Newton vs Leibniz
Now if you don’t know what Newton vs Leibniz was, it was the argument between two Mathematical scholars who both claimed they had discovered Calculus. Isaac Newton and English Physicist of the royal society and Gottfried Leibniz a German mathematician.
Who Was Newton?
Now I would be amazed if most of you didn’t know who newton was as his laws of motion are taught in physics classes around the world. He is credited with many things such as the discovery of gravity, the establishment of classical mechanics in his book Principia Mathematica. He was well linked to the other scientific minds of the day, which gave him an advantage in the argument between him and Leibniz, England and Germany, Protestant and Catholic. As you see this had a massive impact on the time.
Who was Leibniz?
Leibniz was a German polymath, whose main field of expertise was pure maths. He is far less well known as newton, but that did not stop him arguing against newton’s well-established position. He is credited with a famous series for pi, matrices and many ideas in topology.
What is calculus?
In simple, calculus is the study of change. It studies how gradient changes over infinitely small time scales (derivatives), areas under curves (integrals) and values that can be calculated by approaching them and getting ever closer to the true value (limits). Calculus is in modern times used in everything from calculating the way a population is affected by a pandemic to working out how much fuel needs to be put into a rocket to make it get to mars. But the calculus developed by Leibniz or newton is far more basic and relies on far less out of the box thinking. Let us learn about each of their ways of developing calculus separately.
Newton’s Calculus
Newton’s idea was what if there was a way of forming an equation to describe how something changed with time, like a projectile in motion. He called this equation a fluxion, and this is now known as the derivative. He decided to use the general equation for a gradient that had been known for thousands of years and try to just make it work for a curved line such as y=t².
By using a large difference between two points that gave him a rubbish approximation, but as he continued to decrease the difference, he started to see that it was approaching something.
In this case 2!?
But how could you prove it was 2, algebra was the answer. Newton was willing to do what thousands before him had been too scared to do. He was able to embrace the limit.
By stating the formula for the gradient newton realised that he could take a function and simplify and simplify it until he got a nice expression that he could take the ‘limit’ of. This is shown below for the example of t².
This gave newton a reason why the gradient was what it was, and he was willing to keep quiet as many others of the day would also and use his method to solve problems to earn himself fame.
Leibniz’s Calculus
Leibniz was in his prime a bit later than newton and so by the time he came across differentiation Newton was already well known. He though was willing to take it one step further and try to calculate the area under the curve. He started out by drawing lots of rectangles under the graph that roughly estimated the area under the curve.
He decided that the width of one of these rectangles would be a little change in the x axis Δx (above shown as 1/3). He could then calculate the area of one of these rectangles as the height (x²) times the width (Δx). He could then sum all of these up to get an expression for the area., by then looking at by how much each of these rectangles changes as you increase the x values it is fairly easy to see that the little change in area.
This can berearranged to show:
Now Leibniz had the insight to think, doesn’t this look like a gradient?
He imagined doing the inverse derivative of this function to try to return and get back the function for the area!
The notation used here dy/dx is often used when talking about derivatives, this notation was first created by Leibniz.
So, who invented it first?
Well, it may seem obvious to you at first glance from what I have told you, but now I shall tell you, what people at the time heard.
Newton: Nothing.
20 years later… Leibniz: Calculus.
In modern times whoever publishes something first is credited with the Discovery but at the time this was not the case. Also, the only proof we have that Newton was actually telling the truth that he had discovered Calculus first was his own word!
The Inquiry
The royal society of the time set up a committee to decide who had discovered it first. They asked both Leibniz and Newton to give accounts of what they knew and had been told by the other. They found that newton had sent Leibniz a letter describing calculus a few years prior to him publishing his Calculus. This all backed up newton and finally in the autumn of 1713 they declared Newton the true invertor of Calculus. This was then sent to Leibniz. Funny how a committee decided that the head of that very committee had not lied and had indeed invented the thing there was no proof of him ever inventing.
My Views
Now don’t take my views for fact, but I personally do believe it was Leibniz that discovered calculus and that Newton as was the way in the 1700s decided to increase his popularity by cheating, again. Not that this is an attack on newton, on the contrary he was an excellent physicist and created such things that were far outside the grasp of Leibniz or anyone else of the day. But in this specific case I believe that Leibniz should get more recognition. Leibniz created some great ideas as displayed here and he created some great notation that makes a difficult topic seem intuitive and wonderous at the same time. We now almost always use Leibniz’s notation for derivatives and Integrals (except when dealing with derivatives according to time). Please comment below your views on this topic and as Always.
Have fun and never stop solving.
