Principia Mathematica (Motte Translation, 1848) — part 45 of 45

" 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 aU the planets, is at rest, con- firmed 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 planetaiy 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 hemis- phere; 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 zodiac, 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 neighbour- hood of the sun, 4ST

" " their wonderful rarity, 490

" " their origin and nature, 463

'• " in what space of time they ascend from their heads, . . . ^©0

576 INDEX TO THE PRINCIPIA.

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 de- scribed 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

" " itsevoluta, 185

Cylinder, the attraction of a cylinder composed of attracting particles, whose forces are recip- rocally 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 A'crwoocZ, by Picari, and by Cassmi, 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 or&is mag-nMs demonstrated, 498

" the eccentricity thereof how much, ' 452

" the motion of its aphelion how much, 404

Ellipses, 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

INDEX TO THE PRINCIPIA. 577

FoECE, 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 force 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 dis- tances 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

Hbat, an iron rod increases in length by heat, ....••... 412

" of the sun, how 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 cor- poreal fluid whatever, 355, 356

" suffer light to pass through them without any refraction, . . • • • . 485

Hydrostatics, the principles thereof deUvered, . . ...... 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 centripetal force tending to the focus of the figure it is described

by a revolving body, 118

Hypotheses of what kind soever 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 bodies 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, , . . 445

Light, 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

27

578 INDEX TO THE PRINCIPIA.

Mars, its periodic time, 388

" its distance from the sun, 389

" the motion of its aphelion, 405

Matter, its quantity of matter defined, 73

" its vis insita defined. ... 74

" its impressed force defined, 74

" its extension, hardness, impenetrability, mobility, vis inertice, gravity, how discovered, 385

" subtle matter, of Descartes inquired 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, 95

" of transforming figures into others of the same analytical order, .... 141

" of fluxions, 261

" difi"erential, 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, . 45:i

" its quantity of matter, 453

" its mean distance from the earth, how many greatest semi-diameters of the earth con- tained therein, 453

" how many mean serai-diameters, ; . . 454

" its force to move the sea how great, 449

" not perceptible in experiments of pendulmns, 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 apoggeon is in the syzygies with the sun ;

and more swiftly, in a contracted orbit, when the apoggeon 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^pmputed, .... 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 apogseon moves more slowly when the earth is in its aphelion, more swiftly in the peri- helion, 414,445

" its apogason goes forward most swiftly when in the syzygies with the sim ; and goes back- ward in the quadratures, ■■ 414, 446

" its eccentricity greatest when the apogason 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 peri- helion, 414,445

" its nodes are at rest in their syzygies with the sun, and go back most swiftly in the quad- ratures , 414

INDEX TO THE PRINCIPIA. 579

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 serni-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 duiDlicate 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 xmiform force of gravity, 252,265,2-81,283

'* of bodies projected in resisting mediums, and acted on by an uniform force of gravity, 255, 268

" of bodies revolving in resisting mediimis, 287

" of funependulous bodies in resistmg mediums, 304

.^" 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 ^ven velocity according to a given right line, when the centripetal force is recipro- cally 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, then- 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

580 INDEX TO THE PRINCIPIA.

PuiNETS, their periodic times, 388

" their distances from the sun, 389

" the aphelia and nodes of their orbits do almost rest, 405

" their orbits determined, 406

" the way of finding their places in their orbits, . . . . . . , 347 to 350

" their density suited to the heat they receive from the sun, ...•,. 400

" their diurnal revolutions equable, 406

" their axes less than the diameters that stand upon them at right angles, . . , 406

Planets, Pkimakv, surround the sun, 387

" " move in ellipses whose focus is in the sun's centre, 403

" " by radii drawn to the sun describe areas proportional to the times, . 388, 403 " " revolve in periodic times that are in the sesquiplicate proportion of the dis- tances from the sun, 387

" " are retained in their orbits by a force of gravity which respects the sun,

and is reciprocally as the square of the distance from the sun's centre, 389, 393 Planets, Secondaey, move in ellipses having their focus in the centre of the primary, . 413 " " by radii drawn to their primary describe areas proportional to the

times 386,387,390

" " revolve in periodic times that are in the sesquiplicate proportion of their

distances from the primary, 386, 387

Pkoelem Keplerian, solved by the trochoid and by approximations, .... 157 to 160 " " of the ancients, of four lines, related by Pappus, and attempted by Car-

tesius, by an algebraic calculus solved by a geometrical composition, . 135 Projectiles move in parabolas when the resistance of the medium is taken away, 91, 115, 243, 273

" their motions in resisting mediums, . . 255, 268

Pulses of the air, by which sounds are propagated, their intervals or breadths determined, 368, 370 " these intervals in sounds made by open pipes probably equal to twice the length of the

pipes, • ... 370

Quadratures general of oval figures not to be obtained by finite terms, 153

Qualities of bodies how discovered, and when to be supposed universal, .... 384

Resistance, the quantity thereof in mediums not continued, 329

" in continued mediums, 409

" in mediums of any kind whatever, • . 33i.

