Did Newton Base His Laws of Motion on the Tria Prima?

	Inertia	F = ma	FAB = -FBA
	salt	sulphur	mercury
	rest	motion	balance
	tamas	rajas	sattva

It is by now well-recognized that Isaac Newton had a lifelong interest in alchemy. However, the possible influence of alchemy upon the development of his physics seems to be mostly unexplored territory. It has been noticed by many students that the 1st and 3rd laws seem to be mathematical trivialities, both can be derived from the 2nd. So why did Newton choose to base his system on three laws, as opposed to two, or one?

Newton was actively engaged in alchemical pursuits at the time he was writing the Principia. He had employed his relative Humphrey Newton as secretary and copyist during his composition of the Principia, who writes:

He very rarely went to Bed, till 2 or 3 of the clock, sometimes not till 5 or 6, lying about 4 or 5 hours, especailly at spring & ffall of the Leaf, at which Times he us'd to imploy about 6 weeks in his Elaboratory, the ffire scarcely going out either Night or Day, he siting up one Night, as I did another till he had finished his Chymical Experiments, in the Performances of which he was the most accurate, strict, exact: What his Aim might be, I was not able to penetrate into but his Paine, his Diligence at those sett times, made me think, he aim'd at somthing beyond the Reach of humane Art & Industry.

Newton was comfortable with the idea of an esoteric subtext, intended to be understood only by a select audience, as he was heavily engaged in Hermetic exegesis of scripture and myth:

He that would understand a book written in a strange language must first learn the language... Such a language was that wherein the Prophets wrote, and the want of sufficient skill in that language is the main reason why they are so misunderstood.

...even going so far as to find verification of his own system of mechanics in the writings of the ancients:

By What proportion gravity decreases by receding from the Planets the ancients have not sufficiently explained. Yet they appear to have adumbrated it by the harmony of the celestial spheres, designating the Sun and the remaining six planets, Mercury, Venus, Earth, Mars, Jupiter, Saturn, by means of Apollo with the Lyre of seven strings, and measuring the intervals of the spheres by the intervals of the tones. [...] For Pythagoras, as Macrobius avows, stretched the intestines of sheep or the sinews of oxen by attaching various weights, and from this learned the ratio of the celestial harmony. Therefore, by means of such experiments he ascertained that the weights by which all tones on equal strings... were reciprocally as the squares of the lengths of the string by which the musical instrument emits the same tones. But the proportion discovered by these experiments, on the evidence of Macrobius, he applied to the heavens and consequently by comparing those weights with the Weights of the Planets and the lengths of the strings with the distances of the Planets, he understood by means of the harmony of the heavens that the weights of the Planets towards the Sun were reciprocally as the squares of their distances from the Sun. But the Philosophers loved so to mitigate their mystical discourses that in the presence of the vulgar they foolishly propounded vulgar matters for the sake of ridicule, and hid the truth beneath discourses of this kind. In this sense Pythagoras numbered his musical tones from the Earth, as though from here to the Moon were a tone, and thence to Mercury a semitone, and from thence to the rest of the Planets other musical intervals. But he taught that the sounds were emitted by the motion and attrition of the solid spheres, as though a greater sphere emitted a heavier tone as happens when iron hammers are smitten. And from this, it seems, was born the Ptolemaic system of solid orbs, when meanwhile Pythagoras beneath parables of this sort was hiding his own system and the true harmony of the heavens

Now, given his evident disposition, and considering the attacks he was already facing for (re)introducing the 'occult' action at a distance into mechanics, is it too unreasonable to suggest that Newton may have hidden what he considered his grasp of the prisca sapientia in mechanistic clothes? He considered all movement to be vegetative (active) or mechanistic (passive) and the Principia, at least its surface reading, was his definitive statement on only one half of the matter.

A correlation between the gunas of Samkhya and Newton's three laws of motion has been pointed out several times (e.g., usually given as evidence of the validity of Samkhya philosophy) but I have not yet managed to come across any discussion explicitly linking his formulation of the laws of motion to the tria prima, other than an assertion by William Law, which I noted in a previous post. Evidence for or against this hypothesis would be welcome in the comments.