Philosophiae Naturalis Principia Mathematica

Sir Isaac Newton
Philosophiae Naturalis Principia Mathematica

Outline of Contents: Principia

Author’s Preface to Reader (translation by Andrew Motte, 1729)

Definitiones (text: see below with notes)  (notes)
Axiomata sive Leges Motus (text: see below) (notes)
Liber Primus: De Motu Corporum [without resistence]

Sectio I: De methodo rationum primarum & ultimarum, cujus ope sequentia demonstrantur.
Sectio II: De inventione virium centripetarum
Sectio III: De motu corporum in conicis sectionibus excentricis.
Sectio IV: De inventione orbium ellipticorum, parabolicorum & hyperbolicorum ex umbilico dato.
Sectio V: Inventio orbium ubi umbilicus neuter datur.
Sectio VI: De inventione motuum in orbibus datis.
Sectio VII: De corporum ascensu & decensu rectilineo.
Sectio VIII: De inventione orbium in quibus corpora viribus quibuscunque centripetis agitata revolvuntur.
Sectio IX: De motu corporum in orbibus mobilibus, deque motu apsidum.
Sectio X: De motu corporum in superficiebus datis, deque funipendulorum motu reciproco.
Sectio XI: De motu corporum viribus centripetis se mutuo pententium.
Sectio XII: De corporum sphaericorum viribus attractivis.
Sectio XIII: De corporum non sphaericorum viribus attractivis.
Sectio XIV: De motu corporum minimorum, quae viribus centripetis ad
singulas magni alicujus corporis partes tendentibus agitantur.

Liber Secundus: De Motu Corporum [with resistence]

Sectio I: De motu corporum quibus resistitur in ratione velocitatis.
Sectio II: De motu corporum quibus resistitur in duplicata ratione velocitatum.
Sectio III: De motu corporum quibus resistitur partim in ratione velocitatis, partim in ejusdem ratione duplicata.
Sectio IV: De corporum circulari motu in mediis resistentibus.
Sectio V: De densitate & compressione fluidorum, deque hydrostatica.
Sectio VI: De motu & resistentia corporum funependulorum.
Sectio VII: De motu fluidorum & resistentia projectilium.
Sectio VIII: De motu per fluida propagato.
Section IX: De motu circulari fluidorum.

Liber Tertius: De Mundi Systemate
Preface (translation by Andrew Motte)
            Regulae Philosophandi (text: see below) (notes)
Phaenomena.
            Propositiones.  (Tides: notes)

Scholium Generale

Characteristics of Newton’s Latin:

*use of indirect statement (e.g. the Laws and corollaries are in indirect statement)

*a/ab + ablative thing used to express means: “by”

*dative gerund(ive) to express purpose (“for _____ing; to _____”)

4th and 5th declension nouns (motus, status)
3rd and 4th conjugation futures (-am, -es, etc.)
comparative adverbs (-ius)

Word Order:

*adjective-preposition-noun word order
*demonstrative adjectives and adjectives of quantity may come after the noun
(corpus omne; vis illa; vis aliqua; corpus aliud)
*initial est, etc. used as linking verbs, not “there is,” etc.

Vocabulary:

*quatenus: to the degree that; as much as; to the extent that
*tam . . . quam (so . . as; both . . . and)
*ut/uti + individual words as examples: “as ______”
-ce added to pronouns, etc.
*pro: for; in proportion to
*secundum (prep.)
*vel . . . vel: “either . . . or”
vis, (gen. vis, dat. vi, acc. vim, abl. vi; pl. nom./acc: vires, gen.virium, dat./abl. viribus) (3rd declension “super” i-stem)
pronouns/adjectives: aliquis; quivis; idem; uterque; quisquis

Word Formation:

Nouns:
*-trix, (tric-) 3rd decl. adjective suffix (= English “-ive”)
    *-tūdō, -tūdinis, f.  (altitudo)
*-tiō, tiōnis, f.  (coniunctio, oppositio)

Adjectives:
*-āris, -e (lunaris, solaris, luminaris)
*-icus, -a, -um  (Atlanticus, Aethiopicus)
*-ālis, e
 (orientalis)
*-ōsus, -a, -um 
(uadosus)
-ēnsis, -e (Chilensis)
*-ānus, -a, -um  (meridianus, Peruvianus)

Verbs
    *-scō, -scere  (intumesco)

Adverbs
*-(t)im
    *-ē  (distincte

Philosophiae Naturalis Principia Mathematica

Title Page (3rd edition, 1726)
Notes:
Philosophiae Naturalis:  natural (i.e. relating to nature; physical) science

Liber I:

Notes:

Definitio I:

orta: < orior, oriri, ortus sum arise
conjunctim (adverb; *note –(t)im suffix): together, jointly (i.e. multiplied together)

Definitio II

line 21 *summa, -aesum
line 22 duploby two, doubly (adverbial ablative)
duplus est [motus]: duplus, -a, -um (adjective)
duplā: what does this modify?

