1. Keith Rodgers
Dr Avinash Sathaye
MA 330
3-7-14
Pre-Newtonian Calculus
Calculus, the method used to obtain approximations of areas, curve lengths and instantaneous
rates of change, is probably the most important process utilized in modern mathematics and has,
perhaps surprisingly given its sophistication, a long history. Most people, if they know anything
about the discovery of calculus, credit Sir Isaac Newton; fewer people are aware that Newton’s
contemporary, Gottfried Wilhelm von Leibniz, also developed a system of calculus. Even
mathematicians frequently say that Newton and Leibniz invented calculus and countless students
are taught this as fact. However, this is an oversimplification which ignores the work of several
earlier cultures which not only understood both derivatives and integrals, the two mainstays of
calculus, but even devised formulas to produce them.
Most of the early forms of calculus were more abstract than practical, in comparison to the
applications of modern calculus which span multiple fields as diverse as the natural sciences,
physics, engineering, construction and economics. Calculus has produced many of the advances
in astronomical knowledge, ranging from calculating the orbits of planets to computing the
necessary amount of fuel for rocket flight (Whiteoak). In physics, the velocity of an object can be
obtained from the object’s position function and its acceleration from the velocity function.
Conversely, if the acceleration function is known, then the velocity can be obtained and also the
position from the velocity function. Other natural sciences use calculus to construct models of
chemical reactions, population growth and radioactive decay. Engineering, too, utilizes calculus
for such tasks as determining distances, such as the length of a hanging cable, or the volume and

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surface area of multiple shapes, including domes. Social sciences also depend on calculus; for
instance, in economics, instantaneous rates are used to ascertain how fast a company’s cost
increases as its output increases. In addition, economists can establish how to minimize a
company’s cost or maximize its profit. In manufacturing, profit can be increased by knowing
how to minimize the surface area of a product with a given volume or how to maximize the area
of shape with a certain amount of material. With the wide range of applications of calculus, it is
safe to say it permeates all areas of modern life, from the production levels of a business to the
construction of bridges to interplanetary exploration.
Sir Isaac Newton and Gottfried Wilhelm von Leibniz both devised calculus independently,
although Newton discovered it first, in order to be able to determine instantaneous rates of
change of a function and to find the area under a given curve over a certain interval. However,
methods which demonstrate basic concepts used in calculus appeared before the work of these
two mathematicians. Their European contemporaries already had some ideas pertaining to
calculus and, indeed, calculus was an active area of research (Grabiner 218). Some related
concepts had actually emerged as far back as ancient Greece, and later in China, the Arabic
world and India. As Newton famously said, “If I have seen further than others, it is because I was
standing on the shoulders of giants,” which is certainly applicable to the invention of calculus
(“Isaac Newton Quotes”). However, the fact that their methods were predated by others does not
lessen the genius of Newton and Leibniz as can be seen by their work in multiple fields.
Neil DeGrasse Tyson, director of the Haden Planetarium and renowned science popularizer,
has said of Sir Isaac Newton, who lived from 1642 to 1727 (“Isaac Newton”), that “He
discovered the laws of optics, the laws of motion and the universal law of gravitation and
invented integral and differential calculus. Then he turned twenty-six” (“Neil DeGrasse Tyson”).

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Newton also won induction in 1671 into the newly formed Royal Society for his invention of the
reflecting telescope and he eventually became the president of the organization (“Isaac
Newton”). He was given a knighthood by Queen Anne in 1705, the first scientist to receive this
honor (Morley).
Newton’s co-discoverer of calculus, Gottfried Wilhelm von Leibniz of Germany (1646-
1716), also made contributions in many subjects (Burnham). He improved both Blaise Pascal’s
calculating machine and the existing binary system and expressed π/4 as an infinite series (Roy).
In addition, his work on logic resulted in the principle of non-contradiction which states that if an
assumption results in a contradiction, then the given assumption must be false (Burnham), an
idea which is still frequently used in mathematical proofs. Leibniz also invented a water pump
powered by windmills and speculated the Earth was formed in a molten state (Burnham). At the
time of his death, Leibniz was the leading German intellectual (Burnham).
EUROPE BEFORE NEWTON
In order to understand the history of European pre-Newtonian calculus, it is first necessary to
define derivatives and integrals, the foundation of calculus. The derivative of a function, f(x), is a
function itself, often denoted by f’(x), which gives the slope of the tangent line at a specific point
on the original function. The tangent line at a point on a curve, touches the curve only at that
point and travels in the same direction as the curve; therefore, the slope of the tangent line is the
instantaneous rate of change of the curve at that point. Integration is the inverse process of
differentiation and yields the area under the curve. An integral of a function, f(x), is represented
by ʃf(x)dx and is also a function F(x), such that F’(x) is equal to f(x). The integral of f(x) over an
interval [a,b], is expressed as ʃa
bf(x)dx and equals F(b) – F(a), where F(x) is an integral, or

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antiderivative, of f(x). This so-called ‘definite integral’ gives the area under the curve f(x) from a
to b.
Even though Newton formulated calculus in the mid -1660s and published his discoveries in
Philosophiae Naturalis Principia Mathematica in 1687 (“Isaac Newton”), in the 1630s two
French mathematicians, Pierre de Fermat and Gilles de Roberval, had determined that the area
under the curve y=xk, for k an integer and x ranging from 0 to a, is equal to ak+1/(k + 1). They
calculated the area by dividing it into an increasing number of rectangles, a method which was
later perfected by Newton and Leibniz (Katz 123). Another discovery about integration came
from Blaise Pascal, of Pascal’s Calculator fame, who wrote in 1658 the equation which in
modern notation becomes the integral ʃa
b sin(x)dx = cos(a) - cos(b) (Roy 120).
Work on differentiation was also occurring, originating from the efforts of mathematicians
trying to find maxima and minima of functions. In 1659, Dutch mathematician Johann Hudde,
who also served as the mayor of Amsterdam and governor of the Dutch East India Company,
stated that for any polynomial, ∑n
k=1 akxk, a maximum or minimum occurs whenever
∑n
k=1 kakxk-1 = 0 (Grabiner 220). The second sum is now known to be the derivative of the
polynomial. Hudde was correct in that whenever local maxima and minima occur, the derivative
of a polynomial is zero; however, the converse is not always true and sometimes the derivative is
simply zero because the curve is flat at a particular spot. Around that time, mathematicians,
including Fermat, realized that in order to find extrema of functions they should find the slope of
the function’s tangent line and set it to zero (Grabiner 220). In fact, the theorem that states if a
function, f(x) has a local maximum or minimum value at c and f’(c) exists, then f’(c) = 0, is still
known as Fermat’s Theorem (Stewart 207). Fermat and his European contemporaries also
discovered that in order to find the slope of the tangent line of a function, they should take the

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slope between two points on the function and move those points infinitely close to each other.
Fermat and others, such as the renowned mathematician and philosopher Rene Descartes and
Isaac Barrow, one of Newton’s teachers at Cambridge University, managed to find tangent lines
for certain functions (Grabiner 220). The great achievement of Newton and Leibniz was to
devise systematic rules to compute both integrals and derivatives turning calculus into a cohesive
body of knowledge. However, ideas relating to calculus extend much further back than the 17th
century: over 2,000 years earlier, the ancient Greeks invented a form of the concept which is the
cornerstone of calculus and which Newton, Leibniz and other Europeans used for their work in
calculus.
ANCIENT GREECE
The kernel of calculus is the idea of a limit, or using approximations to obtain a value that is
very close to some true value. For a function, f(x), a limit can be defined in the following way:
limx→cf(x) = A on the condition that given any є > 0, there exists δ > 0 such that 0 < |x-a |< δ
implies |f(x)-A|< є (Abbott 104). In other words, it is possible to create an interval centered at A
for which the dependent variable is as close as desired to A by restricting x to a sufficiently small
interval centered at a.
δ,є limit (“Advanced Calculus”)

