in 1881 A.D. It has a second approximation for √(a^{2}+r). This
result can also be obtained from the rule found in all the Indian
arithmetical
works.

·
Geometrical
problems were first analyzed as problems in
arithmetic and then expressed in geometrical language. Area of a cyclic
quadrilateral, Pythagorean Theorem and others were analyzed as
arithmetical
problems first.

·
The
next stage after Sulva Sutra is the age of Aryabhata,
Varahamihira, Brahmagupta, and the unknown author of Suryasiddhanta
(400-650
A.D). Aryabhata is placed by tradition at the head of the Indian
mathematicians
but he appears to be the last of the earlier school. His work appears
to be the
first to give form and an individuality to the mathematical knowledge
existed
before him. His treatises were translated into Latin, French, and
English. He
was the first to in India to hint the daily rotation of the earth on
its axis.
In his work one can find new alphabetic notation adopting both
consonants and
vowels to denote large numbers in a concise way. His admirer, Lalla
abandoned
it in favor of the more popular and the more ancient word numerals.
Vedanga
Jyotisha, Surya Siddhanta, and Varahamihira use the numeral words.
Brahmagupta
was the first to explicitly enunciate this system.

·
A list
of important mathematical topics on which there are
contributions from the ancient and the medieval Hindus. Among the
contributions
of ancient Hindus to higher mathematics, the most outstanding, is on
Indeterminate Analysis. Aryabhata was the first to indicate a method of
getting
integral solutions for* mx + a = ny + b* (m, n, a, b being
integers). He
proceeds by successive reductions to simpler equations until one
reaches a
solution. Brahmagupta has given a clearer exposition of the same.
Bhaskara and
others followed with improvements, alterations and extension.
Brahmagupta was
the first to enunciate the principle of composition of roots to solve
the
indeterminate equations of the second degree. This is similar to the
modern
principle of compositions of quadratic forms. It may be noted that the
so
called Pellian Equation *x ^{2} – Ny^{2} = 1 *should
be
called the Brahmagupta Equation.

·
Spherical
trigonometry was developed earlier than Plane
trigonometry to be used in Astronomy. The semi-chord and the arrow gave
rise to
the sine and the versed-sine functions. Hindus divided the
circumference of a
circle into 21600 minutes. Sine is considered as a length, and not a
ratio, and
expressed in terms of the arc. Hindus attributed to the arc all the
properties
of the angle. They established fundamental formulae of trigonometry and
managed
their astronomical formulae with the help of three functions: sine, the
versed-sine, and the cosine. The works on astronomy from Suryasiddhanta
to
Siddhanta Siromoni give interpolation and extrapolation formulas for
the
tabulation of successive sines.

·
Vedic
hymns contain description of the motions of Sun and
the Moon, the seasons, the number of days in a year, the two ayanas
(Devayana
and Pitriyana), the intercalary month etc. There are sufficient data in
the
Vedas for postulating the phenomenon of precession of the equinoxes and
determining its rate. After the Vedas we have the twenty or so
Siddhantas.
These were the earliest efforts at crystallizing, in a scientific form,
the
knowledge of astronomy. Of these twenty, Suryasiddhanta and Brahma
Siddhanta
are the most popular and have been revised by other writers starting
from
Aryabhata. Siddhantas mention the number of revolutions of the sun,
moon, and
the planets, and their nodes. They assumed the motions of the planets
have mean
circular orbits during a Kalpa and all of them travel the same number
of
Yojanas. This constant is described as Khakaksha the circumference of
the
sphere to which solar rays extend. Bhaskara’s value for this constant
is
187120692 x 10^{8} while Suryasiddhanta gives value as
18712080864 x 10^{6}.
Indian astronomers give the moon’s distance from the earth but erred in
the
distances of the sun and the planets as they were calculated from the
assumption of constant velocity. There is enough evidence that the
Indian
astronomers actually verified the astronomical elements by personal
observations of the heavenly bodies.

·
Names,
after Bhaskara, of other astronomers are given.

