Isaac Newton, Gottfried Wilhelm Leibniz mathematics |

## Calculus

- Newton started with differentiation;
- Leibniz started with integration.
- Newton called it
- the “science of fluents and fluxions”;
- Leibniz called it
- the “calculus.”
- Newton was the first to apply it;
- Leibniz was the first to publish it.
- Newton focused on motion
- and magnitudes;
- Liebiz focused on the tangent
- and on the idea of change.

## Foundations

- Newton’s calculus used infinitesimals
- to solve the problem of planetary motion,
- but infinitesimals were ridiculed
- so Newton gave geometric proofs
- in
*Principia Mathematica.* - Newton understood the concept of a limit,
- which “can approach so closely
- that their difference
- is less than any given quantity,”
- and proposed limits as an alternative to infinitesimals;
- however, it took another hundred and fifty years
- for mathematicians to provide
- a formal and rigorous definition of calculus.

## Convergence

- Newton knew something about the convergence
- of the infinitesimally small
- and the infinitely large.
- Zeno really messed people up with his paradoxes.
- It took genius to say,
- “You can actually run to the finish line.”
- Zeno said that motion and change are illusions,
- but calculus, the mathematic of motion and change,
- shows us why Zeno was wrong.
- For his first contribution to mathematics
- Newton generalized the binomial theorem
- using infinite series.
- Infinitesimals are arbitrarily small;
- choose any small number;
- we can choose a number that is smaller.
- Infinities are arbitrarily large;
- choose any large number;
- we can choose a number that is larger.
- However, the sum of an infinite series
- isn’t necessarily infinite,
- but can converge to a finite number.

## Steps

- Except for geniuses,
- whom I have known,
- people do learn this stuff
- one step at a time.
- Everything is related
- to everything else,
- pretty much, so you
- can relate to it.
- Should we take little steps
- and risk boring you,
- or big steps
- and risk losing you?

The derivative of a function

`f(x) = y`

is the slope of the tangent of the curve at the point`(x,y)`

. Thetangenttrigonometric function, defined as the ratio of thesineover thecosine, represents the slope of the circle at the point where the radius touches the circle. Did I get that right?See also in

The book of science:Infinite and infinitesimal—John WallisFundamental theorem of calculus—James Gregory, Isaac BarrowPrincipia Mathematica—Isaac NewtonReadings on wikipedia: