This the actual document in facsimile. You can see the scratched out first hypothesis and replace. You can also see there's not a lot of changes in this first page. You've already seen these figures. He compared two different circles to get the result for uniform circular motion. You can see hypothesis four is written on the edge of the page.
He really didn't have any room for it, so he came back. Obviously, he didn't realize he needed it right away. Then at some point realized he needed it, and squeezed it in on the side. Why's he x-ing everything out at some point, I don't know because it ended up not being x-ed out.
It ended up being copied by his assistant at Cambridge. More striking here, when he get's down to problem one, which is this circle problem, it's striking he's using the word gravitas. Here it's gravitatis, all through the original draft of this, he's using some variant on the word gravitas.
The variants for those who don't know Latin has case index depending on whether it's accusative, nominative, dative, ablative, possessive, degenative, etc. So the case endings don't necessarily matter to us. The key point is he's got the word gravitas in there, in some six or seven places, that he had originally gravitas and every place he removes it and puts in v, in this case, vis centripeta.
Vis centripeta is just centripetal force. Vis is the nominative singular of the word. De vi centripeta would be on centrifugal force, just a different ending. Initially, as he worked this out, he's writing it out in terms of gravity, and then he decides to replace it with centripetal force.
Why? I don't know. The most likely reason I can see is somewhere along the way as he wrote this, he came to realize that what he actually had was not an account of gravity and inverse gravity, etc. He had an account of motion under centrifugal forces. He had a generic theory going, of a fragment of a theory, of motion under centripetal forces and what happens in Book 1 of The Principia is that fragment of a theory runs 69 propositions long.
It gets developed into something really rich but he has noticed that what he really has is an abstract theory of motion under a kind of force that nobody had done before. It is, of course, as I said earlier, a tractable, and maybe the only tractable, straightforward generalization of circular motion because when you go to arbitrary curvilinear forces, we no longer have anything like a theory.
We really need something to constrain it to have a large number of propositions. It's also looking more and more, not the handwriting, of course, like a Huygens' theory. Now, here's a theory of pendular motion. Here's a theory of motion under centrifugal forces. Here's a theory of conical pendulum motions.
Here's a theory of motion under centrifugal forces. What should we expect if it's like Huygens? We should expect some surprising results to test things along the way. Again, I'm showing the figures building down to, this is problem three. You notice in his original, he did problem two and problem three with the same figure which made it a nightmare.
Believe me, it's a nightmare. Nobody does that now. Everybody who publishes some version of gives you two separate figures for the two the way I did but that's not what Newton did, nor did he do it in the first edition. It's one figure for both in the first edition, it takes awhile for him to change.
You can see his figures are pretty well drawn actually given that he's working with a pen. By the way, Newton made his own ink and he was apparently really meticulous from the records of his buying the ingredients. It was a very big deal to him to get the particular form of ink.