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Mathematical Principles of Natural Philosophy
(Philosophiæ Naturalis Principia Mathematica)
by Sir Isaac Newton
The original, squashed down to read in about 20 minutes

(Cambridge, 1687)

Isaac shone at Grantham Grammar School, had a brief (and final, he never married) fling with an apothecary's stepdaughter, and went off to Cambridge University to produced tracts on literal interpretations of Scripture, to make major discoveries in optics and mathematics, to dabble in alchemy and, later, be made master of the Royal Mint.
The Principia was a true revolution in human thinking- not only did it provide the famous explanation of gravity, but a method by which almost any physical event could be described in numbers. This was the beginning of modern science and engineering, and Newton's Laws are still the rules by which we build bridges and fly spacecraft.
Sadly, the story of his being inspired to study gravity by seeing an apple fall is almost certainly made-up, but it is probably true that, among his many great achievements, he invented the cat-flap - a device which, like everything else, can be understood through Newton's numbers, as its swinging is described by his Third Law.
Abridged: GH

Mathematical Principles of Natural Philosophy

An Ode
to the Splendid Ornament of Our Time and Nation:
the Treatise by the Eminent Isaac Newton
by Mr. Edmond Halley

Behold! the compass of the skies,
Jove's firm foundation for his works
The Sun in his throne moves the starry greats,
The comet's bearded star, and wandering Diana
Such vexed the philosophers of old,
And caused the schools to shout and shake,
Here, by simple numbers is cleared of dust and haze.
Celebrate now with nectars and with song,
NEWTON! Has opened the treasure-chest of truth!
NEWTON! Soars with Phoebus and the muses!
And has touched most near the mind of God.

Since the ancients made great account of the science of mechanics in the investigation of natural things; and the moderns have endeavoured to subject the phænomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. For all the difficulty of philosophy seems to consist in this - from the phænomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phænomena.

I heartily beg that what I have here done may be read with candour; and that the defects in a subject so difficult be not so much reprehended as kindly supplied, and investigated by new endeavours of my readers.

ISAAC NEWTON. Cambridge, May 8, 1686.


The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

THUS air of double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. And the same is known by the weight of each body.

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.

The vis insita, or innate force of matter, is a power of resisting, by which every body endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.

An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.

A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as a centre.

Of this sort is gravity, by which bodies tend to the centre of the earth; magnetism, by which iron tends to the load-stone; and that force, whatever it is, by which the planets are perpetually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone whirled about in a sling, endeavours to recede from the hand that turns it; and by that endeavour, distends the sling.

The absolute quantity of a centripetal force is the measure of the same proportional to the efficacy of the cause that propagates it from the centre, through the spaces round about.

Thus the magnetic force is greater in one load-stone and less in another according to their sizes and strength of intensity.

The accelerative quantity of a centripetal force is the measure of the same, proportional to the velocity which it generates in a given time.

Thus the force of the same load-stone is greater at a less distance, and less at a greater: also the force of gravity is greater in valleys, less on tops of exceeding high mountains; and yet less at greater distances from the body of the earth; but at equal distances, it is the same everywhere; because (taking away, or allowing for the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small.

I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices.


Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top does not cease its rotation, otherwise than it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

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

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed 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 subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone.

IN the preceding Books I have laid down the principles of philosophy, principles not philosophical, but mathematical: such, to wit, as we may build our reasonings upon in philosophical inquiries. It is enough if one carefully read the Definitions, the Laws of Motion, and the first three Sections of the first Book. He may then pass on to this Book, and consult such of the remaining Propositions of the first Books as his occasions shall require.


We are to admit no more causes of natural things than are both true and sufficient to explain what is observed.

To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.

Therefore to the same natural effects we must, as far as possible, assign the same causes.

Such as respiration in a man being as that in a beast; the falling of stones in Europe being as in America; the light of our cooking fire and of the sun; the reflection of light in the earth, and in the planets.

The qualities of bodies, which admit neither intension nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

We are certainly not to relinquish the evidence of experiments for the sake of dreams and vain fictions of our own devising; nor are we to recede from the analogy of Nature, which uses to be simple, and always consonant to itself. If it universally appears, by experiments and astronomical observations, that all bodies about the earth gravitate towards the earth, and that in proportion to the quantity of matter which they severally contain, that the moon likewise, according to the quantity of its matter, gravitates towards the earth; that, on the other hand, our sea gravitates towards the moon; and all the planets mutually one towards another; and the comets in like manner towards the sun; we must, in consequence of this rule, universally allow that all bodies whatsoever are endowed with a principle of mutual gravitation. For the argument from the appearances concludes with more force for the universal gravitation of all bodies that for their impenetrability; of which, among those in the celestial regions, we have no experiments, nor any manner of observation. Not that I affirm gravity to be essential to bodies: by their vis insita I mean nothing but their vis inertiæ. This is immutable. Their gravity is diminished as they recede from the earth.

In experimental philosophy we are to look upon propositions collected by general induction from phænomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phænomena occur, by which they may either be made more accurate, or liable to exceptions.

