As I understand it, the two formulas can be written:
Newton's:
F=ma
and Einstein's:
F=ma(1/sq.rt(1-(v^2/c^2))
That difference between the two is the Lorentz factor which is almost 1 for any "ordinary" speed. So at any speeds that Newton might encounter the formula was:
Force equals mass times acceleration times 1
This wikipedia page has a nice chart of the Lorentz factor and when it starts to matter:
https://en.wikipedia.org/wiki/Lorentz_factor
Even at a speed one tenth of light speed, the formula is:
Force equals mass times acceleration times 0.995
It is unfortunate that Newton was not correct. If it was always F=ma, we could theoretically build a spaceship that went faster than light. (That is of course JMO, that FTL travel being impossible is unfortunate.) Instead at the light speed limit, additional force applied does not result in additional acceleration. The energy ends up adding to the conserved energy of that system, most simply thought of as mass increasing, as you phrased it in the OP, but that simplified thinking is
TOO simple and should be avoided.
Holden Kenne said:
The reason I was reading about relativity is because I read that, at the time of the publication of Philosophae, Newton did not take account for speeds reaching that of near-c. I just want to understand why they don't.
I think Newton probably considered his equations to be valid under all circumstances. There was no reason to guess that speeds near light mattered. As a bad analogy, I might say that I think chocolate would taste good on Pluto, because the law that chocolate tastes good seems to validly generalize. We generalize rules until they break down. There just was no way for Newton to encounter a situation where they broke down enough to not consider them likely to be universally applicable.
So it is not that Newton did not take into account speeds near light, but that he had no reason to expect that special consideration needed to be taken for any speed. I have no reason to expect chocolate to taste different on Pluto, so I regard the chocolate taste rule as a general one.