Prolegomena · Text of the first edition, 1687 · pp. 1–12

Definitiones

Definitions

Latin complete · fully annotated

Headnote

Newton opens with eight definitions and then a long scholium. That is an odd way into a book about planets, but he had a real problem to solve. In 1687 the words force, motion and quantity of matter were used loosely by everybody. Descartes measured "quantity of motion" by speed times size and ignored direction. "Force" could mean a push, an impetus a body carries inside it, or a tendency. You cannot prove theorems with words that wobble, so before any mathematics Newton fixes their meaning. He defines mass (I), momentum (II), inertia (III), impressed force (IV), and the centripetal force that will govern the whole book (V–VIII), split into three separate measures. The Scholium then says what he will not define (time, space, place and motion), because everyone thinks they know them, and it argues that true motion can be told apart from merely apparent motion. Its proof is a spinning bucket of water.

Sentence by sentence 131 Latin sentences on this page, each paired with one English sentence. Select a Latin sentence and its English comes alongside; select an English sentence and the page brings the Latin to the centre, then the English follows. Arrow keys step, Esc clears. Where the 1687 page sets a word apart (roman in italic text, italic in roman text), the English sets its counterpart apart too; hover either to light the other.

PHILOSOPHIÆ
NATURALIS
Principia
MATHEMATICA.

Definitiones.

MATHEMATICAL
PRINCIPLES
of Natural
PHILOSOPHY.

Definitions.

The title, again

The gist, in plain English

Mathematical Principles of Natural Philosophy. Definitions.

Words

philosophia naturalis: natural philosophy: what we now call physics, together with astronomy and much else
principia: “first principles”, “foundations” (neuter plural of principium)
definitiones: definitions

Commentary

The title repeats on the first page of the text. Mathematica carries weight here. Descartes had written Principia Philosophiae (1644), a natural philosophy of vortices argued in words. Newton’s title is a correction: the principles of nature are to be mathematical, so they can be stated exactly and either proved or refuted.

Def. I.

Quantitas Materiæ est mensura ejusdem orta ex illius Densitate & Magnitudine conjunctim.

Aer duplo densior in duplo spatio quadruplus est. Idem intellige de Nive et Pulveribus per compressionem vel liquefactionem condensatis. Et par est ratio corporum omnium, quæ per causas quascunq; diversimode condensantur. Medii interea, si quod fuerit, interstitia partium libere pervadentis, hic nullam rationem habeo. Hanc autem quantitatem sub nomine corporis vel Massæ in sequentibus passim intelligo. Innotescit ea per corporis cujusq; pondus. Nam ponderi proportionalem esse reperi per experimenta pendulorum accuratissime instituta, uti posthac docebitur.

Definition I.

The quantity of matter is the measure of it that arises from its density and its bulk taken together.

Air twice as dense in twice the space is four times as much. Understand the same of snow and of powders condensed by compression or liquefaction. And the reasoning is the same for all bodies that are condensed in different ways by any causes whatever. Of a medium, if there is one, freely pervading the interstices between the parts, I take no account here. It is this quantity that I mean everywhere in what follows by the name of body or mass. It becomes known through the weight of each body. For I have found, by experiments with pendulums made with the greatest accuracy, that it is proportional to the weight, as will be shown later.

Def. I · Quantity of matter (mass)

The gist, in plain English

The quantity of matter is its measure, arising from its density and its bulk taken together. Air of double density in double the space is four times as much. The same goes for snow or powders condensed by squeezing or melting, and for any body condensed in any way. I take no account of any medium, if there is one, that passes freely between the parts. From here on I mean this quantity whenever I say body or mass. It is known from the body’s weight: by very careful pendulum experiments I have found it proportional to weight, as I will show later.

Words

quantitas materiæ: quantity of matter, i.e. mass
conjunctim: “jointly”: multiplied together
magnitudo: size, volume (bulk), not “magnitude” in the abstract
massa: Newton’s new technical noun, the ancestor of our “mass”
innotescit: “becomes known”
posthac docebitur: “it will be shown later”: Book III, Prop. VI, the pendulum tests with gold, silver, lead, glass, sand, salt, wood, water and wheat

Commentary

The problem. Newton needs a quantity that belongs to a body itself and does not depend on where the body is. Weight fails that test: a body weighs less on a mountaintop (he says so in Def. VII), and Book III will show it weighs less still at the Moon. Volume fails too, because snow can be packed. So he defines a third thing.

The idea. Mass = density × volume, so doubling the density and doubling the space gives four times the matter. Critics since Ernst Mach have called this circular, since density is surely mass per volume. Newton is taking density as the more intuitive notion (how tightly packed the stuff is) and making mass the derived one. The definition does the job a definition has to do. Mass is additive, it is independent of place, and it becomes measurable once the last sentence is added.

The empirical punch line. The striking claim is the last one. Mass, defined as amount of stuff, is proportional to weight, and Newton checked this with pendulums to about one part in a thousand. That is the equivalence of inertial and gravitational mass. It is an experimental fact, not a definition. Einstein built general relativity on it two centuries later.

m = ρ V, W \propto m (at a given place) .

Def. II.

Quantitas motus est mensura ejusdem orta ex Velocitate et quantitate Materiæ conjunctim.

Motus totius est summa motuum in partibus singulis, adeoq; in corpore duplo majore æquali cum Velocitate duplus est, et dupla cum Velocitate quadruplus.

Definition II.

The quantity of motion is the measure of it that arises from the velocity and the quantity of matter taken together.

The motion of the whole is the sum of the motions in the individual parts; so in a body twice as large, with equal velocity, it is double, and with double the velocity, fourfold.

Def. II · Quantity of motion (momentum)

The gist, in plain English

The quantity of motion is its measure, arising from velocity and quantity of matter taken together. The motion of the whole is the sum of the motions of its parts. So in a body twice as large, moving with equal speed, it is double; with double the speed as well, it is quadruple.

Words

quantitas motus: quantity of motion, our momentum
adeoq;: = adeoque, “and so”, “therefore” (the q; is the printer’s abbreviation for -que)

Commentary

Why define this? Descartes had also said quantity of motion is "size × speed" and that God conserves its total in the world. His rules of collision built on that came out badly wrong, because he ignored direction. Newton uses the same words but means a directed quantity: in the Laws (Corol. III) motions in opposite directions subtract.

Modern form.

p = m v .

"Additive over parts" sounds trivial, but everything else rests on it. Additivity lets Newton treat a planet as one body, and it lets him prove (Leges, Corol. IV) that internal forces cannot move the centre of mass.

Def. III.

Materiæ vis insita est potentia resistendi, qua corpus unumquodq;, quantum in se est, perseverat in statu suo vel quiescendi vel movendi uniformiter in directum.