This
paper is a
re-examination and re-assessment of the Bhakshali Manuscript (BM). G.
R. Kaye
has written introduction and notes after the earlier work of Hoernie.
G. R.
Kaye’s biased view of Greek influence of anything Indian prompted
others to
re-examine his statements. This article examines the manuscript under
the
following headings:

1.
Method
of presentation

2.
Peculiar
terminology, abbreviations, and the cross symbol
for subtraction

3.
Decimal
Notation and the absence of word numerals

4.
Symbol
for unknown

5.
Rule
of *Regula Falsi*

6.
Square-root
rule and the process of reconciliation, and

7.
Series
and sequences.

·
Rules
and examples are presented in verse in Sloka meter and
the explanations are given in prose. Verification method is
incorporated to
prove a mathematical problem, and is common among Indian mathematical
commentators as early as 9^{th} century A.D. It seems as though
the
Bhakshali Manuscript was a teaching notes by a tutor.

·
The
mathematical terminology adopted in BM, generally
follows other Hindu mathematical works, but contains a few exceptions.
There
are several abbreviations like ** yu
(yutam)** for addition and

·
Special
numerals are used and the decimal notation employed
in BM but not the word numerals. Absence of word numerals, used by
Varahamihira
in the sixth century, is a sure index that it belongs to an earlier
period. The
use of the decimal notation is another indication of the early period
of the
Christian era.

·
The
dot (**∙**)
symbol is used with two different kinds of significance. It was
primarily a
symbol of ‘emptiness’ and secondarily has become a symbol for the
unknown or
absent quantity. In BM the dot symbol is used simultaneously for
several
unknowns. The symbol for ‘zero’, ‘negative’, and for the ‘unknown’ seem
to have
a common ancestry and their nebulous beginnings are reflected in BM.

·
The
‘Rule of Regula Falsi”
is the same as that of analytical geometry to find the
coordinates of
the point of intersection of the line joining two given coordinate
points with
the *x*-axis. One can say it is a
precursor of the interpolation theory. This rule should not be confused
with
the Indian ** ‘ishtakarma’** which is an
operation with an assumed number, used
in cases where the final result, arrived at by a series of
manipulations, is
proportional to the number originally assumed.
The use of Regula Falsi in BM is a fanciful deduction based on a
misunderstanding of an ingenious method of generalized

·
A
striking feature in BM is the chapter on the square root
rule, also found in the Arab work of the 12^{th} century. The
rule is
merely a corollary of the general square root rule found in all Indian
mathematical works from Aryabhata onwards. The earliest evidence of
concrete
numerical application of this rule even to a higher order is found in
Sulva
Sutras. The BM rule gives only the second approximation but is
suggestive of a
process permitting repeated application to any desired order of
approximation.
Brahmagupta gives a similar rule in connection with the square root of
a number
in sexagesimal notation.

·
The
problems on series and sequences are very original,
varied and interesting. In most of the problems, the sum to *n
*terms happens to be proportional to
the first term and therefore Ishtakarma can apply. This paper discusses
eight
types of problems using modern notation. The identity

T_{n
} = *(1-a _{1} )(1-a_{2}
) . . . .
(1-a_{n-1} ) a_{n}*

is
common in Hindu
mathematical works from at least the 8^{th} century onwards.
The
Arithmetical Papyrus of Akhim (9^{th} century) contains such a
problem.
Mr. Kaye infers from this that BM must belong to the tenth century or
later.

__GENERAL
REMARKS. __

The
Bhakshali text,
apart from peculiarities, is more or less a replica of other Hindu
mathematical
works such as Ganitasara Sangraha. It contains in common with them the
following:

1.
Practical
and commercial problems

2.
Problems
of income and expenditure

3.
Motion-problems

4.
Profit
and loss

5.
Interest

6.
Bills
of exchange or hundika, and

7.
Miscellaneous
problems involving literary and social
references.

In BM
solutions are
given in general form as to be nearly algebraic character without
employing
adequate symbols. This is the characteristic of all Indian works. There
are
omissions, such as expressions for sums of squares and cubes,
indeterminate
equations of the first and second degree, shadow problems, permutation
and
combination. The omissions may indicate an early date of composition,
probably
prior to tenth century. The BM may be placed near the times of
Sridhara,
Mahavira, and Chaturveda with whose works the BM has many points in
common. The
systematic presentation of the working steps and methods of
verification shows
it was written for the benefit of the students. It has a great value
for a
practical teacher since Bhakshali was near to Taxila, the renowned
University
centers in Ancient India.

This
describes the
rule of classifying the yearly calendars into fourteen distinct types
(seven
ordinary and seven leap years) which recur in a given century.

