Orbit is esspetially free falling past the side of the planet, fast enough that the force of gravity doesn't completely pull you in. So you kind of fall past the planet, but get curved towards it enough you don't go flying out away either. The minimum velocity to orbit around a planet with radius r and gravity g is Vorbit = √(rg) I believe

An orbit is analogous to position. Trying to predict the position of planets proved difficult to do accurately. People had figured out velocity a long time ago, but no one had ever explored the concept of describing an orbit just from the accelerations of the planets. It turns out acceleration has a fairly simple equation, and in order to get back to the observable thing of orbits, you need to integrate it twice. Newton was the first person to describe acceleration and to use it to predict the motion of planets. Especially for predicting Mars, it was essential because Jupiter is massive enough and close enough to alter its orbit, so simple Kepler's Laws aren't sufficient to describe the orbit.

Orbits, it turns out, are mostly just rates of change.

So, the heart of the issue is that each object's path changes continuously, and the forces involved change in kind. Even worse, the objects interact with each other, again continuously - it's not one-sided.

If you imagine trying to do it pre-Calculus, some kind of "just map it all out into a grid, etc.", you can see the problems this continuous change imposes (exercise left for the reader).

By involving the Stravinsky Interpretation, it quickly becomes clear that the dimorphic superposition destabilizes. The clever reader might object "but what if you fold in all the noodly surfaces to recohere the manifold?"

And that clever reader would be right! But we didn't know that until old Dr. Isaac "Zeke" Newton came along and made it that way.

Some say the devil himself taught him how it's done, because no one else can read his notes! So keep your eye on old Zeke when you run into him.

In the best of all possible worlds.

In the grocery store, buying cookies

Newton's laws, including gravity and motion, can be expressed in terms of differential equations.

Differential equations pretty much requires calculus, which just hadn't been formalized yet.

Newton's laws also provide convincing reasons for the necessity and legitimacy of calculus, by being able to derive orbits from his simple laws.

Yeah, just the one. They weren't as confused back then.

I mean, just in general he was a genious, but also like many geniouses, he fell prey to a shit ton of nonsense. From how I understand it, he spent equal time and believed in equal importance on his study of physics, alchemy and finding hidden codes in numerology of the bible.

Sense of humor and personality, no clue, don't think that generally is super strongly carried in surviving literature.

My math prof told me he just paved over the small, itsy bitsy problem that all of his equations fell apart if any value was zero... so he just ignored it. Then Leibniz seen it and fixed everything to solve for that pesky little issue of dividing by zero.