. 



I'Kl.NViriA. 



II'I.V. 



rio 



\l*o, in time* which arc M the fint velocities directly and the liret 

 tiairtiiMM inversely, they lose the some fraction* of tlioir v 

 and oWribo spaces juiutly proportional to the time* and the fint vlo 

 cities, i Cor. Lemma 2 ; which answers to finding the fluxion* (callw 

 !*n> momenta) of simple algebraical quantities, a Cor. and Scholium 

 ION*.] (8) Whn a particle descends or aKendi by gravity, the 

 whole force* (gravity and nrtirtaiv* compounded) at the taiginni^g ] 

 equal successive spaces, are in geometrical progression. 8 Cor. (0 

 DoUrmination of the proportions of tho timea of accent and descent it 

 the hurt. 7 Cor. <10) The law of resistance being jointly as the density 

 and the aquare of the velocity, required the law of density so thai 

 (gravity acting) a given curve may be described ; as also the law ol 

 Telocity. 2 Cor., followed by 4 Examp. and SchoL, and also by 8 rules. 



.Srrtion 3. When the resistance is partly as the velocity, partly as its 

 square. (11) No force acting, ami times being taken ill arithmetical 

 I impression, the reciprocals of the velocities, increased by a certain 

 constant quantity, will be in geometrical progression. 2 Cor. (12) But 

 if space* be taken in arithmetical progression, the velocities increased 

 by a constant quantity will be in geometrical progression. 8 Cor. 

 (13) Gravity acting, the relation between the time and velocity in the 

 ascent or descent is shown. Cor. and SchoL (14) Relation connecting 

 the space described with the preceding. Cor. and SchoL 



Scrti'nt 4. On spiral motiun in a resisting medium. Lemma 3. A 

 property of the equiangular spiral. (15) The density being inversely 

 as the distance from the centre, and the centripetal force inversely as 

 its square, the particle can revolve in on equiangular spiral 9 Cor. 

 (16) And, other things remaining, the same can be when the force is 

 inversely as any power of the distance. 3 Cor. and SchoL (17) To find 

 the force and law of resistance, by which a body may move in a given 

 spiral, with a given law of velocity. (18) Given the law of force in the 

 last, to find the density of the medium. 



Section 5. On the density and compression of fluids, and on hydro- 

 statics. Definition of a fluid. (19) A homogeneous fluid compressed 

 in a close vassal (gravity, Ac., apart) is everywhere equally pressed, and 

 at rest. 7 Cases and Cor. (20) If a solid sphere form the nucleus of a 

 fluid mass bounded by a concentric sphere, and the parts of tho fluid 

 gravitate equally to the centre at equal distances, the pressure sus- 

 tained by the sphere is the weight of a cylinder which has the super- 

 Gees for its base and the height of the incumbent fluid for its altitude. 

 9 Cor. (21) The density being proportional to the compression, and 

 the centripetal force of particles inversely as their distance from the 

 centre, then at distances in geometrical progression, the densities will 

 be also in geometrical progression. Cor. (22) But if the force be 

 inversely as (dist) 1 , then at distances in harmonica! progression the 

 densities will be in geometrical progression. Cor. and SchoL (23) If 

 the particles of the fluid repel each other, the density is as the com- 

 pression (and then only) when the repellent force of two particles is 

 inversely as the distance of their centres. Scholium. (In consequence 

 of the particles being supposed to repel only their nearest, Newton 

 treats this only as a purely mathematical result) 



Section G. On the resisted motion of pendulums.* (24) The quan- 

 tity of matter in pendulums of the same length, is in a ratio compounded 

 of their weights and of the duplicate ratio of their times of oscilL.ti. m 

 in vacuo. 7 Cor. (25) A pendulum which moves in a medium in which 

 the resistances ore as the moment* of the times, and another moving 

 unresisted in a medium of the same specific gravity, make their 

 cycloidal oscillations in the same timea, and describe'proportional parts 

 of their arcs together. Cor. (26) Resistance being as the velocity, 

 cycloidal oscillations are isochronous. (27) Resistance being as the 

 (velocity) 1 , the difference between the time of cycloidal oscillation in a 

 resisting medium and a non-resisting one of the some specific gravity, 

 will be very nearly as the arcs of oscillation. 2 Cor. (28) The resist- 

 ances being as the moments of the time, the resistance is to the force 

 of gravity as the excess of on arc of descent (cycloidal) over the 



difference of an arc of descent and ascent. Cor. (31) If the resistance 

 be altered in a given ratio, the difference of the arcs of ascent and 

 descent is altered in the same ratio. Scholium Generate, containing 

 many experimental results. 



Stcti'in 7. On the motion of fluids and resistance of projectileut. 