Hæc semper proportionalis est suo corpori, neq; differt quicquam ab inertia Massæ, nisi in modo concipiendi. Per inertiam materiæ fit ut corpus omne de statu suo vel quiescendi vel movendi difficulter deturbetur. Unde etiam vis insita nomine significantissimo vis inertiæ dici possit. Exercet vero corpus hanc vim solummodo in mutatione status sui per vim aliam in se impressam facta, estq; exercitium ejus sub diverso respectu et Resistentia et Impetus: Resistentia quatenus corpus ad conservandum statum suum reluctatur vi impressæ; Impetus quatenus corpus idem, vi resistentis obstaculi difficulter cedendo, conatur statum ejus mutare. Vulgus Resistentiam quiescentibus et Impetum moventibus tribuit; sed motus et quies, uti vulgo concipiuntur, respectu solo distinguuntur ab invicem, neq; semper vere quiescunt quæ vulgo tanquam quiescentia spectantur.

Definition III.

The innate force of matter is a power of resisting, by which every body, so far as in it lies, perseveres in its state either of resting or of moving uniformly straight forward.

This force is always proportional to its body, and differs in nothing from the inertia of the mass except in the way it is conceived. Because of the inertia of matter, every body is dislodged from its state of resting or of moving only with difficulty. Hence the innate force may also be called, by a most telling name, the force of inertia. But a body exerts this force only when its state is changed by another force impressed on it, and its exercise is, seen from different sides, both resistance and impetus: resistance, in so far as the body, to keep its state, struggles against the impressed force; impetus, in so far as the same body, yielding with difficulty to the force of an obstacle that resists it, endeavours to change that obstacle’s state. Common people attribute resistance to bodies at rest and impetus to bodies in motion; but motion and rest, as commonly conceived, are told apart only by point of view, and things commonly regarded as at rest are not always truly at rest.

Def. III · The innate force of matter (inertia)

The gist, in plain English

The innate force of matter is a power of resisting, by which every body, as far as in it lies, perseveres in its state of resting or of moving uniformly straight ahead. This force is always proportional to its body and differs from the inertia of mass only in the way we think of it. Because of inertia a body is only with difficulty knocked out of its state, so this innate force may be given the very telling name force of inertia. A body exercises it only when some other force acts on it to change its state. Seen one way the exercise is resistance (the body struggles to keep its state); seen another it is impetus (the body, giving way with difficulty to an obstacle, tries to change the obstacle’s state). Common people credit resistance to things at rest and impetus to things in motion, but motion and rest, in the ordinary sense, differ only relatively, and things commonly thought at rest are not always truly at rest.

Words

vis insita: “implanted force”, innate force: Newton’s phrase for what we call inertia
quantum in se est: “as far as in it lies”, “left to itself” (a phrase from Lucretius and Descartes)
in directum: “straight on”, in a straight line
difficulter deturbetur: “is dislodged with difficulty”
vis inertiæ: “force of inertia”, literally “force of inactivity, sluggishness”
vulgus: “the common people”: Newton’s recurring foil for unreflective usage

Commentary

The difficulty. For Aristotle, and in everyday experience, a moving thing needs something to keep it moving. Medieval impetus theory said a thrown stone carries an inner push that slowly wears out. Newton keeps the old word force but turns its meaning around. The vis insita does not keep the body going. It makes the body hard to change. Uniform straight-line motion now needs no cause at all, which is Law I.

A confusing word. To a modern reader "force of inertia" sounds like a contradiction, because forces change motion. Newton is clear that this "force" is exercised only when another force acts, and that it is proportional to mass ("differs from the inertia of mass only in the way we conceive it"). In modern terms it is the $m$ in $F = m a$ , seen as resistance to acceleration. The resistance/impetus pair is Law III in miniature: the same interaction looks like a push or like a resistance depending on which body you stand on.

Why the last sentence matters. "Motion and rest differ only relatively" sets up the Scholium, where Newton argues that true motion is not relative, though uniform motion cannot be told from rest (Corol. V of the Laws).

Def. IV.

Vis impressa est actio in corpus exercita, ad mutandum ejus statum vel quiescendi vel movendi uniformiter in directum.

Consistit hæc vis in actione sola, neq; post actionem permanet in corpore. Perseverat enim corpus in statu omni novo per solam vim inertiæ. Est autem vis impressa diversarum originum, ut ex ictu, ex pressione, ex vi centripeta.

Definition IV.

Impressed force is an action exerted on a body to change its state either of resting or of moving uniformly straight forward.

This force consists in the action alone, and does not remain in the body after the action. For a body perseveres in every new state by the force of inertia alone. Impressed force, moreover, has various origins: a blow, pressure, centripetal force.

Def. IV · Impressed force

The gist, in plain English

Impressed force is an action exerted on a body to change its state, either of resting or of moving uniformly in a straight line. It consists in the action alone and does not remain in the body afterwards, for the body keeps any new state by its force of inertia alone. Impressed force has various origins: a blow, pressure, centripetal force.

Words

vis impressa: “impressed force”: force in our modern sense, the agent of change
ictus: a blow, an impact
pressio: pressure, steady pushing

Commentary

The key line is neque post actionem permanet in corpore: the force does not stay in the body after the action. This is where Newton breaks with impetus theory. A kicked ball does not carry the kick inside it. The kick changed its state, and the new state then persists on its own. Force is an interaction, not a property of a body.

Newton lists a blow (instantaneous), pressure (continuous contact) and centripetal force (action toward a centre, possibly at a distance) as kinds of one thing. That puts collisions and gravitation in a single category. The proof of Prop. I in Book I takes advantage of this: it builds a continuous force out of a series of sudden blows.

Def. V.

Vis centripeta est qua corpus versus punctum aliquod tanquam ad centrum trahitur, impellitur, vel utcunq; tendit.

Hujus generis est gravitas, qua corpus tendit ad centrum Terræ: Vis magnetica, qua ferrum petit centrum Magnetis, et vis illa, quæcunq; sit, qua Planetæ perpetuo retrahuntur a motibus rectilineis, et in lineis curvis revolvi coguntur. Est autem vis centripetæ quantitas trium generum, absoluta, acceleratrix et motrix.

Definition V.

Centripetal force is that by which a body is drawn, driven, or in any way tends towards some point as to a centre.

Of this kind is gravity, by which a body tends to the centre of the Earth; magnetic force, by which iron seeks the centre of a magnet; and that force, whatever it may be, by which the planets are continually drawn back from straight-line motions and made to revolve in curved lines. The quantity of centripetal force is of three kinds: absolute, accelerative and motive.

Def. V · Centripetal force

The gist, in plain English

Centripetal force is that by which bodies are drawn, pushed, or in any way tend toward some point as to a centre. Of this kind is gravity, by which bodies tend to the centre of the Earth; magnetic force, by which iron seeks the magnet; and that force, whatever it is, by which the planets are continually drawn back from straight-line motion and made to revolve in curved lines. The quantity of centripetal force is of three kinds: absolute, accelerative and motive.

Words

vis centripeta: “centre-seeking force”: Newton’s coinage, modelled on Huygens’s vis centrifuga
trahitur, impellitur, vel utcunq; tendit: “is pulled, pushed, or tends in whatever way”: Newton leaves the mechanism open on purpose
quæcunq; sit: “whatever it may be”

Commentary

The reframing. Huygens had analysed circular motion through the centrifugal tendency of a body to fly outward. Newton turns this around. To understand an orbit, ask what pulls the body in, since by Law I it would otherwise fly off along the tangent. One word from Newton, centripeta, shifts attention from the body’s tendency to the agent acting on it.

