The Untold Story of Indian Origins of Claculus

What is calculus? Historiography Origins of calculus in India References
On Indian origins of calculus
Dr V N Krishnachandran
(24 January 2024)

Outline
1 What is calculus?
2 Historiography of Indian origins of calculus
Idea gets proposed
Idea gets traction
Idea gets approval
3 Origins of calculus in India
Limits of sequences
Differentiation
Integration
4 References

Context
Context of the present talk:
National Education Policy 2020
Indian Knowledge Systems (IKS)
UGC Guidelines for Incorporating IKS in Higher Education
Curricula

What is calculus?

Calculus: A narrow definition
Narrow definitions - Examples:
1 Merriam-Webster Dictionary: “The mathematical methods
comprising differential and integral calculus—often used with
‘the’.”
2 Encyclopaedia Britannica: “Calculus, branch of mathematics
concerned with the calculation of instantaneous rates of
change (differential calculus) and the summation of infinitely
many small factors to determine some whole (integral
calculus).”

Calculus: A broad definition
Broad definition (P. P. Divakaran in “The First Textbook of
Calculus: Yuktibhās
.ā” (2007)):
Contemporary mathematics almost unanimously holds the view
that the central concern of calculus is with the properties of
‘functions’. In its early avatars, in India or in Europe, calculus
provided a set of conceptual and technical tools, essentially
geometric, with which to approach and find answers to an array of
questions regarding simple functions. . . . It seems then fair to
characterise the discipline of calculus primarily as this collection of
principles and methods formulated in essence by the pioneers.

Idea gets proposed
Historiography of Indian origins
of calculus

Idea gets proposed
The idea gets proposed

Idea gets proposed
Bapu Deva Shastri (1821—1900)
Bapu Deva Shastri,
a Professor of Mathematics and Astronomy at
the then Government Sanskrit College, Benares
published a paper in 1858 titled Bhāskarā’s
knowledge of the Differential Calculus
in the Journal of the Asiatic Society of Bengal.
(See next slide.)

Idea gets proposed
Bapu Deva Shastri
According to Shastri: “Bhaāskarācārya says that the difference
between the longitudes of a planet found at any time on a certain
day and at the same time on the following day is called its rough
motion during that interval of time; and its Tātkālika motion is its
exact motion.”
Shastri interpreted the term Tātkālika as meaning “instantaneous”
and based on this Shastri claimed that Bhāskarācārya was referring
to the velocity of a planet. He further claimed that Bhāskarācārya
was fully acquainted with the principle of the Differential Calculus.
However, the claims of Shastri were rejected by the then
established European scholars.

Idea gets proposed
Sir Brajendranath Seal (1864–1938)
A few decades later, Sir Brajendranath
Seal (1864–1938), the first chair of the
newly constituted Department of Philosophy at
Calcutta University, in his book “The Positive
Sciences of the Ancient Hindus” published
in 1915 tried to address some of the objections
raised against the claims of Bapu Deva Shastri.

Idea gets proposed
P. C. Sen Gupta (1876—1962)
The observations of Bapu Deva
Shastri was elaborated with more evidences
by P. C. Sengupta of Calcutta University
in a paper titled Infinitesimal Calculus in Indian
Mathematics - Its Origin and Development
published in the year 1932 in the Journal of the
Department of Letters (Calcutta University).
(See next slide.)

Idea gets proposed
P. C. Sen Gupta (1876—1962)
P. C. Sen Gupta observed:
“The idea of the calculus originated among the ancient Indian
astronomers, Āryabhat
.a and his pupils, Brahmagupta, Muñjāla and
Bhāskara in their attempt to determine (i) the ‘instantaneous’, or
‘tātkālika’, daily motion of planets; (ii) the position-angle of the
ecliptic at any secondary to the equator; (iii) the surface and
volume of a sphere; (iv) the value of π.”

Idea gets traction
The idea gets traction

Idea gets traction
C. K. Raju
C. K. Raju in his Cultural
Foundations of Mathematics (p.14) (2007):
“The calculus historically originated in India in
the process of calculating precise trigonometric
values, and the length of the circumference of a
circle, using both infinite and indefinite series.”

Idea gets traction
David Bressoud
David
Bressoud, recipient of Mathematical Association
of America Award for Distinguished Teaching in
1994, asserted in a paper titled “Was calculus
invented in India?” published in 2002:
“No. Calculus was not invented in India.”
(See next slide.)

Idea gets traction
David Bressoud

Idea gets traction
P. P. Divakaran
P. P. Divakaran, a former professor at the Tata
Institute of Fundamental Research, Mumbai
and author of Mathematics in India writes::
“How did this isolated community on the Nila
. . . arrive at such deep insights and forge such
powerful tools as to have anticipated the fine
fruits of European calculus by two centuries and
more? The logically inescapable answer is that
they had invented calculus - since there is
no other route to these results - and this is borne out by the
texts from the period, . . .”