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The precise δ, є definition of limits is relatively recent, devised in the mid-19th century by the
French mathematician Augustin-Louis Cauchy, 1789-1857, who was also a great innovator in set
theory (Stewart 91). Another older and less exact way to represent a limit is geometrically, for
instance by taking the slope between two points as the distance between them becomes infinitely
small. Another geometric method of describing limits is by dividing a certain shape, u, into a
number of other shapes, v. As the number of copies of v increases, the shape u is replicated more
closely. The figure below shows a polygon with an increasing number of sides inscribed in a
circle and one circumscribed outside the circle. In this case, u is the circle’s circumference and v
is the side of the polygon. This is the root of integration which Newton and Leibniz used to solve
the area problem. However, long before the time of Newton and Leibniz, the ancient Greeks also
used this form of geometric limits to solve problems of areas and volumes and lengths.
Geometric Form of a Limit (“The method of Exhaustion Pi”)
This practice was known as the Method of Exhaustion, first discovered by Antiphon, of
whom there is little historical information, in the late 5th century B.C. while he was attempting to
devise a formula for constructing a square with the same area as a given circle (“The Method of
Exhaustion and Limit”). This occurred during the age of Classical Greece (600-300 B.C.) which

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produced such mathematical greats as Pythagoras (Joseph 19). Many Greek mathematicians used
the Method of Exhaustion, including the man who is frequently viewed as the greatest
mathematician of all time.
Archimedes, born on the Greek island of Sicily in 287 B.C., was variously referred to as “the
wise one,” and “the great geometer” (“Archimedes”). In his youth, he attended school in
Alexandria and was taught by the followers of Euclid (“Archimedes”). Archimedes made many
contributions to mathematics and physics including explaining the principle of the lever which
he then used to invent the Archimedian screw for transferring water, approximating the square
root of three and even attempting a calculation of the number of grains of sand which could fit
into the known universe (“Archimedes”). He is perhaps most famous for using the principle of
displacement to prove that King Hiero’s jeweler had given the king a crown made partially of
silver instead of gold, keeping some of the apportioned gold for himself (“Archimedes”).
Archimedes’ passion for mathematics proved to be too ardent when, at age seventy-five in 212
B.C., Romans invaded Sicily and Archimedes refused to come with a soldier who captured him
until he finished a problem (“Archimedes”). Instead of waiting, the soldier simply killed him.
Archimedes used the Method of Exhaustion to find the volume of a cylinder and a sphere, and
to show that there exists a triangle in any parabola with exactly three quarters of the parabola’s
area (“The Mathematical Achievements”). He also used geometric approximations to estimate
the value of pi (“Archimedes’ Method of Exhaustion”). His approach is illustrated in the picture
on the next page. One polygon with identical sides is inscribed in the circle and another
circumscribed outside the circle. The polygon shown is an octagon, but the number of sides, n,
can be arbitrarily large. Thus, upper and lower bounds for the circumference of the circle are
achieved and the accuracy of the estimates improves as the number of sides of the polygons

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increases. In the case shown, the diameter is one, so the circumference equals pi. To find the
circumferences of the polygons, it is necessary to find a formula for the side of each polygon.
Example of Archimedes’ Limit for the Calculation of Pi (“Archimedes’ Exhaustion”)
Archimedes first created a right triangle using the length of a side, SI, of the inscribed
polygon. The circle has center (0,0), in modern Cartesian coordinates, so x2 + y2 = r2. Therefore,
(AC)2 + (BC)2 = (x + r)2 + y2 + (x-r)2 + y2 = 2(x2 + y2) + 2r2 = 4r2 = d2, which satisfies the
Pythagorean Theorem, meaning the triangle is indeed a right triangle.
Right Triangle for a Section of Inscribed Polygon (“Archimedes’ Exhaustion”)

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Then Archimedes created another triangle within the right triangle using the central angle of
the polygon, 2α, and SI. The point B is in the center of the circle. The two remaining angles of
BDC are identical since they are both opposite a side of length r. Angle ADC equals 90° as it is a
right angle, so angle ADB is 90° - β. Angle ABD clearly measures 180° - 2α, as the neighboring
angles form a line. The unknown angle equals 90° - β as, again, both angles are opposite a side
of length r. Summing the angles of the triangle ADB, 2α + 2β = 180°, meaning 90° - β = α.
Therefore, the unknown angle is α, where α is 180°/n, and sin(α) = SI/1 = SI. Therefore, the
circumference of inscribed polygon is nsin(α).
Formula for Angles within the Triangle (“Archimedes’ Exhaustion”)
Rotating the polygon circumscribing the circle, as shown on the next page, with side length
Tn, (1/2)Tn forms a right angle with the radius, so tan(α) = (1/2)Tn/(1/2) = Tn. Therefore, the
upper bound for pi is ntan(α). Using n = 12, 24, 48 and, eventually, 96, Archimedes concluded:
“the ratio of the circumference of any circle to its diameter is less than 31/7 but greater than
310/71”(“Archimedes’ Exhaustion”). Using decimals approximations, 3.1408 < π < 3.1457. Both
estimates are accurate to two decimal places.

