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Opus in profectus

# Mathematical Principlesof Natural Philosophy

## Definitions

### Definition I.

Quantitas materiæ est mensura ejusdem orta ex illius densitate et magnitudine conjunctim.   The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

Aer densitate duplicata, in spatio etiam duplicato, sit quadruplus; in triplicato sextuplus. Idem intelige de nive & pulveribus per compressionem vel liquesactionem condensatis. Et par eft ratio corporum omnium, quæ per caufas quascunque diversimode condensantur. Medii interea, si quod fuerit, interstitia partium libere pervadentis, hic nullam rationem habeo. Hanc autem quantitatem sub nomine corporis vel masse in sequentibus passim intelligo. Innotescit ea per corporis cujusque pondus: Nam ponderi proportionalem esse reperi per experimenta pendulorum accuratissime instituta, uti posthac docebitur.   Thus air of double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any caused whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.

### Definition II.

Quantitas motus est mensura ejusdem orta ex velocitate et quantite materiæ conjunctim.   The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly.

Motus totius est summa motuum in partibus singulis; ideoque in corpore duplo majore, æ quali cum velocitate, duplus est, & dupla cum velocitate quadruplus.   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.

### Definition III.

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

Hæc semper proportionalis est suo corpori, neque differt quicquam ab inertia massæ, nisi in modo concipendi. Per inertiam materiæ fit, ut corpus omne de statu suo vel quiescendi vel movendi difficulter deturbetur. Unde etiam vis insita nomine significatissimo vis Inertiæ dici possit. Exercet vero corpus hanc vim solumodo im mutatione status sui per vim aliam in se impressam facta; estque exercitium exercitium illud sub diverso respectu & resistentia & impetus:Resistentia, quatenus corpus ad conservandum statun suum reluctatur vi umpresse; impetus, qutenus corpus idem, vi resistentis obstaculi difficulter cadendo, conatur statum obstaculi illius mutare. Vulgus resistentiam quiescentibus & impetum moventibus tribuit: sed motus & quies, uti vulgo concipiuntur, respectu solo distinguuntur ab invicem; neque semper vere quiescunt quae vulgo tanquam quiescentia spectantur.   This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertiæ, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition; and the exercise of this force may be considered both as resistance and impulse; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.

### Definition IV.

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

Consistit hæc vis in actione sola, neque post actionem permanet in corpore. Perserverat enim corpus in statu omni novo per solam vim inertiæ. Est autem vis impresa diversarum originum, ut ex ictu, ex pressione, ex vi centripeta.   This force consists in the action only; and remains no longer in the body when the action is over. For a body maintains every new state it acquires, by its vis inertiæ only. Impressed forces are of different origins as from percussion, from pressure, from centripetal force.

## AxiomsorLaws of Motion

### Law I.

Corpus omne perſeverare in ſtatu ſuo quieſcendi vel movendi uniformiter in directum, niſi quatennus illud a viribus impreſſi cogitur ſtatum suum mutare.   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.

Projectilia perſeverant in motibus ſuis, niſi quatenus a reſiſtentia aëris retardantur, & vi gravitatis impelluntur deorſum. Trochus, cujus partes cohærendo perpetuo retrahunt ſeſe a motibus rectilineis, non ceſſat rotari, niſi quatenus ab aëre retardantur. Majora autem planetarum & cometarum corpora motus ſuos & progreſſivos & circulares in ſpatiis minus reſiſtentibus factos conſervant diutius.   Projectiles continue 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, whose parts by their cohesion are continually drawn aside from rectilinear motions, does not cease its rotations, otherwise than it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in freer spaces, persevere in their motions both progressive and circular for a much longer time.

### Law II.

Mutationem motus proportionalem eſſe vi motrici impreſſæ, & fieri ſecundum lineaum rectam qua vis illa imprimitur.   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.

Si vis aliqua motum quemvis generet; dupla duplum, tripla triplum generabit, ſive ſimul & ſemel, ſive gradatim & ſucceſſive impreſſa fuerit. Et hic motus quoniam in eandem ſemper plagam cum vi generatrice determinatur, ſi corpus antea movebatur, motui ejus vel conſpiranti additur, vel contrario ſubducitur, vel obliquo oblique adjicitur, & cum eo ſecundum utriuſque determinationem componitur.   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.

### Law III.

Actioni contrariam ſemper & æqalem eſſe reactionem: ſive corporum duorum actiones in ſe mutuo ſemper eſſe æqualis & in partes contrarias dirigi.   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.

Quicquid premit vel trahit alterum, tantundem ab eo premitur vel trahitur. Si quis lapidem digito premit, premitur & hujus digitus a lapide. Si equus lapidem funi alligatum trahit, retrahetur etiam & equus (ut it dicam) æqualiter in lapidem: nam funis utrinque diſtentus eodem relaxandi ſe conatu urgebit equum verſus lapidem, ac lapidem verſus equum; tantumque impediet progreſſum unius quantum promovet progreſſum alterius. Si corpus aliquod in corpus aliud impingens, motum ejus vi ſua quomodocunque mutaverit, idem quoque viciſſim in motu proprio eandem mutationem in partem contrariam vi alterius (ob æqualitatem preſſionis mutuæ) ſubibit. His actionibus æquales fiunt mutationes, non velocitatum, ſed motuum, (ſcilicet in corporibus non aliunde impeditis). Mutationes enim velocitatum, in contrarias itidem partes factæ, quia motus æqualiter mutantur, ſunt corporibus reciproce proportionales.   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; for the distended rope, by the same endeavor to relax or unbend itself, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of one another as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these action equal, not in the velocities but in the motions of the bodies (that is to say, if the bodies are not hindered by any other impediments). For, because the motions are equally changed, the changes of the velocities made towards contrary parts are inversely proportional to the bodies.

Isaac Newton, 1686, 1713, 1726   Translated by Andrew Motte, 1729