Idea gets approval
The idea gets approval

Idea gets approval
Luke Hodgkin
Luke Hodgkin in his book A History of
Mathematics: from Mesopotamia to Modernity
(p.168) published in 2005 remarks thus:
“ . . . To suppose that some version
of ‘calculus’ underlay the derivation of the series
[Gregory series] must be a matter of conjecture.
The single exception to this generalization
is a long work ... known as Yukti-bhasa
by Jyesthadeva... it does give explanations of
how the formulae are arrived at which could be taken as a
version of the calculus.”

Idea gets approval
Roger L. Cooke
Roger L. Cooke, in his History
of Mathematics: A Brief Course (p.201), whose
third edition was published in 2013, writes:
“The infinite occurs in three forms
in calculus: the derivative, the integral, and
the power series. . . . In India, mathematicians
had used infinite series a few centuries before
the Europeans began to use them, to solve
geometric problems via trigonometry.” Cooke
further observes that, after Brahmagupata and
Bhaskara, “mathematics continued to flourish in India even
anticipating some parts of the calculus”

Idea gets approval
Craig Smoryński
Craig Smoryński in his MVT:
The Most Valuable Theorem (p.208) (2016):
“There is no doubt that the Calculus began
to arise in India long before the European
development got under way . . . However,
if we take the rules of differentiation to be
the ‘principle of the Differential Calculus’, then
we must deny the Indians credit for inventing
the Calculus. They did not have these rules.”

Idea gets approval
David Bressoud, again!
David
Bressoud, who emphatically asserted in 2002
that calculus was not invented in India, wrote
thus in his book Calculus reordered: A history
of the big ideas (pp.55–56) published in 2019:
“Today we write Aryabhata’s relationship as
lim
∆θ→0
∆(sin θ)
∆θ
= cos(θ).
and now we recognize as the rule for the
derivative of the sine.”’
(See next slide)

Idea gets approval
David Bressoud, again!
“One could claim that the first function to be differentiated
was the sine, it happened in India, and it occurred well over
a thousand years before Newton or Leibniz were born.”

Limits of sequences
The origins of calculus in India

Limits of sequences
Limits of sequences

Limits of sequences
How limits are presented
At several places in Yuktibhasha one can see statements of the
following form:
f (n) ≈ a when n is large. (1)
This can be definitely interpreted as an informal way of stating the
following fact:
lim
n→∞
f (n) = a.
(See next slide for examples)

Limits of sequences
Examples of limits
We give below, some examples of statements of the form Eq.(1)
given in Yuktibhās
.ā.
When n is large,
n + (n − 1) + (n − 2) + · · · + 2 + 1
n2
≈
1
2
When n is large,
n2 + (n − 1)2 + (n − 2)2 + · · · + 22 + 12
n3
≈
1
3
.
For any arbitrary positive integer k, when n is large,
nk + (n − 1)k + (n − 2)k + · · · + 2k + 1k
nk+1
≈
1
k + 1
.

Limits of sequences
Examples of limits
Let
V1(n) = n + (n − 1) + · · · + 1
and for k > 1, let
Vk(n) = Vk−1(n) + Vk−1(n − 1) + · · · + Vk−1(1).
.
When n is large,
Vk(n)
nk+1
≈
1
1 · 2 · 3 · · · (k + 1)
.

Differentiation
Differentiation

Differentiation
Aryabhata (476-–550)
According Bressoud, Āryabhat
.a must have been aware of the the
relationship
∆(sin θ) ≈ (∆θ) cos θ.
Bressoud now points out that, in contemporary mathematical
notations, this may be written in the form
lim
∆θ→0
∆(sin θ)
∆θ
= cos θ.
or, equivalently,
d
dθ
(sin θ) = cos θ.

Differentiation
Manjula (fl. 932)
In Laghumānasa, Manjulā states a result which in modern
notations is equivalent to
sin w′
− sin w = (w′
− w) cos w, when w′
− w is small (2)
which can be written using differentials in the following form:
d(sin w) = cos w dw (3)
The result appears also in the works of Āryabhat
.a II (950 CE),
Bhāskara II (1150 CE) and several later writers.

Differentiation
Bhaskara II (1114—1185)
There are further evidences in Bhāskara II’s work to show that he
did actually know the basic principles of differential calculus.
Bhāskara II has explicitly stated the following results related to
differential calculus.
When a variable attains the maximum value its differential
vanishes.
When a planet is either at its furthest from the Earth or at its
closest, the equation of the centre vanishes. Hence for some
intermediate position the differential of the equation of centre
is equal to zero.
In the second of the two results, one can see shadows of the Mean
Value Theorem.