Definitio III:

line 2: vis insitainnate force (vs. vis impressa):  The phrase was old (Horace, Cicero, Kepler, etc.). but Newton was redefining it.
3 quantum in se est: as much as in it lies; as much as possible (cf. Magna Carta; Descartes)
3-4 vel . . . vel
uniform-iter: What part of speech?

9-10 solummodo: (adv.) only [L&S s.v. solus, adv. A2]
11 factā
est:
 linking verb (not there is; see “Charcteristics” above).
11-12 & resistentia & impetus both . . . and (does not connect these with respectu)
13 vi: dative (with cedendo)
14 obstaculum, -i, nt.: obstacle, hindrance [L & S]
17 ab invicemfrom each other

Definitio IV:

20 exercitaexerted

25 vi centripeta
: see Definitio V

Definitio V:

versus: (preposition + acc.) towards
punctumpoint
utcumquein some way
Note: Newton says there are three kinds of quantities of centripital force: absoluta (Definitio VI), acceleratrix (Definitio VII), and motrix (Definitio VIII).

Definitio VI:

quantitas absolutaabsolute quantity
maior vel minor: i.e. proportional

magnes, magnetis, m. 
magnet, loadstone [L&S]
intensione
intensity
virtutisforce

 

Definitio VII:

quantitas acceleratrix (-trix adj. suffix): accelerative quanitity
eadem [vis]
propterea quod [propterea sets up “quod” because; cf. eo quod] for the reason that; because
sublata aeris resistentia: ablative absolute
sublata (< tollo): taken away
resistentia: noun (not participle): resistence

Definitio VIII. (1st paragraph)

quantitas motrixmotive quantity
uti: variant form of the conjunction “ut” (here: as)

Axiomata sive Leges Motus (text)

Notes:

Note: the laws are written in indirect statement

Lex I

line 6-7 *a viribus impressis: ablative of means with preposition “a.”
(cf. “a resistentia aeris” below.)  See Definitio IV.
line 8 quatenusto the extent that; by as much as; in as much as
line 9 deorsumdownwards
line 10 Trochus: a child’s toy hoop
line 11 recti-lin-eis: what part of speech?  (-eus suffix)
line 12 majora: what does this modify?
lines 12-13 motūs . . . in spatiis minus resistentibus factos:  motions . . .made in spaces that are less resisting/resistent, i.e. the vacuum of outer space
line 13 diutius: comparative of diū.

Lex II

(line 16) motus: genitive
vi motrici impressae: dative (with proportionalem)
*Note the adjective suffix: -tric-  (“mot-ive”). cf. generatrice (“generative”) below.
line 17 qua: in the direction in which; where
line 18 *vis (illa): noun
line 18 dupla [vis] duplum [motum]
line 19 sive . . . sivewhether  . . . or
lines 19-20 impressa fuerit: tense? what would the classical Latin form be?
line 20 plagamregion, quarter, tract; direction
line 21 determino (1): fix, settle; bound; direct
line 22 conspiranti (dative): (breathing together); agreeing, combining, harmonizing; in the same direction

Lex III

line 5 tantun-dem: (just so much) by the same amount
line 7 funi < funis, -is, m.rope
line 9 utrin-quefrom both sides
line 9 eodem relaxandi se conatu
lines 9-10 versustowards
lines 10-11 *tantum . . . quantum: . . . as much as
(line 12) (motus) eius: i.e. of the other body
lines 12-13 quomodo-cunquein some way; somehow
line 13 idem [corpus]: i.e. the first body
vicissim: in turn
line 15 sub-i-bit (sub-eo) undergo; suffer
line 16 line 16 non velocitatum, sed motuum: see Definitio II
 aliund-eotherwise
line 17 it-idemlikewise
line 18 factae: modifies mutationes
lines 18-19 reciproceinversely
line 20 scholio:  note

Corollarium I

lines 22-23 eodem tempore . . . quoin the same time . . . as/in which 
quo [corpus] latera [viribus] separatis [describere]
line 25 uniformi: what does this modify?
line 31 haec vis: force N
nihil: adverb
line 32 vi altera: force M

Liber I:  De Motu Corporum

Liber II:  De Motu corporum

Liber III: De Mundi Systemate
Regulae Philosophandi (text)

Notes:

Regula I

phaenomenīsphenomena; appearances (in the sky or air)
explicandis: case?
sufficiantare sufficient; suffice
utiquecertainly; at any rate
natura nihil agit frustra” cf. Aristotle (Politics I.2)
Frustra fit per plura quod potest fieri per pauciora” a version of “Ockham’s    razor” (William Ockham, c. 1285-?1349, English philosopher).  Another         version was:  “Pluralitas non est ponenda sine necessitate
Cf. also Copernicus I.x lines 52-55. For a reaction against a multitude of seemingly unneccesary rules, see also Descarte, Dissertatio II.ii lines 1-10.

