QA
27
14K3
UC-NRLF
B M 2m> 355
INDIAN MATHEMATICS
G
R.
KAYE
CALCUTTA & SIMLA
THACKER, SPINK & CO
'915
D^J-
INDIAN MATHEMATICS
G
R.
KAYE
CALCUTTA & sniLA
THACKER, SPINK & CO
1915
8A
PRINTED BY
THACKER, SriNK & CO. CALCUTTA
CONTENTS.
/i
I.
The
1.
literature
orientalists
who
exploited Indian:
-histC'Ty
a 'Ul
about a century ago were not always perfect in
and consequently promulgated
many errors. Gradually, however, sounder methods have
obtained and we are now able to see the facts in more
their
methods
of investigation
correct perspective.
In particular the early chronology has
been largely revised and the revision in some instances has
important bearings on the history of mathematics and allied
According to orthodox Hindu tradition the Surya
subjects.
Siddhanta, the most important Indian astronomical work,
was composed over two million years ago
!
Bailly,
towards
the end of the eighteenth century, considered that Indian
astronomy had been founded on accurate observations made
thousands of years
before the
Christian
era.
Laplace,
basing his arguments on figures given by Bailly considered
that some 3,000 years B.C. the Indian
astronomers had
recorded actual observations of the planets correct
second
one
to
Playfair eloquently supported Bailly's views
;
;
William Jones argued that correct observations must
have been made at least as early as 1181 B.C. and so on
but with the researches of Colebrooke, Whitney, Weber,
Sir
;
;
Thibaut, and others more correct views were introduced
and it was proved that the records used by Bailly were quite
modem and
original
It
that the actual period of the composition of the
Surya Siddhanta was not
earliar
than A.D. 400.
may, indeed, be generally stated that the tendency
of the early orientalists
tendency
is
was towards antedating and
this
exhibited in discussions connected with two
notable works, the Sulvasutras and the Bakhshali arithmetic,
the dates of which are not even yet definitely fixed.
1
INDIAN MATHEMATICS.
2
j
r
2.
In the 16tli century, A.D.,
Hindu
tradition ascribe
the invention of the nine figures with the device of place
to make them suffice for all numbers 'to the beneficent
Creator of the
univtirf^e
'
;
and
this
was accepted as evidence
of
This is a particular
the very great antiqiiitj of the system
quite
general, for early
was
that
i>^'an;attituile
.illtistr^ltipA
!
Indian works' claim either to be directly revealed or of divine
One consequence of this attitude is that we find
origin.
absolutely no references to foreign origins or foreign influence.*
We
have, however, a great deal of direct evidence
that proves conclusively that foreign influence was very real
indeed Greek and Roman coins, coins with Greek and
—
Indian inscriptions, Greek technical terms,
etc.,
etc.
;
and
the implication of considerable foreign influence occurs in
certain classes of literature and also in the archseological
remains of the north-west of India.
to foreigners
is
One of the few references
who acknowledged
given by Vahraha ^Mihira
that the Greeks knew something of astrology
;
but although
he gives accounts of the Romaka and the Paulisa siddhdntas
he never makes any direct acknowledgment of western
influence.
* It may be noted that beyond the vague pseudo-prophetic references in the Puratia.^, no early Indian writer mentions the invasion of
Alexander the Great.
II.
For the purpose
3.
history of
of discussion three periods in
Hindu mathematics may be considered
(I)
The
:
with upper
period
S'ulvasiitra
the
limit
c.
A.D. 200
(II)
(Ill)
The astronomical period c. A.D. 400—600.
The Hindu mathematical period proper, A.D.
600—1200.
Such a division into periods does not,
represent the facts, but
it is
of course, perfectly
a useful division and serves the
purposes of exposition with sufficient accuracy.
have prefixed an
earlier, or Vedic,
We
might
period but the literature
Vedic age does not exhibit anything of a mathematical
nature beyond a few measures and numbers used quite inforof the
remarkable fact that the second and third
of our periods have no connection whatever with the first
It is a
mally.
or
S'lilvasutra
period.
The
Indian mathematicians
later
completely ignored the mathematical contents of the S'ulva-
They not only never refer
sutras.
utilise
and
century
is
no Indian writer
known
to
them but do not even
We
the results given therein.
state that
to
can go even further
earlier
than the nineteenth
have referred to the S'ulvasutras
as containing anything of mathematical value.
connection will be illustrated as We proceed and
seen that the works of the third period
may
This disit will
be
be considered as
a direct development from those of the second.
The S'ulvasutra
4.
means
'
period.
the rules of the cord
'
and
—The
is
the
term
S'lilvasutra
name given
to the
supplements of the Kalpasutras which treat of the construction of sacrificial altars.
tras
The period
in
which the
S'ulvasu-
were composed has been variously fixed by various
INDIAN MATHEMATICS.
4
Max Muller
authors.
500 and 200 B.C.
places the
origin
gave the period as lying between
R. C.
;
somewhere within the
Dutt gave 800
B.C.
Apastamba school
of the
Buhler
;
as
probably
last four centuries before the Christian
and Baudhayana somewhat earlier Macdonnell gives
the limits as 500 B.C. and A.D. 200, and so on. As a
matter of fact the dates are not known and those suggested
era,
by
;
authorities
different
circumspection.
contents of
It
must
must be used with the greatest
also be borne in mind that the
the S'ldvasutras
,
known
as
to
us, are
taken
and that in matters of
from quite modern manuscripts
extensively
been
edited.
probably
The
have
detail they
editions of Apastamba, Baudhayana and Katyayana which
;
have been used for the following notes, indeed,
each other to a very considerable extent.
from
differ
The SulvasTdras are not primarily mathematical but
are rules ancillary to religious ritual they have not a mathe-
—
matical but a religious aim.
No
proofs or demonstrations
are given and indeed in the presentation there
mathematical beyond the bare
is
nothing
Those of the rules
facts.
that contain mathematical notions relate to (1) the construction of squares and rectangles, (2) the relation of the
diao'onal to the sides, (3) equivalent rectangles
(4)
equivalent
5.
theorem
number
circles
In connection with
is
of
stated
and squares,
and squares.
(1)
and
quite generally.
examples which
may
(2)
the
Pythagorean
It is illustrated
be summarised thus
Apastamba.
3H 4-= 52
122 + 16-' -202
Baudhayana.
32+ 4-= 52
= 252
+ 152=172
72 + 242=252
122 + 358 = 372
152 + 352=392
152+20-'
52+12^=132
152
+ 352=^392
82
+ 152=172
122
+ 352=372
52^-122=132
82
by a
:
INDIAN MATHEMATICS.
Katyayana gives no such rational examples but gives (with
Apastamba and Baudhayana) the hypotenuse corresponding
to sides equal to the side and diagonal of a square, i.e., the
triangle a, a\/2,
2^
+ 6'=: 40.
and he alone gives
rt/y/S,
There
no indication
is
Incidentally
may
-
^
This has been
that the
be represented by
+3+
i
3 4
3.4.34^
much commented upon
but, given a scale of
measures based upon the change ratios
Baudhayana actually
an
it is
that
3,
measurement
;
and
accurate to about ith of an inch
the
result
^a
\/W+b = a-{-
4,
and 34 (and
gives such a scale) the result
expression of a direct
feet
and
Sulvasfitra
given an arithmetical value of the diagonal
is
which
of a square
+ 3' = 10,
obtained from any general rule.
examples were
rational
1^
was
obtained by
;
is
only
for a side of six
or
it is
possible
approximation
the
by Tannery's R- process, but
it is
quite
'J,
no such process was known to the authors of the
The only noteworthy character of the fracthe form with its unit numerators. Neither the
certain that
Sttlvasutras.
tion
is
value
nor this form of fraction occurs in any later
itself
Indian work.
There
is
one
other
Pythagorean theorem
of
an
indication
of
point
the
formation
of
with
the
occurrence
the
viz.,
square by the
a
The text
successive addition of gnomons.
is as follows
connected
be noted,
to
relating to this
:
" Two hundred and
twenty-five
of
these
bricks
constitute the sevenfold agni with aratni
pradesa.
To these
and
'
sixty-four
more are
these bricks a square
is
to be added.
formed.
The
square consists of sixteen bricks.
With
side of the
Thirty-three
INDIAN MATHEMATICS.
6
bricks
still
remain and these
are placed on
all
round the borders."
sides
This subject
is
never again referred to in Indian mathe-
matical works.
The questions
(a)
whether the Indians
completely realised the
theorem, and
irrational
(b)
of this period
the
generality of
whether they had a sound notion
have been much discussed
;
had
Pythagorean
but the
of
the
ritualists
who coinposed the Sulvasutras Avere not interested in the
Pythagorean theorem beyond their own actual wants, and it
is
quite certain that even as late as the
mathematician gives evidence
of the
irrational.
of a
1
2th century no Indian
complete understanding
Further, at no period did the Indians
develop any real theory of geometry, and a comparatively
modern
Indian work denies the possibility of any proof of
the Pytuacjorean theorem other than experience.
The
fanciful suggestion of
goras obtained
his geometrical
Burk
that possibly
Pytha-
knowledge from India
supported by any actual evidence.
is
not
The (liinese had acquain-
tance with the theorem over a thousand years B.C., and the
Egyptians as early as 2000 B.C.
().
Problems relating to equivalent squares and rectangles
are involved in the prescribed altar constructions
and conse-
quently the Sulvasutras give constructions, by help of the
Pythagorean theorem,
of
(1)
a square equal to the sum of two squares
(2)
a square equal to the difference of two squares
(3)
a rectangle equal to a given square
(4)
a square equal to a given rectangle
(5)
the decrease of a square into a smaller square.
;
;
;
;
Again we have to remark the significant fact that none
any later Indian
of these geometrical constructions occur in
Work.
The
first
two are
direct geometrical applications of
INDIAN MATHEMATICS.
c-=ar~h^
the rule
the third pives in a geometrical
;
^
ay 2 and —
the sides of the rectangle as
gives a geometrical
{''—)
7
and corresponds
the fourth rule
;
=
«6
construction for
to Euclid, II, 5
;
form
(6
the
—
-j
—
^^j
fifth is
perfectly clear but evidently corresponds to Euclid, II,
not
4.
Circle.
— According to the altar building ritual
of the period it was,
under certain circumstances, necessary
7.
The
to square the
circle,
and
its
and consequently we have recorded
attempts at the solution
in the Sulvasutras
of this
problem,
connection with altar ritual reminds us of the cele-
brated Delian problem.
The
solutions
crude although in one case there
is
offered are
very
pretence of accuracy.
Denoting by a the side of the square and by d the diameter
of the circle whose area is supposed to be a^ the rules given
may
be expressed by
(a)
d—a-\-^{ax/2—a)
(/3)
a=d-l^d
Neither of the
first
two
rules,
which are given by both Apas-
tamba and Baudhayana, is of particular value or interest.
The third is given by Baudhayana only and is evidently
obtained from (a) by utilising the value for a' 2 given in
paragraph 5 above.
a_
We
thus have
3
_
_
:\
^-2+v^- " ^ 577
12;24
139:^
408
=
_ i4--i
s
I
^8.:i9
j_
8.-29.6
1
^ 8.2i).6.8
,
41
S.-jg.e.S.lSQH
which, neglecting the last term,
is the value given in rule
(7).
This implies a knowledge of the process of converting a
fraction into partial fractions with unit numerators, a knowledge most certainly not possessed by the composers of the
Svlmsfifras
for as Thibaut says there is nothinf^ in
;
INDIAN MATHEMATICS.
8
these rules which would justify the assumption that they
were expert in long calculations
;
and there
is
no indication
in any other work that the Indians were ever acquainted
with the process and in no later works are fractions expressed in this manner.
It is
worthy
of note that later Indian
record no attempts
at
the
solution
of
mathematicians
the
problem
of
squaring the circle and never refer to those recorded in the
S'ulvasutras.
III.
A.D. iOO TO 600.
There appears to be no connecting link between the
8.
mathematics
S'ulvasutra
•of
the
astronomical
nomical works then
of the five
Varaha Mihira's
collection
account we have of what
astronomy
most important
which
astro-
Of these the Surya Siddhdntu,
in use.
afterwards
400,
Siddhdntika
Pahclia
his
which was probably composed in
than A.D.
of
developments
In the sixth century A.D.
ideas.*
Varaha Mihira wrote
gives a summary account
ment
"not
Indian
recorded until the introduction into India of
is
western
later
Subsequent to the Sulvasutras nothing
subject.
further
and
its original
form not
earlier
became the standard work.
is the earliest and most authentic
may
be termed the
'•
in India.
scientific treat-
Although," writes Thibaut,
Hindus learned from the
Greeks, he at any rate mentions certain facts and points of
doctrine which suggest the dependence of Indian astronomy
directly stating that the
on the science
from
his
of
Alexandria
undoubted Greek origin."
Varaha Mihira
Siddhantas
—the
writes
:
Paulisa, the
Saura and the Paitamaha
Paulisa
and, as
;
astrological writings, he freely
is
accurate,
near to
—
*
'
* This
employs terms
of
There are the following
Romaka, the Vasishtha, the
The Siddhanta made by
it
stands the Siddhanta pro-
claimed by Romaka, more accurate
The two remaining ones are
we know already
far
is
the Savitra (Surya)*
from the truth."
somewhat important bearing on the date of the
example, the date of their composition were accepted
as 500 B.C. a period of nearly 1,000 years, absolutely blank as far as
mathematical notions aie concerned, would have to be accounted for.
S'nlcasvtras.
has
a
If, for
INDIAN MATHEMATICS^.
10
Tho
5).
Pancha
Siddhdntikd
contains
material
of
considerable mathematical interest and from the historical
point of view of a value not surpassed by that of anv later
The mathematical
Indian works.
Siddhanta
is
Paulisa
of the
section
perhaps of the most
and may
interest
be
considered to contain the essence of Indian trigonometry.
It is as follows
' (1)
:
The square-root
of the tenth part of the square
of the circumference,
which comprises 360
the diameter.
Having assumed the
parts,
is
four parts of a circle the sine of the eighth
part of a sign
" (2)
[is
Take the square
constant.
to be found].
of the radius
The fourth part
of
and
it is
call it
the
[the square
The constant square is to be
lessened by the square of Aries.
The squareroots of the two quantities are the sines.
