Calculus in Indian Mathematics

Calculus is usually said to begin with Newton and Leibniz in the 17th century. Yet Indian astronomers had met some of its central ideas much earlier: the speed of a planet at a single instant, the rate at which the sine of an angle changes, a curved surface cut into thin strips, and infinite series. On this page we follow these ideas from Manjula (10th century) and Bhaskara II (12th century) to Madhava and the Kerala school (14th–16th centuries).

On this page: Why astronomers needed it · Instantaneous motion · The differential of sine · Bhaskara's proof · Maxima and later results · The surface of a sphere · Madhava's series · The priority controversy

Why Astronomers Needed It

Indian astronomers had to predict eclipses and conjunctions of planets accurately. Eclipses were occasions for fasting and religious rites, so a correct prediction was a public test of the astronomer's skill.

For such predictions it is not enough to know where a planet is today and where it will be tomorrow. One needs its true speed at a given moment, because the Moon and the planets do not move uniformly across the sky.

The astronomers described the motion of a planet with three quantities: the mean longitude $v$ (where the planet would be if it moved uniformly), the mean anomaly $w$ (an angle that measures how far the planet is along its orbit from the apogee), and the true longitude $u$ (where it really is). To a good approximation

${u}={v\pm e\sin w}$,

where $e$ is a constant of the orbit (the largest correction, called the greatest equation of the centre) and the sign depends on how the anomaly is measured. If the three quantities change to $u'$, $v'$ and $w'$ a little later, the planet's true motion in that interval is

${u'-u}={v'-v}\pm e\left(\sin w'-\sin w\right)$. (1)

Aryabhata (born 476) and Brahmagupta (7th century) already used this formula. The difficulty is the sine difference ${\sin w'-\sin w}$. Their sine tables gave values only every ${3^\circ 45'}$, and for angles in between they had to interpolate, which was not accurate enough.

Instantaneous Motion

The Sanskrit term for the true speed at a moment is tatkalika-gati, "motion at that time". Bhaskara II explains it in his Siddhanta Shiromani (1150) by comparing it with the ordinary daily motion:

The difference between a planet's longitudes at the same hour on two consecutive days is its rough motion (sthula-gati) for that day.
The fine or instantaneous motion (sukshma tatkalika-gati) is the motion the planet would have if its speed stayed the same as at the chosen moment.

To get the fine motion, Bhaskara says, the interval must be taken very small. He links it to a kshana, which Indian texts describe as an immeasurably small instant of time. This is close to the modern idea of instantaneous velocity: the speed at one moment, obtained from smaller and smaller time intervals.

In modern language, if a quantity $u$ changes by a small amount $\delta u$ during a small time $\delta t$, the instantaneous speed is the limit of ${\frac{\delta u}{\delta t}}$. Bhaskara had no limits, but the idea of a "speed at an instant" is clearly there.

The Differential of Sine

The astronomer Manjula (932) was the first known Indian author to replace the sine difference in formula (1) by a simple product. For a small change of the angle he used

${\sin w'-\sin w}\approx{\left(w'-w\right)\cos w}$

(with the angles measured in radians). With it formula (1) becomes

${u'-u}={v'-v}\pm{e\left(w'-w\right)\cos w}$.

In the notation of calculus this is ${\delta u}={\delta v\pm e\cos w\,\delta w}$, and the rule behind it is the derivative of sine:

${\delta(\sin\theta)}={\cos\theta\,\delta\theta}$, that is, ${\left(\sin\theta\right)'}={\cos\theta}$.

We do not know how Manjula found the rule. It appears again in the works of Aryabhata II (around 950), Bhaskara II and later astronomers.

Example. Let ${\theta=30^\circ}$ and ${\delta\theta=1^\circ}$. One degree in radians is ${\frac{\pi}{180}\approx 0.0174533}$.

Exact difference: ${\sin 31^\circ-\sin 30^\circ}\approx{0.515038-0.5}={0.015038}$.

Manjula's rule: ${\cos 30^\circ\cdot 0.0174533}\approx{0.866025\cdot 0.0174533}\approx{0.015115}$.

The two values differ by about 0.5%. The smaller the step, the better the agreement, which is exactly what the astronomers needed.

Bhaskara's Proof

Bhaskara II shows where the rule comes from with a geometric argument. Let a point move counterclockwise on a circle with centre $O$ and radius $R$. Indian astronomers worked with the Rsine ${R\sin\theta}$, the half-chord in a circle of radius $R$, rather than with the sine of a unit circle.

A quarter circle with centre O, the points P and Q on the arc and the point T on the tangent at P

The point moves from $P$ to $Q$. $PT$ lies on the tangent at $P$ and has the same length as the arc $PQ$.

