C. K. OGDEN .S ON S AR ASTRO N ( )M Y . Translated ASTRONOMY. By Sir JOHN F. W i Engravings. 8vo. price IS*. 1] . LSTBONOMl B S : JOHN F H js.ed. METEOROLOGICAL ESSAYS. Translate< r:er and V.P.R.*. IARD OWEN'S LECTURES on the Y and PHYSIOLOGY of the INVERTEBRATE AN - [CTIONARY //('jo n, revised aii 1 c 'TR1CITY, 'HEORY and . In 2 vols. witb many DiscoverK vols. crown 8vo LECTURES Translated ,.Hvo.21a. ORGANIC I'.'HES on LIGHT in its of nil the known Photographic Pro- TURKS on POLARISED LIGHT, i left by the Author; and edited by the Rev. B, Pow BLL, ; 0,0 L^ ANALYTICAL SIR ISAAC NEWTON'S PRINCIPIA. BY HENRY LORD BROUGHAM, F.R.S. MEMBER OF THE NATIONAL INSTITUTE OF FRANCE, AND OF THE ROYAL ACADEMY OF NAPLES; E. J. ROUTH, B.A. FELLOW OF ST. PETER'S COLLEGE, CAMBRIDGI LONDON: LONGMAN, BROWN, GKEEN, AND LONGMANS; C. KNIGHT EDINBURGH : A. AND C. BLACK ; GLASGOW : B. GRIFFIN. 1855. LONDON: Printed by SPOTTISWOODE & Co. New-Street-Square. TO PROFESSOR THE BARON PLANA, OF TURIN, THIS WOBK IS INSCRIBED, AS A 8MALL TOKEN OF RESPECT FOR THE GREAT TALENTS AND PROFOUND LEARNING OF THAT EMINENT MATHEMATICIAN. 1C97S20 CONTENTS. General remarks. Division of the work into Three Books, 1. State of Physical Astronomy and Dynamics before Sir Isaac New ton , 2. General law of gradual discovery, ib. Exam- ples Logarithms, ib Fluxions, ib. History of this Cal- culus, 3. Calculus of Variations, 5 Euler, Lagrange, Bernouilli, Emerson, 7. Copernican Theory, ib. Galileo's discoveries, 8. Kepler's laws, ib. Huygens, ib. Borelli, ib. Hooke, ib. Halley, ib. Peculiar maturity of the New- tonian theory as at first delivered, 10. Nothing since sup- plied to its demonstration which Sir Isaac Newton originally had left imperfect, ib. note. Three services beside the disco- very of Gravitation, performed by this work to science, ib. Prodigious merit, even if gravitation were struck out of it, 11. Reception of the Principia slow even in England, ib. Editions, ib. Maclaurin and Voltaire, 12. Difficulty of read- ing it from its Conciseness and Synthetical form, ib. Jesuits' edition, 13. Submission to papal authority, ib. Pius VII. 's liberality, Sorbonne and Buffon, ib. note. I. Definitions of the Principia, 14. Two remarks on them, ib. Early view given of the Great Discovery to which the whole work leads, 15. Three laws of motion, ib. Six corollaries ~, to them, ib. Summary of dynamics, as it existed before Sir Isaac Newton, 16. Scholium to the laws of motion, upon uniform and accelerated motion, 17- Laws and formulas on velocity, space, and time, ib. note. (SECTION I. Principia.) Method of prime and ultimate ratios, 19. Treatise on Fluxions, 20. Fundamental principle of the generation of quantities, ib. Generation of curves, 21. A 3 ri CONTENTS. Nomenclature, 22. Notation, ib. Advantages and disad- vantages of the two notations, 23. Finding fluxion (or diffe- rential) of a rectangle, 24. Square, ib. Solid, ib. Quantity of any power by analogy, ib. Deduction of the rules from other principles, 25. Finding fluents (or integrals), ib. Method of drawing tangents, 26. Normals, ib. Exemplified in the conic sections, 2?. Problems of maxima and minima, ib. Example, ib. Quadrature of curves, 28. Example : Parabola, ib. Rectification of curves, ib. Example : Cir- cular arcs, ib. Measurement of solids, 29- Example : Cone, sphere, and cylinder, ib. Finding radius of curvature, ib. Example : Parabola, 30. Addition of constant quantity in integration, ib. Method of investigation used by Sir Isaac Newton, ib. Subjects of the Three Books, 31. (SECTION II. Principia.) Areas proportional to the times, round a centre of forces, ib. Empirical discovery of Kepler, ib. Proposition and its converse proved, 32. Corollaries to this fundamental law of centripetal forces, 33. Law of circular motion, the force as the square of the arc, and inversely as the distance, 34-. Demonstration, ib. Importance of this propo- sition, 35. Consequences in showing the laws of motion, ib. Demonstrates the general law, of which Kepler's rule of the sesquiplicate ratio is one case, 36. Demonstrates the law of the inverse square of the distance, 3?. Law extended to other curves, ib. Consequence that bodies fall through portions of the diameter, proportional to the squares of the times in which they describe the corresponding arcs, 38. Moon being deflected from the tangent of her orbit by gravitation proved from hence, 40. Reference to other proofs of it, 41. note Investigation of General Expressions for Centripetal Force, 42. Five for- mulas given, 43. Herrman's, 44. Laplace's, 46. Maclau- rin's, ib. J. Bernoulli's, ib. Proof that this is taken from Prop. VI. B. I., Principia, 47. Keill's imperfect acquaintance with this subject, ib. Herrman's mistake, 48. Formulae exemplified in the case of the parabola, 49. Ellipse and hyperbola, ib. Centrifugal forces. Formulae of Huygens, 50. Subject of Centripetal forces divided into four heads, ib. i. The force required to describe given conic sections ii. The drawing conic sections from points or tangents being given; 1. When one focus is given ; 2. When neither is given. iii. The find- ing the motion in trajectories that are given. iv. The finding trajectories generally when the forces are given. i. The first head is treated of in the remainder of the Second, and the whole of the Third Sections of the Principia. Central force in a circle, when the centre of forces is the centre of the circle, or any other point in the diameter, or in the circumference re- CONTENTS. Vli spectively, 51. Central force in an ellipse when the centre of force is the centre of the ellipse, 53. Converse of the pro- position, 54. Equality of periodic tiroes in concentric similar curves, when the law of the force is as the distance, ib. Con- sequence of the sun being in the centre of the system, 55. (SECTION III. Principia.) Law of forces when the centre of forces is in the focus of the curve, ib. General theorem that in each of the three conic sections the law is the inverse square of the distance, ib. Converse of the proposition proved, 57. J. Ber- nouilli's objection to Sir Isaac Newton's proof, 58. Shown to be groundless, ib. His objection to Herrman's demonstration, 59. Refuted, ib. Motion in concentric conic sections, the centre of forces being in the focus, ib. Demoivre's theorem, 60. Demonstration of Kepler's law of sesquiplicate ratio generally, 6l. Inverse problem of finding the orbit from the force being given, ib. Determination of the nature of the orbit from the forces, 62. Sir Isaac Newton's observations on the investigation of disturbing forces, ib. Anticipates La- grange's investigation, 63. note. Importance of Perpendicular to the Tangent and Radius of Curvature in all these inquiries, 63. i. (SECTIONS IV. V. Principia.) General ohservations on these sections, 64. Illustration of their use in Physical Astronomy, 65. Further illustration from their application to the problems on comets, ib. Comparison of theory with observation by Newton, 66. By Halley, 6?. Comets of 1680, 1665, 1682, 1683, ib. General remarks on the importance of these sections, 68. iii. Motion (1) in given conic sections, (2) in straight lines, ascending or descending. iii. (1.) (SECTION VI. Principia.) Method of determining the place of a body in a given trajectory, being a conic section, at any given time, 69. Solution for the parabola, 70. Method conversely of finding the time, the places being given, 71. Solution for the ellipse, or Kepler's problem, ib Difficulty of the problem, 72. Sir Isaac Newton's proof that no oval is quadrable, ib. Class of curves returning into themselves and quadrable, beside the class mentioned by him of ovals connected with infinite branches, 73. Demonstration respecting the ellipse, 74. Observations, ib. Sir Isaac Newton's solution of Kepler's problem indirectly by the cycloidal, ib. Another solution directly by a cycloidal curve, 75. Astronomical No- menclature, 76. ii (2.) (SECTION VII. Principia. Motion ascending and descending in straight lines, ib. Determination of times of descent and ascent, ib. Determination of velocities in case of parabolic lines, 77. Time of moon falling to the earth, 79. A 4 Vlii CONTENTS. Analogy of the case of planets falling into the sun, to the structure of bees' cells, ib. note. General solution of the pro- blem for all kinds of centripetal force and orbit, 79. iv. (SECTION VIII. Principia.) Observations upon the general inverse problem of centripetal forces, or finding the orbit, the force being given, 80. Sir Isaac Newton's solution, though geo- metrical, is less synthetical than usual, 82. Determination of the trajectory generally by the method of quadratures, ib. Re- marks on that method, 85. The subject illustrated in the case of the inverse cube of the distance, ib. Another solution given by a polar equation, 86. Conclusion of the subject of centri- petal forces in fixed orbits, and round an immoveable centre, ib. Of motion in moveable orbits divided into two heads, 87- i. When the orbit and centre are in the same plane. ii. When the orbit's plane is eccentric. i. (SECTION IX. Principia.) Determination of the motion of the apsides, ib. Proportion of force to distance, which make the axis or apsides advance and retire respectively, 88. Deter- mination of motion of apsides from the force and conversely, 8p. Gravitation the only force by which the line of apsides can coincide with the fixed axis, 90. Motion of the apsides with different centripetal forces, ib. Application of the theory to the motion of the moon's apsides, 91. To the motion of the earth's apsides, 92. Sir Isaac Newton did not reconcile the theory with observation, as regards the moon, ib Misstaie- ment of Bailly on this subject, ib. History of the question respecting the agreement of the theory with the observation, 93. Euler, D'Alembert, Clairaut, ib. Clairaut's error, and his discovery of the agreement between the theory and fact, 94. Laplace's solution and discoveries, ib Reference to the papers of the three mathematicians on the problem of these bodies, ib. note. Bailly's further erroneous statement respecting Sir Isaac Newton, 95. Proof of that error, ib. General opinion of Bailly on the Newtonian lunar theory erroneous, 96. Testimony of Laplace, 97- Error of Laplace respecting Sir Isaac Newton's assumption as to the perigeal motion, ib. ii. (SECTION X. Principia.*) Determination of trajectories in a given plane, when the centre is out of that plane, 98. Of trajectories on a curve surface, 100. Example of the circle and cylinder, ib. Motion of pendulums, 101. Properties of hypercycloids, and hypocycloids, ib. Isochronism of the cycloid, 1 02. General solution for all curves by the evolutes, ib. Peculiarity of cycloid and logarithmic spiral in being their own evolutes, 103. Reason why Sir Isaac Newton took the case of hypercycloids and hypo- cycloids, and not cycloids, ib Measurement of gravity by the pendulum, deduced from these propositions, ib. Conclusion CONTENTS. IX of the subject of motion where the centre of forces is immoveable, ib. (SECTION XI. Principia.) Motion in orbits where the centre is dis- turbed, or where other forces disturb the motion divided into three heads, i. Disturbance produced by the mutual action of two bodies revolving round one another, 104. Demonstration of their motion round each other, and round the common centre of gravity, 105. Motion referred to a body in the centre of gravity, 10?. Amount of the body which must be in the im- moveable centre that it may act there, as the bodies would act on each other, were one to be in the fixed centre, ib. Deter- mination of their absolute trajectories in space, ib. Application to the earth and moon, 108. ii. Disturbances produced by the action of the whole bodies of any subordinate system on each other, and by the bodies of other systems on any given subordinate system, illustrated from Laplace, ib. Remarks on Newton's investigations, and the problem of three or more bodies, 109- Comparative disadvantages under which he laboured, ib. Improvement, first, of the calculus itself, and secondly, by the introduction of that of variations, peculiarly fitted to facilitate these inquiries, 110. How the latter especially bears on the subject, ib. Motion of the moon's apsides and nodes, 112. Variation in the rate of both their motions, ib. Acceleration of the moon's motion, ib. The cause discovered by Laplace from the algebraical expression, 113. Connexion between the transverse axis and the mean motion, ib. Kepler's law demonstrated, 114. Proved by the mere exami- nation of the algebraical expression only to be true if there are no disturbing forces in action, 115. Same inspection likewise shows the retardation of the apsides and nodes to be caused like the moon's acceleration by the decrease of the earth's eccentricity, ib. Confirmation of the calculus by actual observation, ib. Slow secular inequality of the moon discovered by Laplace, in dimi- nution of her secular acceleration, 11 6. Irregularity of other orbits and motions, ib. Motion of earth's apsides produced by the disturbing forces of the greater planets, 117- Variation of orbits of other planets, ib. Disturbances at first seem not reduc- ible to any fixed rule, 118. Euler's attempt and errors, ib. His important discovery, ib. Discovery by Lagrange and Laplace of the stability of the system, and universal operation of the rule, 118. Mean motions of Jupiter and Saturn commen- surable, 119 Proportion of motion and distances of Jupiter's satellites, ib. Laplace's remarks on Jupiter and Saturn, ib. No satellite but the moon disturbs its primary, 120. The greater axes of the planetary orbits do not vary from one long period to another, 121. The period of their change being X CONTENTS. short, the mean motions of the planets undergo no secular varia- tion, 122. General law of stability of the system, 123. Gene- ral reflexion, 124. No resistance of an ethereal medium, nor any transmission of gravity in time, ib. iii. Marvellous powers of Sir Isaac Newton in discussing the subject of disturbing forces, 125. Great superiority to all his successors, ib. Determination of the disturbances arising from a third body's action upon other two, and theirs upon the third and each other, or problem of three bodies, 126. Attraction as the dis- tance, alone preserves all motion undisturbed, ib. Produces im- mense velocities, 127. The small actual derangement shows the inverse square of the distance not to be much departed from, 128. Investigation of the general problem, ib. Case of moving bodies and proportion of masses to forces, 131. Accelerations and retardations at different parts of the orbit : quadrature and syzygies, ib. Different planetary variations deduced by Sir Isaac Newton from the solution, 132 Extraordinary generalization of the problem to precession and tides, ib. Sixty-sixth proposition and its corollaries embrace all that has been done on the subject, 134. Error of Laplace, ib. note. Attraction under two heads, i. that of spherical ; ii. that of non- spherical bodies. i. (SECTION. XII. Principia.) Attraction of spherical snrfaces, 135. Remarkable inferences showing the solidity of the earth, 136. Attraction of spheres on particles beyond their surface, 137. On particles within their surface, 1 39. Five general theo- rems, ib. Corollary comparing corpuscular attraction with cen- tripetal forces, 1 40. Peculiarity of the actual law of gravi- tation,^. General solution for all other laws of attraction, 141. Reduced to the quadrature of a curvilinear area, ib. Solution of this quadrature, 142 Remarkable result when the force is inversely as the cube, or any higher power of the distance, 144. Attraction of spherical segments, ib. ii. (SECTION XIII. Principia.) Attraction of bodies not sphe- rical, 145. Proportion of attraction to homologous sides of similar bodies, 146. General theorem for attraction of all bodies as related to the centre of gravity, the force being gravitation, 14-7. Attraction according to any power of the distance in any symmetrical solids, ib. General solution, 148. Laplace's formula for attraction, 149. Motions of infinitely small bodies like light, 150. (SECTION XIV. Principia.) Proportion of angles of incidence, refraction and reflection, ib. Inflection and deflection, 151. Subsequent ex- periments on the coloured fringes by flexion, ib. General remark on the perfection of Newton's discoveries, 152. Solution of Descartes' focal problem, ib. Newton's optics, ib Dates CONTENTS. XI of the publication of Lectiones Opticae, Principia, and Optics, ib. note. General conclusions from the Newtonian discoveries relating to attraction, 153. Universal prevalence of gravitation, ib. Further proof of this from Herschel's discoveries in double stars, 164. Those observed by Cassini, ib. Not understood till the elder Herschel's time, ib. Their revolutions round each other and periods, 155. Apparently follow the law of sesqui- plicate proportion, ib. note. Three other important applications of the Newtonian theory of attraction, ib. 1. To find the weights of bodies at the sun and different planets, ib. 2. To find the masses of the heavenly bodies, and their densities, 156. Singular confirmation of the Newtonian theory by Lap- lace's calculation on Jupiter derived from different sources, 157- 3. Application of the theory to the figures of the heavenly bodies. ib. The earth's figure as determined by Newton, ib. Of Jupiter, compared with observations, 158. Measurements and pendulum experiments show the earth not to be homo- geneous, ib. Newton's computation respecting the earth's figure, on the supposition of its being homogeneous, 159- His frac- tion for the excess of the quotient diameter, ^-y, still used as accurate, ib. Remarks upon the wonderful completeness of the theory at its first establishment deduced from hence, ib. No improvements whatever upon that theory since his time, nor any defect found in its proofs, 160. Note on structure of the Earth, l6l. II. Introductory remark, 162. Resistance of media in all motions except those of the heavenly bodies, ib. Resistance of air, ib. Water, ib. Pressure and motion of fluids, ib. Hydrostatics, Hydraulics, Pneumatics, ib. Arrangement of the Second Book of the Principia under five heads. I. Motion of bodies in media which resist in different proportions to the velocity. i. Where the resistance is as the velocity. (Section I. Prin- cipia.') ii. Where the resistance is as the square of the velocity. (Section II. Principia.} iii. Where the resistance is partly in one proportion to the velocity, partly in another. (Section III. Principia.) II. Spiral motion in resisting media. (Section IF. Principia.) III. Motion of pendulums in resisting media. (Section V. Prin- cipia.) Xli CONTENTS. Page IV. Motion of projectiles in resisting media. (Section VII. Prin- cipia, in part.) V. Pressure and motion of fluids. i. Statics of fluids, or hydrostatics. (Section V. Principia.) ii. Motion of fluids, or hydraulics. (Section VII. Principia, in part.) iii. Motion propagated through fluids, whether elastic or non- elastic, including the pulses and waves, or acoustics. (Sec- tion VIII. Principia.) iv. Circular motion of fluids, or vortices. (Section IX. Prin- cipia.) Two general remarks on Book II., 163. Much done before Newton, ib. Summary of former discoveries, ib. Archimedes, ib. Galileo, ib. Pascal, ib. Torricelli, 164. Much left to succeeding inquirers, ib. Bernouilli, ib. Clairaut, ib. Laplace, 165. Value of this portion still great, and only less than that of the rest of the Principia, ib. ANALYTICAL VIEW. PKINCIPIA. BOOK SECOND. CHAPTER I. THE ELEMENTARY PRINCIPLES OF HYDROSTATICS, AND THE LAWS OF DENSITY OF AN ELASTIC FLUID COLLECTED ROUND A CENTRE OF FORCE. I. Elementary Principles of Hydrostatics - 16? 1 . What a fluid is, the terms viscosity, solidity, &c., - ib. 2. What the foundation is on which the theory of Hydrosta- tics is built. Newton, xix. - - 168 3. The fundamental equation by which we know the pro- perties of a fluid in equilibrium. Note I. - - 3 1 5 4. Three consequences of this equation. Note I. - ib. (1.) That there must in all cases be a certain relation among the forces - _ ifr. (2.) Level surfaces are surfaces of equal density - 318 (3.) Level surfaces are surfaces of equal temperature - 319 5. Newton, xx., fluids under the action of gravity only _ 1 70 II. The Law of Density in a compressible Fluid under the action of a central force - . - - 1 70 CONTENTS. Xlll Page 1. Solution of the question ; defect in the application to physical questions. Newton, xxi. xxii. &c. 2. First application measurements of heights. Note I i. 320 Second application form of our atmosphere, the zo- diacal light not part of the sun's atmosphere. Note II. - 322 CHAPTER II. THE FIGURE OF THE EARTH. 1. Newton's calculation of the ellipticity of the earth; its defects - - - 177 2. An accurate investigation of the ellipticity on the supposi- tion that the earth is homogeneous; the form thus found proved to be stable - - 178 Note III. - - - 324 3. Newton's calculation of the law of variation of gravity - 179 4. Newton's application of his theory to the planet Jupiter - 181 5. The figure of the earth considered as heterogeneous, Clai- raut and Laplace's results. Note IV. - - 328 a. The form of the strata - - 328 /3. The law of variation of gravity - 329 6. The law of density in the interior of the earth. Note IV. 330 7. Whether the interior of the earth is solid or fluid. Note IV. 333 8. Measures to determine by observation the ellipticity of the earth's surface. Note IV. - 337 a. Measurement of degrees - - 337 /3. Observations on the pendulum - 338 y. Astronomical observations - - 341 CHAPTER in. THE MOTION OF A PARTICLE IN A RESISTING MEDIUM. 1. The object and mode of conducting the inquiry - 183 2. When the resistance varies as the velocity - 186 a. Rectilinear motion - - ib. ft. Curvilinear motion. Section I. - ib. 3. When the resistance varies as the square of the velocity - 196 a. Rectilinear motion - - ib. (3. Curvilinear motion. Section II. - - - . - ib. 4. When the resistance varies partly as the velocity and partly as the square of the velocity, and the motion is rectilinear. Section III. - 205 5. When the resistance varies as any power of the velocity and the motion is rectilinear consideration of an analytical difficulty in the solution the terminal velocity and instances - - - 210 xiv CONTENTS. Page 6. The motion of a particle in a resisting medium round a centre of force - 212 a. The method used by Newton ... t '6. ft. The method supplied by the Planetary Theory. Section IV. - -21? CHAPTER IV. THE MOTION OF FLUIDS, AND THE RESISTANCE TO BODIES MOVING IN THEM. SECTION VII. 1. Newton's investigation of the law of resistance to similar bodies - - 221 2. The manner in which the resistance depends on the form. of the body. Newton, xxxiv. and xxxv. - - 226 3. Their resistance when the body is a surface of revolution 228 4. The surface of least resistance, its properties and form Scholium, Prop, xxxiv. - 229 5. The law of resistance deduced from experiment. Prop. XL. and Scholium ' - - - 233 NOTE V. THE RESISTANCE MADE TO BODIES MOVING IN FLUIDS DEDUCED FROM THE GENERAL PRINCIPLES OF DY- NAMICS. 1 . The equation of motion - 344 a. The ordinary hydrodynamics - - ib. ft. How changed when " internal friction" is taken into account - - 345 2. The ordinary law of resistance - 346 a. How deduced from the equations of motion - 347 ft. The general results of experiments m ade since New- ton's time compared with the law _ - 349 y. The resistance should be deduced from a rigorous solution of the equations of motion adapted to the case under consideration, case of the pendulum - 357 3. The resistance to a pendulum - - ib. a. Bessel's mode of expressing the resistance - - ib. ft. The careful experiments of Sabine, Baily, Cou- lomb, &c. - 360 y. Poisson deduces the nature of the motion and the re- sistance from the ordinary Hydrodynamic equations - 362 S. On comparing the theory with experiment they are found not to agree - 363 e. Professor Stokes takes into account the effects of internal friction _ 365 . The results agree with experiment - ? ' - 366 CONTENTS. XV Page 4. The resistance to floating bodies - 369 a. The phenomenon of emersion - - 6. ft. Waves are excited in the Fluid - - ib,. y. Strange variations of the resistance as the body changes - 370 CHAPTER V. THE MOTION OF A PENDULUM. 1. Some general considerations - - 240 2. Motion in vacuo. Newton, xxiv. Note VI. - 372 3. The properties of a pendulum. Note VI. - 375 4. Motion in a resisting medium ; modern method of con- sidering the perturbations of a pendulum ; the New- tonian method - - 242 5. Newton's experiments to discover the law of resistance - 252 CHAPTER VI. MOTION OF FLUIDS RUNNING OUT OF SMALL ORIFICES. 1. Newton's solution of the question, without limiting the orifice - - 257 2. Newton's corollary to deduce the resistance made by a fluid to a body moving in it - 260 3. The fallacy of Newton's reasoning How others attempted to pursue the investigation. Note VII. - - 381 4. The velocity as given by the Equations of Motion. Note VII. - ... 381 5. The efflux of an elastic fluid through a small orifice strange conclusions to be deduced from the formula St. Venant and Wantzels' experiments. Note VII. - 381 CHAPTER VII. THE MOTION OF WAVES. 1. General consideration on the nature of Waves - - 266 2. Waves in air sound - - 269 a. The nature of sound deduced from the phenomena. Scholium, Prop. t. - 269 ft. Examination of the case considered by Newton - 271 y. Velocity of sound Newton's error - 274 3. The manner in which sound spreads after entering through an orifice. Newton, XLH. - 277 E. The notes sounded by different pipes. Scholium, Prop. t. - 278 ri CONTENTS. Page Waves in water - - - 281 a. The motion is of the vibratory kind .- . . - 281 ft. Newton's reasoning on this subject. Newton, XLIV. - 282 y. Velocity of waves. Newton, XLV. & XLVI. - - 283 e. The nature of the motion of waves as given by a strict hydrodynamic theory. Note VIII. - - 389 '(,. Waves caused by the motion of a boat - - 3p4 p. Cause of breakers over sunken rocks - 396 0. How the wind raises the waves - - 397 CHAPTER VIII. THE THEORY OF THE TIDES. I. Newton's investigation on the tides - 285 a. The tides considered as a question in the motion of fluids, deduced from the Lunar Theory - - 286 ft. General explanations of eight phenomena of the tides 288 y. The calculation of the height of the lunar and solar tides - - - 291 L The tides in the moon - - 292 II. The theories that have been proposed since Newton's time. Note IX. - - 398 1. The Equilibrium Theory - - ib. a. Its fundamental hypothesis - 399 ft. The results of calculation made according to this theory, three kinds of tides - - 401 y. Airy's opinion of this theory - 403 The Hydrodynamic theory - - ib. a. Its fundamental assumptions - ib. ft. The results of calculation, three kinds of tides - 404 3. The Wave Theory - - 405 a. Consideration of ocean tides - - 406 ft. River tides, results of calculation and explanation of the chief phenomena - - 408 Where this theory fails - 412 III. Some results of observation - - 413 CHAPTER IX. THE CIRCULAR MOTION OP FLUIDS. 1. The hypothesis of the Cartesian theory - - 296 2. Newton's hypothesis as to the law of internal friction in fluids, the motion of a cylindrical vortex, Ber- noulli's objections to this result. Prop. LI. - - 297 CONTENTS. XV11 Page Some difficulties of the Cartesian theory which are considered by Newton - ' . - 301 a. The Sun's rotation. LII., Cor. 4. - 30.3 /3. The third law of Kepler. Scholium LIII. - ib. y. The two first laws of Kepler, Scholium LIII. - ib. . The density of the planets, LIII. - - ib. e. The disturbance of the sun's motion by the planetary vortices, LII., Cor. 5. &c. - 304 CHAPTER X. THEORY OF COMETS. 1. The comets are planets moving in very eccentric conic sections - - 306 2. To determine the particular orbit of a given comet - 307 APPENDIX. Note I. Galileo's persecution, 417. Note II. Another demonstration to p. 58,, 420. Note III. Force varying inversely as the distance, 421. Note IV. Central forces to more than one point, 424. Note V. Leibnitz's Dynamical tracts, 437. INTRODUCTION. THE object of this Analytical View is two-fold; first, to assist those who are desirous of under- standing the truths unfolded in the Principia, and of knowing upon what foundation rests the claim of that work to be regarded as the greatest monu- ment of human genius secondly, to explain the connexion of its various parts with each other, and with the preceding and the subsequent progress of the science. I. It cannot be denied that fully to comprehend the propositions, to follow the demonstrations throughout, requires the reader to be well ac- quainted with the Mathematics; and even one so prepared, will find the task far from easy in conse- quence of the synthetical method almost every- where adopted, the geometrical form of the inves- tigation in many cases where the algebraical would be more convenient, and the extreme conciseness very generally studied, often to the omission of XX INTRODUCTION. many steps in the process.* But it is equally un- deniable that with very moderate mathematical acquirements, a distinct and accurate knowledge may be obtained of the fundamental truths un- folded, of the reasoning by which some of them, and these the most important, are sustained, and of the nature of the proof on which the others rest. There is not much difficulty, indeed, in learning those truths, in comprehending the propositions without going further. But this is in every way a most imperfect knowledge, and neither can give satisfaction though retained, nor is likely to be re- tained without a knowledge also of the demonstra- tions. A great advantage, however, is gained, if the learner can not only follow the demonstration of the more important propositions, so as to be satis- fied of their truth, but can comprehend the nature of the proof in the other instances. He has both made solid progress in mastering the principles of the science, and has become able to judge for himself the merits of the great work which first taught it to the world. Thus two classes of readers may benefit by this Analytical View ; those who only desire to become * No one needs scruple to confess how difficult he has found the reading of the Principia, when so consummate a geometrician as Clairaut has made a like observation, (Mem. Acad. 1745, p. 329.), though somewhat qualified. INTKODUCTION. XXI acquainted with the discoveries of Newton, and the history of the science, but without examining the reasoning, and those who would follow the reasoning to a certain extent, and so far as a knowledge of the most elementary parts of geometrical and ana- lytical science may enable them to go. It has been found upon trial that readers of both descrip- tions have been able to peruse the work with advantage, even readers of the second description. These have easily followed, not only the commen- tary upon the gradual progress of discovery, and the state of the science before Newton*, but passing over the exposition of the differential calculus f have pursued the demonstration of the fundamental law of gravitation J, and even apprehended the proof of its universal action according to the inverse proportion of the squares of the distances. Pass- ing over the detailed discussion of central forces ||, the illustration of its application to planets and comets is involved in little difficulty ^f; and the manner of finding the place of these bodies at given periods of their revolution may be under- stood without entering into the details of the in- vestigation.** The ascertainment of the force * First 20 pages. ^ F. 20. to p. 31. t P. 31. to 41. P. 41. to 50. || P. 50. to 64. ^f P. 64. to 69. * * P. 69. to 76. INTRODUCTION. answering to a given orbit is much easier than the converse of finding the orbit from knowing the force ; accordingly the subject of the inverse problem may be passed over.* So may the great head of disturbing forces f, but the interesting his- tory of the problem of three bodies may without difficulty be comprehended. J The investigation, however, of Lagrange and Laplace of the prin- ciple upon which the stability of the system depends, must be taken upon the results without entering into the steps of the process. But the wonderful anticipation by Newton of sub- sequent discoveries may be generally understood and appreciated. || The subject of the attraction of masses, spherical and others, may be regarded as not coming within this elementary view of the work.^f The application of dynamical principles to the rays of light **, and the general statement respecting the Newtonian discoveries, and the con- stitution of the universe, may easily be followed. ff Nor can there be great difficulty in understanding the explanation given under the Third Book, of the effects of attractive forces upon the figures of the heavenly bodies, the motions of comets, so far * P. 76. to 87. t P. 87. to 108. J P. 93, 94. 108. to 112. P. 112. to 124. || P. 132. to 134. 1[ P. 134. to 150. * * P. 150. to 153. f f P. 153. to 160. INTRODUCTION. XX111 as that subject has not already been dealt with, and the doctrine of the tides, and in tracing the course of reasoning by which these important subjects are investigated.* The theory of motion in resisting media, and generally whatever relates to fluids, whether their pressure or their movements, forms the next sub- ject of inquiry ; but in the elementary view with which we are now occupied, it may be properly passed over, with only this remark, that the matters contained in the Second Book are con- veniently described at the end of the account given of the First, although it has been found expedient to follow a different plan of arrange- ment in the Analytical View of the Second from that originally laid down. The great progress which has been made in hydrodynamics since the time of Newton, has rendered it necessary to enter more minutely into the investigations connected with the Second Book, than into those of the First and the corresponding portions of the Third. Hence this portion of the present work cannot fail to be found less elementary than the former. An acquaintance with the subject, too, is less indispensable towards obtaining a knowledge of the Newtonian philosophy as exposing the system * P. 285. et seq. a 4 INTRODUCTION. of the universe, its structure and motions, al- though it must be sedulously studied by all who would become acquainted with physical science, and all who desire to understand the whole of the Principia. It has been deemed expedient in giving the pro- positions of the First Book, to anticipate in some de- gree their application to the motions of the heavenly bodies, which form the subject of the Third. This course was naturally suggested by the circumstance that the greater part of these propositions bear a dis- tinct reference to the heavenly motions. But it is truly gratifying to find, as we now do, from Sir D. Brewster's valuable Life of Sir Isaac Newton (one of the most precious gifts ever made both to scientific history and physical science), that the illustrious author himself considered this the best method of teaching the Principia to those not thoroughly conversant with mathematics. Applied to by the celebrated Dr. Bentley, who was desirous of so far understanding the book that he might be able to lecture upon its uses in behalf of Natural Religion, he laid down a plan of reading closely resembling that sketched in the beginning of this Introduction ; recommending that after the earlier propositions of the First Book, the Third should be taken so far as to perceive its scope, and then such INTRODUCTION. XXV parts of the First as " he should have a desire to know, or the whole, in order, if he should think fit." Newton also requires in his correspondent a much more moderate provision of geometrical and algebraical knowledge than another mathematician laid down as requisite, to whom application had been made for advice, and who gave Dr. Bentley so formidable a list of books as necessary to be read, that he at once appealed to Sir Isaac Newton him- self, who prescribed three or four instead of above thirty. (Yol. i. p. 464.) If it has been made manifest that a very limited acquaintance with mathematics may suffice for attaining a competent notion of the general scope of Newton's discoveries and of the great work which revealed them to the world, it is no less certain that the knowledge thus acquired must be superficial, except as regards the fundamental doctrine of gra- vitation, the foundation of the system ; and that in order well to understand the dynamical researches which have exercised so mighty an influence upon the whole of Natural Science, a much more full and minute study of the Principia is required. It is to be hoped, therefore, that readers of the two classes referred to, those of the second especially, may be encouraged to pass into the third, for whose use this Analytical View is designed ; may make XXvi INTRODUCTION. themselves, in some measure, masters of the Cal- culus, the help of which is required in most of the investigations ; and may follow these so as to under- stand the whole of the propositions, by satisfying themselves of their demonstrated truth. That this also can be done with only a previous knowledge of elementary geometry and algebra, has been proved upon trial, that knowledge sufficing to attain an acquaintance with the rules explained in those parts of this treatise above recommended to be passed over by the more general reader. It is by no means intended to affirm that a complete know- ledge of the Principia can be attained without much further study. An intimate familiarity with the Calculus, or with the analogous method of Limits, is required by those who desire to follow the whole of the demonstrations, and to perceive the connexion between the different steps as clearly as they can trace those of any elementary process in geometry or algebra. Other helps than this work are required, and are not wanting, to facilitate the entire mastery of the subject by such as would thoroughly understand the Principia in all its parts. The present treatise is not designed for their use, further than as it may aid them in the earliest part of their studies. It is intended for those who may not be able, or may not be disposed, to go beyond acquiring such a knowledge of the INTRODUCTION. XXV11 subject and of the Book, as can be attained by a moderate degree of labour, and an acquaintance with only elementary mathematics. II. The accomplishment of the other object of this Treatise, examining the connexion of the dif- ferent parts of the Principia with each other, and with former and subsequent discoveries, showing its transcendent merits, and removing the objections, or rather the criticisms, that have sometimes been offered upon a few comparatively unimportant por- tions of the great work, will, beside performing that service, also afford additional help to the study of it, and tend to promote the taste for understand- ing it, so as to judge of its unparalleled excellence. It is satisfactory to find that many of the propo- sitions are capable of demonstration by a process different from that employed by Newton, especially when this process is more easily followed. In many cases the analytical substituted for the syn- thetical method, is interesting as a matter of curi- osity, independently of its more didactic character. This may also be predicated of the occasional pre- ference of algebraical to geometrical reasoning. The greatest interest, however, belongs to observ- ing the mutual bearings of the propositions, per- ceiving sometimes how one arises out of another, sometimes how the two are so connected that toge- ther they exhaust the subject, sometimes how the XXViii INTRODUCTION. establishment of one mere general truth furnishes the proof of others less general which had been pre- viously reached by a different route ; often to mark the diversities as well as resemblances of propo- sitions, and the particular circumstances upon which these severally depend ; not rarely to note in what way others had imperfectly obtained the knowledge of these truths, or altogether had failed to observe them ; frequently to find them deriving new support from things afterwards brought to light, and to see them explain phenomena subse- quently for the first time observed ; above all, to see, and as we see to marvel how, beside those doc- trines, the teaching of which forms the main object of the work, which are expounded with an exhaustive fulness, and are at their first discovery established in absolute perfection, so that scarce any addition has, in the vast majority of instances, been found either possible or required, there are also the foundations laid of new discovery in other direc- tions, the rudiments provided of other systems, and the very course plainly. pointed out by which these unthought of truths should in remote ages be explored. On some few points differences of opinion having arisen as to other men's claims to the discovery, all controversial matters are purposely avoided. But although it must be confessed that philoso- INTRODUCTION. XXIX pliers, as well in foreign countries as among our- selves, have shown no reluctance to allow New- ton's title to the first place, there have occa- sionally been criticisms hazarded rather than objections made, touching several parts of his great work ; and these in most cases have origi- nated rather in inadvertence than in any unworthy prejudice. It became necessary to correct such errors, in justice to the illustrious author, who could riot have been aware of the statements, except perhaps in one instance. It is possible he might have known the groundless remarks of J. Bernouilli in that particular; the equally erro- neous statements of Bailly and Laplace of course he could not have been aware of, and we may confidently add, never could have foreseen; in- deed they could only be ascribed to oversight in those eminent authors. The error detected by F. Boscovich respecting the comet's path belongs to another class, and arose entirely from a ne- glect of that careful examination of the limits of a problem peculiar to the ancient analysis, that exhaustive process by which the prolixity some- times complained of finds ample compensation in its precluding the possibility of mistake. Thus it is hoped that, partly by the account of the work and partly by discussions connected with the subjects of which it treats, the study XXX INTRODUCTION. of it may be both promoted and facilitated ; and this kind of service towards the progress of science is not to be altogether contemned. They who are incapable themselves of advancing it by the discovery of new truths, may usefully employ themselves in helping others to a knowledge of what the great masters have done ; and they may best do this if they shall not disdain the office of elementary explanation and discussion. The wisest of the ancients was said to have brought Philosophy down from heaven to earth ; he cer- tainly valued himself chiefly on his unceasing ef- forts to stir up in men's minds the desire of knowledge. What he found necessary with regard to the nature of the subject, we in our day may perceive to be equally necessary because of the clouds in which writers of vast and original ge- nius almost unavoidably involve the records of their inquiries after unknown truths. But whatever brings men acquainted with those profound researches, raises their minds to con- templations far more sublime than any which are connected merely with worldly science. To survey the most wonderful works of creation, to compre- hend the laws by which the system of the universe is governed, the principles which everywhere per- vade it, and bear irrefragable testimony to the unity as well as the power of the divine Author and Dis- INTRODUCTION. XXXI poser of all, is the most impressive lesson that un- assisted reason can teach our species. It is an ob- servation of Paley, marked with his wonted sagacity, that though Physical Astronomy, until well under- stood, presents less striking proofs of design to the mind than the other branches of science, yet when fully apprehended, it very far exceeds all the other evidences of Natural Religion. We must recollect, too, that Newton himself regarded this as the most precious part of his philosophy; declared that in framing it he had been moved by a desire to in- culcate religious belief, expressed his gratification in finding that his efforts had not been vain, and closed the exposition of its principles with a com- mentary upon the nature and attributes of God. His followers may be permitted to indulge the hope that he would have prized their humble at- tempts at diffusing a knowledge of his immortal labours, rather as falling in with his pious wishes for the happiness of others, than as contributing to the illustration of a fame which is imperishable, nor admits of any increase. ERRATA. Page 33. In the figure between S and c, pnt C at end of line V. Page 42. In the figure at end of PS, put V. Page 47. line 9, for Y P read Y S ; and after " circle," transpose, " and joining VF " to before " we." Page 48. line 10, add "and the Fathers fall into the same error." I. 109. NEWTON'S PRINCIPIA, ANALYTICAL VIEW. THIS work is justly considered by all men as the greatest of the monuments of human genius. It contains the exposition of the laws of motion in all its varieties, whether in free space or in resisting media, and of the action exerted by the masses or the particles of matter upon each other, those laws demonstrated by synthetic reason- ing ; and it unfolds the most magnificent discovery that was ever made by man the Principle of Universal Gravitation, by which the system of the universe is go- verned under the superintendence of its Divine Maker. Two of the three Books into which the treatise is divided are chiefly composed of mathematical investigations, con- ducted by the most refined and profound, but at the same time the most elegant application of geometry, and of a calculus which is only a particular form of the fluxionary method invented by the illustrious author in his early years. The Third Book contains an explanation of the motions of the heavenly bodies, deduced chiefly from the first portion of the former part, and grounded upon the 2 NEWTON'S PBINCIPIA. phenomena observed by astronomers. This concluding portion, however, of the great work, is also interspersed with geometrical reasoning of the same admirable descrip- tion as characterized the former, and applied to the so- lution of problems respecting the heavenly motions. Before Sir Isaac Newton appeared to enlighten man- kind, and to found a new era in the history of physical science, the eminent men who had preceded him had made, during the century immediately preceding his birth, very important steps in furthering the advancement of our knowledge ; and they had approached exceedingly near that point which forms the most important of all his dis- coveries, according to a kind of law which seems to regulate the progress of human improvement a law of continuity, which apparently prevents any sudden, and, as it were, violent change, from being made in the in- tellectual condition of the species, and prescribes the unfolding of all great truths by slow degrees, each mighty discovery being preceded by others only less considerable than itself, and conducting towards it. The great discoveries in pure mathematics afford striking examples of this truth. That of Logarithms by Napier is, perhaps, the instance in which the most considerable deviation has been made from the rule ; but even here there had been some curious methods of mechanical calculation invented before, and the discoverer of lo- garithms himself had reached the point very nearly by other most ingenious contrivances, before he actually made his great step. But the Fluxionary or Differential Calculus gives a remarkable exemplification of the general principle ; and its subsequent most important extension, the Calculus of Variations, furnishes another not less striking. Ever since Descartes's happy application of Algebra to Geometry had NEWTON'S PRINCIPIA. opened the way to the grand discovery of Newton and Leibnitz, the foundation of modern science, mathematicians had been intent upon the resolution of problems connected with the rectification and quadrature of curves, and the determination of points that possess properties of maxima and minima, as well as the finding of normals, tangents, and osculating circles. These inquiries had led them to consider the laws by which the relations between the or- dinates and abscissae referred to any given axis are go- verned at different points of that axis ; for in truth that implies the nature of the curvature itself, and includes the manner in which the length of the curve line increases or diminishes, as well as the space which it incloses. They were thus led to examine the generation of those curve lines and curvilinear spaces, whether that is conceived to be effected by the movement in the one case of points, and in the other of straight lines, or is supposed to be produced by the constant juxtaposition of indefinitely small straight lines inclined to each other according to a given law, in the one case, and indefinitely small rect- angles in the other. The latter is perhaps the more natural supposition of the two, and not the less easy. For if any one is set to measure the area of a field bounded by a curvilinear outline, as he can at once measure a space inclosed within straight lines, his course will be to divide the given space into rectangles, and then to divide each of the smaller curvilinear spaces into other rectangles, and so on till he has exhausted the whole by a series of rectangles, always decreasing in size as they increase in number, and the last of which seem to coincide nearly or sensibly with the area of the outer or curved line of boundary. Thus he would proceed by trial and actual measurement of the space ; and thus do land-measurers (the lineal descendants of the first B 2 4 NEWTON S PRINCIPIA. geometers, as well as their namesakes) still proceed. But speculative mathematicians being aware of the general properties of the lines they have to examine, and these being regularly formed, which the boundary of the field is not, they could calculate the relations to each other of the sides of the rectangles into which they divided the figure, and could thus form series of rectilinear figures diminishing in size, and which series might be carried to any length so as ultimately to exhaust the curvilinear area. Thus ABC being a semicircle, it was easy to find the area of the semihexagon or three equilateral triangles ADF, FDE, and DEC, and then of the triangles FBExS, and again of the triangles F O B x 6, and so on ; so that the radius AD being called r, there was obtained a series of this form, f r 2 ^3 + f r 2 (2- ^3) +Jt? ^2 (2 Vz2 ^~2\ +* &c. : And thus we have also the approximation to the length of the circle. But the extreme cumbrousness of this calculus, which is still more unmanageable in other curves where the radii are not, as in the case of the circle, equal, made it necessary to find some other method ; and geometricians accordingly examined the laws by which the areas increase in each curve, so that by adding all those innumerable increments together their sum might give the exact space * The three first terms give 3.10582 ; the seven first come very near the ordinary approximation, 3.14159, for they give 3.14144. NEWTON'S PBINCIPIA. 5 required. The same process was attempted with the lengths of the curves, considering them as polygons whose sides diminished while their numbers increased indefinitely. In this way Cavalleri, Fermat, and Wallis, and still more Harriot and Roberval, appear to have come exceedingly near the discovery of the general rule for performing these operations before Newton and Leibnitz, unknown to each other, made the great step. Roberval especially had solved many problems of quadrature and of drawing tangents, by methods extremely similar to the Newtonian. Nor were the ancient methods of Exhaustion and Indi- visibles so far distant as to let us doubt that, had the old geometers been possessed of the great instrument of algebra, and bethought them of its truly felicitous ap- plication according to the idea of Descartes, long before our times they would have anticipated the discoveries which form the great glory of modern science.* The discovery of the Calculus of Variations affords a similar example of gradual progress. When the differential calculus had enabled us to ascertain the maxima and minima of quantities, for example the value of one co-ordinate to a curve, at which the other becomes a maximum or a minimum, or, which is the same thing, the point of greatest and least distance between the curve and a given right line, or, which is the same thing, when the general relation of the co-ordinates being given we were enabled by means of the calculus to examine what that particular value was at which a maximum property belonged to one of them then geometricians next inquired into the maxima and minima of different curves, that is to say, into the general re- lation between the co-ordinates which gave to every * Among other marvels in Galileo's history he seems to have made a near approach to the calculus. See M. Libri's most able and learned work, Hist, de Math, en Italic, torn iv. B 3 6 NEWTON'S PRINCIPIA. portion of the curve a maximum or a minimum value in some respect. Thus, instead of inquiring at what value of x (the abscissa) in a known equation between x and the ordinate y, y became a minimum, or the curve ap- proached the nearest to its axis, the question was what relation x must have to y (or what must be the equation as yet unknown) in order to make the whole curve, for example, of the shortest length between two given points, or inclose with two given lines the largest space, or (having some property given) inclose within itself the largest space, or be traversed in the shortest possible time by a body impelled by a given force between two given points. Here the ordinary resources of the differential calculus failed us, because that calculus only enabled us, by substituting in the differential equation the value of one co-ordinate in terms of the other, to make the whole equal to nothing, as it must be at the maximum or minimum point where there is no further increase or decrease. But here no means were afforded of making this substitution, and the problem seemed, as far as this method went, indeterminate. Various very ingenious resources were employed by Sir Isaac Newton, who in the Principia seems to have first solved a problem of the Isoperimetrical class that is, finding the solid of least resistance; and soon after by the Bernouillis and other continental ma- thematicians, who worked by skilful constructions and suppositions consistent with the data. The calculus called that of Variations has since been invented for the general solution of these and other similar problems. It con- sists in treating the relations of quantities, or of their functions, as themselves varying, but varying according to prescribed rules, just as the differential calculus regards the quantities themselves, or their functions, as varying according to prescribed rules. It bears to the differential NEWTON'S PKINCIPIA. 7 calculus somewhat of the relation which that bears to the calculus of fixed and finite or unvarying quantities. It is wonderful how very near Bernouilli, when he solved the problem of finding the line of swiftest descent, came to finding out this calculus ; if, indeed, he may not be said to have actually employed it when he supposed, not as in the case of the differential calculus, two ordinates of a known curve infinitely near one an- other, but three ordinates infinitely near, including two branches of an unknown curve, each infinitely small ; for he certainly made the relation of these ordinates to the abscissa vary. Euler used the calculus more sys- tematically in the solution of various problems ; but he was much impeded for want of an algorithm. This important defect was supplied by Lagrange, who reduced the method to a system and laid down its general prin- ciples ; but had Euler gone on a little step further, or had Bernouilli been bent on finding out a general method instead of solving particular problems, or had Emerson, who has one or two similar investigations in his book on Fluxions, reduced the method by which he worked them to a system by giving one general rule (which, writing a book on the subject, he was very likely to have done), the fame of that discovery would have been theirs, which now redounds so greatly and so justly to the glory of Lagrange. ^he discovery of Gravitation as the governing prin- ciple of the heavenly motions, is no exception to the rule which we have stated of continuity or gradual progress. When Copernicus had first clearly stated the truth to which near approaches had been made by his pre- decessors, from Pythagoras downwards, that the planets move round the sun, and that the earth also moves on its axis while the moon revolves round the earth, he yet 8 NEWTON'S PRINCIPIA. accompanied his statement with so little proof beyond the agreement with the phenomena, which the Ptolemaic hypothesis could equally boast of*, that for more than half a century afterwards it had no general acceptance, Bacon himself rejecting it ; when Galileo, by his telescopic discoveries, especially of the phases of Venus and the sa- tellites of Jupiter, and by his yet more important dis- coveries in the laws of motion, may be said first to have proved the truth of the Copernican system. Afterwards the satellites of Saturn, added to Kepler's observation of Mercury's transit over the sun, afforded most important confirmation. The great discoveries of this eminent man followed close after those of Galileo: First, the motions of the planets were found to be in ellipses with the sun in one focus ; secondly, lines drawn to the sun from them were found to describe areas proportional to the times of their revolution; and, thirdly, the relation was established be- tween the squares of those times and the cubes of the distances of the bodies from the focus. How near this brought scientific men to the cause or law of the whole is manifest, especially when we regard the connexion thus established between the re- volving bodies and the great luminary in the centre. Although Kepler himself erroneously mingled with the influence which this law of motion led him to ascribe to the sun, a transverse force which he deemed necessary to maintain the projectile motion of the planets round .the centre; yet others formed more correct ideas of the matter. It seems to have been Huygens, who, fourteen years before the " Principia " was published, first showed the true nature of centrifugal forces. Several years earlier, however, Borelli, in treating of the motion of Jupiter's * It is certain that its greater simplicity was, before Galileo's time, the only argument in favour of the Copernican theory against the Ptoleinsean. NEWTON'S PRINCIPIA. 9 satellites, considers the planets as having a tendency to resile from the sun and the satellites from the planets, but as being "drawn towards and held by those central bodies, and so compelled to follow them in continued revolutions." He also most accurately compares the re- ceding (or centrifugal) force with the tendency of a stone whirled in a sling to fly off at every instant of its motion. Hooke, a man of unquestionable genius, and whose partial anticipations of many great discoveries are truly remarkable, about the same time with Borelli, asserted that the at- traction of the sun draws away the planets from moving in straight lines, and that the force of the attraction varies with the distance. He had, as early as 1666, read to the Royal Society a paper explaining the curvilinear motion of the planets by attraction. Halley, as well as others, had even hit upon the inverse duplicate ratio, by supposing that the influence from the sun was diffused in a circle, or rather a sphere, and that therefore the areas proportioned to that influence were as the squares of the radii, and that consequently the intensities, being inversely as those areas, were inversely as the squares of the radii or dis- tances. Finally, Hooke had foretold, that whoever set himself to investigate the subject experimentally would discover the true cause of all the heavenly motions. Such were the near approaches which had been made to the law of Gravitation before its final and complete discovery. But although in this gradual progress it re- sembles almost all the other great improvements in science, in one material respect it differs from them all. The theory was perfect which Newton delivered, and the whole subject was at once thoroughly investigated. It was not merely that the general principle hitherto anxiously sought for, and of which others had caught many glimpses, was now unfolded and established upon appropriate founda- 10 NEWTON'S tions ; but almost every consequence and application of it was either traced, or plainly sketched out; it was pursued into all the details; a systematic account of its operation was given, symmetrical, and in its main branches complete ; so that, however nearly former inquirers had approached the general law, the distance was prodigious between their conjectures, how learned and happy soever, and the magni- ficent work which the genius of Newton had accomplished.* It must be observed, too, that, beside this grand achieve- ment, the Principia performed three other most important services to physical and mathematical science. First. It laid a deep and solid foundation for subsequent discoveries in the science of physical astronomy, both by the general principles of dynamics which it unfolded, and by the ap- plication which it made of these to the heavenly bodies and their motions. Secondly. It gave a complete system of dynamics applicable to all subjects connected with motion and force and statics a system throughout abound- ing in the most important original mathematical truths, expounded and proved with singular beauty, though with ex- treme conciseness. Thirdly. It propounded and showed the application of a new calculus, or method of mathematical investigation, that method by the help of which those truths had been discovered ; and by which others, before resting upon an empirical foundation, were demonstrated. Thus it is no exaggeration to say that, even if the great dis- * The subsequent discoveries of mathematicians by means of the improve- ments in the calculus, have added new illustrations, and traced further consequences of the theory. But there is only one of their improvements which can justly be said to have advanced the evidence of the fundamental principle further than Sir L Newton had carried it, by supplying any de- fect which he had left ; we allude to the reconcilement by Clairatit of the moon's apogeal motion according to the theory with the observations. This is fidly explained in the sequel. It forms one of the most interesting passages in the whole history of science. NEWTON'S PRINCIPIA. 11 co very of the law which governs the universe were taken away from the Principia, it would still retain its rank at the head of all the works of mathematicians, as the most wonderful series of discoveries in geometrical science, and its application to the principles of dynamics. That the reception of this work was not such as might have been expected has frequently been alleged; and al- though an ingenious and well-meant attempt has lately been made by an eminent author* to relieve this country from its share of the imputation, chiefly by showing the estimation the author was held in immediately after its publication; it is, on the one hand, certain that Newton's previous fame was great by former discoveries, and that after its appearance the Principia was more admired than studied. There is no getting over the inference on this head which arises from the dates of the two first editions. There elapsed an interval of no less than twenty-seven years between them; and although Cotes speaks of the copies having become scarce and in very great demand when the second edition appeared in 1713, yet had this urgent demand been of many years' continuance, the re- printing could never have been so long delayed ; nor was the next edition required for thirteen years after the se- cond. So that in forty years the greatest work ever com- posed by man reached only a third edition ; and that third has, during the succeeding hundred years, been the one generally in use ; although translations and excerpts have been published from time to time, and two editions were printed on the Continent, one at Amsterdam and one at Cologne. The doctrines of the work were, however, much more readily embraced and more generally diffused in this country, which had the benefit of Maclaurin's ad- * WhcwelFs History of the Inductive Sciences, TO!, ii. 12 NEWTON'S PRINCIPIA. mirable view of the more general principles of the system, published about the middle of the last century. On the Continent they made their way far more slowly ; nor was it until Voltaire employed his great powers of clear ap- prehension and lucid statement to give them currency, that the Cartesian prejudices of our neighbours gave way, and the true doctrine found a general and a willing acceptance. It must be admitted that the manner in which the truths of the Principia were unfolded, not only added somewhat to the slowness of the world at large in embracing them, but has also contributed to the reluctance with which men have generally undertaken the task of reading that great work, and satisfying themselves of the proofs upon which its doctrines rest. Conciseness is everywhere rigorously studied. Not only does the author avoid all needless pro- lixity and repetition in unfolding his discoveries, but he leaves out so many of the steps of his demonstration, and assumes his reader to be so expert a geometrician, that the labour of following him is often sufficient to deter ordinary students from making the effort. If mathe- matical reading is never the same passive kind of operation with other studies, the perusal of the Principia is emphati- cally an active exercise of the mind. For what, to the intuitive glance of him who could discover the theorem or solve the problem, appeared too plain to require any proof, may well stop common minds in their progress towards the point whither he is guiding them ; the dis- tances which he can stride at once over this difficult path must, by weaker persons, be divided into many portions, and travelled by successive steps. Add to which, that, as the method of proof is throughout synthetical, and as it is geometrical, the helps of modern analysis are thus withheld. Upon the whole, therefore, a most valuable NEWTON'S PRINCIPIA. 