Aries.
of]
" (3)
In order to find the rest take the double of the
deduct
arc,
it
from
the quarter, diminish
the radius by the sine of the remainder and
add to the square
of
half the
sine
square-root of the
of
double
sum
is
the
Aries
"
The
is
indicated the
rules given
may
first
"
Thfr
arc.
the desired sine."
[The eighth part of a " sign " (=30°)
"
square
of half of that the
sign "
is
3° 45'
and by
of .30°.]
be expressed in our notation
(for
unit radius) as
(1)
(3)
7r=VTo"
Sin-^
=
(2)
Sin 30°=|,
(^)
They are followed by a
'
Sin
60°= v^I^T
^(1-Sin(9(^2,)y
table of 24 sines progressing
vals of 3° 45' obviously taken from
by
inter-
Ptolemy's table of chords.
Instead, however, of dividing the radius into 60 parts, as
INDIAN MATHEMATICS.
did Ptolemy, Paulisa divides
——
chord a
a
—^ =
siw
,
,.
.
.
iuto
it
.
120
parts
_
,
,,
11
,
,
;
i
this division oi the radius enabled
^
as
for
i
.
•
nim to
convert the table of chords into sines without numerical
change.
(3438')
Aryabhata gives another measure for the radius
which enabled the sines to be expressed in a sort of
circular measure.
We
thus have three distinct stages
(a)
The chords
(6)
The Paulisa
(c)
The Aryabhata
of
Ptolemy, or
or
sine
ch'da
sin
~
sin
-^
- -^-^^
r
= 6U
A with
3
n
sine or
,
:
with
ch'd
^=120
(I
with r=3438'
To obtain
(c)
the value of
Thus the
earliest
- actually
known
u.sed
was
At one time
the invention of
attributed to el-Battani [A.D.
now know
=3.14136)
record of the use of a sine
function occurs in the Indian astronomical
period.
fSi(
this
works
of this
function
was
877—919] and although we
we must aclaiowledge that
the invention to a much more scientific
this to be incorrect
the Arabs utilised
end than did the Indians.
In some of the Indian works ~of this period an interpolation formula for the construction of the table of sines is
given.
It
=
n
+
This
may
j^
a
1
is
_
be represented by
§15l!!:JL where
Sin a
A " - 8in
».
a— Sin (h— l)a,
given ostensibly for the formation of the table, but
the table actually given cannot be obtained from the formula.
10.
Aryabhata.
—Tradition
places
Aryabhata
A.D. 476) at the head of the Indian mathematicians
indeed he was the
*
first
and
to write formally on the subject.*
Although Arj-abhata's Ganitdy as
generally accepted as authentic, there
matter.
(born
is
published by Kern, i-^
an element of doubt in the
first
12
INDIAN MATHEMATICS.
He was renowned as an
astronomer and as such tried to introduce sounder views of that science but was bitterly opposed
by the orthodox. The mathematical work attributed to
him consists of thirty-three couplets into which is condensed
a good deal of matter.
Starting with the orders of numerals
he proceeds to evolution and
Next comes a semi-astronomical
volumes.
he deals with the
of
involution,
shadow problems,
circle,
and areas and
section in whicli
etc.
propositions on progressions followed by
algebraic
The remaining
identities.
rules
then a set
;
some simple
may
be termed
practical applications with the exception of the
which
very
last
relates to indeterminate equations of the first degree.
Neither demonstrations nor examples are given, the whole
text consisting of sixty-six lines of bare rules so coiidensed
that
it is
some record
it
is
of the state of
of interest chiefly because
knowledge at a
we know,
matics
;
it is
the earliest
and because
it
it
is
critical period in
the intellectual history of the civilised world
far as
As a
often difficult to interpret their meaning.
mathematical treatise
;
because, as
Hindu work on pure mathe-
forms a sort of introduction to the
school of Indian mathematicians that flourished in succeeding
centuries,
Aryabhata's work contains one of the
known
minates of the
The
is
earliest records
to us of an attempt at a general solution of indeterfirst
degree by the continued fraction process.
rule, as given in the text, is
no doubt as to
its
general aim.
hardly coherent but there
It
may
be considered as
forming an introduction to the somewhat marvellous develop-
ment
in
of this
branch of mathematics that we find recorded
Brahmagupta and Bhaskara, Another
given by Aryabhata is the one which contains
the works of
noteworthy rule
an extremely accurate value
of a circle to the diameter,
is
of the ratio of the circumference
viz., tt
= 3 ^^^ (=3-1416)
;
but
it
rather extraordinary that Aryabhata himself never utilised
this value, that
it
was not used by any other Indian mathe-
INDIAN MATHEMATICS,
the
before
matician
12th
century
15
and that no
writer quotes Aryabhata as recording
noteworthy points are the rules relating to volumes
which contain some remarkable inaccuracies,
of a
pyramid
the base
;
is
not
the volume of a sphere
this
uncommon
Other
e.g.,
of solids
the volume
given as half the product of the height and
of the area of a circle (of the
the root of
Indian
value.
this
in
area,
later
stated to be the product
is
same radius
or t-^J^
.
as the sphere)
Similar
Indian works.
as the epanthein occurs in Aryabhata's
The
errors
rule
and
were
known
work and there
is
a
type of definition that occurs in no other Indian work, e.g.,
" The product of three equal numbers is a cube and it also
has twelve edges."
IV.
A.D. (JOO— 1200.
Aryabhata appears to have given a
11.
delinite bias to
Indian mathematics, for following him we have a series of
works dealing with the same
selves
we know very
but some
not
if
been preserved
all
little
indeed beyond the mere names
*
'
*
•
•
•
*'
'•
born A.D. 598.
Bhaskara
•
'
•
born A.D.
•
mathematically and
Brahmagupta's work
is
of
all
matician,
One
possibly sounder
same
topics
—with
a
Brahmagupta's work appears to have been
the others.
Bhaskara mentions another mathe-
Padmanabha, but omits from
of the chief points of difference
of geometry.
is
much more importance historically.
•Generally these writers treat of the
used by
991.
born A.D. 1114.
the most renowned of this school, probably
is
so, for
—and
9th century.
?
S'ridhara
difference
writers them-
:
Mahavira
Bhaskara
tlie
the works of the following authors have
Brahmagupta
undeservedly
Of
topics.
Brahmagupta
•cyclic quadrilaterals
this subject until
deals
his list Mahavira.
is
fairly
in the
treatment
completely with
while the later writers gradually drop
by the time
of
Bhaskara
it
has
ceased
to be understood.
The most
interesting characteristics of the works of this
period are the treatment of
(i)
:
indeterminate equations
[(n) the
rational
;
right-angled
triangle;
and (m) the perfunctory treatment of pure geometry.
INDIAN MATHEMATICS.
Of these topics
it will
15
be noted that the second was dealt
with to some extent in the Sulvasutras
mination seems to show that there
is
no
;
but a close exa-
real connection
and
that the writers of the third period were actually ignorant
of the results achieved
by Baudhayana and Apastamba.
Indeterminate
Equations. The
12.
names and dates connected with the early history
minates in India are
:
interestini;
of indeter-
INDIAN MATHEMATICS.
16
They were naturally disappointed but the
532).
may
their visit
effect of
have been far greater than historical records
show.
state of knowledge
The
1.3.
regarding indeterminate
equations in the west at this period
Some
is
works of Diophantus and
of the
are lost to us
;
not definitely known.
all
those of Hypatia
but the extant records show that the Greeks
field of this analysis so far as to
had explored the
achieve
rational solutions (not necessarily integral) of equations of
the
and second degree and
The Indian works record
first
deoree.
Greek analysis.
of the
is left
solutions
integral
distinct advances
on what
For example they give rational
of
=c
+1=
± by
(A)
ax
{B)
Du'
solution of (A)
The
certain cases of the third
is
t-
only roughly indicated by Arya-
bhata but Brahmagupta's solution
(for the positive sign) is
same
±cq-bt,
y= + cp + at
zero or any integer and ^V? i^ the penultimate con-
practically the
as
x=
where
t
is
vergent of
ajb.
The Indian methods
for the solution of
Du''
may
+1=
be summarised as follows
If
where
r
«2
:
Da^+h=c- and Da-+ft=y' then will
D{caT ya)^+ h^= (c y± a a Df
(a)
is
any suitable
integer.
Also
where n
is
any assumed number.
The complete integral solution is given by a combination
given by Brahma(a) and (6) of which the former only is
INDIAX MATHEMATICS.
17
gupta, while both are given by Bhaskara
The
later).
tion
and
'
latter designates (a) the
(b)
the
method.
cyclic
'
the history of mathematics.
all praise
:
it is
(a)
and
Of the
(6)
theory of numbers before
)
Greek
cyclic
'
Hankel
says,
method {i.e.,
" It is beyond
'
He
Lagrange."
in the
attributed its
mathematicians, but the opinions
the best modern authorities
Heath)
These solutions are alone
'
certainly the finest thing achieved
invention to the Indian
of
centuries
Indian works an important place in
sufficient to give to the
the combination of
(five
method by composi-
'
Tannery, Cantor,
{e.g.,
are rather in favour of the hypothesis of ultimate
origin.
The following consjjectus of the indeterminate problems
by the Indians will give some idea of their work
this direction
and although few of the cases actually
dealt with
in
;
occur in Greek works
now laiown
to us the conspectus signi-
ficantly illustrates a general similarity of treatment,
*(1)
±hy
(2)
ax -^ hy
(3)
X ^^
a,
^^ c
+
(5)
(6)
Du'
(7)
(8)
(9)
(10)
(11)
(12)
(13)
Ax
cz
Mod.
+ By
Z)w- + 1 =
(4)
*
ax
-{-
6i
••.
I
^ =
...
Cxy
-{-
=
rt^
Mod.
h^
=D
^2
- l=t^
Du^ ±s ^ t'
Dhi' ±s = f'
+ s = at^
Du' ± au =
s - Du^ = f
Du + s =
x ±a =
X ±b =^t^
I*"
t^
^2
s'',
Of these only numbers
1
—
-">,
7, S,
12
— 14
occur before the twelfth
century.
2
INDIAN MATHEMATICS.
18
+a=
- f) + 3 =
_ ^8) + 3 = s\ 3
2
+ I -= f
+ btf = s\ ax' x' - if ±1 ^
x'-^ if ±1 = s\
+ =
1
(14) aa;
(15)
bx
s'\
t^
{x'^
{'x'
by''
(16) ax''
(17)
t-
-
(18) x^
(19) aa;^
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
a^x' - b^o Mod.
+
6
=
Mod.
c
c
xy = u^
X — y =
+y^
x^ + y^ = s\
+ ^2 ^
X —y ^s\
+ =
+ ^^ =
+^=
+ ^) 5 4. 1
+ ^2 ^y =.
as' + 1 =
ax + 1 = s\
X
wxyz = a {w
y
— a ^ o Mod. b
X
t'\
s"-,
<3
a;2
s«,
cc
t^
a;3
_(_
0)8
^'^
?/2
a;2
^2^
(^.
:^
^2
t'
-\-
-{-
-\-
z)
a;3
a;
+
?/
+
3
= S-, a;
+
12
=
s-\-t-\-u
+
v
-
a;2
;
t'
I/'-!
-v^?!^
^;^
+
^
+ 3=
^8,
a;"'
+
?/-
-
4 =. M^
^ +y = w\
vj
+
+ V^MT' +
'2
=
z'
+
i/a;+?/+2
i/x-^^-^
+
^x'-y^ + ^ = f
(30)
w +
2
=
a^
+
a;
2
=
6-,
^
+
2
=
c",
+ 18 =: e», a;^ + 18 = P, ^2
+ 6 + c + + e + /+ + 11 = 13.
w'a;
ft
14.
+ 2 =
+ 18 =
2
(Z
(^^
g\
5r
Rational right-angled triangles.
— The Indian
mathematicians of this period seem to have been particularly
attracted by the problem of the rational right-angled triangle
and give a number of rules for obtaining integral solutions.
The following summary
of the various rules relating to this
problem shows the position
of the Indians fairly well.
INDIAN MATHEMATICS.
19
INDIAN MATHEMATICS.
20
Other problems connected with the rational right-angled
by Bhaslcara are of some historical interest
of the sides is 40 and the area 60, (2) The
The
sum
(1)
sum of the sides is 56 and their product 7 X 600, (3) The area
hypotenuse, (4) The area is
is numerically equal to the
triangle given
e.g.,
numerically equal to the product of the sides.
The geometry of this period
(1) Lack of definitions, etc.
15.
characterised
is
by
:
;
(2)
Angles are not dealt with at
(3)
There
(4)
Traditional inaccuracies are not
(5)
A
is
no mention
of
all
parallels
and no theory
of proportion
uncommon
;
gradual decline in geometrical knowledge
is
noticeable.
On
the other hand, we have the following noteworthy
rules relating to cyclic quadrilaterals
{i)
q=.^{s—a) (s—b) {s-c) (s-d)
(m) x'=^{ad-\-hc) (ac-^bd) (ab-^cd)
y-={ab-\-cd) {ac-\-bd)l{ad^bc)
where x and y are the diagonals of the cyclic quadrilateral
This {ii) is sometimes designated as Brahma{a, b, c, d).
*
gupta's theorem'.
16.
The absence
of definitions
order sufficiently differentiate the
that of the early Greeks
;
and
indifference to logical
Indian geometry from
but the absence of what
may
be
termed a theory of geometry hardly accounts for the complete
absence of any reference to parallels and angles.
Whereas
on the one hand the Indians have been credited with the
invention of the sine function, on
the other
there
is
no
evidence to show that they were acquainted with even the
most elementary theorems
The presence
with correct ones
of triangles
and
of a
(as such) relating to angles.
number
is significant.
quadrilaterals,
of incorrect rules side
The one
viz.,
by
side
relating to the area
the area
is
equal to the
21
INDIAN MATHEMATICS.
product of half the sums of pairs of opposite
enough occurs
By
Boethius and Bede.
—that the area
iVryabhata gives an
emphasized.
volume
pyramid
of a
common
sphere are
For cones
Mahavira, the idea on which
a function of the perimeter
is
all
—
it is
fvirther
is
rule
incorrect
the
for
volume of a
and Mahavira,
incorrect rules for the
;
to Aryabhata, S'ridhara
the rules assume that 7r=3.
incorrect rules for the circumference
and
Mahavira,
Ahnies, Brahmagupta,
well as in the works of
based
sides, strangely
a Chinese work of the 6th century as
in
Mahavira gives
and area
an
of
ellipse
so on.