Let ${\angle BOP=w}$ and ${\angle BOQ=w'}$. The perpendiculars $PM$, $QN$ and $TS$ fall on $OB$, and the points $n$ and $r$ on $QN$ and $TS$ are at the height of $P$.

The real increase of the Rsine is ${Qn}={QN-PM}={R\left(\sin w'-\sin w\right)}$. Bhaskara calls such a difference bhogya-khanda.
If the point did not turn but went on along the tangent for the same distance, it would reach $T$. The increase of height would then be ${Tr}={TS-PM}$. Bhaskara calls it the instantaneous sine difference (tatkalika bhogya-khanda).

Bhaskara noticed that the real differences ${R\sin A-R\sin 0}$, ${R\sin 2A-R\sin A}$, ${R\sin 3A-R\sin 2A}$, … for equal steps $A$ get smaller and smaller, "on account of the bending of the arc". The tangent segment $PT$ does not bend, and this makes $Tr$ easy to compute.

The tangent $PT$ is perpendicular to the radius $OP$, so the triangles $PTr$ and $OPM$ are similar. Their sides are in proportion:

${\frac{Tr}{PT}}={\frac{OM}{OP}}={\frac{R\cos w}{R}}$, so ${Tr}={PT\cos w}$.

The length $PT$ equals the arc $PQ$, that is, ${PT}={R\left(w'-w\right)}$. When $Q$ is close to $P$, the point $T$ is close to $Q$ and ${Tr\approx Qn}$. Therefore

${R\left(\sin w'-\sin w\right)}\approx{R\left(w'-w\right)\cos w}$.

Dividing by $R$ gives Manjula's rule. Bhaskara used it, among other things, to compute the ayana-valana, an angle needed in eclipse calculations.

Note the clever idea: the instantaneous sine difference is an exact length for any arc, however large. Bhaskara uses a proportion between finite lengths and only at the end remarks that the result is true when the change is very small. He avoids infinitely small quantities, while the idea of an instantaneous rate is clearly present.

Maxima and Later Results

Bhaskara II also made two remarks that we now connect with the theory of maxima:

When a quantity reaches its greatest value, its instantaneous change is zero. Today we say that the derivative is zero at a maximum.
The equation of the centre (the correction ${\pm e\sin w}$) is zero both at the apogee and at the perigee, and between them there is a position where its instantaneous change is zero.

Some historians see in the second remark an early form of Rolle's theorem: if a smooth function takes equal values at two points, its derivative is zero somewhere between them. Others warn that this reads too much into a remark about one particular motion.

Later astronomers of the Kerala school went further. Nilakantha Somayaji (1444–1544), in his commentary on the Aryabhatiya, proved with similar triangles the exact formulas

${\sin(\theta+\delta\theta)-\sin\theta}={2\sin\frac{\delta\theta}{2}\cos\left(\theta+\frac{\delta\theta}{2}\right)}$,

${\cos(\theta+\delta\theta)-\cos\theta}={-2\sin\frac{\delta\theta}{2}\sin\left(\theta+\frac{\delta\theta}{2}\right)}$.

They hold for every $\delta\theta$, large or small. For a small $\delta\theta$ we have ${2\sin\frac{\delta\theta}{2}\approx\delta\theta}$, and the first formula gives Manjula's rule again. In the example above it gives the exact difference: ${2\sin 0.5^\circ\cos 30.5^\circ}\approx{0.015038}$.

Nilakantha also studied second differences, the differences of the differences. In modern form his result for the sine is

${\sin(\theta+h)-2\sin\theta+\sin(\theta-h)}={-4\sin^2\frac{h}{2}\cdot\sin\theta}$.

So the second difference of the sine is proportional to the sine itself with a minus sign. This is the discrete form of the modern fact ${\left(\sin\theta\right)''}={-\sin\theta}$.

The Kerala astronomers also used, in modern notation, the derivative of an inverse sine,

${\delta\left(\arcsin(e\sin w)\right)}={\frac{e\cos w}{\sqrt{1-e^2\sin^2 w}}\,\delta w}$,

and Acyuta Pisharati (about 1550–1621) used what amounts to the derivative of a quotient:

${\delta\left(\frac{e\sin w}{1\pm e\cos w}\right)}={\frac{e\cos w\pm\frac{(e\sin w)^2}{1\pm e\cos w}}{1\pm e\cos w}\,\delta w}$.

Both agree with the rules of modern calculus. As always in these texts, the results are stated for particular astronomical quantities, not as general rules for any function.

The Surface of a Sphere

Integral calculus finds an area by cutting a figure into many thin pieces and adding them up. Bhaskara II used this idea to explain why the surface of a sphere equals circumference × diameter, that is, ${S=2\pi r\cdot 2r=4\pi r^2}$.

