Episode Eleven – Newton’s Achievements

Calculus

One of his greatest works for which he is best known, is his invention of calculus.

Now, I am aware that many people debate whether Newton or Leibniz developed calculus first. While I believe that in fact Newton may have developed it first, I am not sure whether this will ever be known with complete certainty.

And really, it does not matter much. It seems quite likely that Newton and Leibniz developed calculus independently, although Newton seems to have begun his work first.

It would not be the first time two different people independently developed important and major scientific advances at much the same time. Another classic example would be Wallace and Darwin. Both of whom developed somewhat similar theories of evolution.

Regardless of which of them developed it first or if they each discovered it independently, Newton certainly developed calculus and so he deserves great credit for that.

What is calculus? It is a branch of mathematics which is essentially composed of two aspects.

Differential calculus, which studies patterns of continuous change.

And integration, which amounts to adding up infinitesimally small values and is the mathematical reverse of differentiation. Calculus is the underpinning of much of modern mathematics and without it much of modern mathematics would not be possible.

As differential calculus studies things in motion, it is a fundamental underpinning of the physics of moving objects. Or indeed any quantity which changes continuously over time.

The original calculus textbook.

In fact, much of Newton’s physics depends on and was derived using differential calculus. For instance, you can use calculus to derive an equation for acceleration from an equation for velocity.

Now, just how important is calculus? Well, it is very hard to overstate the importance of calculus. It is one of the most important, widely applicable branches of mathematics ever invented.

Why is this? Because it can be used to describe the behaviour of virtually anything that moves or changes over time. Which is to say, virtually anything at all.

You can use calculus to describe the velocity of a space rocket. Or how stock markets change over time. It can be used to study the equations and or graphs that describe phenomena. You can use calculus to find various properties of these equations and graphs. Such as the rates of change and optimal values and the list goes on.

Calculus is used in countless optimization problems. In such problems you take equations that describe relationships between certain variables and you find the value or values of those variables that give you the optimal results.

Suppose you have an equation that describes the amount of material used to create containers of a certain volume. You can use calculus to find the dimensions of a container that will hold 1.5 litres but which will minimize the amount of material used.

Thatâ€™s litres (leeters) not litters, though I guess you could find out how many litters of kittens you can fit in a container, using calculus too!

A great variety of problems where you want to maximize or minimize some quantity can be solved using calculus. For example, problems which are very frequent in business and/or design where efficiency must be maximized and cost minimized.

That is differential calculus.