**REMARKS ON
BHASKARA’S APPROXIMATION TO THE SINE OF AN ANGLE**^{}

Bhaskara’s
rational
approximation to sin (π/n) can be written as 16(n-1) / (5n^{2}-4n+4),
which is the best rational approximation. This can also be expressed as
[n^{2}
– (n-2)^{2}] / [n^{2} + (1/4) (n-2)^{2}] which
is
Ganesa’s (16^{th} century astronomer) variant.

One of
the most
prominent and scientific writers is Aryabhata of Kusumapura born in the
year 476
A. D. There is another Aryabhata who is known by his work
Mahasiddhanta. There
is a confusion of recognizing between the two. Sudhakara Dwivedi gives
dubious
references from Bhaskara to show that Bhaskara did not know of the
older
Aryabhata but only the younger one. This may be due to incomplete and
erroneous
manuscripts of Aryabhatiyam being in circulation.

Aryabhata’s
treatise on Algebra has been translated into several European
languages. Many
works have been attributed to him but the Aryabhatiyam is the only work
which
can be called his. It consists of Dasagitika Sutra, Ganita, Kalakriya,
and Gola
dealing respectively with astronomical tables, mathematics, the measure
of
time, and the spherics. He was one of the first to expound the
principles of the
Indian astronomical system in a condensed and technical form. His was
the
original statement that the daily rotation of earth was on its axis but
could
not assert and maintain this against the then popular geocentric
theory. He
goes against the prevailing orthodox notions: in his theory of the
eclipses and
in the subdivision of chatur-yuga into four equal parts. He was an
innovator in
astronomy and he attempted to reform some of the prevailing notions and
doctrines. In the eyes of the orthodox teachers he was a heretic. His
high
originality can be seen in the mathematical works in Dasagitika and in
the
Ganitapada. He can be considered as the father of Indian mathematics by
looking
at the later writings by Brahmagupta, Bhaskara, Mahaviracharya, and
Sridhara.

Aryabhata
gives in
his Dasagitika, a peculiar notation for expressing numbers in terms of
the
letters of the Sanskrit alphabet. The twenty-five varga letters of the
alphabet
represent the numbers one to twenty-five respectively in the square or
odd
places while the avarga letters represent the numbers 30, 40, ….., up
to 100
occupying the even or non-square places. The nine vowels attached to
any
consonant indicate that the value of the consonant is multiplied by 1,
100, 100^{2}.
. . 100^{8} respectively. This system was used merely for
mnemonic
purposes and not followed in the Ganitapada. His notation and
numeration
indicate that the Hindus of that age were acquainted with the principle
of the
position system in the decimal or the centesimal scale. He defines the
square
and the cube of a number and gives rules for finding the square-root
and the
cube-root. Sridhara, Mahaviracharya, Bhaskara and Brahmagupta give more
or less
identical rules for the extraction of the square-root and the
cube-root, while
no method of extracting the cube-root is given by an early Greek
writer. He has
given some mensuration formulas, some of which are obviously wrong. The
area of
the triangle is given correctly but the volume of a solid with six
edges is
given to be equal to the product of half the area of the base and the
height.
This is because he considered the solid of six edges as the analogue of
the
triangle. There is a general direction for determining the area of any
figure
by decomposing it into trapeziums.

The
value of π
is given by the rule “when the diameter is 20,000, the circumference
will be
62,832 approximately”. It is remarkable that Aryabhata should have
given it
while nothing like it occurs in the Greek works. This rule occurs
before his
rule for the formation of the sine-tables. This leads one to suppose
that the
above value of π was used only for the construction of the sine-tables
at
intervals of 3.75^{o} and that the less approximate values,
such as √10,
were used elsewhere. He gives the rule for deriving successive
sine-differences,
in the modern notation, {*d ^{2}(Sin
x)/dx^{2}} = - Sin x. *The ‘Sine” is equivalent to the
modern sine
multiplied by 3438. According to the rule, each sine-difference
diminished by
the quotients of all the previous differences and itself by the first
difference gives thee next difference. The differences given in
Dasagitika are:
225, 224, 222, 219 ....... 37, 22, 7. Same results are also given in
Suryasiddhanta with the same rule. Mr. Kaye holds that this rule may be
an
attempt at the enunciation or application of Ptolemy’s Theorem, but the
trigonometry of Ptolemy does not give it. The author (AAK) gives a
possible
reasoning how the rule may have been developed. As there were some
discrepancies in some differences it is possible corrections were made
by
comparing the results with the actual ones obtained by direct
calculation for
the common angles 30