13 service was rendered to students by the able and learned commentary of the Fathers, Le Seur and Jacquier, who, in 1739 and 1742, published the Principia, with very copious illustrations, although it is to be regretted that they resort far less frequently to analysis than was de- sirable. It is remarkable enough, and affords an addi- tional proof of the slow progress which truth had then made in some parts of Europe, that these excellent authors deemed it necessary to accompany their publication of the Third Book, which treats of the heavenly motions, with a declaration in these words : " Newtonus in hoc tertio libro Telluris motas hypothesim asserit. Autoris propositiones aliter explicari non poterant, nisi eadem quoque facta hypothesi. Hinc alienam coacti sumus gerere personam; caeterum latis a summis Pontificibus contra Telluris motum Decretis nos obsequi profitemur." This edition is dated, as might be supposed, at Rome.* The Principia begins with a definition of terms, and a compendious statement of the science of dynamics as it existed previous to Newton's discoveries. The definitions, eight in number, comprise that of quantity of matter, which is in the proportion of its bulk and density, the density being the proportion of its mass to its bulk the quantity of motion, which is in proportion to the velocity and quantity of matter jointly the vis inertia, which is * It must, however, be observed, that such bigotry and intolerance was not confined to Rome. As late as 1769, Buffon was compelled, by the interference of the Sorbonne, to publish a recantation of some portion of his fantastical theory of the earth, comprehending, as it happened, the very few things in it which had any reasonable foundation. We ought also to mention, for the credit of the Papal Government, that a late pontiff (Pius VII.) procured a repeal of the decree against the Copernican system. 14 NEWTON'S PRINCIPIA. the force or power of matter to persist in any given state, whether of rest or of motion in a straight line, and to resist any external force impressed upon it to change that state centripetal force, which is the power that draws towards a given point or centre bodies at a distance from it finally, the three kinds of amount of centripetal force ; the absolute amount, in proportion to the intensity of the power exerted in drawing towards the centre ; the accelerating, in proportion to the velocity generated in a given time; and the moving, in proportion to the motion generated in a given time towards the centre.* Two things are worthy of remark in these definitions : first, that, as if foreseeing the cavils to which his doc- trines would give rise, he guards, in a scholium, against the supposition that he means to give any opinion respect- ing the nature or cause of centripetal force, much less that he ascribes any virtue of attraction to mere centres or mathematical points ; whereas he only means to express certain known and observed facts : secondly, that, in illustrating his definition of centripetal forces, he really anticipates his great discovery ; for, after giving the examples of magnetic action, and of a stone whirled in a sling, he proceeds to the motion of projectiles, and shows how, by increasing the centrifugal force, they may be made to move round the earth, as may also, he says, the moon, if she be a heavy body, or in any other way be deflected towards the earth, and retained in her orbit. That force, he adds, must be of a certain amount, neither more nor less ; and the business of mathematicians is to find this necessary amount; or, conversely, having the amount * There are eight definitions in the book, though we have only given them under seven heads, not having made a separate definition of the force impressed, which is here mentioned under the important head of the vis inertia. NEWTON'S pmNCiriA. 15 given, to find the curve in which it makes the body move. The connexion between the inquiries which form the main subject of the two first books of the Principia and Physical Astronomy, the subject of the third, is thus explicitly stated ; but a plain indication is also here afforded of the great discovery in which the whole inves- tigation is to end. The doctrines of dynamics, known previously to his discoveries, are then given in the form of corollaries to the three general Laws of Motion. The first law is that of the vis inertia, already explained ; and it is to be observed here that a steady and clear conception of the tendency of all moving bodies to proceed in a straight line unless deflected from it, is, perhaps, more than anything else, that which distinguished the Newtonian from the immediately preceding doctrines, mixing up as these did more influences than one proceeding from the centre with a view to explain the composite motion of the planets. The second law is, that all changes in the motion of any body, or all changes from rest to motion, are in proportion to the moving force impressed, and are in the straight line of that force's direction. The third law is, that reaction is always equal and opposite to action ; or that the mutual actions of any two bodies are always equal to one another, and in opposite directions. From these laws the six corollaries which are added deduce the fundamental principles of dynamics; and there is a scholium to the whole, which states the applica- tion of those principles to the descent of heavy bodies, and the parabolic motion of projectiles. Of all the prin- ciples, the most important is that of the Composition and Resolution of forces. As by the first law a body always 16 NEWTON'S PRINCIPIA. in the straight line it moves in, unless in so far as some other force alters its direction ; and as by the second law any new force impressed tends to move it in its own direction, it follows that, if two forces, not in the same nor in directly opposite directions, act at one time, and by an instantaneous impulse, on any body at rest, it must move in such a direction as that it shall be found both in a line parallel to the direction of the one force, and in a line parallel to the direction of the other ; that is to say, in the diagonal of a parallelogram whose two contiguous sides are in the directions of the two forces, and are respec- tively equal to the space each force would carry it through in its own direction. Moreover, as each force separately would have carried it to the end of the line of its direc- tion in the given time, it must move through the diagonal in the same time which it would have taken to move through either side if either force had acted alone. Thus the direction of every motion occasioned by any two forces acting at an angle to each other, may always be found by completing the parallelogram of which the direc- tions of those forces are the contiguous sides ; and so of any motion occasioned by any number of forces whatever acting angularly. And, conversely, every motion of a moving body may be resolved into two, of which the one is in any given direction whatever, and the other is found by completing the parallelogram, whereof that given direc- tion is one of the sides, and the direction the body moves in is the diagonal. From this resolution of forces it is easily shown, that if any weights or other powers acting in parallel lines are applied to the opposite ends of a lever moving on a centre or fulcrum, the effect of each will be directly as its distance from that centre, in other words, as the length of the contiguous arm of the lever ; consequently, that if the NEWTON'S PRINCIPIA. 17 weights or powers are made inversely as those lengths, the whole will be in equilibrio or balanced. This is the well known and fundamental principle of the lever, the founda- tion of mechanics ; and it applies also to the wheel and axle and the pulley. The fundamental properties of the screw, the wedge, and the inclined plane are deduced in like manner from this important proposition. So may all the properties of the centre of gravity, and the method of finding it ; for, in fact, the fulcrum of the lever is the common centre of gravity of two bodies equal to the two weights, and placed at the opposite ends of the lever; and the line joining the bodies is divided in the inverse proportion of those bodies. It also is easily shown that the common centre of gravity of two or more bodies is not moved, nor in any way affected, by their mutual actions on each other, but it either remains at rest, or moves forward in a straight line. So are the relative mo- tions of any system of bodies, whether the space they occupy is at rest, or moves uniformly in a straight line. The Scholium to the Laws of Motion first considers very briefly the motion of falling bodies which descend with a velocity uniformly accelerated, that velocity which is given to them by the attraction of the earth during the first instant continuing and having at each succeeding instant a new impulse added. The acceleration, therefore, is as the time ; and they move through a space propor- tional to the velocity and the time jointly, consequently proportional to the square of the time, since the velocity is itself proportional to the time.* * Velocity is as time, i. e., v is as m t ; space is as velocity x time, or s as v x t ; therefore space is as time x time, or as square of time, that is, * is as m t x t, or m t 2 . The proportion of the space fallen through by the force of gravity (or moved through by any body uniformly acce- lerated) to the square of the times, is also demonstrated thus. Let the C 18 NEWTON S PRINCIPIA. The Scholium next, with equal brevity, states the projectile motion of heavy bodies. If a body be impelled in one direction by a force producing a uniform motion, and in another direction at any angle with the former by a force not uniform but accelerated, the diagonals which it will move through will at every instant change their direction towards the quarter to which the acce- lerating force tends. But a series of such diagonals is a polygon of an infinite number of sides, infinitely small : in other words, a curve line. Now in the case of a projectile, this continued or accelerating force is such as to make the body, if no other force acted on it, fall through spaces proportional to the square of the times. velocity acquired at any moment P of the time A P be P M, and because the velocity uniformly increases, or as the time, PM:AP;:BC :AB, and therefore the line A C is a straight line, and the triangles A P M, A B C, are similar. But if q N is infinitely near P M, or P q represents the smallest conceivable time, the motion during that time may be conceived to be uniform and not accelerated. Now the space through which any body moves is as the velocity multiplied by the time (s = v 0, therefore the space moved through in the time P q is as P q x q N. So the space moved through in the time A B will be as the sum of all the small rectangles P q x N q, or as the triangle ABC. But the triangle A B C is to any oter of the triangles APM as AB 2 : AP 2 ; therefore the spaces are as the squares of the times. The great general importance of this proposition which Galileo first proved, makes it necessary to have the demonstration clearly fixed in the reader's recollection. NEWTON'S PRINCIPIA. 19 The other force acting once for all would make it, were there no gravity acting, move in spaces proportioned to the times simply. The latter or projecting force would make it move through AB uniformly, or in spaces proportional to the times ; the force of gravity would make it move through AP with a motion proportioned to the square of the times ; therefore it will move in a curve passing through M, if P M is equal, and parallel to AB ; and AP will be as the square of AB or PM, which is the property of the conic parabola m . A P = P M 2 , m being the parameter to the point A. The Scholium concludes by stating some consequences of the equality of action and reaction, the third Law of Motion, with respect to oscillation and impact, and also with respect to mutual attractions ; of which conse- quences the most important is that the attraction or weight of heavy bodies in respect of the earth, and of the earth in respect of them, is equal. The great work itself, after these preliminary though essential matters, proceeds to its proper subject. But in order to show how the demonstrations are conducted, a short treatise is prefixed upon the method of Prime and Ultimate Ratios, in eleven Lemmas, with their corollaries. This method consists in considering all quantities as c 2 20 NEWTON'S PRINCIPIA. generated by the uniform progression or motion of other quantities, and examining the relations which the smallest conceivable spaces thus generated by this motion bear to one another, and to the spaces generated at the moment of their inception, or when they are nascent, which is termed their prime ratio, and at the moment of their vanishing, or when they are evanescent, which is termed their ultimate ratio. Thus a point moving along in a straightforward direction generates a straight line ; a line moving parallel to itself, or two lines moving at right angles to one another, generate a rectangle : one line moving, while a point in it moves along it so that its progress on the moving line always bears a given ratio to the progress the line has made (m.AP = PM), describes a triangle; the same motion, if the progress of the point bears a variable relation to that of the line (x.AP = PM'; x.xAP being some function of A P), describes a curve line and curvilinear area ; and so of solids, which are generated by the motion of planes. It follows from this mode of generation that if the length of any curve line be divided into an infinite number of lines, the sum of these will not differ from the curve line by any assignable quantity, nor will each differ from a straight line ; and if its area be divided into an infinite number of smaller areas by lines drawn parallel to the NEWTON'S PBINCIPIA. 21 line whose progressive motion generated the curvilinear area, the sum of these infinitely narrow areas will differ from the area of the curve by a difference less than any assignable quantity, nor will each differ from a rectangle ; in other words, the ratio of the nascent curve line and nascent curvilinear area will be that of equality with the small lines and small rectangles, and the ultimate ratio of the sums of the lines and rectangles to the whole curve line and curvilinear area, respectively, will be that of equality : Or to put it otherwise, if the axis of the curve be divided into parts P P, &c., and the area into spaces PMRP, &c., by ordinates PM, PR, &c., and the num- ber of these spaces be increased, and their breadth PP be diminished indefinitely, which is the operation of the generative motion of PM, the size of each of the small spaces M N R O (by which the curvilinear a"reas differ from the rectangles) diminishes indefinitely, and the ultimate ratio of all the curve areas PMRP, and all the rectangles PNRP, becomes that of equality, and therefore the sum of evanescent differences NM OR, NROR, &c., whereby the whole curvilinear area differs from the whole amount of the rectangles P NRP, becomes less than any assignable quantity, or the curvilinear area coincides with the sum of the rectangles. And so of the sum of all the diagonals MR, RR, &c., which becomes the curve line MR A. Hence we infer that the amount of these small spaces c 3 22 NEWTON'S PRINCIPIA. or quantities N M O R, formed by multiplying together two evanescent quantities, is as nothing in comparison with the rectangles P M O P formed by only one evanescent quantity multiplied into a finite quantity, and may be neglected in any equation that expresses the relations of those rectangles with each other. But if some other quantities be found which are, in comparison with these small ones, themselves infinitely small, the areas formed by multiplying this second set of small quantities may be rejected in any equation expressing the relations of those first small quantities. Thus we have the .origin and constitution of quantities which in the Newtonian scheme are called fluotions of different orders, because conceived to express the manner of the generation of quantities by the motion of others, and in Leibnitz's language are called infinitesimals or differences, because conceived to express the constant addi- tion of one indefinitely small quantity to another. Ob- taining the fluxions, or the differences, from the quantity generated by the motion or by the addition, is called the direct method; obtaining the quantity generated from the fluxions, or finding the sum of all the differences, is called the indirect method. The one theory calls the direct method that of 'finding fluxions, the indirect that of finding fluents ; the other theory calls the former differentiation, or finding differentials, the latter integration, or finding integrals. The two systems, therefore, in no one respect whatever differ except in their origin and language; their rules, principles, applications, and results, are the same. A different symbol has been used in the two systems ; Newton expressing a fluxion by a point or dot, and the fluxion of that fluxion, or a second fluxion, by two dots, and so on. Leibnitz prefixes the letter d, and its powers NEWTON'S PRINCIPIA. 23 d 2 , d 3 , &c., instead, to express the differentials. In like manner f for sum is used by the latter to express the integral, and f by the former for the fluent. Although the continental method of notation is now generally used, and is on the whole most convenient, yet it has its inconve- nience, as the d is sometimes confounded with co-efficients of the variable quantities ; it is in some respects, too, not very consistent with itself; as by making d x 2 mean the square of the fluxion, or differential of x ; whereas it, strictly speaking, appears to denote the differential of x 2 . There can be no doubt, however, which notation is the most convenient in the extension of the system to the calculus of variations, where the symbol is ; for, although the varia- tion of a fluxion or differential may perhaps even more conveniently be expressed by 8 x than by d x, yet the fluxion of a variation can with no convenience be expressed by^, or otherwise than by dx. The expression of second fluxions undeveloped is also far less convenient bv the Newtonian notation. Thus the fluxion of -f is dx sometimes required to be expressed without developement, as in the expression for the radius of curvature, where it is often expedient not to develope it in the general equation, but to find -^ in terms of x or y before taking its fluxion ; yet nothing can be more clumsy than to place a dot over the fraction, whereas d (-- is perfectly con- venient. Several important considerations arise out of the nature and origin of these infinitesimal quantities as we have described them ; and to these considerations we must now shortly advert, as they give the rules for finding the 24 NEWTON'S PBINCIPIA. fluxions or differentials of all quantities, and, conversely, lead to those for investigating or finding the fluents or integrals of fluxional or differential expressions. A rectangle A M being generated by the side P M moving along A P while the side N M moves along A N, the movement or fluxion or differential of A M, or of A P x P M, is P S + M O, part of the gnomon P S ) R > M + S O, because the rectangle M V is evanescent compared with the other two, and is to be rejected. Therefore the differential of AP x PM = PM x PT + NM xNO, orPMxPT+APx MR. Calling A P = x, and PM = y, and P T = dz, and MR = dy, we have the differential of x y = x dy + yd x. But if the figure be a square, and A P = P M, or x = y, then the differential is 2 x d x. So if we would find the diffe- rential of a parallelepiped whose sides are x } y, and z, we shall in like manner find that it is x y d z + x z d y -f y zdx\ if a: = z, then it is 2 y x d x + x* dy, and if x = y = z, or the figure be a cube, it is 3 # 2 d x. From hence, although the geometrical analogy serves us no further (as there are only three dimensions in figures), we derive by analogy the rule that the differential of x m is m x 1 d x. Also there is no dimension of figure less than unity ; but by the same analogy we obtain the dif- ferential of x~ m , or [ , namely, m x~ m - 1 dx, or NEWTON'S PEINCIPIA. 25 x and of -^>OTx m xy- n =mx nl - } y- n dxnx m y- n - } dy, Consistently with the same principles, we may deduce this rule otherwise and more strictly. Let x + d x be the quantity when increased by the differential. This multiplied by itself, or its square when completed, is x 2 4- 2 x d x + (d .r)' 2 ; but to have the mere increment or dif- ferential we must deduct # 2 , and we must also reject (d #) 2 as evanescent compared with the function 2 x d x, which leaves 2 x d x for the differential. So the cube is x 3 + 3 * a rfar + 3ar(rfj? a ) + (rfjr) 8 , and rejecting, in like manner, we have 3 x 1 dx ; and by the binomial theorem (x + d x) m is x m +mx m ~ l d x, + &c. + (d x) m , of which only the second term can upon the same principles be retained; that is mx- 1 dx: And the same rules apply to the differentials of surds; so that the differential of (x + yf is dx+ It also follows that the fluent or integral is a quan- tity such that, by taking its fluxion or differential according to the foregoing principles, you obtain the given fluxional or differential expression. Thus if we have to integrate any quantity as x m d x, we divide by m + 1, and increase the exponent by unity, and erase x m+l the differential quantity ; so that - - is the integral required. But as every multiplication of any two quan- tities whatever gives a finite product, and every involution a finite power, while we can only divide so as to obtain a finite quotient, or extract so as to obtain a finite root, where the dividend or the power operated upon happens to be a perfect product or a perfect power; so in like 26 NEWTON'S PRINCIPIA. manner we can only obtain the exact integral where the expression submitted to us is a complete differential. Thus, though such an expression as is integrable, such an expression as is not integrable, for want of the x in the numerator; and various approximations and other contrivances are resorted to in order to ac- complish or, at least, approach this object, of which the methods of series, of logarithms, and circular arcs are the most frequently used. The simplest case of integration by series may be understood in examples like the last; for if the square root be extracted by a series, we may be able to integrate each term, and so by the sum of the integrals to approach the real value of the whole. From the doctrine as now explained, and the original foundations of the method as traced above, it follows that a variety of the most important problems may be solved with ease and certainty, which by the ancient geometry could only in certain cases, or by a happy accident, be investigated. Thus the tangents of curves may be found. For as the subtangent SP:PM::MN:TN, S p = :^ : And so the perpendicular may always be drawn ; for the subnormal RP = ==. There- SF ydx dx fore we have only to insert the one of these quantities in terms of the other from the equation between x and y (the equation to the curve), and we get the expressions for the NEWTON'S PRINCIPIA. 27 subtangent and subnormal. Thus in the common parabola, whose equation is y 2 = a x y the subtangent = y y or 2 x ; and in the hyperbola, whose equa- tion is xy = a i , the subtangent is x. So in the circle y 2 = 2 r x # 2 , " (the subnormal) = r x (r being the radius); all which we know from geometrical demon- stration to be true. Next, it is evident that when a quantity increasing has attained its maximum, it can have no further increment ; or when decreasing it has attained its minimum, it can have no further decrement; consequently in such cases the differential of the quantity is equal to nothing.* Hence a ready solution is afforded of problems of maxima and minima. Thus would we know the proportion which two sides of a rectangle must have to each other, in order that, their sum being given, they may form a rectangle con- taining the greatest space possible ; the differential of the rectangle must be put equal to nothing. Thus their sum being = a, the quantities are x and a #, and their rectangle is a x x 2 , its differential adx 2xdx t and this being put =0, we have adx 2xdx, ora:=-; therefore the figure must be a square. So would we know the point of the parabola (bx}' 1 = a(y c') where the curve comes nearest the line b, the ordinate y must be a minimum, and dy =0. Now y = <*^' + e , and Ay = ^=^ x * Sir I. Newton's own statement of the method is here followed. Me- thodus Fluxionum Opuscttla, torn. i. p. 86. edit. Geneva, 1 744. It has, however, been since universally admitted that the more accurate view is to regard the change of the sign as the criterion, both as to maximum and minimum values, and as to points of contrary flexure. 28 NEWTON'S PRINCIPIA. dx, which being put = gives us # = i; or, at the extremity of the line b> the curve approaches the nearest ; and that whatever be its parameter; for a has vanished from the equation. Again, we have seen that the ultimate ratio of the sum of all the rectangles M P x P Q, contained by the ordinates and the increments of the abscissa to the curve's area A P M is that of equality ; or, in other words, that the differential of a curvilinear area being the rectangle con- tained by the ordinate and the differential of the abscissa, or y dx, the integral of this, or the sum of all those small rectangles, is equal to the area. In this expression, then, let y be inserted in terms of x, and the integral gives the area. Thus in the parabola y= A/a x; therefore d x Va x is the differential of the area, and its integral, or which is 2 v 2 d v 2 y^ 2 z/ 2 the same thing, the integral of -* *-, is - x , or x a o a o a, 2 xy, that is, - xy, or two- thirds of the rectangle of the co- ordinates ; as we also know from conic sections. Next, we have seen that the ratio of the infinitely small rectilinear sides into which a curve line may be divided (each of those small lines being the hypothenuse of a right-angled triangle, the sides of which are the differentials N T, M N of the co-ordinates), to the infinitely small portions of the curve itself is that of equality ; therefore the differential of the curve is equal to the square root of the sum of the squares of the differentials of the ordinate and abscissa, and that differential is equal to /dx 2 + d ?/ 2 . Hence in the circle, an arc whose cosine is x and radius r is equal to the integral of _ And an arc whose v r 2 x* cosine is r x t is equal to the integral of JL x NEWTON'S PRINCIPIA. 29 Again, because solids may in like manner be con- sidered as composed of infinitely thin solids or plates, one placed upon the other, their differential is the area of the surface multiplied by the differential of the axis. Thus the base of any solid generated by the revolution of a surface rectilinear or curved must be a circle, and the proportion of the radius to the circumference being taken as r : c, y being the ordinate to the line bounding the vertical section, the surface will be -~ and the differential of the axis x being dx, the differential of the solid will be Cy ( X - t in which y in terms of x being inserted from the boundary line's equation, the integral gives the solid con- tent. Thus if the line which bounds is straight and parallel to the axis, or the solid is a cylinder, its content is the circle multiplied by the axis ; and if the line is drawn to a point in the axis, or the solid is a cone, then its content is one-third of the same product, or one-third of the cy- linder well-known properties of those two figures, proved by ordinary geometry. So in like manner we find the sphere to be two-thirds of the circumscribing cylinder, the celebrated discovery of Archimedes, of which he caused the diagram to be inscribed on his tomb. Lastly, it may in like manner be shown that the radius of the osculating circle at any point of any curve, that is, the circle touching it at such point, and having the same curvature with it at that point, is equal to y where dy being found in terms of x, the differential of ^ is to be taken, so that there will in the result in each case be no differential at all. 30 NEWTON'S PRINCIPIA. Thus in the parabola y* = 2ax, the radius of curvature is In all these operations, however, it must be observed, that as constant or invariable quantities have no dif- ferentials, so when we reverse the operation and find in- tegrals from given differential expressions, we never can tell whether a constant must not be added in order to com- plete that quantity, by taking whose differential the given expression was originally obtained. The determining of this constant quantity, and the finding whether there be any or not, depends upon the particular conditions of each problem. It is always added as a matter of course. Thus when we integrate d x + dy,we cannot tell whether this quantity arose from taking the differentials of x and y only, or from taking the differential of x + y + c ; and it must depend upon the nature of the question whether c is to be added to the integral or no ; and if to be added, how it shall be ascertained. Having explained this important method of investigation, by the help of which Newton was enabled to make his greatest mathematical discoveries, and by the principles of which he demonstrates them in the Principia, it only remains, before proceeding to the analysis of those dis- coveries, that we should remark the preference which he gives to the geometrical methods, improved and adapted to his purpose by the doctrine of Prime and Ultimate ratios. He uses this doctrine similar in principle to, and the foun- dation of, the noble and refined calculus which we have been considering; but he does not at all employ that calculus. The First book treats of the motion of bodies with- NEWTON'S PRINCIPIA. 31 out regard to the resistance of the medium that fills the space in which they move ; and it is principally devoted to the consideration of motions in orbits determined by centripetal forces, and to examining the attraction of bodies. The Second book treats of the resistance of fluids chiefly as affecting the motions of bodies that move in them. The Third book contains the application of the principles thus established to the motions, attractions, and figures of the heavenly bodies. I. The fundamental proposition, as it may justly be termed, of the whole system, is one which Newton's predecessors may be said to have nearly reached; which Kepler, had he been more inclined to trust demonstration than em- pirical observation, probably would have attained ; and which Galileo would certainly have discovered had he con- templated the facts discovered by Kepler, particularly his second law* : The proposition is this. If a body is driven by any single impulse or force of projection, and is also drawn continually by another force so as to revolve round a fixed centre, the radius vector, or line drawn from the body to that centre, describes areas which are in the same fixed plane, and are always proportional to the times of the body's motion ; and conversely, if any body which moves in any curve described in a plane so that the radius vector to a point either fixed or moving uniformly in a straight line, describes areas proportional to the times of the body's motion, that body is acted on by a centripetal force tending towards and drawing it to the point. To prove this, we have to consider that if a body moves equably on in a straight line, the areas or triangles * See the historical notice above respecting this second law, viz., that the planets describe areas proportional to the times by their radii vectores. 32 NEWTON'S PRINCIPIA. which are described by a line drawn from it to any point are proportional to the portions of the straight line through which the body moves, (that is to the time, since, moving equably, it moves through equal spaces in equal times,) because those triangles, having the same altitude, are to one another in the proportions of their bases. S being the point and AO the line of motion, SAB is to SB c as AB to B c. If then at B a force acts in the line S B, drawing the body towards S, it will move in the diagonal B C of a parallelogram of which the sides are B c and B V, the line through which the deflecting force would make it move if the motion caused by the other force ceased. Cc therefore is parallel to VB, and the triangle SBC is equal to the triangle SBc; consequently the motion through A B and B C, or the times, are as the two triangles SAB and SBC: and so it may be proved if the force acting towards S again deflects the body at C, making it move in the diagonal CD. If, now, instead of this deflecting force acting at intervals A, B, C, it acts at every instant, the intervals of time become less than any assignable time, and then the spaces AB, BC, CD will become also indefinitely small and numerous, and they will form a curve line ; and the straight lines drawn from any part of that curve to S will describe curvilinear areas, as the body moves in the curve ABCD, those areas being proportional to the times. So conversely, if the triangles NEWTON'S PRINCIPIA. 33 SBc and SBC are equal, they are between the same parallels, and cC is parallel to SB, and T>d to SO; consequently the force which deflects acts in the lines S B and SC, or towards the point S. It is equally manifest that the direction of the lines Be, C d, from which the centripetal force deflects the body, is that of tangents to the curve which the body describes, and that consequently the velocity of the body is in any given point inversely proportional to the perpendicular drawn from the centre to the tangent; the areas of the triangles whose bases are equal, being in the proportion of their altitude, that is, of those perpendiculars, and those areas being by the pro- position, proportional to the times. There are several other corollaries to this important proposition which deserve particular attention. B c and D e are tangents to the curve at B and D respectively; B C and D E the arcs described in a given time; C c and E e lines parallel to the radii vectores S B and S D respectively ; and C V, E d parallel to the tangents. The centripetal forces at B and D must be in the proportion of V B and d D (being the other sides of the parallelograms of forces) if the arcs are evanescent, so as to coincide with the diagonals of the parallelograms V c and d e. Hence the centripetal forces in B and D are as the versed sines D 34 NEWTON'S PRINCIPIA. of the evanescent arcs ; and the same holds true if instead of two arcs in the same curve, we take two arcs in dif- ferent but similar curves.* Frgm these propositions another follows plainly, and its consequences are most extensive and important. If two or more bodies move in circular orbits (or trajectories) with an equable motion, they are retained in those paths by forces tending towards the centres of the circles ; and those forces are in the direct proportion of the squares of the arcs described in a given time, and in the inverse pro- portion of the radii of the circles. First of all it is plain, by the fundamental proposition, that the forces tend to the centres S, s, because the sectors A S B and PBS being as the arcs A B, B P, and the sectors a s b, p b s, as the arcs a b, b p, which arcs being all as the tunes, the areas are proportional to those times of describing them, and therefore S and s are the centres of the deflecting forces. Then, drawing the tangents A C, a c, and completing the parallelograms D C, d c, the diago- nals of which coincide with the evanescent arcs A B, a b, we have the centripetal forces in A and a, as the versed sines A D, a d. But because A B P and a b p are right angles (by the property of the circle), the triangles A D B, A P B, and a d b, a p b, are respectively similar to one another. Wherefore A D : A B :: A B : A P and AD AB 2 , . ,.. , ab" 2 = ~t^n > and m "ke manner a d or, as the evan- AF ap escent arcs coincide with the chords, AD = arc -- and arf=arc . Now these are the properties of any arcs de- scribed in equal times ; and the diameters are in the pro- * If BC, DE, are bisected, the proportion is found with the halves of V B, D d ; and that is the same proportion with the whole versed sines. NEWTON'S PRINCIPIA. 35 portion of the radii ; therefore the centripetal forces are di- rectly as the squares of the arcs, and inversely as the radii. It is difficult to imagine a proposition more fruitful in consequences than this ; and therefore it has been de- monstrated with adequate fulness. In ihejirst place, the arcs described being as the velocities, if F,/ are the centripetal forces, and V, v the velocities, and R, r the radii, F :/ :: V 2 : u 2 ; and also :: r : R ; orF : V 2 u 2 f :: ^- ' . Now as in the circle V and R, v and r Jti r are both constant quantities, the centripetal force is itself constant, which retains a body by deflecting it towards the centre of the circle. Secondly. The times in which the whole circles are described (called the periodic times) are as the total cir- cumferences or peripheries ; T : t : : P : p : But the pe- ripheries are as the radii or :: R : r. Therefore T : P v t :: R : r; also V : v :: : therefore inversely as the radii, or T : t :: ? : and V 2 : v 2 :: ~ : -*. But the centripetal forces F :/::=-:; substituting for the 36 NEWTON'S PRINCIPIA. ratio of V 2 : v 2 , its equal the ratio of ~ 2 : , F :/:: =2 : ; or the centripetal forces are directly as the distances and inversely as the squares of the periodic times ; the forces being as the distances if the times are equal; and the times being equal if the forces are as the distances. It also follows that if the periodic times are as the distances, then F : f :: =- : - ; that is, R 2 ?- 2 :::-, or inversely as the distances. In like man- j\ r ner if the periodic times are in proportion to any power n, of the distance, or T : t :: R n : r n , we shall have T* : ? :: R- : r" and F : / :: ^ : ^ that is :: n _ l ' ^TJ-; and conversely if the centripetal force is in the inverse ratio of the (2n l) th power of the dis- tance, the periodic time is as the n th power of that dis- tance. Likewise, as the velocities of the bodies in their orbits or V : v ::? : T -, if we make T : t :: R* : r*, R r 11 thenV:t; ::- : -, or :: : ,. Thus, sup- o pose n is equal to ^ we nave f r tne velocities V : v :: - : =, or they are in the inverse subduplicate pro- portion of the distances; and for the centripetal forces we have F : / :: ^r, : ^ ''- g- 2 : '*'> or the attraction to the centre is inversely as the square of the distance. NEWTON'S PRINCIPIA. 37 Now if =|, T : t:: R^ : r*, or T 2 : P :: R a : r } ; in other words the squares of the periodic times are as the cubes of the distances from the centre, which is the law discovered by Kepler from observation actually to prevail in the case of the planets. And as he also showed from ob- servation that they describe equal areas in equal times by their radii vectores drawn to the sun, it follows from the fundamental proposition, first, that they are deflected from the tangents of their orbits by a power tending towards the sun; and then it folio ws,secondly, from the last deduction re- specting it, (namely, the proportion of F : f "pa ' ">) ^at this central force acts inversely as the squares of the distances, always supposing the bodies to move in cir- cular orbits, to which our demonstration has hitherto been confined.* The extension, however, of the same important pro- position to the motion of bodies in other curves is easily made, that is to the motion of bodies in different parts of the same curve or in curves which are similar. For in evanescent portions of the same curve, the osculating circle, or circle which has the same curvature at any point, coincides with the curve at that point; and if a line is drawn to the extremity of that circle's diameter, A M B and a m b may be considered as triangles ; and as they are right angled at M and m, A M 2 is equal to A P x A B and a m 2 to a pxa b; and where the curvature is the same as in corresponding points of similar curves, those squares are proportional to the lines A P, or a p ; or those versed * We shall afterwards show, from other considerations^ that this sesqui- plicate proportion only holds true on the supposition of the bodies all moving without exerting any action on each other, when we come to con- sider Laplace's theorems on elliptical motion. D 3 38 NEWTON'S PRINCIPIA. sines of the arcs A M and a m are proportional to the squares of the small arcs. Hence if the distances of two bodies from their respective centres of force be D, d, the deflecting force in any points A and a being as the versed sines, those forces are as A M 2 : a m* ; and from hence follows generally in all curves, that which has been demon- strated respecting motion in circular orbits. The planets then and their satellites being known by Kep- ler's laws to move in elliptical orbits, and to describe round the sun in one focus areas proportional to the times by their radii vectores drawn to that focus, and it being further found by those laws that the squares of their periodic times are as the cubes of the mean distances from the focus* they are by these propositions of Sir Isaac Newton which we have been considering, shown to be deflected from the tangent of their orbit, and retained in their paths, by a force acting inversely as the squares of the distances from the centre of motion. But another important corollary is also derived from the same proposition. If the projectile or tangential force in the direction A T ceases (next figure), the body, instead of moving in any arc A N, is drawn by the same centripetal force in the straight line A S. Let A n be the part of A S, through which the body falls by the force of gravity, in the same time that it would take to describe the arc A N. Let A M be the infinitely small NEWTON'S PRINCIPIA. 39 arc described in an instant; and A P its versed sine. It was before shown, in the corollaries to the first pro- position, that the centripetal force in A is as A P, and the body would move by that force through A P, in the same time in which it describes the arc A M. Now the force of gravity being one which operates like the centri- petal force at every instant, and uniformly accelerates the descending body, the spaces fallen through will be as the squares of the times. Therefore, if A n is the space through which the body falls in the same time that it would describe A N, A P is to A n as the square of the time taken to describe A M to the square of the time of describing A N, or as A M 2 : A N 2 , the motion being uniform in the circular arc. But A M, the nascent arc, is equal to its chord, and A M B being a right angled triangle as well as A P M, A B : A M :: A M : A P and AM 2 A P = r^n~- Substituting this in the former proportion, we have 4^r : A n :: A M 2 : A N 2 , or A ; A N 2 AM 2 : AM 2 , that is :: 1 : AB. Therefore A N 2 = A n x A B, or the arc described, is a mean propor- tional between the diameter of the orbit, and the space through which the body would fall by gravity alone, in the same time in which it describes the arc. D 4 40 NEWTON'S PRINCIPIA. Now let A M N B represent the orbit of the moon ; A N the arc described by her in a minute. Her whole periodic time is found to be 27 days 7 hours and 43 mi- nutes, or 39,343 minutes ; consequently A N : 2 A N B :: 1 : 39,343. But the mean distance of the moon from the earth is about 30 diameters of the earth, and the diameter of her orbit, 60 of those diameters ; and a great circle of the earth being about 131,630,572 feet, the circumference of the moon's orbit must be 60 times that length, or 7,897,834,320, which being divided by 39,343 (the num- ber of minutes in her periodic time), gives for the arc A N described in one minute 200,743, of which the square is 40,297,752,049, or AN 2 , which (by the propo- sition last demonstrated) being divided by the diameter AB gives A n. But the diameter being to the orbit as 1 : 3.14159 nearly, it is equal to about 2,513,960,866. Therefore A n = 16.02958, or 16 feet, and about the third of an inch. But the force which deflects the moon from the tangent of her orbit, has been shown to act inversely as the square of the distance ; therefore she would move 60 x 60 times the same space in a minute at the surface of the earth. But if she moved through so much in a minute, she would in a second move through so much less in the proportion of the squares of those two times, as has been before shown. Wherefore she would in a second move through a space equal to 16^ nearly (16.02958). But it is found by experiments frequently made, and among others by that of the pendulum *, that a * It is found that a pendulum, vibrating seconds, is about the length of 3 feet 3i inches in this latitude ; and the space through which a body falls in a second is to half this length as the square of the circumference of a circle to that of the diameter, or as 9.8695 : 1, and that is the proportion of the half of 3 feet 3 inches to somewhat more than 16 feet. NEWTON'S PRINCIPIA. 41 body falls about this space in one second upon the surface of the earth. Therefore the force which deflects the moon from the tangent of her orbit, is of the same amount, and acts in the same direction, and follows the same proportions to the time that gravity does. But if the moon is drawn by any other force, she must also be drawn by gravity ; and as that other force makes her move towards the earth 16 feet ^ inch, and gravity would make her move as much, her motion would therefore be 32 feet f inch in a second at the earth's surface, or as much in a minute in her orbit ; and her velocity in her orbit would therefore be double of what it is, or the lunar month would be less than 13 days and 16 hours. It is, therefore, impossible that she can be drawn by any other force, except her gravity, towards the earth.* Such is the important conclusion to which we are led from this proposition, that the centripetal forces are as the squares of the arcs described directly, and as the distances inversely. The great discovery of the law of the universe, therefore, is unfolded in the very beginning of the Principia. But the rest of the work is occupied with tracing the various consequences of that law, and first of all in treating generally of the laws of curvilinear motion. The demonstration of the moon's deflection has been now anticipated and expounded from the Third Book, where it is treated with even more than the author's accustomed conciseness. But there seemed good ground for this anticipation, inasmuch as the Scholium to the Fourth Proposition refers in general terms to the con- * The proposition may be demonstrated by means of the Prop. XXXVI. of Book I., as well as by means of the proposition of which we have now been tracing the consequences (Prop. IV). But in truth the latter theorem gives a construction of the former problem (Prop. XXX VI.), and from it may be deduced both that and Prop. XXXV. 42 NEWTON'S PRINCIPIA. nexion between its corollaries, and the Theory of Gravi- tation. The versed sine of the half of any evanescent arc (or sagitta of the arc) of a curve in which a body revolves, was proved to be as the centripetal force, and as the square of the times ; or as F x T 2 . Therefore the force F is directly as the versed sine, and inversely as the square of the time. From this it follows that the central force may be measured in several ways. The arc being Q C, we are to measure the central force in its middle point P. Then the areas being as the times ; twice the triangle S P Q, or Q L x S P is as T in the last expression ; and, therefore, Q R being parallel to L P, the central force at P is as So if S Y be the perpendicular upon the tangent P Y, because P R and the arc P Q, evanescent, coincide, twice the triangle S P Q is equal to S Y x Q P ; and the NEWTON'S PRINCIPIA. 43 O "R central force in P is as ^^ ~ p 2 . Lastly, if the revo- lution be in a circle, or in a curve having at P the same curvature with a circle whose chord passes from that point through S to V, then the measure of the central force will be -^Ya py- By finding the value of those solids in any given curve, we can determine the centripetal force in terms of the radius vector S P ; that is, we can find the proportion which the force must bear to the distance, in order to retain the body in the given orbit or trajectory ; and conversely, the force being given, we can determine the trajectory's form. This proposition, then, with its corollaries, is the foun- dation of all the doctrine of centripetal forces, whether direct or inverse ; that is, whether we regard the method of finding, from the given orbit, the force and its proportion to the distance, or the method of finding the orbit from the given force. We must, therefore, state it more in detail, and in the analytical manner, Sir Isaac Newton having delivered it synthetically, geometrically, and with the utmost brevity. It may be reduced to five kinds of formulEe. 1. If the central force in two similar orbits be called F and f, the times T and t, the versed sines of half the arcs S and s, then F : / :: : i an d generally F is as S fi- 2. But draw S P to any given point of the orbit in the middle of an infinitely small arc Q C. Let T P touch the curve in P, draw the perpendicular S Y from the centre of forces S to P T produced, draw S Q infi- nitely near S P, and Q R parallel to S P, Q o and R o 44 NEWTON'S PRINCIPIA. parallel to the co-ordinates S M, M P. Then P being the middle of the arc, twice the triangle S P Q is propor- tional to the time in which C Q is described. Therefore QP x SYorQL x PS is proportional to the C O time; and QR is the versed sine of -~, therefore O f\ T> F as r -jr 2 becomes F as^ j-Qg a p 2 '> an< ^ if S M = x, M P = y, and because the similar triangles Q R o and S M P give Q R = QQ s*VI F> and because A M bein S the first differential of S M, o Q is its second differential (negatively), therefore Q R = ~- xy (taken x with reference to d t constant), and F is as But L Q 2 = Q p2 - Lp2 and n x x L Q 2 x L P is the differential of S P or Vx> + ^ Therefore (xdy- y dx) z y \y L Q 2 = v - ^5 *4 ' = , y . , and F is as 1 * 2 - 2 ' But as the differential of the time (L Q x PS) may be made constant, Q R will represent the centripetal force ; and that force itself will therefore be as , * taken with reference to d t constant. * Of these expressions, although I have sometimes found this, which was first given by Herrman, serviceable, I generally prefer the two, which are in truth one, given under the next heads. But the expression first given ^ is without integration an useful one. NEWTON'S PRINCIPIA. 45 3. The rectangle S Y x Q P being equal to Q L x S P , we have Fas QR QR _ QR OR O P 2 4. Because F = g y 2 x Q P 2 and is e(1Ual t0 chord P V of the circle, which has the same curvature with Q P O in P, and whose centre is K (and because Q P 2 = Q R x P V by the nature of the circle and the equality of the evanescent arc Q P with its sine, and thus P V = -||- 2 , - therefore |^ = p^ ), Fisas s - -- * n like manner if the velocity, which is inversely as S Y, be called t?, F is as py. Now the chord of the osculating circle is to twice the perpendicular S Y as the differential of S P to the differential of the perpendicular ; and calling S P the radius vector r, and S Y=p, we have PV = , and F is as Tj? and also F is as ^. In these formulas, substituting for p and r their values in terms of x and y, we obtain a mean of estimating the force as proportioned to r, which is ^ / >+y s . 5. The last article affords, perhaps, the most obvious methods of arriving at central forces, both directly and inversely. Although the quantities become involved and embarrassing in the above general expressions for all curves, yet in any given curve the substitutions can more easily be made. A chief recommendation of these expres- 46 NEWTON'S PRINCIPIA. sions is, that they involve no second differentials, nor any but the first powers of any differentials. But it may be proper to add other formulas which have been given, and one of which, at least, is more convenient than any of the rest. One expression for the centrifugal force (and one some- d s z times erroneously given for the centripetal) * is p~, s be- ing the length of the curve and B the radius of curvature. This gives the ready means of working if that radius is known. But its general expression involves second diffe- ds* rentials, the usual formula for it being , * consequently we must first find -^- = X (a function of ;r), and then there are only first differentials. Another for this radius of curvature is ds* and this is USed b ^ La P lace 5 and ano- ther is -j , which, with other valuable formulas, is to be obtained from Maclaurin's Fluxions. But the for- mula generally ascribed to John Bernouilli (Mem. Acad. des Sciences, 1710), is, perhaps, the most elegant of any, F = r - 3 - ^-; and this results from substituting 2 R for its value 5 - , in the equation to F, deduced above from Newton's formula, namely, F = ? . 2p 3 dr * This error appears to have arisen from taking the case where the radius of curvature and radius vector coincide, that is, the case of the circle, in which the centrifugal and centripetal forces are the same. _ See Mrs. SomervUle's truly admirable work on the Mec. Cel., where the error manifestly arises from this circumstance. NEWTON'S PRINCIPIA. 47 But the proposition is so important, that it may be well to prove it, and to show that it is almost in terms involved in the third corollary to Prop. VI. Book I. of the Principia, By that corollary F = T~ (C being the osculating circle's chord which passes through the centre of forces). But drawing S Y, the perpendicular to the tangent, and P C F through the centre of the circle, and joining V F, which is, therefore, parallel to Y P, we have VP:PF::SY:SPorC:2K:^:r 2 T? n and C = ^, which substituted for C in the above equation, gives F=^-^. It is remarkable that the circumstance of this formula being thus involved in that of Sir Isaac Newton seems never to have been observed by Keill, who, in the Philosophical Transactions, xxvi. 74, gives a demonstration of it much more roundabout, and as of a theorem which Demoivre had communicated to him, adding, that Demoivre also informed him of Sir Isaac Newton having invented a similar method before. In fact, he had, above 20 years before, given it in substance, though not in express terms, in the Sixth Proposition, the addition of two lines to which would at once have led to this formula. But, again, when John 48 NEWTON'S PRINCIPIA. Bernouilli, two years afterwards, wrote his letter to Herr- man (Mem. Acad. des Sciences, 1710), he gives it as his own discovery, and as such it has generally been treated, with what reason we have just seen. He is at much pains to state, p. 529., that he had sent it in a letter to Demoivre in February, 1706 ; but the Principia had been published nineteen years before. Herrman, in his Phoro- nomia, erroneously considers the expression as discovered by Demoivre, Grandi, and Bernouilli. (Lib. I. Prop. XXII.) In all these cases p is to be found first, and the expres- sion for it (because, pp. 42, 43., TP:PM::TS:SY i m r< ydx xdy -\ -r^n\ V / \ andTS=- - 7 S and P T = -f- Vdy* + dx*} is p ay ay / x = */x*+y\ Then the radius of curvature R = . a ,y (X being -r- in terms of x, and having no differential in it when the substitution for dy is made). Therefore, the expression for the centripetal force becomes , in which, when y and d y are put in terms of x, as both numerator and denominator will be multiplied by d z 3 , there will be no differential, and the force may be found in terms of the radical that is, of r, though often complicated with x also. It is generally advisable, having the equation of the curve, to find p, r, and R, first by some of the above formulas, and then sub- stitute those values, or d p and d r, in either of the expressions for F, NEWTON'S PRINCIPIA. 49 To take an example in the parabola, where S being the focus, and O S = a, f = 4 a x, and T M = 2 x, and p = Y S = V (a + x) a ;r=SP = a + #, and R = r -jLr = 2 (a + x) A J ( LJL; we have therefore Fas F75, a + x x 2 (a + x) O S (th 2 a (a + *? 2.08.. SP" meter) is constant, inversely as the square of the dis- tance : And the other formula F as 3 ^ gives the same result for the law of force, or 4SP 2 ' Again, in the ellipse, if a be half the transverse axis, and b half the conjugate, and r the radius vector, we have / r = b A / s-- - , V 2a-r' and d p = ~z b d 5 therefore dp the formula ^^ becomes a b d r jr. a> ft* Vr x r^ * ar v the force is inversely as the square of the distance. * This result coincides with the synthetical solution of Sir Isaac Newton in Prop. XIII. 50 NEWTON'S PRINCIPIA. Lastly, as the equations are the same for the hyper- bola, with only the difference of the signs, the value of the force is also inversely as r 2 , or the square of the distance. In the circle a = the radius = r = p ; hence -^P becomes ~, which, being constant, the force is every- where the same. But if the centre of forces is not that of the circle, but a point in the circumference, the force is asi. Respecting centrifugal forces it may be enough to add, that if v is the velocity and r the radius, the centrifugal force f, in a circle, is as . Also if R be v 1 the radius of curvature, f for any curve is = ^ When a body moves in a circle by a centripetal force directed to the centre, the centrifugal force is equal and opposite to the centripetal. Also the velocity in uniform motion, like that in a circle, being as -, the space divided by the time, and the ere being as the s 2 r radius r, f is as or as -3. If two bodies moving in different circles have the same centrifugal force, then the times are as */r. It is to the justly celebrated Huygens that we owe the first investigation of centrifugal forces. The above propositions, except the second, are abridged from his treatise.* .The rest of the investigation of centripetal forces is an expansion of the formulas above given, and their appli- cation to various cases, but chiefly to the conic sections. It may be divided into four branches. First, the rules * Horologium Oscillatorram, ed. 1673, p. 159, App. NEWTON'S PRINCIPIA. 51 are given for determining the central force required to make the body move in a given orbit of one of the four conic sections. Secondly, the inquiry becomes material how curves of a given kind, namely, the conic sections, may severally be found by merely ascertaining certain points in them, or certain lines which they touch, because this enables us to ascertain, among other things, the whole of a planet's orbit, from ascertaining certain points by actual observation. This branch of the subject is purely mathematical, consisting of the rules for drawing those curves through given points, or between, or touching given straight lines ; and it is subdivided into two heads according as one or neither focus is given. The third object is to ascertain the motion, place, and times of bodies moving in given trajectories generally ; and, among others, also of bodies descending, or retarded in ascending, by gravity. The fourth branch treats of the converse inquiry into the figures of the trajectories, and the places, times, and motion, when the nature of the centripetal force is known. It is thus manifest that the great importance of motion in the Conic Sections made Sir Isaac Newton consider those curves in particular, before discussing the general subject of trajectories. In exemplifying the use of the formulas we have E 2 52 NEWTON'S PRINCIPIA. shown the proportion of the force to the distance in the conic sections generally, their foci being the centres of forces. Let us now see more in detail what the pro- portion is for the circle. Let S Jbe the centre of forces and K of the circle, P T a tangent, S Y a perpendicu- lar to it, KM and M P co-ordinates, S K = 6, KO = a, P M = y, and M K = x. Then, by similar triangles, T K P C rr\ T7-p and TSY, we have SY= xK ' or ( because the sub-tangent M T = , and a 2 = # 2 -fy 2 ) - - or / >2a? + 2bx\ . I ^ J ; also S P = v a 2 + 2 b x + b 2 , and because by the property of the circle OS x SB or (a + b) (a )= 2 5 2 = P S x S V; therefore S V = Now by the formula already stated as Bernouilli's, but really Sir Isaac Newton's, the centripetal force in SP P is as -y3 -D* R being the radius of curvature, and in the circle that is constant being = a, the semi- diameter ; therefore the force is as a* +2bx + b z ^ a(2a* + 2bx) 3 8 a 3 B O 2 x S P B O 2 x S P 3 ,' xSP 2 ' BO 2 SP 3 * =rt = P V. Therefore the central NEWTON'S PRINCIPIA. 53 force is as BO 2 P V 3 x S P 2 ' or (because O B* is constant) the central force is inversely as the square of the distance and the cube of the chord jointly. Of consequence, where S is in the centre of the circle and b = o, the force is con- stant, SP becoming the radius and P V the diameter ; and if S is in the circumference of the circle as at B, or a = b, then the chord and radius vector coinciding, the force is inversely as the fifth power of the distance, and is also inversely as the fifth power of the cosine of the angle PSO. By a similar process it is shown that in an ellipse the force directed to the centre is as the distance. Indeed, a property of the ellipse renders this proof very easy. For if S Y is the perpendicular to -the tangent T P, and N P (the normal) parallel to SY, and S A the semi-conjugate axis ; S A is a mean proportional between S Y and PN, A S 2 and therefore S Y = ^prvr j also the radius of curva- ture of the ellipse is (like that of all conic sections) 4P N 3 equal to p2 "^ being the parameter. Therefore we have to substitute these values for S Y and the radius of curvature, R, in the expression for the central NEWTON'S PRINCIPIA. and we have 2 A A P6 x S P ; so that, neglecting the constant the centripetal force is as the distance directly From hence it follows, conversely, that if the centripetal force is as the distance, the orbit is elliptical or circular, for by reversing the steps of the last demonstration we arrive at an equation to the ellipse ; or, in case of the two axes being equal, to the circle. It also follows that if bodies revolve in circular or elliptical orbits round the same centre, the centre of the figures being the centre of forces, and the force being as the distance, the periodic time of all the bodies will be the same, and the spaces through which they move, however differing in length from each other, will all be described in the same time. This proposition, which sometimes has appeared paradoxical to those who did not sufficiently reflect on the subject, is quite evident from considering that the force and velocity being increased in proportion to the distance, and the lengths of similar curvilinear and concentric figures being in some proportion, and that always the same, to the radii, the lengths are to each other as those radii, and conse- quently the velocity of the whole movement is increased in the same proportion with the space moved through. Hence the times taken for performing the whole motion must be the same. Thus, if V and v are the velocities, R, and r the radii, S and s the lines described in the times T and t, by two such bodies round a common centre, V : v :: R : r, and S : s :: R : r ; and because V = -|r and v = j, ~ '. j :. R : r, and S : s :: TR : NEWTON'S PEINCIPIA. 55 t r ; or B, : r :: T K : t r ; and therefore T = t. Hence if gravity were the same towards the sun that it is between the surface and centre of each planet, or if the sun were moved but a very little to one side, so as to be in the centre of the ellipse, the whole planets would revolve round him in the same time, and Saturn and Uranus would, like Mercury, complete their vast courses in about three of our lunar months 'instead of 30 and 80 years, a velocity in the case of Uranus equal to 75,000 miles in a second, or nearly one-third that of light. It also follows from this proposition that, if such a law of attraction prevailed, all bodies descending in a straight line to the centre would reach it in the same time from whatever distance they fell, because the elliptic orbit being indefinitely stretched out in length and narrowed till it became a straight line, bodies would move or vibrate in equal times through that line. This is the law of gravity at all points within the earth's surface, and Sir I. Newton has adapted one of his investigations to it, when treating of the pendulum. Another consequence of this proposition is, that if the centre of the ellipse be supposed to be removed to an infinite distance, and the figure to become a parabola, the centripetal force being directed to a point infinitely remote, becomes constant and equable ; a proposition dis- covered first by Galileo. Sir Isaac Newton having treated of the centripetal force in conic sections, where the centre of forces is the centre of the figure (and generally whatever be the centre in the case of the circle), proceeds to treat of that force where it is directed towards the focus of one or other of those curves, and not to the centre. It is easy to demonstrate a compendious theorem, that which forms the subject of his three first propositions, in which he determines the 56 NEWTON'S PRINCIPIA. law of the force for the three curves (parabola, hyper- bola, and ellipse) severally. For this purpose a simple reference to the formulas already stated will suffice; indeed our illustration of those formulas has already anti- cipated this. If O P A be a conic section whose parameter is D, S Y the perpendicular to the tangent T P, PR the radius of curvature at P; then SY : SP :: ^D : P N (the normal), and S Y = 1 P N 3 ^; also PR = Substitute these values of S Y and P R (p u and R) in the expression formerly given for the central force SP and we have D 3 . S P 3 4 P ;, which is (D being invariable) as the inverse square of the distance. Therefore any body moving in any of the conic sections by a force directed to the focus, is attracted oy a ceniripetal force inversely as the square of the distance from that focus. This demon- NEWTON'S PRINCIPIA. 57 stratiori, therefore, is quite general in its application to all the conic sections. It follows that if a body is impelled in a straight line with any velocity whatever, from an instantaneous force, and is at the same time constantly acted upon by a cen- tripetal force which is inversely as the square of the distance from the centre, the path which the body describes will be one or other of the conic sections. For if we take the expression ~r 2 and work backwards, multiplying the numerator and denominator both by S P, and then mul- 8 D 2 P N 3 tiplying the denominator by g -pa ' p va* we obtain the expressions for the value of S Y, the perpendicular, and for B, the radius of curvature. But no curves can have the same value of S Y and R, except the conic sections ; because there are no other curves of the second order, and those values give quadratic equations between the co-ordinates. By pursuing another course of the same kind alge- braically, we obtain an equation to the conic sections generally, according as certain constants in it bear one or other proportion to one another. The perpendicular S Y and the radius of curvature are given in terms of the normal ; and either one or the other will give the equation. d \d x which gives D 2 d x 3 = 4 r/ 3 x ((P y d x d? x d y) an equation to the co-ordinates. Now whether this be resol- vable or not, it proves that only one description of curves, of one order, can be such as to have the property in question. The former operation of going back from the expression of the central force, proves that the conic sec- 58 NEWTON'S PRINCIPIA. tions answer this condition. Therefore no other curves can be the trajectories of bodies moving by a centripetal force inversely as the square of the distance.* It may be remarked that J. Bernoulli! objects (Mem. Acad. des Sciences, 1710) to Sir Isaac Newton that he had assumed the truth of this important proposition without any demonstration. But this is not correct. He certainly gives a very concise and compendious one ; but he states distinctly that the focus and point of contact being given, and the tangent given in position, a conic section may be described which shall at that point of contact have a given curvature; that the curvature is given from the velocity and central force being given; and that two orbits touching each other with the same centripetal force and velocity cannot be described. This is in substance what we have expounded in the above demonstration. But it must also be observed, as Laplace has remarked, that Newton has in a subsequent problem shown how to find the curve in which a body must move with a given velocity, initial direction, and position ; and since, when the centripetal force is inversely as the square of the distance, the curve is shown to be one or other of the conic sections, he has thus demonstrated the proposition in question ; so that if he had not done so in the corollary to one problem, he has in the solution of another, f J. Bernouilli objects also to a very concise and elegant * The equation may be resolved and integrated ; there results, in the first instance, the equation d x= J V ^ = and therefore the integral is this quadratic, c* a~-2 cf-2 cC JT + C* + 1^=0. Another demonstration is given in the Appendix, No. 2. t Systeme du Monde, liv. v. chap. 5. It is to he observed, that the Seventeenth Prop. Book L, is exactly the same in the first as in the subse- quent editions, except the immaterial addition of a few lines to the demon- stration. Consequently, Bernouilli must have been aware of it when he wrote in 1710. NEWTON'S PRINCIPIA. 59 solution of the inverse problem given by Herrman in the same volume of the Memoires, and which had been com- municated to him before it was presented to the Academy. This solution proceeds upon his general expression for the centripetal force, -- V x *-' r y 2 ; and the objection made is that he works the problem (as he does in a few lines) by multiplications and divisions which show that he was previously aware of the solution in the case of the conic sections. But this is no objection to a solution which being of a problem already known, can only be regarded as a demonstration that the former solution was exact. It is an objection which, if valid, applies certainly to the de- monstration which we have just given of the proposition ; but so it does to all the demonstrations of the ancient geometrical analysis. It is a more substantial objection that Herrman omitted a constant in his integration ; but by adding it, Bernouilli shows that the equation which Herrman found, when thus corrected, expresses the conic sections generally. This truth, therefore, of the necessary connexion be- tween motion in a conic section and a centripetal force inversely as the square of the distance from the focus, is fully established by rigorous demonstration of various kinds. If we now compare the motion of different bodies in concentric orbits of the same conic sections, we shall find that the areas which, in. a given time, their radii vectores describe round the same focus, are to one another in the subduplicate ratio of the parameters of those curves. From this it follows, that in the ellipse whose conjugate axis is a mean proportional between its transverse axis and parameter, the whole time taken to revolve (or the periodic time) being in the proportion of the area (that is in the proportion of the rectangle of the axes) directly, and in 60 NEWTON'S PRINCIPIA. the subduplicate ratio of the parameter inversely, is in the sesquiplicate ratio of the transverse axis, and equal to the periodic time in a circle whose diameter is that axis. It is also easy to show from the formula already given re- specting the perpendicular to the tangent, that the velocities of bodies moving in similar conic sections round the same focus, are in the compound ratio of the perpendiculars in- versely and the square roots of the parameters * directly. Hence in the parabola a very simple expression obtains for the velocity. For the square of the perpendicular being as the distance from the focus by the nature of the curve (the former being a 2 + a x, and the latter a + x), the velocity is inversely as the square root of that distance. In the ellipse and hyperbola where the square of the per- pendicular varies differently in proportion to the distance, the law of the velocity varies differently also. The square of the perpendicular in the ellipse (A being the transverse axis and B the conjugate, and r the radius vector) is B 2 x r B 2 x r -v -- ; in the hyperbola, -r - , or those squares of JuL 7* jCX ~|~ 7* the perpendicular vary as T and , in those A r curves respectively, B 2 being constant. Hence the ve- locities of bodies moving in the former curve vary in a greater ratio than that of the inverse subduplicate of the distance, or ^_, and in a smaller ratio in the latter curve, vV while in the parabola = is their exact measure. V r To these useful propositions, Demoivre added a theorem of great beauty and simplicity respecting motion in the * By parameter is always to be understood, unless otherwise mentioned, the principal parameter, or the parameter to the principal diameter. NEWTON'S PRINCIPIA. 61 ellipse. The velocity in any point P is to the velocity in T, the point where the conjugate axis cuts the curve, as the square root of the line joining the former point P and the more distant focus, is to the square root of the line joining P and the nearer focus. It follows from these propositions that in the ellipse, the conjugate axis being a mean proportional between the transverse and the parameter, and the periodic time being as the area, that is as the rectangle of the axes directly, and the square root of the parameter inversely, t being that time, a and b the axes, and p the parameter, t = =, and Vp b* = ap\ therefore ab = a*Sap *So?**Sp\ and t = Va 3 , and 2 2 = a 3 ; or the squares of the periodic times are as the cubes of the mean distances. So that all Kepler's three laws have now been demonstrated, a priori, as mathematical truths ; first, the areas proportional to the times if the force is centripetal second, the elliptical orbit, and third, the sesquiplicate ratio of the times and dis- tances, if the force is inversely as the squares of the dis- tances, or in other words if the force is gravity. Again, if we have the velocity in a given point, the law of the centripetal force, the absolute quantity of that force in the point, and the direction of the projectile or centrifugal force, we can find the orbit. The velocity in the conic section being to that in a circle at the given distance D as m to n, and the perpendicular to the tangent being p, the lesser axis will be , and the v 2ft 2 m 2 greater axis ^ ^, the signs being reversed in the denominator of each quantity for the case of the hyperbola. Hence the very important conclusion that the length of 62 NEWTON'S PRINCIPIA. the greater axis does not depend at all upon the direction of the tangential or projectile force, but only upon its quantity, the direction influencing the length of the lesser axis alone. Lastly, it may be observed, that as these latter pro- positions give a measure of the velocity in terms of the radius vector and perpendicular to the tangent for each of the conic sections, we are enabled by knowing that ve- Jocity in any given case where the centripetal force is inversely as the square of the distance, and the absolute amount of that force is given, as well as the direction of the projectile force and the point of the projection, to determine the parameters and foci of the curve, and also which of the conic sections is the one described with that force. For it will be a parabola, an hyperbola, or an ellipse, according as the expression obtained for/) 2 (the square of the perpendicular to the tangent) is as the radius vector, or in a greater proportion, or in a less proportion. This is the problem above referred to, which John Bernouilli had en- tirely overlooked, when he charged Sir Isaac Newton with having left unproved the important theorem respecting motion in a conic section, which is clearly involved in its solution. Before leaving this proposition, it is right to observe that the two last of its corollaries give one of those sa- gacious anticipations of future discovery which it is in vain to look for anywhere but in the writings of this great man.* He says, that by pursuing the methods indicated in the investigation, we may determine the variations im- pressed upon curvilinear motion by the action of disturbing, or, what he terms, foreign forces; for the changes intro- * See a singular anticipation respecting dynamics, by Lord Bacon, in De Aug. Lib. ILL, under the head Translation of Experiments. It was pointed out to me by my learned friend B. Montague. NEWTON'S PRINCIPIA. 63 duced by these in some places, he says, may be found, and those in the intermediate places supplied, by the analogy of the series. This was reserved for Lagrange and La- place, whose immortal labours have reduced the theory of disturbed motion to almost as great certainty as that of untroubled motion round a point by virtue of forces di- rected thither.* We have thus seen how important in determining all the questions, both direct and inverse, relating to the centri- petal force, are the perpendicular to the tangent and the radius of curvature. Indeed it must evidently be so, when we consider, first, that the curvature of any orbit depends upon the action of the central force, and that the circle coinciding with the curve at each point, beside being of well-known properties, is the curve in which at all its points the central force must be the same; and, secondly, that the perpendicular to the tangent forms one side of a triangle similar to the triangle of which the differential of the radius vector is a side; the other side of the former triangle being the radius vector, the proportion of which to the force it- self is the material point in all such inquiries. The difficulty of solving all these problems arises from the difficulty of obtaining simple expressions for those two lines, the per- pendicular p and the radius of curvature R. The radius vector r being always Vx' 2 +y 2 interposes little em- barrassment; but the other two lines can seldom be con- cisely and simply expressed. In some cases the value of F, the force, by d r and dp may be more convenient than in others; because p may involve the investigation in less difficulty than R; besides that/? 3 enters into the expression which has no differentials. But in the greater number of * Laplace (Mec. Cel. lib. XT. ch. i.) refers to this remarkable passage as the germ of Lagrange's investigations in the Berlin Memoires for 1786. 64 NEWTON'S PRINCIPIA. instances, especially where the curve is given, the for-^ mula -=T will be found most easily dealt with. ii. The next branch of the inquiry relates to the de- scribing the conic sections severally, where certain points are given through which they are to pass, or certain lines which they are to touch. The subject is handled in two sections, (the fourth and fifth,) the first of which treats the case where one of the foci is given; the second the case where neither focus is given. This whole subject is purely geometrical; and exhibits a fertility of resources in treating these difficult problems, as well as an elegance in the manner of their solution, which has few parallels in the history of ancient or modern geometry. This portion of the Prin- cipia, however, is incapable of abridgment; and there is no advantage whatever in resolving the problems analytically, but rather the contrary; for with the exception of one of the lemmas, in demonstrating which Sir Isaac Newton himself has recourse to algebraical reasoning in order to shorten the proofs, the geometrical process is in almost every instance extremely concise, in all cases much more beautiful, and less encumbered than the algebraical. The superiority of the former to the latter method of in- vestigation in such solutions is apparent on trying al- gebraically some simple case, as that of describing a circle through three points, or through two points and touching a line given in position ; no little embarrassment results from the number and entanglement of the quantities in the solu- tion. Even so great a master of analysis as Sir Isaac Newton, in solving the problem of describing a circle through two points, and touching a given line, could find no better ex- - + e* a* - d* a pression than x = - 2 _ 2 - > although geometrically the construction is easy by drawing a circle NEWTON'S PRINCIPIA. 65 on one segment of the line joining the given points, and another on the given line.* These are comparatively simple problems ; in the more difficult cases of the conic sections this embarrassment is often inextricable, f To illustrate the application of these important pro- blems, let us suppose that by observation we obtain three points in the orbit of any planet, and would ascertain from those points the position of the greater axis, and the focus in which the sun is placed, the eccentricity of the orbit or distance of the focus from the centre of the ellipse, and the aphelion, or greatest distance to which in its course the planet ever is removed from the sun ; this is easily done by means of Prop. XVIII. (Book I.), for that enables us to find the elliptical and hyperbolical trajectories, which pass through given points, when one focus and the transverse axis are given ; and thus to find the other focus, and the centre of the curve, and the distance from the given focus to the further extremity of the axis, which is the aphelion. In like manner the problem which Sir Isaac Newton calls by far the most difficult of any, and says that he had tried to solve in various ways:}:, that of finding the tra- jectory of a comet from three observations, supposing it to move in a parabolic orbit, is reduced by an elaborate and difficult process of reasoning to describing a parabola through two given points, which are found in its own orbit from the observations. Now Prop. XIX. of Book I. gives an easy solution of this problem. It is only to * The above algebraical solution is that of Prop. 43. of the Arith. Univ., vrhere Props, 59, 60, and 61. are also solutions of the three first problems of Sect. V. of the Principia, B. I. f Maria Agnesi's Instituzioni Analitiche abounds in elegant alge- braical investigations of geometrical problems, but affords no grounds for modifying the above remark. J Problema hocce longe difficillimum multimode aggressus (Lib. III. Prop. 41.)- Several other propositions are given in the first book for the purpose 66 NEWTON'S PRINCIPIA. describe from each of the given points a circle, with the dis- tance of that point from the given focus as a radius, and the straight line touching these two circles will be the directrix of the parabola, and the perpendicular to it from the focus, its axis ; the principal vertex being the middle point of that perpendicular. The coincidence of the very eccentric elliptical orbits of the comets with the parabola, makes this parabolic hypothesis answer for determining their places and times in the general case. The correction of the orbit thus found is reduced to finding the orbit of an ellipse which shall pass through three given points, and this is done by the 21st propo- sition of Book L, or rather by the 16th lemma, to which it is a corollary, for inflecting three straight lines from three given points, the differences, if any, between the lines, being given. Sir Isaac Newton tried the accuracy of the methods thus found upon several comets, and particularly on the celebrated one of 1680, called Halley's comet, from the great labour which that mathematician, in aid of his illus- trious friend and master, bestowed upon the calculation of its orbit. The following is a short statement of the general result of a comparison between the places computed from the theory, and the places found by actual observation, in the cases tried. of facilitating the solution of this difficult problem by another method; but the author informs us that he subsequently fell upon the method which he has given in the third book, and which he prefers for its greater sim- plicity. It is, however, very remarkable that he overlooked the important circumstance of there being a porism connected with his solution, or a case in which the problem becomes indeterminate and has an infinite number of solutions ; and what is still more singular that the case of the comet is that of the porism, so that the solution is wholly inapplicable. This was first discovered by F. Boscovich in 1749; it being found that the solution had thrown the comet upon the wrong bide of the sun. (See Life of Simson, Appx.) NEWTON'S PRINCIPIA. 67 First, as regards the comet of 1680, or Halley 's comet. In comparing four observations with the geometrical com- putation, Sir Isaac Newton found an error of 6' 3" on an average in the latitude, and about 1' in the longitude. But Halley, having afterwards made the computations with greater accuracy by arithmetical operations, found the average error, on sixteen observations, in the latitude only about 52", and in the longitude 1' 28". The average error found on a comparison of the theory with twenty- one observations made abroad, was found by Halley only to be 50" in the latitude, and 51" in the longitude.* Secondly, as regards other comets. In the computations of the comet 1665, the error was, on an average of eighteen observations, 8" in the latitude, and in the longitude I' 25". In the latitude the errors by excess nearly balance those by defect, the one being to the other as 40 to 49. In the longitude, supposing the observation of December 7 accurately stated (which, from the error amounting to 7' 33", seems very doubtful), the errors by excess are sixteen times more considerable than those by defect. In the comets of 1882 and 1683, on comparing the observations of Flamstead with the theory, the error was I' 31" in latitude, and 45" in longitude, for eleven observations of the former comet, and for seventeen of the latter comet, 1' 10" in latitude, and 1' 29" in lon- gitude. But the comet of 1723 came nearer its computed place ; the average error of latitude on fifteen observations of Bradley, compared with the same number by Halley himself, and Pound (his uncle), was only 21"^ in the la- titude, and somewhat under 25" in the longitude. It is to be remarked that this is apparently the case in which the * This omits the observation made 26th December, as there is mani- festly an error in the figures of that observation. F 2 68 NEWTON'S PRINCIPIA. observations were the most accurate, three eminent obser- vers checking each other, and no one observation differing from the computation much more than by the average of the rest, while great differences occur in all the other cases, and give rise to a suspicion of error. For in the comet of 1683, there was one day (Aug. 15) in which the latitude differed between three and four times, and the longitude three times more than the average; and in the observations of the comet of 1665 there are several errors in longitude of twice, and one error of no less than five times, above the average. These particular observations, and not the theory, then, were probably at fault in those instances ; but they affect the general average materially. The intimate connection between the purely geometrical parts of the Principia, the Fifth and Sixth Sections of the First Book, and the most sublime inquiries into the motions of the heavenly bodies, those motions, too, which are the most rapid, and performed in spaces the most prodigious, may suffice to show the student how well worthy these mathematical investigations are of being minutely followed. Were they wholly unconnected with such important spe- culations in Physical Astronomy, and only to be regarded as a branch of the Higher Geometry, they would deserve the deepest attention, for their interesting development of general relations between figures so well known as the conic sections, for the marvellous felicity of the expedients by which the solutions are obtained, and for the inimitable elegance with which the reasoning is conducted. As a mere matter of mathematical contemplation, beginning and ending in the discovery of the relations which subsist be- tween different quantities and figures, they afford matter of lasting interest to the geometrician. But it certainly heightens that interest to reflect that the same skilful and simple construction which enables us to describe a para- NEWTON'S PEINCIPIA. 69 bola through given points, or touching given lines, be- side gratifying a curiosity purely geometrical, leads us to calculate within 20" of the truth the place of bodies revolving round the sun in orbits so eccentric that the el- lipse which they describe coincides with a parabolic line, instead of being nearly circular like the path of our globe, although our own distance from that luminary is near a hundred millions of miles. iii. We are next to consider the motion of bodies in conic sections which are given, and ascending or de- scending in straight lines under the influence of gravity ; that is, the velocities and the times of their reaching given points, or their places at given times. This branch of the subject, therefore, divides itself into two parts, the one relating to motion in the conic sections, the other to the motion of bodies ascending or descending under the in- fluence of gravitation. The Sixth Section treats of the former, the Seventh of the latter. (1.) In order to find the place of a revolving body in its trajectory at any given time, we have to find a point such that the area cut off by the radius vector to that point shall be of a given amount ; for that area is proportional to the time. Thus, suppose the body moves in a parabola, and that its radius vector completes in any time a certain space, say in half a year moves through a space making an area equal to the square of D ; in order to ascertain its position in any given day of that half year, we have to cut off, by a line drawn from the centre of forces, an area which shall bear to D 2 the same proportion that the given time bears to the half year, say 3 to w 2 , or we have to cut off a section A S P =~ D 2 , AP being the parabola and S the focus. This will be done if A B F 3 70 NEWTON'S PRINCIPIA. be taken equal to three times A S, and B O being drawn perpendicular to A B, between B O, B A asymptotes, a A G S M rectangular hyperbola is drawn, H P, whose semi-axis or semi-parameter is to D in the proportion of 6 to m\ it will cut the parabolic trajectory in the point P, required. For calling A M = x and P M = y and A S = a ; then A B = 3 a and y x (x + 3 a) = half the square of the hyperbola's semi-axis, which axis being 6 D 36 D 2 18 D 2 / x \ equal to , y (x + 3 a) = g ^ 2 = ^-, or y ^ - + a) 6D 2 fx a\ 3 D 2 /2 1 1 x -.-- rru r 2 I , , Therefore - x y ^ (x a) y 3 D 2 2 211 and - , A M x P M = - x y ; and 5 (x a) y = - S M . P M = SMP; therefore the sector A S P = 3D 2 so that the radius from the focus S cuts off the given area, and therefore P is the point where the comet or other O body will be found in ^ parts of the time. If the point is to be found by computation, we can easily find the value of y by a cubic equation, y 3 + 3 2 NEWTON'S PEINCIPIA. 71 18 2 D 2 , . . y = - g > an( * making B L = y, L P parallel to A M, cuts A P in the point P required. Sir Isaac Newton gives a very elegant solution geometrically by bisecting A S in Gr, and taking the perpendicular Gr R to the given area as 3 to 4 A S, or to S B, and then describing a circle with the radius R S ; it cuts the para- bola in P, the point required.* This solution is infinitely preferable to ours by the hyperbola, except that the demonstration is not so easy, and the algebraical de- monstration far from simple. It is further to be observed, that the place being given, either of these solutions -enables us to find the time. 3 D 2 Thus, in the cubic equation, we have only to find %~- //3 _J__ ^ /7^ I* It is equal to - - ^ - ; and as D 2 is the given integer, or period of e. g. half a year, the body comes to the point P in a time which bears to D 2 the proportion of unity to 6a 2 D 2 Sir Isaac Newton proceeds to the solution of the same important problem in the case of the ellipse, which is that of the planetary system, and is termed Kepler's problem from having been proposed by him when he had discovered by observation that the planetary motions were performed in this curve, and that the areas described by the radii were proportional to the times. In the parabola which is quadrable and easily so, the area being two- thirds of the rectangle under the co-ordinates, the solution of this problem is extremely easy. But the ellipse not * The most singular relation subsists between the hyperbolas and pa- rabolic areas, giving rise to very curious Porisms connected with Quadra- tures. Sec Phil. Trans. 1798, part ii. F 4 72 NEWTON'S PKINCIPIA. admitting of an expression for its area, or the area of its sectors, in finite terms of any product of straight lines, the problem becomes incapable of a definite solution. Newton accordingly begins his investigation by a lemma, in which he endeavours to demonstrate that no figure of an oval form, no curve returning into itself and without touching any infinite arch, is capable of definite quadrature. It is rarely, indeed, that the expression " endeavour," can be applied to Sir Isaac Newton. But some have ques- tioned the conclusiveness of his reasoning in this instance. The demonstration consists in supposing a straight line to revolve round a point within the oval, while another point moves along it with a velocity as the square of the portion of the revolving line between the given centre and the oval, that is, as the radius vector of the oval from the given centre. It is certainly shown, that the moving point describes a spiral of infinite revolutions ; and, also, that its radius is always as the area of the oval at the point where that radius meets the oval. If then the relation between the area and any two ordinates from the oval to any axis is such as can be expressed by a finite equation, so can the relation between the radius of the spiral and co-ordinates drawn parallel to the former, or the co- ordinates to the same axis. Therefore it will follow, that the spiral can be cut only in a finite number of points by a straight line, contrary to the nature of that curve. Indeed, its co-ordinates being related to each other by an algebraical equation is equally contrary to its nature; consequently the possibility of expressing the relation be- tween the area of the oval and the co-ordinates leads to this absurd conclusion, and therefore that possibility cannot exist ; and hence it is inferred that the oval is not quadrable. Sir Isaac Newton himself observes that this demon- stration does not apply to ovals which form parts of curves, NEWTON'S PRINCIPIA. 73 being touched by branches of infinite extent. But it does not even apply to all cases of ovals returning into them- selves, and unconnected with any infinite branches. There is, for example, a large class of curves of many orders, those whose equation is y m = n m %("-v x (a n x n ) ; and when m is even these curves are quadrable ; and in every case where m and n are whole positive even num- bers, it is the equation to a curve returning into itself. This is manifest upon inspection : forfy d x = fn x n ~ l (a n x n y d x is integrable because the power of x with- out is one less than that of x within the radical sign ; and because there is no divisor there can be no asymptote ; while it is plain that the root of a n x n is impossible when either +x or a: is greater than a, n and m being both whole even numbers. Wherefore the curve re- turns into itself; and as y = 0, both when x = Q, and when x = + a, or a, therefore the figure consists of two ovals meeting or touching in the origin of the abscissae. These two ovals admit of a perfect quadrature; the in- m m + l tegral being C -- -, -- r^ (a n - x n ) ' Thus if ^ m = n 2 the area is C - (a 2 a; 2 ) f , the latter quantity being one-half of an area that has to one-third the rectangle of the co-ordinates the same proportion which the difference of the squares of the diameter and abscissa has to the square of the abscissa ; for f (a 2 # 2 ) ~* = \ x y x The particular inquiry respecting motion in the ellipse did not perhaps require the proposition to be proved in the very general form in which Sir Isaac Newton has 74 NEWTON'S PRINCIPIA. given it. That the ellipse cannot be squared might per- haps be sufficiently proved from this consideration, founded upon a reasoning analogous to that on which the lemma in question proceeds. If a curve be described such that its co-ordinates, or the rectangle contained by the co-or- dinates, shall always bear a given proportion to the areas of the ellipse on the same axis, this curve cannot be alge- braical, not merely because of its equation involving quan- tities not integrable (for that may be said to be the ques- tion), but because it will stop short at a given line, which no algebraical curve can do. It will have no branch ex- tending beyond the perpendicular at the end of the axis : and moreover its equation is known to be that of a tran- scendental curve. This reason cannot be applied to all curves returning into themselves ; because, as we have seen in one class, the equation to the curve, whose co-ordinates should express their areas, is algebraical ; and also because, in that class, the secondary curve is found to have two branches which meet in cusps, and so do not stop short. If described by the proportion of areas they would seem to stop short, that property only belonging to one of their branches ; but their equation discloses the second branch. It is one of many instances of a truth perhaps not suf- ficiently remarked by geometricians, that curves sometimes have particular portions to which certain properties belong exclusively, no other part of the curve having them. As the area of the ellipse cannot be found by alge- braical quantities, or by the description of algebraical curves, the problem of Kepler cannot be solved otherwise than by transcendental curves, logarithms, circular arcs, or approximation. Sir Isaac Newton gives a solution by means of the cycloid described on an axis at right angles to the transverse axis of the ellipse, at a distance from its vertex which is a fourth proportional to half the trans- NEWTON'S PRINCIPIA. 75 verse axis, the focal distance, and the eccentricity, and with a generating circle whose radius is the distance of this perpendicular from the centre. A parallel to the cy- cloid's axis, at the point whose abscissa is to the periphery of the generating circle in the proportion of the given time to the periodic time, cuts the ellipse at the place required. This solution requires a construction beside that of the curve described : but a cycloid may be described which shall cut the ellipse directly at the point required. If a circle is described on A B the transverse axis, and its quadrant A h is cut in O, in the given ratio of the times in which the elliptical area is to be cut ; and then a cycloid is described, whose ordinate P M is always a fourth pro- portional to the arch O Q, the rectangle of the two axes and the distance between the foci ; or to O Q, A B x 2 . C F, and 2 . C S, this cycloid cuts the ellipse in the point required, P. The equation to this curve G P is simple enough, and the construction easy ; for the ordinate is in a given proportion to the arc Q O of the quadrant. As, however, an arithmetical approximation by means of series is required in practice, Sir Isaac Newton gives two me- thods, both of great elegance and efficiency. It may be proper here to note the names given by astro- nomers to the lines and angles in the ellipse connected mainly with the investigation of this problem. The sun being in the focus S, and P the planet's place, the aphelion 76 NEWTON'S PEINCIPIA. of the planet is B; the perihelion A; the arch BP, or angle BSP is the true anomaly; BO being to the whole circumference as the time in B P to the whole periodic time, BO, or O S B, is the mean anomaly, and Q B, or Q C B, is the eccentric anomaly, C being the centre of the ellipse : A and B are likewise called the apsides (or apses), and AB, the transverse axis, is called the line of the ap- S C sides ; S C, or more generally -J-Q is the eccentricity. (2.) The next subject of inquiry is the comparison of bodies moving in a straight line towards the centre of forces, with those moving by the same centripetal force in the conic sections whose axis is that straight line. If the projectile force by which a body revolves in any of those curves round the focus as a centre, suddenly ceases, and the body falls towards the centre of the curve, it is shown that its place at any given time, will be the point at which the line of descent is cut by a perpendicular from the point of the curve where the radius from the vertex makes its area proportional to the time consumed in the fall. For take the parabola whose area is x y, and let the distance of the point where the body begins to descend in a straight line be C ; the parabolic sectors, which are as the times, are expressed by y x f- - = (C x) or - x (x + 3 C) ; and if another parabola NEWTON'S PRINCIPIA. 77 with the same vertex, and with a smaller parameter, b, is <\/fo x drawn nearer the straight line, its sectors are - (x + 3C). Now the times in the first parabola, or the areas, at any two points referred to the abscissae x and z, being ^ (x + 3 C), and ~ (z + 3 C), the times or areas in the second parabola will be ~ (x + 3 C), and ^ (z + 3 C), respectively ; and therefore it is evident that the areas at the distances x and z, in the one curve are in the same proportion to one another with the areas in the other curve at those distances. If the parameter be continually diminished of the second curve, until that curve coincides with the axis, the same proportion holds ; and the times, therefore, in falling through the axis, will be as the areas of the first curve, corresponding to the points of that axis. And so it may be shown in the ellipse and hyperbola. Hence it follows, that in the case of the parabola, the velocity of the falling body in any given point is equal to that with which the body would, moving uniformly, de- scribe a circle having for its centre, the centre to which the body is falling, and for its diameter the distance of the given point from that centre. In the circle, the ve- locity at the given point is to the velocity in the circle described from the centre, with the distance of the given point for the radius, as the square root of the distance fallen through to that of the whole distance of the point where the fall begins. Thus let d be the distance of the given point to which the body has fallen, D the distance of the point at which it began to fall ; the velocity in the case of a para- bola is equal to that of the body moving in a circle, whose 78 NEWTON'S PRINCIPIA. radius is d ; in the case of a circle, it is to that of a body moving in a circle whose radius is d, as V D d : V D. And the like proportion subsists in the case of the hyperbola. Further, a rule is thus deduced for determining, con- versely, the time of descent, the place being given. A circle is to be described on A S = D, as the diameter, and another from S the centre, towards which the body falls, with the radius -_-. P being the point to which it has fallen, if the area S X B be taken equal to S C A, the time taken to fall through A P is equal to the time in which the body would move uniformly from B to X. Hence the periodic times being in the sesquiplicate ratio of the distances (t = d?) and because 2 f = 2 V~2 f the time taken to fall through the whole distance to the centre is to the periodic time of a body revolving at twice that distance round the same centre as 1 to 4 V~2 ; and thus we can calculate the time (supposing the planetary orbits to be circular) which any one would take to fall in a straight line to the sun, or any satellite to its principal planet, if the projectile motion were suddenly to cease. NEWTON'S PRINCIPIA. 79 The moon in this way would fall to the earth in about four hours less than five days.* The inquiry is closed with a solution of the general pro- blem, of which the preceding solutions for the conic sec- tions, and for the force inversely as the squares of the distances, are only particular cases ; and the times and velocities are found from the places, or the places from the times and velocities, where a body ascends from or de- scends to the centre, influenced by a centripetal force of whatever kind. On the given straight line of ascent or descent a curve is to be described whose co-ordinates are the centripetal force at each point of the axis, or whose equation is y=X, X being a function of x, the distance from the beginning of the motion. The area of the curve at each point is / y d x=f X d x ; and if that integral is equal to Z 2 , Z is as the velocity at the distance a x, from the centre. Another curve described on the same axis, , 1 i. . A i*dx and whose equation is u = -^-, gives by its areas / -,=- , the time taken to move through the distance a x ; it is equal to . This is easily demonstrated ; for, first, if the velocity be v, and the time d t, the space being d x, we have the forcer/ = -5; andasc?= , there- fore y = -j , and y d x v d v, and f y d x ~ \ but ci x T^ fydx\ therefore Z = =, and the velocity is as the area Z. Again ; for the time in the other curve ; * It is comparing the greatest with the smallest things, to observe that the time of the revolution of a planet round the sun, or the planetary year, bears the same proportion to the time in which the planet would fall to the sun, which the square of the side of a bee's cell does to one of the six tri- angles, or to the sixth part of the rhomboidal plate. (See Appendix to vol. i., Paley Dlustrated.) 80 NEWTON'S PRINCIPIA. u = ^, and v = A/2 . Z ; also d t = = ^ * Therefore t = Z -77- = v/ 2 . , or the time is as the area g. In these expressions, therefore, to find Z and I we have to substitute the values of X and Z in terms of x, and integrate. It is hardly necessary to add, that if, instead of the velocity and the time being sought (Z and ), these are given, and the place reached by the body be sought, we find it by the same construction ; and ascertaining what value of x gives the value of Z, the square root of the area. But it may be well to note here, that if O M be the curve, whose ordinate P M or y X, the centripetal force at P in terms of A P or x, or the gravitation of any particle of a homogeneous fluid towards S at the point P ; then the column of that fluid whose altitude is A P will press at P, as the area A P M O, or as u 2 , the square of the velocity acquired by a body falling through A P. iv. The next object of research is to generalise the pre- ceding investigations of trajectories from given forces, and of motion in given trajectories, applying the inquiry to all kinds of centripetal force, and all trajectories, instead of confining it to the conic sections, and to a force inversely NEWTON'S TRINCIPIA. 81 as the square of the distance. This forms the subject of the Eighth Section, which therefore bears to the Third, Fourth, Fifth, and Sixth, the same relation that the con- cluding investigation of the Seventh Section (on rectili- near motion influenced by centripetal force) bears to the rest of that section. The length at which we before went into the solution of the problem of central forces (inverting somewhat the order pursued in the Principia) makes it less necessary to enter fully into the general solution in this place. We formerly gave the manner of finding the force from the trajectory in general terms, and showed how, by means of various differential expressions, this process was faci- litated. It must, however, be remarked, that the inverse problem of finding the trajectory from the force, is not so satisfactorily solved by means of those expressions. For example, the most general one at which we arrived of Vv* + ( x - of xdx*.dJi. C - - the force inversely as the square of the distance, presents an equation in which it may be pronounced impossible to separate the variables so as to integrate, at least while d X, the differential of --, remains in so unmanageable a form; for then the whole equation is 7 x . ^ Z(dx-(x- = - - , and thus from hence no equation to (y 2 + (x - a) 2 ) 2 the curve could be found. It cannot be doubted that Sir Isaac Newton, the discoverer of the calculus, had applied all its resources to these solutions, and as the expressions for the central force, whether - ^, or -s-ji or z p a . t p n r G 82 NEWTON'S PRINCIPJA. __ __^ - i_ (in some respects the simplest of all, being taken in respect of d t constant, and which is intcgrable in the case of the inverse squares of the distances, and gives the general equation to the conic sections with sin- gular elegance), are all derivable from the Sixth Propo- sition of the First Book, it is eminently probable that he had first tried for a general solution by those means, and only had recourse to the one which he has given in the Forty-first Proposition when he found those methods un- manageable. This would naturally confirm him in his plan of preferring geometrical methods ; though it is to be ob- served that this investigation, as well as the inverse pro- blem for the case of rectilinear motion in the preceding section, is conducted more analytically than the greater part of the Principia, the reasoning of the demonstration conducting to the solution and not following it synthe- tically. A is the height from which a body must fall to acquire the velocity at any point D, which the given body moving in the trajectory V I K (sought by the investigation) has at the corresponding point I ; D I, E K, being circular arcs from the centre C, and C I = C D and C K=C E. It is shown previously that, if two bodies whose masses are as their weights descend with equal velocity from A, and being acted on by the same centripetal force, one moves in V I K and the other in A V C, they will at any cor- responding points have the same velocity, that is at equal distances from the centre C. So that, if at any point D, D b or D F be as the velocity at D of the body moving in A V C, D b or D F will also represent the velocity at I of the body moving in V I K. Then take D F=y as the centripetal force in D or I (that is, as any power of the distance D C, or a x, VC being a, and C D, x] NEWTON'S PRINCIPIA. 83 V D F L will befydx. Describe the circle V X Y with C V as radius. Let V X = z, and Y X will be d z, and N K = . Then I C K being as the time, and d t being constant, that triangle, or , is constant, and K N is as a constant quantity divided by I C, or as If we take to ^A V L B (proportioned to the force at any one point V and therefore given), as K N to I K, therefore this will in all points be the proportion ; and the O 2 squares will be proportional, or fy d x : -^::I K 2 , or K N 2 + I N 2 , to K N 2 ; and therefore / y d x - Therefore = NEWTON S PRINCIPIA. 3? dz ', and multiplying by x, a (twice Qdx x the sector I C K)= 7 == == 5a. Again adz: - ^ 7 ; and a d z = x*dz a a: 2 = twice the sector Y C X. Hence results this construction. Describe the curve a b Z, such that (D b = M) its equation shall be u = and the curve a c a: such that (D c = or the sectors which are the differentials of the areas VIC and V X C, the areas themselves are equal to those areas ; and therefore from V X C being given (if the area c D V a be found), and the radius C V being given in position and magnitude, the angle V C X is given ; and from C X being given in position, and C V in magnitude and position, and also the area CIV, (if V D b a be found), the point I is found, and the curve V I K is known. This, however, depends upon the quantities made equal to u and p NEWTON'S PRINCIPIA. 85 severally being expressed in terms of x, for this is necessary in order to eliminate y from the equations to these curves ; and then it is necessary to integrate these expressions; for else the angle V C X, and the curve V I K, are only ob- tained in differential equations. Hence Sir Isaac Newton makes the quadrature of curves, that is, first the inte- gration offy d x, to eliminate y, and then the integration of the equations resulting in terms of u and x,

the mass of the sun, and e the eccentricity of the earth's orbit. Consequently, as e" decreases, increases, the term being negative; and therefore a itself decreases as e x decreases ; in other words, the moon's orbit is diminished, and her velocity augmented, in consequence of the earth's eccentricity decreasing. But if the diminution of the greater axis is not admitted as necessarily lessening the orbit, we may recollect the relation between the times and the mean distances, the squares of the former being as the cubes of the latter; and the mean motion is, of course, in- versely as the periodic time. However Laplace fur- nishes us with a still closer reason, and illustrates the i 114 NEWTON'S PRINCIPIA. use of the calculus, as it were, by a new triumph, in another part of the Mecanique Celeste.* For the equation t ^ M. of the mean angular motion is shown to be n = ~ t a? t being the time, a the transverse axis, and p the sum of the masses of the two bodies, in this case the moon and the earth. Therefore n, the mean motion, must neces- sarily be accelerated as a, the axis, is diminished. And here in passing, we also observe how Kepler's law of the sesquiplicate ratio may be anew proved, but only if we make ft = S (the sun), and neglect the mass of the planet. For take two planets whose mean motions are n p, and their mean motions be- and because (2 TT being 360), n t = 2 TT, therefore t = , and ?= ~, or t = n n 2 * ,_ **, and ? = ^f ; consequently t* : t* :: a 3 : a >3 V p, V /t x being Kepler's law, which is thus demonstrated. But it is only demonstrated and is only true if V p, is the same to both planets, that is, if p. = S in each case. Now, this may be assumed in the case of those bodies revolving round the sun, or of the satellites of Jupiter and Saturn revolving round those primary planets, because of the great dispro- portion between the central body and the others, (the largest of them, Jupiter, being less than a thousandth part of the sun.) But the law would not hold true if p were taken, which in strictness it ought to be, as S + P, the sum of the masses of the central and the revolving body ; for then p. would differ in each instance, and the sesquiplicate pro- * Liv. ii. ch. 3. NEWTON'S PRINCIPIA. 115 portion would be destroyed. Hence, we arrive through the calculus at this important conclusion, that the law only holds, if the mutual actions of the planets on each other are neglected, and that, therefore, the law is not rigorously true where, as in the case of the earth and others, the actions of the other planets are sensible. Again, the inspection of the algebraical expressions shows that the variation in the eccentricity of the earth's orbit produces, likewise, the retardation of the apsides and nodes ; and this discovery was also made, apparently, by the mere inspection of the expressions which the calculus had furnished. Thus the expression for the motion of the perigee (or apsides) involves the integral / eP d v (y being the true anomaly);* and this quantity is positive. There- fore the decrease of the eccentricity of the earth's orbit, causes a decrease, also, of the perigeal motion of the moon. And one of the terms of the equation to the motion of the nodes contains the same integral f e' 2 d v ; consequently the same eccentricity is likewise the cause of the variation in the period of their revolution.! Now we have seen how extremely small these irregu- larities in the moon's motion are which the theory gives by this analytical process, and that they are hardly sensible in a whole century ; yet it is found that the deductions of the calculus are in a remarkable manner confirmed by actual observation. Practical astronomers, for example, wholly ignorant of Laplace's discoveries, have ascertained that the secular variation in the motion of the moon's apsides, ascertained by comparing the eclipses in the Greek, Arabian, and Chaldean astronomy, with those of the last * Angle of the radius vector with the axis of the orbit. f Mec. CeL liv. vii. ch. 1. This wonderful chapter is a mere series of integrations, and contains, from the inspection of the equations, those extraordinary discoveries respecting the laws of the universe. 116 NEWTON'S PRINCIPIA. century, is about 3.3, or 33 tenths of the moon's mean motion ; and this is the exact result of the calculus. Laplace also discovered, chiefly by similar means, a very small secular inequality in the moon's motion never before sus- pected, and produced by the sun's attraction.* It was found by observing, that the divisor of some of the frac- tional terms of the equation which shows the inequality is extremely small, and that, consequently, the irregu- larity may become sensible. A correction of the tables was thus introduced by this great geometrician, in which the theory approaches, on an average, to within -^^ of the actual observation. The sign of this inequality being negative, it is a retardation of the mean motion, and is to be set against the secular acceleration. It must be observed, moreover, that the errors of the theory, as compared with the observation, are half of them by excess and half by defect ; so that they may be said to balance each other. The maximum of this inequality is little more than 15", and its period is 184 years. Hitherto of the moon; but we are, in like manner, con- ducted by the same refined, though complicated, analysis to the variations in the orbits, and consequently in the motions of the earth and of the other planets, as well as of the satellites of Jupiter and of Saturn. The most remark- able variations produced upon these orbits are the changes in their eccentricity and in their aphelion; the former being constantly, though slowly, shortened the latter moving round in slow revolutions, as the line of the moon's apsides revolves, but revolves much more swiftly. The expressions obtained in the case of any one planet for the eccentricity and perihelion longitude (revolving motion of the axis), are mainly composed of the masses, * Mec. Cel. liv. vii. cli. 5. NEWTON'S PRINCIPIA. 117 distances, eccentricities, and perihelion longitude of the disturbing bodies, with the known eccentricity and longi- tude of the planet in question at a given epoch. Hence we perceive that on these circumstances depends the varia- tion of the eccentricity and the revolution of the axis of the planet. Thus the secular variation of the eccentricity of the earth's orbit is 0.000045572 of e, the eccentricity which at the epoch (1750) was 0.016814 of the semi-axis major of its orbit ; and it has the negative sine in the expression ; consequently the eccentricity is on the decrease, as we before observed. This diminution of the eccentricity amounts to about 18" 79'" yearly (or about 3900 miles). We have already observed that the annual revolving mo- tion of the axis of the earth's orbit is 11" 53'", and its period 109,060 years. The examination of the expres- sions for these irregularities shows, as might be expected, that Mars, Venus, and Jupiter bear the most considerable share in producing the variations.* But it is a truly re- markable circumstance that the direct action of those planets upon the moon's motion is hardly sensible com- pared with their indirect, or, as it is sometimes called, reflected action upon the same body, through the medium of the sun and the earth. For these planets, Mars, Venus, and Jupiter, by altering the eccentricity of the earth's orbit, very sensibly affect the motions of the moon, as we have seen, while directly their action is incomparably less perceptible. The perihelion longitudes of all the other planets are increasing, or their orbits advancing, except Venus, whose apsides are retrograde; and the eccentricities of Venus, Saturn, and Uranus, are decreasing, like that of the earth, whilst those of the other planets are on the * Mec. Gel. liv. ii. eh. 6, 7, S.; liv. vi. ch. 7- 1 3 118 NEWTON'S PRINCIPIA. increase. These variations are greater in Saturn than in any of the others, considerably greater than the varia- tions of Mars; which comes the nearest to them. The variation in the eccentricity of Jupiter's orbit is nearly three times as great as in the Earth's ; that of Saturn between five and six times greater than the Earth's; while the variation in the perihelion longitude of the former is about five-ninths of the Earth's variation; and Saturn's exceeds the Earth's in the ratio of about 25 to 18, and exceeds that of Mars only somewhat more than as 49 to 48. When the attention of mathematicians and astronomers was first directed closely to examine the disturbances of these planets, it appeared hardly possible to reconcile such vast and numerous irregularities, as were found to exist, with the theory of gravitation, or indeed to reduce them under any fixed rule whatever. The case seemed to be- come the more hopeless when so consummate an analyst as Euler, the great improver of the calculus, failed in repeated attempts at investigating the subject, committing several important errors which for a time were not de- tected, but which showed, or seemed to show, a wide dis- crepancy between the theory and the observations. By one discovery, indeed, to which his researches led him, he may be said to have laid the foundation of the most ex- traordinary step which has been made in the knowledge of the planetary system. We allude to his theorem on the periodicity of the eccentricities and aphelia of Jupiter and Saturn. But in most other respects his attempts signally failed. D'Alembert made little progress in this inquiry ; but at length Lagrange, and still more Laplace, by apply- ing all the resources of the calculus, in its last stage of improvement, and after the method of Variations had been systematised, succeeded in reducing the whole to order, NEWTON'S PRINCIPIA. 119 and discovered, while investigating these motions, the great law of the stability of the universe. The circumstance which mainly contributes to render the irregularities in the motions of two planets great, and which especially augments the disturbance of Jupiter's satellites, is that their mean motions should be commen- surable, which those of Jupiter and Saturn are after a very remarkable manner. Five times the mean motion of Saturn are equal to nearly twice that of Jupiter; and the three first satellites of Jupiter are so related to each other, that the mean motion of the first, added to twice the mean motion of the third, is equal to three times that of the second; while the longitude of the first added to twice that of the third, and subtracted from three times that of the second, makes up exactly 180. Laplace showed, that this proportion, if it was not originally fixed between those satellites, must have been established by the action of the attractive and disturbing forces * ; and it is a truly remark- able thing, that when the theory had given a value for the three mean motions, M 3 m + 2 ju, = 0, the comparison of the eclipses for a century was found to make the expres- sion only 9", and consequently to tally with the theory within that very small difference. The observation of the effects which were produced upon the equations which resulted from the analysis, by the proportions above stated between the mean motions of Jupiter and Saturn, induced Laplace to suspect that this made quantities become of importance, which from the high powers of the denomi- nators might otherwise have been insignificant. For one of the terms in the equation to 8 r (variation of the radius vector of the first satellite), for example, had for its deno- * Mec. Cel. iiv. vi. ch. 1. 2. 12. 13.; also for the analytical investiga- tion, see Iiv. viii. throughout, and Iiv. ii. ch. 8. s. 65. I 4 120 NEWTON'S PRINCIPIA. minator 4 (n' w) 2 N 2 in which n and n' are the mean motions of the first and second satellite, and N a composite quantity not materially differing from ri, which differs hardly at all from -, inasmuch as n 2 n' , while n' 2 n" (n" being the mean motion of the third satellite) ; and hence the above denominator becoming little or nothing, the term is of large amount ; and BO of 8 v, the variation of the anomaly.* He accordingly undertook the laborious task of examining this complicated subject by considering all these quantities ; and he arrived at the discovery of, among other inequalities, a retardation of Saturn's motion of about 3" 6'" yearly, and an acceleration in Jupiter's motion of about 1" 18"'. Another irregularity in Saturn's motion with respect to the vernal and autumnal equinox had been observed by astronomers in the last century, and could not be explained. Laplace found this, like all the rest, to follow from the Newtonian theory. In short, when summing up the subject in one of his concluding books, he naturally and justly exclaims, " Tel a ete le sort de cette brillaute decouverte, que chaque difficulte qui s'est elevee, est devenue pour elle uu nouveau sujet de triomphe ; ce qui est le vrai caractere du vrai systeme de la nature." f There is no sensible disturbance produced by any of the satellites, except the moon, upon the motion of their primaries, from the extreme smallness of their masses compared with those of the sun and of their primaries; for 8 r is equal to a series in which m m l m M' M' "M ' &C " *** factors of each term * m > m '> &c " * Mec. Cel. liv. viii. ch. 1. 4. f Ibid. liv. xv. ch. 1 Syst. du Monde, liv. v. ch, 3. J Ibid. liv. vi. ch. 4. NEWTON'S PRINCIPLE. 121 being the masses of the satellites, and M that of the planets. Now, in the case of Jupiter ^ = ^ ; =^- m if and -^f- are somewhat greater; but the greatest of the four factors ^- = only. But in the case of the earth this factor amounts to about ^7. ; so that 8 r and o(J 8 v become sensible; and will be so, even if, in- stead of ^. we take the factor ^-f , which is more JM M. + m correct.* When Laplace began his celebrated investigations of the orbits of Jupiter and Saturn, he found that, on substi- tuting numerical values for the quantities in the expres- sion of the mean movement of the one body as influenced by the action of the other, the sums destroyed one ano- ther, and left the whole effect of this disturbing force equal to nothing, or the mean motion of neither planet at all affected by the other. The formulas could be in each case reduced to terms only involving two co-efficients ; and these destroyed one another, f He soon found that the same principle applies to all the heavenly bodies ; that their mean motions and mean distances (the great axis of their orbits) are not affected by any changes other than those which occur within limited periods of time ; that conse- quently the length of the solar year is precisely the same at any one period of time, as it was at a period so far distant as to enable the changes which are produced within those moderate limits to be effected. This impor- tant proposition he demonstrated upon the supposition, * Mec. Cel. iv. vi. ch. 10. 30. f Ibid. liv. ii. ch. 7. ; liv. xv. ch. 1. 122 NEWTON'S PRINCIPIA. that the squares of the masses, and the fourth powers of the eccentricities, and the angles of the orbits, are neg- lected in the calculus.* But Lagrange afterwards showed, that the theorem holds true, even if these quantities be taken into the account. The examination of the moon's motion demonstrates the same important fact, with respect to the permanency of the greater axis and mean motion of the planets ; for if the solar day were now 7 ^ of a second longer than it was in the age of Hipparchus, the moon's secular equation would be augmented above 42 per cent, or would be in that large proportion greater than it now is known to be. Therefore there has not even been the smallest change of the mean movement of the planets. The other changes which take place in the orbits and motions of the heavenly bodies, were found by these great geometricians to follow a law of periodicity which secures the eternal stability of the system. The motion of the earth's orbit we have already seen is so slow, that its axis takes above 109,060 years to perform a complete revolution ; but after that time it occupies precisely the same position in space as it did when this vast period of time began to run. So the eccentricity of the earth's orbit has been for ages slowly decreasing, and the decrease will go on, or the orbit will approach nearer and nearer to a circle, until it reaches a limit which it never can pass. The eccen- tricity will then begin slowly to increase until it again reaches its greatest point, beyond which the orbit never can depart from the circular form. The same principle extends itself to all the planets. Thus, the time of the secular variation of Jupiter's eccentricity is 70,400 years. All these deductions are the strict analytical conse- * Mec. Cel. liv. ii. ch. 7. and 8. (sects. 54 and 63.) NEWTON'S PRINCIPIA. 123 quences of the equations to the eccentricity of the pla- netary orbits, obtained by the investigation of the total effect of the mutual actions of the heavenly bodies. There results from that analysis this remarkable theorem. That if the eccentricities of the different planets be called e, (f, e", &c., their masses m, m' t ml', &c., and their transverse axes a, of, a", &c., and if the integration be made of the dif- ferential expression for the relation between the differentials of the eccentricities multiplied by the sines of the longitude and the differentials of the time, and for the relation be- tween the differentials of the eccentricities multiplied by the cosines of the longitudes and the differentials of the fdesin. -or , d sin. &* decos. & , de^cos.is / tlme - --> and &c., we obtain the equation e 1 . m . ^a + e^, m\ e! '. m x . Va\ &c., is less than C ; and sup- pose at any one period the whole eccentricities e, ef, ef', &c., to be very small, which is known to be true, C, which at that period was the sum of their squares, must be very smallj the other quantities m, m', &c., being wholly con- stant, and /a, V a', &c., being invariable in considerable periods of time. Therefore, it is clear that the varia- tion in any one of those eccentricities, as e, never can exceed a very small quantity, namely, a quantity propor- tional to VC e' 2 e" 2 , &c. The whole possible amount of the eccentricity is confined within very narrow limits. It never can for any body, whose eccentricity is e, exceed a quantity equal to * Mec. Cel. liv. ii. ch. 6, 7. (sects 53. 55. 57, 58). 124 NEWTON'S PEINCIPIA. Vet - e"* . m" . Va" - &c. So that the eccentricities never can exceed a very small quantity. Thus the changes which are constantly taking place in the planetary orbits are confined within narrow limits; and the other changes which are the consequences of this alteration of the orbits, as, for instance, the acceleration of the moon which we before showed arose from the varia- tion of the eccentricity of the earth's orbit, are equally con- fined within narrow limits. Those changes in the heavenly paths and motions oscillate, as it were, round a given middle point, from which they never depart on either hand, beyond a certain small distance ; so that at the end of thousands of years the whole system in each separate case (each body having its own secular periods) returns to the exact position in which it was when these vast succes- sions of ages began to roll. For similar theorems are deduced with respect to other revolutions of the system, whose general destiny is slow and constant change, but according to fixed rules, regulated in its rate, confined in its quantity, limited within bounds, and maintaining during countless ages the stability of the whole universe by appointed and immutable laws. Laplace examined in the last place the possible effects upon the celestial motions of the resistance of a subtle ethereal medium, and of the transmission of gravity or attraction not being instantaneous, but accomplished in a small period of time. The result of his analysis led him to disbelieve in both these disturbing causes. He found that in order to produce its known effects, the trans- mission of gravity, if effected in time, must be seven NEWTON'S PRINCIPIA. 125 millions of times swifter than that of light, or 147 thou- sand millions of miles in a second.* iii. The great system of most interesting truths which we have now been contemplating is the work of those who diligently studied the doctrines unfolded by Sir Isaac New- ton, respecting the motions of bodies which act upon each other, while they are moving around common centres of attraction. He laid down the principles upon which the investigations were to be conducted ; he showed how they must lead to a solution of the questions proposed, touching the operation of disturbing forces ; and he exemplified the application of his methods by giving solutions of these questions in certain cases. Although his successors, tread- ing in his steps, have reaped the great rewards of their learning and industry, and are well entitled to all praise for the skill with which they both worked and improved the machinery that he had put into their hands, at once improving the calculus invented by him, and felicitously ap- plying it to advance and perfect his discoveries, yet the distance at which his fame leaves theirs is at least equal to that by which a Worcester and a Watt outstripped those who, in later times, have used their mechanism as the means of travelling on land and on water, in a way never foreseen by those great inventors. Strict justice requires that we should never lose sight of the truth repeatedly confessed by Euler, Clairaut, Delambre, Lagrange, Laplace, that all the advances made by them in the use of analysis, and in its application to physical astronomy, are but the conse- quences of the Newtonian discoveries; so that we are guilty of no exaggeration, if we regard the most brilliant achieve- ments of those great men only as corollaries from the pro- positions of their illustrious master. Let us briefly see * Mec. Ca liv. vii. ch. 6 ; liv. x. ch. 7. 126 NEWTON'S PEINCIPIA. how he Laid the deep and solid foundations of the fabric which we have been surveying. After examining the motions of a system of two bodies with respect to one another, and their common centre of gravity, and in space, as those motions are affected by the mutual attractions of the two bodies themselves (in the manner which we have already described), Newton pro- ceeds to the great problem of the Three Bodies, as it has been termed, because the solution is so difficult, that gene- rally the attempt has been confined to the case of three only, this also being sufficient for determining the more impor- tant disturbances of the moon's motions. The inquiry, however, is general in the Principia ; and its subject is, the motion, produced by the mutual actions upon one ano- ther of the bodies in a system. Thus, for example, the inquiry already analysed regards the effect produced upon their motion in space, by the mutual attractions of the earth and moon; that to which we now are proceeding regards their motion, as also influenced by the disturbing force of the sun, and indeed, even by the smaller but not evanescent disturbing forces of the other planets. So as the former inquiry may be extended on the same prin- ciples to the motions of Jupiter and Saturn, and their satellites; this new inquiry applies also to the disturbances of their systems by ours, and of our system by theirs. Newton begins by showing that if the attracting force increases as the distance of the bodies from each other, any two, M and E, will revolve round their common centre of gravity, G, in an ellipse having G for its centre. This is plain from what was formerly proved when treat- ing of the conic sections, and also more lately respecting the centre of gravity. If, then, each of these is attracted, in the same manner, by a third body S, this force, being resolved into two, one parallel to the line joining M and NEWTON'S PEINCIPIA. 127 E, the other parallel to the line joining E and G, the former force will only accelerate the motion of M and E round G by an addition to the mutual attraction of M and E ; the latter force will draw the centre G towards S or towards G', the common centre of gravity of the three bodies, and combined with the action of M and E upon their centre G will make G revolve in an ellipse round G', the common centre of the three, round which also, in like manner, S will describe an ellipse, G' being the centre of those two ellipses. Thus the bodies M and E will de- scribe an ellipse round the centre G, and the centre G and body S will describe ellipses round the centre G', both G and G' being the centres of these ellipses ; and so of any greater number of bodies. Moreover, the ab- solute amount of the attractive force in each centre will be as the distance of the centre from the bodies or centres of gravity severally, multiplied by the masses of the bodies. So that E and S are attracted to G by a force as (M + E + S) multiplied by their respective distances from G. Lastly, the times in which these ellipses are described by the bodies and the centres, are all equal by what was before proved respecting motion when the force varies as the distances. This law of the centripetal force is the only one which preserves the entire ellipticity of the orbits, notwithstand- ing any mutual disturbances; but it produces, at great distances, motions of enormous velocity. Thus we have seen that Saturn would move at the rate of 75,000 miles in a second (or a third of the velocity of light itself), were there no disturbance from the other bodies ; but the dis- turbance might greatly accelerate this rapid motion. If the law be the inverse square of the distance, there will be a departure from the elliptical form of the orbits and from the proportion of the areas to the times, indicating 128 NEWTON'S PRINCIPIA. that the several resulting forces are not directed towards the several centres. But this departure will be less con- siderable in proportion as the body in the centre of any system, or in the common centre of any number of sys- tems, is of a magnitude exceeding that of the revolving bodies, or systems of bodies, because this will prevent the central body moving far from its place, or much out of a straight line; and also the departure will be less in proportion as the bodies, or systems revolving, are at a great distance from the centres or from the common centre, because the diminution of this distance increases the inclination of the lines in which the disturbing forces act, and thus disturbs the motions of the bodies among them- selves. Again, if the law of the attraction varies from the inverse square of the distance in some, and not in others, the disturbing effect will be increased. So that we may infer the universality of the law and also the small amount of the disturbing force, and its acting in nearly parallel lines, if we find the ellipticity of the orbits not much de- ranged, and the proportions of the areas to the times not greatly interrupted. Newton proceeds to examine more minutely the disturb- ances caused in a system of Three Bodies, of which two smaller ones move round a third larger one, and all attract one another by forces inversely as the squares of the dis- NEWTON'S PEINCIPIA. 129 tances. Let S attract M with a force inversely as the square of the distance ; call the mean distance = 1 ; the mean force will be ^ = 1. Let the distance from S, successively taken by M in moving round E, or its true distance, be S M ; thence the force at M is -. Take S L = g^p, and drawing L N parallel to M E, the forces at M are L N 4- Q (Q being a quantity that varies as \ and S N. Now L N : M E :: , T , T S L . M E ME ,, f S L : S M; and L N = - -- = -a- Therefore AT "F 1 the force acting upon M towards E is as ME + ^^ ; con- sequently it will increase the attraction of E, but it will not be inversely as the square of the distance ; and there- fore M will not describe an ellipse round E, and the force N S does not tend towards E, nor does the force resulting from compounding L N, or ME, or L N + M E, with N S, tend to E. So that the areas will not be proportional to the times. Therefore, also, this deviation from the ellip- tical form and from the proportional description of the areas will be the greater, as the distances L N and N S are smaller. Again, let S attract E with a force as if this were equal to S N, it would, by combining with SN, that is, with the attraction of S on M, produce no alteration in the relative motion of M and E. There- fore, that alteration is only caused by the difference S between SN and ; wherefore the nearer SN is to the proportion of ~ ,-, 2> that is (because of the 130 NEWTON'S PKINCIPIA. proportion of S L = -X the nearer S N is to unity, the mean force upon M ; and the nearer the forces exerted by S on M and on E approach to equality, the less will the elliptical orbit be disturbed, and the more nearly will the areas be described proportionally to the times. If the disturbing force of S acts in a plane different from that in which M and E are, M will be deflected from the plane of its orbit; because the force SN ST Vf ^ not P ass through E; consequently this deflection will be greater or less in proportion as this difference is greater or less, and will be least when o^ * s near ty e q ua l to the mean force of S upon M. We have hitherto been supposing S, the greater body round which M and E revolve, to be at rest while they revolve round each other (the case of the earth and of other planets having satellites). If we now suppose E to be the greater and central body, and that M and S both move round E (the case of the planets round the sun), a similar proposition may be demonstrated with respect to the dis- turbances : And it is further clear in this case that if S moves round G, the centre of gravity of M and E, the orbit of S will be less drawn from the elliptical form, and its radius vector will describe areas more nearly proportional to the times than if it moved round E. This appears clearly from observing that the direction of the centripetal force towards G, that is S G, must be nearer E than M ; that the attractive forces by which S is drawn are as S G ; and also that S M varies, while S E remains the same, or nearly so. NEWTON'S PRINCIPIA. 131 In all these cases the absolute attractive forces are as the masses of the attracting bodies; and if there are a num- ber of these, A, B, C, E, &c., of which A attracts all the rest with forces as =, -> &c., (D, d, &c., being the distances from A,) and B also attracts A, C, E, &c., with forces as ^, -^, the absolute attraction of A and B towards each other are as the masses A and B. Hence in a system, as of a planet and its satellites, if the latter revolve in ellipses, or nearly so, and describe areas pro- portional, or nearly so, to the times, the forces are mutually as the masses of the bodies ; and conversely, if the forces are proportional to the masses, and ellipses are described and the areas as the times, the mutual attractions of all are inversely as the squares of the distances. It is proved, by reasoning of the same kind, that the disturbing force of S is greatest when M is in the points C and D of the orbit (or the quadratures), and least when M is in A and B (or the line of conjunction and opposition called the syzygies). When M is moving from C to A and from D to B, the disturbing force accelerates the motion of M, which then moves along with the disturbing force. When M moves from A to D, and from B to C, its mo- tion is retarded, because the disturbing force acts against the direction of M's motion. So M moves more swiftly in syzygy than in quadrature, and its orbit is more curved in quadrature than in syzygy. But it will recede further from E in quadrature, unless the eccentricity of the orbit should be such as to counterbalance this recession: for the operation of the combined forces is twofold ; it both makes the line of apsides move forward in one point of the body's revolution and backward in another, but more for- ward than backward, and so upon the whole makes it ad- 132 NEWTON'S PRINCIPIA. vance somewhat each revolution (a8 we before saw) ; and it also increases the eccentricity of the orbit between qua- drature and syzygy, and diminishes that eccentricity be- tween syzygy and quadrature. So of the inclination of the orbit, which is always diminished between quadrature and syzygies, and increased between syzygy and qua- drature, and is at the minimum when the nodes are in quadrature and the body itself in syzygy. We found before that the force L N was as C^TS- The forces L N and N E are directly as the mass S, and when S is very distant, the forces L N and N E vary as c ^, or inversely as the squares of the periodic times ; and if at a given distance the absolute disturbing force be as the magnitude of the disturbing body, whose dia- (T 3 meter is d, these forces are as ^-^3 ; or as the cube of the apparent diameter of S. Also if instead of one sa- tellite, M, moving round E, we have several whose orbits are nearly of the same form or inclination (like the first three of Jupiter), the mean motion of their apsides and nodes each revolution are directly as the squares of their periodic times, and inversely as the squares of the planet's time, and the two motions (apsides and nodes) are to one another in a given ratio. We now have one of those extraordinary instances which abound in his writings, of Sir Isaac Newton's matchless power of generalization ; of apprehending remote analogies, and thereby extending the scope of his discoveries. Having shown how the disturbing forces of bodies in a system act upon their motions with respect to each other, he now examines the effect of such forces upon the constitution of the bodies themselves. He supposes, for example, NEWTON'S PEINCIPIA. 133 that a number of masses of a fluid revolve round E at equal distances from it by the same laws of attraction by which M moves round E, and that these masses are thus formed into a ring ; then it follows that the portions of this ring will move quicker in syzygy than in quadrature, that is, quicker at A and B than at C and D ; also, that the nodes of the ring, or the intersections of its plane with the plane S E, will be at rest in syzygy, and move quickest in quadrature, and that the ring's axis will oscillate as it re- volves, and its inclination will vary, returning to its first position, unless so far as the precession of the nodes carries it forward. Suppose now E to be a solid body with a hollow channel on its surface, and that E increased in diameter until it meets the ring, which now fills that channel, and suppose E to revolve round its own axis the motion of the fluid, alternately accelerated and re- tarded (as we have shown), will differ from the equable rotatory motion of the solid on its axis, being quicker than the globe's motion in syzygy, and slower in quadrature. If S exerts no force, the fluid will not have any ebbs and flows, but move as round a centre that is at rest ; but if the varying attraction of S operates, being greater when the distance is less, the disturbing force acting in the direction S L, and being as ,. 2> will raise the fluid in A and B, or in syzygy, and from thence to quadrature, C and D, while the force L N will depress it in quadrature, C and D, and from thence to syzygy, A and B, If we now suppose the ring to become solid, and the size of E to be again reduced, the inclination of the ring will vary, and oscillate ; and the precession of its nodes will continue the same and so would the globe, if, without any ring at all, it had an accumulation of matter in the equator, or had matter of greater density there than elsewhere, and K 3 134 NEWTON'S PRINCIPIA. at the poles. If, on the other hand, there is more matter at the poles, or matter of a less dense kind at the equator, the nodes will advance instead of receding. So that by knowing the motion of the nodes, we can estimate the constitution of the globe; and a perfectly spherical and homogeneous globe will move equally and with a single motion only round its axis. No other will. The Sixty-sixth Proposition, or rather its twenty-two corollaries, constitute perhaps the most extraordinary por- tion of the Principia. We have seen that Sir Isaac Newton here deduces most of the leading disturbances in the motions of three bodies, for example, the moon, earth, and sun, from the propositions which had been before demonstrated. We perceive in succession the mo- tion of longitude and latitude ; the various annual equa- tions, motion of the apsides (in which, however, by omitting the consideration of the tangential force, he calculated the amount at one half its true value), the evection*, the alteration, and inclination ; the motion of the nodes. Even the doctrine of the tides, and thb precession of the equinoxes, are all handled clearly, though concisely, in this pro- position. The greater part of the Third Book is occupied with the application of these corollaries to the actual case of the moon, earth, and sun; and it is not any exaggeration to affirm that the great investigations which have been undertaken since the time of Sir Isaac Newton, and of which we have just been surveying the principal results, are an application of the improved calculus to continue the inquiries which he thus here began. The propositions respecting the masses of the attracting bodies which we considered before the corollaries to the * Laplace has erroneously stated that Newton overlooked the Evection ; but it forms, though not by name, the subject of the ninth corollary to this Sixty- sixth Proposition. NEWTON'S PRINCIPIA. 135 Sixty-sixth Proposition (although they come later in the Principia), and the latter of those corollaries, naturally lead to the subject of the next two sections, the one upon the attraction of spherical bodies, the other upon that of bodies not spherical. i. The attraction exerted by spherical surfaces and by hollow spheres is first considered. If P be a particle si- tuated anywhere within A B D C, and we conceive two lines A D, B, C, infinitely near each other drawn through P to the surface, and if these lines revolve round a P b, which passes from the middle points a and b, of the small arcs D C, and A B, through P, there will two opposite cones be described ; and the attraction of the small circles D C, A B upon P, will be in the lines from each point of those circles to P, of which lines C P, D P, are two from one circle, and A P, B P, two from the other circle. Now this attraction of the circle C D is to that of the circle A B, as the circle C D to the circle A B, or as C D 2 to A B (the diameters), 'and by similar triangles C D 2 : A B :: P C 2 : P A' 2 . But by hypothesis, the attraction of C D is to that of A B as A P 2 : P C 2 ; therefore the attraction of D C is to the opposite attraction of A B as A P 2 , to P C 2 , and also as P C 2 to AP 2 , or as AP 2 xPC 2 to AP 2 x P C 2 , and therefore those attractions are equal ; and 136 NEWTON'S PRINCIPIA. being opposite they destroy one another. In like manner, any particle of the spherical surface on one side of P, acting in the direction of a P, is equal as well as opposite to the attraction of another particle acting on the opposite side, and so the whole action of every one particle is de- stroyed by the opposite action of some other particle ; and P is not at all attracted by any part of the spherical sur- face ; or the sum of all the attractions upon P is equal to nothing. So of a hollow sphere ; for every such sphere may be considered as composed of innumerable concentric spherical surfaces, to each of which the foregoing reasoning applies ; and consequently to their sum. We may here stop to observe upon a remarkable in- ference which may be drawn from this theorem. Sup- pose that in the centre of any planet, as of the earth, there is a large vacant spherical space, or that the globe is a hollow sphere ; if any particle or mass of matter is at any moment of time in any point of this hollow sphere, it must, as far as the globe is concerned, remain for ever at rest there, and suffer no attraction from the globe itself. Then the force of any other heavenly body, as the moon, will attract it, and so will the force of the sun. Suppose these two bodies in opposition, it will be drawn to the side of the sun with a force equal to the dif- ference of their attractions, and this force will vary with the relative position (configuration) of the three bodies ; but from the greater attraction of the sun, the particle, or body, will always be on the side of the hollow globe next to the sun. Now the earth's attraction will exert no in- fluence over the internal body, even when in contact with the internal surface of the hollow sphere ; for the theorem which we have just demonstrated is quite general, and applies to particles wherever situated within the sphere. Therefore, although the earth moves round its axis, NEWTON'S PBINCIPIA. 137 the body will always continue moving so as to shift its place every instant and retain its position towards the sun. In like manner, if any quantity of movable particles, thrown off, for example, by the rotatory motion of the earth, are in the hollow, they will not be attracted by the earth, but only towards the sun, and will all accumulate towards the side of the hollow sphere next the sun. So of any fluid, whether water or melted matter in the hollow, provided it do not wholly fill up the space, the whole of it will be accumulated towards the sun. Suppose it only enough to fill half the hollow space ; it will all be ac- cumulated on one side, and that side the one next the sun; consequently the axis of rotation will be changed and will not pass through the centre, or even near it, and will constantly be altering its position. Hence we may be assured that there is no such hollow in the globe filled with melted matter, or any hollow at all, inasmuch as there could no hollow exist without such accumulations, in con- sequence of particles of the internal spherical surface being constantly thrown off by the rotatory motion of the earth. If A H K B be a spherical section (or great circle), PRK and PIL lines from the particle P, and infinitely near each other, SD, S E perpendiculars from the centre, and I q perpendicular to the diameter; then, by the similar triangles PIE,, P p D, we find that the curve surface bounded by I H, and formed by the revolution of IHKLI round the diameter AB, and which is 138 NEWTON'S PRINCIPIA. I P 2 proportional to I H x I q, is as p yx pg* and if tne attrac ' tion upon the particle P is as the surface directly, and the square of the distance inversely, or p~p> that attrac- tion will be as -=5 ^^. But if the force acting in the >xTb line P I is resolved into one acting in P S and another acting in S D, the force upon P will be as p-?, or (because of the similar triangles P I Q, P Sp) as pf . The attraction, therefore, of the infinitely small curvilinear surface formed P 1 by the revolution of I H is as p p- or as p ^ ; that is inversely as the square of the distance from the centre of the sphere. And the same may be shown of the sur- face formed by the revolution of KL, and so of every part of the spherical surface. Therefore the whole attraction of the spherical surface will bs in the same inverse du- plicate ratio. In like manner, because the attraction of a homogeneous sphere is the attraction of all its particles, and the mass of these is as the cube of the sphere's diameter D, if a particle be placed at a distance from it in any given ratio to the diameter, as in. D, and the attraction be inversely as the square of that distance, it will be directly as D 3 , and also as . 2 |y2 , and therefore will be in the sim- ple proportion of D, the diameter. Hence if two similar solids are composed of equally dense matter, and have an attraction inversely as the square of the distance, their at- traction on any particle similarly placed with respect to them will be as their diameters. Thus, also, a particle NEWTON'S PRINCIPIA. 139 placed within a hollow spheroid, or in a solid, produced by the revolution of an ellipsis, will not be attracted at all by the portion of the solid between it and the surface, but will be attracted towards the centre by a force proportioned to its distance from that centre. It follows from these propositions, first, that any par- ticle placed within a sphere or spheroid, not being affected by the portion of the sphere or spheroid beyond it, and being attracted by the rest of the sphere, or spheroid in the ratio of the diameter, the centripetal force within the solid is directly as the distance from the centre ; secondly, that a homogeneous sphere, being an infinite number of hollow spaces taken together, its attraction upon any particle placed without it is directly as the sphere, and inversely as the square of the distance ; thirdly, that spheres attract one another with forces proportional to their masses directly, and the squares of the distances from their centres in- versely ; fourthly, that the attraction is in every case as if the whole mass were placed in the central point ; fifthly, that though the spheres be not homogeneous, yet if the density of each varies so that it is the same at equal distances from the centre of each, the spheres will attract one another with forces inversely as the squares of the distances of their centres. The law of attraction, however, of the particles of the spheres being changed from the inverse duplicate ratio of the distances to the simple law of the distances directly, the attractions acting towards the centres will be as the distances, and whether the spheres are homogeneous or vary in density according to any law connecting the force with the distance from the centre, the attraction on a particle without will be the same as if the whole mass were placed in the centre ; and the attraction upon a particle within will be the same as if the whole of the body comprised within the spherical 140 NEWTON'S PKINCIPIA. surface in which the particle is situated were collected in the centre. From these theorems it follows, that where bodies move round a sphere and on the outside of its surface, what was formerly demonstrated of eccentric motion in conic sections, the focus being the centre of forces, applies to this case of the attraction being in the whole particles of the sphere; and where the bodies move within the spherical surface, what was demonstrated of concentric motion in those curves, or where the centre of the curve is that of the at- tracting forces, applies to the case of the sphere's centre being that of attraction. For in the former case the cen- tripetal force decreases as the square of the distance in- creases; and in the latter case that force increases as the distance increases. Thus it is to be observed, that in the two cases of attraction decreasing inversely as the squares of the central distance (the case of gravitation beyond the surface of bodies), and of attraction increasing directly with the central distance (the case of gravitation within the sur- face), the same law of attraction prevails with respect to the corpuscular action of the spheres as regulates the mutual action of those spheres and their motions in re- volution. But this identity of the law of attraction is con- fined to these two cases. Having thus laid down the law of attraction for these more remarkable cases, instead of going through others where the operation of attraction is far more complicated, Sir Isaac Newton gives a general method for determining the attraction whatever be the proportions between the force and the distance. This method is marked by all the geo- metrical elegance of the author's other solutions; and though it depends upon quadratures, it is not liable to the objections in practice which we before found to lie against a similar method applied to the finding of orbits and forces ; NEWTON'S PRINCIPIA. 141 for the results are easily enough obtained, and in con- venient forms. If A E B is the sphere whose attraction upon the point P it is required to determine, whatever be the proportion according to which that attraction varies with the distance, and only supposing equal particles of A E B to have equal attractive forces ; then from any point E describe the circle E F, and another e f infinitely near, and draw E D, e d ordinates to the diameter A B. The sphere is composed of small concentric hollow spheres E an( l if we call P E, or P F = r, E D = y, and D F = x, D d will be dx, and Er = - ; and the ring generated by the re- volution of r E is equal to r E x ED, or r E xy; therefore this ring is equal to rdx, or the attraction proportional to the whole ring E e will be proportional to the sum of all the rectangles P D x D d, or (a x)dx; that is, to the integral of this quantity, or to - ~ - ; which by the property of if the circle is equal to ^-. Therefore the attraction of the solid E e/F will be as y* x F /, if the force of a particle F/onPbe given; if not, it will be as y 2 x F/x/that force. Now d x : F / : : r : PS, and therefore F/ = and the attraction of E efF is as or taking y = r" (as any power of the distance P E), then the attraction of E e/F is as P S . r^y* dx. Take D N 142 NEWTON'S PRINCIPIA. (=w) equal to P S. r"- 1 y 2 , and let B D=z, and the curve B N A will be described, and the differential area N D dn will be ndz= (by construction) PS. r"' 1 y 1 dx; consequently u d z will be the attractive force of the differential solid E e ,/F; and fudz will be that of the whole body or sphere A E B, therefore the area ANB = /wJz is equal to the whole attraction of the sphere. Having reduced the solution to the quadrature of A N B, Sir Isaac Newton proceeds to show how that area may be found. He confines himself to geometrical methods; and the solution, although extremely elegant, is not by any means so short and compendious as the algebraical process gives. Let us first then find the equation to the curve A N B by referring it to the rec- tangular coordinates D N, AD. Calling these y and x respectively, and making P A = J, AS (the sphere's radius) = a and P S, or a + b, for conciseness, = -. Then D E 2 = 2 a x - PE= V (b + 2 ax-z NEWTON'S PRINCIPIA. 143 = V2 + 2 (a + V)x = Vb* + fx i and D N = ;/ = (by construction) - *-* -^ - i s the attractive force of the particles being supposed as the -th power of the dis- n_ tance, or inversely as (b 2 + f x} 2 . This equation to the curve makes it always of the order ^-^ . If then the force is inversely as the distance, A N B is a conic hyper- bola ; if inversely as the square, it is a curve of the fifth order; and if directly as the distance, it is a conic parabola ; if inversely as the cube, the curve is a cubic hyperbola. The area may next be determined. For this pur- i. r J /^ T , pose we have fy a x = I * ^ . .Let J 2(a+/*)~ r 2 (af + S 2 ) = h, this integral will be found to be 1 x (2 a , + C ; and the constant C is 4 (Q,-\-D) (2a + b} 3 b*- n h . \ _, . . _l_ v - i -- - - ft*-* I . This in every case gives 1 n o n / an easy and a finite expression, excepting the three cases of n 1, n 3, and n 5, in which cases it is to be found by logarithms, or by hyperbolic areas. To find the attraction of the whole sphere, when x = 2 a, we have x (2 " + br " ~ x 144 NEWTON'S PRINCIPIA. for the whole area A N B, or the whole attrac- 3 nj tion. If P is at the surface, or A P = b = 0, and n = 2, then the expression becomes as a, that is, as the distance from the centre directly. We may also perceive from the form of the expression, that if n is any number greater than 3, so that n 3 = m, the terms b z ~ n become inverted, and b is in their denominator thus : ,l 20+ /i )2 - Hence, if n > 3 and A P = b = 0, or the (1 n)b m particle is in contact with the sphere, the expression involves an infinite quantity, and becomes infinite. The construction of Sir Isaac Newton by hyperbolic areas leads to the same result for the case of n = 3, being one of those three where the above formula fails. At the origin of the abscissae we obtain, by that construction, an infinite area ; and this law of attraction, where the force decreases in any higher ratio than the square of the distance, is applicable to the contact of all bodies of whatever form, the addition of any other matter to the spherical bodies having manifestly no effect in lessening the attraction. By similar methods we find the attraction of any por- tion or segment of a sphere upon a particle placed in the centre, or upon a particle placed in any other part of the axis. Thus in the case of the particle being in the centre S, and the particles of the segment R B G attracting with forces as the - power of the distance S O or SI, the curve ANB will by its area express the attraction of the spherical segment, if D N or y be taken = -^pr- NEWTON'S PRINCIP!A. 145 ( x _ a \i c 2 = -7 - , S O being put = c, and A D == x, and (x a) A S = a, as before ; fy d x may be found as before by (x of d x c 2 - d x _, _ integrating - -^ - . The fluent is 2 c 3 - and the whole attraction of the segment upon the particle . 1 a 3 ~ n c 2 a 1-re 2 c 3 -* at the centre S is equal to - -- - - + -5 ; - - 3 n 1 n n 2 4 n + 3" Thus, if n = 2 the attraction is as - - - , or as OB directly, and as S B inversely ; and if c = 0, or the at- traction is taken at the centre, it is equal to a ; and if the attraction is as the distance, or n = 1, then the attractive force of the segment is - (a 2 c 2 ) 2 . ii. Our author proceeds now to the attractions of bodies not spherical ; an inquiry not perhaps, in its greater generality, of so much interest in the science of Physical Astronomy, where the masses which form the subjects of consideration are either spherical, or very nearly spherical, to which our examination has hitherto been confined. But this concluding part, nevertheless, contains some highly important truths available in astro- nomical science, because it leads, among other things, to determining the attraction of spheroids, the true figures of the planets. The attractions of two similar bodies upon two similar particles similarly situated with respect to them, if those attractions are as the same power of the distances -, are to one another as the masses directly, and the w th power of the distances inversely, or the w th power of the homo- L 146 NEWTON'S PRINCIPIA. logous sides of the bodies ; and because the masses are as the cubes of these sides, S and s, the attractions are as S 3 . s n : s 3 . S", or as s n ~ 3 : S"' 3 . Therefore, if n = 1, the attraction is as S 2 : s 2 ; if the proportion is that of the inverse square of the distance, the attraction is as S : s ; if that of the cube, the attraction is as 1 : 1, or equal ; if as the biquadrate, the attraction is as s : S ; and so on : and thus the law of the attractive force may be ascertained from finding the action of bodies upon particles similarly placed. Let us now consider the attraction of any body, of what form soever, attracting with force proportioned to the distance towards a particle situated beyond it. Any two of its particles A B attract P, with forces as A x A P and B x B P, and if G is their common centre of gravity, their joint attraction is as (A-f B) x G P, because B P, being resolved into B G and G P, and A P into A P and G P, and (by the property of the centre of gravity) GPxA=APxG, therefore the forces in the line A P destroy each other, and there remain only P G x B and P G x A to draw P, that is (A + B) x P G ; and the same may be shown of any other particles C and the NEWTON'S PRINCIPIA. 147 centre G' of gravity, of A, C, and B, the attraction of the three being (A + B + C) x G' P. Therefore the whole body, whatever be its form, attracts P in the line P S, S being the body's centre of gravity, and with a force proportional to the whole mass of the body multi- plied by the distance P S. But as the mutual attractions of spherical bodies, the attraction of whose particles is as their distance from one another, are as the distances between the centres of those bodies, the attraction of the whole body A B C is the same with that of a sphere of equal mass whose centre is in S, the body's centre of gravity. In like manner it may be demonstrated that the attraction of several bodies A, B, C, towards any particle P, is directed to their common centre of gravity S, and is equal to that of a sphere placed there, and of a mass equal to the sum of the whole bodies A, B, C ; and the at- tracted body will revolve in an ellipse with a force directed towards its centre as if all the attracting bodies were formed into one globe and placed in that centre. But if we would find the attraction of bodies whose particles act according to any power n of the distance, we must, to simplify the question, suppose these to be sym- metrical, that is, formed by the revolution of some plane upon its axis. Let A M C H G be the solid, M G the diameter of its extreme circle of revolution next to the particle P ; draw P M and p m to any part of the circle, and infinitely near each other, and take P D = P M, and P o = P m ; D d will be equal to o M (d n being infi- nitely near D N), and the ring formed by the revolution of M m round A B will be as the rectangle A M x M m, or (because of the triangles A P M, m o M, being similar, and D d = o M) P M x D d, or P D x D d. Let D N be taken y force with which any particle attracts at the distance P D = P M = x , that is as x n ; and if 148 NEWTON'S PRINCIPIA. by A P = b, the force of any particle of the ring .is as and the attraction of the ring, described by M m, is as b -- x D d x P D, or as b y d x, and the whole attrac- d b tion of the circle whose radius is A M, being the sum of all the rings, will be as bfy d x, or the area of the curve L N I, which is found by substituting for y its value in x, that is x n . This fluent or area is therefore = bfx n d x + C; and C Also, making P b ~ n+I n + 2 = P E in order to have the whole area of L N I, which measures the attraction of the whole circle whose radius is F A, we have (x being = P b = c) n+1 n+2 for NEWTON'S PEINCIPIA. 149 that attraction. Then taking D N' in the same proportion to the circle D E in which D N is to the circle A F, or as equal to the attraction of the circle D E, we have the curve R N T, whose area is equal to the attraction of the solid L H C F. To find an equation to this curve, then, and from thence to obtain its area, we must know the law by which D E increases, that is, the proportion of D E to A D ; in other words, the figure of the section A F E C B, whose revolution generates the solid. Thus if the given solid be a spheroid, we find that its attraction for P is to that of a sphere whose diameter is equal to the spheroid's shorter axis, as ^ ~ to d -f- A. ^~ u, -^ , A and a being the two semi-axes of the ellipsoid, d the distance of the particle attracted, and L a constant conic area which may be found in each case ; the force of attraction being supposed inversely as the squares of the distances. But if the particle is within the spheroid, the attraction is as the distance from the centre, according to what we have already seen. Laplace's general formula for the attraction of a spherical surface, or layer, on a particle situated (as any particle must be) in its axis, is- -ffdfxfdfF, in which f is the distance of the particle from the point where the ring cuts the sphere, r its distance from the centre of the sphere, or the distance of the ring from that centre, d u consequently the thickness of the ring, it the semicircle whose radius is unity, and F the function of f representing the attracting force. The whole attraction of the sphere, therefore, is the integral taken from f = r u to f L 3 150 NEWTON'S PRINCIPIA. 2 TT. udu = r 4 , and the expression becomes -- ffdf xfdf F with (r + n) (r-tt), substituted for /, when / results from this integration. Then let F =-^ or the attraction be that of gravitation; the expression 2 TT . u d u df 2 it . u d u p f becomes ~ J fdf x ~ = / 2 1 2 TT . u d u (r + u)(r u} _ > / = ~7~~ ~2~~ *2 It U d U o9j 1 j-r m - x u = 2 ir U? d u x. ; and the coem- r r cient of d r, taking the differential with r as the variable, is + - 2 - > consequently the attraction is inversely as the square of the distance of the particle from the centre of the sphere, and is the same as if the whole sphere were in the centre.* The First Book of the Principia concludes with some propositions respecting the motion of infinitely small bodies through media, which attract or repel them in their course, that is to say, of the rays of light, which, according to the Newtonian doctrine, are supposed to be bodies of this kind, hard and elastic, and moving with such rapidity as to pass through the distance of the sun from the earth, or 95 millions of miles, in seven or eight minutes, that is, with a velocity of above 211,000 miles in a second. Sir Isaac Newton shows that, if the medium through which they pass attracts or repels them from the perpendicular uniformly, they describe a parabola, according to Galileo's law of projectiles ; but if the attraction or repulsion be * Mec. Cel. liv. ii. ch. 2. The expression is here developed; but it coincides with the analysis in 11. NEWTON'S PRINCIPIA. 151 not equable, another curve will be described ; yet, that in either case the sine of the angle of incidence (or that made with the plane where they enter the medium), is to the sine of the angle of refraction (or that made with the plane they emerge from) in a given ratio ; that the velocities before incidence and after emerging are inversely as the sines of incidence and refraction ; and that if the velocity after incidence is retarded, and the line of incidence inclined more towards the plane of the refracting medium, the small bodies will be reflected back at an angle equal to that of incidence. He then remarks on the inflexion and deflexion which light suffers in passing, not through, but by or near bo- dies, as discovered by Grimaldi *, and as confirmed by his own experiments. He shows that the rays are bent most probably in curve lines, the nearest rays towards the bend- ing body, the furthest rays away from it ; and he infers that, in refraction and reflexion, a similar curvilinear bend- ing takes place somewhat before the actual point of re- fraction and reflexion. He further mentions the colours formed by flexion, as three coloured fringes or bands, "tres colorum fascias." I, however, long ago showed (Phil. Trans. 1797, Part II.) f that this is not the real fact; having found that a much greater number of these fringes are formed by flexion, and that they are, like the pris- matic spectrum, images of the luminous body. This ex- periment has been repeated by Sir David Brewster and others ; nor can any doubt be entertained that there are innumerable fringes decreasing in breadth, and in the breadth of the dark intervals between them, until they become evanescent. But as if it were the fate of all this * Grimaldi termed it diffraction. f In Phil. Trans., 1850, and Mem. Inst. de France, 1854, are my other papers on Inflexion, showing the same phenomenon, as well as the different flexibility of the rays. 152 NEWTON'S piuNciriA. great man's discoveries, that nothing should ever be added to them but by the use of means which he had himself furnished, it was only by applying a form of experiment which Sir Isaac Newton had used in examining the colours of thick and thin plates, that this important fact was ascertained, he not having subjected the phenomenon first observed by Grimaldi to that mode of investiga- tion.* The Fourteenth Section concludes with an elegant so- lution of a local problem in Descartes's Geometry, for finding that form of refracting glasses which will make the rays converge to a given focus, a problem, the de- monstration of which Descartes had not given. The brilliant discoveries made by Sir Isaac Newton upon the refrangibility and colours of light, not belonging to dy- namics, he pursues the subject no further in this place, having reserved the history of those inquiries for his other great work, the Opticsf, perhaps the only monument of human genius that merits a place by the side of the Principia. The truths which we have been contemplating respect- ing the attractions of bodies are fruitful in important consequences respecting the constitution of the universe. We have seen that the law of attraction which makes it de- crease as the squares of the distances increase, and the law which makes it increase as the distances decrease, are the * The Undulatory Theory of light, towards which philosophers have of late years appeared to lean, is no exception to this remark; for the princi- ples of that Theory may be found in the Eighth Section of the Second Book of the Principia, and the Scholium which concludes that Section seems to anticipate the application of its principles to Optical Science. f An abstract of these discoveries had been given in the Lectiones Opticaj at Cambridge seventeen years before the publication of the Prin- cipia in 1687. The Optics only appeared in 1704. NEWTON*S PKINCIPIA. 153 only laws which preserve the proportions between the force and the distance, the same for the attraction of the particles of bodies, and for the attraction of the masses in which those particles may be distributed the only laws which make the attraction of bodies the same with that of their mass placed in the centre of gravity. Now these two laws regulate the actions of bodies gravitating towards each other, the one being the law of gravitation beyond the surface of attracting bodies, the other, the law of gra- vitation between the surface and the centre. Thus, then, there is every reason to believe that this law pervades the material world universally, acting in precisely the same manner at the smallest and at the greatest distances, alike regulating the action of the smallest particles of matter, and the mightiest masses in which it exists. This action, too, is everywhere mutual ; it is always in direct proportion to the masses of the attracted and attracting bodies at equal distances; where the masses are equal, it is inversely as the squares of the distances beyond the bodies, and within the bodies, as the distances from the centre ; and where the masses and distances vary, it is as the masses divided by the squares of the distances in the one case, and as the masses multiplied by the dis- tances in the other. This law then pervades and governs the whole system. The discoveries which astronomers have made since the death of Newton, upon the more remote parts of the universe, by the help of improvements in optical instru- ments, have further illustrated the general prevalence of the law of gravitation. The double fixed stars, many of which had long been known to astronomers, and which were believed to retain at all times their relative posi- tions, have now been found to vary in their distances from each other, and to move with a velocity sometimes 154 NEWTON'S PRINCIPIA. accelerated, sometimes retarded, but apparently round one another, or rather round their common centres of gra- vity. A course of observations continued for above twenty years, led Herschel to this important conclusion about the year 1803 ; his son has greatly added to our knowledge of these motions ; and Professor Struve, of Dorpat, applying geometrical reasoning to the subject, cal- culated the orbits in which some of the bodies appear to move. One of the most remarkable is the star y Vir- ginia, on which Cassini had made observations in 1720.* It has now been found that one of the stars of which it is composed is smaller than the other ; that the revolv- ing motions of the two during the first 25 years had a mean annual velocity of 31' 23"; during the next 21 years, of 29' 11" ; during the next 17 years, of only 2' 42"; and during the last two years (1822, 23) of no less than 52' 51". The elder Herschel calculated the time of their whole revolution, the pe- riodic times of those distant suns, at 708 years ; it is now supposed not to exceed 629. Another pair of stars are found to revolve round one another in between 43 and 44 years, while a third pair take 12 centuries to accomplish their revolution. f Although our observations are far too scanty to lay as yet the ground of a system- atic theory of these motions, they appear to warrant us in assuming that the law of attraction which governs our solar system extends to those remote regions, and as their suns revolve round one another, each probably carrying about with it planets that form separate systems, we shall probably one day find that equal areas are there as here described in equal times, and that the orbits are ellip- tical ; or, which would come to the same thing, that the * Mem. Acad. des Sciences, 1720. t Phil. Trans. 1803, p. 339; ih. 1824, Part IIL NEWTON'S PRINCIPIA. 155 sesquiplicate proportion of the periodic times and mean distances is observed *, from whence the conclusion would of necessity follow, that the centripetal force followed the rule of the inverse square of the distance, and that gravitation such as we know it in our part of the uni- verse, likewise prevails in these barely visible regions. Thus additional confirmation accrues to the first great deduction drawn from the theorems respecting attraction in the Principia. But other interesting corollaries are also to be deduced from these propositions. They enable us to ascertain, for example, the attractions, the masses, and the figures of the heavenly bodies. Sir Isaac Newton boldly and happily applied them to determine these important par- ticulars, apparently so far removed beyond the reach of the human faculties. 1. The weights of bodies at the surface of the different planets were thus easily determined. The law by which the attractive force of spherical bodies decreases as the square of the distance increases, whether those bodies be homogeneous or not, provided their densities vary in the same proportion, and the other law regulating the pro- portion between the periodic times and the distances of the planets, enabled him to compute the attraction of each planet, for equal bodies at given distances from their centres, by comparing the observed distances and periodic times of each ; and he was thus also enabled, by knowing their diameters, to ascertain the weights of bodies at their surfaces. He found in this manner, that the same body which at the surface of the Earth weighs 435 pounds, at * It may even seem that already the observed axes of those remote orbits, when compared with their periodic times, approach the sesquiplicate ratio. Thus one has its axis 7"'9, and time 58 years ; and another its axis 30"-8, and time 452 years. 156 NEWTON'S PRINCIPIA. that of the Sun weighs 10,000, at that of Jupiter 943, and at that of Saturn 549. 2. So too the masses of matter in each planet and in the satellites may be ascertained. The motions of the satellites of Jupiter and Saturn afford the easiest means of determining the masses of those planets ; and the motions of the other planets round the Sun enable us to solve the problem, though not so accurately, as to them. The mass of Jupiter compared with that of the Earth may be easily supposed to be prodigious, when we find all his satellites revolve round him so much more rapidly than the Moon does round the Earth, although all of them but one have much larger orbits. Thus the second satellite revolves in a seventh of our lunar month, though its path is half as long again : and hence, its velocity is between 10 and 11 times as great. Sir Isaac Newton ascertained the masses of Jupiter, Saturn, and the Earth to be to that of the Sun as T ^ 7 , 3^, T^FZ* to 1 respectively. In like manner the densities are found, being as the weights (first found) divided by the axes. Thus he determined the relative densities of Jupiter, Saturn, and the Earth to be as 94 , 67, and 400, to 100, the density of the Sun. Laplace has ascertained the masses of the heavenly bodies by an entirely different calculus, founded upon the comparison of numerous observations with the formulae for determining the disturbances. The result is extremely remarkable in one particular. It agrees to a fraction, as regards Jupiter, with the calculation of Newton, making the mass of the planet yoVr* -^ u * * ne observations of Pound respecting Saturn's axis, on which Newton had estimated Saturn's mass, were subject to considerable uncertainty; so at least Laplace explains the difference of his own results; but he admits* that * Mec. Cel. liv. vii. ch. 16, s. 44. NEWTON'S PRINCIPIA. 157 even in his day there prevailed considerable uncertainty respecting this planet's mass, while that of Jupiter, being well ascertained, agrees perfectly with Sir Isaac Newton's deduction. Laplace gives the masses of the four great planets thus, that of the Sun being unity : Venus ^j^Wa 5 Mars ^j^^aoJ Jupiter 106 V 09 (differing by ^ only from Newton's, who indeed did not insert decimals at all) ; and Saturn A_* The Moon's mass he makes T - - 3 5 3 4*0 8 o 8*5 / that of the Earth being unity, while the greatest of Jupiter's satellites is only 0,0000884972, Jupiter being unity. This great geometrician's observations upon Sa- turn's ring are peculiarly worthy of attention. The ex- treme lightness of the matter of which the planet consists, has already been shown ; it is six times lighter than the mean density of the Earth; or, if the mean specific gravity of the latter be taken as 5 f, that of water being as 1, the matter of which Saturn is composed must be only 3^ times heavier than cork, and lighter than India rubber. But Laplace has satisfactorily shown that his rings must be composed of a fluid, and that no other con- struction can account for their permanence.^ 3. Sir Isaac Newton, lastly, by the principles which we have been explaining in the latter part of our Analysis, investigated the figures of the heavenly bodies. Thus he especially examined that of the Earth. This planet, in revolving round its axis, gives those particles the greatest tendency to fly off which move with the greatest velocity, that is, those which are furthest from their centres of ro- tation ; in other words, those which are nearest the equa- tor ; while those near the poles, describing much smaller circles, move much slower and have far less tendency to * Mec. Cel. liv. x. ch. 8, 9 ; correcting liv. vi. ch. 6. f The mean of Maskelyne and Cavendish's experiments. t Mec. CeL liv. iii. ch. 6. 158 NEWTON'S PRINCIPIA. fly off. Hence there is an accumulation of matter towards the equator, which is raised, while the poles are depressed and flattened, and the equatorial axis is longer than the polar. By comparing the space through which heavy bodies fall in a second in our latitudes with the centrifugal force at the equator, he found that the gravity of bodies there is diminished ^-g at least, or that the equatorial axis is, at least, ^^ longer than the polar. But he con- sidered this estimate as below the truth, because it does not make allowance for the effect produced on gravitation by the increase of the distance at the equator from the centre. Accordingly, by a skilful application of the method of false position, he corrected this calculation, and ultimately brought out the proportion to be that of 229 to 230, making the equatorial axis about 34J- miles longer than the polar, the whole axis being about 7870 miles. He also estimated the two axes of Jupiter to be as 11^ to 10, supposing the density of the body to be the same throughout ; but if it is greater towards the equator, our author observed that the difference between the axes might be decreased as low as 13 to 12, or even 14 to 13 ; which agreed well enough with Cassini's observations in those days, and still more nearly with Pound's. But more accurate observation has since shown that the dif- ference is considerably less, the disproportion being not more than that of 1074 to 1000 ; so that the planet must be very far from homogeneous and its equatorial density greatly exceed its polar. Thus, too, accurate measure- ments of a degree of latitude in the equatorial and polar regions, and experiments on the force of gravity, as tested by the length of the pendulum vibrating seconds in those different parts of the globe, have led to a similar inference respecting the Earth, its axis being now ascertained to bear the relation, not of 230 to 229, as NEWTON'S PRINCIPIA. 159 Newton at last concluded, nor even that of 289 to 288, according to his first approximation, but only that of 336 to 335 *, being an excess of little more than 23| miles. The calculation of Newton was formed on the supposition of the Earth being homogeneous; and it is worthy of remark, that although the later observations, by proving the flattening at the poles to be less than he, on this hypothesis, assigned it, have shown the Earth not to be homogeneous, no correction or improvement whatever has been made on his theory in this respect. We find Laplace, on the contrary, in the very passage to which we are now referring, assuming his precise fraction ^^ as the one given by the theory upon the supposition of the globe being homogeneous, and reasoning upon that fraction, t Now it is fit that we here pause to contemplate perhaps the most wonderful thing in the whole of the Newtonian discoveries. The subject of curvilinear motion, or mo- tion produced by centripetal forces, was certainly in a great measure new, and Sir Isaac Newton's treatment of it was in the highest degree original and successful. But the laws of attraction, the principles which govern the mutual actions of the planets, and generally of the masses of matter, on each other, was still more eminently a field not merely unexplored, but the very existence of which was unknown. Not only did he first discover 4his field, not only did he invent the calculus by means of which alone it could be explored, and without which hardly a step could be made across any portion of it (for the utmost resources of geometrical skill in the hands of the Simsons and the Stewarts themselves, who in other inquiries had performed such wonders by ancient ana- * Moc. Col. liv. iii. ch. 5. f Ibid - liv - & ch - 5 > s - 4L 160 NEWTON'S PRINCIPIA. lysis, would have failed to do anything here), but the great discoverer actually completed the most difficult in- vestigation of this new region, and reached to its most inaccessible heights, with a clearness so absolute, and a certainty so unerring, that all the subsequent researches of his followers, and all their vast improvements on his calculus, have not enabled them to correct by the fraction of a cipher his first results. The Ninetieth and Ninety- first Propositions of the First Book, containing the most refined principles of his method, are applied by him in the Nineteenth of the Third Book to the problem of the Earth's figure; his determination of the ellipticity, sup- posing the mass homogeneous, is obtained from that appli- cation. A century of study, of improvement, of dis- covery has passed away ; and we find Laplace, master of all the new resources of the calculus, and occupying the heights to which the labours of Euler, Clairaut, D'Alem- bert, and Lagrange have enabled us to ascend, adopting the Newtonian fraction of ^^, as the accurate solution of this speculative problem. New admeasurements have been undertaken upon a vast scale, patronized by the muni- ficence of rival governments ; new experiments have been performed with improved apparatus of exquisite delicacy ; new observations have been accumulated, with glasses far exceeding any powers possessed by the resources of optics in the days of him to whom the science of optics, as well as dynamics, owes its origin ; the theory and the fact have thus been compared and reconciled together in more perfect harmony ; but that theory has remained un- improved, and the great principle of gravitation, with its most sublime results, now stands in the attitude, and of the dimensions, and with the symmetry, which both the law and its application received at once from the mighty hand of its immortal author. NEWTON'S PKINCIPIA. ]61 NOTE. The argument in page 136. is succinctly and po- pularly stated respecting the supposition of a hollow in the centre of the Earth, and several steps are omitted. One of these may be here mentioned in case it should appear to have been overlooked. Suppose a mass m de- tached from the hollow sphere M, and impelled at the same time with that sphere by an initial projectile force, then its tendency would be to describe an elliptic orbit round the sun, the centre of forces, and if it were detached from the earth it would describe an ellipse, and be a small planet. But as the accelerating force acting upon it would be different from that acting on the earth, the one being S + M . , , S + m , . , ,. as pp , and the other as ^ (D being the dis- tance and S the mass of the sun), it is manifest that, sooner or later, its motion being slower than that of the hollow sphere, if m be placed in the inside, it must come in con- tact with the interior circumference of the sphere, and either librate, or, if fluid, coincide with it, as assumed in the text. Where parts of the spherical shell come off by the centrifugal force, of course no such step in the reasoning is wanted; nor is it necessary to add that neither those parts nor any other within the hollow shell can have any rotatory motion. 162 NEWTON'S pRiNCiriA. II. HITHERTO we have considered all motion as performed in vacua, or in a medium which offers no resistance to the action of forces upon bodies moving in any direction. It was necessary that the subject should first be discussed upon this supposition ; and the hypothesis agrees with the fact as far as the motions of the heavenly bodies are concerned. But all the motion of which we have any experience upon or near the surface of the earth, is per- formed in the atmosphere that surrounds our globe ; and therefore, as regards all such motion, a material allow- ance must be made for the resistance of the air when we apply to practice our deductions from the theory. It is also obvious that a still greater effect will be produced upon moving bodies, if their motion is performed in a denser fluid, as water. Further, the pressure and motion of fluids themselves form important subjects of considera- tion, independent of any motion of bodies through them and impeded by them. These several matters form the subject of the sciences of Hydrostatics, Hydraulics, and Pneumatics; the first treating of the weight and pressure of watery fluids, the second of their motion, the third of aeriform or elastic fluids. They are discussed in the Second Book of the Principia. It consists of Nine Sec- tions ; of which the First Three treat of the motion of bodies to which there is a resistance in different propor- tions to the velocity of the motion ; the Fourth treats of circular or rather spiral motion in resisting media; the Sixth, of the motion and resistance of pendulums; and part of the Seventh discusses the motion of projectiles ; while the rest of the Seventh, and the whole of the four NEWTON'S PRINCIPIA. 163 remaining sections, treat of the pressure and motion of fluids themselves and propagated in pulses, or otherwise, through fluids. "We shall arrange the subjects under these Five heads, instead of following the precise order of the work itself.