Keith RodgersDr Avinash SathayeMA 3303-7-14

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 ʃabf(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 ʃab 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, ∑nk=1 akxk, a maximum or minimum occurs whenever

∑nk=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=1nf(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=1nf(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

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height h. Zu reasoned that “If volumes are constructed of piled up blocks [areas], and

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:

∑ni=1 i = n(n + 1)/2

∑ni=1 i2 = n(n + 1)(2n + 1)/6

∑ni=1 i3 = [n(n + 1)/2]2 (Stewart A37).

Others, including Archimedes, had used these series before, but Alhazen found the

summation of ∑ni=1 i4 (Katz). To achieve this result, he first demonstrated that

(n + 1)∑ni=1 ik = ∑n

1=1 ik + 1 + ∑np=1(∑n

i=1ik), and then solved for ∑n1=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

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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:

(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=1n-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-1i=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) = ʃ0α cos(x)dx and

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cos(α) = 1 - ʃ0αsin(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

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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”)

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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.

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Works Cited

Abbott, Stephen. Understanding Analysis. New York: Springer, 2010. Print.

“Advanced Calculus.” Mercyhurst College. Web. 1 Mar. 2014.

“Archimedes.” ancientgreece. Web. 10 Feb. 2014.

Bressoud, David. “Calculus Before Newton and Leibniz.” CollegeBoard. Web. 19 Feb. 2014.

Bressoud, David. “Was Calculus Invented in India?” Sherlock Holmes in Babylon and Other Tales of Mathematical History. Ed. Marlow Anderson, Victor Katz and Robin Wilson. Washington: Mathematical Association of America, 2004. 131- 7. Print.

Burnham, Douglas. “Gottfried Leibniz: Metaphysics.” Internet Encyclopedia of Philosophy. Web. 14 Feb. 2014.

Carothers, Neal. “Archimedes’ Method of Exhaustion.” Bowling Green State University. Web. 2 Feb. 2014.

Cohen, Marc. “Anaximander.” University of Washington Philosophy 320. 9 Sept. 2002. Web. 20 Feb. 2014.

Girish. T.E., Nair. C. Radhakrishnan. “Ideas of Physical Forces and Differential Calculus in Ancient India.” Cornell University Library. 1 Jan. 2011. Web. 9 Feb. 2014.

Grabiner, Judith V. “The Changing Concept of Change: The Derivative from Fermat to Weierstrass.” Sherlock Holmes in Babylon and Other Tales of Mathematical History. Ed. Marlow Anderson, Victor Katz and Robin Wilson. Washington: Mathematical Association of America, 2004. 218- 27. Print.

“Hyun Numbers.” Math0001. 12 Mar. 2012. Web. 1 Mar. 2014.

“Isaac Newton.” Biography. Web. 7 Feb. 2014.

“Isaac Newton Quotes.” Thinkexist. Web. 25 Feb. 2014. “Iterative Methods.” Camosun College. Web. 7 Feb. 2014.

Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics. Princeton: Princeton University Press, 2011. Print.

Katz, Victor J. “Ideas of Calculus in Islam and India.” Sherlock Holmes in Babylon and Other Tales of Mathematical History. Ed. Marlow Anderson, Victor Katz and Robin Wilson. Washington: Mathematical Association of America, 2004. 122-30. Print.

25

“Lui Hui’s π Algorithm.” wikipedia. Web. 3 Mar. 2014.

Mastin, Luke. “Indian Mathematics – Brahmagupta.” The Story of Mathematics. 2010. Web. 2 Mar. 2012.

Morley, John. “First Scientist to be Knighted.” Holisticresearch. 2002. Web. 22 Feb. 2014.

“Neil DeGrasse Tyson: My Man Sir Isaac Newton.” youtube. 3 June 2011. Web. 3 Feb. 2014.

Ramasubramanian, K., Srinivas M.D. “Development of Calculus in India.” Avinash Sathye Home Page msc.uky.edu/sohum. Web. 15 Feb. 2011.

Roy, Ranjan. “The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha.” Sherlock Holmes in Babylon and Other Tales of Mathematical History. Ed. Marlow Anderson, Victor Katz and Robin Wilson. Washington: Mathematical Association of America, 2004. 111-21. Print.

Saladino, Andrew. “History of Calculus.” andrewsaladino. 2008. Web. 20 Feb. 2014.

Salerno, Vincenzo. “Archimedes.” Best of Sicily. 2003. Web. 28 Feb 2014.

Stewart, James. Calculus. 6th ed. Belmont: Thomson Brooks/Cole, 2008. Print.

Straffin, Philip D. Jr. “Liu Hui and the First Golden Age of Chinese Mathematics.” Sherlock Holmes in Babylon and Other Tales of Mathematical History. Ed. Marlow Anderson, Victor Katz and Robin Wilson. Washington: Mathematical Association of America, 2004. 69-82. Print.

“The Mathematical Achievements and Methodologies of Archimedes.” Mathematical Database. Web. 7 Feb. 2014.

“The Method of Exhaustion and the Limit Process.” Illinois Institute of Technology mypages. Web. 17 Feb. 2014.

“The Method of Exhaustion Chocolate Pi(e).” Teamkellon. 4 Mar. 2010. Web. 1 Mar 2014.

Whiteoak, Hannah. “How Calculus is Used in Astronomy.” Bright Hub. 29 Jul 2011. Web. 23 Feb. 2014.