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Formula for Side of Circumscribing Polygon (“Archimedes’ Exhaustion”)
In Newtonian calculus, a process analogous to the Method of Exhaustion is used to find
the area under a curve given by a function, f(x), in a certain interval by approximating that area
with a number of rectangles. The x-axis over the region is separated into equally sized segments
by a series of points: x1, x2 … xn. The width of the ith rectangle is xi+1 – xi, or Δx, as the width is the
same for all the rectangles. The height of the rectangle is taken to be f(xi+1) if right endpoints are
used, f(xi) if left endpoints are used and the average of the two if midpoints are used. Thus, if left
endpoints are used, the estimated area is ∑i=1
nf(xi) Δx, where n is the number of rectangles, with
equivalent formulas for the other endpoints (Stewart 301).
Dividing an area into rectangles (“Iterative Methods”)

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The estimate of the area becomes more accurate as the number of rectangles increases and,
eventually, becomes infinitely close to the true area just as Archimedes’ method produces values
which become closer to pi as the number of sides in the polygons increase. More precisely, if the
given area is A, limn→∞∑i=1
nf(xi) Δx = A. It does not matter which type of endpoints are used as
the y values over a given rectangle get infinitely close together as the rectangles’ width becomes
vanishingly small. If the area of interest is on the interval [a,b] along the x-axis, the limit of the
sum is F(b) – F(a), where F(x) is an antiderivative of f(x), in other words the infinite sum
converges to the integral of f(x) from a to b. Integrals can also be used for calculating the length
of a curve, as Archimedes did with the circumference of a circle. This is achieved with forms of
integrals called, somewhat misleadingly, line integrals which can easily be derived once the area
formula is known.
Archimedes is not only considered by some historians to be the “father of integral calculus”
(Salerno), but was also the first known person to use tangent lines (“The Mathematical
Achievements”). His rules for constructing a tangent line to a spiral state that for any point P on
the first revolution of the spiral (he gave other rules for the later revolutions), make a circle
centered at the origin of the spiral with radius OP. Then construct a line segment, OT,
perpendicular to OP which connects the origin to the tangent line at P and connect a pole from
the origin to the point where the spiral has completed a full revolution. The arc length between P
and K is the same as the length of OT (“The Mathematical Achievements”). Thus, there are two
known points on the line, T and P, which, by definition, completely describe that line, shown on
the next page. Archimedes’ work on tangent lines was mainly limited to spirals, whereas Newton
and Leibniz were able to find rules for determining the derivatives of most basic functions, but
Archimedes still foreshadowed their ideas by almost two millennia.

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Archimedes’ Tangent Line to a Point on a Spiral (“The Mathematical Achievements”)
CHINA
The ancient Chinese also used a method similar to the Method of Exhaustion, representing a
limit. Liu Hui, one of ancient China’s greatest mathematicians, lived in the northern Wei
kingdom in the third century A.D. and developed the earliest known Chinese mathematical
proofs (Straffin 69). He is best known for his written treatise in 263 A.D. on a major
mathematical work that contained 246 problems, now known as “The Nine Chapters on the
Mathematical Art” (Straffin 69, 71). Lui used a 192-sided polygon to find the area of a circle and
approximate pi, double the 96 sides of Archimedes’ polygon (Straffin 76). Unlike Archimedes,
Liu only used an inscribed polygon to estimate pi. He approximated the area between the
polygon and the circle with triangles and then transposed those triangles to the outside of the
circle to obtain an upper bound for the circumference (Straffin 76).
First, Lui used a hexagon, with side length M, inscribed within the circle, as shown on the
next page, to obtain the lower bound for a circle’s circumference (Straffin 75-6). Doubling the
number of sides of the polygon, each section of the new polygon is divided into two right
triangles, OAP and APC. The red triangle has side lengths r, G and M/2, while the green triangle
has lengths m, M/2 and j, where j is the circle’s radius minus G. Using the Gou-gu Theorem,

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equivalent to Pythagoras’ Theorem, G2 = r2 – (M/2)2, so G = sqrt(r2 – (M/2)2). The Gou-gu
Theorem also gives m2 = (M/2)2 + j2, where j = r – G = r – sqrt(r2 - (M/2)2). Thus, if the length of
a side of a polygon inscribed within a circle is known, it is possible to find the length of a side of
an inscribed polygon with twice as many sides (Straffin 75 - 6). When a hexagon is inscribed in a
circle, any triangle formed by a side of the polygon and the center of the circle, such as the
triangle OAB, is an equilateral triangle with side length r. Doubling the number of sides five
times to create a polygon with 192 sides, Lui obtained the following bounds: 3.1410 < π <
3.1427 (Straffin 76).
Liu Hui’s π Algorithm (“Liu Hui”)
Two centuries after this estimate, Zu Chongzhi (429 – 500 A.D.) used a polygon with 24,576
sides to obtain the values: 3.1415926 < π < 3.1415927 (Straffin 76). His son, Zu Gengzhi, also
made an important contribution to integration by developing the formula for the volume of a
sphere (Straffin 79). Interestingly, the framework for the sphere problem was laid by Lui Hui
when he considered two perpendicular cylinders inscribed in a cube with side length d (Straffin
79). Both cylinders also have diameter d and a sphere with diameter d is simultaneously
inscribed in both cylinders, as shown on the next page. Lui called the intersection of the

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cylinders a “double box-lid” (Straffin 79). He proved that the volume of the sphere is π/4 times
the volume of the double box-lid (Straffin 79), a fact that Zu Gengzhi would later use to find the
sphere’s actual volume by finding the box-lid’s volume.
Cross Sections of a Sphere within a Double Box-lid inside a Cube (Straffin 79).
Zu’s method was to take one-eighth of the box-lid inscribed within the new cube with length
d/2 = r, as shown in the figure on the next page (Straffin 79). Zu then bisected the cube with a
plane at height h so that a square with length r was formed. At the same time a square, with
length s, was created from the intersection of the plane and the box-lid. Forming a right triangle
from h, r and s, and using the Gou-gu Theorem, s2 + h2 = r2, or h2 = r2 – s2 (Straffin 79). Thus,
the volume of the L-shape, or gnomon, outside the box-lid has area h2.
Zu then compared the eighth of the box-lid to an inverted yangma, or square pyramid, with
height equal to a side of the base, pictured on the next page. If a plane at height h is passed
through the yangma, the resulting cross-section also has area h2
, as it forms the base of a new
yangma with height h. Thus, a cross-section of the two solids have the same area at any given
height h. Zu reasoned that “If volumes are constructed of piled up blocks [areas], and

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corresponding areas are equal, then the volumes cannot be unequal” (Straffin 80). This statement
foreshadows the later idea of triple integrals. This idea would be reproduced by the 17th century
Italian, Bonaventura Cavalieri, and is now known as Cavalieri’s Principle (Straffin 80).
Yangma (Straffin 80). Cross Section 1/8 of the Box-lid Inscribed in
the Cube with Length r (Straffin 80).
By Zu’s reasoning, the volumes of the yangma and the space outside the box-lid are
identical. The volume of the yangma is (1/3)r3, so the volume outside the box-lid is also (1/3)r3
and the volume within the box-lid must then be (2/3)r3 (Straffin 80). Combining all eight parts of
the box-lid, its total volume is (16/3)r3. Using Liu Hui’s discovery that the volume of the sphere
is π/4 times the volume of the box lid, the sphere’s volume is (4/3)πr3 (Straffin 80). The work of
Liu Hui and Zu Gengzhi shows that the ancient Chinese used integration in similar ways to the
ancient Greeks, obtaining two of Archimedes major discoveries: the value of pi and the volume
of a sphere. However, unlike Archimedes, they do not appear to have used any form of
differential calculus.