Differentiation
Nı̄lakan
.t
.ha Somayājı̄ ( 1444 – 1545)
The differential coefficient or derivative of the inverse sine function
is given by
d
dx
sin−1
x =
1
√
1 − x2
.
As demonstrated below there are reasons to believe that
Nı̄lakan
.t
.ha Somayājı̄ of the Kerala school of mathematics and
astronomy was aware of this result.

Differentiation
Nı̄lakan
.t
.ha Somayājı̄ ( 1444 – 1545)
The instantaneous velocity of planets is the rate of change of θ
given by
θ = θ0 − sin−1
r0
R
sin(θ0 − θm)

where θ, θ0 and θm are functions of time t and r0 and R are
constants. We have
dθ
dt
=
dθ0
dt
−
r0 cos(θ0 − θm)d(θ0−θm)
dt
q
R2 − r2
0 sin2
(θ0 − θm)
In the case of planets, the rate of change in θm is very small and so
dθm
dt can be neglected. (See next slide)

Differentiation
Nı̄lakan
.t
.ha Somayājı̄ ( 1444 – 1545)
Thus, in the case of planets
dθ
dt
=
dθ0
dt

1 −
r0 cos(θ0 − θm)
q
R2 − r2
0 sin2
(θ0 − θm)


Nı̄lakan
.t
.ha Somayājı̄ has stated this formula precisely in this form
in Tam
. trasam
. graha Chapter 2 verses 43–44 thereby demonstrating
his knowledge of the differential or derivative of the inverse sine
function.

Differentiation
Acyuta Pis
.ārat
.i (1550–1621)
Acyuta Pis
.ārat
.i has used implicitly the quotient rule of
differentiation:
d
dx
u
v

=
v
du
dx
− u
dv
dx
v2
.
(See next slide.)

Differentiation
Acyuta Pis
.ārat
.i considers another model for the instantaneous
velocity of planets. In this model, the instantaneous velocity is the
rate of change of θ given by
θ = θ0 −
r0
R
sin(θ0 − θm)
1 −
r0
R
cos(θ0 − θm)
(See next slide.)

Differentiation
Acyuta Pis
.ārat
.i (1550–1621)
Acyuta Pis
.ārat
.i has given the following expression for the
instantaneous velocity:
dθ
dt
=
dθ0
dt
−
r0
R
cos(θ0 − θm) +
r0
R
sin(θ0 − θm)
2
1 −
r0
R
cos(θ0 − θm)
1 −
r0
R
cos(θ0 − θm)
d
dt
(θ0 − θm)
which is obviously equal to the expression for dθ
dt obtained by
applying the quotient rule of differentiation.

Integration
Integration

Integration
Integration as limit of sum
To state this representation, let us assume that f (x) is a real
valued function of a real variable x satisfying some smoothness
conditions. Let a, b be real numbers with a b and n be a
positive integer. Then the the following are the “limit of a sum
representations” of the definite integral
R b
a f (x) dx:
Z b
a
f (x) dx = lim
n→∞
n−1
X
j=0
f (xj )h = lim
n→∞
n
X
j=1
f (xj )h
where h = (b − a)/n and xj = a + hj.

Integration
Integration
Computation of the surface area of a sphere

Integration
Jyes
.t
.hadeva’s computation of the surface area of a sphere
Figure: Upper hemisphere cut into n strips

Integration
Jyes
.t
.hadeva’s computation of the surface area of a sphere
In the figure, C is the circumference of a great circle of a sphere of
radius R. Jyesthadeva expresses the surface area of the sphere as
S = 2

C
R

lim
n→∞




n−1
X
j=0
Jj

 δ


where δ = 5400′
n , and then evaluates it correctly as S = 2CR. While
doing so he was evaluating the integral 2CR
R π
2
0 sin t dt as the limit
of a sum.

Integration
Integration
Computation of the volume of a sphere

Integration
Jyes
.t
.hadeva’s computation of the volume of a sphere
Figure: Jyes
.t
.hadeva’s derivation of the formula for the volume of a sphere

Integration
Jyes
.t
.hadeva’s computation of the volume of a sphere
In the notations of the figure, Jyeshthadeva expressed the volume
of a sphere of radius R as
V = 1
2
C
R

lim
n→∞
n
X
j=1
1
2
h
(2R)2
−

2R
n j
2
+ 2R
n (n − j)
2
i
2R
n

and evaluated it correctly as V = 2
3CR2. While doing so, he was
expressing the volume of the sphere as the integral
Z 2R
0
1
2

(2R)2
− x2
+ (2R − x)2

dx
and then evaluating the integral as the limit of a sum.

Selected references

Thanks.

The Untold Story of Indian Origins of Claculus

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