This rule we must follow, that the argument of induction may not be evaded by hypotheses.


That the forces by which the moons around Jupiter are drawn off from a straight course and made to orbit the planet tend to Jupiter's centre; and are reciprocally as the squares of the distances of the places of those planets from that centre.

That the forces by which the primary planets are drawn off from a straight course and made to orbit, tends towards the sun; and are reciprocally as the squares of the distances of the places of those planets from the sun's centre.

That the force by which the moon is retained in its orbit tends to the earth; and is reciprocally as the square of the distance of its place from the earth's centre.

That the moons around Jupiter gravitate towards Jupiter; those around Saturn, towards Saturn; those around the Sun, to the Sun; and by the forces of their gravity are drawn off from straight motions, and retained in curved orbits.

The force which retains the celestial bodies in their orbits we shall hereafter call gravity.

That all bodies gravitate towards every planet; and that the weights of bodies towards any the same planet, at equal distances from the centre of the planet, are proportional to the quantities of matter which they severally contain.

Cor. 1. The weights of bodies do not depend upon their forms and textures.

Cor. 2. Universally, all bodies about the earth gravitate towards the earth; and the weights of all, at equal distances from the earth's centre, are as the quantities of matter which they severally contain.

Cor. 5. The power of gravity is of a different nature from the power of magnetism; for some bodies are attracted more by the magnet; others less; most bodies not at all.

That there is a power of gravity tending to all bodies proportional to the quantity of matter which they contain.

Cor. 1. Therefore the force of gravity towards any whole planet arises from, and is compounded of, the forces of gravity towards all its parts.

Cor. 2. The force of gravity towards the several particles of any body is reciprocally as the square of the distance of places from the particles.

That the force of gravity, considered downward from the surface of the planets, decreases nearly in the proportion of the distances from their centres.

That the motions of the planets in the heavens may subsist an exceedingly long time

It is shewn that at the height of 200 miles above the earth the air is more rare than it is at the superficies of the earth in the ratio of 30 to 0,0000000000003998, or as 75000000000000 to 1 nearly. And hence the planet Jupiter, revolving in a medium of the same density with that superior air, would not lose by the resistance of the medium the 1,000,000th part of its motion in 1,000,000 years. In the spaces near the earth the resistance is produced only by the air, exhalations, and vapours. When these are carefully exhausted by the air-pump from under the receiver, heavy bodies fall with perfect freedom, and without the least sensible resistance; gold itself, and the lightest down, let fall together, will descend with equal velocity.

That the centre of the system of the world is immovable.

This is acknowledged by all, while some contend that the earth, others that the sun, is fixed in that centre.

That the common centre of gravity of the earth, the sun, and all the planets, is immovable.

For if that centre moved, the centre of the world would move also.

That the sun is agitated by a perpetual motion, but never recedes far from the common centre of gravity of all the planets.

The theory of swirling vortices, we cannot accept, for comets are carried with very eccentric motions with a freedom that is incompatible with the notion of a vortex.

Bodies projected in our air suffer no resistance but from the air. Withdraw the air, as is done in Mr. Boyle's vacuum, and the resistance ceases; for in this void a feather and a piece of solid gold descend with equal velocity. And as reason must place the celestial spaces above the earth's atmosphere, where there is no air to resist their motions, all bodies will move with the greatest freedom; and the planets and comets will constantly pursue their revolutions in orbits according to the mere laws of gravity, yet they could by no means have first derived their orbits from those laws.

This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God pantokratwr, or Universal Ruler.

God is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him are all things contained and moved; yet neither affects the other. As a blind man has no idea of colours, so have we no idea of the manner by which the all-wise God perceives and understands all things. He is utterly void of all body and bodily figure, and can therefore neither be seen, nor heard, or touched; nor ought he to be worshiped under the representation of any corporeal thing. We have ideas of his attributes, but what the real substance of any thing is we know not. In bodies, we see only their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savours; but their inward substances are not to be known either by our senses, or by any reflex act of our minds: much less, then, have we any idea of the substance of God. We know him only by his most wise and excellent contrivances of things, and final cause: we admire him for his perfections; but we reverence and adore him on account of his dominion: for a god without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind metaphysical necessity, which is certainly the same always and every where, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to Natural Philosophy.

Hitherto we have explained the phænomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, without suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts, but according to the quantity, of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always in the duplicate proportion of the distances. Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed; but I have not been able to discover the cause of those properties of gravity from phænomena, and I frame no hypotheses; for whatever is not deduced from the phænomena is to have no place in experimental philosophy. To us it is enough that gravity does really exist, and acts according to the laws which we have explained.

And now we might add something concerning a certain most subtle Spirit which pervades and lies hid in all gross bodies; by the force and action of which Spirit the particles of bodies mutually attract one another, and cohere, and electric bodies operate, light is emitted, reflected, refracted, inflected, and heats bodies; and the members of animal bodies move at the command of the will by the vibrations of this Spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles.

● Copyright © 2014 Glyn Hughes.
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