Newton Isaac, 1687- Mathematical principles of natural philosophy

That the matter of the heavens is fluid, . . 511
The principle of circular motion in free spaces, 512
The effects of centripetal forces, . . 512
The certainty of the argument, ... 512
Wh .t follows from the supposed diurnal motion of the stars, 514
The incongruous consequences of this supposition, . . 514
That there is a centripetal force really directed to the centre of every planet, 515
Centripetal forces decrease in duplicate proportion of distances from the centre of every planet, 515
That the superior planets are revolved about the sun, and by radii drawn to the sun describe areas proportional to the times, 517

That the force which governs the superior planets is directed not to the earth, but to the sun, . 518
That the cit cum-solar force throughout all the regions of the planets decreaseth in the duplicate proportion of the distances from the sun, 519

That the circum-terrestrial force decreases in the duplicate proportion of the distances from the earth proved in the hypothesis of the earth’s being at rest, 519
The same proved in the hypothesis of the earth’s motion 520
The decrement of the forces in the duplicate proportion of the distances from the earth and planets,
proved from the eccentricity of the planets, and the very slow motion of their apses, . 520
The quantity of the forces tending towards the several planets: the cireum-solar very great, . 521
The circum-terrestrial force very small, 521
The apparent diameters of the planets, . 521
The correction of the apparent diameters, 522
Why the density is greater in some of the planets and less in others; but the forces in all are as their quantities of matter, 524
Another analogy between the forces and bodies, proved in the celestial bodies, 525
Proved in terrestrial bodies, 525
The affinity of those analogies, 526
And coincidence, . 526
That the forces of small bodies are insensible, 527
Which, notwithstanding, there are forces tending towards all terrestrial bodies proportional to their quantities of matter, 528
Proved that the same forces tend towards the celestial bodies, 528
That from the surfaces of the planets, reckoning outward, their forces decrease in tin duplicate; but, reckoning inward, in the simple proportion of the distances from their centres, . 528
The quantities of the forces and of the motions arising in the several cases, .... 529
That all the planets revolve about the sun, 529
That the common centre of gravity of all the planets is quiescent That the sun is agitated with a very slow motion. This motion defined, 531
That the planets, nevertheless, are revolved in ellipses having their foci in the sun; and by radii drawn to the sun describe areas proportional to the times, 531
Jf the dimensions of the orbits, and of the motions of their aphelions and nodes, . . . 532
All the motions of the moon that have hitherto been observed by astronomers derived from the foregoing principles, 532
As also some other unequable motions that hitherto have not been observed, .... 533 And the distance of the moon from the earth to any given time, 533
The motions of the satellites of Jupiter and Saturn derived from the motions of our moon, . 534
That the planets, in respect of the fixed stars, are revolved by equable motions about their proper axes. And that (perhaps) those motions are the most fit for the equation of time, 534
The moon likewise is revolved by a diurnal motion about its axis, and its libration thence arises, 535
That the sea ought twice to flow, and twice to ebb, every day; that the highest water must fall out in the third hour after the appulse of the luminaries to the meridian of the place, . 535
The precession of the equinoxes, and the libratory motion of the axes of the earth and planet j 535
That the greatest tides happen in the syzygies of the luminaries, the least in their quadratures; and that at the third hour after the appulse of the moon to the meridian of the place. Bat that out of the syzygies and quadratures those greatest and least tides deviate a little from that third hour towards the third hour after the appulse of the sun to the meridian, 536
That the tides are greatest when the luminaries are in their perigees, 536
That the tides are greatest about the equinoxes, 536
That out of the equator the tides are greater and less alternately, 537
That, by the conservation of the impressed motion, the difference of the tides is diminished; and that hence it may happen that the greatest menstiual tide will be the third after the syzygy, 538
That the motio is of the sea may be retarded by impediments in its channels, 538
That from the impediments of channels and shores various phsenomena do arise, as that the sea may flow but once every day, 539
That the times of the tides within the channels of rivers are more unequal than in the ocean, 540
That the tides are greater in greater and deeper seas; greater on the shores of continents than et isla eds in the middle of the sea; and yet greater in shallow bays that open with wide inlets to the sea, 540
The force of the sun to disturb the motions of the moon, computed from the foregoing principles, 542
The force of the sun to move the sea computed, 543
The height of the tide under the equator arising from the force of the sun computed, 543
The height of the tides under the parallels arising from the sun’s force computed, 543
The proportion of the tides under the equator, in the syzygies and quadratures, arising from the joint forces of both sun and moon, 545
The force of the moon to excite tides, and the height of the water thence arising, computed, . That those forces of the sun and moon are scarcely sensible by any other effect beside the tides which they raise in the sea, 546
That the body of the moon is about six times more dense than the body of the sun, 547
That the moon is more dense than the earth in a ratio of about three to two, 547
Of the distance of the fixed stars, 547
That the comets, as o ten as they become visible to us, are nearer than Jupiter, proved from their parallax in longitude, 548
The same proved from their parallax in latitude, 549
The same proved otherwise by the parallax, 550
From the light of the comets’ heads it is proved that they descend to the orbit of Saturn, 550
And also below the orb of Jupiter, and sometimes below the orb of the earth, 551
The same proved from the extraordinary splendor of their tails when they are near the sun, 553
The same proved from the light of their heads, as being greater, cceteris paribus, when they come near to the sun, 553
The same confirmed by the great number of comets seen in the region of the sun, 555
This also confirmed by the greater magnitude and splendor of the tails after the conjunction of the heads with the sun than before, 555
That the tails arise from the atmospheres of the comets, 556
That the air and vapour in the celestial spaces is of an immense rarity ; and that a small quantity of vapour may be sufficient to explain all the phenomena of the tails of comets, 558
After what manner the tails of comets may arise from the atmospheres of their heads. 559
That the tails do indeed arise from those atmospheres, proved from several of their phenomena, 559
That comets do sometimes descend below the orbit of Mercury, proved from their tails, 560
That the comets move in conic sections, having one focus in the centre of the sun, and by radii drawn to that centre do describe areas proportional to the times, 561
That those conic sections are near to parabolas, proved from the velocity of the comets, in what space of time comets describing parabolic trajectories pass through the sphere of the orbis magnus, 563
At what time comets enter into and pass out of the sphere of the orbis magnus, With what velocity the comets of 1680 passed through the sphere of the orbis magnus,564
That these were not two, but one and the same comet. In what orbit and with what velocity this comet was carried through the heavens described more exactly, 564
With what velocity comets are carried, shewed by more examples, 565
The investigation of the trajectory of comets proposed, Lemmas premised to the solution of the problem, 567
The problem resolved, 570
EquuiNoxEs, their precession—the cause of that motion shewn, 413
the quantity of that motion computed from the causes, 453
Air, its density at any height, collected by Prop. XXII, Book II, and its density at the height of one semi-diameter of the earth, shown, 489
its elastic force, what cause it may be attributed to, 302
its gravity compared with that of water, 489
its resistance, collected by experiments of pendulums, 315
the same more accurately by experiments of falling bodies, and a theory, 355
Angles of contact not all of the same kind, but some infinitely less than others, 101
Apsides, their motion shewn, 172, 173
Areas which revolving bodies, by radii drawn to the centre of force describe, compared with the times of description, 103, 105, 106, 195, 200
As, the mathematical signification of this word defined, 100
Attraction of all bodies demonstrated, 397
the certainty of this demonstration shewn, 384
the cause or manner thereof no where defined by the author, 507
the common centre of gravity of the earth, sun, and all the planets, is at rest, confirmed by Cor. 2, Prop. XIV, Book III, 401
the common centre of gravity of the earth and moon goes round the orbis magnus, 402
its distance from the earth and from the moon, 452
Centre, the common centre of gravity of many bodies does not alter its state of motion or rest by the actions of the bodies among themselves, 87
of the forces by which revolving bodies are retained in their orbits, how indicated by the description of areas, 107
how found by the given velocities of the revolving bodies, 110
Circle, by what law of centripetal force tending to any given point its circumference may be