" of mediums is as their density, cceteris -paribus, . . 320, 321, 324, 329, 344, 355 " is in the duplicate proportion of the velocity of the bodies resisted, cceteris -pari- bus, 258,314,324,329,344,351

*♦ is in the duplicate proportion of the diameters of spherical bodies resisted, cceteris

paribus, 317,318,329,344

" of fluids threefold, arises either from the inactivity of the fluid matter, or the te- nacity of its parts, or friction, 286

" the resistance found in fluids, almost all of the first kind, .... 321, 354

" cannot be diminished by the subtilty of the parts of the fluid, if the density remain, 355

" of a globe, what proportion it bears to that of a cylinder, in mediums not continued, 327

" in compressed mediums, , . . 343

" of a globe in mediums not continued, ..,..,... 329

•' in compressed mediums, 344

" how found by experiments, 345 to 355

" to a frustum of a cone, how made the least possible, 328

" what kind of solid it is that meets with the least, ...... . 329

Resistances, the theory thereof confirmed by experiments of pendulums, . . . 313 to 321

" by experiments of falling bodies, 345 to 356

Rest, true and relative, 78

BuLEs of philosophy, 384

Satellites, the greatest heliocentric elongation of Jupiter's satellites, 387

*' the greatest heliocentric elongation of the Huygenian satellite from S aturn's centre, 398

" the periodic times of Jupiter's satellites, and their distances from his centre, . 386, 387

" the periodic times of Saturn's satellites, and their distances from his centre, 387, 388

" the inequalities of the motions of the satellites of Jupiter and Saturn derived from

the motions of the moon, 413

Sksquiplicate proportion defined, 101

/

INDEX TO THE PRINCIPIA. 581

Saturn, its periodic time, 388

" its distance from the "siin, 388

" its apparent diameter, 388

" its true diameter, . 399

" its attractive force, how great, . . • 398

-" the weight of bodies on its surface, 399

" its density, ... .399

" its quantity of matter, . . . . . . ' 399

" its perturbation by the approach of Jupiter how great, 403

" the apparent diameter of its ring, 388

Shadow of the earth to be augmented in lunar eclipses, because of the refraction of the at- mosphere, 447

Sounds, their nature explained, . . . . . .360, 363, 365, 366, 36r, 368, 369

" not propagated in directum, ....... ... 359

" caused by the agitation of the air, 368

" their velocity computed, 368, 369

" somewhat swifter by ihe theory in summer than in winter, 370

" cease immediately, when the motion of the sonorous body ceases, .... 365

" how augmented in speaking trumpets, 370

Space, absolute and relative, 78, 79

" not equally full, ' ... 396

Spheroid, the attraction of the same when the forces of its particles are reciprocally as the

squares of the distances, 239

Spiral cutting all its radii in a given angle, by what law of centripetal force tending to the

centre thereof it may be described by a revolving body, .... 107, 287, 291 Spirit pervading all bodies, and concealed within them, hinted at, as required to solve a great

many phsenomena of Nature, 508

Stars, the fixed stars demonstrated to be at rest, ... 404

" their twinkling what to be ascribed to, 487

" new stars, whence they may arise, . . '. 502

Substances of all things unknown, 507

Sun, moves round the common centre of gravity of all the planets, 401

" the periodic time of its revolution about its axis, 405

" its mean apparent diameter, • . . 453

" its true diameter, 398

" its horizontal parallax, ....••• 398

" has a menstrual parallax, • 403

" its attractive force how great, '....... 398

" : the weight of bodies on its surface, . . • • 399

" its density, 399

" its quantity of matter, , 399

" its force to disturb the motions of the moon, , 391, 419

" its force to move the sea, 448

Tides of the sea derived from their cause, 415, 448, 449

Time, absolute and relative, 78, 79

" the astronomical equation thereof proved by pendulum clocks, and the eclipses of Jupiter's

satellites, 79

A Vacuum proved, or that all spaces (if said to be full) are not equally full, .... 396

Velocities of bodies moving in conic sections, where the centripetal force tends to the focus, . 121

Velocity, the greatest that a globe falling in a resisting medium can acquire, . . . 344

Venus, its periodic time, 388

" its distance from the sun, 388

" the motion of its aphelion, 405

Vortices, their nature and constitution examined, 504

Waves, the velocity with which they are propagated on the superficies of stagnant water, . 361 Weights of bodies towards the sun, the earth, or any planet, are, at equal distances from the

centre, as the quantities of matter in the bodies, 394

" they do not depend upon the forms and textures of bodies, 395

" of bodies in different regions of the earth found out, and compared together, . . 409

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