* Two observations are applicable to this branch of the subject, and to the treatment of it in the Principia ; and these observations lead to our distinguishing this portion of that great work from the rest. First. Much more had been accomplished of discovery respecting the dynamics of fluids before the time of Sir Isaac Newton, in proportion to the whole body of the science, than in the other branches of Mechanics. The Newtonian discoveries, therefore, effected a less consider- able change upon this department of Physics than upon Physical Astronomy and the general laws of motion. As early as the time of Archimedes the fundamental principle of the general or undequaque pressure of fluids had been ascertained ; many of the easier problems, and even some of the more complicated, had been investigated by its aid. When dynamical science was newly constructed by the illustrious Galileo, the progress which he made may almost be said to have formed Hydrostatics and Hydraulics into a system ; and Pascal's original and inventive genius, soon afterwards applied to it, enabled him clearly to perceive the hydrostatic paradox, and even led him to a plain an- ticipation of the hydrostatic press.f Torricelli about the same period reduced the atmosphere under the power of weight and measure, making it the subject of calculation by the beautiful experiment which first ascertained its gravity, * For the arrangement, see the Summary of Contents, t He calls a box of water " a new mechanical principle by which we may multiply force ad libitum." (Equil. of Fluids, 1653.) M 2 164 NEWTON'S PRINCIPIA. which had long been suspected but not proved. Pascal first extended the Torricellian experiment to all the perfec- tion, indeed, which it has ever attained, by showing the con- nexion between the height of places on the earth's surface, and that of the mercurial column; thus demonstrating satisfactorily the pressure of the atmospherical column. Torricelli had also, from experiments on the spouting of water, inferred that the velocity of the spouting column, or jet, is as the square root of the height of the reservoir of fluid whose pressure causes the flow. So that the fun- damental principles being ascertained, considerable progress was also made in their systematic application, when Sir Isaac Newton came to treat the subject as a branch of his general dynamical theory, and to investigate the laws of fluids by means of those profound principles which he had established with respect to all motion. Thus more was done before his time, and less consequently left for him to do here, than in the other branches of the general subject. Secondly. It is also true that the work which he pro- duced upon this branch of science, did not attain the same perfection under his hands, as the rest of the Principia, Although he treated it upon mathematical principles, he left considerably more to be done by his successors than he left to be added by those who should follow him in the field of Physical Astronomy. A great step was almost immediately made by J. Bernouilli, in ascertaining the effects of the air's resistance upon the motion of projectiles; and an error so apparent was pointed out in one of the Propositions in the Principia (Book II. Prop. 37*), that the correction coming to the author's knowledge, he struck it out of the second edition, then in the press. His ori- ginal solution of the problem as to spouting columns, * First Edition, published in 1687. NEWTON'S PRINCIPIA. 165 having differed from the rule which Torricelli had deduced experimentally, Newton again investigated the question by a different and an admirable process ; but even now the subject remains in a very unsatisfactory state. Nor can it be said that the science of hydrodynamics generally has attained the perfection of the other branches of Mechanical philosophy; while it is certain that the application to it of the calculus by Euler and D'Alembert*, and still more by Clairaut, has greatly added to the theorems left by Sir Isaac Newton; and the researches of Laplace upon ca- pillary attraction form a department of science almost unknown before the latter part of the eighteenth century. The statement of these particulars was necessary in order to place the relative merits of the different branches of the Principia in their true light. That a great improve- ment was accomplished in natural knowledge by this por- tion of Sir Isaac Newton's discoveries, none can doubt. That the Second Book displays at every step the profound sagacity and matchless skill of its author, is undeniable. That it would have conferred lasting renown upon any one but himself, had it been the only work of another man, is certain. Nor can we forget that in rating its importance as we have ventured to do, we only undervalue this portion of the Principia, by applying to it the severest of stand- ards, comparing it with the discovery of the laws which govern the system of the universe, and placing it in con- trast with the other parts of that unrivalled effort of human genius. * Their invention of the Calculus of Partial Differences was connected with this subject. (See Life of D'Alembert.) ML 3 ANALYTICAL VIEW. PRINCIPIA. BOOK SECOND. CHAPTER I. THE ELEMENTARY PRINCIPLES OF HYDROSTATICS, AND THE LAWS OF DENSITY OF AN ELASTIC FLUID COLLECTED ROUND A CENTRE OF FORCE. I. Elementary Principles of Hydrostatics. 1. What a fluid is, the terms viscosity, solidity, &c. 2. What the foundation is on which the theory of Hydrostatics is built. Newton, xix. 3. The fundamental equation by which we know the properties of a fluid in equilibrium. Note I. 4. Three consequences of this equation. Note I. (1.) That there must in all cases be a certain relation among the forces. (2.) Level surfaces are surfaces of equal density. (3.) Level surfaces are surfaces of equal temperature. 5. Newton, xx., Fliuds under the action of gravity only. II. The Law of Density in a compressible Fluid under the Action of a central Force. 1. Solution of the question defect in the application to physical ques- tions, Newton, xxi. xxn. &c. 2. First application. Measurements of heights. Note II. Second application. Form of our atmosphere, the Zodiacal light not part of the Sun's atmosphere. Note II. I. 1. " HYDROSTATICS" is that part of statics which treats of the equilibrium of fluids. A fluid is any body whose parts M 4 168 NEWTON'S PRINCIPIA. yield to any force impressed on it } and by yielding are easily moved among themselves. This is Newton's definition. It includes gases and aeriform bodies, as well as those to which we, in ordinary conversation, apply the terms "fluid" or " liquid." The fundamental idea of a fluid is, that of a body whose particles may be moved amongst each other on the appli- cation of the slightest possible force. It is therefore directly opposed to a rigid body, whose definition is that its particles cannot be moved amongst each other, no matter how great a force is applied. It is evident that no substance that we meet with in nature is strictly either a fluid or a rigid body ; but they approach more or less to the one or the other. When they partake more of the fluid than the rigid nature, they are called " viscous ;" when the contrary, they are called " solid." These two are therefore indefinite terms, and no clear boundary can be drawn between them. 2. The science of Hydrostatics is divided into two parts. In one we assume certain general principles as the grounds of all our reasoning. We may consider these as established either by experiment, or as truths which it is the office of the other part of the science to demonstrate. In the other we make certain general assumptions as to the constitution of a fluid, and then we attempt to deduce from these the general principles on which all the rest of hydrostatics is founded. This division occurs in most mechanical sciences. Thus, in Geometrical Optics, we assume the laws of reflection and refraction ; it is the part of Physical Optics to establish their truth. It is not here our office to enter into the science of Molecular Hydrostatics; we must post- pone, therefore, such consideration to a future chapter. The mathematical theory of Hydrostatics is founded upon two laws. NEWTON'S FRINCIPIA. 169 1. The pressure of the fluid upon any element of a surface exposed to it is normal to that surface. 2. Any pressure communicated to a fluid mass in equilibrium is equally transmitted through the whole fluid in every direction. Consider any point in a fluid, and let an indefinitely small plane pass through it ; by the second law the pres- sure is the same, whatever be the inclination of the plane to the horizon ; by the first it is normal, and proportional to the area of the plane. . Let this area be ; then the pressure may be represented by p . This quantity p is therefore what we seek to find. It is what the pressure would be if the area were unity, and the pressure constant over that area. It is therefore called the " pressure referred to a unit of area." The law expressing the equality of pressures in all direc- tions is true in viscous as well as perfect fluids. The dif- ference is this, that in the latter the transmission of the pressure is effected in a moment, in the former it takes time. During this interval the law is not true ; but when a short time has been allowed to pass, the fluid takes up its form of equilibrium, and the pressure becomes equal in all directions. These two laws are not independent. The first con- tains the second. For, let the fluid contained within the pyramid O AB C in the interior c of the fluid become solid. This is allowable, for, the fluid being in equilibrium, the pressure on the solidified element will be borne and resisted in exactly the same B way that it was while still fluid. Let p be the pressure referred to a unit of area on the plane C O A at O, q that on a plane parallel to B C A through O. Let the 170 NEWTON'S PRINCIPIA. pyramid diminish without limit, the pressures on the two sides C O A and B C A will be normal and respectively equal to p x area C O A, g x area B C A. Kesolving these, parallel to O B, we have p area C O A = area C O A q area B C A ~~ area BOA' because this latter ratio expresses the cosine of the incli- nation of the two planes ; hence P = ? Similar equations hold by symmetry for the other sides. And therefore the pressure is equal in all directions. This includes Prop. xix. of Section V. Note I. 5. Newton proceeds to consider the equilibrium of a sphe- rical mass of fluid, like our atmosphere, resting upon a spherical concentric bottom, and gravitating towards the centre of the whole. The object is to determine the pres- sure on any point A of the bottom. Divide the fluid into concentric orbs of equal thickness d x. Now any part of a fluid at rest may be supposed to become rigid ; for it will then resist and be resisted by the remainder of the fluid in exactly the same manner as before. Draw, then, any cylindrical canal from the point A to any point B in the surface of the fluid, and suppose its superficies to become rigid. This canal will be divided into elements by the concentric orbs. Let d s be the length of any one of these elements, and F the force of gravity ; then the weight of that element is Fds. This acts directly to- wards the centre, that is, along dx. Eesolving along the canal, the force with which this element tends to press the bottom of the canal is Fdx. The same is true for NEWTON'S PKINCIPIA. 171 all the elements ; hence the whole pressure on the bottom of the canal is Cpdx, that is to say, it is equal to the weight of a cylinder of fluid, whose base is the area of the part A of the bottom, and whose altitude is the same as that of the superincum- bent fluid. The bottom is not pressed by the whole iceight of the incumbent fluid, but only that part which is described above ; and it will be the same whether the fluid rises per- pendicularly above A in a rectilinear direction, or whether it be contained in crooked cavities and canals, whether these passages be regular or irregular, wide or narrow. If a body of the same specific gravity as the fluid, and incapable of condensation, be immersed in the fluid, it will neither acquire motion by the pressure of the fluid, nor any change of figure. Any portion of a fluid at rest may clearly be supposed to become solidified without affecting the equilibrium. Let a part of the fluid equal and similar to the body about to be immersed become solid ; removing it we may replace it by this body, and the equilibrium will still subsist. It also follows that the resultant of all the pressures exerted by the fluid on the solid is a force equal to the weight of the fluid displaced acting upwards through the centre of gravity of the volume of the body. If, therefore, a solid be immersed in a liquid, it will remain at rest if it be of the same density as the fluid. But if it be of greater density, it will be no longer sustained by the resultant pressures, and will sink to the bottom. If it be of less density, it will rise to the surface, being acted on upwards by a greater force than its own weight. Hence, Newton concludes, bodies placed in fluids have a twofold gravity ; one true and absolute, the other apparent, vulgar, 172 NEWTON'S PRINCIPIA. and comparative. Absolute gravity is the whole force with which the body tends downwards. Relative gravity is the excess of gravity with which the body tends down- wards more than the ambient fluid. The bodies, therefore, which we call light, and which appear to fall so slowly, or even seem to rise in the air, are light only in comparison with the air. If there were no air, their apparent gravity would be their real gravity, and all bodies are found to fall when placed in a vacuum. II. 1. Having discussed some of the fundamental proper- ties of fluids, and obtained the equations of equilibrium, we can proceed to apply them to some of the great problems that Nature presents us with. The first case which Newton considers is the law of density in a compressible fluid which is attracted according to any law by a force tending towards a given centre. He does not consider this pro- blem in its most general form, nor would there be any advantage in doing so. The only forces which present any interest are those which vary according to some power of the distance. Let us assume that the attraction upon any particle whose mass is a unit, and distance from the centre x, is, JL x n where p is some constant quantity, of n -f 1 dimensions. The fluid will manifestly arrange itself symmetrically round the centre of force. We may therefore consider only those particles that lie in the axis of x. Take therefore a small rectangular element at a distance x from the centre of the earth, and whose sides are dx dy dz. This element must be at rest under the action of the fluid pressures on its sides and its own gravity. If p be the pressure referred to a unit of area at this point, these two pressures will be clearly NEWTON'S PEINCIPIA. 173 p dy dz and (p + -r- dx) dy dz acting along the axis of x. And the weight will be -p.^-.dxdy dz; and since there is equilibrium the sum of these must be zero. The equation of fluid equilibrium is then <*P= -?** --- (I)' To solve the problem we require the relation between p and p. In fluids generally we have p = xp - - --- (2). This is the law which Newton takes for granted in the two cases which he has worked at length. He also states the results that would be arrived at if we had assumed other laws ; and, as we shall see, Laplace has been led to believe that the above is far from being true within the earth. Substituting from the second equation the value of p in the first, Hence dividing by f> and integrating, where C is some unknown constant. Hence jU, 1 where D is the density at the centre or at an infinite dis- tance, according as n is less or greater than unity, and can only be determined by some of the given conditions of the fluid. Generally, we conclude from the above, that when the 174 NEWTON'S PRINCIPIA. reciprocals of the (w I)* powers of the distances are in arithmetical progression, the densities at those points will be in geometrical progression. Two cases of the above are worthy of notice, when n = 2 and when n = 0. In the former the force attracts inversely as the square of the distance, and the density at any point is given by p = D . e" 1 "*'*' that is, if the distances be in harmonical progression, the densities will be in geometrical progression. In the latter case the force is constant and equal to ju,, and the density is given by that is, if the distances decrease in arithmetical progression the densities will decrease in geometrical progression. These cases we might suppose to bear some analogy to the state of our atmosphere, the former holding when the changes of elevation are great, the latter when they are small. There is one case, especially considered by Newton, in which the preceding general formula fails, viz. when n=l, for then log p in equation (3) appears to be always infinite; but this is not really the case, for C is also infinite and negative. The form of the integral has changed, and by merely repeating the process, we get X log p = C ]U, log X The preceding investigations are not, however, of any very great practical utility. They are all founded on the supposition that the compression varies as the density. NEWTON'S PRINCIPIA. 175 Now this is only true when the temperature is constant. When it is not, we have seen that the true law is that p = * p (1 + * f> If, however, we attempt to use this equation, we require to know the law according to which t, the temperature, varies as we ascend into the air and descend into the earth. We can have but little assistance in determining this from observation. As Humboldt* has remarked, our experimental knowledge of the interior of the earth is limited in the extreme. The greatest depth below the surface of the sea that has yet been obtained, is probably that of the salt-works of New-Salzwerk, near Minden, in Prussia; yet this was only 1993 feet, or less than sirtjo o P art f earth's radius. The observations even on these small depths are liable to serious errors, as the different periodic variations of temperature caused by the diurnal or annual heating of the surface, the greater exposure to the surface air, &c. The temperature of water at the bottom of the salt mine was 90*8 Fahren- heit, giving a mean decrease of 1 Fahrenheit for every 5 3 - 8 feet. If we tried to make our observations on the law of density instead of that of temperature, for the knowledge of either would enable us to integrate the equations, we can succeed no better. The dippings of strata beneath the surface, which rise again at known distances, only reach some twelve thousand feet below the surface of the sea ; and if to this we added the height of the highest mountain, we have only a knowledge of 5-i^th part of earth's radius. We have also observations made on the temperature of the air at the summits of mountains, and in balloon * Kosmos, i. 150. 176 NEWTON'S PRINCIPIA. ascents. The former will not furnish us with the re- quired law, because the presence of the mountain will affect the temperature of the air by its radiation of the solar rays. Gay Lussac, in his celebrated aerostatic ascent of 3816*12 fathoms, found the temperature at the upper station 14*9, giving a depression of 1 for every 95*14 fathoms. A great variety of observations have been made, and many empirical laws invented to suit them. To mention only one : Mr. Atkinson, in the second volume of the Transactions of the Astronomical Society, asserts, that at an altitude of h feet the depression in temperature will be given by nearly. We might make use of these results, and by repeating our calculation in the manner indicated, obtain various formulas to determine the density at any point. But such results can never be very trustworthy. II. 2. Note II. 177 CHAPTER II. THE FIGURE OF THE EARTH. 1. Newton's calculation of the ellipticity of the earth its defects. 2. An accurate investigation of the ellipticity on the supposition that the earth is homogeneous the form thus found proved to be stable. Note III. 3. Newton's calculation of the law of variation of gravity. 4. Newton's application of his theory to the planet Jupiter. 5. The figure of the earth considered as heterogeneous, Clairaut and Laplace's results. Note IV. a. The form of the strata. & The law of variation of gravity. 6. The law of density in the interior of the earth. Note IV. 7. Whether the interior of the earth is solid or fluid. Note IV. 8. Measures to determine by observation the ellipticity of the earth's surface. Note IV. a. Measurement of degrees. y3. Observations on the pendulum. y. Astronomical observations. 1. IN the eighteenth proposition of the third book, New- ton considers why the earth and planets are protuberant at their equator. He does not investigate the form of the earth, but merely shows that if it had been originally fluid, the matter, by its ascent towards the equator, would enlarge the diameters there, and by its descent towards the poles it will shorten the axis. And even if the earth had not been originally fluid, yet if the earth were not higher at the equator than at the poles, the seas would subside about the poles, and rising towards the equator, would lay all things there under water. Taking for granted that the true form of the earth is 178 NEWTON'S PRINCIPIA. a spheroid, Newton proceeded to calculate its ellipticity. This he does nearly as follows : (1.) From Picart's andCassini's measures of a degree, he finds, supposing the earth spherical, that its radius must be 19,615,800 Paris feet. From some observations on falling bodies at Paris, he calculates that the force of gravity at that place is such, that a body will fall 2174 lines in the first second of its descent. Knowing the earth's radius, and its time of rotation, it is easy to cal- culate the centrifugal force at the equator ; viz., such that under its action, a body would describe 7.54064 lines in the first second. Since the resolved part of the centri- fugal force perpendicular to the earth varies as the square of cosine of the latitude we can calculate the centrifugal force at Paris, and then adding it to the force of gravity, calculate as above, we find the whole undiniinished force of gravity at that place to be such, that a body would describe 2177.267 lines in the first second of its descent. The undiminished force of gravity at the equator will differ from this by a very small quantity ; hence rejecting small quantities of the second order, the ratio of centrifugal force at the equator to equatorial gravity is as 1 to 289. This ratio is still in use. (2.) If we took a spheroid, whose axes are as 101 to 100, by a simple application of Prop. XCI. Book L, Newton shows that the force of gravity at the pole is to that at the equator as 501 to 500. Take now two canals, from the surface to the centre; let one meet the surface at the pole, the other at the circumference. That there may be equilibrium the weights of these two canals must be equal. Conceive these divided by transverse parallel equidistant surfaces into parts proportional to the wholes ; the weights of any number of parts in the one leg will be to the weights of the same number of parts in the NEWTON'S PBINCIPIA. 179 other as their magnitude and the accelerative forces of their gravity conjunctly ; that is, as 101 to 100, and 500 to 501, or as 505 : 501. The difference, viz., four parts, must be supported by the centrifugal force. Hence the ratio of the centrifugal force bears to gravity the ratio 4 : 505. (3.) Newton now brings in the rule of proportion. If a centrifugal force j^j cause a difference of elevation of the two legs T ^, what difference will a centrifugal force g|-g make ? The calculation gives a result -%?-, or the diameter of the earth at the equator is to its diameter at the pole as 230 to 229. The ratio of the difference of these diameters to the equatorial diameter, is called the ellipticity of the planet. This investigation of Newton is manifestly altogether defective. He assumes not only that the spheroid is a form of equilibrium, but that the ellipticity is always proportional to the ratio of the centrifugal force to gravity. These two assertions are indeed true, but they are not self-evident. It was Maclaurin who first demonstrated their truth. It is very remarkable in how wonderful a manner Newton often arrives at correct results by means the most inadequate. Of this there are many other instances besides the present one. He guessed the mean density of the earth he determined by analogy that the velocity of waves varied as the square root of their length. Another analogy led him to a curious result in regard to the tides. 2. NOTE III. 3. Newton remarks that the force of gravity will not be- the same at all points of the earth. For draw any radius Q P = r from the centre to any point P in the circum- N 2 180 NEWTON'S PRINCIPIA. ference. Then the earth, being considered homogeneous, the attraction of the spheroid on any point Z in OP, resolved along this radius, will be proportional to its distance O Z from the centre. The same is true for the centrifugal force. Hence altogether the resolved part of gravity at any point Z in O P resolved along the radius varies as O Z. Let it be represented by co z, where z= OZ, and co may vary with the position of P. The whole weight of the column is therefore and this must be the same for all radii ; hence o> varies inversely as r 2 , and therefore the attraction varies as 2= -, when Z lies on the surface. Hence the force of gravity towards the centre must vary on the sur- face of the same planet reciprocally as the distance of the bodies from the centre of the earth. As the earth is very nearly spherical, this must be also very nearly true for the whole force of gravity. Let G then be the force of gravity at the equator, g that at the point P, whose latitude is X and radius r, then = G { 1 +e sin 2 A} nearly. We shall show in another chapter that the force of gravity is proportional to the length of the seconds' pen- dulum ; hence if I and L be the lengths in the latitude X and at the equator, 7=L{1 +esin 2 A}. The value of I was known in the latitude of Paris, whence that of L can be found, and thence the length in NEWTON'S PRINCIPLE. 181 any latitude. Newton now refers to a number of obser- vations on the length of the seconds' pendulum in various latitudes, as a means of testing the truth of his theory. It appeared that the length of the seconds' pendulum does decrease as we approach the equator in the ratio of the square of the sine of the latitude, and so far observation confirms the theory. But it also appeared that the decrease of gravity was greater than that given by the theoretical expression. Whence Newton concluded that the value of e, as given by theory, was a little too small. Here, however, he was wrong; for if the earth be con- sidered as heterogeneous, an exactly opposite conclusion will follow from Clairaut's theorem. The planet Jupiter, owing to its great angular velocity, is very protuberant at its equator, and thus the difference of the lengths of its two axes could be determined with tolerable accuracy. This planet, therefore, furnished Newton with a good test of the truth of his theory. Since the centrifugal force varies as the square of the angular velocity, and inversely as the radius; and gravity in different planets varies as the radius and as the density conjointly, hence the ratio of the centrifugal force to gravity varies as the square of the angular velocity directly and the density inversely. The ellipticity by what precedes has the same proportion. On substituting the known numerical values of these quantities, we find for the ellipticity of Jupiter -^7. Cassini observed in the year 1691, that the diameter of Jupiter from east to west is greater by about tjth part than the other diameter. Mr. Pound with his 123 feet telescope and an excellent micrometer, measured the diameters of Jupiter in the year 1719, and found them on four occasions to have the ratios 12 to 11, 13| to 12|, 12f to llf, 14 to 13. 182 NEWTON'S PKINCIPIA. Thus Newton found a great discrepancy between the results of observation and theory. He accounts for this by remarking that there are two causes whose effects have not been allowed for. First, the diameters of .Jupiter will appear in the telescope greater than they really are. The magnitude of the correction to be applied depends on the size of the telescope, and the care that has been taken in its construction. The greater diameter of Jupiter being 37", the lesser, according to the above ratio, will be 33" 25"', add thereto 3" for the effects of aberration, and the apparent diameters of the planet will be 40" and 36" 25'", which are to each other in the ratio of 11 to 10, very nearly. Secondly, Newton remarks that the theory assumed Jupiter to be of uniform density. But this is not likely to be true. 5. NOTE IV. 8. 183 CHAPTER III. THE MOTION OF A PARTICLE IN A RESISTING MEDIUM. 1 . The object and mode of conducting the enquiry. 2. When the resistance varies as the velocity. a. Rectilinear motion. ft. Curvilinear motion. Section I. 3. When the resistance varies as the square of the velocity. o. Rectilinear motion. ft. Curvilinear motion. Section II. 4. When the resistance varies partly as the velocity and partly as the square of the velocity, and the motion is rectilinear. Section HL 5. When the resistance varies as any power of the velocity and the motion is rectilinear consideration of an analytical difficulty in the solution the terminal velocity and instances. 6. The motion of a particle in a resisting medium round a centre of force. a. The method used by Newton. ft. The method supplied by the Planetary Theory. Section IV. 1. PART of the theory of the motion of a body in a resist- ing medium is contained in the first four sections of the second book. The manner in which a medium resists the motion of a body moving in it is not the subject of this inquiry. It manifestly depends on a great many circum- stances which we shall presently consider. At present we shall assume that the changes of resistance throughout the motion depend only on the changes of velocity. Again, the resistance will greatly depend on the form of the body, and will change, therefore, as the body during its motion opposes different faces to the resistance of the fluid. If the resultant of the resistances on the several parts of the 184 NEWTON'S PKINCIPIA. body does not pass through the centre of gravity, the resistance itself will tend to cause rotation in the body. To simplify our analysis we shall consider the bodies to be indefinitely small particles. The principle on which we proceed is, that the motion of the particle will be the same as if the resisting medium were removed, and that, first, a retarding force is substi- tuted in its stead, which acts along a tangent to the mo- tion of the particle, and depends only on the density of the fluid and the velocity of the particle at the moment under consideration, and that, secondly, all the impressed forces are, by the buoyancy of the fluid or other causes, diminished in a known ratio. In considering the motions of bodies in vacuo we exa- mined the effects of various laws of gravitation besides that which we know to exist in nature. So we may now examine the consequences of supposing the resistance to vary according to different functions of the velocity. It is to Newton and Wallis * that we owe the first researches on the theory of the motion of bodies in resist- ing media. Wallis, in the same year that Newton pub- lished his " Principia," communicated his reflections on this subject to the Royal Society, who published them in their Transactions for 1687. Wallis, however, does not go so deeply into the subject as Newton did. A little after Newton's book appeared, Leibnitz asserted that he had already discussed these subjects, and that he had commu- nicated his opinions twelve years previously to the Academy of Sciences at Paris. Huygens also considered some points in this theory at the end of Traite de la Pesanteur, which appeared in 1690. Finally, everything which had been either proved, or stated without proof, was demon- strated by Varignon by the aid of the modern calculus. * Montucla, Part IV., Lib. VII. 6. NEWTON'S PRINCIPIA. 185 The process that is now used to determine the motion is founded on the following reasoning. Let us suppose the particle moving in any curved line ; let s be the arc described measured from any point at the time t. The time is supposed to be measured from any epoch anterior to the commencement of the motion. Then in the small time 8 t, the particle will, according to the notation of the differential calculus, describe a small arc 8 s, hence the mean velocity of the particle during this interval will be . Now let 8 t dimmish without limit, the mean velocity will become the actual velocity (u) at the instant t, and hence ds V = dt' Similarly, the velocity being v at the time t, that at the time t + $t will be v + $v ; hence the acceleration is such that in time 8 t a velocity 8 v has been added to the motion ; hence the mean acceleration in that interval, measured by the velocity that would have been added in a unit of time if it had remained constant during that time, will be r-- Now let 8 t diminish without limit, and the mean accelera- tion becomes the actual acceleration (f) at the instant, and dv f ~dTt But an accelerating force is measured by the quantity of velocity it would add to the body in a unit of time, if it remained constant during that interval, so that we have merely to equate the accelerating force as given by the question, to the acceleration as given by the preceding 186 ^NEWTON'S PRINCJPIA. formula, and we shall have an equation to determine the motion. If x and y be the co-ordinates of the particle at any time t, it will follow by similar reasoning applied to the second law of motion that the velocities parallel to the axes are dx , dy d~t and 37 and the accelerations are d 2 x , d?y a and -di parallel to the same axes. 2. PROBLEM. To determine the motion of a particle moving in a straight line in a medium, and thereby resisted in the ratio of its velocity, and acted on by a uniform force. Let V be the velocity of the particle at any time which we shall take as an epoch to measure our time from, and the place of the body at this time, as our " origin " from which to measure the distance of the particle at any other time. Take as our direction of measurement the direction in which the particle is moving at that epoch. Let x and v be the distance and velocity of the body at any time t. Let m be its mass, and let x v be the re- sistance when the body is moving with a velocity v ; that is to say, let x v be the moving force which would be required to keep the body moving, with this velocity in the medium. Let / be the uniform accelerating force measured in the usual way. NEWTON'S PRINCIPIA. 187 Then the whole moving force on the body will clearly be m f K v. Also we know that for any particle the accelerating force is v-, and therefore the moving force is m j-. Equating these two dv ^ = mf K v do This equation contains the whole of the motion. First. We know that v = =-, substituting, we have dv _ x. dx dt * m~Tt . - . integrating throughout the time t, If x were nothing, or the medium did not resist the particle, we should have, v - V -ft - - - - (3.) Hence the motion lost by resistance is XX that is, it is proportional to the space gone over. This is Newton's first proposition. A similar proof will apply if fbe not constant. Secondly. Suppose/=0; or that the particle moves by its " vis insita " only. Then, dv x Tt = ' m ' V 188 NEWTON'S PKINCIPIA. integrating throughout the motion ... t,=V.~ii' - - - (4.) That is, when the -times are in arithmetical progression, the velocities are in geometrical progression. Also, we have already proved that or the velocity lost in passing over any space varies as that space. As soon as the value of x is known, the above formula? may be submitted to accurate calculation. As its value depends on the form of the body, and the density of the medium, it can only be found by experiment in any par- ticular case. We may, however, learn some curious facts from these formula?. From the formula for v, in terms of t, we see that though v continuously decreases as t increases, yet it never vanishes. The particle will then never stop, though constantly retarded. A little consideration will show that this is just what we should expect. For the resistance, varying as the velocity, takes away from the velocity in any small time d t, a certain fraction - of the velocity that the particle has left. And as by taking away continually the halves of any quantity no one can remove the whole, so neither can this resistance ever destroy the whole velocity. From the second formula we learn, that since v can NEWTON'S PEINCIPIA. 189 never be zero, it is always positive, and therefore m Tr x < - V. K The resisted particle can never reach a point distant - V from the origin, and it takes an infinite time to de- scribe this space. We may also represent the motion of the particle by the several parts of an hyperbola. Construct an hyperbola Y PA, whose asymptotes are the per- pendicular straight lines, OX, O Y. Then P being any point, and PN parallel to OY, we know that O N . P N is constant, and equal to one quarter the sum of the squares of the axes. Let the hyperbola be such that this is equal to c 2 . Then take O B = V, and O N = v. By (5), we have, B N = - s. m Hence the velocity being represented by O N, the space described will be proportional to B N. Also, P N = y m 2# X C 2 y or the time is proportional to the area P N B A. If the hyperbola be so drawn that the number of units of area 190 NEWTON'S PEINCIPIA. in c z is equal to -, then the number of units of area in P N B A will equal the number of units of time elapsed. Thirdly. We may now proceed to consider the more general case. We have Hence by division we have, -dv = -- at fm m 1J X Integrating throughout the motion, fm_ / _ _ y X ...._= (v-/?). Hence, if from the velocities there be subtracted the constant quantity ^*, then these differences are in geometrical progression, when the times are in arithme- tical progression. Again, since v = , --, this equation gives integrating throughout the motion, -^.-sr-^f"*'- 1 ) which gives x in terms of t. This equation is the same as the more simple one, x = ^ (ft-v+V) already established. NEWTON'S PRINCIPIA. 191 From these equations we learn by interpretation several facts. 1. Let the velocity of projection be in the direction in which the accelerating force / acts. Hence / is here positive. . Since e ~ m can never change sign, the quantities fm , Tr fm v J - and V J X X have always the same sign. Hence the velocity is always greater or always less than - according as the velocity of projection is greater or less than that quantity. jS. Since e ""* continually diminishes as time goes on, but never vanishes for any finite value of t, hence the ve- locity v continually approaches - m , but never actually equals it. 7. If Fine velocity at any one point should equal - , the velocity is always equal to the same quantity. 8. Since v continually approaches a finite quantity, the expression for x shows that the space described continually increases and finally becomes infinite in an infinite time. f. The velocity continually approaches the limit , hence x is the mass of that body whose limiting velocity is/. 2. Let the velocity of projection be in a direction op- posite to that in which the force/ acts. Here /is negative, and our formulae become 192 NEWTON'S PRINCIPIA. As time goes on, the second factor on the right hand of the first equation continually decreases. Hence v decreases, until a time comes when then v = or the body comes to rest. After this time, v becomes negative, or the body begins to move in the opposite direction. This case has been already considered. 2. PROBLEM. Supposing the force of gravity to be uni- form and to tend perpendicularly to the plane of the horizon, to determine the motion of a projectile in a medium which resists in the ratio of the velocity. Suppose the body projected with a velocity V and in a direction making an angle with the plane of the horizon. Take this also as the epoch from which we measure the time, and let v be the velocity and the angle the direction of motion makes with the horizon and s the arc described at any other time t. Let the position of the particle be defined by two co-ordinates x and y, giving respectively its distance from the point of projection measured along a ho- rizontal line, and its altitude at the time t. Our first ob- ject is manifestly to find x and y in terms of t. By the second law of motion each force produces its own effect as if the particle were at rest and it the only acting force. In considering then the motions parallel to any di- NEWTON'S PRINCIPIA. 193 rection, we may wholly omit those perpendicular. The only force acting along the axis of x is the resolved part of the resistance in that direction. This is But we know thatw= -= and cos = -r . Hence the moving force is dt By similar reasoning that parallel to y is dy mg x-r^- Hence just as in Art. 2., the equations of motion are d* x dx dy dt - (I-) We have met with both these equations before. The mo- tion parallel to x is manifestly the same us that in the second case of Art. 2. We can therefore write down our results at once, viz. d x -r-, - t v COS = -= = V COS a . e a t and The motion parallel to y is the same as that of the third case of Art. 2. Hence o 194 NEWTON'S PRINCIPIA. dy am f,, . qm\ --* in 6 = -~ = - -- h ( V sm a + 2 1 e m at x \ x / 9 m \ f-i ~- 'N * J (1 s ) <7 ?W m f-i-r - M f V sm (3). When any one of the five quantities x, y t v, 0, t, are given, these four equations determine the other four. It is therefore reduced to be a mere matter of arithmetic calculation to determine the position of the particle at any time. It may be laborious and tedious, but there is no difficulty in it. We shall now trace the curve the particle describes. Find t from the second equation and substitute in the fourth, we have y = ( tan COS a/ + f-)V - log (l ----). - - (4.) \x/ 9 ^ \ m Vcosa/ x m V cos a/ Let O be the origin, O B the direction of projection, A O C a horizontal. Take O C = . V cos a, and draw C B ver- tical. Then from the above equa- tion it is manifest that B C is an asymptote to the curve. Take B A = [ J g, a quantity, it will be observed, that is independent both of V and . Join O A, and let = L A O C - (5.) NEWTON'S PBINCIPIA. 195 If P be any point in the curve, and N P M vertical MN=(ten + ~- * ).* V x V cos / ... N P = - ()V log ( l - * . Vx/ y g V TO put N A = , N P = , , m cos a TT where a = ^ V. x cos /3 This is a very simple form of the equation to the curve and enables us to investigate many of its properties with ease. We learn that if the successive values of N A are in geometric progression those of N P will be in arithme- tical progression. This is Newton's second corollary. It is also manifest from the manner in which we eli- minated t, that we have or the particle moves in such a manner that its distance from O A, measured parallel to any fixed straight line, varies as the time. This is Newton's first corollary. Since O B . cos = O C, and O C = V cos a and since any point may be considered as the origin of projection, we learn that the velocity at P is always pro- o 2 196 NEWTON'S PRINCIPIA. portional to the tangent P T. This is Newton's sixth corollary. If I be the latus rectum of the parabola that would be described under the same circumstances of projection, if the medium offered no resistance, then /- 2 V 2 cos 2 9 0~C| 2 3. PROBLEM. To determine the motion of a particle moving in a straight line in a medium resisting in the ratio of the square of the velocity and acted on by a uniform force. Let the symbols V, v, x, m, t, x, have the same meaning that they had in the corresponding problem in which the resistance varied as the velocity. Then the whole moving force upon the particle will clearly be mfK v* also we have the accelerating force on a particle moving , a v * i d x , in any manner equal to -7- ; which, since v ~r^.> ma y a* 80 be put under the form v -7. Taking both these forms, we have dv m = m f x t> 2 dv m v -T- = mf x tr which are identical equations. The first equation gives v in terms of t, the second v in terms of x. First. Suppose / = 0, or that the body moves by its tf vis insita " only. NEWTON'S PRINCIPIA. 197 Then since d V K & m integrating throughout the motion Again 1 - - t - v V ~ m dv x o V -r = U dx m dv x , . = d x v m . . integrating throughout the motion V K ^ V " " m X .-. = V . e~*. - - (2.) Since v = -=-. this equation is the same as e *'dx = Vdt integrating throughout the motion From these equations we may gather every circumstance of the motion. From (1) we learn that if the times are in Arithmetical Progression the velocities are in Harmonica! Progression ; and that the velocity varies inversely as the time when counted from an era ^..units of time before xV o 3 198 NEWTON'S PKINCIPIA. the epoch at which the velocity is V. Also from (2) we learn that when the spaces are in Arithmetical Progression the velocities will be in geometrical Progression. A mere inspection of these equations shows that the particle will continue to move for ever with a continually decreasing velocity, and that it will pass over an infinite space. We may represent, as Newton has done, the motion by the several parts of an hyperbola. Construct the hyper- bola B P, whose asymptotes are the perpendicular straight lines O X, O Y. Then P being any point in it and P N an ordinate, we have ON. PN=c 2 Take O A to contain ^ units of space, and A N to con- tain t units of space, then A N represents the time. Since ON =*+^ ... B = JZL . PN, xc 2 or the ordinate represents the velocity. Since x = 1 vdt putting y for P N, hence the area A P represents the space described. Secondly. We may proceed to consider the more general case. We have dv v. v -T- f -- v 2 dx m v d v - - - - =/<** 1 - , 2 mf NEWTON'S PRINCIPIA. 199 integrating throughout the motion, f Log "~ -,V 2 m/ which is true, whether the body is moving in the direction in which / acts or the opposite, provided only we give / its proper sign. Again we have dv ,. x = f v 2 dt J m f _ m/ mf providedybe positive, but if negative, say f, tan- 1 v/Z - tan-^Z V = - ^2 . * mf mf m the first or second equation being true, according as the particle moves in the direction in which f acts or the opposite. From these equations we can infer the nature of the motion. (1.) Let the velocity of projection be in the direction in which the force facts. Then fis positive. a. Since s m x can never change sign, the quantities V 2 J X o 4 200 NEWTON'S PRINCIPIA. have always the same sign. Hence the velocity is always greater or always less than -v/^L/ X _2_? jr /3. Since s m * continually decreases as x increases, v continually becomes more and more equal to \/ r !2. n y. As t increases and finally becomes infinite, the equation connecting v and t shows that v continually approaches and finally becomes equal to A --^ a f ac ^ which we could not infer from the first equation. This velocity is called the " terminal velocity." Let us repre- sent it by the letter u. The equations may then be put under the simple forms Suppose the particle to begin to fall from rest, then we have V = and/ -g. x = u t log 2 + log (1 + e ) 9 9 (2.) Let the velocity of projection be in a direction opposite to that in which f acts. Here f is negative, write / for it, we learn that v 0, or the particle will come to rest after having described a space given by the equation or, * = %j log (l + X*) ; NEWTON'S PRINCIPIA. 201 and the time of describing this space is to be found from or, In a preceding section Newton had determined the path of a projectile when the resistance varied as the velocity, and here was the place to give the solution of the cor- responding problem, when the resistance varied as the square of the velocity. But this is a far harder question ; we are even now unable to find quite accurately the path described. Newton considered the problem in an indirect manner. He determined the law of density that a given curve may be described, but he could not thence deduce the curve that gave the density uniform. He even made several mistakes, which were corrected at the suggestion of John Bernoulli, in the edition of 1713.* In 1718 Keill, in the course of the quarrel between the supporters of Newton and Leibnitz, dared the foreigners to attempt this question. Bernoulli was the first who gave a solution, and challenged the proposer to furnish his own solution within a certain time. This, however, Keill was unable to do. Meantime Nicholas Bernoulli, of Padua, supplied a solution ; and seventeen days after the time fixed had elapsed, Taylor vindicated the honour of England by a tardy solution. The problem we shall now consider is somewhat more general than that enunciated by Newton, and it is as follows : * Montucla, Part IV. Liv. VII. 6. 202 NEWTON'S PRINCIPLE. The resistance of the air being supposed to vary as the square of the velocity and as the density conjointly, and the forces to tend to directly to the plane of the horizon, to determine what must be the law of density of the medium that the particle may describe a given path, and to find the velocity at any point. Let the axis of x be taken horizontal and that of y vertical, let x, y be the co-ordinates of the particle at any time t, and s the arc described. Let p be the density of the medium at the point (x, y\ and v the velocity of the particle, V the velocity, and the angle of projection. Then the resistance of the medium may be taken as Resistance = x p v 2 . Let Y be the force acting on the particle parallel to the axis of y. The equations of motion will be d 2 x x 2 d x d~P ~ ~m P V d~s ^L! = y _ JL 1,2 dy d t* m ' d s which may be put in the form d*x _ x_ d_x_ &y _v * , d y Multiply these equations by -=&- and -^ and subtract, dx d* y _ d_y d 2 x _ djz dt d t* d t' d t*~ d t By the theorem in the differential calculus for changing the independent variable, we have, therefore, t NEWTON'S PRINCIPIA. 203 But from (1) we have ~dlc\- 1 ,dx Tt\ d dTt dx d s, whence eliminating -=- , we have an equation from which either p or Y may be found when the relation between x and y, which determines the curve is given. The velocity at any point of the curve is that due to one- fourth the chord of curvature. For looking at equation (3), the left-hand side is the denominator of the expression for the radius of curvature R, whence * 2 dx Since-y- is the cosine of the angle, the normal makes ft.' S with the axis of y, the quantity in brackets is one fourth the chord of curvature. Whence the proposition follows. The equation (4) will also enable us to determine the equation to the path when the law of density and the force is given; as an instance take p constant, and Y = g the force of gravity. Then ~J - 9 d x* an equation which can be only approximately integrated. Newton takes several examples to illustrate his reason- ing. For instance, if the path be a semicircle and the force gravity, we have 204 NEWTON'S PRINCIPIA. y 2 = a 2 1 =- J where a is the radius of the semicircle, whence, by a simple substitution, we get 3 x ? ~ 2 * a ' y' so that the density of the medium at any point varies as tangent of the angle a radius through the point makes with the vertical. Newton also determines the law of density when the particle describes an hyperbola with one asymptote vertical, chiefly with the view of finding an approximation to the curve which a particle will describe in a uniformly resisting medium. This was a problem which Newton was unable to solve, except in this imper- fect and indirect manner. We shall not therefore dwell on this, but will proceed at once to indicate the manner in which the question is now answered. Taking the equation dx* V 2 C08 2 ' multiply both sides by V I + p*dx, where p ~^ x -> integrate S 2 ^_ i V 2 / ' .\?SJTp* 2* m COS A/I In gunnery p is usually small, let us reject the powers of p higher than the second, we get s m =1-1 --- cos 2 a (tan a ) mg NEWTON'S PRINCIPLE. 205 2 * > and integrating, we get, after correction, log this will enable us to find / in terms of v, and, by solving the equation, v in terms- of t. If the particle be moving in the direction opposite to that in which f acts, and if, also the above integral becomes imaginary ; the true expression will manifestly be, if Again, we have dv x K v z v -7 f v . - ax m m a for the sake of brevity put o 2 mfa. c T+ ~^~ 4 x 208 NEWTON'S PRINCIPIA. and integrating we have ' & c a, ma. V + -- C where the quantity C is obviously equal to what the left hand side of this equation becomes when V, the initial velocity, is substituted for v. If the particle move in the direction opposite to that in which f acts, and if - .. ,,.. ...... this expression becomes imaginary. It is obvious, how- ever, that if we put that the true integral will be a. -\ V 2 2 f <4* -L V * I v + - + o } i \ ^ 2 J & *. tan m where C is obviously equal to what the left hand side becomes when V is put for v. The quantity * in these formulae is the mass of that particle whose terminal velocity is the quantity , therefore, represents a number p , thus we 01 see that the preceding expressions are perfectly homo- geneous. In exactly the same manner we may proceed to de- termine the motion of a particle in a medium resisting ac- cording to any other function of the velocity. NEWTON'S PRINCIPIA. 209 If the resistance vary as v n and the particle move by its own "vis insita" only. Then since d dv -= = V 7 = . IT d t d x m . . integrating both equations throughout the motion. equations which never become nugatory except when w=l or n=2, both which cases have been already con- sidered. From these equations we may learn several remarkable facts. First. Suppose n greater than 2. Then both the right hand members of the above equations are positive ; hence v can never vanish, and the body will continue moving for ever, with an ever diminishing velocity, and will pass over an infinite space. Secondly. Suppose n greater than unity but less than 2. Then v vanishes only when t is infinite, but then m x = Y. 2- n So that the particle t continues to move always with an ever diminishing velocity, and will pass over a finite space. Thirdly. Suppose n less than unity, then when v vanishes, we have - m - ' that is, the particle moves on with a diminished velocity, p 210 NEWTON'S PRINCIPLE and finally stops after a time given by the first equation, having described a space given by the second. It is ma- nifest that the particle remains at rest, until it is disturbed by some new force. But here we have a remarkable singularity in the equa- tions ; for according to them, as t increases v l ~ n does not remain equal to zero, but becomes negative. What is the explanation of this ? It must be sought for in the nature of a differential equation. There are always two species of integrals. One called the "general integral," which contains the full number of arbitrary constants, and ano- ther, called the "singular solution," not included in the former, and which does not contain the full number of ar- bitrary constants. These latter in dynamical problems are usually of little value, because they do not agree with the initial conditions of motion. But, if by any chance they should satisfy these conditions, it is possible that they may be the true representatives of the subsequent motion. The choice between them and the general integral must be founded on 'extrinsic considerations. The differential equa- tion we started with, is a mere statement of the forces, and must be true throughout the motion. This motion must therefore be represented either by the general or the sin- gular solution. We have seen that the general solution only represents the motion up to a certain time ; after that we must have recourse to the singular solution. If we proceed to find this, by the usual methods, we arrive at the solution f =0, which we see represents the motion subsequently to the above mentioned time.* One of the most remarkable facts connected with motion in a resisting medium is the existence of a " terminal ve- * Duhamel, Cours de Mecanique. NEWTON'S PRINCIPIA. 211 locity" when the body moves in a straight line, under the action of a uniform force. This may be defined to be that velocity which makes the resistance equal to the moving force acting on the particle. Let

