AXIOMA I: Omnia corpora vel moventur vel quiescunt.
Translated as,
All bodies either are moved or are at rest.
AXIOMA II: Unumquodque corpus jam tardius jam celerius movetur.
Translated as,
Each and every body is either moved slowly or quickly.
LEMMA I: Corpora ratione motus et quietis, celeritatis et tarditatis et non ratione substantiæ ab invicem distinguuntur.
Translated as,
Bodies are distinguished by reason of motion and rest, of fast and slow and not by reason of substance from one another.
DEMONSTRATIO: Primam partem hujus per se notam suppono. At quod ratione substantiæ non distinguantur corpora, patet tam ex propositione 5 quam 8 partis I sed clarius ex iis quæ in scholio propositionis 15 partis I dicta sunt.
Translated as,
I place the first part of this as known through itself. But for what bodies may not be distinguished by reason of substance, it is clear from IP5 as well as IP8 but more clearly from things which are said in IP15S.
LEMMA II: Omnia corpora in quibusdam conveniunt.
Translated as,
All bodies occur in certain ways.
DEMONSTRATIO: In his enim omnia corpora conveniunt quod unius ejusdemque attributi conceptum involvunt (per definitionem 1 hujus), deinde quod jam tardius jam celerius et absolute jam moveri jam quiescere possunt.
Translated as,
For in these ways all bodies occur which involve the concept of each and the same attribute (by IID1), next which absolutely are able to now be moved or now be at rest and now slowly and now quickly.
LEMMA III: Corpus motum vel quiescens ad motum vel quietem determinari debuit ab alio corpore quod etiam ad motum vel quietem determinatum fuit ab alio et illud iterum ab alio et sic in infinitum.
Translated as,
A body motion or rest ought to have been determined for motion or rest from another body which also has been determined for motion or rest from another and that again from another and so into infinity.
DEMONSTRATIO: Corpora (per definitionem 1 hujus) res singulares sunt quæ (per lemma 1) ratione motus et quietis ab invicem distinguuntur adeoque (per propositionem 28 partis I) unumquodque ad motum vel quietem necessario determinari debuit ab alia re singulari nempe (per propositionem 6 hujus) ab alio corpore quod (per axioma 1) etiam vel movetur vel quiescit. At hoc etiam (per eandem rationem) moveri vel quiescere non potuit nisi ab alio ad motum vel quietem determinatum fuisset et hoc iterum (per eandem rationem) ab alio et sic in infinitum. Q.E.D.
Translated as,
Bodies (by IID1) are singular things which (by IIL1) are distinguished from one another by reason of motion and of rest to such an extent (by IP28) that each and every one ought to have been necessarily determined for motion or rest by another singular thing, indeed (by IIP6) by another body which (by IIA1) also is moved or rests. But this also (by the same reason) was not able to be moved or rest unless it has been determined to motion or rest and this again (by the same reason) from another and thus to infinity.
COROLLARIUM: Hinc sequitur corpus motum tamdiu moveri donec ab alio corpore ad quiescendum determinetur et corpus quiescens tamdiu etiam quiescere donec ab alio ad motum determinetur. Quod etiam per se notum est. Nam cum suppono corpus exempli gratia A quiescere nec ad alia corpora mota attendo, nihil de corpore A dicere potero nisi quod quiescat. Quod si postea contingat ut corpus A moveatur, id sane evenire non potuit ex eo quod quiescebat; ex eo enim nil aliud sequi poterat quam ut corpus A quiesceret. Si contra supponatur A moveri, quotiescunque ad A tantum attendimus, nihil de eodem affirmare poterimus nisi quod moveatur. Quod si postea contingat ut A quiescat, id sane evenire etiam non potuit ex motu quem habebat; ex motu enim nihil aliud sequi poterat quam ut A moveretur : contingit itaque a re quæ non erat in A nempe a causa externa a qua ad quiescendum determinatum fuit.
Translated as,
From this it follows that a body is moved to so long a motion until it is determined by another body to rest and a body at rest also is at rest so long until it is determined by another to motion. Which is also known through itself. For since I supply a body an example A to rest and I do not attend to other moving bodies, I will be able to say nothing concerning body A unless because it rests. Which if later it happens that body A is moved, that was not able to happen reasonably; from this for nothing is able to follow unless it might be body A rests. If on the other hand it is supposed that A is moved, as much as we attend only to how much A, we will be able to affirm nothing concerning the same unless that it was moved. Which if it happens after that A rests, it also will be not happen reasonably from the motion which it had; from motion for nothing else is able to follow than that A was moved; thus it happens from a thing which was not in A indeed from an external cause by which it has been determined to rest.
AXIOMA I: Omnes modi quibus corpus aliquod ab alio afficitur corpore, ex natura corporis affecti et simul ex natura corporis afficientis sequuntur ita ut unum idemque corpus diversimode moveatur pro diversitate naturæ corporum moventium et contra ut diversa corpora ab uno eodemque corpore diversimode moveantur.
Translated as,
All modes - by which some body is affected by another body - follow simultaneously from the nature of the affected body and from the nature of the affecting body, in such a way that the same exact body may be moved differently according to the diversity of the natures of the bodies moving [the same exact body] and conversely that diverse bodies may be moved differently by the same exact body.
AXIOMA II: Cum corpus motum alteri quiescenti quod dimovere nequit, impingit, reflectitur ut moveri pergat et angulus lineæ motus reflectionis cum plano corporis quiescentis cui impegit, æqualis erit angulo quem linea motus incidentiæ cum eodem plano efficit. Atque hæc de corporibus simplicissimis quæ scilicet solo motu et quiete, celeritate et tarditate ab invicem distinguuntur : jam ad composita ascendamus.
Translated as,
When a moving body strikes another resting [body] which is unable to move away, [the moving body] is reflected back so that it proceeds to be moved and the angle of the line of the motion of reflection with the plane of the resting body which it struck, will be equal to the angle which the moving line of incidence makes with the same plane. And this concerning the simplest bodies which of course are distinguished from one another only by motion and by rest, by speed and slowness: now we ascend to composite bodies.
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