Neutrality about cause. "Drawn, pushed, or tends in whatever way" commits Newton to nothing about how the force acts: pull at a distance, push from an invisible medium, or something else. Book I is pure mathematics of such forces. Book III identifies the planets’ force with terrestrial gravity, and the General Scholium refuses to guess its cause (hypotheses non fingo). The phrase "that force, whatever it is" anticipates the end of the book.

Def. VI.

Vis centripetæ quantitas absoluta est mensura ejusdem major vel minor pro efficacia causæ eam propagantis a centro per regiones in circuitu.

Uti virtus Magnetica major in uno magnete, minor in alio.

Definition VI.

The absolute quantity of a centripetal force is its measure, greater or less according to the efficacy of the cause that propagates it from the centre through the regions round about.

As the magnetic power is greater in one magnet and less in another.

Def. VI · Absolute quantity of centripetal force

The gist, in plain English

The absolute quantity of a centripetal force is its measure, greater or less according to the efficacy of the cause that propagates it from the centre through the regions round about. For example, magnetic power is greater in one magnet and less in another.

Words

efficacia causæ: “the effectiveness of the cause”: the strength of the source
in circuitu: “round about”, in the surrounding space

Commentary

Newton splits "how strong is this force?" into three separate questions (Defs. VI–VIII), because the planets force the distinction on him. Absolute strength belongs to the source: a stronger magnet, a more massive Sun. In Newtonian gravity it corresponds to $GM$ , the source’s gravitational parameter, which Book III compares for the Sun, Jupiter, Saturn and the Earth by timing their satellites.

Def. VII.

Vis centripetæ quantitas acceleratrix est ipsius mensura Velocitati proportionalis, quam dato tempore generat.

Uti Virtus Magnetis ejusdem major in minori Distantia, minor in majori: vel vis gravitans major in Vallibus, minor in cacuminibus præaltorum montium (ut experimento pendulorum constat) atq; adhuc minor (ut posthac patebit) in majoribus distantiis a Terra; in æqualibus autem distantiis eadem undiq; propterea quod corpora omnia cadentia (gravia an levia, magna an parva) sublata Aeris resistentia, æqualiter accelerat.

Definition VII.

The accelerative quantity of a centripetal force is its measure proportional to the velocity that it generates in a given time.

As the power of the same magnet is greater at a smaller distance and less at a greater; or the force of gravity is greater in valleys and less on the peaks of very high mountains (as experiments with pendulums establish), and less still (as will appear later) at greater distances from the Earth, but at equal distances the same everywhere, because it accelerates all falling bodies (heavy or light, large or small) equally once the resistance of the air is taken away.

Def. VII · Accelerative quantity of centripetal force

The gist, in plain English

The accelerative quantity of a centripetal force is its measure proportional to the velocity it generates in a given time. So the power of the same magnet is greater at a small distance and less at a greater one. Gravity is greater in valleys and less on the tops of very high mountains (as pendulum experiments show), and less still (as will appear) at greater distances from the Earth. At equal distances it is the same everywhere, because it accelerates all falling bodies equally, heavy or light, large or small, once air resistance is removed.

Words

acceleratrix: “accelerative”: per unit of mass, belonging to a place
cacumina: summits, peaks
præalti montes: very high mountains
sublata Aeris resistentia: “with air resistance taken away” (ablative absolute)
gravia an levia: “heavy or light”

Commentary

Accelerative strength belongs to a place. It is the velocity gained per unit time by any body placed there. In modern terms it is the field strength $g (r)$ , an acceleration. Newton can define a property of places, independent of the body put there, only because of Galileo’s fact that all bodies fall alike in a vacuum. The definition contains an empirical law.

a = \frac{Δ v}{Δ t} (at a given place, for any body) .

The parenthesis "less still, as will appear, at greater distances from the Earth" points ahead to the Moon test of Book III, Prop. IV: the Moon’s fall in one minute matches terrestrial gravity diluted by $1/6 0^{2}$ .

Def. VIII.

Vis centripetæ quantitas motrix est ipsius mensura proportionalis motui, quem dato tempore generat.

Uti pondus majus in majori corpore, minus in minore; inq; corpore eodem majus prope terram, minus in cælis. Hæc vis est corporis totius centripetentia seu propensio in centrum & (ut ita dicam) pondus, & innotescit semper per vim ipsi contrariam & æqualem, qua descensus corporis impediri potest.

Hasce virium quantitates brevitatis gratia nominare licet vires absolutas, acceleratrices & motrices, & distinctionis gratia referre ad corpora, ad corporum loca, & ad centrum virium: Nimirum vim motricem ad corpus, tanquam conatum & propensionem totius in centrum, ex propensionibus omnium partium compositum; & vim acceleratricem ad locum corporis, tanquam efficaciam quandam, de centro per loca singula in circuitu diffusam, ad movenda corpora quæ in ipsis sunt; vim autem absolutam ad centrum, tanquam causa aliqua præditum, sine qua vires motrices non propagantur per regiones in circuitu; sive causa illa sit corpus aliquod centrale (quale est Magnes in centro vis Magneticæ vel Terra in centro vis gravitantis) sive alia aliqua quæ non apparet. Mathematicus saltem est hic conceptus. Nam virium causas & sedes physicas jam non expendo.

Est igitur vis acceleratrix ad vim motricem ut celeritas ad motum. Oritur enim quantitas motus ex celeritate ducta in quantitatem Materiæ, & vis motrix ex vi acceleratrice ducta in quantitatem ejusdem materiæ. Nam summa actionum vis acceleratricis in singulas corporis particulas est vis motrix totius. Unde juxta Superficiem Terræ, ubi gravitas acceleratrix seu vis gravitans in corporibus universis eadem est, gravitas motrix seu pondus est ut corpus: at si in regiones ascendatur ubi gravitas acceleratrix fit minor, pondus pariter minuetur, eritq; semper ut corpus in gravitatem acceleratricem ductum. Sic in regionibus ubi gravitas acceleratrix duplo minor est, pondus corporis duplo vel triplo minoris erit quadruplo vel sextuplo minus.

Porro attractiones et impulsus eodem sensu acceleratrices & motrices nomino. Voces autem attractionis, impulsus vel propensionis cujuscunq; in centrum, indifferenter et pro se mutuo promiscue usurpo, has vires non physice sed Mathematice tantum considerando. Unde caveat lector ne per hujusmodi voces cogitet me speciem vel modum actionis causamve aut rationem physicam alicubi definire, vel centris (quæ sunt puncta Mathematica) vires vere et physice tribuere, si forte aut centra trahere, aut vires centrorum esse dixero.

Definition VIII.

The motive quantity of a centripetal force is its measure proportional to the motion that it generates in a given time.

As weight is greater in a greater body and less in a smaller; and in the same body greater near the earth and less in the heavens. This force is the centripetence, or propensity towards the centre, of the whole body, and (so to speak) its weight; and it is always known by the force, opposite and equal to it, by which the body’s descent can be prevented.