so that dao 2 x an . _=-__ 8mH , 2 xa .'. C0j CO Q . COS n t, where co , co t are the values of co at the beginning and end of the interval t. Thus we learn that the changes in co are very much greater than those in e, but that by the end of one revolution both have returned to their original values. NEWTON'S PRINCIPIA. 221 CHAPTER IV. THE MOTION OF FLUIDS, AND THE RESISTANCE TO BODIES MOVING IN THEM. SECTION VII. 1. Newton's investigation of the law of resistance to similar bodies. 2. The manner in which the resistance depends on the form of the body. Newton, xxxiv. & xxxv. 3. Their resistance when the body is a surface of revolution. 4. The surface of least resistance, its properties and form. Scholium, Prop, xxxiv. 5. The law of resistance deduced from experiment. Prop. xl. and Scholium. 1. SUPPOSE we have two systems of an equal number of particles similarly placed, and proportional each to each both in density and volume. Let them begin to move, the particles of one system amongst themselves, and those of the other amongst themselves, with like motions and in proportional times. If no action ever took place between the particles, by the first law of motion the similarity between the systems will always exist. It is also clear that any collisions or reflexions among the particles will not affect this similarity of motion ; if any collision occur in one system an exactly similar collision will occur in the other ; similar changes of motion will be thereby produced in the two systems. Next, suppose the particles attract or repel each other with accelerating forces, which are as the squares of the velocities directly and the diameters inversely of the cor- responding particles in the two systems. Consider two homologous particles, one in each system, the attractions 222 NEWTON'S PRINCIPIA. of the rest on these being each in the above ratio, the resultants will also be in the same ratio, and the attracting particles at the beginning of the motion being similarly placed, and the forces in each system proportional, the directions of the resultants will be parallel. Now we know that two similar particles beginning to move in parallel directions will describe similar orbits in proportional times, when at the end of those times the directions of the forces are parallel and proportional to the squares of the velo- cities and the reciprocals of any homologous sides of their orbits. Hence these two particles begin to move similarly under the action of such forces as tend to preserve the similarity of their motions. And the same is true for all homologous particles in the two systems. Hence all the particles of the one system at the end of any small time T, are placed similar to those of the other at the end of the small proportional time T' and are moving in a similar manner. Hence the same thing will again be true at the end of the next proportional intervals, that is, at the end of the proportional times 2 T and 2 /. Therefore the particles will continue always to move among themselves with like motions and in proportional times. Let there be two fluids or systems such that the particles of the one are similar to those of the other ; let the diameters and distances of any two particles in one system be d times the diameters and distances of the cor- responding particles in the other, and let the density of these particles in one system be p times that of the cor- responding ones in the other system. Let the particles begin to move from similar positions, and if we suppose the forces in the two systems to be always proportional to the squares of the velocities directly and the diameters of the corresponding particles inversely, the several particles will describe similar orbits in similar times. Let the velo- NEWTON'S PKINCIPIA. 223 city of any particle in one system be v times that of the corresponding particle in the other when at the correspond- ing part of its orbit. Let two large bodies which are similar to each other in the same manner that two cor- responding particles are similar, be similarly projected into these two systems. They will then describe similar orbits in proportional times. The diameter of one body is d times that of the other, and the velocity of one will be v times that of the other. Let us consider the resistances to these bodies : it will arise partly from the centripetal forces with which the particles and the body act on each other, and partly from the collisions and reflexions of the particles and the body. The resistances of the first kind are, by hypothesis, as the squares of the velocities directly and the diameters of the corresponding particles inversely, and the masses of those particles directly, that is, the ratio of the resistances in the two systems is The resistances of the second kind are as the number of reflexions and the forces of those reflexions. The number of the reflexions in the two systems are as the velocities of the corresponding particles directly, and the spaces between their reflexions inversely, hence the ratio is -> The forces of the two systems are as the velocities and masses of the corresponding particles, hence their ratio is v. d 3 . p ; hence the ratio of the resistances is joining these two ratios, the ratio of the whole resistance in the two systems will be t; 2 . d*. p. In such fluids, and under such conditions as those we 224 NEWTON'S PKINCIPIA. have just been considering, the resistances vary as the square of the velocity, the square of the diameter, and the density of the fluid, If we have two fluids whose particles when at a distance do not act with any force on each other, such fluids come under the description of the similar systems just considered. Let the particles of the two fluids be equal, then the resistances to equal similar bodies moving in them are accurately as the squares of the velocities of the bodies and the densities of the fluids. Next, suppose the bodies not equal. Because the motion of the fluid varies continu- ously from point to point, and because the force of collision due to two equal particles moving in the same manner is equal to that of one particle of double size, the forces of collision will be the same if we divided the fluid into elements, and considered them as particles. Let the equal fluids be divided into elements, which are proportional to the volumes of the similar bodies moving in them. Then the resistances will vary as the square of the diameter, the square of the velocity of the body, and the density of the fluid. But how far are we justified in applying these conclusions to the fluids we meet with in nature? The forces to which collision and reflexion are due, are those which are sensible only at distances which are indefinitely small com- pared with the average distances between the particles. Are these the only forces which exist between the par- ticles of a fluid? Incompressible fluids are the nearest approach to such a state of things. In elastic fluids the particles have a tendency to recede from each other, and our previous reasoning cannot therefore apply to them. Let there be three fluids A, B, C ; let them consist of similar and equal particles regularly disposed at equal dis- tances, and let the parts of A and B have a tendency to NEWTON'S PBINCIFIA. 225 recede from each other with forces that are as T and V, and let the particles of the medium C be entirely destitute of such forces. Let four equal bodies move in these media, viz. D in the medium [A,] and E in [B] F and G in [C], and let vel. of D = vel. of F _ /T_. vel. of E. vel. of G ~ "V V ' then since the forces are as the squares of the velocities, and the diameters of the particles are equal, therefore the resistances in the two fluids are as the squares of the ve- locities, that is Res, to D _ Res, to F T ,. , Kes. to E " Kes.toG = ' V Let us suppose also that vel. of D = vel. of F .-. vel. ofE = vel. of G augment the velocities of D and F in any ratio, and di- minish the force V of the particles in the medium B in the duplicate of that ratio, the medium B will approach to the form and condition of the medium C, and therefore the resistances to the equal and equally swift bodies E and G moving in those media will approach equality. Hence by (2.) the bodies D and F, when they move with great swift- ness, meet with resistances nearly equal. Hence the re- sistance to a body moving very swiftly in an elastic fluid is almost the same as if the parts of the fluid were destitute of their centrifugal forces and did not tend to fly from each other. So that the resistance to similar bodies moving very swiftly in an elastic medium vary as the squares of the velocities and the squares of the diameters. Q 226 NEWTON'S PRINCIPIA. This reasoning requires that the velocity should be so great that the forces of the particles will not have time to act. 2. The preceding investigation has led us, on certain as- sumptions, to the law of resistance to similar bodies, but it now remains to discover what change in the resistance would be caused by a change of form in the body. A new assumption becomes necessary. Let us suppose the par- ticles to be so rare that their distances are infinitely greater than their diameters, so that each particle may be able to give its blow to the body and then to make its escape without affecting the particles which have not yet given their stroke. It is manifest that to find the resistance according to this principle we have to divide the surface into elements, find the resistance on those elementary planes separately, and by integration add the results. It is necessary to find the resistance on a small plane in- clined at an angle to the direction of motion. Let the area of the plane be A, then the number of particles that will strike it will be proportional to A cos. and to the velocity v conjointly, and each particle will strike the plane with a normal velocity v cos. 0. The mass of each particle is supposed the same. Hence the whole normal resistance will be proportional to A cos. 0j 2 u 2 , and resolving this along the direction of motion the resistance will be pro- portional to A~cosT0| 3 v 1 . Hence if x A u 2 be the resist- ance on the plane when perpendicular to the direction of motion, the resistance when inclined at an angle will be x A y 2 cos. flf. It will be observed that this reasoning is true whether the particles be elastic or not. Any change of elasticity affects the resistance by changing x. Let a cylinder be made to advance in the direction of its NEWTON'S PRINCIPIA. 227 axis with a uniform velocity v in a medium, and let us suppose that the particles of the fluid are perfectly elastic. They will then rebound with the same velocity relatively to the cylinder as that with which they struck it. There- fore the cylinder, on striking each particle, gives it a velocity twice its own, and in moving forwards a length half its axis communicates a motion to the particles which is to the whole motion of the cylinder as the density of the medium to the density of the cylinder. Hence the cy- linder meets a resistance which is to the force by which its whole motion may be taken away in the time in which it describes half its axis as the density of the medium is to the density of the cylinder. If Z be the length of the axis, the time of describing the half axis will be \ -, and the ac- celerating force that would generate a velocity v in this 2 v z time is ~j- ; hence the moving force, which is the re- sistance, is where A is the area of the base, and p the density of tne fluid. Next, let us suppose the particles perfectly inelastic ; they will not be reflected, and the cylinder will merely communicate its own simple velocity to the particles it strikes against. The resistance is therefore only half as great as before, that is Resistance = Av 2 p. Thirdly. If the particles be imperfectly elastic, the par- ticles will rebound from the cylinder with a less velocity than if they were elastic, and a greater velocity than if they were inelastic ; hence Resistance = x A v 2 p, where is some quantity lying between 1 and 2. Q 2 228 NEWTON'S PRINCIPIA. 3. Let us now apply this to find the resistance on a sur- face of revolution moving in the direction of its axis. Let v be the velocity of the body relative to the fluid. We shall then suppose the fluid in front of the body to be at rest. Take the axis of revolution of the surface as the axis of x, let y be the ordinate and s the arc of the gene- rating curve. By what has been already said, the resist- ance or pressure on the annulus 2 it y d s when resolved along the axis will be because this latter factor expresses the cosine of the angle the normal makes with the line of motion of the body. Hence the whole resistance will be Let the surface be terminated by a plane perpendicular to the axis of x, the section will be a circle ; let r be its radius. Let y >j . r, and s and The value of a can be found from those of A and B, for the weight of a globe of water one inch in diameter is 132.645 grains in air, and therefore 132.8 grains in vacuo. And since the weights of globes vary as the cubes of their diameters, hence a a A-B = where A B is measured in grains, and a in inches. We are now therefore able to calculate the value of x when t is given. If t be not very small, the last term may be neglected. For since when is small log. (1 + ) = a nearly, the value of that term is very nearly M 2 -U't . 9 and the exponent being large, this term will be insensible. We may therefore take The space deduced from this formula requires a correction depending on the narrowness of the wooden vessel in which the experiments were made. Now if a globe descend in a cylinder, the area of whose section is L, the resistance will be increased in the ratio NEWTON'S PRINCIPIA. 235 ^.(-J^JW. Hence w 2 will be decreased in the same ratio, and .*. u decreased in the ratio of ju. : 1. In all the following ex- periments the area of the greatest circle of the sphere is small compared with the section of the cylinder ; the value of ft, is therefore nearly unity. In the expression for x, the first term is usually a hundred times the second ; hence it will be sufficiently accurate to reduce the space x by simply multiplying it by /*. This will be evident on an inspection of the numbers in the following experiments. Experiment 1. A globe whose weight was 156^ grains in air, and 77 grains in water, described the whole height of 112 inches in 4 seconds. On repeating the experiment, the globe again spent the very same time of 4 seconds in falling. Here A = 156^-f grains, 2 a .84224 inches, . . u 29.03 1 1 inches per second. The principal term in # is 1 16.1245, ~| > The second is 3.0676, J ,-. x= 113.0569, and /*= .9914. Hence the space which the globe falling in water de- scribes in four seconds is, by theory - - 112.08, experiment - - 112. Experiment 2. Three equal globes, whose weights were severally 76^ grains in air, and 5 r ^ grains in water, were let fall successively. Every one fell through the water in 15 seconds of time, describing in its fall 112 inches. Here A = 76fV grains, 2a= .81296 inches, .-. w = 7.712 inches per second. 236 NEWTON'S PKINCIPIA. The principal term in x is 115.678, The second is 1.609, .-.a: =114.06 9, and /* = .895. Hence the space which the globe falling in water de- scribes is, by theory 113.1741 experiment - 112. J Experiment 3. Three equal globes, whose weights were severally 121 grains in air and 1 grain in water, were suc- cessively let fall, and they fell in water through a height of 112 inches in the times 46", 47", and 50". By theory these globes ought to have fallen in about 40". The weight of the globe in water is so small, that any errors in the weighing, or produced by the rarefaction of the wax, by the heat of the hand, or by the weather, or by bubbles of air adhering to the globes, become sensible. That the experiment may be certain, the weight of the globe in water should be several grains. Newton began the foregoing experiments to investigate the resistances of fluids, before he had discovered the theory laid down in the preceding sections. In order to examine the theory after its discovery, he undertook another course of experiments. He procured a wooden vessel, whose breadth on the inside was 8f inches and depth 15^ feet. He made four globes of wax with enclosed lead, each weighing 139^ grains in air, and 7-jj grains in water. These he let fall, measuring the times of their falling in the water with a pendulum oscillating half seconds. The globes were cold, and had remained so for some time, both when they were weighed and when they were let fall, because warmth rarefies the wax, and by rarefying it diminishes the weight of the globe in water, NEWTON'S PRINCIPJA. 237 and wax when rarefied is not instantly reduced by cold to its former density. Before they were let fall, they were totally immersed in water, lest, by the weight of any part of them that might chance to be above water, their descent should be accelerated in the beginning. Then, when after their immersion they were perfectly at rest, they were let go with the greatest caution, that they might not receive any impulse from the hand that let them down. At Newton's first experiments the weather was a little colder than when the globes were weighed, and therefore he repeated the experiments another day. On making the experiments several times over, he found that the globes fell mostly in the times of 49^ to 50 oscillations, the times varying, however, from 47^ to 53 oscillations. On com- puting from the theory the time of descent, it was found to be 50 oscillations, very nearly. Newton tried eight other sets of experiments, in all of which there was a tolerable accordance between theory and experiment. The times in which the balls descended varied, sometimes as much as one-fifth of the whole time of descent ; but the errors were generally as much on one side as on the other of the theoretical result. Whence, for such swift motions as these, we may conclude that the resistance on a globe is very nearly represented by the preceding theory. Those globes which fell with a slower motion were found to agree best with the theory, and the reason of this is as follows. When the globes were first let fall, it was found that they oscillated about the centre, that side which is heaviest tending to descend first. The globe in conse- quence of these oscillations communicated a greater motion to the fluid than it otherwise would, that is, it met with a greater resistance. In the heavier and larger globes this oscillation was not checked by the water until after several oscillations, and hence these were more resisted than the 238 NEWTON'S PRINCIPIA. lighter ones, which fell slower. Every care was taken to diminish these oscillations, but they could not be alto- gether prevented. From the top of St. Paul's, in June, 1710, two glass globes were allowed to fall through the height of 220 English feet. One was full of quicksilver, the other con- tained air. The two globes rested on a wooden table, which turned round iron hinges on one side, the other side being supported by a wooden pin. The two globes were let fall together by pulling out the pin by means of an iron wire reaching from thence quite down to the ground, so that the pin being removed, the table, which had then no sup- port but the iron hinges, fell downwards, and turning round, the globes dropped off it. At the same instant with the same pull of the iron wire that took out the pin, a pendulum, oscillating seconds, was set in motion. The wooden table was not found to turn so quickly on its hinges as it ought to have done. Hence the times of descent were prolonged 3". This must be taken into account, as in that time the ball full of mercury would have described 37 feet. The difference between theory and experiment was found to be very small. To mention one instance, one globe of air fell 220 feet in 8.2"; according to theory it should have fallen in that time 225 feet 5 inches. The resistances to the globes of mercury were found to be so small in comparison with their weight, that they fell in nearly the same time which they would have taken to fall through the same height in vacuo. The differences could not be observed with sufficient accuracy to furnish a test of the theory, the difference being sensibly the same for globes of all weights. In July, 1719, Dr. Desaguliers made some other experiments of the same kind. The globes let fall were formed of hogs' bladders blown full of air. They were NEWTON'S PRINCIPIA. 239 let fall from the lantern at the top of the cupola of the same church, a height of 272 feet. The agreement be- tween theory and experiment was even more striking than in the last experiments. The theoretical results are calculated on the supposition that the resistance is proportional to the square of the velocity, the density of the fluid, and the surface of the sphere. The comparison with experiment serves to test all these laws. We may, therefore, conclude that for such velocities as those here experimented on, varying from 6 to 30 feet per second, and for spheres of radii varying from to 3 inches, and for the fluids of air and water, those three laws are tolerably correct. Newton remarks that these experiments are more accu- rate than those he made with pendulums ; for the vibrations not being very small, the body always excited a motion of the fluid contrary to the motion of the pendulum on its return, and thus the whole resistance appeared greater than it really was. [See NOTE V.] 240 NEWTON'S PRINCIPIA. CHAPTER V. THE MOTION OP A PENDULUM. 1. Some general considerations. 2. Motion in vacuo, Newton xxiv. 3. The properties of a pendulum. 4. Motion in a resisting medium Modern method of considering the perturbations of a pendulum, the Newtonian method. 5. Newton's experiments to discover the law of resistance. 1. THE importance of the pendulum can hardly be over- rated. It ministers to our comforts in a variety of ways. But what is more to our present purpose, it is a powerful engine of discovery. It can be made to test the laws of impact, it teaches us the law of resistance by which fluids retard the motion of bodies moving in them. It enables us to note the variations of gravity over the earth, and thus reveals to us not only the form of the world, but also the force with which it attracts external objects. It has even been applied lately to prove ocularly the rotation of the earth. There is no end to its applications. It is the great accuracy with which the time of oscillation may be observed that renders this instrument so useful. By noting the time of any great number of oscillations, and dividing that time by the number of oscillations, we can find the time of any one with great accuracy. It is, there- fore, highly necessary for us to consider carefully the pro- perties of a pendulum. A pendulum is any solid body which oscillates about a fixed horizontal axis. A simple pendulum consists of a NEWTON'S PRINCIPIA. 241 material particle suspended from a fixed point by an in- flexible inextensible string without weight. The length of this string is called the length of the simple pendulum. A perfect simple pendulum is only a mathematical idea ; we may approximate to such an instrument, but we can-? not accurately construct it. It will, therefore, in many cases be necessary to have the means of determining, when any compound pendulum is given, the length of the equi- valent simple pendulum. The first person who solved this generally was Huygens in his " Horologium Oscillato- rium," 1673. The principle on which he proceeded was not so simple as that to which this and such like problems are now referred. But the result is that if h be the distance of the centre of gravity from the axis of suspension, and mi 2 the moment of inertia about an axis through the centre of gravity parallel to the axis of suspension, the length of the equivalent simple pendulum will be 7 _ i* + A 2 ~~h~ This is on the supposition that the body moves in vacuo under the action of gravity. If it move in a medium resisting according to any function of the velocity, the above will still be the length of the equivalent simple pen- dulum. For the resistance on a unit of area being supposed to be x v n , according to the usual theory of resistance, the moment of the whole resistance on the surface will be where A depends on the form of the surface, and not on the nature of the motion. Hence to make the simple pendu- lum move in the same manner, we have merely to suppose that the weight of the particle is equal to the weight of the pendulum, and that it experiences a resistance which R 242 NEWTON'S PRINCIPIA. follows for variations of velocity the same law that any element of the pendulum experiences, but whose magni- tude is such that the whole moments of the resistance on the pendulum and particle about the axis of suspension are equal. In reasoning then on the pendulum we may always consider it as a simple pendulum. For when we have determined its motion, the preceding formulae enable us to determine that of the compound pendulum. Newton does not confine himself to the case in which the particle describes a circle. Supposing the string flexible, he has given in the first book a way of making it describe any given curve. Of all these the cycloid is that which possesses the most important properties. The motion of a particle in a cycloid is discussed, because it gives us a deeper insight into the laws of pendulous movements than that in any other curve. The motion of a particle in a circle is considered, because in practice most pendulums are so constructed that any point in them describes a circle. [See NOTE VI.] A particle constrained to move in a cycloid whose axis is vertical is acted on by gravity and resisted by a constant force. To determine the motion. Newt. xxv. Let / be twice the radius of the generating circle, s the distance of the particle at any time t from the lowest point (C) of the cycloid, v the velocity, and m the mass of the particle. Let/* be the constant resistance. Then the moving force along the tangent is supposing the particle to be descending. NEWTON'S PRINCIPIA. 243 d' 2 s But this is also m -j -% > hence we have dt* / m This equation being of a standard form, we can write down its integral, s L cos A /? ( t A) -f A/ * w # where L ancf A are constants depending on the initial con- ditions of motion. Take a point O in the arc, so that CO = , and let sf be mg' the arc when measured from this point. Then s' = L cos (t - A). Suppose the particle began its descent from D, then A is clearly the time at which the particle was at D, and L = arc O D. It is manifest the greatest velocity will be at O. If the particle had been undisturbed by/, the same equation would have given the arc measured from C. If then the particle when resisted by / be at P at any time, and if another particle not resisted by/ be at Q at the same time, then DP bears to PQ the constant ratio DO to DO. The effect of any constant resistance is to diminish the arc continually in an arithmetical progression, but not to affect the time of oscillation, R2 244 NEWTON'S PRINCIPIA. A particle constrained to move in a cycloid whose axis is vertical, is acted on by gravity and resisted by the medium in which it moves in the ratio of the velocity. To deter- mine the motion. Newt. xxvi. Let / be twice the radius of the generating circle. Let s be the distance of the particle at any time t from the lowest point of the cycloid, and v be the velocity, and m the mass of the particle. Let x be the coefficient of resistance. Then the moving force along the tangent will be the particle being supposed ascending its arc. But this is also m -j-2- Hence i~1> 2 quantity ^T= Tel* Let us now consider the con- struction by which Newton re- "* n presented the resistance. Let a straight line B be drawn equal to the arc of the cycloid which an oscillating body describes, and at each of its points D draw the perpendicular D K equal to part of the resistance at D, then a and a l being the arcs de- scribed in the descent and subsequent ascent, then (a, ) -^- - 9 = area of curve a K B. NEWTON'S PEINCIPIA. 251 This admits of a very short proof, for the equation of motion being we have by one integration and at the limits of integration s= and *= +a p we have v o which is what we had to prove. We have now to consider the nature of the curve KB. We have accurately true s = a sin (n t + b) v = a n cos (n t + b) ~\ (n t -f &) J calling y the ordinate and supposing the resistance to be equal to k v m .:y = xa m n m -' i cos" 1 (nt+b). Let x be the abscissa measured from the middle point O of a B, and x + a ^ = a sin (n t + b). First, let the resistance vary as the velocity, then eli- minating t, If we neglect the variations of a with t, this is the equation to an ellipse. The terms in y thus neglected are of the order x 2 . Secondly, let the resistance vary as the square of the velocity. Eliminating t 252 NEWTON'S PRINCIPIA. If we neglect the variations of a with t this is the equation to a parabola. The terms in y thus neglected are of the order x 2 . Newton takes these figures to be accurate enough for practical purposes, and we might now proceed to deduce the resistances at O from the difference of the arcs. But we have said enough to illustrate Newton's method. It is not so convenient for use as the more modern formula?. But it is remarkable that the two methods are of equal degrees of approximation. Thus Newton arrived at as accurate a result as that which we now use, the only difference being that he expressed his result, according to the custom of the age, in a geometrical form. We must now deduce the true law of resistance from a combination of theory and experiment. It is clear from what precedes, that the observations must be made on the decrements of the arcs. Newton suspended a wooden ball weighing 57^ ounces troy, its diameter being 6|j- London inches, by a fine thread on a firm hook, so that the distance between the hook and the centre of oscillation of the globe was 10 feet. He marked on the thread a point 10 feet 1 inch distant from the centre of suspension, and even with this point he placed a ruler divided into inches, by the help whereof he observed the lengths of the arcs described by the pendulum. Then he numbered the oscillations in which the globe would lose part of its motion. In the following table the first column represents the first arc, or space the pendulum was drawn aside from the perpen- dicular ; the second column the number of oscillations ; the third column the last arc, which is always less than the first ; and the fourth column the difierence between the first and last arcs ; NEWTON'S PRINCIPIA. 263 2 164 If i T 4 121 3 i 2 8 69 7 1 16 35 14 2 32 18 28 4 64 91 56 8 The difference between the arc of descent and the sub- sequent arc of ascent will be above differences divided by the number of oscillations. In the greater arcs these are very nearly as the squares of the arcs described, and in lesser oscillations in a ratio somewhat greater. But theory taught us that if the resistance vary as the m ih power of the velocity, the above difference of arcs should vary as the ra th power of the arc. Hence the resistance when the globe moves very swift is in the duplicate ratio of the velocity nearly, and when it moves slowly, somewhat greater than in that ratio. The points to be determined by experiment are how the resistance depends on the velocity, how on the surface of the body, and how on the density of the fluid. To deter- mine the first, we must assume some law of resistance : let us suppose that the resistance varies partly as the velocity, partly as the square of the velocity, and partly in some intermediate ratio. This being the result sug- gested by the last-mentioned experiments ; let the differ- ence of the arcs be represented by where V is the greatest velocity in any oscillation. Then, according to the notation of the preceding theory, we have hence substituting, the difference of the axes will be repre- sented by 254 NEWTON'S PRINCIPIA. A'. 2~a + B'. 2~$ + C 7 ^* a form well adapted for a comparison with the results of experiments. Taking the second, fourth, and sixth expe- riments of the set discussed above, we get three equations to determine the three quantities A', B', C'. A' = .0000916, B' = .0010847, a = .0029588. But the resistance will be expressed by .0000583 a' + .0007593 of * + .0022169 a' 2 j where W is the weight of the body. Thus for such swift motions in air as those varying from 4 to 120 inches per second, the resistance varies as the square of the velocity. Since a! in the second case represents 1, in the fourth 4, in the sixth 16, the resistance will be to the weight of the globe, in the second case, as .0030345 to 121, in the fourth .041748 to 121, in the sixth .61705 to 121. The next point to be determined is the manner in which the resistance depends on the surface. For this purpose Newton suspended a leaden ball of 2 inches diameter, weighing 26 ounces troy, by the same thread that he suspended the former ball, the length of the simple pen- dulum being 10J feet. He found the resistance to the ball was 7^ times that on the former ball. But the ratio of the squares of the diameters was llyf to 1 nearly. " Therefore the resistance of these equally swift balls was in less than a duplicate ratio of the diameters. But the resistance of the thread has not yet been considered, which was certainly considerable. This could not be accurately NEWTON'S PRINCIPIA. 255 determined, but it was found to be greater than a third part of the whole resistance of the lesser pendulum. When this third part is subtracted, the ratio 7^ to 1 be- comes 7 to f or 10 to 1, a ratio not very different from llyf to 1." Since the resistance of the thread is of less moment in greater globes, Newton also tried the experi- ment with a globe whose diameter was 18| inches. He found the resistance on this globe to be 7 times that on the first globe, whose diameter was 6| inches ; but the squares of the diameters are in the ratio of 7.438 to 1. The difference of these ratios is scarcely greater than what might arise from the resistance of the thread. Therefore the parts of the resistances which are in swift motions when the globes are equal as the squares of the velocities, are also when the velocities are equal as the squares of the diameters of the globes. We shall find occasion to modify this conclusion in another chapter. In order to determine the manner in which the resist- ance depended on the density of the fluid, Newton calcu- lated the resistance made to a body oscillating in water and in air, and found that that part which is proportional to the square of the velocity (and which alone it is neces- sary to consider in swift motions) is proportional to the density of the medium. ** This is not perfectly accurate ; for more tenacious fluids of equal density will undoubtedly resist more than those that are more liquid, as cold oil more than warm, warm oil more than rain water, and water more than spirits of wine. But in liquids which are fluid enough to retain for some time the motion im- pressed upon them by an agitation of the vessel, and which being poured out are easily resolvable into drops, the rule will be pretty accurate, especially for large bodies moving with a swift velocity. " Lastly, since it is the opinion of some that there is a 256 NEWTON'S PKINCIPIA. certain aetherial medium extremely rare, which freely pervades the pores of all bodies, and from such a medium some resistance must needs arise, in order to try whether the resistance which we experience in bodies in mo- tion be made upon their outward superficies only, or whether their internal parts meet with any considerable resistance upon their superficies, I thought of the follow- ing experiment." Newton suspended a round deal box by a thread 1 1 feet long, and compared the resistance made to it when empty and when filled with lead. When full of air its weight was 7 y h of the weight when filled with metal. But since the decrement of the arc varies in- versely as the weight, the box when filled with lead should make 78 oscillations before the arc was decreased by the same quantity that when filled with air it lost in one oscillation. But Newton counted 77 oscillations. Assuming that the greater resistance of the full box arises not from any other latent cause, but only from the action of some subtile fluid on the included metal, we may suppose that this resistance in equally swift bodies will be as the num- ber of particles that are resisted. Let A be the resistance on the external and B on the internal superficies of the box when empty. Then A and 78 B will be these re- sistances when the box is full. By the preceding ex- periment the resistance on the full box A + 785 is to the resistance on the empty box A B as 78 to 77. Solving this simple equation, it follows that the resistance on the internal parts of the empty box (B) will be j^^- th part of the resistance (/4) on the external superficies. NEWTON'S PRINCIPIA. 257 CHAPTER VI. MOTION OF FLUIDS RUNNING OUT OF SMALL ORIFICES. 1. Newton's solution of the question, without limiting the orifice. 2. Newton's corollary to deduce the resistance made by a fluid to a body moving in it. 3. The fallacy of Newton's reasoning How others attempted to pursue the investigation. Note vii. 4. The velocity as given by the Equations of Motion. Note vii. 5. The efflux of an elastic fluid through a small orifice strange conclu- sions to be deduced from the formula St. Venant and Wantzels' experiments. Note vii. 1. THE problem to determine the motion of water running out of any vessel through a hole is so difficult that it has not yet been completely solved. We cannot there- fore be surprised if Newton's solution of the question is not very satisfactory. He begins by considering a case in which he is able to give some account of the motion. Let us suppose that we have a circular cylinder filled with fluid, the surface of which is retained at a constant level in such a manner that the velocity of descent of all its parts is also uniform. In order to aid our conceptions, Newton supposes this effected by the descent of a cylinder of ice APQB, of the same breadth with the cavity of the vessel, and having the same axis. The motion is supposed quite uniform, and its parts, as soon as they touch the su- perficies A B, are supposed to dissolve into water and flow down by their weight into the vessel. Let u be the uni- form velocity of descent of the cylinder, and let k be the 258 NEWTON'S PRINCIPIA. height through which a body must fall to acquire this velocity, Let there be a hole E F in the centre of the bottom, and let the whole cavity of the vessel which encompasses the falling water be full of ice, so that the water may pass through the ice as through a funnel. The particles of water are supposed to cohere a little, that in falling they may approach each other, and thus instead of being divided into several, they will form a single cataract. We shall also suppose the form of the funnel such that the particles of water fall freely down. The lines of motion will form surfaces of re- volution whose axis is the axis of the funnel, and the forces of cohesion act- ing on any particle have a resultant normal to these surfaces. The velocity of the particle will not therefore be affected by this force, and if z be the depth of any particle below the plane A B and v its velocity we have each particle arrives with the same velocity at the aperture E F. The form of the funnel was supposed such as to admit of this motion; hence the area of any section must be inversely as the mean vertical velocity of the fluid for all parts of that section. Now, if the particles of water did not exert any action on the icy funnel through which it flows we might suppose this funnel dissolved into water without changing the cir- cumstances of the motion. But the descending fluid does act on the dissolved ice, and the whole nature of the motion is changed. But Newton argues that nevertheless the NEWTON'S PBINCIPIA. 259 efflux of water, as to its velocity, will remain the same as before. It will not be less, because the ice now dissolved will endeavour to descend ; it will not be greater, because the ice now water cannot descend without hindering that of other water equal to its own descent. The same force ought always to generate the same velocity in the effluent water. Newton does not mean that the same quantity of water flows out, but that the same velocity is generated. This is not, however, the vertical velocity at the orifice ; " for the particles of water do not all of them pass through the hole perpendicularly, but flowing down on all parts from the sides of the vessel and converging towards the hole, pass through it with oblique motions, and in tending downwards meet in a stream whose diameter is a little smaller below the hole than at the hole itself." The particles of water at the actual orifice are not all moving in the same direction. Those in the centre are descending vertically while those near the circumference have a lateral motion, but at the vena contracta the whole fluid is descending vertically. If therefore B, B' be the areas of the sections of the vena contracta and orifice, and v the velocity of the fluid at the vena contracta, the mean velocity of the fluid perpendicular to the orifice will be clearly B B' u The quantity of water that flows past any horizontal plane is proportional to the product of the area by the mean perpendicular velocity. Thus the discharge is equal to q =B V<2 ff (h + k) where h is the depth of the vena contracta below the sur- face of the fluid in the vessel. T> 1 Newton found the ratio = r^ nearly, whence the 260 NEWTON'S PRINCIPIA. mean velocity perpendicular to the orifice is nearly the orifice being at the side of the vessel : thus the mean perpendicular velocity at the orifice is that due to half the T> depth below the surface. This ratio p, however, depends on the nature of the orifice and the thickness of its sides ; accordingly different experimentalists have given different values. Thus for the ratio* of the diameters (the square 21 11 root of the above ratio), Newton found ., Poleni , '- > i > Bernoulli ^, Du Buat =, Bossut ^-, Michelotti ^, Ven- turi -, Bidone ^ , Eytelwein . The velocity of efflux will be the same whether the hole be at the centre or the side of the vessel ; for though a heavy particle of water will take a longer time in descend- ing to the same depth by an oblique curve than by a straight line, yet in both cases it acquires in its descent the same velocity. The velocity also is independent of the form of the hole, for it merely depends on the depth below the surface. And if the orifice be immersed in water the velocity of efflux is that due to the height of the water in the vessel above that of the surrounding fluid. 2. Newton proceeds to deduce as a corollary from this theory the law of resistance of a fluid to the motion of a body in it. The case of a discontinuous fluid has already been discussed, and it has been shown that if v be the Encyc. Brit., Hydrodynamics. NEWTON'S PKINCIPIA. 261 velocity, and A the area of a great circle of a sphere moving in a fluid of density p, that the resistance will be A y 2 p. The globe and particles are supposed perfectly elastic, and thus endued with the utmost force of reflexion. But if, on the contrary, they are perfectly hard, and with- out any reflecting force, the above expression for the resistance must be diminished one half. " But in continued mediums the cylinder as it passes through them does not immediately strike against all the particles of the fluid that generate the resistance made to it, but presses only the particles that lie next to it, which press the particles be- yond, which press other particles, and so on, and in these mediums the resistance is diminished one other half." If in the centre of the hole E F a small circle P Q, of area C, is placed, the weight of water which it sustains will be greater than the weight of a cone whose base is P Q and alti- tude H G, and less than that of a spheroid on the same base and of the same altitude. For let HP and H Q be the boundaries of the cataract. The cataract falls freely, and therefore there is no pressure -' on the sides of the mass of still water H Q P. The pressure on the circle P Q is the weigH of water H Q P. And this will still be equal to the pressure if the ice which forms the sides of the cataract be dissolved, and the whole water be left to flow out of the orifice in any manner whatever. A little con- sideration will show that the mass of fluid H P Q must have its boundaries meeting in a point at H, and being convex to G, they also meet the sides of P Q at an acute angle. Therefore these boundaries lie without the surface 8 3 2(J2 NEWTON'S PRINCIPIA. of the cone and within the surface of the spheroid described on the base PQ, and with an altitude HG. The weights of these two solids are respectively 1 2 -^Chffp, and - C h g p, where p is the density of the fluid, and h the altitude HG. Therefore the weight supported by the little circle lies between these two quantities. If the circle be very small, both these will be small, and we may take the weight sup- ported as being very nearly equal to their arithmetic mean, that is where P is the pressure on the little circle. If, however, C be not very small, compared with B, let us assume that the pressure is ft P = C* where B is the area of the orifice EF. Then when C is very small, this must agree with the pre- vious result, hence /3 = 1, and when C = B the weight supported is that of a cylinder whose base is C and altitude h, hence = ^. Moreover, so long as C is less than half B, the expression makes P lie between the limits assigned above. Newton next proceeds to point out the analogy between the pressure on the circle and the resistance to a circle moving in a still fluid. Let the vessel touch the surface of NEWTON'S PKINCIPIA. 263 stagnant water with its bottom CD, and let the water run out of the cylindrical canal EFTS perpendicularly to the horizon, and let a small circle P Q be placed anywhere in the middle of the canal with its surface horizontal. Let U be the velocity of the fluid at the surface, V that at the orifice, A the area of a section of the cylinder, B and C that of the orifice and little circle P Q, h the altitude of the cylinder. Then V* = U 2 +2gh, V(B - C) = UA. If we suppose A to be infinitely greater than B, this last equation shows that U is indefinitely small, and therefore and the pressure on the little circle is P = . ~ where is some quantity that becomes unity when C 1 ^- diminishes without limit. -D Cv* ..P= p Now let the orifice of the canal E F S T be closed, and let the little circle ascend with such a velocity that the rela- tive motion of the circle and fluid which is compelled to rush past it may be the same as when the water fell from the height HGr, and the circle was at rest. The pres- sure on the circle will be the same as before. The r\ velocity of the fluid will be R _ ^V, and that of T> the plane ^ _ ~ V. Let us suppose B infinitely greater than C, then the resistance on a plane moving with a velocity V in still water is C u 2 p. If a cylinder, 264 NEWTON'S PRINCIPIA. a sphere, and a spheroid, of equal breadth be placed suc- cessively in the middle of a cylindric canal, so that their axes may coincide with the axis of the canal, these bodies will equally hinder the passage of the water through the canal. The resistances will then be equal. The resistance, therefore, on a sphere will be C v 2 p, where C is the area of one of its great circles. 3. The investigation of this question, as given by Newton in his first edition, was very erroneous. He had totally neg- lected -the contraction of the vein after the fluid had passed the orifice ; hence he had deduced that the velocity of the efflux was that due to only half the height of the water in the vessel. This mistake he afterwards corrected, but the investigation still remains open to very serious objections. For it is quite certain that the first case considered by Newton in which the water descends by a funnel bears no resemblance to the actual state of the motion. The water is not found to flow out in a cataract, leaving a mass of unmoved water supported by the bottom. Each particle, whether vertically over the hole or near the circumference of the cylinder, descends in a nearly vertical direction, acquiring or losing velocities in nearly the same ratio. Those particles which are once in a horizontal section remain very nearly in the same horizontal plane. When the particles approach very close to the bottom, they acquire a considerable horizontal motion, and, in consequence, the issuing stream continues to contract after it leaves the orifice. The manner in which Newton deduces the law of resistance from the velocity of efflux is also erroneous. It is very ingenious and wonderful, but at the same time very uncertain. The proposition being false in principle, we cannot expect a corollary founded on that principle to be altogether correct. The reasoning by which the resis- tances to a sphere and cylinder are shown to be equal can NEWTON'S PRINCIPIA. 265 only be correct on the assumption that all the water above the cylinder, sphere, or spheroid, whose fluidity is not necessary to make the passage of the water the quickest possible, is congealed. This Newton himself admits. But such an assumption is by no means a legitimate one. It is also certain, by experiment, that the amount of the resistance depends very materially on the form of the surface of the body. [See NOTE VII.] 266 NEWTON'S PRINCIPIA. CHAPTER VII. THE MOTION OF WAVES. 1. General consideration on the nature of Waves. 2. Waves in air sound. a. The nature of sound deduced from the phenomena. Scholium, Prop. L. ft Examination of the case considered by Newton. 7. Velocity of sound Newton's error. 8. The manner in which sound spreads after entering through an orifice. Newton xlii f. The notes sounded by different pipes. Scholium, Prop. L. 3. Waves in water. o. The motion is of the vibratory kind. . Newton's reasoning on this subject. Newton, xliv. y. Velocity of waves. Newton xlv. & xlvi. . The nature of the motion of waves as given by a strict hydro- dynamic theory. Note viii. Waves caused by the motion of a boat, /t. Cause of breakers over sunken rocks. 6. How the wind raises the waves. 1. To one who has never considered the nature of the motion that occasions the appearance of a wave, the idea is at first difficult. John Bernouilli the younger * declared that he could not understand Newton's proposition on this sub- ject. The best illustration is that of a field of standing corn, because it clearly shows that, in some waves at least, there is no actual transfer of a quantity of matter. When the wind blows on the field a hollow will be seen travelling along it. This is a wave. There can manifestly be nothing * Whewell, History of Inductive Sciences, vol. ii. p. 310. quotes Prize Dis. on Light, 1736. NEWTON'S PRINCIPIA. 267 moving across the field, because everything is fastened to the ground ; but there is the appearance as if something were moving along. Each particle of corn in turn descends and rises again, and, as each ear is a little later in its mo- tion than the one in rear, the state or position in which the first was, has travelled to the second. Our first idea of a wave is, that it is a state of motion which travels along, the particles themselves only oscillating. Our second, that this state of motion may gradually change, and either increase or die away. Suppose we have a series of particles in a straight line, let them begin to move up and down, either in this straight line, or in any parallel straight lines, so that if one be taken whose mean position, measured along the straight line is distant x from the origin of measurement, its distance at the time t from its mean position is represented by the formula y =af(nt m x), where f is any functional symbol. Then here, so long as n t m x is constant, y remains the same ; that is, a " state " travels along in such a manner that its distance x at any time t is given by this formula. Differentiating we have but ~-y- is the velocity ; hence the " state " travels with the velocity - ; this, therefore, is the velocity of the wave. Suppose we call this v, then we have y = af m (v t #). 268 NEWTON'S PRINCIPIA. The " length " of a wave is the distance between two par- ticles in a similar state of motion. If A be this length, y must be the same for x = x and x = x + A. ~No\vf(z) will be a periodic function : suppose its values recur at intervals of c, . . m A = c, c or m = - A and therefore, This is the type of a wave that travels in any one direction. But it can also be demonstrated that it is the general expression for all waves propagated with a uniform velocity in one direction without change of form. There are a great variety of waves; we may have waves transmitted through elastic fluids, where a state of condensation travels along. We can have them propa- gated along the surface of an inelastic fluid, as water in the form of an elevation. We can have them in solid bodies, the particles of which are supposed to oscillate about their mean position, and to act on each other with forces different from those which would act if the body had been fluid. We shall briefly consider these in turn. These waves may not travel at all parts in the same direc- tion. They may spread themselves out from a centre. In all cases a surface passing through all points in a similar state of motion is called a front of the wave. The phase of a wave at any point is the situation of the particle of that point considered as aifecting its displacement and motion. Thus, two particles are in the same phase when their displacements are equal, and motions the same. They are in opposite phases when the displacement and motion of one are equal but opposite to those of the other. NEWTON'S PRINCIPIA. 269 2. The true nature of sound can only be discovered by observation. We must know something about what we have to explain. The experimental facts divide themselves into three species. First, those relating to the manner in which the sounding body makes the sound. Secondly, those relating to the manner in which the sound when formed is conveyed to the ear. Thirdly, those relating to the manner in which, when the sound is thus conveyed, we hear it. In regard to the first, it is universally true that all sounding bodies are tremulous bodies. All vi- brating bodies, however, do not give sound. In regard to the second, it is observed, (1.) that sound cannot be con- veyed through a vacuum. Hanksbee suspended a bell in the receiver of an air-pump ; the sound died away gradually as the air was removed (Phil. Trans. 1705.) (2.) Sound is not instantaneously conveyed ; the report of a gun is not heard until after the flash has been seen. But all sounds travel with equal velocity. Thus the various notes of any piece of music played at a distance reach us in perfect order. The velocity is found to vary slightly with the temperature, and at 62 Fahrenheit travels at the rate of 1125 feet per second. (3.) Sound, unlike light, does not travel necessarily in a straight line, but on entering through an orifice, as a window, spreads out in all directions, but, nevertheless, it is heard with greatest distinctness in front of the 1 orifice. Thus the sound of a carriage can be heard round a corner, but a change in the loudness is perceptible the moment when the carriage goes behind the intervening walls. (4.) Sound can be conveyed by solid bodies, and other fluids besides air, but in passing from one to another a considerable portion of its intensity is lost. (5.) The intensity of sound diminishes as the distance from the sounding body increases. But when sound travels along a tube, as a speaking pipe, the gradual diminution of the sound 270 NEWTON'S PRINCIPIA. is very slow. (6.) Sound travelling through a dusty atmosphere produces no perceptible motion in it. In re- gard to the third species of facts, we know but very little ; it is sufficient for our present purpose that we find in the ear vibrating bodies prepared to receive the sound. These are the more obvious phenomena. They will serve as a guide to the true theory. The true test of that theory will be in its exact explanation of more refined and less obvious phenomena. Since sound is caused by a body vibrating in air, before we invent any new theory we should prove that none of the natural consequences of such a motion will account for the phenomena of sound. Now we know that the motion of a tremulous body will disturb the air near it, and that moving air the air beyond it and so on. Thus a disturbance will be propagated on all sides in the elastic medium which surrounds the body. Also we find tremulous bodies in the ear fitted to receive such vibrations. What is more natural than to suppose that these vibrations are themselves what we call sound ? To verify this it is ne- cessary to examine strictly the nature of the disturbance in the air caused by a vibrating body. The investigation cannot be given here, but the general result is as follows. Suppose a disturbance propagated from a centre in all directions, then, first, the intensity of the disturbance will decrease as the square of the distance increases. Thus as the disturbance was originally small, the motions at some little distance from the sounding body will be quite insensible to any of our senses except that one which was expressly adapted to receive its impressions. Secondly, the velocity with which any wave travels is independent of the nature of the disturbance, and agrees very closely with the observed rate at 'which sound travels. Thirdly, the nature of the motion of the particles is such NEWTON'S PRINCIPIA. 271 that at some distance from the sounding body, the directions of oscillation of the particles will pass through the centre of disturbance supposing the propagation symmetrical in all directions, or more generally, whatever be the nature of the disturbance, the vibrations are normal to the front of the wave. In fluids no other vibrations are possible. Of course Newton could not investigate completely the motion of sound. That was far beyond the power of the mathematics of his day. But, in a wonderful manner he solved to a certain degree the simpler case of the motion of the air in a tube. The principle he used, and the present mode of reasoning on this subject, are both illustrated in the following analytical view of the investigation. Whatever the motion of the air may be, we suppose the tube so small that we need not consider any motion ex- cept that which is along the length of the tube, and this motion will be the same for all particles in the same per- pendicular section of the tube. Suppose, then, that at the time t, the particle which when the air was at rest was at a distance x from the origin of measurement is at the dis- tance x + . Then an element of air whose length had been dx, is now dx + d%, and as the mass must remain the same, the density which was D is now T -j dx The square of may be neglected, for we know that that particular motion of the air which we call sound, whatever it may be, is very small. Also the pressures on the two sides of the element are, when reduced to a unit of area, p and p + d x, 272 NEWTON'S PRINCIPIA. and since the pressure varies as the density we have P =*p, and therefore the moving pressure on the element is d Pj K -~ a x. dx But the mass moved is D dx ; hence on substitution for p, the accelerating force on the element is .ii dx* Thus the force depends on the displacement, and the displacement, in its turn, on the force. If we assume a form for the displacement, and then show that this dis- placement leads to a force that will produce this exact displacement, we have discovered a possible motion. And if this displacement also agree with all the other conditions of the question, we have discovered the actual motion. For it is clear that from a given disturbance under given circumstances, only one kind of motion can result. Newton assumes accordingly that = a. sin (nt mx). The accelerating force is then, by two differentiations, x. m z .a sin (ntmx) = - Km?. that is, the force varies as the distance from a fixed point' and urges the particle towards that point. This force is well known to lead to the very form for that we started with, (Prop, xxxviii. of Book I.) provided w 2 = x w 2 , or the velocity ( ) with which the wave is propagated is *s/iT This, therefore, is the velocity of sound. NEWTON'S PRINCIPIA. 273 As an example of a case in which the motion is actually represented by this law, let us take an infinite tube, and suppose the air in a small part of it to be set in motion, so that it will begin to move according to the form we have assumed for . Let the extremities of this part of the tube be A and B, the particles A and B and all the remainder of the air in the tube is supposed to be at rest. Let us now consider the motion during a small time 8 1. The particles between A and B will, by the above reasoning, continue to move according to the law assumed for . The velocity and condensation, therefore, being represented by ~ and ~ will be respectively at d x an sin (nt mx) and a m sin (M 1 m x). Let x be measured in the direction from A to B, which we shall suppose from left to right. The particle A is at rest, the condensation at that point and for all points on the left of A is zero. Take a particle a on the right of A, where A a = 8 x, the condensation at that point is indefinitely small and is decreasing. The particle A will, therefore, remain at rest. The particle a will come to rest, that is, be in the same situation that A is in, at the end of the time 8 1, where n 8 t = m 8 X) that is, the left end of the pulse travels onward with a velocity , leaving the air behind it undisturbed. By similar reasoning, it can be shown that the right end of the pulse travels onward with the same velocity. At the end of the time 8 t, the pulse will merely have advanced a space 8 x, and the circumstances of the motion will be the same as before. The same motion will, therefore, be T 274 NEWTON'S PRINCIPIA. repeated in the next interval, and so on throughout all time. Thus we see a single pulse can be propagated along the tube without any change in magnitude, form, or time of vibration. In order to form these pulses let us imagine a plate to be placed at one end of the tube, vibrating accord- ing to the law = a sin n t, then each pulse of the plate as formed will be propagated unaltered along the tube, and the motion at any distance x from the plate will be given by the law = a sin (n t m or). The velocity of sound is independent of the values of m and n. These are the only constants on which the differ- ence between two notes could depend. Whatever, then, is the pitch of a sound, it will travel with the same velocity. The numerical value of the velocity of sound depends on the constant x. This expresses the constant ratio of the elastic force or pressure of the air to its density. The density of the air Newton calculated to be about th part of the density of quicksilver, and the pressure of the air is equal to the weight of a column of mercury about 30 inches high. Hence the ratio of these two is equal to OA g x x 11890, the units being feet and seconds of time. The square root of this, which is the velocity of sound, is 979. Sound, therefore, should travel at the rate of 979 feet per second. The elastic force p for a given density being increased by an increase of temperature, the result thus obtained should be too small in hot air and too great in cold. But this NEWTON'S PRINCIPIA. 275 result should be true about the temperature of spring and autumn. When Newton completed this investigation the true velocity of sound was not accurately known. From some rough experiments, conducted by himself, he believed that this result was really near the truth. But subsequent experiments showed that it was erroneous by 163 feet. This was a very serious error, and Newton tried to explain it away, by saying that no allowance had been made either for the crassitude of the solid particles of the air, or the presence of vapours. He even attempted to show that on taking these into account, the calculated and observed velocities were in close agreement. Such explanations are, however, unsatisfactory, and unless some other explanation had been found, the theory would stand in direct opposition to experiment. The theory of sound continued to advance by the labours of the great mathematicians who followed Newton ; the motion of sound in a tube was investigated without any assumption as to the nature of the motion, but the discrepancy still remained unexplained, This was reserved for Laplace, who remarked that in rapid vibrations, the sudden rarefactions and condensations of the air must affect its temperature, and therefore its elasticity. The amount of this must be determined by experiment, and it was shown that if s represent the condensation, the form to be used must be, not as heretofore p = * D (1 + s} y but p = K (1 + /3) D . (1 + *), where /3 lies between '3748 and '4. The velocity of sound, therefore, is not \/ x, but A/ x (1 + /3). The agreement 276 NEWTON'S PRINCIPIA. of the observed and calculated velocity was now found to be very close. The value of x depends on the general temperature of the air. By Amontem's law, we know that if x be the value at any temperature, the value at any other exceeding this by 6 will be X = X (1 + 0), where a = -. If, therefore, the temperature of the air be above 60, the velocity of sound will be 1124 The demonstration we have given of Newton's propo- sition of the motion of sound in a tube may easily be extended into a vigorous demonstration. For the accele- rating force on any element having been shown to be the equation of motion must be putting a 2 for x (1 +/3). This equation must be true for the motion of sound in a tube under all circumstances. Thus, the tube may be finite, open or closed, and the dis- turbance may be caused in any manner. The complete integral of this equation is known to be =/(*-*) + t(*t + x), where / and fy are unknown functions depending on the nature of the disturbance and the other circumstances of the tube. This investigation of the motion of sound in the air depends on the assumption that the medium is a perfect fluid. But this is not the case. The effect of internal NEWTON'S PRINCIPIA. 277 friction is found to consist partly in a diminution of the velocity of propagation, and partly in a more rapid dimi- nution of the intensity than would correspond to the increase of distance from the centre of divergence. The diminution of velocity is found on calculation to be so small, that it is less than one foot in 1,578,000 miles. The change of intensity, though not so utterly insignificant as the change of velocity, is found to be still insensible. Such small differences as these may be altogether neglected. It is well known that sound does not always travel in a rectilinear direction. "Sounds," Newton remarks, " may be heard though a mountain be interposed ; and if they come into a chamber through the window, dilate themselves into all parts of the room, and are heard in every corner, not as reflected from the opposite walls, but as directly propagated from the window, as for as our senses can judge." Let us imagine a series of waves to be advancing directly through an orifice, and let the aper- ture be divided into small elements. Each element may be considered as the origin of a series of disturbances. The disturbance in any part of the room will be made up of those propagated from all the several elements. Now, the motions being all small, it is a well known principle that the actual disturbance will be the sum of all those that would have been propagated from each separate element on the supposition that there was no other motion at the same time in the medium. Each separate disturbance may be calculated by the known rules according to which any motion spreads itself on all sides. The actual disturbance, on which the sound depends, will be represented by a definite integral. It is needless to go through the work, but the result is, that, provided the length of a wave be not very small compared with the orifice and this is the case in sound, and the waves on the surface of water the 278 NEWTON'S PRINCIPIA. motion will be sensible at other points of the room, as well as directly in front of the orifice. Thus Newton's expectation was confirmed, though he was ignorant of the condition on which it depended. But if the length of the waves are indefinitely small compared with the size of the orifice, and this is the case in the Undulatory Theory of Light, then it is found that, excepting directly in front of the orifice, the definite integral is altogether insensible, the condensations and forward motions of one wave being superimposed on the rarefactions and backward motions of another in such a manner that there is no sensible disturb- ance. Such waves must be considered as being propa- gated in rectilinear directions. The first step in the theoretical explanation of the sounds produced in pipes was made by Newton. He remarks that Sauveur found by experiment that an open pipe about five Paris feet in length gives a sound of the same tone with a viol string that vibrates a hundred times in a second. Therefore, he argues, there are near one hundred pulses in a space of one thousand and seventy Paris feet, which a sound runs over in a second of time ; therefore one pulse fills up a space of about 10 T 7 n Paris feet, that is twice the length of the pipe. From whence it is probable that the lengths of the pulses in all sounds made in open pipes are equal to twice the length of the pipes. Newton did not examine any further into the subject, but leaves it for others to carry out the theory. Lagrange and Bernoulli were the first to give more minute explanations of the leading facts. Since that time, Euler, Lambert, Poisson, have developed the subject still further. The theory is to a great extent included in the equations to the motion of sound in a tube which we have already given. It can be shown from these that the motion is made up of two waves continually travelling along the tube NEWTON'S PRINCIPIA. 279 in opposite directions. Of these one may often be con- sidered as the reflection of the other at the open or closed end of the tube. These two waves will "interfere" as it is called, and there will be in consequence two sets of points in the tube, which possess remarkable properties. At any point of one set there is no motion in the air, but only a condensation. At any point of the other set there is no condensation, but only motion. The first are called "nodes," the other "loops." These points are placed in regular order, alternately a node and loop at distances one fourth the length of a wave. Suppose a tube closed at one end to be sounding a note in unison with a vibrating plate at the other. Then clearly, since the air must remain in contact with the tube at the closed end, there can be no motion there. That point must be a node. Since the air moves in consonance with the vibrating plate, there must be no "action or reaction" between the air and plate. If there were, the sounds would not be in unison, and the note of both the tube and plate would begin to change. The pressure of the air must therefore be the same on each side of the plate; therefore there is no condensation in front of the plate. That point must be a loop. One end of the tube is therefore a loop, the other a node. Hence the length ( / ) of the tube must be an odd multiple of j, where X is the length of a wave. That is, That note which has the largest value of X which can be sounded from a given pipe is called the "fundamental" note of that pipe. For this note n 0, and we have The pitch of a note being determined by the length of the T 4 280 NEWTON'S PRINCIPIA. wave which forms it, all the notes that can be sounded from the same closed tube will form the series 1111, 1 3 5 7 9' &C ' for we have merely to give n all its values from nothing to infinity, to get all the possible values of A. Suppose now the tube to be open at the end opposite to that by which the air is set in motion. It is assumed that at the open end the density of the air must be the same as that of the surrounding air, a fact that is not exactly true. Taking it for granted, it will follow that both ends of tube must be loops. The length of the tube must therefore be an even multiple of -, that is A '="8- The fundamental note corresponds to n = 1 or A = 27, and all the notes that can be sounded from the same open tube will form the series 1111 1 2 3 4 5' &C ' We also learn that the note of the pipe being measured by the value of A, it will be proportional to the length of the pipe, and that to get the same note from two pipes, one closed and the other open at the end opposite to that at which the air is set in motion, the second must be double the length of the first. The theory of stringed instruments is in many instances remarkably similar to that of wind ones. The equations of the oscillations of a stretched string are exactly the same as those of a sounding pipe. Much progress has been made by the experimental re- searches of Hopkins, Chaldni, Savart, Willis, and others. NEWTON'S PRINCIPJA. 281 But we have no space to do more than merely allude to their labours. There are a great many other waves propagated through the air besides those which can be heard. Our ear is a musical instrument that vibrates in consonance with all notes whose periods lie between two limits. All other sound waves therefore, though their theory is exactly the same, are not heard. Besides these sound waves, there are other great waves which traverse the air in all directions, and whose passage is indicated by the variations of the barometer. These waves are very large, and pass over whole continents in their course, and seem to recur with some regularity. Their theory is different from that of sound, and as yet imperfectly understood. The examination of the " facts " connected with them was begun by Herschel, and has been since continued with much success by William Birt. So also the waves that produce sound can travel through other substances besides air. The theory is in some points different from that we have been considering. 