# Did Newton already have F= ma (or equivalent) before finding his gravity equations?

Newton's gravity equations are in an F = ma configuration. Multiply the acceleration formula by m, you get the force formula. So did he have F = ma (or some equivalent - I think the concept of acceleration may not have been a direct quantity in those days), before he got those two gravity equations? Did he get to one of them from the other, via F = ma?

• Universal gravity is studied in book 3 of Principia after the laws of motion are formulated in book 1. But the order of exposition does not follow the historical order, Kepler's laws were part of the mix that led Newton to formulating the modern concept of force and the laws of motion. And neither they, nor Huygens's formulations in terms of speeds and accelerations, require either forces or the second law. Once Newton had formulated the second law, rewriting the inverse square law in terms of force was, of course, straightforward. – Conifold Jun 29 '19 at 21:46

It is abundantly clear that in Newton's time F=ma was already firmly established.

One way to illustrate that is to llook at the way the three laws of motion are described in the Principia.

Here is how F=ma is dealt with:

The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impress'd altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force) if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joyned, when they are oblique, so as to produce a new motion compounded from the determination of both.

Clearly, Newton felt no need to convince the reader of F=ma. That in itself shows that in Newton's time F=ma was firmly established.

The purpose of writing down the first second and third law right at the beginning of the Principia was not to introduce them. In a sense it was a formality.

Like other scientific books of the time the structure of the Principia was modeled after the 'Elements' (The 'Elements' written by the greek mathematician Euclid). Thoughout centuries of science Euclid's Elements have been tremendously influential.

Euclid had demonstrated his mastery of his subject by identifying the smallest set of statements that as a set of statements is sufficient to serve as the logical basis of an entire body of geometrical mathematics.

Newton presented the three laws as a statement of logic: according to Newton those three axioms were sufficient to serve as the logical basis of all of mechanics

So: by the looks of it: in Newton's time F=ma was firmly established.

• Thank you. The question was about whether it is known that the work on gravity came later than the work on the second law. They were all published together, with the laws of motion coming early in the book. So it may well be that the work on gravity came later, but is it known to have come later? – David Jun 29 '19 at 21:42
• The law of gravity was also "known" and widely discussed before the time when Principia was published. The contribution of Newton is that he proved it by showing that this is the only law of attraction consistent with Kepler's laws. – Alexandre Eremenko Jun 30 '19 at 16:39
• @David Please note: triggered by the question you had submitted on physics.stackexchange I had asked a question of my own here on hsm.stackexchange, asking about the origin of F=ma Conifold argued to me that at the time the very concept of accelerating force was not used in the same way by everyone. My best guess: Newton's contemporaries recognized the second law, and as far as I know were not particularly likely to object to it, but not in the sense that it was an already existing concensus. – Cleonis Jun 30 '19 at 17:18