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| PHILOSOPHIAE NATURALIS
PRINCIPIA MATHEMATICA
AUCTORE
ISAACO NEWTONO, Eq. Aur.
Editio tertia aucta & emendata
LONDINI:
Apud Guil. & Joh. INNYS, Regiae Societatis typographos.
MDCCXXVI.
AXIOMATA,
SIVE
LEGES MOTUS.
LEX I.
Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare.
PROJECTILIA perseverant in motibus suis, nisi quatenus a resistentia aëris8 retardantur, & vi gravitatis impelluntur deorsum. Trochus,12 cujus partes cohaerendo perpetuo retrahunt sese a motibus rectilineis, non cessat rotari, nisi quatenus ab aëre retardatur. Majora autem planetarum & cometarum corpora motus suos & progressivos & circulares in spatiis minus resistentibus factos conservant diutius.
LEX II.
Mutationem motus proportionalem esse vi motrici inpressae, & fieri secundum lineam rectam qua vis illa imprimitur.
Si vis aliqua motum quemvis generet; dupla duplum, tripla triplum generabit, sive14 simul & semel, sive gradatim & successive15 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 adicitur, & cum eo secundum utriusque determinationem componitur.
LEX III.
Actioni contrariam semper & aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales & in partes contrarias dirigi.
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 ita dicam)16 aequaliter 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 aequalitatem pressionis mutuae) subibit. His actionibus aequales fiunt mutationes, non velocitatum, sed motuum; scilicet in corporibus non aliunde impeditis.
Mutationes enim velocitatum, in contrarias itidem partes factae, quia motus aequaliter mutantur, sunt corporibus reciproce proportionales.
Obtinet etiam haec lex in attractionibus, ut in scholio proximo probabitur.
COROLLARIUM I.
Corpus viribus conjunctis diagonalem parallelogrammi eodem tempore describere, quo latera separatis.
Si corpus dato tempore, vi sola M  in loco A impressa, ferretur uniformi
cum motu ab A ad B; & vi sola N in
eodem loco impressa, ferretur ab A ad C: compleatur parallelogrammum ABCD, & vi utraque feretur corpus illud eodem tempore in diagonali ab A ad D. Nam quoniam vis N agit secundum lineam AC ipsi BD parallelam, haec vis per legem II nihil mutabit velocitatem accedendi ad lineam illam BD a vi altera genitam. Accedet igitur corpus eodem tempore ad lineam BD, sive vis N imprimatur, sive non; atque ideo in fine illius temporis reperietur alicubi in linea illa BD. Eodem argumento in fine temporis ejusdem reperietur alicubi in linea CD, & idcirco in utriusque lineae concursu D reperiri necesse est. Perget autem motu rectilineo ab A ad D per legem I.
COROLLARIUM II | | Natural Philosophy Mathematical Principia Author Isaac Newton, Sir, Third Edition Reviewed and Incremented LONDON: At Guil. & Joh. Innys, Royal Printing Company MDCCXXVI
AXIOMS OR RATHER THE LAWS OF MOTION LAW I Every object perseveres in its own status of rest or of uniform motion in a straight line, until, and unless, external forces impressed upon it compell it to change its status. PROJECTLILES persevere in their own motion until, and unless, retarded by an external air resistance or impelled downwards by the force of gravity. A hoop,12 whose parts being cohesively held together, will conduct itself perpetually in a rectlinear motion; its rotation never stops, until, and unless, retarded by [the resitance of]13 external air. On the other hand, the large bodies of planets and comets with their own progressive and circular motion in spaces of lower air resistance, by that very fact preserve it for much longer. LAW II A change of the motion is proportional to the motive force that is impressed, and according to the straight line which that force imparts. If some force should generate a motion; twice [the force] will generate double [the motion], thrice [the force] will generate triple [the motion], whether [that force] will have been applied altogether and at once, or whehter it will have been applied gradually and successively. And this motion (because it always strikes along the same way with the force that has determined it) will add itself to that of the body already in motion, or if contrary17, will subtract itself from that of the body already in motion, or if sideways, it will be added to the body already in motion, obliquely, according to the compounded determination of the two. LAW III To every action there is always an equal and opposite reaction: or rather, the actions of two mutual bodies on each other are always equal and directed to opposite parts. Whatever presses or pulls on one thing, is as much as pressed, or pulled, by another thing. If someone presses on a stone with the finger of the hand, the finger is also pressed by the stone. If a horse pulls a stone tied to a rope, the horse ( so to say) will be equally pulled back towards the stone: in fact, the action of the rope, at both ends either pulled or equally loosened, will try to push the horse towards the stone, or the stone towards the horse; thus impeding the progress of the former as much as it promotes that of the latter. If a body impinges (collides) with another body, and by its force change its own motion in any way, in turn, the same changes in motion (due to the equal and mutual pressure) will be suffered by each one of them but in contrary parts. It is in these equal actions that changes are created, not to the velocity but to the motion;18 of course, provided that the bodies are not impeded by something else. In fact, the change of the velocities are equal on the parts because the motions are equally changed, being reciprocally proportional to the bodies. This law is also maintained in attractions, as will be proved in a further scholium [comment]. COROLLARY I A body joined by two forces is determined19 at the same time by the diagonal of the parallelogram, that which the sides separate. If a body, at a given time, is impressed by a single force M at point A, it would be transported with uniform motion from A to B; and if a single force N is impressed at the same point, but, [say] it would be transported from A to C: complete the parallelogram ABCD and each of the two forces will carry, at the same time, that body along the diagonal from A to D. For, since the force N acts along the line AC, identical to the parallel line BD, this force, by Law II, will not affect the velocity of the other force [M] moving towards the line BD. The body therefore will arrive to the line BD at the same time, whether the force N is applied or not; and for this reason, at the end of that time it will be found somewhere alaong the line BD. By the same argument, at the end of that time it will be found somewhere along the line CD, and for this reason each of the two will be found at point D where they both meet. On the other hand, it will direct itself in a rectilinear motion from A to D by Law I. | |
| NOTES The text was taken from a reprinted edition published in the year 1871 in Glasgow by James Maclehose, publisher to the University. This edition is freely available on the internet as a pdf file, and is based on the revised Third Edition dated 1726, when Newton was still alive, as the title above shows. He died in 1727. About this title: Newton’s most famous work Principia (1687) explains the laws governing the motion of physical objects (heavenly and otherwise). Principia rests on the new branch of mathematics that Newton invented simultaneously with Leibniz (1646-1716), calculus, a tool that allowed mathematicians to move beyond the work done by the ancient Greeks [and Roman Engineers] for the first time in almost two thousand years. The book is made available by Liberty Fund, Inc., a private educational foundation established to encourage the study of the ideal of a society of free and responsible individuals. Copyright information: The text is in the public domain. - The text was typed by myself and checked against the Liberty Fund edition, but the figures were redrawn by myself taking care to maintain an approximate identical aspect ratio to the originals.
