741 



PRINCIPIA. 



PRINCIPIA. 



712 



bodies, and is proportional to the quantity of matter in them. 2 Cor. 

 (8) If each of two globes be everywhere of one density at one distance 

 from the centre, the attraction of each on the other is inversely as the 

 square of the distance of their centres. Cor. 4. (9) In descending to 

 the centre of a planet, gravity diminishes as the distance from that 

 centre. (10) The motion of the planets can continue for an immensely 

 long time. 



lliifiothea* 1. The centre of the solar system is at rest. (Newton 

 takes the universally admitted hypothesis, and shows what the long 

 disputed centre of the system is.) (11) The centre of gravity of the 

 whole system is at rest. (12) The sun is perpetually in motion, but 

 never far from the centre of gravity of the whole. Cor. (13) The 

 planets move in ellipses, having their focus in the sun's centre, and 

 they describe equal areas in equal times about that focus. (14) The 

 nodes and aphelia of the planets are at rest : 2 Cor. and Schol. modify- 

 ing the proposition by considerations of perturbation. (15) To find 

 the axes of the orbits. (16) To find the eccentricities and aphelia. 

 (17) The diurnal motion of the planets is uniform, and the libration 

 of the moon arises from the diurnal motion. (18) The figures of the 

 planets are oblate. (19) To find the proportions of the axis of a 

 planet. (20) To compare the weights of bodies at different parts of the 

 earth. (21) The equinoctial points must regress, and the axis of the 

 earth must have a nutation twice in each year. (22) All the lunar 

 motions and inequalities follow from the preceding principles. (23) 

 The inequalities of other satellites may be derived from those of the 

 moon. (24) The tides of the sea arise from the actions of the sun and 

 moon. (The Jesuits' edition inserts in this place the treatises of 

 Daniel Bernoulli, Maclaurin, and Euler, on the tides.) (25) To find the 

 disturbing force of the sun upon the moon. (26) To find the horary 

 increment of the moou's area about the earth. ('27) From the moon's 

 horary motion to find its distance from the earth. 2 Cor. (28) To 

 find the diameters of the orbit in which the moon would move, but 

 for eccentricity. (29) To find the variation of the moon. (30) To 

 find the horary motion of the moon's nodes in a circular orbit. 2 Cor. 

 To find the horary motion of the moon's nodes in an elliptic orbit. 

 Cor. (32) To find the mean motion of the moon's nodes. (33) To 

 find thetrue motion of the moon'snodes. Cor. (Newton, in the third 

 edition, here adds Machin's method of finding the motion of the 

 moon's nodes.) (34) To find the horary variation of the moon's 

 inclination. 4 Cor. (35) To find the moon's inclination at a given 

 time. Schol. giving an account of several other peculiarities of the 

 lunar motions, and completing the lunar theory. (36) To find the 

 force of the sun upon the sea. ' Cor. (37) The same for the moon. 

 10 Cor. (38) To find the figure of the moon. Cor. Lemma, 1, 2, 3. 

 On the effect of a ring of matter at the equator, disturbed by the sun, 

 upon the earth's rotation. lfi/pntke*is 2." The effect of such a ring in 



i^ precession is the same whether the ring be fluid or solid. (89) 

 To find the precession of the equinoxes. Lemma 4. Comets are 

 the moon, and in the planetary regions. 3 Cor. (40) Comets 

 revolve in cunic sections, having the sun in a focus, and describe equal 

 areas in equal times. 4 Cor. Lemma 5. To find a curve of the para- 

 bolic kind, which shall pass through any number of given points. Cor. 

 Lemma 6. From any given places of a comet to find its place at any 

 intermediate time. Lemmas 7, 8, 9, 10, 11. On the parabola, pre- 

 paratory to the next propositions. (41) To find the parabolic orbit of 



t, from three observations. Example, the comet of 1680, from 

 various observations, and a long discussion of the physical characters 



nets. (4-2) To correct the orbit of a comet ; with other example 

 and discussions. It is to be understood that throughout this third 

 book continual comparison with observation occurs, which it is unne- 

 cessary to repeat as to each particular case, since the purpose of the 

 book itself is the comparison of the results of theory with observation. 

 The ' Scholium Oeneralc,' at the end of the Principia, has been so 

 often quoted, alluded to, attacked, or brought forward as Newton's 

 final explanation of the metaphysics (if that be a proper word) of his 

 system, that even as a matter of reference we may be justified in trans- 

 lating it entire. It is not found in the first edition. Two notes, as 

 marked, are Newton's. 



" The hypothesis of vortices is encumbered with many difficulties. 



Since a planet's radius describes areas proportional to the times, the 



irte of the vortex should be in the duplicate ratio of 



their distances from the sun. And since the planets' periods are in the 



plicate ratio of their distances, the periods of the parts of the 

 vortex should also be in that ratio. Since the minor vortices about 

 Jupiter, Saturn, Ac., preserve their rotation, and swim quietly in the 

 vortex of the sun, the periodic times of the parts of the solar vortex 

 should be equal. The revolutions of the sun and planets about their 

 axes, which should agree with the motions of the vortices, differ from 

 all these proportions. The motion of comets is highly regular, follows 

 the same laws as that of the planets, and cannot be explained by 

 vortices. They are carried most excentrically to all parts of the 

 heavens, which could not be unless the vortices were removed. 



In the Brat edition this Is a lemma, which prob.ibly appeared to Newton 

 to nerd no proof, as none in given. In the third it i>, as we see, made an 

 Mnmption. Laplace has confirmed it, by showing that the precession and 

 natation would remain unaltered, if the whole sea were at any one moment to 

 he MUdifted without alteration of its speellic gravity. 