Another
topic is on
the mathematics of Sun-dial and the Shadows. In discussing this topic,
constructions are given for drawing a circle, a triangle (equilateral!)
given a
side and a rectangle (square!) given a diagonal. Directions are given
for
determining experimentally the horizontal and vertical planes by means
of water
and plumb-line respectively. Pythagorean rule is used for finding the
radius of
the gnomon-circle given the height of the gnomon and the length of the
shadow.
Two rules are given for determining (1) the lengths of the shadow of a
gnomon
of given height and (2) the height of the source of light and its
distance from
two equal gnomons casting known shadows. The rules are:

*(1)
**Shadow
= (height of gnomon x the distance of the light
from the gnomon) / (the difference between and the heights (the gnomon & the light)*

*(2)
**The
distance between the end of a shadow and the base of
the light, when two gnomons are considered = (the length of the shadow
x the
distance between the ends of the shadows) / (the difference)*

It is
not clear
which difference is meant in (2).

Aryabhata
in an
Eclipse Problem a property of the circle is enunciated which is ‘that
in a
circle, the product of the arrows is the square of the semi-chord of
the arc’.
He gives a theorem, derived from this property, to use in the
calculation of
eclipses. The next topic in the Ganitapada is the arithmetical
progression. A
general formula is given for the sum of the terms beginning with the *(p+1)*^{th}* * term.

*n = [a
+ ((n-1)/2 + p) d]*

where *a* is the first term, *d* the increment. An
alternative form is
also given as ‘add the beginning and end terms and multiply by the sum
by half
the number of terms’. These are followed by another formula determining
the
number of terms. Next follows the contents of a triangular pile and a
square
pile as

*n(n+1)(n+2)/6 and
n(n+1)(2n+1)/6*

The
formula for the
sum of the cubes is given by ∑ *n ^{3}*
=
(∑

A pair
of
semi-geometrical identities to solve simultaneous equations of the
types x ± y
= p, x y = q;

x ± y
= p, x^{2}
+ y^{2} = q; x y = p, x^{2} + y^{2} = q. Next
topic is
his rule for finding the interest on the Principle (P), the interest on
the
amount (A) and the time (t). Solving the quadratic equation P r^{2}
t^{2}
+ Prt = A determines the rate of interest. After giving this rule he
proceeds
to enunciate the principle of the rule of three and the usual rules for
the
division of one fraction by another and for reducing all fractions to a
common
denominator.

Rules
for reversing
the steps in a mathematical process are enunciated by Aryabhata. This
principle
is useful for verification purposes and for the solution of equations
where one
has to clear the variable from all the ramifications in which it is
involved.
He gives an identity. In modern notation it is

∑ [∑(X^{i}_{r}
– X_{r}) / (n-1)] = ∑
X_{r} , i = 1, . . . ,n; r = 1, . . . ,n

From
the fact that two
particular cases of this theorem occur in Diophantus it is argued that
the
problem is of Greek origin. Mahaviracharya gives many more varieties of
problems not found in Diophantus or in any other ancient Greek works.
Hence one
cannot infer the Hindus copied from Greek works. After the Indian
version of
the *Epanthem* follows the ordinary
method for the solution of a simple equation where both sides are
linear
functions of the variable. This is succeeded by a discussion of the
relative
velocity of one moving body with respect to another, when both are
moving (i)
in the same direction, and (ii) in opposite directions.

Aryabhata’s
solution of the linear indeterminate equation is the crown of his
mathematics.
He writes the problem as **“ to find a number
which leaves residues n, n’ with respect to the
modulii m, m’
respectively”**

*We
have to find x and y to satisfy the equation:*

*29x +
15 = 45y + 19.*

*The
method is to express x in terms of y as*

*x = y
+ (16y + 4) / 29.*

*Since
(16y +4)/9 should be an integer let it be equal
to z.*

*Then y
= z + (13z – 4)/16. Again let p = (13z – 4) /16
leading to z = p + (3p + 4)/13. At this stage, since the coefficient is
small,
choose p = 3 so that 3p +4 is divisible by 13. To find the value of x,
we
retrace the steps and get z = 4, y = 7, x = 1. Aryabhata calls p as
‘mati’.*

Thus
Aryabhata’s
rule is a method of detached coefficients for carrying out the backward
process
for evaluating successively z, y, and x. This process was called
“Vallikakuttaakaara” by later mathematicians. Bhaskara’s method the
‘creeper’
(valli) is extended to its utmost length. Mahaviracharya gives exactly
Aryabhata’s rule using Aryabhata’s nomenclature.

