1. The history of calculus
Intro to Mathematical Reasoning Math 2971W
Zihan Yu
02/13/2016

2. I. Introduction
Today, we are learning calculus in a succinct way. But most students are still busy
remembering the equations of calculus in textbooks but forget to look back and learn the
origin of it. To better understand calculus, it is worthwhile to dig into more details about
the creators of calculus, from ancient Greek, to the age of Newton and Leibniz.
II. The Pioneers of Calculus
At present, as we shall usually see in today’s textbooks, calculus begins with
differentiation. However, around 200 B.C., the ancestors of calculus were more focused
on integration; determining volumes or length of arcs were the first problems arising in
calculus history. These problems were mainly solved by Greek mathematicians,
especially by Archimedes, whose profound and outstanding achievements marked the
peak of all ancient mathematics and also the beginning of the theory of integration. He
was the first one who determined an approximate value of π, the surface area of sphere
and of cylinders. Moreover, he invented the formulas for the areas of an ellipse, of
parabolic segments, and also of sectors of a spiral (Rosenthal, 1951). His had far more
achievements than just listed. And the major ideology behind his achievements was the
method of exhaustion, that is, a method of finding the area of a shape by inscribing a
sequence of polygons with increasing number of edges, and whose areas converge to the
area of the containing shape.
In calculating areas of figures in two dimensions, Archimedes’ method of exhausting
maybe easily understandable. But, in order to find the volume of a solid, we need to

3. divide the solid into infinite thin cylindrical polygons and add those polygons’ areas up. A
direct approximation of this limit would require substantial works. In order to reduce the
workload of such a computation, Archimedes created the mechanical infinitesimal
method. With the aid of the principle of the lever, this method allows us to calculate any
triangle’s area only using one of its side. And this method could also be extended to all
other polygons. In such a way, the computation of areas had been significantly simplified,
thus reducing the difficulty of calculating volumes of solids.
Considering Archimedes’ magnificent explorations in geometry, it is astonishing that
he found almost no successors in continuing his work. In fact, after Archimedes’ death,
Greek mathematics declined and took a very different direction called trigonometry.
Again, Archimedes’ method of exhaustion was an essential step in the early development
of integration. Moreover, it is his masterpieces in geometry that enlightened the path for
calculus.
Another important mathematician prior to Newton and Leibniz was Pierre de Fermat.
He received a Bachelor of Civil Laws from the University of Law at Orleans in 1631
(McGraw Hill Higher Education, 2013). His job as a jurist left him substantial time for
mathematical researches. During the first part of the 17th century, he was the greatest
mathematician, not only in general but especially in pre-evolved calculus.
One of the most outstanding discovers from Fermat was his methods of finding
tangents. As early as in 1629, he solved this problem by applying the determination of
maxima and minima. For calculating the maxima or minima of an expression, one should

4. replace the unknown x by x+Δx. Both expressions are approximately the same when Δx
is close enough to zero. Then it’s necessary to cancel on both sides all that is possible to
cancel. If we write F(x) for the given expression, then the cancellation would leave the
expression as F(x+Δx) – F(x). After subtraction, only terms containing Δx are left. In
order to eliminate Δx, one need to divide the expression by Δx and drop all terms still
containing Δx. This last step would generate an equation containing the value of x which
produces the desired maxima or minima. Here is an expression of the whole process:
[
𝐹( 𝑥 + 𝛥𝑥) − 𝐹(𝑥)
𝐹(𝑥)
]
𝛥𝑥=0
= 0.
Fermat also gave a geometrical approach to find the tangent in the same year. For a
given curve C, let PT (T is on the horizontal axis) be the tangent line of C and the point of
tangency is P (cf. The Following Figure), let P1 be a point of C in the neighbor of P,
denote the projections of P and P1 on the horizontal axis as Q and Q1, and let T1 be the
point on the tangent line whose projection on the horizontal axis is Q1. To determine the
subtangent A = TQ, in which E represents its increment QQ1, Fermat uses the similarity
between triangles QTP and TQ1T1 and uses P1 to approximate T1. Now, approximately:
A:QP≈E : (Q1P1 – QP). If we write the expression of curve C as y=F(x), then A: F(x) =