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MEDIEVAL ISLAMIC WORLD
Work on calculus also occurred later in the medieval Islamic world and one great
mathematician used even more sophisticated methods of integration than the ancient Greeks and
Chinese in order to evaluate the volume of a paraboloid (Katz 125). Abu al-Hasan ibn al-
Haytham, who is frequently referred to by his Latinized name Alhazen, was born in 10th century
Persia at a time when the Arabic world was ∙experiencing an age of artistic and scientific
innovation (Bressoud, “Newton”). Alhazen moved to Egypt around the year 1000 and worked in
connection with the recently founded University of Al-Azhar where he was productive in many
fields including optics, astronomy and, especially, mathematics ((Bressoud, “Newton”). He
developed formulas for the sums of infinite series (Katz), a procedure which later became the
basis of one of the fundamental theorems of calculus. These formulas are for the sums of the
series of the first integer powers shown below:
∑n
i=1 i = n(n + 1)/2
∑n
i=1 i2 = n(n + 1)(2n + 1)/6
∑n
i=1 i3 = [n(n + 1)/2]2 (Stewart A37).
Others, including Archimedes, had used these series before, but Alhazen found the
summation of ∑n
i=1 i4 (Katz). To achieve this result, he first demonstrated that
(n + 1)∑n
i=1 ik = ∑n
1=1 ik + 1 + ∑n
p=1(∑n
i=1ik), and then solved for ∑n
1=1 ik + 1. He showed this to be
true for k = 3 and n = 4 by assuming the statement was true for n = 3 and showing it must then be
true for n = 4. The following proof, for k = 3 and n = 4, is generalizable to any integers k and n
even though Alhazen did not calculate the summation for any k greater than 4:

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(4 + 1)(13 + 23 + 33 + 43)
= 4(13 + 23 + 33 + 43) + 13 + 23 + 33 + 43
= 4(43) +4(13 + 23 + 33) + 13 + 23 + 33 + 43
= 44 + (3 + 1)(13 + 23 + 33) + 13 + 23 + 33 + 34 (Katz 124).
Since it is given that the equation for the sum is valid for n = 3, then (3 + 1)(13 + 23 + 33) = 14 +
24 + 34 + (13 + 23 + 33) + (13 + 23) + 13. Substituting the result into the summation equation, the
formula is derived for n = 4 (Katz). Alhazen had used an early form of proof by induction, one of
the main types of proof in modern mathematics: assume a statement is true for the nth case and
then show it must be true for the (n + 1)th case.
Armed with these formulas, Alhazen evaluated an integral of a fourth degree polynomial,
quite likely the first person in history to do so ((Bressoud, “Newton”). He examined the volume
of the paraboloid formed by rotating the region bounded by the polynomial x = ky2, the x-axis
and the line x = kb2 around the same line (Katz 125). The resulting solid is also pictured.
Region bounded by x=ky2 and x-axis (Katz 125)

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Alhazen’s Paraboloid (Bressoud, “Newton”)
Alhazen’s approach was to divide the region into disks and take the volume of each disk
using the formula V = πr2h, where r is the disk’s radius and h is the thickness, a technique still
common in modern integration. y goes from 0 to b, so if there are n disks, h = b/n. The radius of
a disk in the paraboloid is the constant x-value, kb2, minus the x-value on the curve at a
particular height, k(ih)2 for the ith disk. Thus, the ith disk has volume πh(kh2 – ki2h2)2 which
simplifies to πk2h5(n2 – i2)2. Alhazen then compared the volume of the paraboloid with that of a
cylinder with radius kb2 and height b (Katz 125). Dividing the cylinder into n disks and
substituting variables, the volume of a typical slice of the cylinder is πk2h5n4. Therefore, the total
volume of the cylinder becomes πk2h5n5 and the volume, less than the top slice, is πk2h5(n-1)n4.
Similarly, using his summation formulas, the first n-1 disks of the paraboloid have the volume: V
= πk2h5∑i=1
n-1(n2 – i2)2 = πk2h5(8/15n*n4 -1/2n4 -1/30n). Subtracting negative terms, Alhazen
discovered that πk2h5(8/15(n-1)n4 < πk2h5∑n-1
i=1(n2 – i2)2 < 8/15n*n4. In other words, the volume
of n-1 of the paraboloid’s disks is between 8/15 of the cylinder with n-1 slices and 8/15 of the
whole cylinder. As n get infinitely large, removing the nth disk does not affect the volume of
either shape, so the volume of the paraboloid has 8/15 of the volume of the cylinder (Katz 126).