- The language form used by Newton is a kind of modern Latin, the late mediaeval Latin common in the XVII century: Newton wrote the book in Cambridge around the year 1686 and earlier, since his book preface is dated 8 May 1686. The edition in question was therefore reviewed and updated by Sir William Thomson, LL.D., Fellow of St. Peter's College, Cambridge and Hugh Blackburn, M.A., Fellow of Trinity College, Cambridge, and Professor of Natural Philosophy and Mathematics in the University of Glasgow.
- I have chosen a section of the Principia which deals with the three Laws of Motion. There are two reasons for this choice: 1) It is one that is most familiar with readers as the three laws are taught at school, and 2) it is one of the sections that contain the least mathematical treatment, at least in the opening paragraphs.
- It is interesting to note that Newton used, in my view, an approach that is more down to earth and understandable to people non so familiar with scientific language and concepts, despite knowing that his target audience were other scientists rather than the common public.
- The abbreviations Eq. and Aur. stand for: Eq.=Equites, hence knight, hence Sir; Aur. stands for: aurati, (singular: auratus) meaning golden in relation to the wearing of spurs, or shields, or robes, ornated with gold, by knights.
- The laws are translated literally and may sound to you slightly different from the standard sentences learnt at school.
- The dieresis sign (the two dots over the ë) means that the two vowels, ae, do not form a diphtong, hence they are pronounced individually. English example: naïve.
- The format of each proposition is given in the form: Axiom followed by a Corollary. This was, then, the common way of expressing scientific truths whereas today the jargon gives more emphasis to experimental results. Einstein, too, stated axiomatic propositions. The experiments proved his statements much later! Today, Scientific Law (Leges) cannot be called Laws unless they are first proven by experiment.
- It should be noted that Newton's mathematical treatment uses the application of Calculus, a joint, but separate discovery, of both Newton and Gottfried Wilhelm Leibnitz (1646 – 1716) who, also, wrote in Latin and French, the trendy "vulgate" from Latin! His name was German, implying that he may have been born in the region of Alsace, where they grow some very good white wines, and that periodically switched sides between those two great countries of France and Germany. Today we use Leibnitz's notation, dy/dx, (read: dy by dx) instead of Newton's, small y with a dot on top, ẏ.
- Upon cross checking, Leibnitz was born in Saxony; they grow no good wines there, that may be why he went to France!
- Trochus is derived from the Greek word "trochos" and it was an ancient game consisting in rolling a hoop along the ground and pushing it with a stick. When I was a boy it was called "hoop rolling", and the hoop was made of plastic and sold to play the "hula hoop"; so, we played the hula hoop and the hoop rolling. I feel that Newton might have had the hoop rolling game in mind, and, possibly, the vision of a hoop skirt which, in those days, being very fashionable as ladies dresses, the so-called Farthingale dress, were made of wire, wicker, willow cuttings and from 1580 even whalebone. Hence the clause:"cujus partes cohaerendo". I don't think he was thinking of the binding energies that hold together the nuclei of the billions of atoms in the farthingale dress hoops.
- "The resistance of" is implied as it was mentioned in the preceding sentence of the same paragraph. Newton just did not wish to repeat it.
- sive...sive = or...or; whether...or
- successive = this is a medieval Latin word derived from successus meaning 1) success and 2) approaching as in the example: successus hostium - approaching of the enemy. Most Latin dictionaries will not include the word successive because it is medieval rather than classical Latin.
- ut ita dicam - expression used by Cicero, Quintilian and Tacitus: so to say, so to speak.
- contrary must not be understood as "hostile" but as "contrarian direction" or more simply, "opposite direction".
- motion must be understood as acceleration. Newton is explaining that the body has already a velocity and that a change of this velocity, due to the influence or impact of other forces, will affect its motion, i.e. its acceleration. The term acceleration was coined later from accelero, -as, -are = to hurry up. It is formed by accelero + tio to give: acceleratio, 3rd Decl. noun, nom. sing. and accelerationis gen, sing.
- determined: this sentence is rather difficult to translate correctly as Newton uses the present infinitive active voice of descrivere, meaning: to describe, to define, to determine, whereas I was expecting a past tense and a passive voice. Did he make an error or am I missing something? I will come back to it.
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