Brahmagupta
17.
gives a fairly complete set of rules
dealing with the cyclic quadrilateral and either he or the
mathematician from
end
definite
in
view
whom
—
^the
he obtained his material had a
construction of a cyclic quadri-
with rational elements.
—The
fully appreciate the theorems,
some
lateral
commentators did not
of
the works of Mahavira and S'ridhara
Bhaskara they had ceased to be
which are given
in
and by the time of
understood. Bhaskara
;
indeed condemns them outright as unsound. "How can
a person " he says " neither specifying one of the perpendiculars, nor either of the diagonals, ask the rest
questioner
is
a blundering devil and
still
more
so
Such a
?
is
he who
answers the question."
Besides the two rules
graph
Brahmagupta
15,
(^)
and
(ii)
already given in para-
gives rules corresponding to
the
formula
2r=-^-^
SlU A
(in)
^
'
(iv)
and
a^-\-b'—c' and
If
teral {ay,
etc.,'
cjj,
hy,
+ /5^ = y' then the quadrilaand has its diagonals at right
a^
Co) is cyclic
angles.
This
figure
trapezium."
is
From
sometimes
the triangles
termed " Brahmagupta 's
(3, 4, 5)
commentatoi obtains the quadrilateral
and
(39,
(5,
12, 13) a
60, 52, 25),
INDIAN MATHEMATICS.
22
with diagonals 63 and 56,
of
etc.
He
introduces a proof
also
Ptolemy's theorem and in doing this follows Diophantus
19) in constructing
{in,
new triangles
from triangles
and
(a,
/3,
y,)
and uses the actual
examples given by Diophantus, namely (39, 52, 65) and (25,
60,
{ay, by, cy,)
and
{a, b, c)
{ac, (3c, yc,)
65).
An examination
18.
of the
Greek mathematics
the
of
period immediately anterior to the Indian period with which
we
are
now
dealing shows that geometrical knowledge
in a state of decay.
rical
work
of
much
After Pappus
(c.
was
A.D. 300) no geomet-
value was done.
His successors were,
apparently, not interested in the great achievements of the
earlier
Greeks and
acquainted with
certain that they were often not even
it is
many
of the earlier treatises
of their works.
had ceased to
The high standard
attract, errors crept in,
the style of exposition deteriorated and practical purposes
predominated.
The geometrical work
of
Brahmagupta
is
almost what one might expect to find in the period of decay
in Alexandria.
scientific
It contains
exposition
of the
one or two gems but
subject
obviouslv taken from western works.
it is
not a
and the material
is
V.
We
19.
have, in the above notes, given in outline the
historically
important matters relating to Indian mathematics.
For points
of detail the
works mentioned in the annexed
bibliography should be consulted
;
but we here
briefly indicate
the other contents of the Indian works, and in the following
we
a somewhat
sections
shall refer to certain topics that
have achieved
fictitious importance, to the personalities of the
Indian mathematicians and to the relations between
the
mathematics of the Chinese, the Arabs and the Indians.
Brahma-
Besides the subjects already mentioned
20.
GUPTA
deals very
briefly
operations, square
with
the
and cube-root,
ordinary
arithmetical
rule of three, etc,
;
interest,
mixtures of metals, arithmetical progressions, sums of the
squares of natural numbers
but
also
including
ties
geometry as already described
elementary
elementary mensuration of
and positive
;
solids,
of
the
circle
qualities, cipher, surds, simple algebraic identi-
indeterminate equations
;
notions
shadow problems, negative
of
the
first
and second
and
degree, which occupy the greater portion of the work,
simple
equations of the
receive comparatively but
and second degrees which
first
little
attention.
MahavIra's work
is fuller but more elementary on the
The ordinary operations are treated with more
completeness and geometrical progressions are introduced
whole.
many problems on
is
made
algebra.
of the
It is
(inaccurately).
indeterminates are given but no mention
and it contains no formal
the only Indian work that deals with ellipses
'
cyclic
method
'
INDIAN MATHEMATICS.
2-i
The only extant work by S'ridhara
but shorter
but he
;
is
is like
Mahavira's
quoted as having dealt with quadratic
^^quations, etc.
Bhaskara's Lilavatl
is
based on S'ridharas work and,
besides the topics already mentioned, deals with combinations, while his Vlja-ganita, being a
more systematic expo-
sition of the algebraical topics dealt
with by Brahmagiipta,
is
the most complete of the Indian algebras.
After the time of Bhaskara (born A.D. 1114) no Indian
mathematical work
Even
a
of historical value or interest
is
known.
before his time deterioration had set in and although
" college
Bhaskara
"'
it,
was founded to perpetuate the teaching of
apparently, took an astrological bias.
The Indian method
21.
examples
stating
of
—parti-
cularly those involving algebraic equations—are of sufficient
interest to be recorded here.
and not symbolical at
all
The early works were
and even
in
rhetorical
modern times the
nearest approach to a symbolic algebra consists of abbrevia-
The only
tions of special terms.
negative sign of operation, which
real
is
symbol employed
Sarada
is
used in place of the dot as the latter in the
script is
The
first
employed to indicate cipher or nought.
mention
of special
terms to represent imknown
quantities occurs in Bhaskara's Vlja-ganita which
was written
Bhaskara says
in the twelfth century of our era.
many
the
In the Bakhshali
or at the side of the quantity affected.
Ms., a cross
is
usually a dot placed above
:
"As
and the colours black (kdlaka), blue
(nilaka), yellow (pltaka) and red (lohitaJca) and others besides
these have been selected by ancient teachers* for names of
as
{ydvat
tdvat)
'
'
values of
unknown
The term ydvat
of
quantities."
tdvat
is
understandable and so
colours but the conjunction
The use
of
is
is
the use
not easy to understand.
two such diverse types as ydvat
*Xot Indians.
tdvat
and kdlaka
25
INDIAN MATHE.AIATICS.
(generally abbreviated to yd
and
hd) in one system suggests
the possibility of a mixed origin.
former
connected with
is
unknown
an unlimited number
if
possible that the
definition
monddon aoriston,
number of units.' To
to
'
as
'
many
as
the
of
i.e.,
'
an
pass from
requires little ima-
'
Diophantus had only one symbol for the unlmown
gination.
and
is
iMthos
quantity,
undefined (or unlimited)
'
It
Diophantus'
the use of ydvat tdvat were of Diophantine origin the
Indians would have had to look elsewhere for terms for the
other
unknowns.
With
reference to the origin of the use
of colours for this purpose
we may point out that the very
early Chinese used calculating pieces of two colours to represent positive
and negative numbers.
As neither the Greeks nor the Indians used any
sign for
addition they had to introduce some expression to distinguish
The Greeks used
the absolute term from the variable terms.
M° an
abbreviation for monddes or
used ru for
rujja,
'
units
'
while the Indians
a unit.
The commoner abbreviations used by the Indians are
as follows
:
Ijd
for ydvat tdvat,
kd
,,
kdlaJca,
ru
,,
rupa,
,,
varga,
va
the
unknown.
first
the second unknown.
the absolute quantity.
a square.
gha „ ghana, a cube.
ka
It
is
,,
a
Icara/ia,
surd.
hardly appropriate to discuss Sanskrit mathematical
terminology in detail here but
it
mention a few other terms.
To denote the fourth power
varga varga
is
used but
it
will
not be out of place to
occurs only once within our period.
In more modern times varga ghana gMta-\ denoted the
power, varga ghana, the sixth and so on.
+ G'/(a<a=the product.
fifth
INDIAN MATHEMATICS.
26
Certain Greek terms are used, e.g.,jdimtra (Gk. diametron), Jcendra {Gk. kentron), trikona {Gk.
harija {Gk.
lepte),
{Gk.
denarion),
'
etc.
orizon),
dramma
Many
of
trigonon), lipta {Gk,
{Gk. drachme),
dindra
however, are
these terms,
borrowed from Indian astrological works which contain a
considerable
number
of
Greek terms such as Hridroga {Gk.
(udrochoos) Pdrthona {Gk.
Parthenos), djjoklima {Gk. apok-
lima), etc., etc.
side
with
The curious may compare pdrsva 'a rib,
primarily
means
claw
or
horn
which
a
the Greek pleura ; koti
'
'
'
but
is
used for the perpendicular side of a triangle, with
kdthefos
is
:
jdtija
which means
'
legitimate,'
'
genuine,
'
but
used to denote a right-angled triangle with orthogonia
and so on.
;
I
*?
iff
fir.
^
ST.
^
'^
^
^W
^
ft
r &
^i-
if?
r
r
p.
£»
(£
V
jsr
"^
p- i*
ii p/ ig:
5»..iLi>'.-'.
??rS^tt^^:^ifK-X>:\-
if'i:
tf
cp
H
!? f^
'fe^
t
I
4v
1^
£^
^<6^
^ <^nr
c^ fe 5? £
"
&
IF sy B^ ft
Ft
fif
s^
ft»*
•
^ ^
S.'ff- re
s^"
ir*-
".•
2
p-
'^Is P
rs^
"s^
2
theories as to the origin or these symoois nave vcku. puuu»ucu,
some
of
which
orientalists
still
continue to be recorded.
gave them
place-value,
but
The
earliest
this error
soon
26— (o)
W'r rdiulmlc this section with a few illustrations transliterated from Sanskrit manuscripts.
Indian Forms.
yd
6 ru 300
ya 10 rn 100
yam
yam
l& yd Q ru
16 yo 9 ru 18
yd va va
yd va va
yd_
i/(i
1
yd va 2 yd 400 nl
yiJ
va
197 io 1644
Aw
ka & ka b ka
1
1
'2
)» 9999
f/n
n't
\
r« 630i{
n'l
ka
ru
'i
Equivalents.
VI.
According to the Hindus the modern place-vahie
22.
system of arithmetical notation
is
led the early orientalists to believe that, at
system had been
in use in India
This
divine origin.
of
any
rate, the
from time immemorial
;
but
an examination of the facts shows that the early notations
in
use were
not
ones and that the modern
place- value
place-value system was not introduced until comparatively
modern
termed
niently
(c)
The early systems employed may be conve-
times.
the
(a)
KharoshthI,
Aryabhata's alphabetic notation,
(6)
Brahmi,.
the
the word-symbol
(d)
notation.
(a)
and was
The KharoshthI
script
is
written from right to
at the beginning of the Christian era.
in
the accompanying table.
It was,
from the Aramaic system and has
with the other Indian
are written on the
(6)
notations.
The Brdhml notation
Indian notation for
in fairly
common
apparently,
little
derived
direct connection
The smaller elements
left.
old notations of India.
the
left
and Central Asia
The notation is shown
in use in the north-west of India
It
it
is
the most important of the
might appropriately be termed
occurs in early inscriptions and was
use throughout India for
many
centuries,
and even to the present day is occasionally used. The
symbols employed varied somewhat in form according to
time and place, but on the whole the consistency of form
exhibited
is
remarkable.
They
are written from left to
right with the smaller elements on the right.
Several false
theories as to the origin of these symbols have been published,
some
of
which
orientalists
still
continue to be recorded.
gave them
place-value,
but
The
earliest
this error
soon
VI.
According to the Hindus the modern place-value
22.
system of arithmetical notation
is
system had been
in use in India
This
divine origin.
of
led the early orientalists to believe that, at
any
rate,
from time immemorial
;
the
but
an examination of the facts shows that the early notations
in
use were
place-value
not
and that the modern
ones
was not introduced until comparatively
The early systems employed may be conve-
place-value system
modern
times.
termed
niently
(c)
the
(a)
Kharoshthi,
Aryabhata's alphabetic notation,
(6)
Brahmi,.
the
the word-symbol
(d)
notation.
(«)
and was
The KharoshtJil
script is written
the accompanying table.
It was,
from the Aramaic system and has
with the other Indian
are written on the
(6)
notations.
The Brahml notation
Indian notation for
in fairly
common
apparently, derived
little
direct connection
The smaller elements
left.
old notations of India.
the
left
and Central Asia
The notation is shown
at the beginning of the Christian era.
in
from right to
in use in the north-west of India
It
it
is
the most important of the
might appropriately be termed
occurs in early inscriptions and was
use throughout India for
many
centuries,
and even to the present day is occasionally used. The
symbols employed varied somewhat in form according to
time and place, but on the whole the consistency of form
exhibited is remarkable.
They are written from left to
right with the smaller elements on the right.
Several false
theories as to the origin of these symbols have been published,
some
of
which
orientalists
still
continue to be recorded.
gave them
place-value,
but
The
earliest
this error
soon
INDIAN MATHEMATICS.
"28
disproved
letters of
was then suggested that they were initial
numerical words then it was propounded that the
itself
;
it
;
symbols were aksharas or syllables
that the symbols were
the
corresponding
;
then
it
initial letters (this
These
numerals.
was again claimed
time Kliaroshthi) of
theories
have been
severally disproved.
The notation was possibly developed on different prinThe first three symbols are natural
ciples at different times.
and only differ from those of many other systems in consisting
of horizontal instead of vertical strokes.
No
principle of
formation of the symbols for "four" to "thirty" is now
evident but possibly the " forty " was formed from the
thirty by the addition of a stroke and the " sixty " and
" seventy " and
" eighty "
and " ninety
"
appear to be
The hundreds are (to a limited
extent) evidently built upon such a plan, which, as Bayley
pointed out, is the same as that employed in the Egyptian
but after the " three hundred " the Indian
hieratic forms
system forms the "four hundred" from the elements of
connected in this way.
;
" a hundred " and " four,"
and
exhibited in the table annexed.
so
on.
The notation
is
1
1
INDIAN MATHEMATICS.
30
-
(c)
Aiyabhata's alphabetic notation also had no place-
value and differed from the
smaller elements on the
left.
read from
It
left to right.
Letters
.
Values
..
Letters
.
t
.
.
Values
.
.
2
tli
.
.
Values
.
.
d
5
dh
be exhibited thus
c ch
j jh
6
8
7
n
n
9
10.
dh n
d
th
t
and
:
11 12 13 14 15 16 17 18 19 20.
p ph
Letters
4
3
having the
in
It was, of course, written
may
h kh g gh n
1
Brahml notation
b
bh
m
y
r
v
I
s
sh
s
h
21 22 23 24 25 30 40 50 60 70 80 90 100.
The vowels indicate multiplication by powers of one
The first vowel a may be considered as equivalent
100"*,
The values of
to
the second vowel ^=100^ and so on.
the vowels may therefore be shown thus
hundred.
:
Vowels
Values
.
.
.
.
a
1
i
u
ri
U
o
ai
e
10» 10^ 10^ 10^
10^*^
au
lO'^ 10»* 10^^
The following examples taken from Aryabhata's Gitihd
illustrate the application of the
system
:
M?/M^/in=(2+30). 10^+4.10*5=4320000
caijagiyinumlchIi=6+30 +3.10^ +30.10'+5.10* +70.10*
(50+7).10^=57,753,336
The notation could thus be used
in a sort of mnemonic form.
numbers
referred to in paragraph 9 above
for expressing large
The table
of sines
was expressed by Arya-
bhata in this notation which, by the way, he uses only for
astronomical purposes.
in India,
It did
not come into ordinary use
but some centuries later
it
appears occasionally in
a form modified by the place-value idea with the following
lvalues
:
1
2
INDIAN MATHEMATICS.
vowels are sometimes used as ciphers
Initial
earliest example of this modified system
century.
is
The
also.
of the twelfth
variations occur.
Slight
The word-symbol
{d)
31
notatioti.
extraordinarily popular in
India
— A notation that became
and
is
use was
in
still
introduced about the ninth century, possibly from the East.
In this notation any word that connotes the idea of a
number may be used to denote that number e.g. Two
may be expressed by nayana, the eyes, or Jcarna, the ears,
:
«tc.
seven by asva, the horses
;
the lunar days
(of
hands and
nails (of the
lunar mansions
view
feet)
;
fifteen
;
twenty by
;
by
thirty-two by danta, the teeth
notation.
— The
orthodox
universal was invented in India and until recently
thought to have been
Hindu
in use in India at a
tradition ascribes the invention to
etc.
etc.,
;
that the modern place-value notation that
is
tithi,
tiakha, the
twenty-seven by nakshatra the
modern place-value
The
(e)
;
the sun)
(of
the half month)
now
is
it
was
very early date.
God
According
!
to Ma(;oudi a congress of sages, gathered together by order
of
Brahma (who
king
figures
An
!
reigned 366 years), invented the nine
inscription of A.D. 595
is
supposed to contain
a genuine example of the system.
According to M. Nau,
the " Indian figures " were
in Syria in
known
A.D. 662
;
but his authority makes such erroneous statements about
" Indian "
astronomy that we have no faith
about
says
other
" Indian"
mediaeval works refer to
On
the other hand
'
matters.
Indian numbers
it is
held that there
in
what he
Certain
'
and
is
other
so on.
no sound evi-
dence of the employment in India of a place-value system
earlier
'
than about the ninth century.
The suggestion
of
divine origin' indicates nothing but historical ignorance
Ma^oudi
595
is
is
obviously wildly erratic
;
the inscription of A.D.
not above suspicion* and the next inscription with an
example
of the place-value
*
The
fip;ures
system
is
nearly three centuries
were obviously added at a later date.
INDIAN MATHEMATICS.
32
later,
while there are hundreds intervening with examples
The references
of the old non-place value system.
diseval
me-
in
works to India do not necessarily indicate India proper
but often simply refer to
'
the East
and the use
'
of the
term
with regard to numbers has been further confused by the
misreading by
Woepcke and others
of
hindasi (geometrical, having to do with
the Arabic term
numeration,
etc.)
which has nothing to do with India. Again, it has been
assumed that the use of the abacus " has been universal in
immemorial," but
India from time
based upon
fact, there
use in India until quite
modern
evidence that indicates that the
into India, as
it
times.
(in § 10)
of
some
is
not
of
its
Further, there
is
notation was introduced
was into Europe, from
the circle are briefly described and
a right to left script.
it
has been pointed out
that Aryabhata gives an extremely accurate value
The topic
,
assumption
no evidence
In paragraph 7 above certain attempts at squaring
23.
TT
this
being actually
is
perhaps of sufficient interest to deserve
special mention.
The Indian values given and used
and the subject is wrapped in
are not altogether consistent
some mystery.
Briefly
put—the Indians record an
extremely
accurate value at a very early date but seldom or never
actually use
it.
The following table roughly
the matter stands
Date
Circa.
:
exhibits
how
INDIAN MATHEMATICS.
Date
Circa.
33
INDIAN MATHEMATICS.
34
24.
The mistakes made by the early
naturally misled the historians of
orientalists
have
mathematics, and the
opinions of Chasles, Wcepcke, Hankel and others founded
upon such mistakes are now no longer authoritative. In
spite,
however, of the progress
there are
still
many
made
in historical research
errors current, of which, besides those
already touched upon, the following
may be cited as examples
:
casting out nines " is not of Indian origin
(a) The proof by
and occurs in no Indian work before the 12th century
(6) The scheme of multiplication, of which the following is
"
'
;
an Indian example
of the
16th century,
was known much earlier to the Arabs
and there is no evidence that it is of
(c) The
Regula duorum
Indian origin
jalsorum occurs in no Indian work {d)
The Indians were not the first to give
;
;
double solutions of quadratic equations
Bhaskara was not the discoverer
differential calculus," etc., etc.
;
of the " princijjle
of the
VII.
Of the personalities of the Indian mathematicians
25.
we k]iow very
little
indeed but Alberuni has handed
Brahmagupta's opinion
own
down
Aryabhata and Pulisa* and
We
opinions are worth repeatiiig.
The following notes
inscription.
is
of
his
have also Bhaskara's
contain, perhaps, all that
worth recording.
Alberuni writes (1,376)
'
:
which Brahmagupta relates on
which he himself agrees,
is
Now
his
is
it
own
entirely
evident that that
unfounded
blind to this from sheer hatred of Aryabhata,
And
excessively.
the same to him.
in this respect
I
and with
authority,
but he
;
whom
is
he abuses
Aryabhata and Pulisa are
take for witness the passage of Brahma-
gupta where he says that Aryabhata has subtracted something from the cycles of Caput Draconis and of the apsis of
moon and thereby confused the computation
eclipse.
He is rude enough to compare Aryabhata to
the
of
a
the
worm
which, eating the wood, by chance describes certain characters in
who knows
these things thoroughly
stands
Aryabhata, Srishena and Vishnuchandra
They
gazelles.
'
faces.
'
" He, however,
without intending to draw them.
it,
opposite
to
like the lion against
are not capable of letting
him
see their
In such offensive terms he attacks Aryabhata and
maltreats him.'
Again
:
'
''
Aryabhata
differs
the book Smriti, just mentioned, and he
an opponent."
On
from the doctrine
who
differs
from us
the other hand, Brahmagupta
of
is
praises
Pulisa for what he does, since he does not differ from the
book Smriti.'
*
Again, speaking of Varahamihira, Srishena,
According to Alberuni Pulis'a was an Indian and Paulis'a a
(J
reek.
IXDIAX MATHEMATICS.
36
Aryabhata
a
man
Brahmagupta says
\'ishnuchandra,
and.
'
:
If
declares these things ilhisory he stands outside the
generally acknowledged, dogma,
and that
Of Yarahamihira. Alberuni writes
the Hindus
'
:
is
not
allowed/
In former times,
used to acknowledge that the progress of science
due to the Greeks
is
much more important than
that which
due to themselves. But from this passage of Yarahamihira
alone {see paragraph 2 above) you see what a self -lauding
is
man
he
others:
'
is,
whilst he gives himself airs as doing justice to
but. in
another place
110) Alberuni says
(ii,
the whole his foot stands firmly on the basis of truth
Would
clearlv speaks out the truth
guished
men
God
to
On
'
:
and he
all
distin-
followed his example."
Of Brahmagupta, Alberuni writes
for instance, at
Brahmagupta, who
110)
(ii,
'
:
But look,
certainly the most
is
distinguished of their astronomers .... he shirks the truth and
under the compulsion
lends his support to imposture
some mental derangement,
like a
to rob of his consciousness
whom God
man whom
If
death
Brahmagupta
is
of
about
.... is
one
'
They have denied our signs,
although their hearts knew them clearly, from wickedness
and haughtiness," we shall not argue with him, but only
whisper into his ear "If people must under circumstances
of those of
says,
'
—
give
up opposing the
case),
religious codes (as
why then do you
order people
forget to be so yourself"
the belief that that which
I, for
my
seems to be your
to
be pious
part,
am
if
you
inclined to
made Brahmagupta speak the
above mentioned words (which involve a
sin against con-
was something of a calamitous fate, like that of
Socrates, which had befallen him, notwithstanding the
science)
of his knowledge and the sharpness of his
and notwithstanding his extreme youth at the time.
For he wrote the Brahmasiddhdnta when he was only thirty
If this indeed is his excuse we accept it and
years of age.
abundance
intellect,
lirop the matter.'
INDIAN MATHEMATICS.
Au
inscription found in a ruined temple
deserted village of Khandesh in the
illustrious
Bhaskaracharya whose
like
'
:
refers
the
is
revered
by the
down the law in
metrics,
in the Vaiseshika system,
unto the three-eyed
Patna, a
Triumphant
feet are
wise, eminently learned .... who laid
was deeply versed
at
Bombay Presidency,
to Bhaskara in the followinu terms
a poet,
37
.
.
.
.was in poetics
in the three branches, the
.Bhaskara, the learned,
)nultifarious arithmetic and the rest.
endowed with good fame and religious merit, the root of the
.
creeper
—true
of learning
;
knowledge
whose
feet
Veda, an omniscient seat
of the
were revered by crowds of poets,
The inscription goes on to
'
.
Changadeva, chief astrologer
tell
of
etc.
us of Bhaskara's grandson
King Simghana, who,
spread the doctrines promulgated by the illustrious
to
Bhas-
karacharya, founds a college, that in his college the Siddhanta-
siromani and other works composed by Bhaskara, as well
as other works
by members
of his family, shall be necessarily
expounded.'
Bhaskara's most popular work
which means
"
charming.
'
He
is
uses
entitled the Lildvatl
the
phrase
'
'
Dear
intelligent Llldvati,'' etc.,
and thus have
arisen certain legends
as to a daughter he
supposed
be
is
to
addressing.
The
legends have no historical basis.
Bhaskara at the end
tises
as
''
on algebra
of
too diffusive
" and
substance of them in
oratification
of his V'lja ganita refers to the trea-
Brahmagupta,
"
"'
t^'ridhara
and Padmanabha
states that he has compressed the
a well reasoned compendium, for the
of learners."
VIII.
Chinese Mathematics.
2G.
—There appears to be abun-
dant evidence of an intiniate connection between Indian and
A number
Chinese mathematics.
China and Chinese
and succeeding
visits to
centuries.
Indian
of
The records
of
these
not generally found in Indian works and our
them
is
most cases conies from Chinese
in
no record
in
embassies to
India are recorded in the
matical knowledge.
visits are
knowledge
way indebted
of
and there
authorities,
Indian works that would lead us
that the Hindus were in any
fourth
to suppose
mathe-
to China for
But, as pointed out before, this silence
on the part of the Hindus
is
characteristic,
and must on no
account be taken as an indication of lack of influence.
have now before us a
fairly
complete
We
account of Chinese
mathematics* which appears to prove a very close connection
between
the
two
countries.
connection
This
briefly
is
illustrated in the following notes.
The
questions
earliest
is
Chinese work that deals with mathematical
said to be of the 12th century B.C.
an acquaintance with the
and
the most celebrated Chinese mathematical work
chmig Suan-shu or
was composed at
'
'
is
records
it
Pythagorean theorem.
Perhaps
the Chin-
Arithmetic in Nine Sections
'
'
second century
least as early as the
which
B.C.
known to have
been written in A.D. 263. The "' Nine Sections " is far
more complete than any Indian work prior to Brahmagupta
while
Chang T'sang"s commentary on
(A.D. 628) and in some respects
is
It treats of fractions, percentage,
in
is
advance
of
By Voshio Mikami.
and
and second
plane figures
problems involving equations of the
*
of that writer.
partnership, extraction of
square and cube-roots, mensuration
solids,
it
first
INDIAN MATHEMATICS.
Of particular interest to us are the following
degree.
area of a segment of a circle =1
chord and
a
'
(c
+ «)«.
where
'
in the
;
paragraphs 5 and
are used
with unit numerators
7
above)
;
diameter
the
bhata's strange rule
given by
the volume of the cone=(
;
the Indians
all
pyramid which
a truncated
One
and Sridhara.
of
What
feet high, the
the height of the break
is
)
the following
like
for
""
?
feet
:
end
upper
which being broken reaches to the ground 3
stem.
^
volume
correct
reproduced by Brahmagupta
problems
bamboo 10
a
is
and the
section deals with right angled triangles
and gives a number
" There
is
;
a
of
for Arya-
sphere=^/VrX volume, which possibly accounts
is
the
is
problems dealing with the evaluation
of roots, partial fractions
which
c
The
:
'
the perpendicular, which actually occurs in
'
MahavTra's work
{cf.
39
of
from the
This occurs in
every Indian work after the 6th century.
problem
The
about two travellers meeting on the hypotenuse of a rightangled triangle occurs some ten centuries later in
exactly
the same form in Mahavlra's work.
The Sun-Tsu
of
about the
first
Suan-ching
century.
is
an
arithmetical treatise
It indulges in big
elaborate tables like those contained in
it
Mahaviras work
square-root and
gives a clear explanation of
examples of indeterminate equations of the
The example
unknown.
'
:
'
contains
first
degree.
There are certain things whose number
is 3,
be the number
7th and 9th
and by
?"'
7 the
is
is
2.
2
;
is
by
What
re-appears in Indian works of the
The
given by Brahmagupta and is
centuries.
remainder
;
it
Repeatedly divided by 3 the remainder
5 the remainder
will
numbers and
earliest
Indian
example
is
" What number divided
by 6 has a remainder 5, and divided by 5 has a remainder
of 4 and by 4 a remainder of 3, and by 3 a remainder of 2 ?"
:
Mahavira has similar examples.
In the 3rd century the Sea Island Arithmetical Classic was
written.
Its distinctive
problems concern the measurement
INDIAN MATHEMATICS.
40
of the distance of an island
oiven
tion
occurs
from the shore, and the solu-
Aryabhata's Ganita some two centu-
in
The Wu-fsao written before the 6th century
lies later.
appears to indicate some deterioration.
contains the
It
erroneous rule for areas given by Brahmagupta and Mahavira.