Regula II

effectuum naturalium: with “eaedem . . . causae” translate with assignandae sunt.
quatenusas much as; to the degree that (cf. Lex I line 8 above)
uti [causa] respirationis

Regula III

intendito be stretched, extended, become greater in degree
remittito be relaxed, diminished, reduced in degree [Those which cannot be diminished cannot be completely taken away].  These relate to the medieval concept of “latitude of forms” of qualities which could be quantified, and thus undergo “intension” or “remission” by degrees.
nequeuntare not able
quae-que
competuntbelong (+ what case? why?)
instituereto conduct; (within reach)

The five universal qualities found in the first paragraph are: extensio, durities, impenetrabilitas, mobilitas, and vis inertiae.

51  gravia esseare heavy, i.e. gravitate
52 pro quantitate materiaein proportion to the mass
53 vicissimin turnon the other hand

59 im-penetra-bili-tate: one of the universal qualities
utiqueat any rate, certainly
61 attamenbut yet
62 per vim insitamby inherent force [see Definitio III]

Regula IV

65 non obstantibus contrariis hypothesibus: ablative absolute
notwithstanding contrary hypotheses; with contrary hypotheses not hindering
66 quamproximeas nearly as possible
    *haberibe regarded, considered
67 [propositiones] accuratiores
68 obnoxiae (+ dat.): liable to; subject to
69 tollaturbe destroyed, done away with

    Phaenomena
Propositio xxiv, Theorema xix (text: explanation of tides)

Notes:

Background:
Because of the nature of the Mediterranean Sea, tides are almost imperceptible there.  The first Greek known to describe them was Pytheas, a late fourth century BC navigator, astronomer, and mathematician, who got through the straits of Gibralter (which were usually controlled by the Carthaginians) and sailed up the North Sea.  Contemporaries dismissed many of his observations.  Connections of the tides with the moon were known, but an explanation for the cause of tides was still lacking in Newton’s time, and this physical explanation was one of his major contributions (though details remained to be worked out).

*mare, maris, nt. (-i stem: abl. sing –ī; nom./acc pl. –ia, gen. pl. -ium): sea
line 7 *tam . . quamboth . . . and
    lun-aribus: a lunar day is 24 hours, 50 minutes; hence the tides are 50 minutes later each day.  See below (per horas diei lunaris           . . .)
    sol-aribus: the solar day varies, with an average of 23 hours, 56 minutes.
7-8  in-tum-esc-ereto swell up, become swollen  (i.e. high tide) (Note the -sco suffix; see above)
8  patet: (from Liber I, prop. 66, corollaries 19 and 20).
10 appulsus, –ūsapproach
*luminariumluminaris (as substantive) luminary [body], i.e. of the sun and moon
*meridianummeridian (i.e. line of longitude, either directly overhead or opposite; see below)
*minoriless;  modifies spatio (for the -i ablative of comparative adjective, cf. Harvey).
11 sequi: also dependent on patet
tractus, -us: district, region, area
12 orient-alieastern
13 Chil-ensi
Peruvi-ano
13-14 *lit<t>us, lit<t>oris, nt.: shore
14 secundam: i.e. second after the sun/moon has crossed the meridian. [see second paragraph]
14-15 incidit: Long or short “i” on -cid?

17 numeroāre: (verb)
18 infra horizontem: i.e. away from on the opposite side of the earth.
18-19 vigesimas = vicesimus, -a, -um
22-23  subinde: immediately following
24 –ve: or (cf. –que and vel)

26 *cernō, -ere: discern, see
28 coniunctione: i.e. when the sun and earth are lined up together on the same side of the earth (at new moon);  lined up on the opposite sides (i.e. at full moon) = oppositio, -onis.   This happens twice per month.  (See diagram below)
29  fluxus . . . maximus: i.e. “spring tide.”
    quadratura: the sun and moon at right angles to each other (quadrature); at this time the tide is loweest (“neap tide”).
32 *maior: greater (more, bigger)

32 ceteris paribus
: ablative absolute [still used in English today].
[Greek]: zenith, acme, peak

 

 

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