THE THIRD Boos, or application, styled 'De Syetemate Mundi,' contains forty-two propositions, and preliminaries. It is to be noted that it was the original intention of Newto' that this book should be a popular one ; and the original draft (so it is considered by Mr. ltIgaud) was preserved, and was published in English, in 1728, under the title of The System of the World demonstrated in an Easy and Popular Manner by the illustrious Sir Isaac Newton ; and again in the original Latin. It is Opusculum XVII. in the collection of Castillioneue, who takes it from an edition published in 1731. It is not altogether popular, but, containing the mathematical propositions concerning comets to which Halley alludes in his letter, it may be the third book as it stood at the time when the idea of suppressing it was in Newton's mind. But we think that this work has been too easily admitted, and that its genuineness requires a searching discussion.
Regulie philosophantli. 1. No more causes of natural things are to be admitted than are both true and sufficient to explain their pheno mena. 2. The same causes are to bo assigned to effects of the same kind, as far as that can be done. 3. Those qualities of bodies which can neither be strengthened nor weakened, and which belong to all bodice which aro capable of being tried, are to be considered as universal qualities. 4. In experimental philosophy, all propositiote collected by induction from phenomena are to be held either exactly or approximately true until other phenomena are found by which those propositions can be made either more accurate or subject to exception.
Phenomena. 1. The satellites of Jupiter describe areas proportional to their times about the planet, and their periodic times are in the sesquiplicate ratio of their distances from the planet. 2. The same is true of the satellites of Saturn (five then known). 3. The five primary planets, Mercury, Venus, Mars, Jupiter, and Saturn, revolve about the sun. 4. And their periodic) times, and that of the earth about tho euu, or the sun Meerut the earth, the fixed stars being at rest, are In the sesquiplicate ratio of their maw distances from the sun. 5. And the primary planets are very far from describing equal areas in equal times about the earth; but do so about the sun. 6. The moon describes equal emu in equal times about the earth.
(1) Tho satellites of Jupiter are attracted to the planet by forces inversely as the squares of their distances. (2) Tho same of the primary planets about the sun. (3) The same of the moon about the earth. (4) The force which retains the moon in her orbit is the same force as that which, at the earth's surface, we call gravity. Schol. This is the celebrated teat-proporsition;the failure of which, in the first instence, made Newton lay his theory aside. (5) A. dinner result inferred as to satellites about their primaries, and primaries about the sun. 8 Cor. and Schol. (6) All bodies gravitate towards every planet ; and gravitation towards every planet at a given distance from it, is as the masa of that planet. 5 Cor. (7) Attraction belongs to all 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.
ypothais 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 iucrement of the moon'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 excentricity. (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 the true motion of the moon's nodes. 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. (33) 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. II ypotheri 2.• The effect of such a ring in causing precession is the same whether the ring be fluid or solid. (39) To find the precession of the equinoxes. Lemma 4. Comets are above the moon, and in the planetary regions. 3 Cur. (40) Comets revolve in conic sections, having the sun in a focus, and describe equal areas in equal times. 4 Cur. 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 a comet, from three observations. Example, the comet of 1680, from various observations, and a long discussion of the physical characters of comets. (92) To correct the orbit of a comet ; with other examples 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.