3. That the elevations on the surface of water which we call waves are not really the transmission of a body of water, but merely the movement of a particular state of motion, will be obvious on very little consideration. First, by experiment, a piece of cork floating on water agitated in this manner is found merely to oscillate, and does not advance as the wave moves onwards. Secondly, by theory, such a disturbance being supposed given to the water, as we know by experience will produce the appear- ance of waves, the principles of Hydrodynamics lead us to the conclusion that the motion is really of a vibratory kind. 282 NEWTON'S PRINCIPIA. On this subject Newton has not many propositions. He has shown an analogy between the motion of waves and the oscillation of water in a pipe. Let there be a column of fluid of length / in the U pipe M A m : let the water when at rest stand at the level M m. Then if it be raised above this level in one leg to the height P, it will be depressed an equal amount in the other, say to Q. The weight of water tending to pull it back again will be equal to twice the weight of the column P M ; that is, the force varies as the displacement, and tends to pull the body back to its ori- ginal position. This case of motion has been investigated in the First Book, and the result is, that the motion is vibratory, the time of oscillation will be always the same, and equal to 2 TT divided by the square root of the ratio of the accelerating force at any distance to that distance. But the moving force is twice the column P M, and the mass moved is the column /. Hence the time will be 727 "V-> or it varies as the square root of the length of the column of water. Now, in waves the motion is carried on by the succes- sive ascents and descents of the water ; hence the time that must elapse before any particle now at the top of a wave will again be the top is /27 V 9 where I must here be assimilated to half the length of a wave. We cannot say they are equal to it, but they will increase or decrease together. Newton assumes NEWTON'S PRINCIPIA. 283 /-- ~ 2' and therefore the time of oscillation is where A is the length of a wave, measuring either from top to top or hollow to hollow. This expression for the time is the same as that for the oscillation of a simple pendulum whose length is A. Hence a wave whose length is equal to the length of a simple pendulum should advance a space equal to that length in the time of one vibration of that pendulum. The velocity of the wave will therefore be i ^ Thus the velocity is independent of the magnitude of the waves, or the density of the fluid. It is greater for long waves than short ones. The waves of sound in air travel with equal velocity, whatever be their length. The undu- latory theory of light requires us to believe that, except when light goes through a vacuum, the waves of different length travel with different velocities. Newton does not propose this as more than a first attempt. He says, " These things are true on the suppo- sition that the particles of water ascend and descend in a right line. But, in truth, that ascent and descent is per- formed in a circle, and therefore I propose the time defined by this proposition as only near the truth." In modern times the motions of waves have been more accurately investigated. If h be the uniform depth, A the length of the waves, v the velocity of propagation, 284 NEWTON'S PRINCIPIA. If the depth be very great we have v = v f very nearly, and when the wave is very long, v =. V ' g h very nearly. The former of these results is not the same as Newton's, but the general conclusion will be unaltered. The exact result may, however, be obtained by Newton's reasoning, if we slightly change the value assumed for I and take 1= 2; 7T The second result admits of an easy demonstration, which we shall speak of in another place. [See NOTE VIIL] NEWTON'S PRINCIPIA. 285 CHAPTER VIII. THE THEORY OF THE TIDES. I. Newton's investigation on the Tides. a. The tides considered as a question in the motion of fluids, de- duced from the Lunar Theory. /3. General explanations of eight phenomena of the tides. 7. The calculation of the height of the lunar and solar tides. 8. The tides in the moon. II. The theories that have been proposed since Newton's time. Note IX. 1. The Equilibrium Theory. a. Its fundamental hypothesis. ft The results of calculation made according to this theory, three kinds of tides. y. Airy's opinion of this theory. 2. The Hydrodynamic Theory. a. Its fundamental assumptions. /8. The results of calculation, three kinds of tides. 3. The Wave Theory. a. Consideration of ocean tides. #. River tides, results of calculation and explanation of the chief phenomena. y. Where this theory fails. III. Some results of observation. Note IX. THE cause of the tides in the ocean have always excited the curiosity and wonder of mankind. Their regularity, the magnitude of the scale on which they take place, the difficulty of conceiving what that power could be that can raise twice a day so vast a body of water, all render the question one of the most interesting in the whole range of science. Phitheas, it is said, was the first who remarked that the tides followed the course of the moon. But the " why " was still as great a mystery as before. Galileo 286 NEWTON'S PRINCIPIA. thought he could explain it by a combination of the rotation of the earth about its axis with the annual motion round the sun. The impossibility of deducing an explanation from these premises of the most ordinary phenomena of the tides is evident from what we have said in a previous chapter. Galileo, busy in establishing by new proofs the rotation of the earth, was naturally inclined to find in it the cause of a phenomenon so mysterious, which if it had succeeded would have furnished him with a most powerful argument. Descartes had another equally impossible theory; it was reserved for Newton to suggest the true cause of the motion of the sea. The discoverer of gravi- tation could not be long before he saw that whether or not it was the only cause of the tides, it must certainly be one of them. The attraction of the moon could not be the same in all parts of any extended sea. Motion, therefore, must ensue. Nor could any position of rest be ever assumed, because the earth and moon themselves are in motion. And here Newton showed his superiority over those philosophers who afterwards treated of the same subject. He saw that the motion of the tides was a question of Hydrodynamics, and in his First Book he con- siders it as such, and has even shown that in one particular case the water would be lowest in that part which is immediately under the moon. That he did not do more is no reproach. Even at the present day the theory of Hydrodynamics is in its infancy : how impossible then it must have been in Newton's time, when the simple laws of the motion of a single particle had only just been understood, to have attempted the consideration of a fluid under the action of complicated forces. Newton gives, therefore, merely a general explanation of the tides, and enters into some numerical calculations merely as a first attempt. NEWTON'S PEINCIPIA. 287 Let us imagine that a uniform channel is cut round the earth at the equator and filled with water : let it be sup- posed that this fluid is disturbed by the action of the sun or moon; supposed for simplicity also to move in the equator. Newton endeavoured to discover in what man- ner the water would move in this imaginary channel, Book L, Prop. LXVL, Cor. 19. In considering the motion of a satellite supposed when undisturbed to describe a circle round the earth, Newton has shown that the disturbing force will in two different ways cause the orbit to become slightly oval, the earth remaining in the centre and the greater axis being perpendicular to the straight line joining the sun and the centre of the earth. This is somewhat parallel to the case in point. If each particle of water were kept in its place by its centrifugal force they would tend to rise at the quadratures and sink at the syzygies, they would be swifter at the syzygies and slower at the quadratures ; they would ebb and flow in its channel after the manner of the sea. But the analogy is not perfect, for the water is supported, not by its centri- fugal force, but by the channel in which it flows. We see, however, that an ebbing and flowing of the sea will be produced, though the points of greatest and least height of water may be different. This reasoning applies whether the disturbing body be the sun or moon. Though, therefore, by this reasoning we cannot deduce a perfectly accurate theory of the tides, yet we are able to perceive that the sea ought to rise and fall twice in each day. The moon will form at any place two high tides and two low tides in the interval between leaving the meridian of that place and returning to it again, that is, in a lunar day. The sun will also at the same place form two high tides and two low tides in the interval between leaving the meridian and returning to it again, that is, in a solar day. 288 NEWTON'S PRINCIPIA. These two will not appear separately, but a mixed tide will result from the two. Since both are small, we may consider the actual tide as being the sum of the two tides that would have been formed had the disturbing bodies acted alone. The force of the moon to raise the tides is much greater than that of the sun, as we shall presently see, so that for most purposes we may altogether neglect the solar tide, or rather regard it as a small correction to be applied to results calculated on the supposition that the moon alone acted to raise the tide. First. The lunar tide follows the moon at a given in- terval, called the " establishment." Hence as the moon by her proper motion rises every day about forty minutes later than on the preceding day, the high tide will be as much later every successive day. But more minute investigation shows that the high tide does not follow the moon's passage over the meridian by any constant interval. We must make a small correction for the effect of the solar tide. If the sun follow the moon across the meridian, the solar high tide will follow the lunar high tide, and the actual high water formed by the union of the two will lie between these two tides, and will therefore follow the moon's passage across the meridian at an interval slightly greater than the " establishment" So if the sun precede the moon the interval will be less. Therefore, when the moon is in the first and third quarters, the high water is a little later, when in the second and fourth a little earlier, than the establish- ment would indicate. Secondly. At or about new and full moon, the high tide formed by the sun coincides with that formed by the moon, hence the high tides will be higher and the low tides lower than when the moon is in any other position relative to the sun. These are called Spring Tides. At or about the half moon, the high tide formed by the sun coincides with the NEWTON'S PRINCIPIA. 289 low tide formed by the moon. They tend to neutralise each other. The high tides will be lower and the low tides higher than at other times. These are called Neap Tides. Thirdly. The effects of the luminaries depend on their distance and vary inversely as the cubes of their distances. Hence the respective tides of the sun or moon are greatest when the luminary is in perigee and least when in apogee. In winter the solar tides are therefore greater than in the summer, and the lunar tides have like changes every fort- night. Fourthly. The effects of the disturbing bodies depend on their declinations. If the moon were at the pole, it would attract the water without any daily remissions of its action. The water would assume a position of equilibrium, and there would then be no daily tides. Thus as the luminaries decline from the equator, their effects become less and less. Fifthly. The river tides are formed by the propagation of the tidal disturbance from the seas where they were formed. In these therefore the greatest tides occur later than they should do according to the above statement. Sixthly. The effects of the disturbing bodies depend on the latitude of the place. Suppose the attraction of the moon to raise a tide, one of whose highest points is in la- titude a, therefore the other is in the other hemisphere at the same latitude. This state of tide in one day will travel round the earth, the vertices always remaining at the same latitude a. Then it is clear that the altitude of the tide at any place will depend on its distance from a vertex of this heap of waters. That place will have the highest tide over which the vertex passes, and the height at any other place will be less and less the greater the angular distance from the vertex ; that is, the greater the difference of latitude between the place and one vertex. The tide will therefore be least at the equator and poles, and u 290 NEWTON'S PRINCIPIA. greatest in latitude a ; thus the height of the tide varies with the latitude. Seventhly. But the difference between the latitude of the place and one vertex is not the same as that between it and the other vertex. Hence, though we have two tides every day, these two tides will be of unequal magnitude. The moon's orbit is inclined at but a small angle to the ecliptic ; hence, speaking generally, her orbit will have nearly the same position relative to the equator that the sun's orbit has. A line drawn through the centre of the earth and moon is the axis of the tidal spheroid. The tides which occur on the side next the sun, when the sun has north declination, will be greater than the tide on the op- posite side of the earth. Therefore in summer the day tides are greater than the night tides ; similarly in winter the night tides are greater. " If the pole of the tidal flood follow the moon, say at six hours, the pole will be north from the tune the moon is six hours west of the sun to the time when she is six hours east, that is, from the time when the high tide is at noon to the time when it is at mid- night." In such a case therefore the afternoon tide is greater than the morning tide in summer and less in winter. Similar reasoning will apply when the " age of the tide " is twelve, thirty, &c. hours. Eighthly. If the tide be brought to any place by two un- equal channels, the two tides following the two transits of the luminary across the neighbouring oceans will meet one ano- ther at this place, and form a compound tide. Suppose one tide to be delayed six hours, it will be in exact opposition to the tide which arrives by the other path. If the two were equal, the waters would stagnate, and there would be no tide at this particular point. But if the luminary be not in the equator, the two tides are unequal; hence, compound- ing the two, there will be one tide every twenty-four hours. Newton quotes from Halley, as an example of this, the NEWTON'S PRINCIPIA. 291 port of Batsham, in Tunquin, lat 20 5(X N. The tide arrives by two inlets, one from the seas of China, between the continent and the island of Leuconia ; the other from the Indian Sea, between the continent and the island of Borneo. " The tide begins every successive day later by about three quarters of an hour; so that in fifteen days the time of high water advances from one o'clock in the afternoon, for instance, to twelve at night ; after which it does not advance to one in the morning, but falls back thirteen hours to twelve at noon, and so on perpetually. In this way the high water is always in the afternoon during the summer half year (March to October), and in the forenoon during the remaining half. About the time when the tide falls back thirteen hours, the tides are very small and scarcely perceptible ; at the intermediate times they are greatest." (Phil Tr^ms. 1833, page 224. Whewell.) Let us now follow Newton in his attempt to calculate numerically the forces of the sun and moon to raise the tides. He first refers to his Lunar Theory for a calcula- tion of the force of the sun to draw the moon towards the earth. When in quadratures this force is 638( ^. 6 g, where g is the force of gravity at the surface of the earth. When in syzygy the force is double this quantity. But the disturbed bodies are here the particles of water at the surface of the earth, which are nearer the earth than the moon in the ratio of 60^ to 1. These forces must, there- fore, be decreased in the same ratio. The force in quadra- tures is therefore z ^ itm <7> and that in syzygy double this quantity. The first depresses the water in quadra- ture, the second raises that in syzygy. Hence the two produce the same effect, and the whole force to raise the sea will be the sum of the two, that is, tfrice the lorce in quadrature. u 2 292 NEWTON'S PRINCIPIA. The force of the sun, therefore, to raise the tide, will be 128^200 9' -^ u ' ; *ke centrifugal force has been shown to be 2^9 g> The latter raises the water to a height under the equator exceeding that under the poles by 85,472 Paris feet. Hence, the sun's force is -^th part of the centri- fugal force, and will raise the water ^~> or 1 Paris foot and 11^ inches, about two English feet. If the mass of the moon had been known in Newton's time, he might have made a similar calculation to deter- mine its force to raise the tides. But he was obliged to deduce this mass from the height of the tide itself. " Be- fore the mouth of the river Avon, three miles below Bris- tol, the height of the ascent of the water in the vernal and autumnal syzygies of the luminaries (by the observations of Sturmy), amounts to about 45 feet ; but in the quadra- tures to 25 only. The former of these forces arises from the sum of the forces of the sun and moon, the latter from their difference." If, therefore, L and S are supposed to represent respec- tively the forces of the sun and moon while they are in the equator as well as in their mean distances from the earth, we shall have L -f- S to L S as 45 to 25, or L to S as 7 to 2. Newton remarks that the observations at Plymouth by Colepress gave a ratio 41 to 23, a propor- tion which agrees tolerably well with the former. He, however, prefers the former result, because the observa- tions were made on larger tides. This reasoning proceeds on the supposition that the earth is without rotation, and in that case there would be high water immediately under the luminary. But. this is not the case ; it does not occur until three hours after the transit of the luminary. Nei- ther does the highest tide occur at the syzygy, but about three days after. Newton attributes this to the " force of reciprocation," which the water once moved retain a little NEWTON'S PRINCIPIA. 293 while by their vis insita. The luminaries continue to act with great power for a little while after the moment of their greatest strength, and thus continue to raise the tides. But Laplace remarks (Mec. Cel. xiii., chap. 1.) that "vrai- semblable " as this is, it is nevertheless erroneous, for an accurate investigation shows that, notwithstanding this continued action of the luminaries, the greatest full tides should occur exactly at the syzygies and the least exactly at the quadratures, and that therefore the explanation of the delay must be sought for in the accessory circumstances. It is, in fact, due to friction. Newton then proceeds to make several " corrections" to this result which he conceives to be necessary. He observes that the luminaries are not in the positions of greatest efficiency at the moments of the greatest high tide at the place under consideration; that therefore it is not the whole force of these luminaries that is employed to raise the tide, but this force multiplied by the cosines of certain angles. He corrects, therefore, for the instan- taneous angular positions of the luminaries at the moment of the greatest tide. He also remarks that changes in the distance of the moon are produced by the inequality in her motion called the " Variation," and he adds, therefore, another correction for the distance of the luminary at the moment of the greatest tide. But Laplace points out that this correction also is wrong. In fact it contradicts the previous reasoning, for if the tide be due to the accumu- lated action of the luminaries during a certain instant, we must not consider it as proportional to the force at the end of that interval. But when all corrections are applied, he finds that the sun's force raises the tides by 1 foot 11 ^ inches, and the moon's force will therefore raise the same to the height 8 feet 7^ inches, and the joint action of the two to the height 10 feet, and when tHe moon is* u 3 294 NEWTON'S PRINCIPIA. in its perigee to the height of ]2i feet, and more, especially if the wind sets the same way as the tide. Newton makes some remarks on the observed nature of the tides in different parts of the world. His explanations are not always perfectly correct. To have a full tide raised, an extent of sea from east to west is required of no less than 90 degrees. Hence, he infers, the tides in the Pacific are greater than those in the Atlantic, and those in the North Atlantic than those within the tropics. In some ports, where the water must be forced in and out through narrow channels, the flood and ebb must be greater than ordinary, and this force of efflux, being once given to the water, may, he argues, con- tinue until it raises the tide as much as fifty feet. But on such shores as lie towards the sea with a steep ascent, where the waters may freely rise and fall without that precipitation of influx or efflux, the proportion of the tides agrees with the forces of the sun and moon. The force of the moon being only ^ 7 , 4Uu th part of gravity a^ the surface of the earth, will not be sensible in any statical or hydrostatical experiment, or even in those of pendulums. It is in the tides only that this force shows itself. I have already mentioned that the mass of the moon was unknown in Newton's time. He makes use of the obser- vations on the tides to determine her mass and density. Her force is ^j- 5 th part of the sun's force to raise the tides. And, by lunar theory, these forces vary as the masses of the attracting bodies directly and the cubes of their distances directly; that is, as their densities and the cubes of their apparent diameters. These diameters being 31', 16," and 32% 12", the ratio of the densities is .4-891 to 1. But the density of the sun was known to be one-fourth that of the earth. Hence the moon is denser NEWTON'S PRINCIPIA. 295 than the earth in the ratio of 11 to 9. The true diameter of the moon is to the true diameter of the earth (according to Newton) as 1 to 3-65. Hence the mass of the moon is ~- s (or nearly ^th) that of the earth. It is no w known that the true mass is about -^gth. that of the earth. If the moon's body were fluid like our sea, the force of the earth would raise tides in it. The tide in the moon caused by our earth is to the tide in our sea caused by the earth as the mass of the earth to the mass of the moon. This ratio Newton imagined to be 1 to 39 '788. Hence he concluded the lunar tide was 93 feet. Upon this ac- count the figure of the moon would be a spheroid whose greatest diameter produced would pass through the centre of the earth and exceed the diameters perpendicular by 186 feet. It might be supposed that the tides in the moon must be overwhelming. But as the moon always turns the same face to us, the tide is stationary, and would therefore merely affect the permanent surface of the lunar ocean. But no sea has ever been detected in the moon. If the moon had been originally fluid, or had been created in a form which would have no tendency to break up, her longest diameter must point to the earth. Hence the same face of the moon would always appear to us, and the body could not rest in any other position, but would always re- turn by a slow vibratory motion. [See Note IX.] U 4 296 CHAPTER IX. THE CIRCULAR MOTION OF FLUIDS. 1. The hypothesis of the Cartesian theory. 2. Newton's hypothesis as to the law of internal friction in fluids, the motion of a cylindrical vortex Bernoulli's objections to this re- sult. Prop. LI. 3. Some difficulties of the Cartesian theory which are considered by Newton. a. The Sun's rotation. LIT. Cor. 4. /3. The third law of Kepler. Scholium LIII. 7. The two first laws of Kepler. Scholium LIU. 8. The density of the planets, LIU. e. The disturbance of the Sun's motion by the planetary vortices. LII. Cor. 5, &c. 1. BEFORE the time of Newton the Cartesian theory was believed in by almost every nation of Europe. It was therefore necessary that some notice should be taken of these opinions in a work in which a totally different system was proposed for the first time. Accordingly Newton has devoted a section to the consideration of this hypothesis. The philosophy of Descartes was naturally a very popular one. Its explanations of the general facts of astronomy were so exceedingly simple that it required no previous learning to enable any one to understand them. Men were pleased to think that in a few minutes they could learn the cause of the motion of the planets. The sun was supposed to be the centre of a vast vortex or whirlpool, the density and angular velocity of the various parts of which were different. The planets being placed, each in that NEWTON'S PRINCIPIA. 297 stratum whose density was equal to its own, were thus dragged round the sun. Those planets which had satellites were themselves the centres of smaller whirlpools in which their secondaries revolved according to the same laws by which they themselves were carried round the sun. The ellipticity of the orbits were accounted for by supposing the vortices themselves not circular. Such was the theory as given by Descartes. It was open to so many ob- jections that it was greatly changed in character by his successors. Newton proved that the original theory could not be made to agree with Kepler's laws. Bernoulli* imagined the vortices to be circular, and accounted for the elliptic path of the planet by a combination of an os- cillatory movement with the circular motion of the whirl- pool. Bouguer then showed that the two portions of the curve which the planet would describe in its oscillations from aphelion to perihelion would not be equal or similar. D'Alembert showed that an elliptic vortex was, under the circumstances of the case, impossible. The theory was always unsatisfactory : all sorts of suppositions were made in vain by Huygens, Perrault, Villemot, Mollieres, Ga- maches, &c. They never could explain one phenomenon without contradicting another. Newton of course considers the Cartesian theory as it was originally given by its author. We shall confine our- selves within the same limits. The theory has no longer any adherent, and there can be no advantage in attacking that which no one defends. 2. If the particles of a fluid did not exert any action on each other, a vortex in which the velocity was any function of the distance would be possible. For suppose the whole fluid in any cylinder to be revolving round the axis ; * Montucla, ii. 327. 298 NEWTON'S PKINCIPIA. describe two indefinitely near cylinders with this straight line as their common axis. The fluid between these two cylinders will evidently revolve unchanged whatever be its velocity, and however different from that of the neighbouring fluid, the normal pressure bejng sufficient to counteract the centrifugal force. There is, however, no perfect fluid in nature : if a stream of one fluid be made to pass through another, it will carry the particles of the second" along with it. A true theory of vortices must take account of this " internal friction" of the fluid. Newton starts with the hypothesis that " The resistance arising from the want of lubricity in the parts of a fluid is, cseteris paribus, proportional to the velocity with which the parts of the fluid are separated from each other." This hypothesis, as Newton himself remarks, is probably not altogether correct, but, nevertheless, there can be no doubt that it will give us a general idea of the motion. The first problem to which we shall apply it will be the following: Two infinite cylinders having a common axis revolve in any uniform manner about that axis. Fluid is placed be- tween them, and soon acquires a rotatory motion; it is required to determine what that motion will be after it has become settled or steady. Divide the whole fluid by concentric cylinders whose radii continually differ by 8r. Then we may suppose the fluid between any two of these to solidify, and the circum- stances of the case will not be in the least altered, provided we at the same time make 8 r diminish without limit. Let us then consider the friction between any two of these solidified cylindrical elements. Let and co -f S cu be the angular velocities of two consecutive cylinders, and let the radius of the surface of junction be r. It is manifest that the relative velocity of the two cylinders is NEWTON'S PRJNCIPIA. 299 d co r ^ ir ' and therefore the friction which is proportional both to the velocity and the number of particles that rub against each other will be proportional to ****** dr In order that the motion of any cylindrical element may not change, the friction on its two sides must be equal. Hence the expression >,,; dr must be the same for all values of r ; call this value a ; hence, integrating, we have where a and /3 are constants depending on the angular velocities of the inner and outer cylinders. Supposing these to be co l > 2 , respectively, and r l r z to be the radii, we have the two equations to find a and /3. In order that this state of motion may be " stable," it is necessary that the centrifugal force should increase from the centre to the circumference, unless the fluid be sufficiently tenacious to resist the change of motion that might follow the inequality of pressure. 300 NEWTON'S PRINCIPIA. If the fluid be homogeneous, this condition requires that j 2 r = + 2a/3 + /3 2 r should increase with r. Hence its differential coefficient must be always positive ; that is, on substituting for a. and /3 their values, this gives If there be added or taken away from the fluid and cylinders any constant quantity of angular velocity, the mutual attraction of the particles of fluid will not be changed, and therefore the relative motion will remain the same as before. If we impress on the whole system such a velocity as to make j3=0, we get and this is the case considered by Newton. Unless the fluid be heterogeneous, this state of motion is unstable. Each particle of fluid has a greater tendency to fly from the centre than those beyond it, and hence if any circum- stance occurs to alter the symmetry of the motion, the particles of those strata which have too great a centrifugal force will immediately fly from the centre, and the whole motion will be changed. This state of motion is like the case of a stratum of a heavy fluid resting on the surface of a lighter one. NEWTON'S PRINCIPIA. 301 Bernoulli, in his dissertation, entitled " Nouvelles Pensees sur le Systeme de Descartes avec la Maniere d'en deduire les Orbites et les Aphelies de Planetes," has made two objections to these investigations. First, that Newton in calculating the amount of the friction between two layers of fluid has not considered the pressure between those layers ; and, secondly, that in calculating the effect of the friction, he has not taken into account the arm of the lever at which it acts. Bernoulli then attempted to show that the density, being supposed to vary with the distance, the motion in the vortex will agree with that given by Kepler's third law. But D'Alembert* showed that this does not follow from Bernoulli's own equations, for it appears that he has, in integrating, omitted the lower limit ; thus he considered which is true only when m is positive, whereas he after- wards assumes m= \. The first of Bernoulli's objections has been anticipated by Newton himself. In the scholium to the fifty-second proposition, he says, f ' The matter by its circular motion endeavours to recede from the axis of the vortex, and therefore presses all the matter that lies be- yond. This pressure makes the attrition greater and the separation of the parts more difficult, and by consequence diminishes the fluidity of the matter." D'Alembert re- marks that Mr. Musschenbroek in some very exact expe- riments, found that when the velocity is small, the. friction was proportional to the velocity, and not to the pressure. That similar results have been obtained by later experi- mentalists is evident from what has been said in a pre- vious chapter. * Traite des Fluidcs, p. 408. 302 NEWTON'S PRINCIPIA. Bernoulli's second objection has more force, according to our preceding investigation, the friction being r *irr* r > and the arm of the lever at which this acts being r, we have to be constant for all values of r, this gives If a sphere begin to rotate and communicate an angular velocity to the surrounding fluid, the inner parts of the vortex will move quicker than those without, and these by friction will be continually communicating velocity to those outer strata. The vortex created will, therefore, always grow larger and larger, and the motion of the sphere will be continuously transferred from the centre to the circumference, until it is swallowed up and lost in the boundless extent of space. Hence there must be some active principle which may tend to communicate velocity to the cylinder, otherwise it will move slower and slower and finally lose all its motion. If any motion had been communicated to the infinite fluid also, the friction of the fluid and cylinder will never cease to retard or accelerate that body until the whole fluid and cylinder revolve round with the same angular velocity. This, therefore, ought to be the state of the Cartesian vortex. But it is well known that the planets do not describe their orbits in the same time. Newton next argues that even if we grant the existence of an active principle that will keep up the angular velo- NEWTON'S PRINCIPIA. 303 city of the Sun, yet still it is impossible to explain the existing phenomena by a vortex. The third law of Kepler declares that the squares of the periodic times of the planets are as the cubes of their mean distances. But his preceding propositions have shown that this is not the case in a vortex. It is true that his reasoning is founded on two hypotheses ; first, that the resistance arising from friction varies as the velocity; and, secondly, that the degree of fluidity is the same throughout the vortex. Let us, however, suppose that the friction varies as the mth power of the velocity, and the frictional power of the several strata as the nth power of the distance. Then, according to Newton, we have r n = constant Hence, that Kepler's law may be true, we must have o n + 2 = - m; either, then, m is greater than unity, or n is negative. But Newton considered that if the resistance did not vary as the velocity, it would vary in a less ratio, that is, m less than unity, and that if the parts of the fluid had various degrees of fluidity, those parts that had least fluidity would be heaviest, and would, therefore, be furthest from the centre, that is, n positive; hence he thinks that the theory of vortices cannot be made to explain the third law of Kepler. The motion of the vortex cannot be made to agree with the two first laws of Kepler. The planets move in ellipses which are so placed that their major axes are not parallel. To account for this Descartes supposed the vortices them- 304 NEWTON'S PRINCIPIA. selves elliptical. The fluid between two elliptic curves of motion must always keep between them, and therefore the velocity must be greatest where the distance between the paths is least. But by Kepler's law the velocity is such that the areas described are proportional to the time. These two conclusions can be shown not to agree. As an example, let us take the three orbits of Venus, Earth, and Mars. " At the beginning of the sign of Virgo, where the aphelion of Mars is at present," says Newton, " the distance between the orbits of Mars and Venus is to the distance between the same orbits at the beginning of the sign of Pisces as 3 to 2. Therefore the matter of the vortex between those orbits ought to be swifter at the be- ginning of Pisces than at the beginning of Virgo in the ratio of 3 : 2. Therefore the velocity of the Earth at the beginning of Pisces should be to its velocity in the begin- ning of Virgo in the same ratio. But judging from the Sun's diurnal motion, we know that the Earth is swifter at the beginning of Virgo than at the beginning of Pisces. Hence the hypothesis of vortices is not reconcileable with astronomical phenomena." If a body be carried round a vortex, it must have the same density as the fluid along its path. In that case it may be regarded as a piece of solidified fluid, and its cen- trifugal force is just balanced by the pressures on its two sides. But if the density be different, this balancing no longer takes place. The centrifugal force will be greater or less than the difference of pressures, according as the density of the body is greater or less than that of the fluid. Hence the body will recede from or approach to the centre according as its density is greater or less than that of the fluid. If the strata of fluid be not placed so that their densities increased with the distance, not only would they themselves be in an unstable state, but the globes also, if NEWTON'S PRINCIPIA. 305 displaced, would not return to their former orbits. We should, therefore, expect those planets which are furthest from the sun to be the densest ; but the contrary is the case. Principia, prop, viii., book Hi., cor. 4. The planets that have satellites were supposed by Des- cartes to be the centres of smaller vortices ; but it has been already shown, that every vortex continually grows larger and larger, and thus every part of the fluid will be agitated with a motion resulting from the action of all the globes. Therefore the vortices will not be confined to any certain limits, but by degrees will run mutually into each other, and by the mutual action of the vortices on each other the globes will be perpetually moved from their places. They cannot possibly keep any certain positions among them- selves unless some force restrains them. Forces are ne- cessary, therefore, not only to keep the globes turning, but also to keep them in their places. 306 CHAPTER X. THEORY OP COMETS. 1. The Comets are planets moving in very eccentric conic sections. 2. To determine the particular orbit of a given comet. (1.) THE visit of a comet presents to us a great mys- tery. It comes unexpectedly, remains but a short time, often presenting a magnificent spectacle, and then dis- appears in a manner as wonderful as its appearance. It is one of those great problems that God has set before us. Many of the ancients regarded them * as simple meteors, sent from the Supreme Being as signs of his anger or prognostics of future events. Others, as the illustrious Seneca, had better views on the subject "I do not follow the opinion of our philosophers ; I do not consider the comets to be passing fires, but as one of the eternal works of nature. "f Newton set about the solution of this great problem in his usual logical manner. The first point to be settled was whether they were simple meteors of our atmosphere or not. Their want of diurnal parallax is a convincing proof that they are at least further off than the moon. But how much further? We must have some idea of their distance to determine to what class of forces their * PingrS, Comet ; Bossnt, Hist, de Math, t Seneca, Nat. Qiusst. lib. vii. NEWTON'S PRINCIPIA. 307 motions are subject. Comets are found to move more than ordinarily slow or swift according to the position of the earth. They have, in fact, an annual parallax. There- fore they belong to the region of the planets. This result Newton confirms by some considerations on the brightness of their heads. He proves that they shine by reflected light, and then comparing their light with that of the planets, he shows how much those persons are mistaken who remove the comets almost as far as the fixed stars. If the comets move in the region of the planets, they must be subject to the attractions of the sun and the other planets. Neglecting for a first approximation these latter attractions compared with that of the sun, we know that the comet, being attracted towards the sun by a force which varies inversely as the square of the distance, must move in a conic section with the sun in one focus, and describe round it areas proportional to the times. This conic section may be either an ellipse, hyperbola, or parabola. In the former case the squares of their periodic times are as the cubes of their mean distances. This will furnish us with a complete test of the truth of our argu- ments. By observations made on the latitude and lon- gitude of a comet, we can calculate its orbit with an accuracy depending on the degree of approximation to which we have carried the solution. We can then deter- mine the distance and position of a comet at any moment. If these results agree with observation, there can be no doubt that our theory is correct. The difficulty of the problem is unfortunately as great as its importance. The comets move in such elongated ellipses, that, as a first approximation at least, we may consider their orbits as parabolas. But Newton speaks of the problem, even when thus simplified, as " Problema hocce longe difficillimum." (2.) Newton proposed two ways of determining the 308 orbit of a comit. One of these may be found in his little treatise De Systemate Mundi, and the other in the Third Book of his Principia. The first is by far the simplest of the two. The small part of the comet's path is con- sidered as a straight line, the two parts of which are described by the comet with the velocity it had at the beginning of those parts. These velocities are assumed to be V times that in a circle at the same distance. Let a first approximation to the distance of the comet from the earth at the time of the middle observation be sup- posed known, and let us represent it by the letter x. Newton then shows how to draw a straight line which will be divided by the directions in which the comet is seen at the times of the three observations in the ratio of the intervals between those observations. If this straight line, thus found, can be described in the given time with the velocity given by our assumed value of V, the value of x that corresponds to V has been found ; if not, a se- cond value of x must be assumed ; and then by an applica- tion of the rule of false, a new value may be determined, which is nearer the truth, and by repeating this process, we can arrive at any degree of accuracy. The value of V being therefore supposed known, the velocity, position, and direction of motion of the comet are also known at a given time. By the First Book we can describe the orbit that a body projected in a given manner from a given point will describe round a given centre of force. Begin, then, with an approximate value of V, and construct the orbit. Make a fourth observation on the comet when at a great distance from its position at the three first obser- vations. The longitude thus found should agree with that in the orbit just constructed. If not, a new value of Fmust be found, and by the rule of false, the true value may be continually approximated to. NEWTON'S PRINCIPIA. 309 The other method begins by assuming the orbit to be parabolic. Three observations of a comet are supposed to be made at given times; then the positions of the earth T, t, T, being known, we can draw the directions of the straight lines T A, tB, T C, joining the comet and the earth at those times. Suppose now that by some means a first approxima- tion to the distance of the comet at the times of the second observation has been dis- covered, then we can mark off a distance t B, and B will be one point in the comet's orbit. Suppose also that from some properties of the parabola we can determine approximately, First, the length of the sagitta B E drawn from the point B of a comet's path in a known direction and cut off by the chord A Cjoining the positions of the comet at the first and last observations ; Secondly, the ratio in which the chord A C is divided in the point E: Thirdly, the length of the chord A C. Then, by means of the first two properties, we can construct the chord A C, and if its length be the same as that given by the third property, our assumption as to the value of t B was correct. If not, by making several trials, and by applying the " rule of false," we can correct our assumed distance t B, and, therefore, ultimately obtain three points A, B, C in the orbit. By a simple construction given in the First Book, a parabola can be described which passes through two points and has its focus in the centre of the sun. This will be the comet's orbit. x 3 310 NEWTON'S PRINCIPIA. Both these methods, it will be observed, require the dis- tance of the comet at the time of the observation to be known at least approximately. The expedient* suggested was to consider a small portion of each orbit as rectilinear and described with uniform motion. Then four observa- tions being made at moderate intervals of time, four straight lines are given, across which a straight line is to be drawn so as to be cut in three parts in the same ratios of the intervals of time. This geometrical problem had been al- ready solved by Wallis, Wren, and Newton, but its applica- tion to comets had never led to any satisfactory result. A little consideration will enable us to understand why. If the four straight lines pass through one point, the problem admits of more than one solution. Geometers call this a porismatic case. In fact, a whole family of straight lines can be drawn cutting the four straight lines in the re- quired ratios. When the straight lines do not exactly meet in a point, only one straight line can be drawn ; but, as might be expected, the construction to draw this line is such that any small error in the data will make a very great change in the cutting line. This, Boscovich remarks f, is exactly what occurs in the application to comets ; the arcs of the two orbits, which are considered as straight lines, are necessarily small, and in this case the four straight lines can be shown to meet very nearly in one point. Newton was not a man to be content with merely a general theory of comets ; he at once reduced it to practice. He proceeded to try his method on the comet of 1680. By scale and compass, in a figure in which the radius of the earth was 16^ inches, he determined the elements of the orbit. Halley afterwards again calculated these with * Edin. Trans., vol. iii. f Phil. Recent, a Bcncdicto Stay cum adnot. Boscovich, lib. ii. p. 345. NEWTON'S PKINCIPIA. 311 greater accuracy by means of an arithmetical calculus. Moreover, observing that a remarkable comet had ap- peared four times at equal intervals of 575 years, he tried to find an elliptic orbit whose greater axis should be 138-2957 times the radius of the earth's orbit, so that a comet might revolve in it in 575 years. Taking this ellipse as the orbit of the comet, the observations from the beginning to the end were found to agree as perfectly with the computed places of the comet as the motions of the planets do with the theories from whence they are calculated; and this agreement clearly shows that it was one and the same comet that appeared all that time, and also that the orbit of that comet has been rightly defined. IN T E S. 315 NOTE I. THE reasoning by which the law of density of a com- pressible fluid under the action of a central force was found may be extended to the determination of the con- ditions of equilibrium of any fluid under the action of any forces. This proposition is of course the foundation of the modern science of Hydrostatics, and may be investigated as follows To determine the conditions of the equilibrium of any fluids, acted on by any forces. Choose any rectangular axes of reference, and take any small element whose edges are parallel to the axes and equal to dx, dy, dz respectively. Let xyz be the coordinates of one corner. Let X Y Z be the resolved parts of the acce- lerating forces acting on the element parallel to the axes, and let p be the density of the fluid. If/? be the pressure referred to a unit of area at the corner {xyz), the pressure on the side of the element parallel to the plane of y z will.be pdydz, which by Law I. acts perpendicular to the face. The pres- sure on the opposite face will be and therefore the pressure tending to move the element parallel to the axis of x is -~- dx dy dz. 316 NEWTON'S PRINCIPIA. But the moving force dz tends to do the same, and since the element is in equili- brium, the sum of these two must be zero. By Law II. the pressure referred to a unit of area on all the faces meeting at the corner (xyz) are equal, and as similar reasoning applies to all the sides of the element, we shall have These three equations are necessary and sufficient for equilibrium. These equations may be put under a form which is very useful, and may sometimes be used independently of any axes of coordinates. For draw any curve in the fluid and consider an element ds of its arc. Since no assumption has been made as to the axis of x, take it as tangent to the arc, and if S be the resolved part of the force along the tangent, the equation (1) shows that 1 dp _ -j = o. pas lleverting to the old position of the axes, the resolved parts of X Y Z along the tangent is hence dx ^dy dz J " * 3 "~ ** ^T~ ' as as as - dp = Xd* + Y dy + Ztlz, NEWTON'S PRTNCIPIA. 317 an equation which we shall have frequent occasion to use in illustrating Newton's propositions. Let us now consider some of the consequences of this proposition. Consequence I. In order that there may be equilibrium, the forces and density of the fluid must be such that P Xdx + pYdy + pZdz is a perfect differential, that is, we must have dpX. _ dpY d P X _ dpZ iU v ~dY~-'"dT > ~~d7~-~- ~d^ If we eliminate p from these equations, we get X + Y + Z = 0. \dz dyJ \dx dxj \dy dx ) Hence unless this equation be satisfied there is no fluid which will be in equilibrium under the action of the forces- But if the equation be satisfied, the law of density must still be such as to satisfy one of the equations (5) (6). The only forces we meet with in nature are those which tend to fixed centres, and vary as some function of the distance from those centres. In all such cases the quantity Xdx + Ydy + Zdz is a complete differential. For calling r the distance of any point from one of the centres of force, and * p x (I + cc T) \ a) Integrate this, on the supposition that T is constant, log. p = -- ^3 - r- (z n 5 J + c. x (1 + a. T) V 2 a/ Let AO, h v be the heights of the mercury in the barometer at the lower and upper stations, and T O , T P the temperatures at those stations respectively. Therefore the above expres- sion gives lo - = - The value of T being constant is taken as the mean of the two observed values T O , T X . It is this fact that the decrease of temperature is found by actual observation at every application of the formula that renders the results so trust- worthy. If we neglect z 2 the above leads to = 20177 in yards. The logarithms are in this formula the ordinary tabular ones to base 10, and the temperatures are expressed in degrees above 32 F. If very great accuracy be required, a variety of small corrections for x, , g Q are necessary. The correction for the variation of gravity with the height may be found by substituting the value of z thus found in and this is to be added to the former result. (2.) There is another very interesting application of these 322 NEWTON'S PRINCIPIA. formulae which we find in the second volume of Laplace's " Mecanique Celeste." We are enabled to determine in some measure the form of the atmosphere of the heavenly bodies. We have as yet neglected the effect of the centri- fugal force ; suppose the angular velocity to be cu, 9 the co-latitude of the particle of atmosphere under considera- tion. Then including this force in our equation we have ^ = f^dx + u* x sin. d (x sin. 9), p x 2 where the earth is considered to be a homogeneous sphere whose particles attract according to the law of the inverse square of the distance. Now along the free surface of the atmosphere p is constant, and . . d p = ; hence 2 d x + co 2 x sin. d (x sin. 0) = .-. const. = - + . x 2 sin. 2 0. x 2, This therefore is the equation to the surface of the atmo- sphere. Let us compare the polar and equatorial diameters. Call them 2 R and 2 R'. When x = R we have 0, the co-latitude, a right angle ; hence g + o> 2 R* = const. when x = R', 9 is nothing, hence ^7 = const. Now at the equator the centrifugal force is less than gravity, that is NEWTON'S PRINCIPIA. 323 Laplace deduces from this that the zodiacal light cannot be part of the Sun's atmosphere, for it has the form of a very flat lens, in which the polar diameter is far less than two-thirds the equatorial. Another sufficient reason is, that an atmosphere cannot extend beyond the orbit of a planet which describes its revolution in a time equal to the rotation of the Sun. Hence, as the Sun revolves in twenty-five days and a half, its atmosphere cannot extend so far as Mercury or Venus. We know that the zodiacal light extends much further. It is therefore not part of the Sun's atmosphere. The zodiacal light is a lenticularly shaped envelope which revolves round the Sun. Her- schel ( 897.) conjectures it to be no more than the denser part of that medium which resists the motion of comets, loaded perhaps with the actual materials of the tails of millions of those bodies, of which they have been stripped in their successive perihelion passages. It is an illuminated shower or tornado of stones. According to Professor Thomson, the inner parts of this tornado are always getting caught by the resistance of the Sun's atmosphere, and drawn to his mass by gravitation. They are always approaching the Sun, but very gradually, and he asserts that the mere fall of these aerolithes on the Sun is suffi- cient to account for the permanence of the Sun's heat. 324 NOTE III. THE whole calculation of Newton is founded on the supposition that the earth is homogeneous. Taking this for granted, it is possible to investigate without any long analysis the form of the earth. We cannot easily prove that the spheroid is the only form of equilibrium, but we shall show that it is at least one of the forms of equilibrium. That therefore if the earth had been originally fluid, it might have assumed a spheroidal form ; or if not originally fluid, yet if created in this form that its several parts would have no tendency to break up. A spheroidal mass of homogeneous fluid revolves round an axis with a uniform velocity; to determine if the equi- librium of the fluid be possible, and if so what is the ellipticity of the surface. Let a, b be the axes of the surface, e and s the eccen- tricity and ellipticity of the surface, then by definition Let the axis of revolution (i) be taken as the axis of z, and let the centre be the origin. Let o> be angular velocity, p the density of the fluid. It is well known that the at- tractions of the spheroid on a particle whose co-ordinates are x y z resolved parallel to those axes are respectively NEWTON'S PRINCIPLE. 325 For brevity's sake call the coefficients of x, y, z, A, A, 13. Then, according to the notation of last chapter, we have X=-(A-; 2 )* The equation to the surface is integrating we have O 2 + y 2 ) 4 z 2 = constant, the equation to a spheroid; hence the equilibrium is possible. The eccentricity will be given by or by substitution w 2 3 (1 - * 2 ) (3-2 e 2 ) x/TT^ . + - 5 - - -- -3 - am' 1 e = 0. v e 2 e 3 Q 2 The quantity - is the same as oo 2 a _ centrifugal force at equator 4 equatorial gravity and this we usually denote by the letter m. Hence the equation to find e becomes Y 3 326 NEWTON'S PRINCIPIA. _ 3 ~ e<1 ) (3-2 e 2 ) vr=r^ . , _ 2 e 2 e 3 ~3 W> We have already seen that m is nearly g-go' whence we find 1 ~ 232' If a curve be constructed of which the abscissa is e and the ordinate the left hand side of the above equation, it will be found to resemble from e=0 to e=l, the line OE BC where O C = 1 . Take O G = f m, and draw G E F parallel to the axis of X, then we see that the curve cuts this straight line tivice, and each of these ~ c x points corresponds to a value of e that will satisfy the above equation. It would appear that for the same value of o> there are two values of e, GE and GF, which are consistent with equilibrium. But it does not follow that if a mass of fluid be set in motion with an angular velocity co that it can take either of these forms. There is another condition to be satisfied. There is a certain principle in mechanics, called the conservation of areas, which teaches us that " if any number of bodies revolve round a centre, and are acted on only by their mu- tual attraction and by forces directed to the centre, the sum of the products of the mass of each by the projection on a given plane of the area which it describes round that centre bears a constant ratio to the time." Hence the fluid must take up such a form that this ratio shall be the same as that in the fluid as originally set in motion. It requires but little consideration to perceive that for two forms so different that one is nearly a sphere and the other ex- cessively elliptical, it is impossible that the same angular ve- locity could sweep out the same areas in the same time. This form of equilibrium of the earth is stable, for if the form were, by any chance, to become less spherical, by the principle of the conservation of areas, the angular NEWTON'S PKINCIPIA. 327 velocity would decrease, and therefore the earth would return to its original form. So also, by similar reasoning^ the same would occur if the earth were to become less el- liptical. If = be small, it is easy to see that the preceding equa- tion leads us to since 2e 2 = s, when the powers of e higher than the second are neglected. T 4 328 NOTE IV. 5. THE problem to determine the ellipticity of a planet considered as heterogeneous, is by no means an easy one. It certainly was beyond the powers of an age when the laws that govern the equilibrium of fluids were almost unknown. Newton determines the form of equilibrium from the con- dition that the weights of all columns of fluid, from the centre to the surface, must be equal. This, however, is not sufficient for equilibrium. Huygens added afterwards another condition, that the form of the surface must always cut perpendicularly the direction of the resultant force. But even these two conditions together are not sufficient. Clairaut (Figure de la Terre, Chap. III.) gave an instance in which, under a particular law of gravity, the particles of fluid could be so arranged that both these conditions were satisfied; yet he also showed that, so far from the fluid being in equilibrium, it was actually impos- sible for any fluid to rest in equilibrium under the action of these forces. It was Clairaut who first investigated all the necessary conditions of equilibrium, and showed that both the principles hitherto used were included in the one he proposed. His famous work, " Theorie de la Figure de la Terre" was published in 1743, and in it he applied his theory to determine the form of the earth considered as heterogeneous. Very little has been effected in this subject since his time. The form of the investi- gation has been changed, but all the results remain essen- tially the same. The form of the earth, whatever it may be, must consist of " level " strata of equal density, of which the surface is one. Clairaut assumes all these to be NEWTON'S PRINCIPJA. 329 spheroids, having their minor axes in the same direction, but not necessarily of the same ellipticity. He then shows that when there is a certain relation between the density of any stratum and its ellipticity the fluid will be in equili- brium. Assuming that the density increases with the depth, it follows that the ellipticities must decrease from the surface to the centre, so that the strata are more and more nearly spherical the nearer they are to the centre of the earth. Suppose the earth to consist of a spheroidal nucleus formed of spheroidal strata of different densities, sur- rounded by a very thin layer of fluid (the sea), and sup- pose the laws of the density and the ellipticity of the strata to be any whatever, except that the ellipticity of the outermost stratum is the same as that of the thin layer of fluid upon it, then if G be the equatorial gravity, g the gravity at latitude A, e the ellipticity of the outer- most stratum, m the ratio of the centrifugal force at the equator to the equatorial gravity, g = G (1 + n sin. 2 A) This is a very remarkable proposition : the law of gravity along the surface of the earth is then quite independent of the law of density in the interior. It also furnishes us with a method of determining the ellipticity by observa- tions on the force of gravity in different parts of the earth. It is usually called " Clairaut's Theorem." If the earth was not originally fluid the strata of equal density may not have 'been spheroidal. But merely assuming that they differ but little from spheres, and that the surface is covered by a fluid in equilibrium, Laplace has shown that the changes in the force of gravity at the surface, and for all external points, is quite inde- 330 NEWTON'S PRINCIPIA. pendent of the nature of the internal structure. There is a certain general connection between the form of the surface and the variation of gravity which he establishes on the above suppositions. This general connection has been lately demonstrated without making any hypothesis respecting the distribution of matter in the interior of the earth, but merely assuming the law of universal gravita- tion. The investigations of Laplace lead to the same result as those of Clairaut, but there is this difference between their modes of reasoning. Clairaut assumes that the forms of the strata are spheroidal, and then shows that the whole will be in equilibrium. Laplace merely assumes that they differ but slightly from spheres, and then de- duces from the condition of equilibrium that their forms are spheroidal. The solution of the question would in fact lead to a functional equation : we cannot write down the condition of equilibrium without knowing the attrac- tion, and we cannot find the attraction without knowing the form of equilibrium. The analysis by which Laplace was enabled to prove the strata spheroidal is entirely his own invention. Its power is very great, and in many other investigations it has proved a useful engine of dis- covery. 6. We have seen that the earth cannot be homogeneous; we have also learnt from the investigations of Clairaut and Laplace that it consists of strata of different densities increasing from the circumference to the centre : it becomes an interesting question to determine this law of density. Legendre was the first who ascertained what is very probably the true law. But it is the more finished results of Laplace that we shall now consider. They are entirely built upon one assumption. We know that in gases and fluids the ratio of the change of pressure to the change in density is constant. But in solids and semi-fluid bodies it is more natural to suppose that this ratio increases with NEWTON'S PRINCIPIA. 331 the density. The most simple assumption is to suppose that it varies as the density. Supposing this to be the truth it is not difficult to investigate the density of the strata. But it is an assumption, and must stand or fall according as its results agree with, or differ from, those of observation. Fortunately it enables us to integrate the equation connecting the density and ellipticity of any stratum, and thus the ellipticity of the external stratum furnishes us with a test of the truth of the law. Taking for granted the truth of the law, a very simple calculation will give us the corresponding law of density. Let us consider the earth as a perfect sphere, and let us neglect the effect of the centrifugal force. The strata of equal density will then all be spheres. Let p be the density of that stratum whose radius is x. Let p be the pressure at that stratum referred to a unit of area. We have first to find the attraction on a particle situated in the stratum whose radius is r. The attractions of all the external strata is manifestly nothing. To find the attractions of the internal ones, we have merely to suppose them concentrated into their common centre and attracting according to the usual law. This will manifestly give where p, is the force of attraction of a unit of mass at a unit of distance. The law of fluid equilibrium will then give r - (1.) Also we have by our assumption d p = K p . d f> - - - - (2.) whence we get 332 NEWTON'S PRINCIPIA. * dp i r r -. . -V- = , / px*dz, 47^ dr r 2 _/o r or, which is the same thing, 4- (''- differentiating, d?r Put V (p for the sake of brevity. We have now a K differential equation to find p r. The integral is well known to be p r = A sin (q r + B). Now because the integral in equation (3.) has r = for its lower limit, it will be found on substitution that this value of p r will not satisfy it unless B = 0. Indeed, if B were not zero, we would have the density infinite at the centre, which is manifestly impossible ; for, no matter how great the pressure may be, it must still be finite. We have then This gives a density gradually decreasing from the centre, and therefore not contrary to what is a priori probable in the case of the earth. The values of A and q have yet to be determined. But NEWTON'S PRINCIPIA. 333 the experiments of Cavendish and Maskelyne have revealed to us the mean density of the globe ; and, supposing the density of the superficial stratum to be the same as that of granite, we have two equations to determine A and q. Working out these numerical calculations, we find that 2 Da . /57T r\ = sm CT J' where D is the density of the superficial stratum. There are two results which will serve to verify this law. 1. We know that there is a precession of the equinoxes, because the resultants of the attractions of the sun and moon do not pass through the centre of the earth. The position of the lines of action of these resultants mani- festly depend on the law according to which the density of the several strata varies. This calculation has been made, and it is found that, taking the above law of density, the precession should amount to 51"'3566. The observed precession is 50"' 1. 