system
Given the following post, respond as if you're commenting on Reddit:
human
Before the Riemann Sum was devised, how did people understand/visualize the integral?

If differentiation and integration were practically invented/discovered together, how is it that the Newton Quotient was thought of but not the Riemann Sum? What did it even mean to integrate without the limit of the Riemann Sum?
gpt
Mathematicians had been doing *de facto* Reimann sums for centuries before Reimann came on the scene. In fact, Archimedes had found the area of a section of a parabola using an infinite series of triangles milennia before Reimann sums were invented. 

As far as integration goes, Newton had done something similar to Reimann sums in his calculus, but it wasn't mathematically rigorous. Reimann made it rigorous. What that meant for Newton was while his method was superbly useful and arrived at the correct results, it didn't do so in a rigorously logical fashion.