The arithmetic
Chang-Cli'iu-chien
of
written
the 6th
in
may have
century contains a great deal of matter that
been
Indeed the later Indian
the basis of the later Indian works.
works seem to bear a much closer resemblance to Chang's
arithmetic than they do to any earlier Indian work.
The problem
the hundred hens "
'
of
'
is
of considerable
Chang gives the following example
"A cock
money, a hen 3 pieces and 3 chickens 1 piece.
If then we buy with 100 pieces 100 of them what will be their
respective numbers ?"
interest.
:
costs 5 pieces of
No mention
but
problem
is
made by Brahmagupta,
occurs in Mahavira and Bhaskara in the
it
form
of this
" Five doves are to be had
:
and
drammas
for 5, 9 geese for 7
birds for 100
for 3
3 peacocks for
9.
drammas, "
following
7 cranes
Bring 100 of these
for the prince's gratification."
It is
noteworthy that this problem Was also very fully treated by
Abfi Kamil (Shoga) in the 9th century, and in Europe in the
middle ages
it
acquired considerable celebrity.
Enough has been
considerable
show that there existed a very
between the mathematics of the
said to
intimacy
Indians and Chinese
;
and assuming that the chronology
is
roughly correct, the distinct priority of the Chinese mathematics
gives
is
established.
On
the other hand
more advanced developments
Brahmagupta
of indeterminate equa-
tions than occurs in the Chinese Avorks of his period,
is
and
not until after Bhaskara that Ch'in Chu-sheo recorded
it
(in
A.D. 1247) the celebrated fat-yen cJiin-yi-shu or process of
indeterminate analysis,
I'-hsing
nearly six
maintained
which
centuries
intellectual
is,
however,
earlier.
intercourse
The
with
attributed to
Chinese
had
India since the
41
INDIAN MATHEMATICS.
century and had translated
first
They
works.
many
Indian (Buddhistic)
generally give
(unlike their Indian friends)
the source of their information and acknowledge their indebtFrom the 7th century
edness with becoming courtesy.
Indian scholars were occasionally employed on the Chinese
Mr. Yoshio Mikami states that there
Astronomical Board.
is
no evidence
On
of
Indian influence on Chinese mathematics.
the other hand he says
"
'
the discoveries
made
China
in
niay have touched the eyes of Hindoo scholars."
Arabic Mathematics.
27.
with very
little
justification,
—
It
that
has often been assumed,
owed
Arabs
the
their
knowledge of mathematics to the Hindus.
Muhammad
earliest
Musa el-Chowarezmi
b.
is
the
Arabic writer on mathematics of note and his best
known work
is
The early
the Algebra.
to have been somewhat
for,
(A.D. 782)
orientalists
appear
prejudiced against Arabic scholarship
apparently without examination, they ascribed an Indian
origin to
follows:
M.
b.
Musa's work.
' There
nothing in history,"
is
and CoLEBROOKE repeated
]\lrisa
individually,
The argument used was as
it,
'
respecting
wrote Cossali,
Muhammad
ben
which favours the opinion that he took
from the Greeks, the algebra which he taught to the Muham-
madans.
History presents him in no other
light
than a
mathematician of a country most distant from Greece and
Not having taken algebra from the
contiguous to India
Greeks, he must either have invented
from the Indians.'
as pointed out
As a matter
it
himself or taken
by Kodet, no sign of Indian influence and
upon Greek knowledge and it
practically wholly based
now
well
known that
influence
and
that,
;
the development of mathematics
the Arabs was largely,
if
it
of fact his algebra shows,
not wholly, independent
on the other hand, Indian
of
is
is
among
Indian
writers on
)nathematics later than Brahmagupta were possibly influ-
enced considerably by the Arabs.
11th
century
wrote
:
'
Alberuni early in the
You mostly
find
that even the
42
INDIAN MATHEMATICS.
so-called scientific theorems of the
utter confusion, devoid of
any
Hindus are
order.
logical
in a state of
..
.since they
cannot raise themselves to the methods of strictly
scientific
deduction .... I beyan to show them the elements on which
this science rests, to point out to
them some
deduction and the scientific method of
The
fact
is
all
rules of logical
mathematics,
etc."
that in the time of el-ManiQn (A.D. 772) a
astronomical work (or certain works) was
certain Indian
On
translated into Arabic.
was assumed that
this basis it
the Arabic astronomy and mathematics was wholly of Indian
origin, while the fact that
Indian works were translated
is
really only evidence of the intellectual spirit then prevailing
in
No
Baghdad.
one can deny that Aryabhata and Brahma-
gupta preceded M.
is
b.
Musa* but the
fact
remains that there
not the slightest resemblance between the previous Indian
works and those
of
M.
b. Miisa.
obscured by the publication in
treatise
by M.
As
Indorum.
b.
is
The point was somewhat
Europe
of
an arithmetical
Musa under the title Algoritmi de Numero
well known the term India did not in
mediaeval times necessarily denote the India of to-day and
despite the title there
Indeed
its
contents
Indian origin.
is
nothing really Indian in the work.
prove conclusively that
The same remarks apply to
it
is
not of
several other
mediaeval works.
28.
From the time of M. b. Miisa onwards the Muhammadan mathematicians made remarkable progress. To
illustrate this fact
we need only mention a few
of their dis-
tinguished writers and their works on mathematics.
b.
Qorra
b.
Merwan
"
Tabit
(826-901) wrote on Euclid, the Almagest,
the arithmetic of Nicomachus, the right-angle triangle the
parabola, magic squares, amicable numbers, etc.
LQka
el-Ba'albeki (died
c.
* It should not be forgotten, however, that
was an Arabian, while .Tamblichus,
natives of Syria.
Qosta b.
A.D. 912) translated Diophantus
Nicomachus (A.D. 100)
and Eutocius were
Damascius,
43
INDIAN MATHEMATICS.
and wrote on the sphere and cylinder, the
El-Battani (M.
etc.
a
commentarv
Gabir
b.
rule of
two
errors,
Sinan, A.D. 877-919) wrote
b.
Ptolemy and made notable advances
Abu Kamil Shoga b. Aslam (c. 850-930)
on
in trigonometry.
wrote on algebra and geometry, the pentagon and decagon,
the rule of two errors, etc.
Hipparchus. and M.
circle
and sphere,
'1-Wefa el-Buzgani, born
commentaries on Euclid, Diophaiitus,
A.D. 940, wrote
in
Abu
Musa, works on arithmetic, on the
b.
Abu Sa
etc.. etc.
el-sigzT
'id,
(Ahmed
b.
M.
of
an angle, the sphere, the intersection of the parabola and
b. Abdelgalil,
hyperbola, the
hyperbola and
A.D., 951-1024) ^vTote on the trisection
Lemmata
its
of Archimedes, conic sections, the
asymptotes,
etc.. etc.
(M. b. el-Hasan, 1016 A.D.) wrote
Abu Bekr.
was born
el-Blrunl)
A.D. 973 and
in
besides works on history, geography, chronology
nomy
Chaijami, the
and died
a
and
astro-
wrote on mathematics generally, and in particular on
tangents, the chords of the circle, etc.
He
in-
Alberuni (M. b.
determinate equations after Diophantus.
Ahmed, Abu'l-Rihan
el-Karchi
on arithmetic and
in
celebrated
Omar
b.
Ibrahim
was born about A.D.
poet,
el-
104(>
A.D. 1123 a few years after Bhaskara was born.
Wrote an algebra in which he deals with cubic equations?
commentary on the
on mixtures of metals
difficulties in
;
the postulates of Euclid
and on arithmetical
:
difficulties.
This very brief and incomplete resume of Arabic mathe-
matical works written during the period intervening between
Brahmagupta and Bhaskara indicates at least
considerable intellectual activity and a great advance on the
the time of
Indian works of the period in
except,
perhaps,
all
indeterminate
branches of mathematics
equations.
IX.
That the most
29.
^?ssentially
based upon western knowledge
A somewhat
in this direction
now
established.
sections that
may
is
also established
—but the connection
not very intimate with respect to those
is
be termed Greek,
e.g.,
quadratic indeter-
That the Arabic develop-
minates, cyclic quadrilaterals, etc.
mathematics was practically independent
of
influence
is
The Arab mathematicians based
]\I.
b.
of Indian
also proved.
wholly upon Greek knowledge
to us,
is
of the
Bhaskara are
intimate connection between early Chinese and
Indian mathematics
ment
works
iniportaiit parts of the
Indian mathematicians from Aryabhata to
;
but the
Work almost
them known
their
earliest of
Musa. flourished after Brahmagupta so that the
Arabs could not have been the intermediaries between the
and Indians.
•Greeks
Indeed their chronological position
has misled certain writers to the erroneous conclusion that
they
ol)tained
elements of
their
mathematics from the
Indians.
Other possible paths
of
Indians and Greeks are by
The former
Persia.
is
communication between the
way
of China
not so improbable as
Further information about the early
might possibly throw
light
silk
on the subject.
and by way
it
at
trade Avith China
The
intellectual
communication between India and China at the
period
well
is
known
—there
of
available
the
If
early Chinese mathematical
we might be
but as the evidence
able to
now
critical
being numerous references to
such communication in Chinese literature.
lations
of
seems.
first
draw more
stands there
works
were
definite conclusions,
is
warrant more than the bare suggestion
sound trans-
nothing that would
of a Chinese source.
INDIAN MATHEMATICS.
We
have already mentioned the
45
certain other facts
which at
somewhat
shows a
remarkable
of enlightenment in India that
" The
Crupta period.
translations,
parallelism with the a^e
roughly corresponds with
Sassanian
direct
diffused
Greek
religion
literature
who were
and who,
con-
in their
throughout the
Smith discusses the probability
influence on India
but states that there
is
of
no
evidence.
Although
it
may
to the actual route
be possible to offer only conjectures
by which any
knowledge reached India
period under
the
consideration
mathematical
the
work
of
sculpture, architecture, coinage,
Mr. Vincent
Smith
as^
particular class of Greek
the fact remains that durino- the
,
Greek on India was considerable.
in
the-
real missionaries of culture in the
the west by their
Mr. Vincent
orient."
consideration
period, A.D. 229-652,
Persian empire at this time were the Syrians,
nected with
Greek
and there are
I,
jnstifv the
least
The Sassanid
of the Persian route.
certain
visit of
mathematicians to the Court of C'hosroes
refers
"
intellectual
It
the
is
influence of
evident
not onlv
Indians but
also in
astronomy, astrology, &c.
to the cumulative
proof that
the remarkable intellectual and artistic output of the Gupta
by reason of the contact
India and that of the Roman
period was produced in large measure
between the
Empire
;'"
The
civilization of
and research
almost daily adding to such proof.
flourishing state of the
in India since the
its
is
days of Asoka,
principal rulers gave a
Gupta empire, the gxeatest
and the wise
influence of
great impetus to scholarship of
The numerous embassies to and from foreign
countries which were means of intellectual as well as political
communication no doubt contributed to the same end
all
kinds.
—
—
;
and the knowledge of Greek works displayed by Aryabhata,
Varaha Mihira, and Brahmagupta was one of the natural
results of this renaissance of learning.
APPENDIX
I.
ExTKACTS FROM TexTS,
The Sulvasutras.
*1.
manners
measure out the
difEereut
shall explain
4:
how
to
them.
4«
4:
The cord stretched across a square produces an
size.
Take the measure
square for the length
its
We
2.
4:
area of twice the
4(3.
shall treat of the
circuit of the area required for
^
•45.
we
In the following
of building the agni.
for the breadth, the diagonal of
the diagonal of that oblong
:
side of a square the area of
which
is
the
three times the area of
is
the square.
*
H:
48.
The diagonal
4c
4:
of
the areas which the two
4:
an oblong produces by
itself
sides of the oblong produce
both
sepa-
rately.
49.
and
This
is
seen in those oblongs whose sides are three
four, twelve
and
five,
and
fifteen
eight, seven
and
twenty-four, twelve and thirty-five, fifteen and thirty-six.
4:
He
4:
^
:|c
51.
If
square cut
oflt
a piece from the larger square by making a
mark on the ground with the
which you wish to deduct.
oblong so that
it
there cut
By
square
is
oi¥.
side
touches the other
this line
of the
Draw one
smaller square
of the sides across the
side.
Where
which has been cut
it
off
touches
the small
deducted from the large one.
4c
*
from another
you wish to deduct one square
4:
He
He
4:
These numbers refer to BaudhSi'ana's edition as translated by Dr
Thibaut.
INDIAN MATHEMATICS.
58.
If
47
you wish to turn a square into a
of the cord
stretched
towards the prachl
in
circle
diagonal from
the
draw
half
the centre
Describe the circle together with
line.
the third part of that piece of the cord which stands over.
Aryahhafd's gamta
6.
The area
perpendicular
of
common
a
—
(Circa,
triangle
to the
is
A.D. 520).
the product of the
two halves and
Half the product of this and the height
is
half the base.
the
solid
with
six edges.
^
H:
Add
10.
^
add again sixty-two thousand.
mate value
4:
four to one hundred,
of the circumference
The
4:
multiply by eight and
result
the approxi-
is
when the diameter
is
twenty
thousand.
The
13.
*
4:
4:
circle is
4e
4:
produced by a rotation
and the quadrilateral are determined by
;
the triangle
their hypotenuses
the horizontal by water and the vertical by the plumb
4:
4:
H:
4:
;
line.
4:
29.
The sum of a certain number of terms diminished
hy each term in succession added to the whole and divided
by the number of terms less one gives the value of the
whole.
Brahmagupta
1.
He who
— (Born
distinctly
A.D. 598).
knows addition and the
rest of
the twenty operations and the eight processes including
measurement by shadows
*
45
is
a mathematician.
*
4=
4c
The principal multiplied by its time and divided
by the interest, and the quotient being multiplied by the
factor less one is the time.
The sum of principal and
interest divided by unity added to the interest on unity is the
14.
principal.
INDIAN MATHEMATICS.
48
The number
17.
common
of the
of
terms
and added
Half the
sum
less
one multiplied bv the
term is the amount
and first terms is the
to the first
of the last
*****
*****
last.
amoiuit, and this multiplied by the
mean
is
difference
sum
the
The product
21.
number
of half the sides
and opposite
of the sides set
down
is
40.
is
sum
four times and each lessened by the
being multiplied
product
sides
Half the
the rough area of a triangle or quadrilateral.
sides
terms
of
of the whole.
together
—the
square-root
the
of
the exact area.