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Others in the medieval Middle East also made significant contributions to calculus. These
included another Persian, Sharaf al-Din al Tusi of the twelfth century, a well-known astronomer
and astrologer as well as a mathematician. He discovered that the derivative of a third degree
polynomial ax3 + bx2 + cx + d = 3x2 + 2bx + c (Saladino). Nowadays, this is recognizable as a
specific case of the power rule of derivatives. The Islamic world had surpassed earlier ancient
Greece and China with its use of integration and greatly outstripped the Greeks’ scant use of
differential calculus.
INDIA
There was one civilization that was able to rival the medieval Islamic world in its
understanding of calculus: India. By at least 500 A.D., the Indians were using limits to estimate
pi, with a method similar to that of Archimedes’ Method of Exhaustion, and were also using
limits to derive formulas for various volumes (“Development of Calculus”). Another
achievement at this time, was using limits to find that the sum of the geometric series a + ar + ar2
+ ar3 + … equals a/(1 - r) when the absolute value of r is less than one (“Development of
Calculus”). The Indians also had a great interest in astronomy and used trigonometry for making
measurements of planetary motion (Bressoud, “India” 131).
Stemming from this early use of trigonometry, later studies used limits to divide a section of
a circle into an increasing number of triangles and express trigonometric functions as lengths of
those triangles’ sides. Thus, they created the following infinite series, still in use in modern
mathematics, to express trigonometric functions:
Sin(x) = x – x3/3! + x5/5! – x7/7! + …, where n! = n(n-1)(n-2) …(1).
Cos(x) = 1 –x2/2! + x4/4! – x6/6! + …

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Arctan(x) = x – x2/2 + x3/3 – x4/4 + … (Bressoud, “India” 131).
It is clear that these formulas were known by 1500 A.D. and, unusually for Indian
mathematical works of the time, there are detailed explanations for the derivations (Katz 126).
Incidentally, one of the later innovators of calculus, Leibniz, used the arctangent series,
discovered independently in Europe, to express π/4 as an infinite series, as the arctangent of 1 is
π/4 (Burnham).
In addition to using integration through the application of limits, the Indians developed forms
of derivatives for trigonometric functions at a very early stage and even used them to solve
equations. One of these early pioneers was the mathematician and astronomer, Brahmagupta
(597-668 A.D.), who, in addition to his trigonometric work, also devised rules for performing
arithmetic with the number zero and was the first to clearly state the quadratic formula (Mastin).
Brahmagupta used derivatives to obtain a formula for approximating sin(x + є), enabling the
estimation of the sine of an angle when it was not possible to calculate the exact value (Bressoud,
“India” 134). Brahmagupta used (sin(x + α) – sin(x – α))/2α, where α is as small as possible, to
estimate the derivative of sin(x). He then arrived at the second derivative of sin(x) as (sin(x + α)
– 2sin(x) + sin(x - α))/α2. It is not known how he formulated the second derivative, but it is
believed he used trigonometric identities developed by an even earlier Indian mathematician,
Aryabhata, born in 476 A.D. (Bressoud, “India” 134). He went on to determine that sin(x + є) is
approximately equal to sin(x) + єf ’(x) + (є2/2)f ’’(x).
Formulas for trigonometric integrals were discovered about 1,000 years later in the 16th
century, by the mathematician Jyesthadeva (circa 1500 – 1610), who, unlike most contemporary
Indian mathematicians, wrote many detailed mathematical proofs (Bressoud, “India” 136).
Jyesthadeva wrote what, in modern notation, states sin(α) = ʃ0
α cos(x)dx and