The
Ganitapada ends
with the Indeterminate Equations and the rest of his work is Astronomy.
The
impression left by the Ganita is that it is a collection of working
rules
necessary for solving practical problems such as interest problems,
problems in
astronomy, and problems of survey. The author’s style is business-like,
lacking
the richness of imagination, the zeal in problem-setting, and the
extravagant
poetry characteristic of other Indian authors like Bhaskara and
Mahaviracharya.
Later Hindu astronomer-mathematicians are indebted to Aryabhata. He was
the
first to give a form and individuality to the scattered bits of
mathematical
knowledge that existed before his time.

*Quarterly
Journal of the Mythic Society, Vol. 18, 4,
1928, and Vol.19, 1, 1929*

There
were four
different kinds of numerals in use in

The
Kharoshti
script was written from right to left and the numerals followed this
direction.
These numerals occur in the Saka inscriptions (first century B.C). The
symbols
used are:

(a)
one, two, three
vertical strokes for 1, 2, 3 respectively, (b) an inclined cross for 4,
(c) a
symbol for 10, (d) a cursive combination of two tens for twenty, (e) a
sign
resembling the Brahmi symbol with a vertical strike to its right for
‘one
hundred’. In this notation not more than three repetitions are allowed
of any
symbol and a new symbol is always used to avoid the fourth repetition.
The
Kharoshti numerals with their additive and iterative principles appear
to be
the first stage in the growth of the Hindu notation. These are absorbed
to and
superseded by the Brahmi notation.

Brahmi
notation is
the most important of the early Hindu notations. Some fragments of
these
numerals occur in Asoka’s edicts (300 B.C). They also appear in the
Nanaghat
cave inscriptions of the second century B.C. The next evidence of these
numerals is found in

The
early Hindus
counted in the ten-scale as so many units, tens, hundreds, and so on in
successive powers of ten. The Sanskrit numeration has ‘one and ten’,
two and
ten’ (ekadasa, dwadasa) etc. Large number such as 108 was expressed as
‘ashtottharashata’ (eight above hundred). Later it took a convenient
form where
the number of units, tens, hundreds etc., occurred in a number. For
example:
five, seven and two meant five units, seven tens and two hundreds. The
Hindus
adapted this notation in verses denoting the numbers according to a set
of
rules. For example: one was represented by anything which is unique
such as
earth, moon etc., two by those which occur in doubles (eyes, black
&
white), naught by heaven, twelve by the names of the Sun. Aryabhata
thought of
substituting in its place his scheme of ‘katapayadi’ notation and the
decimal
notation thus combining the advantages of both. This notation was
difficult for
an average person and it was soon forgotten. But this notation
suggested a
method of using the same symbol with variations to denote a multiple
powers of
a hundred.

When
the
word-numeral notation was being used, the Brahmi symbols were at hand
and a
notation for a new symbol had to be invented for zero. The word ‘Sunya’
or
‘aakaash’ denoting the absence of a power of ten in the word-numeration
must
have suggested the symbol ‘0’. An earlier or an alternative form of
this symbol
is the dot (**∙**).

The
Hindus called
the decimal notation ‘*anka palli’*;
the word ‘anka’ means a mark or symbol. In the word-numeral notation
adapted by
Varahamihira (6^{th} century A.D.) and others ‘anka’ used to
represent
the numeral 9, probably because nine not ten numerals were in common
use. By 9^{th}
century, at least, all the ten symbols were in place. In connection
with these
ten numerals of the new notation (date unknown), a rule came (*ankaana vaamathogathi*) that is, the
order of the numerals is from left to right. One does not know whether
this
rule refers to the order in writing or to an arrangement in some form
of abacus.
It is said that Hindu astrologers were using a wooden calculating board
called
‘*paati*’, hence ‘*paati ganita*’ (name for
Arithmetic).

This
paper traces
the history of the Hindu numerals. It establishes that Western writers,
who had
a bias towards Greek mathematicians, are wrong in saying that Hindus
did not
have their own system of numerals and these numerals were spread by
Arab
merchants to their country. From there these numerals were adapted by
others.
Leonardo Fibonacci spread the Hindu numerals in *Liber Abaci.*

*Current
Science, Vol.6, 1937-38*

This
proposes a new
Half Regular Continued Fraction development of quadratic surds,
distinct from
suggested by Minnegerod. Such a development was implied in the method
developed
by Bhaskara for obtaining the integral solutions of the indeterminate
equation

*X ^{2}
– NY^{2} = 1.*

This
article rebuts
the erroneous view that Bhaskara’s method gives rise to the simple
continued
fraction. The full development of this method is described in the
subsequent
articles.