For brevity these quantities of force may be called absolute, accelerative and motive forces, and, to keep them apart, be referred to bodies, to the places of bodies, and to the centre of forces: that is, motive force to the body, as an endeavour and propensity of the whole towards the centre, compounded of the propensities of all the parts; accelerative force to the place of the body, as a certain efficacy spread from the centre through each of the places round about, to move the bodies that are in them; and absolute force to the centre, as endowed with some cause without which the motive forces are not propagated through the regions round about, whether that cause be some central body (such as a magnet at the centre of a magnetic force, or the Earth at the centre of the force of gravity) or some other cause that does not appear. This concept, at any rate, is mathematical. For I am not now weighing the physical causes and seats of forces.

Accelerative force, then, is to motive force as speed is to motion. For the quantity of motion arises from the speed multiplied by the quantity of matter, and the motive force from the accelerative force multiplied by the quantity of the same matter. For the sum of the actions of the accelerative force on the individual particles of a body is the motive force of the whole. Hence near the surface of the Earth, where accelerative gravity, or the force of gravity, is the same in all bodies, motive gravity, or weight, is as the body; but if one goes up into regions where accelerative gravity becomes less, the weight will be diminished likewise, and will always be as the body multiplied by the accelerative gravity. So in regions where accelerative gravity is half as great, the weight of a body half or a third as large will be a quarter or a sixth as great.

Further, I call attractions and impulses accelerative and motive in the same sense. And I use the words attraction, impulse, or propensity of any kind towards a centre indifferently and interchangeably, considering these forces not physically but only mathematically. So let the reader beware of thinking that by words of this kind I anywhere define a kind or manner of action, or a physical cause or reason, or that I attribute forces truly and physically to centres (which are mathematical points), if I should happen to say that centres attract, or that forces belong to centres.

Def. VIII · Motive quantity, and the three measures together

The gist, in plain English

The motive quantity of a centripetal force is its measure proportional to the motion it generates in a given time. So weight is greater in a larger body and less in a smaller, and greater in the same body near the Earth than in the heavens. This is the centripetency, or tendency toward the centre, of the whole body: its weight, so to speak. It is always known by the equal and opposite force that can stop the body from descending.

For brevity these quantities may be called motive, accelerative and absolute forces, and kept distinct by referring them to the body, to the body’s place, and to the centre of force. The motive force belongs to the body, as the striving of the whole toward the centre made up of the strivings of all its parts. The accelerative force belongs to the body’s place, as a kind of efficacy spread from the centre through each surrounding place to move the bodies found there. The absolute force belongs to the centre, as endowed with some cause without which motive forces would not spread through the surrounding regions, whether that cause is a central body (a magnet at the centre of magnetic force, or the Earth at the centre of gravity) or something else that does not appear. This notion is purely mathematical; I am not now considering the physical causes and seats of forces.

So accelerative force is to motive force as velocity is to motion. Quantity of motion comes from velocity times quantity of matter, and motive force from accelerative force times the same quantity of matter. … Hence near the Earth’s surface, where accelerative gravity is the same for all bodies, motive gravity (weight) is as the body. If we go up to regions where accelerative gravity is less, weight decreases in the same ratio, and it will always be as the body multiplied by the accelerative gravity. … Finally, I use the words attraction, impulse or any propensity toward a centre interchangeably and without distinction, considering these forces not physically but only mathematically. The reader should not think that by these words I am defining any kind or manner of action, or a physical cause or reason, or that I am attributing forces truly and physically to centres (which are mathematical points) when I happen to say that centres attract or that there are central forces.

Words

motrix: “motive”: the total force on a body, our $F$
centripetentia: a Newtonian coinage: “centre-seeking-ness”
ut ita dicam: “so to speak”
brevitatis gratia: “for the sake of brevity”
conatus: striving, endeavour (a technical term in 17th-century mechanics)
indifferenter & pro se mutuo: “indifferently and interchangeably”
puncta Mathematica: mathematical points

Commentary

The triad, assembled. For a body of mass $m$ at a place where the accelerative force is $g$ , produced by a source of absolute strength $μ$ :

motive (body) F = m accelerative (place) g (r), g (r) = \frac{absolute (centre) μ}{r ^{2}} for gravity, as Book III will show.

"Accelerative is to motive as velocity is to motion" is Newton’s way of writing $F = m g$ next to $p = m v$ .

Why this is not pedantry. Book III depends on keeping these apart. Accelerative gravity toward the Sun is the same for every planet at a given distance, which is why Kepler’s third law holds across planets of different sizes. Motive gravity (weight) differs from body to body. Absolute gravity distinguishes the Sun from Jupiter. Mix them up and the argument for universal gravitation falls apart.

The famous disclaimer. The last paragraph, with mathematice tantum, matters as much as the formulas. Newton announces that "attraction" is shorthand for a mathematical quantity. He is not claiming that points pull or that matter acts where it is not. Leibniz and the Cartesians accused him of bringing back occult qualities. This paragraph, and later the General Scholium, are his answer.

Scholium.

Hactenus voces minus notas, quo in sensu in sequentibus accipiendæ sunt, explicare visum est. Nam tempus, spatium, locum et motum ut omnibus notissima non definio. Dicam tamen quod vulgus quantitates hasce non aliter quam ex relatione ad sensibilia concipit. Et inde oriuntur præjudicia quædam, quibus tollendis convenit easdem in absolutas & relativas, veras & apparentes, Mathematicas et vulgares distingui.

I. Tempus absolutum verum & Mathematicum, in se & natura sua absq; relatione ad externum quodvis, æquabiliter fluit, alioq; nomine dicitur Duratio; relativum apparens & vulgare est sensibilis & externa quævis Durationis per motum mensura, (seu accurata seu inæquabilis) qua vulgus vice veri temporis utitur; ut Hora, Dies, Mensis, Annus.

II. Spatium absolutum natura sua absq; relatione ad externum quodvis semper manet similare & immobile; relativum est spatii hujus mensura seu dimensio quælibet mobilis, quæ a sensibus nostris per situm suum ad corpora definitur, & a vulgo pro spatio immobili usurpatur: uti dimensio spatii subterranei, aerei vel cælestis definita per situm suum ad Terram. Idem sunt spatium absolutum & relativum, specie & magnitudine, sed non permanent idem semper numero. Nam si Terra, verbi gratia, movetur, spatium Aeris nostri quod relative & respectu Terræ semper manet idem, nunc erit una pars spatii absoluti in quam Aer transit, nunc alia pars ejus, & sic absolute mutabitur perpetuo.

III. Locus est pars spatii quam corpus occupat, estq; pro ratione spatii vel absolutus vel relativus. Partem dico spatii, non situm corporis vel superficiem ambientem. Nam solidorum æqualium æquales semper sunt loci; Superficies autem ob dissimilitudinem figurarum ut plurimum inæquales sunt; situs vero proprie loquendo quantitatem non habent, neq; tam sunt loca quam affectiones locorum. Motus totius idem est cum summa motuum partium, hoc est, translatio totius de ipsius loco eadem cum summa translationum partium de locis suis, adeoq; locus totius idem cum summa locorum partium, & propterea internus & in corpore toto.