 32) Two systems of similar particles similarly placed, with given 

 ratios between their densities, and beginning to move similarly in 

 proportional times, will continue to do so, if there be no contact of 

 particles except at iiisUnU of reflexion, and if there be no attracting 

 forces of particles on one another but such as are as the diameters 

 of corresponding particles inversely and the square of the velocities 

 directly. 2 Cor. (88) And finite parts of these systems ore resisted 

 m a ratio compounded of the duplicate ratio of the velocities and 

 diameters, and the ratio of the densities. 6 Car. (84) The circular 

 end of a cylinder encounters twice as much resistance M a sphere 

 moving with the same velocity. Scholium, containing among other 

 things the construction (without demonstration) of the solid of least 



too* of tkss* proportion! would require explanation to make their 

 mesiuap elear, which *t lure not bete room to fire, 

 t preceding not.. 



resistance, which shows that Newton must have carried his Minion* 

 (before 1687) far enough to solve some problems at least in what in 

 now called the calculus of variations. (86) A medium consisting of 

 equal particles at equal distances, to find the resistance it offers to a 

 sphere. 7 Cor. and SchoL (36) To find the motion of water issuing 

 through an orifice at the bottom of a cylinder. 10 Cor. Leo 

 (87) Resistance to a cylinder moving in a non-elastio fluid. 

 and SchoL Lemma 5, 6, 7. A cylinder, sphere, or spheroid, 

 same circular section, placed in a cylindrical canal of running water, 

 or moving equally in it, is equally urged or resisted. SchoL (38) 

 Resistance to a globe in a non-elastio fluid. 4 Cor. (89) Tho same 

 when the globe is in a cylindrical canal. SchoL (40) The same, 

 showing how to find the resistance experimentally. Scholium, con 

 taining accounts of fourteen experiments. 



Section 8. On motion propagated through a fluid. (41) Pressure is 

 not propagated through a fluid in right lines, unless when the particles 

 lie in right lines. Cor. (42) Every motion propagated through a 

 fluid diverges from the direct path. (43) Every treinul.iUK !<ly 

 excites in an elastic medium pulses in every direction ; but in a non- 

 elastic medium, a circular motion. Cor. (44) 8 Cor. (45 > 

 of water in a bent tube. (46) The velocity of waves. 2 Cor. (47) 

 In pulses, the motion of the particles follow the law of 

 pendulum.* Cor. (48) The velocities of pulses in differm- 

 as the square roots of the elastic force directly and the 

 inversely. 3 Cor. (49) Given the density and elasticity, to tind the 

 velocity. 2 Cor. (60) To find the length of the pulses. Scholium. 



.Section 9. On tho circular motion of fluids. (I: ( Des 



Cartes's vortices on the hypothesis that the resistance which arises 

 from the want of lubricity of the parts of a fluid, is proportional t.. : I,, 

 velocity with which the part* of the fluid are separated. ) (51) It 

 cylinder of infinite length revolve about its axis in a uniform 

 infinite fluid, and create a vortical motion, the periilie i 

 particles of fluid are as their distances from the axis. 6 Cor. 

 But if the revolving body be a sphere, these periodic times are as the 

 squares of the distances from the centre. 11 Cor. and SchoL (53) A 

 body revolving in a vortex so as to return to its place, muni I . of the 

 some density as the parts of the vortex, and move in the name manner. 

 2 Cor. and Scholium, completing the refutation above mentioned. 



THE TiiiiiD BOOK, or application, styled 'De Systemote Mmuli,' 

 contains forty-two pro)>osition8, 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. Uigaud) 

 was preserved, and was published in English, in 172X. under th< 

 of 'The System of the World demonstrated in on Easy and 1'opular 

 Manner by the illustrious Sir Isaac Newton ;' and again in the or 

 Latin. It is Opusculum XVII. in the collection of Caatillioneug, who 

 takes it from on 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 V 

 mind. But we think that this work has been too easily admitted, :uid 

 that its genuineness requires a searching discussion. 



SeguUt phihnopkandi. 1. No more causes of natural things ore to 

 be admitted than ore both true and sufficient to explain their pin-no- 

 lueua. 2. The same causes ore to be assigned to effects of tin 

 tind, as for as that can be done. 8. Those qualities of bodies which 

 can neither be strengthened nor weakened, and which belong to all 

 bodies which are callable of being tried, are to be considered as 

 universal qualities. 4. In experimental philosophy, all propositions 

 collected by induction from phenomena are to be held either exactly 

 or approximately true until other phenomena are found liy which 

 those propositions can be made either more accurate or subject to 

 exception. 



P tunumfna. 1. The satellites of Jupiter describe areas proportional 

 to their times about the planet, and their periodic times ore 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 

 lanets, Mercury, Venus, Mars, Jupiter, and Saturn, revolve about the 

 sun. 4. And their periodic times, and that of the earth about tho 

 sun, or tho sun about the earth, the fixed stars being at rest, are in 

 .he sesquiplicate ratio of their mean distances from the sun. 5. And 

 .he primary planets are very for from describing equal areas in equal 



times about the earth; but do so about the sun. 6. The m 



describes equal areas in equal times about the earth. 



(1) The satellites of Jupiter ore attracted to the planet by forces 

 y as the squares of their distance*. (2) The some of tho 

 primary planets about the sun. (3) The same of the moon about the 

 nrth. (4) The force which retains the moon in her orbit is the same 

 orce as that which, at the earth's surface, we call gravity. Sclml. 

 This is the celebrated test-proposition, 'the failure of which, in the first 

 nstance, nude Newton lay his theory aside. (5) A similar result 

 nferred as to satellites about their primaries, and primaries about the 

 sun. 8 Cor. and Schol. (6) All bodies gravitate towards every 

 >lanet ; and gravitation towards every planet at a given distance from 

 t, is as the mass of that planet. 6 Cor. (7) Attraction belongs to all 



Infringe hu shown ( Hlta. Tiar.') that the method of Newton here only 

 ewU to a concealed identity, which proves nothing, 



