What if Calculus was discovered by the Ancient Greeks

So partly inspired by the other thread.... but...

Archimedes was interested in finding the area under the curves under polynomials. The method he used was similar to the Riemann sum (using rectangles to estimate area.) What if the Greeks realized that the rate of change was the answer to the problem. What if the Greeks were able to discover differentiation and in turn integration. The concept wouldn't be particularly hard. Honestly they would only have to deepen their understatement of the limit, in OTL they had all of the other tools needed.

Of course they probably wouldn't have the tools to examine trig functions and natural logarithmic problems (even though they could calculate e considering that f(x)=e^x f'(x)=e^x....)

But what would change if calculus was discovered nearly two thousand years early?
 
So partly inspired by the other thread.... but...

Archimedes was interested in finding the area under the curves under polynomials. The method he used was similar to the Riemann sum (using rectangles to estimate area.) What if the Greeks realized that the rate of change was the answer to the problem. What if the Greeks were able to discover differentiation and in turn integration. The concept wouldn't be particularly hard. Honestly they would only have to deepen their understatement of the limit, in OTL they had all of the other tools needed.

Of course they probably wouldn't have the tools to examine trig functions and natural logarithmic problems (even though they could calculate e considering that f(x)=e^x f'(x)=e^x....)

But what would change if calculus was discovered nearly two thousand years early?

Probably not much, really.

I suspect that Archimedes probably HAD a functional version of calculus, but that he didn't publish anything unless he could prove it otherwise.

Note that Newton and Leibnitz's calculuses of the late 1600s required 'infinitesimals' and other such fictitious entities. While the system worked in general (obviously), it was in no way rigorous - especially compared to Euclid's Elements, which were the gold standard for math at the time.

Newton and Leibnitz got away with it at the time because it worked so well, and was so useful - you couldn't do gravitational/orbital calculations without it, for instance, but it wasn't until Cauchy published Cours d'Analyse in 1821, about a century and a half later that there was actually a solid underpinning for it.
 
Archimedes also came close to inventing the decimal system.

I don't think it would change the classical world much, because their mind set wasn't based on using these discoveries to produce applications.

However if/when the scientific revolution eventually does happen, it will happen faster. Just as the existence of Euclid gave modern science/maths a headstart, Archimedes producing a version of calculus and/or decimals would give us an even bigger head start.
 
First you must change the Greek paradigm about what a number is: they taught that a number must be expressible as a fraction, that's why they talked about "irrational numbers". Kind of like right now people don't understand the purpose of expanding the set of numbers to complex numbers or vectors and mock on "imaginary numbers". You should expand the Greek definition of a number to at least our definition of real number.

Then, you need Cartesian coordinates, and then Analytic Geometry.

And finally someone really willing to beat Zeno and his paradoxes :D
 
Didn't something similar to calculus get invented in India before Newton?
Running with the Buddhist philosophical principle that facts, fundamentally, only apply to individual points and instants, Indian astronomers did identify an object's instantaneous velocity as a meaningful characteristic like its instantaneous position.
 
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