" Projectiles, in our atmosphere, feel only the resistance of the air ; 

 take that away, as in Boyle's vacuum, and the resistance ceases, since 

 the light feather and the solid gold fall with the same velocity in this 

 vacuum. And such is the case in the celestial spaces which are above 

 the atmosphere of the earth. All bodies in those spaces must move 

 perfectly freely ; and hence the planets move perpetually, according to 

 the laws explained, in orbits of given form and position. They will 

 persevere in their orbits bythe laws of gravity ; but they could by no 

 means originally have taken that regular orbital path by the same 

 laws. 



" The six principal planets revolve about the sun in circles concentric 

 with the sun, in the same direction, and very nearly in the same plaue. 

 The ten moons revolve about the earth, Jupiter, and Saturn, in con- 

 centric circles, in the same direction, and very nearly in the plane of 

 their planets' orbits. And all these regular motions have no origin in 

 mechanical causes * ; for the comets are freely carried in very excentric 

 orbits, and through all parts of the heavens. By such motion comets 

 pass freely and easily through the planetary orbits, and in their aphelia, 

 where they move more slowly and remain longer, they are very distant 

 from each other, and attract each other very little. This most elegant 

 group of sun, planets, and comets can only arise from the design and 

 government of a powerful and intelligent being. And if the fixed 

 stars be centres of like systems, all these constructions of a similar 

 design must be under the dominion of one being : the more since the 

 light of the fixed stars is of the same nature as that of the sun, and all 

 the systems give light to each other. And that the systems of the 

 fixed stars may not fall into each other, he has placed them at 

 immense distances from each other. 



" He rules all things, not as the soul of the wdrld, but as the lord of 

 all things. And from his dominion, God the Ruler is often called 

 no.VTOKpd.Tup. For God is a relative term, which refers to hia servants ; 

 and deity is the rule of God, not over his own body, as those think 

 who call him the soul of the world, but over his servants. The most 

 high God is a being eternal, infinite, absolutely perfect ; but a being 

 however perfect, without dominion, is not God the Ruler. For we say, 

 my God, your God, the God of Israel, the God of gods, the Ruler of 

 rulers; but we do not say, my eternal one, your eternal one, the 

 eternal one of Israel, the eternal one of gods ; we do not say, my 

 infinite one, or my perfect one. These appellations have no relation 

 to servants. The word God always t signifies ruler ; but every ruler 

 is not God. The rule of a spiritual being makes him God, true rule a 

 true god, the highest rule the highest god, a feigned rule a feigned 

 god. And from the reality of his dominion it follows that the true 

 God has life, intellect, and power ; and from his other perfections, 

 that he is the highest or most perfect being. The Eternal is infinite, 

 omnipotent, and omniscient; that is. endures from eternity to eternity, 

 and is present from infinity to infinity ; he rules all and knows all 

 that is or can be. He is not eternity nor infinity, but eternal and 

 infinite ; he is not time nor space, but endures and is present. He 

 endures for ever, and is everywhere present; and by his universal 

 existence, both in time and space, he is the creator J of time and space. 

 Since any particle of space always is, and also some one indivisible 

 moment of duration, certainly the framer and ruler of all things can 

 never be nonexistent at any time or in any place. Every thinking 

 soul is, at different times, and in its different organs of sense or 

 motion, the same indivisible person. There are parts of time suc- 

 cessive, and parts of space co-existing ; but neither in the person of 

 man, that is, in his thinking principle, and much less in the thinking 

 substance of God. Every man, considered as a sentient existence, is 

 one and the same man throughout his whole life, and in every one of 

 his organs of perception. God is one and the same God always and 

 everywhere. He is omnipresent, not only virtually but substantially ; 

 for the first without the second cannot be. In him all things are 

 contained and moved, but without mutual .effect from each other. 

 God no way feels the motion of bodies : they feel no resistance from 

 the omnipresence of God. All men confess that the most high God 

 exists necessarily, and by the same necessity always and everywhere. 

 Whence also he is throughout alike, all eye, all ear, all brain, all arm, 

 all thought, understanding and action, not in the human or corporeal 

 manner, but in one altogether unknown. As a blind man has no idea 

 of colour, so we have no i lea of the modes in which the all-wise God 



* By cauta wrchaiiifa Newton means a material machinery, such as the solid 

 orbs of some of the Ptolcmaists. 



( *' Our countryman Pocock derives the word Deus from the Arabic rtu (some- 

 times rfi), which means ruler. And in this sense the princes are called gods, 

 Psalms, Ixxxiv. 6, and John x. 45. And Moses is called God of his brother 

 Aaron, and of king Pharaoh, Exodus iv. 16, and vii. 1. And in the same sense 

 the souls of departed princes were called gods by Gentile nations, but falsely 

 from want of dominion." (Author's note.) 



J ** Et existendo semper et ubique durationem et spatium constiluit." 



" So thought the ancients, as Pythagoras (Cic., ' De Nat. Door.,' lib. I.) ; 

 Thalcs; Anaxagoras ; Virgil (' Georgics,' iv. 220, ' ALn.,' vi. 721); Philo 

 ('Allegor.,' lih. i. sub init.) ; Aratus (' I'hoon.,' sub init.). So also the sacred 

 writers, as Paul, Acts xvii. 27, 28 j John, Gospel, xiv. 2 ; Moses, Deut. iv. 30, 

 and x. 4 ; David, Psalms, cxxxix. 7, 8, 9 ; Solomon, 1 King sviii. 27 ; Job, xxii. 

 12, 13, 14, Jeremiah, xxiii. 23, 24. The idolaters fulsely imagined the sun, 

 moon, and stars, the souls of men, and other p;irts of the universe to be parts of 

 the moet high God, and therefore to be worshipped." (Author's note.) 