The diameter and the square of the radius respecby three are the practical circumference
square-roots
extracted from ten times the
The
the same are the exact values.
tively multiplied
and
area.
square of
The integer multiplied by the sexagesimal parts of
fraction and divided by thirty is the square of the minutes
62.
its
and
is
101.
The
tion.
solve
added to the square
to be
*
whole degrees.
of the
*
*
*
'
These questions are stated merely for gratificaproficient
by the
102.
*
may
devise a thousand others or
rules taught problems set
As the sun obscures the
by
may
others.
stars so does the expert
an assembly
eclipse the glory of other astronomers in
people by reciting algebraic problems, and
still
more by
of
their
solution.
Mahdvlrcrs Gatdta-Sara-Sangraha
i.
—
(Circa.
The number, the diameter and the circumoceans and mountains
the extensive
13-14.
ference of islands,
dimensions of
A.D. 850).
;
the rows
of habitations
and
halls belonging to
the inhabitants of the world, of the interspace, of the world
of light, of the
world of the gods and to the dwellers in
and miscellaneous measurements
made out by means
of
of all sorts
computation.
—
all
hell,
these are
147.
vi.
least
INDIAN MATHEMATICS.
49
Divide by their rate prices.
Diminish by the
among them and then multiply by the
least the mixed
and subtract from the given number
up (this) into as many (as there are left)
price of all the things
Now split
of things.
and then
These separated from the total price give
divide.
*****
the price of the dearest
purchase.
of
article
[This
is
a
solution of example 36 below.]
169.
vi.
has to be laiown that the products of gold
It
as multiplied
by
their colours
when divided by the mixed
gold gives rise to the resulting colour
(varrja).
**:(:*
[See examples
24 and 25 below.]
Area has been taken to be
4c
two kinds by Jina
namely, that which is for
in accordance with the result
practical purposes and that which is minutely accurate.
of
*****
vii.
2.
—
Thus ends the section
233.
vii,
of devilishly difficult
problems.
— (Circa
Sridhara's Trisatikd
A.D. 1030).
Of a series of numbers beginning with unity and
1.
sum is equal to half the product of the
number of terms and the number of terms together with unity.
increasing
by
one, the
In exchange of commodities the prices being trans-
32.
posed apply the previous rule
(of three).
With
reference to
*****
the sale of living beings the price
is
inversely proportional
to their age.
the gnomon be divided by twice the sum of the
gnomon and the shadow the fraction of the day elapsed or
If
65.
which remains
be obtained.
will
Bhdsham
L.
I
1.
by
delightful
soft
and
propound
its
correct
*
— (Born
this
A.D. 1114).
easy process of
calculation,
elegance, perspicuous with concise terms,
and pleasing
«
to the learned.
4:
*
4i
4
INDIAN MATHEMATICS.
50
A
L. 139.
some
From
is 'put.
side
that multiplied by twice
assumed nunlber and divided by one
This being set aside
is
than the
less
square of the assumed number a perpendicular
obtained.
is
by the arbitrary number
multiplied
*****
and the
put .is subtracted
side as
Such a triangle
hypotenuse.
Thus, "With
L. 189.
termed
is
same
the
'
be the
will
genuine.'
may
sides,
many
be
Yet, though indeterminate,
diagonals in the quakirilateral.
been sought as determinate by Brahma-
have
diagonal's
—the remainder
'
gupta and others.''
-,
*
*
*
*
*
The circumference less the arc being multifirst.'
From the
the product is termed
arc
the
by
plied
quarter of the square of the circumference multiplied by five
L. 213.
'
subtract
first
that
first
product.
By
the remainder divide the
product multiplied by four times the diameter.
The
quotient will be the chord.
In the like suppositions, when the operation,
V. 170.
restriction, disappoints the
owing to
be
intelligent
Accordingly
discovered
it
by the
answer must by the
exercise
ingenuity.
of
iS'skJd T-i'. The 'conditions^—a clear intellect,
assumption of unknown quantities, equation, and the rule
of three
—are means of operation
*
H?.
The
V. 224.
understanding
intelligent
?
V. 225.
is
lii'
rule of three terms
algebra.
*'
analysis.''
-*):
What
is
is
*
arithmetic
there
Therefore for the dull alone
',
unknown
it is,
spotless
to the
set forth.
To augment wisdom and strengthen confidence,
read, read, mathematician, this abridgement elegant in style,
easily understood
by youth, comprising the whole essence
and containing the demonstration
excellence and free from. defect.
calculation
—
full of
,0-
of
of its principles
•'>.'-•
APPENDIX
II.
Examples.
1.
One-half, one-sixth,
and one-twelfth parts of a pole
clay, and sand.
are
immersed respectively under water,
Two
hastas are visible.
—8
Answer
2.
Find the height
'^.
whom
if
dmmma
was given to a beggar by
he asked alms.
Tell
how many
cowries
thou be conversant in arithmetic with the
reduction termed sub-division of fractions
?
—
Answer 1 cowrie.
(1,280 cowries=l dramma).
3.
Out
of a
23.
of a sixteenth of the fifth of three-quar
ters of two-thirds of half a
the miser gave
?
hastas.
The quarter
a person from
of the pole
swarm
L. 32.
on a blossom
of bees one-fifth settled
of Icadamba, one-third on a flower of sillndhri, three times the
difference of those
numbers
flew to a
of kutaja.
One
about in the
air,
bloom
which remained, hovered and flew
bee,
same moment by the pleasing fragrance of a
and pandanus. Tell me, charming woman, the
allured at the
jasmine
number
of bees
?
Answer—lb
L. 54, V. 108.
4.
The third part of a necklace
amorous struggle fell on the ground.
resting on the couch, the sixth part
of pearls
*
Z = the
Lilarati,
was seen
was saved by the lady
and the tenth part was taken up by her
F=Vija Ganita, both by
'S'=Siidhara, C=Chaturvecla.
broken in an
Its fifth part
lover.
Six pearls
Bliaskara, J/=Muliavira,
INDIAN MATHEMATICS.
52
remained on the
was composed
lace
Say, of
string.
how many
pearls the neck-
?
AnsH'er~30.
5.
which
A
is
S. 26.
powerful, nnvanquished, excellent black snake
32 hastas in length enters into a hole 7| aiigulas
^^^.^'' ^^^^ i^i *^^® course of a quarter of a
day
ornament
aiigulas.
of
by
arithmeticians,
its tail grows
2|
tell me by what time this same enters fully into the hole ?
in
j'y
o^
^
Answer
—76i
6.
M.
days.
(24 angulas=l
v,
A certain person travels at the rate of 9 yojanas a day
and 100 yojanas have already been traversed.
Now
a mes-
senger sent after goes at the rate of 13 yojanas a day.
how manv days
he overtake the
will
first
person
Ansiver—25.
7.
31.
hasta.)
?
M.
A white-ant
advances 8 yivas
In
vi,
327,
less one-fifth in a
day
and returns the twentieth part of an angula in 3 days. In
what space of time will one, whose progress is governed by
these rates of advancing and receding proceed 100 yojanas ?
^wswer—98042553
(8
8.
days.
(7.
yavas=l angula, 768000 angulas=l
Twenty men have
283.
yojana).
to carry a palanquin two yojanas
and 720 dinaras for their wages. Two men stop after
going two krosas, after two more krosas three others give up,
after going half the remaining distance five
and
What wages do they
Ansivers— 18,
(4
9.
earn
57,
155,
leave.
M.
490.
vi,
231.
krosas=l yojana).
It is well
loiown that the horses belonging to the
sun's chariot are seven.
*
men
?
i=the
Lilavati,
F=Vija
Si^Sridhara, r=Chaturvecla
Four horses drag
it
along being
Ganita, both by Bhaskara, J/=Mahavira,
INDIAN MATHEMATICS.
They have
harnessed to the yoke.
ijojanas.
How many
times yoked in four
—
Answer
J-J
to do a journey
times are they unyoked and
?
travels
Every 10 yojanas, and each horse
M.
40 yojanas.
10.
VI, 158.
a female slave sixteen years of age brings thirty-
If
two, what will one twenty cost
?
—25|.
L. 76,
Answer
11.
Three hundred gold coins form the price of 9 damsels
What
of 10 years.
is
the price of 36 damsels of 16 years
M.
Answer—1^0.
12.
70
of
how many
The price
of a
hundred
bricks, of
we have
V, 40.
which the length,
thickness and breadth respectively are 16, 8 and 10,
at six dindras,
?
settled
is
received 100,000 of other bricks a
Say,
quarter less in every dimension.
what we ought
to
pay?
Answer—2d^II.
13.
Two
elephants, which are ten in length, nine in
breadth, thirty-six in girth
drona
C. 285.
of grain.
and seven
How much
will
in height,
be the rations of ten other
elephants which are a quarter more
dimensions
in
height and
other
?
Answer
(64
consume one
— 12
dronas, 3 prasthas, 1| kudavas.
kudavas=16 prasthas=l
C. 285.
drona).
One bestows alms on holy men in the third part of
a day, another gives the same in half a day and a third disIn what time, keeping to these
tributes three in five days.
14.
rates, will
*
they have given a hundred
Answer
—174-
L=the
Lilavati.
?
^- '^^^^
r=Vija
S=SrIdhara, C=Chaturveda.
Ganita, both byBhaskara,
J/=Mahavira
INDIAN MATHEMATICS.
54
what are the apportioned
mathematician,
Say,
15.
shares of three traders whose original capitals were respectively 51, 68
and
85,
conducted by them
of
300
which have been raised by commerce
in joint stock to the aggregate
amount
?
A7isn'er—15, 100, 125.
L. 93.
One purchases seven
16.
Eighteen
the profit.
is
Answer— 18-^ {^
17.
-
is
two and
and a pah
dramma and
sells six for three.
the capital
?
Bakhshdli Ms. 54.
1)=24.
a pala of best camphor
If
nishkas.
for
What
may
be had for two
wood for the eighth part
wood also for the eighth
of sandal
half a pala of aloe
of a
of a
dramma, good merchant, give me the value of one nishka in
for I wish to prepare a
the proportions of 1, 16 and 8
;
perfume
?
Answer
—Prices—Drammas 14|,
drammas =1
(16
If three
18.
f,
f + i, V' %"
and a
half
mdnas
of rice
of
(64
be had for
take these
and give me
we must make a hasty meal
companion will proceed onwards ?
beans
and depart, since my
Ansiver— ^^ and
may
price,
quickly two parts
thirteen kdkinls, merchant,
and one
:
for
L. 97, F. 115.
-J^.
kakinis=l dramma).
19.
If
the interest on 200 for a
what time will the same sum
Answer 665 months.
month be
lent be tripled
6 drammas, in
?
—
20.
If
on the
five
thousand,
C. 287.
sum with interest at the rate of
hundred by the month amount in a year to one
the principal
tell
the principal and interest respectively
Answer— 625,
*
^^^
nishka).
one dramma and eight of beans for the same
of rice
-^-
375.
?
L. 89.
Z=tlie Lilavati, T"=Vija Ganita, both by Bhaskara, /)/= Mahavira,
iS'=Sndhara, C'=ChaturYecla.
INDIAN MATHEMATICS.
55
In accordance with the rate of five per cent, (per
21.
mensem) two months
the time for each instalment
is
paying the instalments of 8 (on each occasion) a
What
months.
free in 60
Ansiver
is
the capital
;
and
man became
?
— 60.
M.
vi,
64.
Five hundred drammas were a loan at a rate of
22.
The
interest not
known.
months was
lent to another person at the
accumulated
months to
in ten
on the principal
money
interest of that
same
rate
and
it
Tell the rate of interest
78.
?
Answer— m.
C. 288.
Subtracting from a
23.
for four
sum
lent at five in the
hundred
the square of the interest, the remainder was lent at ten in
the hundred.
amount
The time
—Principal
Answer
There
24.
both loans was
alike
F. 109.
8.
part of
1
varm,
1
part of 2 varms,
I
part
Throwing these into
and then what is the varna
14 varms, and 8 parts of 15 varms.
the
fire
of the
make them
mixed gold
all
into one
?
Answer— 101.
^-
[The term varna corresponds to
'
and the
?
varms, 2 parts of 4 varms, 4 parts of 5 varms, 7 parts
of 3
of
is 1
of
equal
interest
of
'
carat
'
or
^^'
^'^^'
measure of
purity of gold.']
25.
Gold
1,
2,
4 suvarnas,
3,
and
losses 1,
2,
3,
4
mashakas.
The average
loss is
hl+^l±^^+M==^,
Bahhshdli Ms. 27.
*
Z = the
.'s=Sridhara,
Lilavati,
F=Vija
C=Chaturveda.
Ganita, both by Bhaskara,
.V=Mahavira
INDIAN MATHEMATICS.
S6
Of two arithmetical progressions with equal sums
26.
and the same number
terms the
of
first
terms are 2 and
sum
the increments 3 and 2 respectively and the
the number of terms
Answer
A
27.
—
?
Bakhshali Ms. 18.
3.
merchant pays
At the
I,
What was
I shall
was the amount
Answer
A
'
'
:
— 40
Give
'
'
be six times as rich as
3 times as
total
Four
30.
is
?
F. 106, 156.
170.
as A,
—A
:
of their respective capitals
and
be twice
" If you deliver
you. Tell me what
other replies
gives a certain amount,
C gives
C and the
?
me a hundred and I shall
The
!
as
Answer
goods
Bakhshdli Ms. 25.
as rich as you, friend
me
of the
— 40.
One says
28.
29.
on certain goods at three
first
amount
the original
Answer
ten to
octroi
he gives ^ of the goods, at the
and at the third ^. The total duty paid is 24.
different places.
second
3,
Find
15.
much
as B,
B
D
gives twice as
gives 4 times as
much
much
132.
gives
4,
Bakhslidli
etc.
possessing respectively
jewellers
8
Ms.
54.
rubies,
10 sapphires, 100 pearls and 5 diamonds, presented each
from
and
his
own
stock one apiece to the rest in token of regard
gratification at
of stock of precisely
and thus they became owners
the same value. Tell me, friend, what
meeting
;
were the prices of their gems respectively
Answer
— 24,
?
