# The Mathematical Principles of Natural Philosophy (1846)/Index

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INDEX TO THE PRINCIPIA.

Æquinoxes, their præcession—the cause of that motion shewn, | 413 | |

“ | the quantity of that motion computed from the causes, | 458 |

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, shewn, | 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 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 neighbourhood of the sun, | 487 |

“ “ | their wonderful rarity, | 490 |

“ “ | their origin and nature, | 463 |

“ “ | in what space of time they ascend from their heads, | 490 |

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 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 decryption, 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 |

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 |

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 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 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, 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 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 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 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, | 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 |

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 inertiæ, 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 |

“ | 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 semi-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, 144 |

“ | 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 apogæon is in the syzygies with the sun; and more swiftly, in a contracted orbit, when the apogæon 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 apogæon moves more slowly when the earth is in its aphelion, more swiftly in the perihelion, | 414, 445 |

“ | its apogæon goes forward most swiftly when in the syzygies with the sun; and goes backward in the quadratures, | 414, 446 |

“ | its eccentricity greatest when the apogæon 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, | 443 |

“ | 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 on by an uniform force of gravity, | 255, 268 |

“ | of bodies revolving in resisting mediums, | 287 |

“ | of funependulous bodies in resisting 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 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 |

Planets, 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, Primary, 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 distances 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, Secondary, 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 |

Problem Keplerian, solved by the trochoid and by approximations, | 157 to 160 | |

“ “ | of the ancients, of four lines, related by Pappus, and attempted by Cartesius, 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, | 331 |

“ | of mediums is as their density, cæteris paribus, |
320, 321, 324, 329, 344, 355 |

“ | is in the duplicate proportion of the velocity of the bodies resisted, cæteris paribus, |
258, 314, 374, 329, 344,351 |

“ | is in the duplicate proportion of the diameters of spherical bodies resisted, cæteris paribus, |
317, 318, 329, 344 |

“ | of fluids threefold, arises either from the inactivity of the fluid matter, or the tenacity 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 | |

Rules of philosophy, | 384 | |

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

“ | the greatest heliocentric elongation of the Huygenian satellite from Saturn'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 |

Sesquiplicate proportion defined, | 101 |

Saturn, its periodic time, | 388 | |

“ | its distance from the sun, | 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 atmosphere, | 447 | |

Sounds, their nature explained, | 360, 363, 365, 366, 367, 368, 369 | |

“ | not propagated in directum, |
359 |

“ | caused by the agitation of the air, | 368 |

“ | their velocity computed, | 368, 369 |

“ | somewhat swifter by the 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 phenomena 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 planeta, | 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 |