# What are the origins of calculating the area of a v/t graph to determine displacement?

I'm curious as to the origins of thinking of displacement as the area under a v/t curve.

I assume that Newton (and/or Leibniz) was already familiar with the concept and knew that calculating the area was simple for linear curves, but developed his new "calculus" in order to calculate areas for more complex curves.

I am also aware of the Merton rule, developed a few centuries before Newton, which proved more-or-less that the average velocity for an object experiencing constant acceleration would be exactly halfway between the initial and final velocities. I am not sure as to the methods used to prove the Merton rule.

I think that the concept of "area under the curve" couldn't really come about until coordinate plane graphing was invented, which I believe started with Descartes.

So what is the first reference to calculating areas under curves? What was the context of this first use? What is the first reference to doing so on a v/t graph specifically to calculate displacement?

• These are two questions. We try to have only a single question per post. I think the area under the curve graph has a more complex story. The $vt$ graph is about looking for a specific point in time. Choose one. Jan 19 at 9:18
• Yes, the so-called Merton rule amounts to what today is the compèutation of an integral, but at that time there were no concept of integral and the rule worked only for that specific case. Jan 19 at 10:26
• And yes, when Newton (and Leibniz) understood for the first time that derivative and integral can be seen as each-other inverse operations, the geometrical intepretation was intuitive. Jan 19 at 10:29
• @Mauricio - Hi. Ultimately the question I want answered is the title of the post: "What are the origins of calculating the area of a v/t graph to determine displacement?". However, I realize that interesting questions rarely have a single simple answer. I expect that the answer to my question is a 'story', and my follow up questions at the end of my post were intended to indicate the elements of that story that I most care about. Would you like me to edit my question to make that clearer? Jan 21 at 17:55
• @MauroALLEGRANZA The Wikipedia page (Mean Speed Theorem) that you linked to has two images that I find interesting. The first has Nicole Oresme's name on it, but it looks like a modern interpretation of the results of Oresme's proof. Can you direct me to a translation of Oresme's original proof? The Galileo image looks interesting and original. Do you know what work that image is from? Jan 21 at 18:22