96 [These are relative values only].
16, 1,
L. 100.
31.
pearls
*
The quantity
of rubies without flaw, sapphires,
belonging to one person,
i=the
Lilavuti,
F=Vija
-S^Sridhara, C=Chaturveda.
is
five, eight
and
and seven
Ganita, both by Bhaskara, J/=Mahaviia,
57
INDIAN MATHEMATICS.
The number of like gems belonging to another
is seven, nine and six.
One has ninety, the other sixty-two
rupees.
They are equally rich. Tell me quickly, then,
respectively.
who
gem ?
intelligent friend,
of each sort of
Answer
—
14, 1,
[Bhaskara
'
art conversant with algebra, the prices,
assumes
tively are five, three, six
them
two and one.
one,
;
and
eight
the camels belonging
;
and one their mules are eight,
and the oxen owned by them are seven,
;
All are equally rich.
friend, the rates of the prices of horses
Answer— 85,
friend, in
opened one by one, would
what portion
fill
fill
of a
of Pritha,
those of his antagonist
;
angered in combat, shot a quiver
»With
half his arrows he parried
with four times the square-root of
the quiverful he killed his horse
;
;
with six arrows he slew
with three he demolished the umbrella, standard and
bow and with one he
;
cut
off
the head of the foe.
were the arrows which Arjuna
let fly
35.
pams
*
L. 67, V. 133.
For 3 'panas 5
jjalas of
11 ])alas of long pepper
i=the
'S:=Sridhara,
Lilavati,
How many
?
Ansiver— 100.
4
will
L. 95.
of arrows to slay Karna.
Salya
day
?
— yV.
The son
?
a cistern, which,
Answer
34.
rest
one day, half a day, the
it in
and the sixth parts respectively
third
me severally,
V. 157.
four fountains, being let loose together
if
Tell
and the
76, 31, 4, etc.
Say quickly,
33.
156.
to these four persons respec-
are two, seven, four
two, one and three
&
relative values.]
'
The horses belonging
32.
to
V. 105
1, etc.
F=Vija
C=Chaturveda.
ginger are
and
obtained,
for
for 8 paiias 1 pala of
Ganita, both by Bhaskara, J/=.\l;ihavira,
INDIAN MATHEMATICS.
58
By means
pepper.
Ansiver
M.
?
— Ginger 20, long
pepper 44, pepper
cranes for five
;
150.
vi,
4.
Five doves are to be had for three drammas
36.
;
seven
nine geese for seven and three peacocks for
Bring a hundred of these birds for a hundred drammas
nine.
the
for
purchase money of 60 panas
of the
pahs
quickly obtain 68
prince's
gratification
1
V. 158-9;
Answer— VviQeQ
Birds
3, 40,
56, 27,
5,
M.
vi,
152.
21, 36.
12.
was treated fully by Abu KamilSee H. Suter
Das Buck der Selten-
(This class of problem
el-Misri
900 A.D.).
(c.
:
Bibliotheca Mathematica 11 (1910-11), pp. 100-120.
heit, etc.
37.
In a certain lake swarming with red geese the tip
bud
of a lotus
of a
was seen
half a hasta
Forced by the wind
the water.
was submerged at a distance
of
two
above the surface of
gradually advanced and
it
Calculate quickly,
hastas.
mathematician, the depth of the water
?
Answer— \K
38.
upon
ing
of the
If a
level
it is
*
i=the
= Sridhara,
tip of
it
F. 125.
in
one place by the force
meet the ground at sixteen
mathematician, at how
—
;
thirty-two hastas and stand-
ground be broken
broken
Answer
•S
bamboo measuring
wind and the
hastas, say,
root
L. 153
many
hastas
from the
?
12.
Lilavati, T'= Vija Ganita, both
C =Chaturveda.
L. 148.
by Bhaskara, 7l/=Mahavira,
59
INDIAN MATHEMATICS.
A
39.
snake's hole
high and a peacock
at the foot of a pillar
is
at a distance of thrice the pillar, gliding towards his
snake,
Say quickly at how
he pounces obliquely on him.
hole,
many
Seeing a
summit.
perched on the
is
hastas
9
from the snake's hole do they meet, both pro-
hastas
ceeding an equal distance
Answer
—
L. 150.
12.
From
40.
?
a tree a hundred hastas
monkey
a
high,
descended and went to a pond two hundred hastas
distant,
while another monkey, jumping a certain height off the tree,
proceeded quickly diagonally to the same spot.
travelled
by them be
height of the leap,
tion
equal, tell
if
me
the space
If
quickly, learned
man, the
thou hast diligently studied calcula-
?
Answer— m.
L. 155; F. 126.
The man who
41.
of 2 yojanas,
travels to the east
and the other
at the rate of 3 yojanas.
5 days turns to
days
will
man who
The
—
latter
moves
man
at the rate
northward moves
having journeyed for
move along the hypotenuse.
he meet the other
Answer
travels
In
how many
?
M.
13.
vii,
211.
The shadow of a gnomon 12 angulas high is in one
place 15 angulas. The gnomon being moved 22 angulas
further its shadow is 18.
The difference between the tips
42.
of the
shadows
is
of the
shadows
is
25 and the difference between the lengths
3.
Find the height
Answer— im.
of the light
C. 318
;
Ar. 16
;
?
L. 245.
The shadow of a gnomon 12 angulas high being
lessened by a third part of the hypotenuse became 14 angulas.
Tell, quickly, mathematician, that shadow ?
43.
Answer—221.
*
^' 1^^'
Z=theLilavati, r=Vija Ganita, both by Bhaskura, J/'=Mahavira,
<S=Sridhara, C=Chaturveda.
INDIAN MATHEMATICS.
60
Tell the perpendicular
44.
drawn from the
intersection
of strings mutually stretched from the roots to
two bamboos fifteen and ten hastas
ground of unlcnown extent
Answer
—
high
L. 160.
the square
is
and the face two hastas and altitude twelve, the
of four
and
thirteen
what
fifteen,
is
the area
S.
In the figure of the form of a young
46.
is
splitting
and the middle breadth
sixteen
up
it
Ansiver
two
into
is
moon
77.
the middle
By
three hastas.
triangles tell me, quickly, its area
—24.
The
47.
flanks
?
Ansiver—lOS.
length
of
upon
?
6.
Of a quadrilateral figure whose base
45.
summits
standing
?
S. 83.
sides of a quadrilateral
with unequal
sides
13x15, 13x20 and the top side is the cube of 5 and the
bottom side is 300. What are all the values here beginning
are
with that of the diagonals
Answers— U^D,
?
280, 48, 252, 132, 168, 224, 189, 44100.
M.
If
^2
+
aC,
vii,
59.
and a2 4- J2 = f-z then the quadrilateral Ac, Be,
is cyclic and the diagonals are Ah -\- aB and Aa + Bh,
=
^2
W
C2,
(ABc^
the area
is
i
B=
C
=
20,
315, 280
25
;
a
+ ahC^), &c. In the present case ^ = 15,
= 5, b = 12, r = 13. The diagonals are
For
the area 44100.
;
the LVavatl,
full details see
§ 193.
48.
O
friend,
who knowest
the secret of calculation,
construct a derived figure with the aid of 3 and 5 as ele-
ments, and then think out and mention quickly the numbers
measuring the perpendicular
hypotenuse
Answer— IQ,
That
is
the other side and the
M.
30, 34.
construct a triangle of the form 2w«,
where
*
side,
?
ni
=
5,
>i
=
//=the Lilavati, F=Vija
iS=Sridhara, (7=Chaturveda.
vi^
—
)i^,
vii,
94.
iii''^-\-/i.'^>
3.
r4anita,
both by Bhaskara, i)/=Mahavira,
INDIAN MATHEMATICS.
In the case of a longish quadrilateral
49.
perpendicular side
is
What
73.
Answer
55, the base is 48
is
are the elements here
—
what
figure the
and then the diagonal
?
M.
8.
3,
Intelligent friend,
50.
Lllavati, say
is
61
is
if
vii,
121,
thou knowest well the spotless
the area of a circle the diameter of which
measured by seven, and the surface
of a globe or area like
a net upon a ball, the diameter being seven, and the solid
content within the same sphere
Answer
—Area 38
If^gty
?
surface 153
;
|if;;
;
volume 179 \i%l
L. 204.
51.
In a circle whose diameter
cumference
the area
is
what
ten,
thou knowest, calculate, and
is
tell
the cir-
me
also
?
Answer
52.
If
?
—
x/ 1,000,
The measure
the part devoured
Answer
S. 85.
/v/6250.
Rahu
of
is
52, that of the
moon
25,
7.
—The arrow of Rahu
is 2,
that of the
moon
5.
C. 311.
This
an eclipse problem and means that circles of diameters 52
and 2.") intersect so that the portion of the line joining the
two centres common to the two circles is 7. The common
is
chord cuts this into segments of 5 and
53.
The combined sum
ence, the diameter
circumference
diameter
is,
The
*
is
measure
1116.
what the calculated
of the circumfer-
Tell
area,
me what
the
and what the
?
Ansiver— 108, 972,
64)—
of the
and the area
2.
rule
y^ 64
given
36.
is
M.
vii,
32.
circumference = //i^ ^combined sum+
which assumes that 7r=3.
L=the Lllavati, F=Vija
tS=Sridhara, C=Chaturveda.
Ganita, both by Bhaskara, J/=Mahavira,
INDIAN MATHEMATICS.
62
The circumferential arrows
54.
How many
18
are
are the arrows in the quiver
number.
in
?
M.
Ansiver—31.
—^^-^
n =^-
rule given is
The
where
c is
the
289.
number
of arrows in the outside layer.
Tell
55.
me,
thou knowest, the content of a spherical
if
piece of stone whose diameter
Answer
is
a hasta and a half
?
—1|4.
The
rule
56.
A
S. 93.
given
v=d^ (^+^^8^-
is
altar
sacrificial
is
built of bricks 6 afigulas high,
It is 6 hastas long, 3
half a hasta broad and one hasta long.
Tell me rightly, wise
hastas broad and half a hasta high.
man, what its volume is and how many bricks it contains.
Answer—^,
S. 96.
72.
24 aiigulas=l hasta.
mound
hastas
thou knowest,
If
57.
tell
me
quickly the measure of a
grain whose circumference
of
is
36
and height
4
?
Answer
— 144.
S.
102.
The rule used assumes that 7r=3.
58.
In the case of a figure having the outline of a bow,
the string measure
measure
of the
is
bow
12,
is
and the arrow measure
not known.
Find
it,
Answer— ^/'360•
59.
is
M.. vii, 75.
24 in measure, and
its
What is
Answer— y/bim.
arrow
is
taken to be 4
in
the minutely accurate value of the area
measure.
*****
*****
*****
^
rule vised
i=the
is
o
^^-
=
^'*'^'
?
^2.
—
ac/v/10.
Lilavati, F=Vija
S=Sridhara, C=Chaturveda.
*
The
In the case of a figure having the outline of a bow
the string
The
is 6.
friend.
.
(xanita,
both b}' Bhaskara, J/=Mahavira,
INDIAN MATHEMATICS.
63
Multiplier consisting of surds two, three
60.
and eight
multiplicand the surd three with the rational iiumber
Tell quickly the product
?
^wsit'er— V9+>v/450+v75+v54.
What
61.
is
the
having the third
number which
and
;
The
therefore r
—
added gives two
and
five
and the
and a
less
48.
solution
=
than
L. 51.
may
be summarised
this:
)=fiS,
/(3)
= 17 4
//4
The eighth part
of a
grove and
skipping in a
troop of monkeys
on
the
Twelve remaining were
seen
How many
Ansiver—iS or
The
fifth
hill
amused
were there
with
in all
?
V. 139.
part of the troop less three, squared, had
in sight,
Say how many there were
— 50
squared
their sport.
16.
gone to a cave and one monkey was
on a branch.
with
delighted
chattering to each other.
Answer
./'(.'
——— =48.
1
63.
by
?
Answer
62.
multiplied
one-third, one half
quarter of the original quantity
was
V. 32.
part of the product subtracted,
remainder divided by ten
seventy
:
five.
having climbed
?
or 5.
" But,"' Bhaskara
congruous.
saj's,
V. 140.
" the second
is
not to be taken for
People do not approve
a
neirutive
it is in-
alisolute
numVjer."
64.
five
times
Say quickly what the number is which added
itself divided by thirteen becomes thirty ?
Answer—"^-.
to
V. 168.
65.
A certain unknown quantity is divided by anotlier.
The quotient added to the divisor and the dividend is fifty-
three.
What
is
Answer— b,
*
Z=the
the divisor
M.
8,
Lilavati,
?
F— Vija
iS=Sridhara, C=Chaturveda.
vi,
274.
Ganita, both by Bhaskara, .l/=Mah£vira,
INDIAN MATHEi\LA.TICS.
64:
What number
that which
by nought
and added to half itself and multiplied by three and divided
by nought amounts to the given number sixty-three ?
66.
Ansicer
—14.
What
67.
is
multiplied
This assumes that {7=1.
four
numbers
L. 46.
are such that their product
is
equal to twenty times their sum, say, learned mathematician
who
art conversant with the topic of the product of
quantities
unknown
?
Answer— 5.
4.
2,
V. 210-
11.
for three of the quantities
Bhaskara puts arbitrary values
and gets
11 for the fourth.
you are conversant with operations of algebra
tell the number of which the fourth power less double the
sum of the square and of two hundred times the simple
68.
number
If
is
ten thousand less one
?
Ansusr— II.
may be
This
V. 138.
expressed by
.<•*— 2
case in which the fourth
sum
the cube of their
cubes
of the
The square
69.
is
(a;2+200
a:)
= 9999.
It is the
only
power occurs.
sum
equal
of
to
two numbers added to
twice
the
sum
of
their
i
Ansiver—1, 20
70.
Tell
sum of them
when added
numbers
me,
;
if
5,
F. 178.
76, etc.
you know, two numbers such that the
multiplied respectively
to
by four and three may
of the same
two be equal to the product
?
Ansicer— b, 10 and
11,
F. 209, 212.
6.
Sav quickly, mathematician, what is the multiplier
by which two hundred and twenty -one being multiplied and
sixty-five added to the product, the sum divided by a hundred
71.
and ninety-five becomes cleared
Ansu-er—5, 20, 35 &c.
*
?