described, 108, 111, 114
Comets, a sort of planets, not meteors, 465, 486
higher than the moon, and in the planetary regions, 460
their distance how collected very nearly by observations, 461
more of them observed in the hemisphere towards the sun than in the opposite hemisphere ; and how this comes to pass, 464
shine by the sun’s light reflected from them, ... . 464
surrounded with vast atmospheres, .... 463, 465
those which come nearest to the sun probably the least, . 495
why they are not comprehended within a zodia , like the planets, but move differently into all parts of the heavens, 502
may sometimes fall into the sun, and afford a new supply of fire,. 502
the use of them hinted, 492
move in conic sections, having their foci in the sun’s centre, and by radii drawn to the sun describe areas proportional to the times. Move in ellipses if they come round again in their orbits, but these ellipses will be near to parabolas, 466
Comet’s parabolic trajectory found from three observations given, . 472
corrected when found, ..... , 495
place in a parabola found to a given time, . 466
velocity compared with the velocity of the planets, , 466

Comets’ Tails directed from the sun, 489
brightest and largest immediately after their passage through the neighbourhood of the sun, . . 487
their wonderful rarity, 490
their origin and nature, 463
in what space of time they ascend from their heads, 499
Comet of the years 1664 and 1665—the observations of its motion compared with the theory, . 496
of the years 1680 and 1681—observations of its motion, . . 474
its motion computed in a parabolic orbit, 478
in an elliptic orbit, 479
its trajectory, and its tail in the several parts of its orbit, delineated, 484
of the year 1682—its motion Compared with the theory, . . 500
seems to have appeared in the year 1607, and likely to return again after a period of
75 years, 501, 502
of the year 1683—its motion compared with the theory, 499
of the year 1723—its motion compared with the theory,....... 501
Conic Sections, by what law of centripetal force tending to any given point they may be described by revolving bodies, 125
the geometrical description of them when the foci are given, 125
when the foci are not given, . . . 131
when the centres or asymptotes are given, 147
Curvature of figures how estimated, .... 271,423
Curves distinguished into geometrically rational and geometrically irrational, 157
Cycloid, or Epicycloid, its rectification, 184
its evoluta, 185
Cylinder, the attraction of a cylinder composed of attracting particles, whose forces are reciprocally as the square of the distances, 239
Descent of heavy bodies in vacuo, how much it is, 405
and ascent of bodies in resisting mediums, 252, 265, 281, 283, 345
Descent or Ascent rectilinear, the spaces described, the times of decription, and the velocities acquired in such ascent or descent, compared, on the supposition of any kind of centripetal force, 160
Earth, its dimension by Norwood, by Picart, and by Cassini, 405
its figure discovered, with the proportion of its diameters, and the measure of the degrees upon the meridian, 405,409
the excess of its height at the equator above its height at the poles, 407,412
its greatest and least semi-diameter, 407
its mean semi-diameter, 407
the globe of the earth more dense than if it was entirely water, 400
the nutation of its axis, 413
the annual motion thereof in the orbis magnus demonstrated, 498
the eccentricity thereof how much, . 452
the motion of its aphelion how much, 404
Elupbes, by what law of centripetal force tending to the centre of the figure it is described by a revolving body, 114
by what law of centripetal force tending to the focus of the figure it is described by a revolving body,. 116
Fluid, the definition thereof, 108
Fluids, the laws of their density and compression shewn, . . 293
their motion in running out at a hole in a vessel determined, 331
Forces, their composition and resolution, ...... 84
attractive forces of spherical bodies, composed of particles attracting according to any law, determined, 218
attractive forces of bodies not spherical, composed of particles attracting according to any law, determined, 233
the invention of the centripetal forces, when a body is revolved in a non-resisting space about an immoveable centre in any orbit, 103,116
the centripetal forces tending to any point by which any figure may be described by a revolving body being given, the centripetal forces tending to any other point by which the same figure may be described in the same periodic time are also given, 113
the centripetal forces by which any figure is described by a revolving body being given, there are given the forces by which a new figure may be described, if the ordinates are augmented or diminished in any given ratio, or the angle of their inclination be any how changed, the periodic time remaining the same, 116
centripetal forces decreasing in the duplicate proportion of the distances, what figures may be described by them, 120 196
Force, centripetal force defined, 74
the absolute quantity of centripetal force defined, 75
the accelerative quantity of the same defined, 76
the motive quantity of the same defined, . . 76
the proportion thereof to any known force how collected, 109
a centripetal force that is reciprocally as the cube of the ordinate tending to a vastly remote centre of lorce will cause a body to move in any given conic section, . . 114
a centripetal force that is as the cube of the ordinate tending to a vastly remote centre of force will cause a body to move in an hyperbola, 243
centrifugal force of bodies on the earth’s equator, how great, 405
God, his nature, 506
Gravity mutual between the earth and its parts, 94

of a different nature from magnetical force, 397
the cause of it not assigned, 507
tends towards all the planets, 393
from the surfaces of the planets upwards decreases in the duplicate ratio of the distances from the centre, 400
from the same downwards decreases nearly in the simple ratio of the same, 400
tends towards all bodies, and is proportional to the quantity of matter in each, 397
is the force by which the moon is retained in its orbit, 391
the same proved by an accurate calculus, 453
is the force by which the primary planets and the satellites of Jupiter and Saturn are retained in their orbits, 393
Heat, an iron rod increases in length by heat, 412
of the sun, bow great at different distances from the sun, 486
how great in Mercury, 400
how great in the comet of 1680, when in its perihelion, 486
Heavens are void of any sensible resistance, 401,445, 492; and, therefore, of almost any corporeal fluid whatever, 355, 356
suffer light to pass through them without any refraction, 485
Hydrostatics, the principles thereof delivered, 293
Hyperbola, by what law of centrifugal force tending from the centre of the figure it is described by a revolving body, 116
by what law of centrifugal force tending from the focus of the figure it is described by a revolving body, 117
by what law of oe a tripetal force tending to the focus of the figure it is described by a revolving body, 118
Hypotheses of what kind oever rejected from this philosophy, . 508
Jupiter, its periodic time, 388
its distance from the sun, 388
its apparent diameter, 386
its true diameter, 399
its attractive force, how great, 398
the weights of b< dies on its surface, 399
its density, 399
its quantity of matter, 399
its perturbation by Saturn, how much, 403
the proportion of its diameters exhibited by computation, 409
and compared with observations, 409
its rotation about its axis, in what time performed, 409
the cause of its belts hinted at, 415

IgGHT, its propagation not instantaneous, 246
its velocity different in different mediums, . . 245
a certain reflection it sometimes suffers explained, 245
its refraction explained, 243
refraction is not made in the single point of incidence, 247
an incurvation of light about the extremities of bodies observed by experiments, 246
not caused by the agitation of any ethereal medium, . 368
Magnetic force, 94,304, 397, 454
Mars, its periodic time, . . 338
its distance from the sun, 389
the motion of its aphelion, . 495
Matter, its quantity of matter defined, 73
its vis insita define i. . 74
its impressed force defined, . 74
its extension, hardness, impenetrability, mobility, vis inertia, gravity, how discovered, 385 subtle matter of Descartes ij quired into, 320
Mechanical Powers explained and demonstrated, 94
Mercury, its periodic time, . 388
its distance from the sun, 389
the motion of its aphelion, 405
Method of first and last ratios, . £5
of transforming figures into others of the same analytical order, 141
of fluxions, . 261
differential, 447
of finding the quadratures of all curves very nearly true, 448
of converging series applied to the solution of difficult problems, 271,436
Moon, the inclination of its orbit to the ecliptic greatest in the syzygies of the node with the sun, and least in the quadratures, 208
the figure of its body collected by calculation, 454
its librations explained, . 405
its mean apparent diameter, . 453
its true diameter, 453
weight of bodies on its surface, 453
its density, , . . 453
its quantity of matter, . 453
its mean distance from the earth, how many greatest sem>-diameters of the earth contained therein, 453
how many mean semi-diameters, . 454
its force to move the sea how great, 449
not perceptible in experiments of pendulums, or any statical or hydrostatical observations, 452 its periodic time, 454
the time of its synodical revolution, 422
its motions, and the inequalities of the same derived from their causes, . . 413, 444