21. 21
cos(α) = 1 - ʃ0
αsin(x)dx (Bressoud, “India” 136). Evaluating the integrals shows that an integral
of cos(x) is sin(x) and an integral of sin(x) is –cos(x), which is now known to be correct.
As well as taking derivatives of trigonometric functions, Indians also had the concept of
instantaneous velocity in the 6th century B.C. (“Ideas of Physical Forces”). They used the process
of interpolation, creating a curve given known points on that curve, for calculating instantaneous
velocity and for much of their other work with derivatives (“Development of Calculus”).
Equations to determine instantaneous velocity of planets were developed using variables such as
a planet’s longitude and distance from the Earth, foreshadowing the later technique of using the
position function to derive velocity (“Development of Calculus”). With their knowledge of
instantaneous velocity, the Indians used a primitive form of differential equations where
derivatives are treated as variables, a branch of calculus widely used today in engineering and
physics. Mathematicians used differential equations as early as the 6th century A.D. to calculate
positions of planets (“Ideas of Physical Forces”). Instantaneous velocity was also used to
determine when a planet’s motion changes from pro-grade to retrograde. Pro-grade motion is
defined as movement in the direction of some other object whereas retrograde motion is
movement away from that object. Between the two different types of relative motion, there are
points where the planet appears to stand still in the sky. Differential equations with instantaneous
velocity were used to predict these stationary times (“Development of Calculus”).
The twelfth century mathematician, Braskara II, was the head of the astronomical observatory
at Ujjain, the foremost mathematical institute of medieval India due to the interconnection of
mathematics and astronomy (Saladino). He used derivatives quite extensively and understood
them at sufficiently advanced levels to discover what is now known as Rolle’s Theorem after its

22. 22
later European inventor, 17th century Michel Rolle (Saladino). The theorem states if a function
f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) and if
f(a) = f(b), then for some number c in the open interval (a,b), f’(c) = 0 (Stewart 214-5). If f(x) is
a constant function the derivative is always zero. In other cases, the theorem can, nowadays, be
proven using the Extreme Value Theorem which makes the following rather intuitive statement:
if f(x) is continuous on [a,b], then f reaches an absolute maximum and minimum value on the
interval (Stewart 206). Since f is not constant, then either f(x) > f(a) or f(x) < f(a) for some x in
(a,b). If f(x) > f(a), and therefore f(x) > f(b), the absolute maximum must occur in (a,b), so the
function will change from increasing to decreasing. This is the same as saying f has a local
maximum in (a,b), meaning for some c in (a,b), f’(c) = 0 (Stewart 207). Similarly, if f(x) < f(a), f
has a local minimum in (a.,b). Rolle’s Theorem is illustrated in the picture. Although this proof
was almost certainly unknown in India, the knowledge of the theorem shows India had a
continuous history of using calculus extending over many centuries. However, their
achievements were not recognized by historians until the 19th century (Katz 137).
Illustration of Rolle’s Theorem (“Hyun Numbers”)

23. 23
CONCLUSION
It is not uncommon to mistakenly give a person, or a small number of people, credit for
originating an idea when, instead, they revolutionized understanding of that notion. Far-reaching
concepts are rarely spontaneously generated. For instance, it is commonly believed that Charles
Darwin had the original idea of evolution, when in fact the idea of species changing into other
species extends back to the ancient Greek philosopher, Anaximander in the 5th century B.C., over
two millennia before Darwin was born (Cohen). Similarly, around the time of Anaximander, the
ancient Greeks used early forms of integration and the Indians had already developed the concept
of instantaneous velocity. Long before Newton and Leibniz, several civilizations used calculus to
solve many problems that, later, post-Newtonian calculus would be used for, including
determining volumes of solids and lengths of curves. Newton and Leibniz are rightly viewed as
great geniuses as they did merge the concepts of calculus into a precise mathematical art and
established the connection between integration and differentiation. However, previous use of
calculus spanned a couple of millennia and several thousand miles. Therefore, it seems that there
should be a re-examination of the traditional view of the history of calculus, given the copious
and erudite results of earlier mathematicians. The achievements of Newton’s “giants”, such as
Archimedes, Hui Lui, Alhazen, Jyesthadeva and Fermat, should not go unrecognized.

24. 24
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