IV. Motus absolutus est translatio corporis de loco absoluto in locum absolutum, relativus de relativo in relativum. Sic in Navi quæ velis passis fertur, relativus corporis locus est navis regio illa in qua corpus versatur, seu cavitatis totius pars illa quam corpus implet, quæq; adeo movetur una cum Navi; & Quies relativa est permansio corporis in eadem illa navis regione vel parte cavitatis. At Quies vera est permansio corporis in eadem parte spatii illius immoti in qua Navis ipsa una cum cavitate sua & contentis universis movetur. Unde si Terra vere quiescit, corpus quod relative quiescit in Navi, movebitur vere et absolute ea cum Velocitate qua Navis movetur in Terra. Sin Terra etiam movetur, orietur verus et absolutus corporis motus partim ex Terræ motu vero in spatio immoto, partim ex Navis motu relativo in Terra; et si corpus etiam movetur relative in Navi, orietur verus ejus motus partim ex vero motu Terræ in spatio immoto, partim ex relativis motibus tum Navis in Terra, tum corporis in Navi, et ex his motibus relativis orietur corporis motus relativus in Terra. Ut si Terræ pars illa ubi Navis versatur moveatur vere in Orientem, cum Velocitate partium 10010, et velis ventoq; feratur Navis in Occidentem cum Velocitate partium decem, Nauta autem ambulet in Navi Orientem versus cum Velocitatis parte una, movebitur Nauta vere et absolute in spatio immoto cum Velocitatis partibus 10001 in Orientem, et relative in Terra Occidentem versus cum Velocitatis partibus novem.

Tempus absolutum a relativo distinguitur in Astronomia per Æquationem Temporis vulgi. Inæquales enim sunt dies Naturales, qui vulgo tanquam æquales pro Mensura Temporis habentur. Hanc inæqualitatem corrigunt Astronomi ut ex veriore Tempore mensurent motus cælestes. Possibile est ut nullus sit motus æquabilis quo Tempus accurate mensuretur. Accelerari & retardari possunt motus omnes, sed fluxus Temporis absoluti mutari nequit. Eadem est duratio seu perseverantia existentiæ rerum, sive motus sint celeres, sive tardi, sive nulli; proinde hæc a mensuris suis sensibilibus merito distinguitur, & ex ijsdem colligitur per Æquationem Astronomicam. Hujus autem æquationis in determinandis Phænomenis necessitas, tum per experimentum Horologii oscillatorii, tum etiam per Eclipses Satellitum Jovis evincitur.

Ut partium Temporis ordo est immutabilis, sic etiam ordo partium Spatii. Moveantur hæ de locis suis, & movebuntur (ut ita dicam) de seipsis. Nam Tempora & Spatia sunt sui ipsorum & rerum omnium quasi loca. In Tempore quoad ordinem successionis; in Spatio quoad ordinem situs locantur universa. De illorum Essentia est ut sint loca, & loca primaria moveri absurdum est. Hæc sunt igitur absoluta loca, & solæ translationes de his locis sunt absoluti motus.

Verum quoniam hæ spatii partes videri nequeunt, & ab invicem per sensus nostros distingui, earum vice adhibemus mensuras sensibiles. Ex positionibus enim & distantiis rerum a corpore aliquo, quod spectamus ut immobile, definimus loca universa; deinde etiam & omnes motus æstimamus cum respectu ad prædicta loca, quatenus corpora ab iisdem transferri concipimus. Sic vice locorum & motuum absolutorum relativis utimur, nec incommode in rebus humanis: in Philosophicis autem abstrahendum est a sensibus. Fieri etenim potest ut nullum revera quiescat corpus, ad quod loca motusq; referantur.

Distinguuntur autem Quies & Motus absoluti & relativi ab invicem per eorum proprietates, causas & effectus. Quietis proprietas est, quod corpora vere quiescentia quiescunt inter se. Ideoq; cum possibile sit ut corpus aliquod in regionibus fixarum, aut longe ultra, quiescat absolute; sciri autem non possit ex situ corporum ad invicem in regionibus nostris, utrum horum aliquod ad longinquum illud datam positionem servet, quies vera ex horum situ inter se definiri nequit.

Motus proprietas est, quod partes quæ datas servant positiones ad tota, participant motus eorundem totorum. Nam gyrantium partes omnes conantur recedere de axe motus, et progredientium impetus oritur ex conjuncto impetu partium singularum. Igitur motis corporibus ambientibus, moventur quæ in ambientibus relative quiescunt. Et propterea motus verus et absolutus definiri nequit per translationem e vicinia corporum, quæ tanquam quiescentia spectantur. Debent corpora externa non solum tanquam quiescentia spectari, sed etiam vere quiescere. Alioquin inclusa omnia, præter translationem e vicinia ambientium, participabunt etiam ambientium motus veros, et sublata illa translatione non vere quiescent, sed tanquam quiescentia solummodo spectabuntur; sunt enim ambientia ad inclusa ut totius pars exterior ad partem interiorem, vel ut cortex ad nucleum. Moto autem cortice, nucleus etiam, absq; translatione de vicinia corticis, ceu pars totius, movetur.

Præcedenti proprietati affinis est, quod moto loco movetur una locatum, adeoq; corpus, quod de loco moto movetur, participat etiam loci sui motum. Igitur motus omnes, qui de locis motis fiunt, sunt partes solummodo motuum integrorum et absolutorum, et motus omnis integer componitur ex motu corporis de loco suo primo, et motu loci hujus de loco suo, et sic deinceps, usq; dum perveniatur ad locum immotum, ut in exemplo Nautæ supra memorato. Unde motus integri et absoluti non nisi per loca immota definiri possunt, et propterea hos ad loca immota, relativos ad mobilia supra retuli: Loca autem immota non sunt, nisi quæ omnia ab infinito in infinitum datas servant positiones ad invicem, atq; adeo semper manent immota, spatiumq; constituunt quod immobile appello.

Causæ, quibus motus veri et relativi distinguuntur ab invicem, sunt vires in corpora impressæ ad motum generandum. Motus verus nec generatur nec mutatur nisi per vires in ipsum corpus motum impressas: at motus relativus generari et mutari potest absq; viribus impressis in hoc corpus. Sufficit enim ut imprimantur in alia solum corpora ad quæ fit relatio, ut ijs cedentibus mutetur relatio illa in qua hujus quies vel motus relativus consistit. Rursus motus verus a viribus in corpus motum impressis semper mutatur, at motus relativus ab his viribus non mutatur necessario. Nam si eædem vires in alia etiam corpora, ad quæ fit relatio, sic imprimantur ut situs relativus conservetur, conservabitur relatio in qua motus relativus consistit. Mutari igitur potest motus omnis relativus ubi verus conservatur, et conservari ubi verus mutatur; et propterea motus verus in ejusmodi relationibus minime consistit.