5. E: (F(x+E) – F(x)). Thus, the expression of the subtangent A would be :
𝐴 =
𝐹( 𝑥) ∗ 𝐸
𝐹( 𝑥 + 𝐸)− 𝐹(𝑥)
Once again one should divide the denominator by E and then sets E = 0 to obtain the
desired x (Rosenthal, 1951).
One should quickly recognize the importance of Fermat’s findings. Those
expressions mentioned above are called the difference quotient. By taking the limit of it,
we get the derivative of a certain function. That is:
lim
ℎ→0
𝑓( 𝑎 + ℎ) − 𝑓(𝑎)
ℎ
= 𝑓′
(𝑎)
III. Newton and Leibniz
In the late 17th Century, England had many great mathematicians and scientists.
Among all of them, Isaac Newton was undoubtedly the greatest. He had a lot of identity:
physicist, mathematician, astronomer, philosopher, alchemist and theologian. Indeed,
Newton is considered to be one of the most influential men in history(The story of
mathematics, 2010).
During 1665-1666, Newton developed a revolutionary new approach to
mathematics: infinitesimal calculus, or fluxional calculus. Based on the interests in
motion, he named variables (x or y) the fluent and its rate of change the fluxions
(𝑥̇ 𝑜𝑟 𝑦̇). Newton then realized that the fluxions themselves could also be treated as
fluent. Thus, the rate of change of the fluxions, or the second fluxions of x and y, could
be denoted by 𝑥̈ 𝑎𝑛𝑑 𝑦̈ (Cirillo, 2007). To address the inverse relationship between
fluent and fluxions, Newton also introduced a new notion called moment:

6. “This was the ‘indefinitely small’ part by which fluents grew in ‘indefinitely small’
period of time, and was represented by the sign o. The moment of the fluent x would
therefore be 𝑥̇ 𝑜, and of the fluent y, 𝑦̇ 𝑜. In this way, it follows that quantities x and y
will become in an indefinitely small interval 𝑥 + 𝑥̇ 𝑜 and 𝑦 + 𝑦̇ 𝑜.” (Gjertsen, 1986).
These innovations could be applied to implicit differentiation. Suppose one needs to
find the tangent to the curve 𝑥3
− 𝑎𝑥2
+ 𝑎𝑥𝑦 − 𝑦3
= 0. In this case, the function of the
curve could be expressed as (𝑥 − 𝑦)(𝑥2
+ 𝑥𝑦 − 𝑎𝑥 + 𝑦2
), a line and an ellipse. Under
Newton’s ideology, a curve can be treated as a path traced by a moving point; thus,
Newton replaced 𝑥 + 𝑥̇ 𝑜 for x and 𝑦 + 𝑦̇ 𝑜 for y to get
(𝑥 + 𝑥̇ 𝑜)3
− 𝑎( 𝑥 + 𝑥̇ 𝑜)2
+ 𝑎( 𝑥 + 𝑥̇ 𝑜)( 𝑦 + 𝑦̇ 𝑜) − ( 𝑦 + 𝑦̇ 𝑜)3
= 0
The expansion of this equation gives us
Because 𝑥3
− 𝑎𝑥2
+ 𝑎𝑥𝑦 − 𝑦3
= 0, by substitution, we left with
Because o is an infinitely small quantity, we cast out these terms, leaving
3𝑥2
𝑥̇ − 2𝑎𝑥𝑥̇ + 𝑎𝑥𝑦̇ + 𝑎𝑥̇ 𝑦 − 3𝑦2
𝑦̇ = 0.
Newton ended his calculation here, but this method would be more familiar to us if

7. we solve for
𝑦̇
𝑥̇
. By doing so, we get
If we used modern methods to differentiate implicitly, we would get the same result as
above. Notice that the third term of the original equation, 𝑎𝑥𝑦, need to use the product
rule to calculate the differentiation. Thus, Newton’s fluxional method implicitly
contained the product rule (Cirillo, 2007).
Unlike Newton, Leibniz focused more on adequate notations to represent ideas and
ways of combining these. That is, a process of reasoning using symbols. He was the
creator of the product rule and the quotient rule. He also generated other differentiation
rules that we are still using today, such as d(ax) = adx. Moreover, Leibniz used
expressions dx and dy to demonstrate the difference between two infinitely small values x
and y, respectively, and dy/dx, which is known as the derivative of a function, to indicate
the ratio of the two values. He also raised the idea of using the symbol∫ to represent
the operation of the sum as well as the term function. If in the last paragraph, we used his
notations on the problem presented by Newton, then the solution would have looked very
familiar with our modern implicit differentiation. Leibniz’s notations were considered the
standard of calculus’ symbols because he published his complete work in 1684. Thus, it is
his notations, rather than the Newton’s, that is presented in today’s calculus system
(Cirillo, 2007).
IV. Works Cited

8. Rosenthal, Arthur. (1951). The History of Calculus. The American Mathematical
Monthly, 58(2), 75-86.
Cirillo, Michelle. (2007). Humanizing Calculus. The Mathematics Teacher, 101(1), 23-
27.
McGraw-Hill Higher Education. (2013). The History of Calculus. Retrieved Feb. 22,
2016 from http://www.mhhe.com/math/calc/smithminton2e/cd/tools/timeline/.
The Story of Mathematics. (2010). 17th Century Mathematics- Newton. Retrieved Fed.
22, 2016 from http://www.storyofmathematics.com/17th_newton.html.
Gjertsen, Derek. (1986). The Newton Handbook. New York: Routledge & Kegan Paul,
p.214.