L. 253
;
F. 65.
i=theLilavati, F=Yija Ganita, both by Bhaskara, iU=Mahavira,
<S'=Sridhara,
C=Chaburveda.
INDIAN MATHEMATICS.
What number
72.
five,
65
divided by six has a remainder of
divided by five has a remainder of four, by four a remain-
der of three and by three one of two
—
Answer
Br. xviii, 7
59.
What
73.
?
square multiplied by eight and
added to the product
will
be a square
F. 160.
;
having one
?
7. 82.
Here
74.
Stt^
+1=^-2 and m=6,
Making the square
35, etc.
t=\l
,
99, etc.
of the residue of signs
and
minutes on Wednesday multiplied by ninety-two and eighty-
added
three respectively with one
square
;
who does
to the product
an exact
this in a year is a mathematician.
Br. xviii, 67.
92 w2
(1)
+1=^2
83 m2 +1=^2.
(2)
Answer— {I) ^=120, ^=1151.
75.
What
is
«<=9, ^=82.
(2)
the square which multiplied by sixty-
seven and one being added to the product will yield a squareroot
;
and what
is
that which multiplied by sixty-one with
one added to the product
friend,
if
the method of the
will
'
do so likewise
rule of the square
spread, like a creeper, over thy
mind
'
?
Declare
it,
be thoroughly
?
V. 87.
(1)
67 v? +1=«"'.
(2)
Answers-{\) w=5967, ^=48842.
61 u -\-l=t\
(2)
w=226,153,980,
«=1,766,319,049.
76.
Tell
me
quickly, mathematician,
that the cube-root of half the
sum
of their
smaller number, and the square-root
*
L=thQ
Lilavati,
F=Vija
two numbers such
product and the
of the
sum
of
their
Ganita, both by Bhaskara, J/=Mah5vira,
=Sridhara, C=Chaturveda.
5
INDIAN MATHEMATICS.
66
and those extracted from the sum and difEerence
increased by two, and that extracted from the difierence of
their squares added to eight, being all five added together
squares,
may
yield a square -root
—excepting, however, six and
eight?
F. 190.
Answers—x=8
;
1677/4
;
15128, etc.
;
?/=6, 41
;
246, etc.
* 2i=the Lilavati, 7=Viia Ganita, both by Bhaskara, J/=Mahavira,
5=Sridhara, C7=Chaturvecla.
CHKONOLOGY.
BIBLIOGRAPHY.
(For a more complete bibliography see that given in the
Journal of
the Asiatic Society of Bengal,
VII, 10, 1911.)
First Period.
— On the S'ulvasutras, J.A.S.B., XLIV,
—The Baudhayana S'ulvasutra, The Pandit
Thibaut, G.
1875.
(Benares) 1875-6.—The
BiiRK, A.
Katyayana S'ulvasutra,
1882.
76.,
—Das
1901
;
Apastamba-S'ulba-Sutra, Z.D.M.G., 55,
56, 1902.
Second Period.
Burgess, E. and Whitney, G.
Jour.
Am.
—The Silrya Siddhanta,
Or. Soc, VI,
1855.
Bapu Deva Sastri and Wilkinson,
L.
— The
Surya
Siddhdnta and the S'iddhdnta Siromani, Calcutta,
1861.
Thibaut, G. and Sudharkar Dvivedi.
— The
Pancha-
siddhdntikd of Varaha Mihira, Benares, 1889.
RoDET, L.
Kaye, G.
Lerons de Calcul d'Aryahhata, Paris, 1879.
R.— Aryabhata,
J.A.S.B., IV, 17, 1908.
Third Period.
Colebrooke, H.
suration
T.
—Algebra with Arithmetic and Men-
from
the Sanscrit
of
Brahmagupta and
Bhascara, London, 1817.
Rangacarya, M.
viracdrya,
—The
Ganita-Sdra-Sangraha of Mah^-
Madras, 1908.
INDIAN MATHEMATICS.
69
Eamanujacharia, N. and Kaye, G. R.
—The Trisatika
of S'rTdharacharya, Bib. Math., XIII, 3, 1913.
Notations.
—Indische Palreographie, Strassburg, 1896.
E. C. — On
Genealogij of Modern Numerals,
Buhler, G.
Bayley,
the
London, 1882.
WoEPCKE,
F.
—Memoire snr
la
propagation des Chiffres
indiens, Jour. Asiatique,
Kaye,
G. R.
S.B.,
—Indian
Notations, J. A.
Arithmetical
The Use
7,1907.
III,
1863.
of the
Abacus
Ancient India, J. A. S. B., IV, 32, 1908.
in
Old
Indian Antiquary,
Indian Numerical Systems,
1911.
Fleet,
F.
J.
— Aryabhata's
Numbers,
Abacus
/. R. A.
S.,
in India, J. R.
and Karpinski,
Smith, D. E.
system
1911.
A.
of
expressing
The Use
of the
S., 1911.
L. C.
— The Hindu Arabic
Numerals, Boston, 1911.
Other Works.
—Albenmi's India, London, 1910.
G. —Astronomic, Astrologie und Mathematik,
Sachau, E.
Thibaut,
C.
Grundriss
9,
der
Indo-Arischen
Philologie,
III,
1899.
Hoernle, R.
—The Bakhshali Manuscript, Indian Anti-
quanj, XVIII,
Kaye, G. R.
1888.
—Notes on Hindu Mathematical
Bib. Math.,
Methods,
XI,
4,
Indian
Education
—1910-1913.
Source of Hindu Mathematics, J. R. A.
The Bakhshali Manuscript,
1912.
Methods,
1911.—Hindu Mathematical
S.,
The
1910.
J. A. S. B., VII, 9,
INDIAN MATHEMATICS.
70
Heath,
T, L.
—Diophantus
of Alexandria, Cambridge,
1910.
—
Rosen, F. The Algebra
London, 1831.
H.
SuTER,
Die
of
Mathematiker
Mohammed
ben Musa,
und Astronomen
der
Araber und Ihre Werke, Leipzig, 1900.
—
YosHio Mikami. The development of Mathematics
China and Japan, Leipzig, 1912.
in
The general works on the history of mathematics by
Cantor, Gunther, Zeuthen, Tannery and v. Braunmuhl
and the articles by Woepcke, Rodet, Vogt, Suter and
Wiedemann
XI
should also be consulted.
82.
INDEX.
Abacus, 32, 69.
Chasles, 34.
Abbreviations, 24-25.
Chaturveda, 53-55.
Ghin-chang Suan-shU 38.
Chinese mathematics, 6, 38-41, 44,
Abu Bekr, 43.
Abu Kamil, 40,
Abu Said, 43.
—
43, 58.
70,
Abu'1-Wefa, 43.
Chosroes
I, 15,
45.
Alberuni, 35-36, 41-42, 43, 69.
el-Chowarezmi
see
Alexandria,
Circle,
9, 15, 22.
Algoritmi de
Numero Indoriim —-42.
Apastamba,
4, 68.
7,
M.
b.
ilusa,
8, 11, 12,
32-33,
47-48, 61-62.
„
Alphabetical notations, 30-31, 69.
the,
squaring, 7-8.
value of
TT
]],
12, 32-33,
Angles, 20.
47-48.
Arab mathematicians, 41-43,
70.
Colebrooke,
1,
41, 68.
Arithmetical notations, 27-32.
Cossali,
Aryabhata, 11-14, 21, 31, 35-36,
40,47, 68, 69.
Cube-root, 23, 38.
Astrology, 24, 26.
Cyclic quadrilaterals, 20, 22, 48,
41.
Cyclic method, 16, 23.
Astronomy, 1, 9, 68.
Athenean schools, 15.
50, 60.
Damascius, 1.5, 42.
Decimal notations (see Place-
Bailly, 1.
Bakhshali M.s., the
—
1,
24, 55, 56, 69.
value notations).
el-Battani, 43.
Delian problem,
Baudhayana,
Diophantus,
Bhaskara,
4, 46, 68.
7.
15, 16, 22, 25, 70.
12, 14-21, 24, 37, 49-50,
68.
el-Biruni see alberuni,
Brahmagupta,
Egyptians,
6.
Epantheiii, the, 13, 47.
12, 14-23, 35-36, 38,
Equations, 12, 15-18, 23-24, 26(a),
47-48. 68.
Brahmi numerals,
27, 29,
40, 58-66.
Euclid, 7, 19.
Braunmiihl, A. von, 70.
Eutocius, 42.
Biihler, G., 3, 69.
Examples,
24, 2G(a), 51/.
Biirk, A., 6, 68.
Burgess, E., 68.
Fleet, J. F., 69.
Fractions, 5, 51.
Cantor, M., 17, 70.
Casting out nines, 34.
Chang-ch'iu-chien, 41.
Chang T'sang,
38.
Geometry,
14, 20-22, 46, 58/.
Gnomons, 5-6.
Greek influence,
2, 0, 16, 17, 22, 45.
INDEX.
72
Greek terms,
Gunther,
Punchu Siddhantika, the, 9-11, 68.
Pappus, 22.
9, 26.
S., 70.
20.
Parallels,
Paulisa Siddhanta, the,
Hankel, H.. 17, 34.
Heath, Sir T. L., 17, 70.
Hoerule, R.. 69.
Hypatia. 15. 16.
2, 10.
Pellian equation, 16, 17, 65.
Place-value notations, 2, 29, 31-32.
Plato, 15, 19.
Problems, 52/.
Progressions, 23, 48, 49, 56.
Inaccuracies, 20-21, 40.
12, 15-18,
Ptolemy,
40, 65-66.
Pulisa, 35.
Indeterminate equations,
10, 11, 33.
Pj^ramid, volume of, 21.
Pythagorean theorem, 4-6, 38.
Inscriptions, 31, 37.
Interpolation formula, 11.
Interest, 47, 56.
Qosta
Jones, Sir W.,
b.
Luqa, 42-43.
Quadratic equations,
Jambliehus, 42.
16-18, 24,
63-66.
1.
Quadrilaterals, 20-22, 60-61.
Kali^a
iSiitras, 3.
Ramanugacharia, M., 69.
el-Karclu, 43.
C,
Karpinski, L.
Rangacharia, M., 68.
69.
Katyayana, 4, 5. 68.
Kern, 11.
Kharoshthi numerals, 27, 29.
Laplace,
Rational triangles,
4, 18-19, 50,
60.
Regula duorum falsorum, 34.
Right-angled triangles, 4-5, 18-20,
60.
1.
Letter numerals, 30-31.
Llldvatl, the, 24, 37, 49-50, 51-64.
Macdonnell,
4.
Magoudi, 31.
Mahavira, 14,
19, 21, 23,
39; 40,
Rodet, L., 41, 68, 70.
Romaka Siddhanta, 2,
Rosen, 70.
Rule
Rule
9.
of three, 23, 26(a), 50, 53.
of
two
errors, 34.
48-49, 52-63, 68.
Mikami, Yoshio, 38, 41, 70.
Muhammad b. Musa, 41-42,
Muhammad
Rihau
Ahmed, Abu
b.
'aSW Island' Arithmetic, the, 39-40.
70.
'1-
cl-Birunl. 35-36, 41-42,
43, 69.
Max,
Musa {see M.
Miiller,
Simplicius, 15.
Sine function, 9.
—
Sines, table of
10, 11.
•,
Smith, D. E., 69.
Smith, Vincent, 45.
4.
6.
Siddhanta Siromani, the, 37, 68.
Musa),
—
Sphere, volume of
Nau, F., 31.
Nicomaehus,
Square-root,
38-39.
2,
Squaring the
b.
Ibrahim el-Chaijami, 43.
Paimauabha,
14, 37.
,6 7, 46.
Sridhara, 14, 21, 24, 37, 49, 51, 52,
60-62, 69.
Srishena, 35.
Sulvasutras, the,
Omar
—
circle, 7-8, 47.
27-32, 69.
Numerical words, 31.
*
13, 39, 61, 62.
Squares, construction of
42.
''Nine Sections' Arithmetic, the,
Notations,
•,
5, 63.
1,
3-8, 46-47, 68.
S'un-Tsii Suan-ching, the, 39.
Surya Siddhanta,
Symbols, 24.
the, 1, 9, 68.
73
INDEX.
Tabit
b.
Volumes,
Qorra, 42.
13, 21, 61, 62.
Table of siaes, 10.
Tai-yen process, 40.
Tannery, T,, 5, 17, 70.
Whitney, 1, 68.
Weber, 1.
Terminology, 24-26.
Wieddemau, E., 70.
Woepcke, 32, 34, 69,
Thibaut, G., 1, 7,
Triangle, area of
—
9, 68, 69.
•,
20, 21, 47.
Triangles, right-angled, 4, 18-19,
Word
70.
numerals, 31.
Wu-t'sao, the, 40.
50, 60.
Trigonometry, 9-10.
i^avat tdvat, 24-25, '2Q(a).
Trisatika, 24, 27, 69.
Yoshio :Mikami, 38, 41, 70.
Varaha Mihira,
Zero. 64.
2, 9, 35, 36, 68.
Vlja Ganita, the, 24, 37, 50, 51-66.
Vishnuchandra, 35, 36.
Zeuthen, H., 70.
•lMS!Smm:L.
ON^THELASTD^T,^
^^^^^
CIRCULATION DEPARTMENT
202 Main Library
TO—#
LOAN PERIOD
HOME USE
|2
1
^
ALL BOOKS AAAY BE RECALLED AFTER
7 DAYS
Imonth loans m^y be reopwed by calling 642-3405
Ivear loans may be recharged t^' b'.nging the Dooks
to the Circulation Desk
Renewals and recharges may be made 4 days prior to
due date
DUE AS STAMPED BELOW
Jaw
41984
4 t983
^^^n9l^
m
CIR
JUN
5
19B^
r.n
JftN
'.
SWW
aRCULAIION OE^T
ri'^^-
iviA!
mo
d i 1990
CIRCULAR*
0CT3 01988
PODAA NO.
M^
FORM
r^^^.
DD6, 60m,
LD
UNIVERSITY OF CALI FORN A, BERKELEY
/83
BERKELEY, CA 94720
I
1
21A-60m.-4,'64
(E4555sl0)476B
Universiiy
Berkeley
PAMPHLET BINDER
Syrocuse, N. Y.
GENERAL LIBRARY
-
U.C.
BERKELEY
Stockton, Calif.
BDQDafl^filfl