Effectus quibus motus absoluti et relativi distinguuntur ab invicem, sunt vires recedendi ab axe motus circularis. Nam in motu circulari nude relativo hæ vires nullæ sunt, in vero autem et absoluto majores vel minores pro quantitate motus. Si pendeat situla a filo prælongo, agaturq; perpetuo in orbem donec filum a contorsione admodum rigescat, dein impleatur aqua, et una cum aqua quiescat; tum vi aliqua subitanea agatur motu contrario in orbem, et filo se relaxante, diutius perseveret in hoc motu: superficies aquæ sub initio plana erit, quemadmodum ante motum vasis, at postquam, vi in aquam paulatim impressa, effecit vas, ut hæc quoq; sensibiliter revolvi incipiat, recedet ipsa paulatim e medio, ascendetq; ad latera vasis, figuram concavam induens, (ut ipse expertus sum) et incitatiore semper motu ascendet magis & magis, donec revolutiones in æqualibus cum vase temporibus peragendo, quiescat in eodem relative. Indicat hic ascensus conatum recedendi ab axe motus, & per talem conatum & innotescit & mensuratur motus aquæ circularis verus & absolutus, motuiq; relativo hic omnino contrarius. Initio ubi maximus erat aquæ motus relativus in vase, motus ille nullum excitabat conatum recedendi ab axe: Aqua non petebat circumferentiam ascendendo ad latera vasis, sed plana manebat, & propterea motus illius circularis verus nondum inceperat. Postea vero ut aquæ motus relativus decrevit, ascensus ejus ad latera vasis indicabat conatum recedendi ab axe, atq; hic conatus monstrabat motum illius circularem verum perpetuo crescentem, ac tandem maximum factum ubi aqua quiescebat in vase relative. Igitur conatus iste non pendet a translatione aquæ respectu corporum ambientium, & propterea motus circularis verus per tales translationes definiri nequit. Unicus est corporis cujusq; revolventis motus vere circularis, conatui unico tanquam proprio & adæquato effectui respondens; motus autem relativi pro varijs relationibus ad externa innumeri sunt, & relationum instar, effectibus veris omnino destituuntur, nisi quatenus de vero illo & unico motu participant. Unde & in Systemate eorum qui Cælos nostros infra Cælos fixarum in orbem revolvi volunt, & Planetas secum deferre; Planetæ & singulæ Cælorum partes, qui relative quidem in Cælis suis proximis quiescunt, moventur vere. Mutant enim positiones suas ad invicem (secus quam fit in vere quiescentibus) unaq; cum cælis delati participant eorum motus, & ut partes revolventium totorum, ab eorum axibus recedere conantur.

Igitur quantitates relativæ non sunt eæ ipsæ quantitates quarum nomina præ se ferunt, sed earum mensuræ illæ sensibiles (veræ an errantes) quibus vulgus loco mensuratarum utitur. At si ex usu definiendæ sunt verborum significationes; per nomina illa Temporis, Spatij, Loci & Motus proprie intelligendæ erunt hæ mensuræ; & sermo erit insolens & pure Mathematicus si quantitates mensuratæ hic subintelligantur. Proinde vim inferunt Sacris literis qui voces hasce de quantitatibus mensuratis ibi interpretantur. Neq; minus contaminant Mathesin & Philosophiam qui quantitates veras cum ipsarum relationibus & vulgaribus mensuris confundunt.

Motus quidem veros corporum singulorum cognoscere, & ab apparentibus actu discriminare, difficillimum est; propterea quod partes spatij illius immobilis in quo corpora vere moventur, non incurrunt in sensus. Causa tamen non est prorsus desperata. Nam suppetunt argumenta partim ex motibus apparentibus, qui sunt motuum verorum differentiæ, partim ex viribus quæ sunt motuum verorum causæ & effectus. Ut si globi duo ad datam ab invicem distantiam filo intercedente connexi, revolverentur circa commune gravitatis centrum; innotesceret ex tensione fili conatus globorum recedendi ab axe motus, & inde quantitas motus circularis computari posset. Deinde si vires quælibet æquales in alternas globorum facies ad motum circularem augendum vel minuendum simul imprimerentur, innotesceret ex aucta vel diminuta fili tensione augmentum vel decrementum motus; & inde tandem inveniri possent facies globorum in quas vires imprimi deberent, ut motus maxime augeretur, id est facies posticæ, sive quæ in motu circulari sequuntur. Cognitis autem faciebus quæ sequuntur & faciebus oppositis quæ præcedunt, cognosceretur determinatio motus. In hunc modum inveniri posset & quantitas & determinatio motus hujus circularis in vacuo quovis immenso, ubi nihil extaret externum & sensibile, quocum globi conferri possent. Si jam constituerentur in spatio illo corpora aliqua longinqua datam inter se positionem servantia, qualia sunt stellæ fixæ in regionibus nostris: sciri quidem non posset ex relativa globorum translatione inter corpora, utrum his an illis tribuendus esset motus. At si attenderetur ad filum & inveniretur tensionem ejus illam ipsam esse quam motus globorum requireret; concludere liceret motum esse globorum, & tum demum ex translatione globorum inter corpora, determinationem hujus motus colligere. Motus autem veros ex eorum causis, effectibus & apparentibus differentijs colligere, & contra, ex motibus seu veris seu apparentibus, eorum causas & effectus, docebitur fusius in sequentibus. Hunc enim in finem Tractatum sequentem composui.

Scholium.

So far it has seemed right to explain the less familiar words, in the sense in which they are to be taken in what follows. Time, space, place and motion I do not define, as being very well known to everyone. I will say only that common people conceive these quantities in no other way than through their relation to things perceived by the senses. And from this arise certain prejudices, to remove which it is convenient to distinguish these quantities into absolute and relative, true and apparent, mathematical and common.

I. Absolute, true and mathematical time, of itself and by its own nature, without relation to anything external, flows uniformly, and by another name is called duration; relative, apparent and common time is any sensible and external measure of duration by means of motion (whether accurate or unequal), which common people use in place of true time: an hour, a day, a month, a year.

II. Absolute space, by its own nature, without relation to anything external, always remains similar and immovable; relative space is any movable measure or dimension of this space, which our senses define by its position relative to bodies, and which common people take for immovable space: such as the dimension of a space below ground, in the air or in the heavens, defined by its position relative to the Earth. Absolute and relative space are the same in kind and in size, but they do not always remain the same in number. For if the Earth, for example, moves, the space of our air, which relatively and with respect to the Earth always remains the same, will at one time be one part of absolute space into which the air passes, at another time another part of it, and so, absolutely speaking, it will be changing all the time.

III. Place is the part of space that a body occupies, and, according to the kind of space, it is either absolute or relative. I say a part of space, not the situation of the body or its surrounding surface. For the places of equal solids are always equal, whereas their surfaces, because their shapes differ, are for the most part unequal; and situations, properly speaking, have no quantity, and are not so much places as attributes of places. The motion of the whole is the same as the sum of the motions of the parts; that is, the translation of the whole out of its place is the same as the sum of the translations of the parts out of their places; and so the place of the whole is the same as the sum of the places of the parts, and is therefore internal, and in the whole body.