**THEORY OF
THE NEAREST SQUARE CONTINUED FRACTION**

This
paper was
inspired by the remark of Sir Thomas Little Heath “*Indian
Cyclic Method of solving the equation x ^{2} – Ny^{2}
= 1 in integers due to Bhaskara in 1150 is remarkably enough, the same
as that
which was rediscovered and expounded by Lagrange in 1768”. *This
paper
investigates the Indian method of half-regular continued fraction
(h.r.c.f.).
This new continued fraction (the nearest square continued fraction) is
a
natural sequel to Bhaskara’s cyclic method. This theory was developed
with the
help of the simplest mathematics known to Hindus about the fifth
century A.D.

It
starts with the
definition of the quadratic surd of the form [(P + √R) / Q] as a
standard
form if R is a non-square positive integer, and P (≠ 0), Q, [(R – P^{2})
/ Q] are integers having no common factor. If P = 0 it is sufficient (R
/ Q)
and Q are relatively prime integers.

Starting
with some
elementary results goes on to state and prove several Theorems to the
development of a new continued fraction Bhaskara continued fraction (B.
c. f.)
or the nearest continued fraction. This paper gives a table of B.c.f.’s
equal
to the square-roots of non-square integers less than 100.

In
conclusion, the
B.c.f. has a complicated individuality of its own and further
investigation
into the character of the cyclic part, the transformations that convert
the
simple continued fraction into the continued fraction to the nearest
square,
and the associated quadratic forms.

**PEEPS INTO
INDIA’S
MATHEMATICAL PAST**

This paper
traces the ancient
Hindu mathematics from 800 B.C. to 1200 A.D. It has a brief description
of the
mathematics and astronomy developed by Indian mathematicians of that
era. The
following are the topics selected:

1.
Vedanga
Jyothisha (1200 B.C.)

2.
The
Sulva – Sutras (between 800 and 500 B.C.)

3.
Surya
Siddhanta (gone through revisions from 500 A.D. to 100
A.D.)

4.
Aryabhata
(Aryabhatiyam: 499 A.D.)

5.
Varahamihira
(6^{th} century)

6.
Brahmagupta
(598 A.D.)

7.
Sridhara
(probably 8^{th} century)

8.
Mahavira
(period between Brahmagupta and Bhaskara)

**THE
HISTORY OF INDIAN MATHEMATICS**

*The
Tamil Encyclopedia, Vol.3, 1 ^{st}
Edition*

This
is written in Tamil language. It covers more or less the topics on
mathematics covered in some of the above articles.

__CONTENTS__

1.
Ancient
Hindu Mathematics, Educational Review,

2.
The Training of
Indian almanac mathematics, Ibid, 1935

3.
A
misunderstood chapter on Indian mathematics, Ibid , 1940 (to be traced)

4.
The
Hindu sine Table, Journal of Indian Mathematical
Society, 1923-24 (to be traced)

5.
A
classical Indian puzzle problem, Ibid, 1923-24 (to be traced)

6.
Bhaskara and samclishta
kuttaka, Ibid, 1929-30

7.
New proofs of
old theorems – Apollonius and Brahmagupta, Ibid, 1929-30

9.
The Earliest Solution
of the Biquadratic, Current Science, Vol. 7, 1938

10.
The
New Continued Fraction, Current Science, Vol.6,
1937-38

11.
Theory of nearest
square continued fraction, Journal of Mysore University, 1940

12.
Peeps
into India’s mathematical past, Ibid, 1945

13.
Some
glimpses of ancient Hindu mathematics, Mathematics Student, 1933

14.
Fourteen
calendars, Ibid, 1937

15.
The Bhakshali Manuscript,
Ibid, 1939

16.
Remarks on
Bhaskara’s approximation to the sine of an angle, Ibid, 1950

17.
Astronomy
– past and present, Mysore University Magazine,
1930 (to be traced)

18.
Mathematics of
Aryabhata, Quarterly Journal of Mythic Society, 1926

19.
The Hindu – Arabic Numerals,
Ibid, 1928

20.
History
of Indian Mathematics (in Tamil), Kalai Kalanjiyam (Tamil
Encyclopedia, 1st
Edition)

Vol. 3