He divided the circumference of a great circle into 21,600 minutes of arc. Its radius in the same units is ${R\approx 3438'}$, because ${\frac{21600}{2\pi}\approx 3437.75}$. Starting from the top of the sphere, he drew 24 circles at distances of ${225'}$, ${450'}$, …, ${5400'}$ along the surface. The radius of the $n$-th circle is the Rsine ${T_n=R\sin(n\cdot 225')}$, and its length is

${C_n}={\frac{21600}{3438}\,T_n}$.

Between two neighbouring circles lies a narrow ring of width ${225'}$. Bhaskara treated it as a trapezoid with parallel sides $C_n$ and $C_{n+1}$:

${A_n}\approx{\frac{C_n+C_{n+1}}{2}\cdot 225}$.

Adding all 24 rings of the upper half (with ${C_0=0}$ at the top), each inner circle is counted twice with the factor ${\frac{1}{2}}$ and the last one once:

${\frac{S}{2}}\approx{225\cdot\frac{21600}{3438}}\cdot{\left(T_1+T_2+\dots\right.}{\left.+T_{23}+\frac{1}{2}T_{24}\right)}$.

With the traditional table of 24 Rsines (225, 449, 671, …, 3431, 3438) the sum of all 24 values is 54,233. Since ${\frac{1}{2}T_{24}=1719}$,

${T_1+\dots+T_{23}+\frac{1}{2}T_{24}}={54{,}233-1719}={52{,}514}$,

${\frac{S}{2}}\approx{21600\cdot\frac{225\cdot 52{,}514}{3438}}\approx{21600\cdot 3436.8}$.

The second factor is almost exactly the radius 3438. So half the surface is circumference × radius, and the whole surface is circumference × diameter.

Bhaskara's second method cuts the sphere like the segments of a fruit: into narrow lunes from pole to pole. With a circumference of 96 units, each lune has area ${\frac{52{,}514}{1719}\approx 30.55}$, while the diameter of a circle with circumference 96 is ${\frac{96}{\pi}\approx 30.56}$. Since there are 96 lunes, the surface is again 96 × diameter, that is, circumference × diameter.

For the volume Bhaskara imagined the surface covered with tiny squares, each one the base of a thin pyramid whose apex is at the centre. Each pyramid has volume ${\frac{1}{3}\cdot\text{base}\cdot r}$, so the whole sphere has

${V}={\frac{1}{3}S\cdot r}={\frac{1}{3}\cdot 4\pi r^2\cdot r}={\frac{4}{3}\pi r^3}$.

What is missing compared with integral calculus is the passage to the limit: Bhaskara stops at 24 strips and uses the agreement with the known answer as a check.

Madhava's Series

The boldest step was taken in Kerala, in south-west India. Madhava of Sangamagrama (about 1340–1425) and his followers found infinite series for trigonometric functions and for $\pi$:

${\sin\theta}={\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}+\dots}$

${\cos\theta}={1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\frac{\theta^6}{6!}+\dots}$

${\frac{\pi}{4}}={1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots}$

In Europe these series were found by Newton, Gregory and Leibniz in the second half of the 17th century. Madhava's own writings on them have not survived; historians generally credit the series to him on the evidence of later Kerala texts, such as Nilakantha's Tantrasangraha (1501) and Jyeshthadeva's Yuktibhasha (about 1530), which also explains how the series are derived. The details are on the page Madhava's sine and cosine series.

The Priority Controversy

In 1858 the Indian scholar Bapu Deva Sastri drew attention to the formula ${\delta(\sin\theta)}={\cos\theta\,\delta\theta}$ in Bhaskara's Siddhanta Shiromani and claimed that Bhaskara knew the principles of the differential calculus. His short translation did not make clear the role of the instantaneous motion in Bhaskara's argument.

The English mathematician William Spottiswoode replied that the claim went too far: he saw no use of infinitely small intervals and thought that the approximate character of the result had not been understood. He judged from the translation, without the original text.

Later historians took a middle position, which most still hold today:

Indian astronomers clearly used the idea of an instantaneous rate of change and the rule ${\delta(\sin\theta)}={\cos\theta\,\delta\theta}$, and knew that it is exact only for small changes.
Bhaskara II derived the rule with similar triangles, and the Kerala school found infinite series and studied second differences.
They did not build a general theory: there is no general notion of a function, of a derivative for any function, or of a limit, and no link between differentiation and integration. These came with Newton and Leibniz.

So it is fair to say that important ideas of calculus appeared in India centuries before Europe, while calculus as a unified method was created in 17th-century Europe. Whether any Kerala knowledge reached Europe is an open question; no direct evidence of such a transfer has been found.