IV. Absolute motion is the translation of a body from one absolute place into another absolute place; relative motion, from one relative place into another. Thus in a ship under full sail, the relative place of a body is that region of the ship in which the body is, or that part of the whole hollow which the body fills, and which therefore moves together with the ship; and relative rest is the body’s staying in that same region of the ship or part of the hollow. But true rest is the body’s staying in the same part of that unmoved space in which the ship itself, together with its hollow and all its contents, is moving. Hence if the Earth is truly at rest, a body that is relatively at rest in the ship will move truly and absolutely with the velocity with which the ship moves on the Earth. But if the Earth moves too, the true and absolute motion of the body will arise partly from the true motion of the Earth in unmoved space and partly from the relative motion of the ship on the Earth; and if the body also moves relatively in the ship, its true motion will arise partly from the true motion of the Earth in unmoved space and partly from the relative motions both of the ship on the Earth and of the body in the ship, and from these relative motions will arise the relative motion of the body on the Earth. For example, if the part of the Earth where the ship is moves truly eastward with a velocity of 10010 parts, and the ship is carried westward by sails and wind with a velocity of ten parts, while a sailor walks in the ship towards the east with one part of velocity, the sailor will move truly and absolutely in unmoved space with 10001 parts of velocity towards the east, and relatively on the Earth towards the west with nine parts of velocity.

In astronomy absolute time is distinguished from relative time by the equation of common time. For the natural days, which are commonly taken as equal and used as a measure of time, are unequal. Astronomers correct this inequality so as to measure the celestial motions by a truer time. It is possible that there is no uniform motion by which time may be measured exactly. All motions can be sped up and slowed down, but the flow of absolute time cannot be changed. The duration, or persistence, of the existence of things is the same whether motions are fast or slow or there are none; so this duration is rightly distinguished from its sensible measures, and is inferred from them by the astronomical equation. That this equation is needed in determining the phenomena is proved both by the experiment of the pendulum clock and by the eclipses of the satellites of Jupiter.

As the order of the parts of time is unchangeable, so too is the order of the parts of space. Suppose these parts moved out of their places, and they will be moved (so to speak) out of themselves. For times and spaces are, as it were, the places of themselves and of all things. All things are placed in time as to order of succession, and in space as to order of position. It belongs to their essence to be places, and that primary places should move is absurd. These, then, are absolute places, and only translations out of these places are absolute motions.

But since these parts of space cannot be seen, or told apart from one another by our senses, we use sensible measures in their place. For from the positions and distances of things from some body that we regard as immovable we define all places; and then we also reckon all motions with respect to those places, in so far as we conceive bodies to be carried away from them. So instead of absolute places and motions we use relative ones, and with no inconvenience in human affairs; but in philosophy we must abstract from the senses. For it may be that no body is really at rest to which places and motions could be referred.

Absolute and relative rest and motion are distinguished from each other by their properties, causes and effects. It is a property of rest that bodies truly at rest are at rest with respect to one another. And so, since it is possible that some body in the regions of the fixed stars, or far beyond, is absolutely at rest, but it cannot be known from the positions of bodies relative to one another in our regions whether any of them keeps a given position relative to that distant body, true rest cannot be defined from their positions relative to one another.

It is a property of motion that parts which keep given positions relative to their wholes share in the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion, and the impetus of bodies moving forward arises from the combined impetus of the individual parts. So when surrounding bodies are moved, the things that are relatively at rest within them are moved. And therefore true and absolute motion cannot be defined by translation out of the neighbourhood of bodies that are regarded as at rest. The external bodies must not only be regarded as at rest but must truly be at rest. Otherwise all the enclosed bodies, besides their translation out of the neighbourhood of the surrounding ones, will also share in the true motions of the surrounding ones, and if that translation is taken away they will not be truly at rest but only regarded as at rest; for the surrounding bodies are to the enclosed as the outer part of a whole to the inner part, or as the shell to the kernel. And when the shell moves, the kernel also moves, as part of the whole, without any translation out of the neighbourhood of the shell.

Akin to the preceding property is this: when a place is moved, what is placed in it is moved along with it; and so a body that moves out of a moving place shares also in the motion of its place. Therefore all motions that take place out of moving places are only parts of whole and absolute motions, and every whole motion is compounded of the motion of the body out of its first place, and the motion of this place out of its place, and so on, until one arrives at an unmoved place, as in the example of the sailor given above. Hence whole and absolute motions can be defined only by means of unmoved places, and for that reason I referred them above to unmoved places, and relative motions to movable ones; but the only unmoved places are those that all keep given positions relative to one another from infinity to infinity, and so always remain unmoved, and make up the space that I call immovable.

The causes by which true and relative motions are distinguished from each other are the forces impressed on bodies to generate motion. True motion is neither generated nor changed except by forces impressed on the moving body itself; but relative motion can be generated and changed without any forces impressed on this body. For it is enough that they be impressed only on the other bodies to which the relation is made, so that, as these give way, the relation in which this body’s relative rest or motion consists is changed. Again, true motion is always changed by forces impressed on the moving body, but relative motion is not necessarily changed by these forces. For if the same forces are also impressed on the other bodies to which the relation is made, in such a way that the relative position is preserved, the relation in which the relative motion consists will be preserved. Every relative motion, then, can be changed while the true motion is preserved, and preserved while the true motion is changed; and therefore true motion does not consist in relations of this kind at all.

The effects by which absolute and relative motions are distinguished from each other are the forces of receding from the axis of circular motion. For in circular motion that is purely relative these forces are nil, but in true and absolute circular motion they are greater or less according to the quantity of the motion. If a bucket hangs from a very long cord and is turned round and round until the cord is quite stiff with twisting, and is then filled with water and left at rest together with the water; and then by some sudden force it is set turning the opposite way and, as the cord untwists, keeps on in this motion for some time: the surface of the water will at first be flat, as it was before the vessel moved, but after the vessel, by the force gradually impressed on the water, has made the water too begin to revolve perceptibly, the water will gradually recede from the middle and climb the sides of the vessel, taking on a concave shape (as I have found by experiment myself), and, as its motion grows ever quicker, will climb higher and higher, until, completing its revolutions in the same times as the vessel, it is at rest relative to it. This ascent shows the endeavour to recede from the axis of motion, and by such an endeavour the true and absolute circular motion of the water, which here is wholly contrary to the relative motion, is both made known and measured. At the beginning, when the relative motion of the water in the vessel was greatest, that motion produced no endeavour to recede from the axis: the water did not make for the circumference by climbing the sides of the vessel, but stayed flat, and so its true circular motion had not yet begun. But afterwards, as the relative motion of the water decreased, its ascent up the sides of the vessel showed an endeavour to recede from the axis, and this endeavour revealed its true circular motion, continually increasing and at last becoming greatest when the water was at rest relative to the vessel. This endeavour, then, does not depend on any translation of the water with respect to the surrounding bodies, and so true circular motion cannot be defined by such translations. Each revolving body has only one truly circular motion, answering to a single endeavour as its proper and adequate effect; but its relative motions, according to its various relations to external things, are innumerable, and, like relations, are wholly without true effects, except in so far as they share in that one true motion. Hence, even in the system of those who would have our heavens revolve below the heavens of the fixed stars and carry the planets along with them, the planets and the individual parts of the heavens, which are indeed relatively at rest in their nearest heavens, truly move. For they change their positions relative to one another (which does not happen with things truly at rest), and, being carried along with the heavens, share in their motions, and, as parts of revolving wholes, endeavour to recede from their axes.