revolves more slowly, in a dilated orbit, when the earth is in its perihelion; and more swiftly in the aphelion the same, its orbit being contracted, .... 413, 444, 445 revolves more slowly, in a dilated orbit, when the apogaeon is in the syzygies with the sun ; and more swiftly, in a contracted orbit, when the apogaeon is in the quadratures, . 445
revolves more slowly, in a dilated orbit, when the node is in the syzygies with the sun; and more swiftly, in a contracted orbit, when the node is in the quadratures, . . 446
moves slower in its quadratures with the sun, swifter in the syzygies; and by a radius drawn to the earth describes an area, in the first case less in proportion to the time, in the last case greater, 413
the inequality of those areas computed, 420
its orbit is more curve, and goes farther from the earth in the first case; in the last case its orbit is less curve, and comes nearer to the earth, 415
the figure of this orbit, and the proportion of its diameters collected by computation, . 423 a method of finding the moon’s distance from the earth by its horary motion, . . 423
its apogaeon moves more slowly when the earth is in its aphelion, more swiftly in the peri-
helion, 414, 445
its apogaeon goes forward most swiftly when in the syzygies with the sun; and goes backward in the quadratures, 414,44€
its eccentricity greatest when the apogaeon is in the syzygies with the sun ; least when the same is in the quadratures, 414, 446
its nodes move more slowly when the earth is in its aphelion, and more swiftly in the perihelion, 414,445
its nodes are at rest in their syzygies with the sun, and go back most swiftly in the quadratures 414
Moon, the motions of the nodes and the inequalities of its motions computed from the theory of gravity, 427,430,434, 436
the same from a different principle, 437
the variations of the inclination computed from the theory of gravity, 441, 443
the equations of the moon’s motions for astronomical uses, 445
the annual equation of the moon’s mean motion, 445
the first semi-annual equation of the same, 446
the second semi-annual equation of the same, 447
the first equation of the moon’s centre, . 447
the second equation of the moon’s centre, 448
Moon’s first variation, _ 425
the annual equation of the mean motion of its apogee, 445
the semi-annual equation of the same, 447
the semi-annual equation of its eccentricity, 447
the annual equation of the mean motion of its nodes, . 445
the semi-annual equation of the same, 437
the semi-annual equation of the inclination of the orbit to the ecliptic, 444
the method of fixing the theory of the lunar motions from observations, 464
Motion, its quantity defined, 73
absolute and relative, 78
absolute and relative, the separation of one from the other possible, demonstrated by an example * 82
laws thereof 83
of concurring bodies after their reflection, by what experiments collected, 91
of bodies in eccentric sections, 116
in moveable orbits, . . 172
in given superficies, and of the reciprocal motion of pendulums, 183
of bodies tending to each other with centripetal forces, 194
of very small bodies agitated by centripetal forces tending to each part of some very great body, 233
of bodies resisted in the ratio of the velocities, 251
in the duplicate ratio of the velocity, . . 258
partly in the simple and partly in the duplicate ratio of the same, , 280
of bodies proceeding by their vis insita alone in resisting mediums, 251,258,259, 280, 281, 330 of bodies ascending or descending in right lines in resisting mediums, and acted on by an uniform force of gravity, 252, 265, 281, 283
of bodies projected in resisting mediums, and acted ©n by an uniform force of gravity, 255, 268 of bodies revolving in resisting mediums, 287
of funependulous bodies in resisting mediums, • 301
and resistance of fluids, 323
propagated through fluids, 356
of fluids after the manner of a vortex, or circular, 370
Motions, composition and resolution of them, 84
Ovals for optic uses, the method of finding them which Cartesius concealed, . 246
a general solution of Cartesius’s problem, 247,248
Orbits, the invention of those which are described by bodies going off from a given place with a given velocity according to a given right line, when the centripetal force is reciprocally as the square of the distance, and the absolute quantity of that force is known, . 123 of those which are described by bodies when the centripetal force is reciprocally as the cube of the distance, 114,171,176
of those which are described by bodies agitated by any centripetal forces whatever, 168
Parabola, by what law of centripetal force tending to the focus of the figure the same may be described, 120
Pendulums, their properties explained, 186,190, 304
the diverse lengths of isochronous pendulums in different latitudes compared among themselves, both by observations and by the theory of gravity, 409 to 413
Place defined, and distinguished into absolute and relative, . . 78
Places of bodies moving in conic sections found to any assigned time, 153
PLanets not carried about by corporeal vortices, 378