A TOUR OF THE CALCULUSFrom Pythagoras to Newton

TWO BASIC QUESTIONSZENO STYLE

• How can you add an infinite number of things together and not get infinity as an answer?

• How can you divide something that is already infinitely small (nearly zero) and not get something else that is infinitely small (nearly zero)?

WHAT IS CALCULUS?

• Calculus is the arithmetic of infinites.

• What do you do in Calculus?• The bulk of a Calculus I class is concerned with finding the slopes of curves

and the areas underneath them.

• Why is Calculus so useful?• It is the mathematics of change. The area of a rectangle is easy (e.g. 5x4).

But the area under a parabola is very difficult to find. When you throw a stone or a cannonball it follows a parabolic arch. What if you wanted to know the area required to fill in an arch that large? Most real world problems with motion involve calculus.

• Math Models

FROM WHENCE DOES THENAME CALCULUS COME?

• Gottfried Leibniz

• “This was the first page of the first publication on calculus. In the October 1684 issue of the Acta eruditorum, Leibniz presented a ‘new method’ for finding maxima, minima, and tangents and, in the last line of the title, promised ‘a remarkable type of calculus for this.’ The name stuck.” (Dunham 4)

A STROLL DOWN MEMORY LANE

PHILOLAUS, THE PYTHAGOREAN

Nature in the Cosmos is harmoniously composedof the limited and the unlimited,

both the entire cosmos and everything in it.

The Pythagoreans upheld this as the ancient headwaters for music understood as a liberal art.

NATURE IN THE COSMOS

A Greek Lesson• Nature in Greek is physis

• Gr. Kosmos means order

IS HARMONIOUSLY COMPOSED

“Is harmoniously composed” is one word, harmocthe, in Greek• Think ‘Harmonized’

• Harmonies imply proportionalities in Greek thought.

OF THE LIMITED AND THE UNLIMITED

• Of the bounded and the unbounded (Gr. apeiron)

A sequence

of real numbers is called a Cauchy sequence, if for every positive real number ε, there is a positive integer N such that for all natural numbers m, n > N

where the vertical bars denote the absolute value. In a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring

to be infinitesimal for every pair of infinite m, n.

Every Cauchy sequence of real (or complex) numbers is bounded (since for some N, all terms of the sequence

from the N-th onwards are within distance 1 of each other, and if M is the largest absolute value of the terms

up to and including the N-th, then no term of the sequence has absolute value greater than M+1).

HARMONIES OF THE LIMITED AND UNLIMITED

• Harmonies of the Limited (Finite Proportionalities of the bounded)

• Harmonies of the Unlimited

Unlimitedly Small—Proportions of Infinitesimals

Unlimitedly Large—Proportions of Infinities

BOTH THE ENTIRE COSMOS AND EVERYTHING IN IT

𝐴=𝜋 𝑟2F=𝐺𝑚1𝑚2

𝑟2F=𝑚𝑎

KE=12𝑚𝑣2T=2𝜋 √ 𝐿𝑔 F=−𝑘𝑥

SO DID THE GREEKS HAVE CALCULUS?

No!

WHY NOT?

• In the Fourth Century BC, Aristotle banned infinities from Greek mathematics.

• He did this as a knee jerk reaction to Zeno’s Paradoxes.

• Pythagoras, turned over in his grave…

ARCHIMEDES METHOD OF EXHAUSTION

• I’m so tired…

• Question:• How do you measure the area under a parabola?

• Answer:• Lots of…..

Triangles!

NEWTON USED…

Rectangles

THE ANCIENTS DID LITTLE WITH ALGEBRA

• Diophantus, the founder of Greek symbolic algebra was concerned with no higher power than x6

• Why is that?• Solving proportionalities involving volumes requires proportions of cubes.

• Some solutions require multiplying the cubes.

THE EARLY MODERNALGEBRA REVIVAL

• In the 16th century• Italians Tartaglia and Cardano sought solutions to algebraic problems• Francois Viete also revived the Diophantine tradition and extended it.

• In the 17th century• Pierre de Fermat began exploring ‘higher parabolas’• Fermat wrote his “Last Theorem” in the margins of Diophantus’ book• Rene Descartes brought together Algebra and Geometry• Descartes’ fusion is now called the Cartesian coordinate system

FERMAT AND PASCALLEAD THE WAY TO CALCULUS

• Fermat discovered the area under ‘higher parabolas’ as well as tangents to them.

• Pascal, a correspondent with Fermat, discovered a new method to prove Fermat’s algorithm.• Pascal offered a method for solving series of higher orders.

• Archimedes had only solved series of cubes.• Pascal could find sums of any power e.g. ++• Or

• He applied this method to the solution of discovering the area under all kinds of polynomial curves.

• They developed what we now call the “Power Rule”

THESE TWO INSPIREDLEIBNIZ AND NEWTON

• The contribution of Newton and Leibniz is not the area under the curves of polynomials. That was already done.

• Nor is it the slopes of tangent lines to those curves. That was already done.

• Their great work is the Fundamental Theorem of Calculus.

• Newton and Leibniz each independently discovered that an inverse relationship holds between the slope and area of curves.

THE FUNDAMENTAL THEOREM OF CALCULUS

• Let an equation (curve B) describe the slope of a tangent line to curve A. In that case the area under curve B provides the values for curve A.

• Or

• Let an equation (curve A) describe the area of curve B. In that case the slope of curve A at any point is the same as the value of curve B at that location.

THE FUNDAMENTAL THEOREM OF CALCULUS

• Let an equation (curve B) describe the slope of a tangent line to curve A. In that case the area under curve B provides the values for curve A.

• Or

• Let an equation (curve A) describe the area of curve B. In that case the slope of curve A at any point is the same as the value of curve B at that location.

SOME EXAMPLES

SOME EXAMPLES

So What’s Left after theFundamental Theorem of Calculus?

THE MOST BEAUTIFUL EQUATION

“Read Euler. Read Euler. He is the master of us all.”

Questions?