Relative quantities, then, are not the very quantities whose names they bear, but those sensible measures of them (true or erroneous) that common people use in place of the quantities measured. But if the meanings of words are to be fixed by usage, then by the names time, space, place and motion these measures will properly be understood; and the language will be unusual and purely mathematical if the quantities measured are meant here. So those who interpret these words in Holy Scripture as meaning the quantities measured do violence to the Scriptures. And those who confuse true quantities with their relations and common measures corrupt mathematics and philosophy no less.

To discover the true motions of individual bodies, and actually to tell them apart from the apparent ones, is indeed very difficult, because the parts of that immovable space in which bodies truly move do not come before the senses. Still, the case is not altogether hopeless. For there is evidence to be had partly from the apparent motions, which are the differences of the true motions, and partly from the forces, which are the causes and effects of the true motions. For example, if two globes, joined by a cord at a given distance from each other, were revolving about their common centre of gravity, the endeavour of the globes to recede from the axis of motion would become known from the tension of the cord, and from that the quantity of the circular motion could be computed. Then, if any equal forces were impressed at once on the alternate faces of the globes to increase or decrease the circular motion, the increase or decrease of the motion would become known from the increased or decreased tension of the cord; and from this, finally, one could find the faces of the globes on which the forces should be impressed to increase the motion most, that is, the rear faces, the ones that follow in the circular motion. And once the faces that follow, and the opposite faces that lead, were known, the direction of the motion would be known. In this way both the quantity and the direction of this circular motion could be found in any immense vacuum, where there was nothing external and perceptible with which the globes could be compared. If now some distant bodies were placed in that space, keeping a given position among themselves, like the fixed stars in our regions, it could not indeed be known from the relative translation of the globes among the bodies whether the motion should be attributed to the globes or to the bodies. But if one attended to the cord and found that its tension was exactly the tension that the motion of the globes required, one could conclude that the motion belonged to the globes, and only then infer the direction of this motion from the translation of the globes among the bodies. How to infer true motions from their causes, effects and apparent differences, and, conversely, their causes and effects from motions either true or apparent, will be taught more fully in what follows. For it was to this end that I composed the following treatise.

Scholium · Absolute time, space, place and motion

The gist, in plain English

So far I have explained how less familiar words are to be taken. Time, space, place and motion I do not define, since everyone knows them. But people conceive them only through their relation to sensible things, and that breeds prejudices. To remove them it helps to divide each into absolute and relative, true and apparent, mathematical and common.

I. Absolute, true, mathematical time flows equably of itself and by its own nature, without relation to anything external; it is also called duration. Relative, apparent, common time is any sensible, external measure of duration by motion (an hour, a day, a month, a year), whether accurate or not. II. Absolute space, by its own nature and without relation to anything external, always stays similar and immovable. Relative space is any movable measure of it, fixed by our senses through its position relative to bodies (for instance the space of our air, defined relative to the Earth). III. Place is the part of space a body occupies, absolute or relative. IV. Absolute motion is the carrying of a body from one absolute place to another; relative motion, from one relative place to another. A sailor on a ship sailing west at 10 parts while the Earth turns east at 10010 parts and the sailor walks east at 1 part moves truly 10001 parts east and, relative to the Earth, 9 parts west.

Astronomers separate absolute time from common time by the equation of time, since natural days are unequal; perhaps no motion is perfectly uniform, but the flow of absolute time cannot change. Since the parts of space cannot be seen, we use sensible measures instead, which is fine for daily life; in philosophy we must abstract from the senses. True and relative motion are distinguished by their properties, causes and effects. Cause: true motion is generated or changed only by forces impressed on the moving body itself. Effect: in true circular motion there are forces of receding from the axis. The bucket: hang a bucket by a long cord, twist the cord, fill the bucket with water, and let go. At first the bucket spins and the water does not, and the surface stays flat although the relative motion is greatest. Gradually the water takes up the motion and climbs the sides into a hollow, as I have myself tried, and the hollow is deepest when the water turns with the bucket and has no relative motion at all. So the endeavour to recede from the axis measures true rotation, not rotation relative to surrounding bodies. The two globes: even in an empty universe, two globes joined by a cord and whirling about their centre would reveal their rotation by the tension in the cord, and by pushing on opposite faces we could find its direction. To gather true motions from their causes, effects and apparent differences, and the reverse, "is what I composed the following treatise for."

Words

æquabiliter fluit: “flows evenly, uniformly”
duratio: duration
similare & immobile: “homogeneous and immovable”
Æquatio Temporis: the equation of time: the correction from apparent solar time to mean time
situla: a bucket, pail
filum prælongum: a very long cord
a contorsione admodum rigescat: “becomes quite stiff from being twisted”
conatus recedendi ab axe: “endeavour of receding from the axis”: centrifugal tendency
ut ipse expertus sum: “as I have tried myself”: one of Newton’s rare first-person experiments in the text
cortex … nucleus: shell (rind) … kernel
vim inferunt Sacris literis: “do violence to Holy Scripture”

Commentary

The problem. Law I says a body left alone moves uniformly in a straight line. But straight relative to what? On the Earth? The Earth turns. Relative to the Sun? Newton will argue the Sun itself moves. If motion only means change of position relative to other bodies (Descartes defined it that way), then Law I has no fixed meaning: the same body is both accelerating and not, depending on the frame chosen. Newton needs a background against which "uniform and straight" means something definite, and he needs a way, at least in principle, to detect motion against it.

His answer has two parts.

Postulate absolute time and space as the stage, flowing evenly and staying put whether or not anything happens.
Detect true motion by its dynamical effects rather than by looking. Rotation produces a measurable outward endeavour (curved water, a taut cord), whatever the neighbouring bodies are doing.

The bucket, read carefully. The experiment cleverly separates relative rotation from effects. Stage 1 has maximal relative rotation and no effect. Stage 3 has zero relative rotation and maximal effect. So whatever the concavity tracks, it is not rotation relative to the immediate surroundings, which is Descartes’s definition of motion. Newton’s target is Descartes, and on that point he wins.

What later physics kept. The two halves have fared differently.

Rotation is detectable: correct, and the basis of gyroscopes, the Foucault pendulum and the flattening of the Earth (Book III, Prop. XIX).
Absolute uniform velocity is detectable: Newton himself proves it is not (Leges, Corol. V): physics in a uniformly moving ship is the same as in a ship at rest. Absolute rest plays no role in his mechanics.

In modern language the Scholium picks out the inertial frames, in which

F = m r ¨

holds with no "fictitious" forces. Rotation and acceleration relative to these frames are absolute. Velocity is not. Mach later asked whether the bucket’s inertial frame is fixed by the distant stars rather than by empty space, and that question helped lead Einstein to general relativity. Newton’s two globes in an empty universe pose the problem Mach would take up.

A note on the ship. The sailor’s 10010 parts is an illustrative number, not a measured speed of rotation. The point is that velocities add (Leges, Corol. I): $10010 - 10 + 1 = 10001$ .