2. The ellipticity of the earth is caused by its rotation, and depends also on the law of density of its strata. The calculated result is s = 30 j. 3iy The result of geodetic measures is s = ^^. It is upon the remarkable agreement of our supposition with observation in these two cases that our belief in the law is founded. 7. But though we have thus investigated the law of den- sity of the strata, it is not to be therefore concluded that the earth is solid throughout ; on the contrary, the rate at which the temperature increases with the depth is such that if continued for 25 miles, the heat would be sufficient, under a pressure of one atmosphere, to melt a stratum of granite. Is, then, the interior of the earth a vast mass of molten strata whose densities obey the law already inves- tigated ? and if so, what is the thickness of the crust ? 334 NEWTON'S PRINCIPIA. Mr. Hopkins has performed some very laborious calcula- tions with a view of determining this question. He begins with some general remarks. There are two ways in which a body may cool, by conduction or convection. The earth being at first fluid, would begin to cool by convection. Now the temperature and pressure will both be greater at the centre than near the circumference. Because the temperature is greater at the centre, the body will solidify first at the outer parts, and the earth would become a crust containing a heterogeneous fluid. But because the pressure is greatest at the centre, the body will tend to solidify first at the centre, and thus on cooling it would become solid throughout. We cannot tell which is the predominating cause, and the investigation of the earth's refrigeration leaves the point uncertain. But there may be other tests whereby we can determine this question. The precession of the equinoxes is caused by the attrac- tion of the heavenly bodies on the ring of matter surrounding the earth's equator. One consequence of this attraction we have already seen to be the recession of the nodes in which it cuts the ecliptic. But this ring is fastened to the earth : its nodes cannot, therefore, recede as fast as they would do if the ring were left to itself. The earth is a heavy load which it has to pull round with it. The less this load the greater would be the precession. If the interior surface of the solid crust be spherical, then neg- lecting the friction between it and the interior fluid, the ring of matter surrounding the equator will only have to pull round with it the solid crust ; the fluid will not turn with it. Hence the precession will be greater than if the earth were solid throughout. But the interior surface of the crust cannot be supposed spherical ; it is most probably spheroidal. Supposing it so, there will be pressures between its interior surface and the contained fluid, caused partly by the motion of the spheroid and partly by the tidal actions in the fluid caused NEWTON'S PKJNCIPIA. 335 by the attractions of the Moon and Sun and by the centri- fugal force. These pressures must of course be taken into account. It would be uninteresting to follow step by step the process of the investigation : they are very long, and I shall therefore confine myself to stating the results. Mr' Hopkins * first considers the earth as a homogeneous spheroidal shell filled with a homogeneous fluid of equal density. The two surfaces of the shell are supposed to have equal ellipticities. On these suppositions he cal- culates the disturbing forces, forms the differential equa- tions for the motion of the pole, integrates them, and by interpretation arrives at the following results. " 1. The precession will be the same, whatever be the thickness of the shell as if the whole earth were homoge- neous and solid. " 2. The lunar nutation will be the same as for the homogeneous spheroid to such a degree of approximation that the difference is inappreciable to observation. " 3. The solar nutation will be sensibly the same as for the homogeneous spheroid, unless the thickness of the shell be very nearly of a certain value, something less than one-fourth of the earth's radius, in which case the nu- tation might become much greater than for the solid sphe- roid. " 4. In addition to the above motions of precession and nutation, the pole of the earth would have a small circular motion, depending entirely on the internal fluidity. The radius of the circle thus described would be greatest when the thickness of the shell would be least, but the inequality thus produced would not for the smallest thickness of the shell exceed a quantity of the same order as the solar nutation, and for any but the most inconsiderable thick- ness of the shell be entirely inappreciable to observation." Thus it appears that the effect of these pressures between * Phil. Trans. 1839, 1840, 1842. 336 NEWTON'S PKINCIPIA. the shell and the contained fluid is that the general effect is the same as if the whole were solid. This method there- fore fails to tell us anything of the thickness of the crust. But when we proceed to consider the case in which both the solid shell and the inclosed fluid are of variable density, we arrive at a different result. The disturbing forces will of course depend on the law of density : taking the law which we have already investigated, we can calculate these forces and compare them with those obtained in the case of a homogeneous shell. The required alterations can then be made in differential equations of the motion of the pole. We can thus find the precession. In order that the result thus found may agree with that found by observation, there must be a certain relation between the ellipticity of the internal surface of the solid part and the mean thickness of the crust. This internal surface is a surface of equal soli- dity. If we knew, then, what function the solidity of a body is of the temperature and pressure, we should be able to express the ellipticity of a surface of equal solidity as a function of the depth. By equating the two values thus found, we should have an equation to determine the thick- ness of the crust. But we do not know the law of the ellipticities of the surfaces of equal solidity. If heat did not affect solidification, they would be the same as the surfaces of equal density. If density did not affect solidification, they would be the same as the surfaces of equal temperature. But both are acting causes. Hence the ellipticity of a surface of equal solidity passing through any point must lie between those of the surfaces of equal density and equal temperature passing through the same point. The ellipticity of the former decrease, those of the latter increase, from the surface to the centre. Hence, in order that the numerical value of precession may be accounted for, it is sufficient that the thickness of the crust shall not be less than a certain value, found to be about one-fourth to one-fifth the radius. This reasoning fails therefore to give more than an NEWTON'S PRINCIPIA. 337 inferior limit to the thickness of the crust. We learn that it must exceed 1000 miles. Mr. Hopkins has lately undertaken a series of experiments with a view to ascer- taining the temperature at which bodies liquefy under great pressures. An account of these was given at the meeting of the British Association. When finally com- pleted they will throw great light on the thickness of the earth's crust. 8. It is not to be supposed that no measures were under- taken to discover by actual observation the ellipticity of the globe : such measures would be an excellent test of the truth of the theory. There are three methods by which this may be effected. The earliest of these is by measure- ment of the length of a degree in different latitudes. ^The length of a degree varies as the radius of curvature of the elliptic meridian, and therefore increases from the equator to the poles. Now when the ellipticity is given, we can express in terms of it the latitude and equatorial radius, the length of a degree at that place. By two measure- ments of degrees we get two equations, and therefore can eliminate the equatorial radius and find the ellipticity. Some of these measures were effected before Newton's time, and their results, as we have seen, were used by him in determining the ellipticity. In 1684 Cassini measured an arc of eight degrees,* and found, to the astonishment of every one, that the length of the degree shortened as he approached the poles. This was in direct opposition to Newton's theory. It was objected that the difference be- tween the measures of two consecutive degrees was so small, that it was possible that the errors of observation might make that which was really the lesser appear to be the greater. To settle this doubt it was determined that three arcs should be measured. Godin, Bouguer and La Condamine were sent to Peru in 1735 to measure an * Encyc. Met., "Figure of the Earth," Z 338 NEWTON'S PRINCIPIA. arc near the equator, and in the succeeding year Mauper- tuis, Clairaut, Camus and Le Mourner went to the Gulf of Bothnia, while the French arc was measured by the Cassinis and Lacaille. The result was decisive, that the degrees shortened from the poles to the equator. But the three values of the ellipticity thus determined were very different. The observations of Peru and France gave ^ 7 , those of Peru and Sweden ? ^, those of France and Sweden T y T , but on making some necessary corrections they gave T ^. So great differences would seem to imply that the earth differed considerably from a spheroidal form. This, how- ever, may be partly accounted for, because two of the measures were in very mountainous countries. A great many measures of degrees, both along arcs of the meri- dian of a parallel of latitude, have been since undertaken. It would detain us too long to consider them, individually. The most probable value of the ellipticity as deduced from them is e -003352. The second method has already been alluded to. By Clairaut's theorem we can determine the force of gravity at any place in terms of the latitude, the eccentricity and the force of gravity at the equator. Two observations of the force of gravity will therefore enable us to determine the ellipticity. The force of gravity, as we shall show when we come to speak of the pendulum, may be de- termined by the use of that instrument. It may also be found by comparing the weight of a body with the strength of a spring, or indeed any force that does not vary with gravity. The former is the most accurate. By a comparison of a great number of pendulum ob- servations Airy has deduced e -003535. In the article, " Figure of the Earth," in the " Encyclopaedia Metropoli- tana," the Astronomer Royal has formed a table of seventy- nine pendulum observations. Thirty of these he has set apart as being " useless for the investigation of the earth's NEWTON'S PKINCIPIA. 339 form." If g be the force of gravity in latitude A, then by Clairaut's theorem ^7 = G [1 +n sin 2 A}. The values of n and G that are found to suit best with the forty-nine " first-rate observations " are w = - 005133 G = 7r 2 X 39-01677 Now if m be the ratio of the centrifugal force at the equator to equatorial gravity, 5 n = - m - e, whence since m '0034672, the ellipticity is easily seen to be e= -003535. The above expression for g enables us to find the force of gravity in any latitude. From a comparison of the results of the forty-nine observations with those given by this formula Airy has deduced. 1. That, c&teris paribus, gravity is greater on islands than continents. 2. That in high north latitudes the formula gives too small, and about latitude 45 too large a value of g ; near the equator the errors are about equally balanced. 3. There is no reason to think gravity to be different in different longitudes, as the irregularities on different meridians do not appear greater than those at places near one another. Nor does it appear that there is any differ- ence between the northern and southern hemispheres. The observations were all reduced to the level of the sea by Dr. Young's rule, which makes an allowance for the attraction of the earth above the level of the sea. But it has been pointed out by Professor Stokes, in the " Transactions of the Cambridge Society" for 1849, that this rule does not take into account all the effects of the irre- gular distribution of land and sea. " Besides the attrac- tion of the land lying immediately under a continental 340 NEWTON'S PBINCIPIA. station, between it and the level of the sea, the more dis- tant parts cause an increase in gravity, since the attraction they exert is not wholly horizontal, on account of the curvature of the earth. Again the horizontal attractions due to the neighbourhood of a continent would cause a plumb line to point slightly towards it, and since a level surface is everywhere perpendicular to the vertical, the level of the sea must be higher than it would be if the continent did not exist. The correction therefore reduces the observation to a point more distant from the centre of the earth than if the continent were away ; and therefore on this account gravity is less on a continent than on an island. The investigation shows this latter effect more than counterbalances the former ; so that, on the whole, gravity is greater on an island than on a continent." It is also probable that the ellipticity deduced by Airy is a little too great, owing to the decided preponderance of oceanic stations in low latitudes among the group where the observations were taken. On looking at the expression for g, we see that, in consequence, the calcu- lated values of gravity would be a little too small, par- ticularly for places near the pole. This will enable us to show that Airy's second conclusion is a mere repetition of the first ; for that in high north latitudes, the iformula should give too small a result is no more than what we should expect ; while about 45, the places of observation being all continental, the formula naturally gives too large a result. Thus we are now enabled to account, at least in great measure, for the anomalies that Airy has noticed. In considering the earth as an irregular figure, we must define what we mean by the ellipticity. Let a be the mean radius of the earth, r the radius of any point whose colatitude is 0. Then since the earth is nearly a spheroid, - cos/ 2 6 is very nearly constant. Its variations, as we go from NEWTON'S PBINCIPIA. 341 place to place, are irregular, and always very small. The mean of all the values of the above fraction may be de- fined to be the ellipticity. We cannot observe directly the value of r. In practice, therefore, we replace this fraction by another which contains g instead of r. If we could make a vast number of observations in all parts of the world, no further correction would be necessary than merely taking the mean of all the observed values of e. But as all our observations are made on land, and are few in number, our errors may all tend in one direction : it is therefore necessary to reduce our results to the surface of the spheroid. The third method of determining the ellipticity is founded upon astronomical observations. If the earth were a perfect sphere and homogeneous, the elliptic orbit of the moon would only be affected by the known disturb- ances of the other heavenly bodies. But the earth is neither spherical nor homogeneous. Hence arise other inequa- lities in the moon's motion, and conversely, when these are observed, they will enable us to discover the ellipticity of the earth. Without making any assumption as to the law of density, the theory of the " Figure of the Earth " en- ables us to find its attraction on the moon : substituting these in the equations of the moon's motion, we can deduce two inequalities. The chief of these is the inequality in latitude, and is about 8", hence we get e = -003370. The other is the inequality in longitude, and lies between 6" -8 and 7", the former gives e = -003360, the latter e = -003407. It is not difficult to see what will be the general nature of the more important inequality. The consequence of the attraction of the ring of matter surrounding the earth at the equator, will manifestly be to pull the moon nearer the equator. The chief disturbance will therefore be that produced by a small force acting on the moon perpen- dicular to the plane of her orbit, and tending towards the earth's equator. The effect of such a force is easily seen. 342 NEWTON'S PKINCIPIA. Let fl M be plane of the lunar orbit when the moon is at M. And suppose that in the small time T the moon, if undisturbed, would describe the arc M M'. But if the moon were at rest at M, suppose that the disturbing force would pull it through half M m towards the plane of the equator & E. Then, by the second law of motion, if we complete the parallelogram M N, the true direction of the moon's motion at the end of the very small time T will be M N. That is, the orbit has been changed from i2 M M' to &' M N ; the node 12 has receded, and the inclination of the orbit has been decreased. By similar reasoning it may be shown that if the moon had been approaching the equator in the direc- tion M li the node would have advanced, and the inclina- tion would have increased. During a quarter of a month the moon approaches the equator ; during another quarter it recedes ; and the forces being similar in each movement, the whole effect at the end of a month would be zero. But, owing to other disturbances, the node of the moon's orbit recedes nearly at a uniform rate along the ecliptic, while the inclination to the ecliptic remains nearly the same. Hence the incli- nation of the lunar orbit to the equator has changed in that half month ; the disturbing forces in the two quarter- months, though they still tend to counteract each other, will not be quite equal in magnitude, hence a small resi- dual effect will be left which soon mounts up till it becomes apparent. By a little consideration of the preceding figure it will become apparent that the greater the angle of inclination at fl the less will be the difference between the consecutive positions H M, &' M of the lunar orbit, i, e. the less the inclination (i) to the equator will be changed. So that when by the motion of the nodes along the ecliptic i is decreasing, the effect of the disturbing force will be greater NEWTON'S PRINCIPIA, 343 in the second quarter-month than the first ; that is, the effect will be greater while the moon is approaching the equator, than while it is receding from it, that is, by what precedes, the whole effect of the disturbing force is to lessen the diminution of z. Similarly, when z* is increasing, this increase is lessened by the action of the disturbing force. This is exactly what would take place if we neglected this disturbing cause and supposed the obliquity of the ecliptic to be less than it really was. Hence we arrive at this conclusion, which we shall state in the words of Laplace : " The non-sphericity of the earth produces in the lati- tude of the moon but one sensible inequality. We can represent its effect by supposing the orbit of the moon instead of moving in the plane of the ecliptic with a con- stant inclination, to move with the same condition on a plane passing through the equinoxes between the ecliptic and equator. This inequality is well adapted for deter- mining the ellipticity of the earth." '/. 4 344 NOTE V. THE RESISTANCE MADE TO BODIES MOVING IN FLUIDS DE- DUCED FROM THE GENERAL PRINCIPLES OF DYNAMICS. 1. The Equation of motion. a. The ordinary hydrodynamics. & How changed when " internal friction " is taken into account. 2. The ordinary law of resistance. a. How deduced from the equations of motion. 0. The general results of experiments made since Newton's time compared with the law. 7. The resistance should be deduced from a rigorous solution of the equations of motion adapted to the case under consideration, case of the pendulum. 3. The resistance to a Pendulum. a. Bessel's mode of expressing the resistance. 0. The careful experiments of Sabine, Baily, Coulomb, &c. % Poisson deduces the nature of the motion and the resistance from the ordinary Hydrodynamic equations. 8. On comparing the theory with experiment they are found not to . Professor Stokes takes into account the effects of internal friction. The results agree with experiment. 4. The resistance to Floating Bodies, o. The phenomenon of emersion, ft Waves are excited in the Fluid. y. Strange variations of the resistance as the velocity changes. (1.) AFTER Newton the chief writers on Hydrodynamics were the Bernoullis, Maclaurin, and D'Alembert. The equations which the latter obtained are the foundation of modern Hydrodynamics. He had previously discovered a general principle whereby every question concerning the motion of bodies may be reduced to another corresponding one concerning their equilibrium. Thus then the science of Hydrodynamics may be reduced to that of Hydrostatics. The simplest case to which we can apply this principle is NEWTON'S PRINCIPIA. 345 that in which the body is a single particle. It then leads us to the three laws of motion. When Newton said that a particle acted on by no external force will remain at rest or move in a straight line with a uniform velocity, he implied that there was no internal tendency in the particle to affect its state of rest or motion. D'Alembert ex- tended this to any system of particles, and his principle asserts that the internal forces of a dynamical system are in equilibrium among themselves during the whole mo- tion. It follows from this that the effective moving forces upon the molecules of a dynamical system, if their directions be reversed, will balance the external impressed forces. Let us apply this to the state of any fluid. Let a small element be taken in any fluid in motion whose coordinates are x, y } z. Let p be the density at this point and p the pressure referred to a unit of area. And let X, Y, Z be the external impressed forces on the element. The effective accelerating forces will be d? x d 2 y d* z Tp 9 TP d7>' Hence the forces d* acting on the element dxdydz, will, when all the elements are considered, balance each other. Hence from the equa- tions of fluid equilibrium, we have _ _ dy d t 2 346 NEWTON'S PBINCIPIA. _ z P dz ~ dt* These three equations are not, however, sufficient to determine the motion. For we have four quantities x,y,z,p to determine in terms of t. A fourth equation is necessary. This D'Alembert supplied from the condition that any portion of the fluid, in passing from one place to another, preserves the same volume if incompressed, or dilates, ac- cording to a given law, if the fluid be elastic, in such a manner that the mass is unchanged. It is usual at present to derive this equation from a principle that in reality is only the above in another form. If u, v, w be the velocities of the fluid at the point xyz in the directions of the axes ; then the result arrived at is d p u d p v d p w dp ~~J -- f~ ~J -- 1 -- ~J - ^~ ~T~L = "' ax ay a z at It will be observed that these equations do not make any assumption as to the molecular constitution of the fluid. All that is required is that the pressure, no matter how transmitted, shall be equal in all directions. These equations are so complicated that hardly anything can be done with them. But there is one general case in which the equations are greatly simplified. This is when X, Y, Z, M, v, w, are such that z ... (A) udx + vdy -\-wdz- - - - (B) are perfect differentials, upon the supposition that the time is constant and the density either also constant or a func- tion of the pressure. " It becomes then of the utmost consequence to inquire in what cases this supposition may be made. Now Lagrange enunciated two theorems by virtue of which, supposing them true, the supposition may be made in a great number of important cases ; in fact, in NEWTON'S PRINCIPIA. 347 nearly all those cases which it is most interesting to in- vestigate. These are : " 1. That (B) is approximately an exact differential when the motion is so small that squares and products of w,y,w, and their differential coefficients may be neglected. " 2. That (B) is accurately an exact differential at all times when it is so at one instant, and in particular when the motion begins from rest. " It has been pointed out by Poisson that the first of these theorems is not true. In fact, the initial motion being arbitrary need not be such as to render (B) an exact differential. " Lagrange's proof of the second theorem lies open to some objections." But it has received two perfectly satis- factory demonstrations.* Supposing the motion to be such that we may put udx + vdy + wdz d$ then it easily follows that d

~J~* ~T~> & c ' a x d y d s a r represent the velocity parallel to axes and along the arc &c. of the curve that the particle in question is describing. At the time t draw curves such that all the particles in them are at that moment moving along tangents to their respective curves. Let s be the arc of any one of these d 2 s curves. The effective accelerating force will be -=- v and the impressed force * Report to the British Association, 1846, on the progress of Hydrody- namics by Professor Stokes. 348 NEWTON'S PRINCIPIA. Hence, by D'Alembert's principle, and the formula for hy- drostatical equilibrium, Let V be the velocity of the fluid, then since lf-v dt ^Lf - v ' 'd t* = ds + dt sV Hr >_ ds ^ dsdt' Hence, substituting and integrating, Zdz-^^- d + C an equation that will be necessary to us further on. This integral is obtained by summing all the elements along any one curve, and C is therefore constant only along one curve, and may vary from curve to curve. To determine its variations we must have data given whereby we can know its value for all points along some surface cutting all the curves. Further it is manifest that these curves may change with the time ; C is then a function of t, The motion is said to be steady when the motion is al- ways the same at the same point. Hence in this case neither

will be , and the normal pressure will therefore be cos

2/* x/== 47^ (1+ ^ Then the resistance on the sphere is 366 NEWTON'S PRINCIPIA. The motion will therefore be the same as if the sphere were resisted by a force *! mf n multiplied by the velocity, and a mass xmf were added to its centre, increasing the inertia without affecting the weight. We are now enabled to account for many of our expe- rimental results, that is, such of them as relate to spheres. We see that neither x nor v! depend on the density of the sphere, but only on the volume ; that both are greater for small than large spheres. The resistance also is independ- ent of the roughness of the surface. One experiment of Sabine showed that x remained the same when p was reduced one half; this would seem to show that for the same fluid, at the same temperature, the value of ja, the coefficient of the friction, varies as the density. But since the value of x was not the same as before, when hydrogen was substituted for air, we see that in different media ^ depends on something else besides the density. We may apply these conclusions to the pendulum, and obtain results which we may test by experiments. The effect of the term depending on x will clearly be to alter the time of the vibration, but not the arc of oscillation ; the term depending on x f (being multiplied by the small factor , whose square may be neglected) will not affect the time of the vibration, but will decrease the arc continu- ally, so that the successive arcs form a geometrical progres- sion. The time will be increased by a fraction of the time equal to x 8 nearly, and the common ratio by which the -5* '8 arcs decrease is 2 nearly. The less the sphere the greater are x and x' ; the more, therefore, is the time altered and the quicker does the arc of vibration decrease. If a sphere move uniformly in a fluid with friction, we NEWTON'S PRINCIPIA. 367 may determine the resistance opposed to its motion by the fluid. The calculation is not brief, but the result arrived at for the resistance is R = 6 TT /// p V, where V is the velocity of the sphere and p.' the constant ratio of p to p. The calculation is founded on the supposition that V is so small that its square may be neglected. The part of the resistance, therefore, which depends on the simple power of the velocity, does not vary as the surface exposed to the fluid, but simply as the radius of the sphere. This becomes important when we apply the above formula to determine the terminal velocity of a very small sphere falling in a fluid under the action of gravity. Let a- be the specific gravity of the sphere, />, as before, that of the fluid ; then, if V be the terminal velocity, we have 6 TT p/ p a V = ... V =|4. r--iy. According to the usual theory, the terminal velocity would have been la. Thus V varies as a 2 instead of Va, and therefore becomes very much smaller, when a is small, than that given by the usual theory. Professor Stokes calculated that for a sphere one thousandth of an inch in diameter, the terminal velocity is 1*593 inches per second; for a sphere one ten- thousandth of an inch in diameter, the velocity is '01593. Those given by the usual theory are respectively 32-07 and 10' 14 inches per second. 368 NEWTON'S PRINCIPIA. The suspension of clouds may, therefore, be explained according to this theory. The minute drops are really falling with very small velocities. The investigations for the motion of cylinders have also been effected, but it will detain us too long to consider all the results. One fact in connexion with the cylinder is remarkable. The motion of the fluid in immediate contact of a sphere moving in a fluid is the same as that of the sphere, and as we go from the surface into the depths of the fluid, the velocity differs more and more from that of the sphere, and finally ends in being zero. The sphere by the friction of its surface tends continually to increase the mass of fluid it drags with it; the friction of the fluid at a distance tends continually to diminish it. These two in the case of a sphere tend to equality, and the motion is ultimately uniform. Not so in the case of a cylinder: the increase on the quantity of fluid carried gains on the de- crease due to the friction of the fluid, and the quantity carried increases continually. The velocity must therefore decrease continually. Professor Stokes has also submitted his results to a comparison with experiment. He first proceeds to obtain ft', the only constant at his disposal : the results of Baily with cylindrical rods give J~ =-116. " It is to be remembered that V~j2 expresses a length di- vided by the square root of a time, and the numerical value above given is adapted to an English inch as the unit of length, and a second of mean solar time as the unit of time." Let us take one instance of his series of comparisons at random : let them be the experiments of Baily on spheres attached to fine wires. Allowance is made for the wire by the theory of the motion of a cylinder. Allowance is also made for the confined space, which is estimated as NEWTON'S PRINCIPIA. 369 being nearly the same as that given by the ordinary Hy- drodynamic theory. Thus in one of Baily's brass 1^ inch sphere, the several parts of n were, for buoyancy 1, for inertia, on the common theory, '5 additional for inertia on account of internal friction '202 correction for wire '012 correction for confined space -032 total 1'746. Kesult of experiment, 1*755, error T ^ . (4.) In the fourteenth volume of the Edinburgh Transac- tions there will be found an interesting account of some experiments by Scott Russell on the resistance experienced by floating bodies in their progress through the water. As the object was to determine the resistance to ships, the experiments were conducted on a large scale, and the bodies used were vessels of 31 to 75 feet long. The ve- locities varied from 3 to 15 miles an hour. Two points are worthy of notice in these experiments. 1. The Emersion of the solid body from the fluid. The ship does not draw as much water when it is in motion as when it is at rest. This is manifestly caused by the re- solved part of the resistance in a vertical direction. 2. The motion of the boat does not excite currents in the water, but generates waves. These waves travel to great distances with a velocity independent of the form of the vessel and, when freely moving, equal to the square root of the product of the depth of the water and gravity. The position of the boat relative to these waves was remarkable. Calling h the depth of the water, the ve- locity of a "free" wave will be ^gh; let v be the velocity of the boat. If v be less than ^gh, the wave will have a tendency to travel a little quicker than the boat, and it was observed that the accumulation of all the waves ge- nerated by the boat formed an elevation at the prow and a depression at the stern. Thus the vessel rode on the posterior side of a " forced wave," with its prow elevated above its stern. If v be greater than */gh, opposite phe- B B 370 NEWTON'S PRINCIPIA. nomena occurred, and the boat rode on the anterior surface of a "forced wave," with its prow depressed below its stern. If the boat were suddenly stopped, the wave became im- mediately "free," and was propagated forwards with the velocity "Sgh. If the velocity of the boat were equal to that of a free wave, the boat rode on the top, with its prow and stern much more out of water than its middle part. By making the vessel move with a velocity = *Sgh, the depth of water is increased by the height of the wave, and it is found that by this artifice boats can be carried without grounding over shallow parts of the canal. Professor Airy has offered an explanation of the phenomena, to which we shall allude when we come to discuss waves. As may be expected, these waves considerably affect the resistance offered to the boat. Accordingly Scott Kussell found that the resistance does not follow the ratio of the square of the velocity, except when the velocity is small and the depth of the fluid considerable. The resistance was found to increase quicker than in the ratio of the square of the velocity, as the velocity ap- proached a certain quantity determined by the depth of the fluid. After this point of maximum, the resistance actually decreases as the velocity increases, until the ve- locity is equal to the velocity of propagation of a free wave, and the resistance is here less than that due to the square of the velocity. After this the resistance increases with the velocity, but in a ratio slower than that due to the square of the velocity. According to the law of pro- gression established, the resistance would reach a second point of maximum when the velocity shall have attained a certain quantity, greater than any obtained in the ex- periments. The best velocity for a boat to travel at in a canal is therefore D 3 406 NEWTON'S PRINCIPIA. or be less than 14 miles, there will be low tide immediately under the luminary. In this instance there was only a semi-diurnal tide, but when the solar path is no longer restricted to the equator, and the channel is any position on the earth, we have both diurnal tides and tides of longer period. The diurnal tide does not exist when either the canal is equatorial, or the luminary is on the equator. When it exists, the phase at the instant of transit depends on whether the depth of the sea be less than 3^ miles, and the luminary on the oppo- site side of the meridian or not. If both these hold or both fail, there will be low water at zenith transit ; if one fails and the other holds, high water. There are also tides of long period ; that is to say, the relative mean level in different parts of some canals is not the same as if the disturbing body did not act. The mean level thus depends on the position of the luminary. The state of the tide will manifestly be affected by fric- tion ; it appears that if we wish to calculate the height of the tide, at any time t, we are to proceed as if calculating the height (on the supposition of no friction) at a later time t + oi, with the luminaries in the position they were in at a preceding time t j3, where and /3 are two quan- tities given by the theory, and depending on the amount of the friction. In considering the effects of two bodies, we see that, according to this theory, they do not produce effects which have the same ratio as in the equilibrium theory. If in that theory, p be the ratio of the lunar to the solar high tide, and p' the ratio given by the wave theory, then where n, n' are the angular velocities of the sun and moon relative to a fixed line on the earth, b the radius of the earth, and k the depth of the sea. This ratio will be different according to the different values of k. This explains the fact, that in different seas the ratio of the NEWTON'S pniNCiriA. 407 lunar and solar tides is not constant, but varies with the depth of the sea. It would take too long to enter into all the applications of this theory to oceanic tides. If we could consider the Atlantic as a canal running nearly north and south, the tides in it would be nearly stationary. This is certainly not the case. The South Sea might be considered as a canal running east and west ; in that case there would be a forced tide wave travelling along it. The only advan- tage of this theory in its application to oceanic tides, is the simplicity of its analytical processes. It is even pos- sible to consider the effect of friction. The seas to which we apply its results are not canals, but we only use them as suggestive of the general characters of the motion. The theory is most successful in determining the tides in rivers. The water at its mouth being disturbed by the tides formed in the open sea, that disturbance will be propagated, with unaltered period, up the river with a velocity proper to the depth, viz., the square root of pro- duct of the mean depth and the force of gravity (32-18 ft.) This disturbance is a long wave ; when the top part of the wave passes any point, it forms high water, and when the lower part, low water at that place. The length of the wave of course depends on the distance up the river travelled by the front part of the wave formed at the mouth of the river, while the other parts are being formed. There may even be many high and low tides on the same river at once. Thus La Condamine observed about twenty places on the Amazon where there was high water with low water at the same time at intermediate places. It is also possible, as this long wave is propa- gated up the river, that the low water at some place a considerable distance up the river may be higher than the high water at the sea. This occurs on the Thames and most rivers. This propagation of the tide is not the transference of a body of water up the river, but the motion of a form. D D 4 408 NEWTON'S PRINCIPIA. We have seen that when a wave travels, along a uniform channel, that the water is moving in the direction in which the wave travels, so long as the water is above its mean level, and in the opposite direction so long as it is below its mean height. This is also the case with the tide, except so far as this circumstance may be modified by the varying depth, breadth, &c., of the river. It is a common error to suppose that the flow up the river ceases when the tide begins to fall. The flow in a uniform canal continues for three hours after tide. But if the depth and width be decreasing, if there be any friction among the particles of water, if there be any impediments to the progress of the wave, as a barrier or a series of bridges, the theory shows that this interval will be much less, and observation confirms the result, for the mean interval at which slack water follows high or low tide at Deptford has been shown* to be thirty -seven to forty minutes. The height of the tide wave also experiences many alterations due to the varying circumstances of the river. If the tide be stopped by a barrier, or if we are consider- ing the tides in a gulf like the Bay of Fundy, theory shows that there will be tide waves reflected from the barrier, and the height of the tide at the extremity may be greatly increased. If the breadth or depth de- crease, the height of the tide will be increased, particularly by any decrease of the former. On entering any river, then, we may expect the tide to be greatly increased. The magnitude of this change depends on the breadth of the wave, and this again depends on whether the tide be formed by the sun or moon ; hence the ratio of the tides formed by the two luminaries will not be exactly the same at all places. But when the tide has entered the river, the friction of the particles of water causes the disturbance to diminish in geometrical progression as it goes up the river. " At the entrance of the Bristol Channel the whole rise * Phil. Trans. 1842. NEWTON'S PKINCIPIA. 409 at spring tides is 18 feet, at Swansea 30 feet, at Chepstow about 50 feet.'" " At Newnhara it is reduced to 18 feet, and it is still less at Gloucester." The form of the wave itself is also changed in its progress up the river, if the channel is so shallow that the height of the wave bears a sensible proportion to the depth. It is in this case necessary to carry our solution of the Hydro- dynamic Equations to a second or third approximation. When this is done we find that the high water travels up the river with a greater velocity than the phase of low water ; the wave becomes steeper in front, and long and gentle in rear. Thus it takes less time for the tide to rise than to fall. " In the St. Lawrence at 40 leagues below Quebec, the rise and fall occupy equal times ; at 6 leagues below Quebec, the rise occupies five hours and the fall seven hours ; at twenty leagues above Quebec, the rise occupies three hours and the fall nine hours." The rule given by theory is, that the excess of the time of the water falling over the time of rising is six times the product of the time occupied by the tide wave in passing from the open sea to the station under consideration, and the ratio of the rise of the tide above the mean depth to the mean depth. In this change of form we find an explanation of the bore. Two things, says Professor Airy, are necessary for its formation. There must be a large tide rising with rapidity, and the channel of the river must be bordered with a great extent of flat sands near the level of low water. The tide rises rapidly in the centre of the river, but the water is not broken. But, the rise being very rapid, the water is elevated above the sands faster than it can cover them, and therefore rushes over them with a great velocity and a broken front. As the tide proceeds up the river, a still more extraordinary change of form takes place ; the middle of the rear becomes less and less steep ; it is at last horizontal, and finally slopes the other way. Thus for every high tide at the mouth there may be two unequal high tides at a distance up the river. In accordance with this, Mr. K-ussell has observed a double 410 NEWTON'S PBINCIPIA. tide on the Dee. On going to a third approximation, triple tides are found possible, and this occurs on the Forth. When there are barriers in the river, as the land at the termination of the Gulf of Fundy, the reflected wave is such that there is no horizontal motion at the bar- rier, and that the tidal elevation at the mouth, caused partly by the sea and partly by the reflected wave, shall altogether be equal to the tide in the sea itself. Hence just as in the case of sound waves in a tube, there will be " nodes " and " loops," if I may so call them. At some places there might be no tide, and at others there will be a rising and falling of the water, but no wave will be pro- pagated along the channel. This is called a stationary tide. The usual case is that which corresponds to the " fundamental node." There will be high water at all parts of the gulf at the same moment, and the tide will be greater the greater the distance from the mouth, and at the termination will be double that at the mouth. But when there is friction in the water, this is not exactly the case. It can be shown that then a tide wave does roll up the river with a velocity different from that proper to the depth, and the end of the flow follows close upon high water. The height of the tide also may or may not in- crease as it travels along, according to circumstances. In canals adjoining two tidal seas in which high water does not occur at the same moment, the motion of the water must be found by the same kind of reasoning. But as the result is complicated, we shall not consider this case. The motion of the tide in channels deeper in the middle than at the sides, as, for instance, the English Channel, has also been considered by Professor Airy. The velocity of the phase is greater in the centre than along the sides. The tide wave will, therefore, become slightly convex, and the velocity being always normal to the front, it will assume such a shape that the part of the normal intercepted between two consecutive positions shall be proportional to the velocity of the wave along that normal. Thus the NEWTON'S PRINCIPIA. 411 tide curve along the shore will assume a position nearly- parallel to the line of coast. The phenomena of the motion in mid-channel will therefore be the same as that in ordinary channels, and the direction of the current will change only when the water is at its mean height. But near the coast there will be a reflected wave from the coast, and the phenomena of motion will more resemble that of the tides in a short gulf; the direction of the current will therefore change at high and low water. At places inter- mediate, intermediate phenomena will take place. Thus the direction of the current will be intermediate between the direction of the current at the centre of the channel and that along the shore. A little consideration will show that on one side, the current at any spot will, in one com- plete period from high to low water, go round every point of the compass in the direction of the hands of a watch, and on the other side in the opposite direction. The for- mer phenomena are found to occur on the English side of the Channel, the latter on the French side. Such are our present theories of the tides. It may be said that we have almost obtained a complete explanation of river phenomena. The great difficulty is with the ocean. In mathematical language we can solve the ques- tion of the motion of the tide in one dimension but not in two. Philosophers have been of late very assiduous in collecting the " facts " of oceanic tides, rightly considering that such knowledge will enable us greatly to advance the theory. Many interesting papers on this subject will be found in the Philosophical Transactions, and to these we must refer our readers. The first step was the construction of what is called " co-tidal lines." These are curves drawn through all the places on the globe which have high water at the same instant. By drawing these lines for the successive hours of the day we can trace the progress of the tide wave over the whole globe. A beautiful map of these will be found in Johnston's Physical Atlas. It appears that the tide wave travels east to west in the great South Seas, as a 412 NEWTON'S TRINCIPIA. forced wave following the moon. On reaching the Cape of Good Hope, one tide wave is sent up the Atlantic north- wards, and another continues its course into the Pacific, and travels westwards and northwards up the west coast of America in a very remarkable manner. The tide wave in the Atlantic travels, therefore, from the south north- wards, and arrives at England fourteen hours after it left the Cape. It here sends branches into the English and other channels that environ England, while the chief wave continues its progress northwards, sending other tide waves, travelling from north to south, down the same channels. Two tide waves arrive, therefore, at the mouth of the Thames, and the London tide is compounded of these two. One has travelled along the south coast in seven hours, and the other has gone all round Scotland and descended the German Ocean in about twenty hours. The difference of these intervals is about twelve hours; thus the two semi-diurnal waves arrive in nearly the same phase and strengthen each other, and the two diurnal waves in oppo- site phases and destroy each other. There is therefore at London no diurnal variation of the tide. These two tides, though they meet at the Thames, do not travel in exactly opposite directions. They both travel a little eastwards, in direct opposition to the motion of the moon. They are both also greatly modified by the many shoals found in those seas. As may be supposed there are points where the two tides meeting in opposite phases, even the semi-diurnal tide will be destroyed. Captain Hewitt has discovered such a point. It is situated about latitude 52^ ; there is simply an alternate tidal current, but no elevation of the water. The tide wave in the Atlantic is probably partly formed in the Atlantic and partly derived from the South Seas. If it were entirely formed in the Atlantic, as the length of that sea from south to north is greater than its breadth, the tide wave would have a tendency to travel partly along its length, with a very irregular velocity, and partly along its breadth. If the tide were entirely derived from the NEWTON'S PRINCIPIA. 413 Southern Ocean, the tide wave would travel along its length with a velocity proper to its depth. This, as may be supposed, is very irregular. The mean velocity of the tide wave would indicate a mean depth of three and a half miles. Again, if we suppose that a tide wave would be re- flected from the northern coasts of America, and regard the Atlantic as a gigantic closed gulf leading out of the South Sea, the velocity of the tide wave would not merely depend on the depth. Until we know more of the nature of the bottom of the sea, we cannot determine theore- tically the motion of the tide wave. By theory, the diurnal tide wave should be very small near the equator and poles. But this is not the case. The reason is manifest. We have neglected, in our theory, the effect of the configuration of the coast. The importance of this is manifest from the slight sketch we have given of the progress of the tide over the ocean. It is impossible for us to do more than merely allude to the effect of wind, shoals, &c. on the tides. Neither can we enter on the question of aerial tides. That such tides must exist is evident, for the attraction of the luminaries will disturb the air as well as the water. The extent of these oscillations have even been determined by very ac- curate observations of the barometer. The variations of pressure, as indicated by the barometer, also affect the ocean tides in a remarkable manner. Daussy, Lubbock, and Birt have investigated the amount, and shown that a rise of one inch of barometer will cause a depression of the tides of 7 inches in London, 1 1 at Liverpool, and 1 3 at Bristol. APPENDIX. APPENDIX. No. I. GALILEO. SOME uncertainty exists respecting the decree or sentence against Galileo, and the Copernican System. The advo- cates of the Romish Church, indeed, deny that it was pronounced against the doctrine of the earth's motion and the sun's rest, and affirm that the sentence was against Galileo personally, on account of his breach of the pro- mise which he had made to Paul V. (Borghese), and the per- formance of which he evaded by giving the doctrine in the form of a dialogue. It is also alleged that the sentence is pronounced by the Inquisition, and not by the Holy See. Of this there can be no doubt ; but it sets forth a previous " declaration of the theological qualifiers, made by de- sire of His Holiness, as well as of the Inquisition," that the Copernican system is "absurd and philoso- phically false and formally heretical, because expressly con- trary to the Holy Scriptures." This seems to go beyond the mere assertion that it is only false as being unscrip- tural. There is, however, no doubt that the alleged infal- libility of the Holy See is confined to matters of faith ; and thus its advocates have some ground for their asser- tion, that the heretical nature of the doctrine was alone set forth both in the sentence on Galileo and in the previous proceedings. The supposition that he was subjected to E E 418 APPENDIX. torture in order to obtain his recantation, appears to rest on no foundation, except the use of the terms rigorous examination (rigoroso esame), said to be the form in which torture is referred to by those sentences. But the story is completely negatived by Galileo's own letter giving an account of the manner in which he had been treated, acknowledging the respect paid to him, and the lenity of his imprisonment, or rather nominal detention, in his friend the Archbishop of Pisa's (Picolomini's) pa- lace. (Tiraboschi, Lett. Ital. torn, viii.* lib. ii. p. 147.) It is probable that the story of his whispering to a friend " E pur si muove," ("And it moves for all that," ) when he rose from his knees, on which he had made the re- cantation, rests upon no better foundation. Nothing can be more unlikely, at least, than his choosing such a moment for this pleasantry a moment of great though forced humiliation, when he had in the most solemn man- ner called God to witness, that " he abjured, cursed, and detested " the positions which he entirely believed ; nothing more unlikely than his exposing himself to the risk of his words being discovered by immediate examina- tion of the person whom he addressed. But it seems still somewhat doubtful how far the sen- tence upon Galileo has been reversed. His " Dialogue " had by decree of the " Congregation of the Index " been put into that list of forbidden books ; and Leo XII. (Genga) nearly thirty years ago ordered it to be expunged from that list. Mr. Drinkwater (Life of Galileo, p. 64.f) states that it had not been erased in 1828. Mr. Lyell (Principles of Geology, p. 56, edition, 1853), considers that this assertion is inconsistent with the account which he received from Professor Scarpellini at Rome, in 1828, * Viviani, who is described as devoted to Galileo more than a son to a father, and who attended his master for the last three years of his life (Montucla, ii. 290.), ; gave not the least countenance to the exaggerated accounts of his treatment. f Published by the Useful Knowledge Society in 1829. APPENDIX. 419 that Pius VII. (Chiaramonte) had assembled the Congre- gation " which repealed the decree against Galileo and the Copernican system," with only one dissentient voice. It is, however, possible that the Index might still retain the name of the book. It must also be remembered that nothing can be more inconsistent with the usual proceedings and policy of the Roman See, than making such an admission of error as is involved in the repeal of a sentence pronounced, and purporting to proceed upon the ground of heresy. There can be no other explanation given of such a man as Benedict XIV. (Lambertini), the friend and patron of Boscovich, the correspondent of Voltaire and other lite- rary men, himself an eminent cultivator of letters, and remarkable for the liberality of his opinions, suffering the sentence on Galileo to continue the disgrace of his church. It must, however, be borne in mind, that probably soon after Galileo's condemnation, certainly for the last century or more, the prohibition to teach the Copernican system was little more than nominal ; nothing else was required than to term it the Hypothesis, and not the doctrine, or theory, of the earth's motion. The declaration of the Fathers (Leseur and Jacquier) accordingly calls it an Hy- pothesis. They regard the proceedings in Galileo's case, (to which it is manifest that their words refer) as a decree against the motion of the earth. But many have sus- pected that this is a covert attack on those proceedings; and certainly, considering the time of the declaration ap- pearing, when Benedict XIV. was in every way promo- ting science and letters, and pursuing generally the most liberal and enlightened policy, this supposition is not without plausibility. It must be added that whenever the outward deference to the decree of the Consistory was shown, the doctrine, under the name of Hjpothesis, was openly taught, with the utmost freedom in giving the proofs whereon it rested. 420 APPENDIX. No. II. (p. 58). ANOTHER DEMONSTRATION. TAKING the expression for the force and making it h 2 r = the general form j-? where h is constant, we have h z r u, - 3 T> = 3 This is a differential equation of the second order, and it can have but one general integral, and that can contain only two arbitrary constants. Such an integral has been found ; for it has been shown that any conic section with the focus in the centre of force satisfies the equa- tion. The nature of the case shows that any singular solution is out of the question. We may also arrive at this result by reversing the process in p. 49. Substituting for R, 5 x dp u, , dp it* dr j . , ,. 1 2 a _^_=-i- and -f = -TO -- and integrating, = = ~ dr r 2 p* h 2 r 2 ' p 2 h 2 h G). This is a well known property of all conic sec- tions; but to show that no other curves possess it, put - = M : then since 9 = w 2 + -5^ therefore -j-A + u 2 = r p d v a 01 2 l . u + C, and therefore 7 = d 6. APPENDIX. 421 -***-'* The integral of this, calling C + ^=L 2 ,is cos ^ - 1 = , or - = u = ^ + L cos , the general equa- tion to a conic section. No. III. FORCE VARYING INVERSELY AS THE DISTANCE. IT is remarkable that what at first sight seems to be the most simple of all the cases, that of the central force vary- ing inversely as the distance, or of m = 1 in ~, should be found so much the most difficult of solution, and that, whether the proportion of - enters into motion related to one centre only or to more centres than one. Herman, in the Phoronomia, turns away from it, merely observing that his formula fails when m = 1. Clairaut, in his excel- lent commentaries on the Principia, his additions to Madame du Chatelet's translation, deduces, chiefly from the Pro- positions of the Second and Eighth Sections (Lib. I.), a general differential equation for the curve described by a body under the influence of a centripetal force as Y, a function of the radius vector ; and the equation is there- fore a polar one. It involves the integration of J'Y dy. Consequently, when Y = - , the case we are now con- sidering, the integral contains an unmanageable logarithm ; B E3 422 APPENDIX. for the equation becomes b*f*dx= / x d ?Ai. V *J y ' = 2 5-. He makes no mention of this y* (2 E - log y*Y case, as, like Herman and most others, he seems unwilling to approach it ; undoubtedly, however, such is the applica- tion of his formula. Keil, in his paper on Central Forces in Phil. Trans. 1708, p. 174., gives the case of the force as - and reduces it to finding - = P, the perpendicular to the tan- (ft-logr)* gent. By one process grounded on Prop. XIA. Lib. I., this result is obtained for the case of -, that is - 1 s*/*2dxd 2 x + 2dyd 2 y r (xdx + i/dy) JJ- d-?- - = J** J+y* ' r X - 7-0 2 log (o: 2 + v 2 ) c = ; and d t* being (y d x - x dy}* . rd x* + d y* = i ^-,the equation becomes / - r^- 2 c 2 J (dxxdy) 2 The process grounded on the formula / = ~ r> z p . i\ is, if possible, more hopeless; for this gives h (y* + (x - c) 2 f x (d x d* y - d y d* x} 2 (y d x x d y + c d y} 3 = r, or h (y* + (x- c) 2 ) (d x d* y - APPENDIX. 423 " 2 (y d x x dy + c d y) 3 The difficulty follows - wherever that proportion en- ters into the investigation. Thus in the problems con- nected with different centres, when it is found that forces varying as - 2 add - 2 , being combined with forces varying as the distance directly, or as r and q, give an elliptic orbit, the resultant of the latter forces passes through the centre, and the locus of that resultant is the opposite semi-ellipse, and so of a circle. But when the proportion is - and , (also if the force towards each centre is as the radius vector to the other centre,) the resultant passes through innumerable points to an opposite curve, sometimes of a different kind, although each result- ant differing in its direction from all the others, and in the case of the circle, from the diameter, is equal to the one passing through the middle point of the line joining the two centres. In this case, therefore, there is no combined action of the forces - 2 and -^ or -^and - 6 or of their seve- ral resultants, with the resultant of and , as there is in the case of - and ?-, but the several forces act wholly in m m the direction of the radii vectores severally. It evidently appears to be a more simple and natural combination that the two sets of forces should diminish E E 4 424 APPENDIX. with the distance increasing, as in -^ and -5 combined with and , than that one set should decrease and ano- r q ther increase with the distance, as in . and - with r and r 2 q* q, in which case there must even be an extinction of force at one point, where (taking the sum of the forces instead of their resultant)- 2 + 7, ~ ~, or r is as in the equation 2 = 7ft 2 . Of course the value of q would be the same ; and the resultant (more accurately taken to measure the increase of the force) would at one value give the two sets of forces as counterbalanced. The younger Euler (J. A. Euler) has a paper in the Berlin Mem. 1760, p. 250, upon the action of a central force decreasing as the distance, in the case of the attracted body's descent towards the centre, and states the reason of this problem being insoluble except by arcs or logarithms. He finds that taking a = the height from which the descent begins, f that at which the centripetal force is equal to the gravity of the attracted boJy, the time of descent towards the centre is = -L C dy ? y ^ the ^_ J^/ a V loor tance from the centre. No. IV. CENTRAL FORCES TO MORE THAN ONE POINT. 1. IT is to be lamented that Sir L JS T ewton did not treat the problem of forces directed to more fixed points than APPENDIX. 425 one, as to two such points, either in the same or different planes from the body acted on. This is the fundamental point in considering disturbing forces when the centres are not fixed, which makes the problem more complicated and difficult. It is, however, sufficiently so even where the centres are fixed. 2. That the subject must have attracted his attention there can be no doubt. He had gone so much into the more difficult inquiries respecting disturbing forces that he must have fully considered the somewhat simpler, what may be termed the fundamental, case of fixed centres. In- deed, a paper communicated to the Royal Society in 1769 (Phil. Trans, p. 74.) contains a demonstration by W. Jones, an intimate friend of Newton, of a proposition on this subject, which Machin had immediately after Sir Isaac's death given .to the translator of the Principia. Machin had observed on the want of some investigation of the motion of forces directed to two centres, as required to explain the motions of planet and satellite, which gravi- tate to different centres, in a word the problem of the Three Bodies. The proposition of Machin and Jones goes but a very little way to supply the defect complained of. It is confined to the case of the line joining the two centres being in different planes from the line of projection ; it is that the triangle formed by the radii vectores and the line joining the two centres or fixed points, describes equal solids in equal times round that line ; and the demonstration is similar to that of the first proposition, of equal areas in equal times when a single force is directed to one centre. It seems reasonable to conclude, that Newton had, upon full consideration, found the full investigation of the subject beyond the powers of the calculus as it then existed. It is at least certain that, though he might have mastered it, he never could have delivered his results synthetically as in the Principia. 3. The solutions on disturbing forces generally consider one force as acting in the one direction, that of the radius 426 APPENDIX. vector, and another in a line perpendicular to that radius vector. Thus Clairaut (Mem. Acad. 1748, p. 435.) gives these equations rd*v + 2 drdv = Yl d x 2 r being the radius vector, v its angle with the axis, dx the differential of the time, II the force to the centre, 2 the disturbing force. So D'Alembert (Mem. Acad. 1745, p. 365.) takes the same course, and obtains an equation to the orbit in question, depending on the integration of fndz, II being the disturbing force acting in a line perpendicular to the radius vector, and z the circular arc described with a radius equal to the distance between the centre of force and the vertex of the orbit. This assumes, however, that the orbit is itself nearly circular. 4. If P = distance of E (Earth) from Moon (M)'s quadrature, s = sin. angle of rad. vec. r with the per- pendicular to a, the distance of E from S, the Sun ; IN. r AT *u i v = velocity of M ; then v a v = supposing the motion of M to be almost uniform. Here one of the forces acting on M is directed towards E and E + M S x M E . . . .. is = , , \- ,. ; the other force is in a line JV1 ill D M. S x S E parallel to S E or a, and is in consequence of this investigation that Clairaut for some time announced, as did also Euler and D'Alembert, that there was a material error in the Newtonian theory of the Moon's motion. The error, which afterwards was found to arise from their having omitted the consideration of certain quantities, was acknowledged by Clairaut three years later (Mem. Acad. 1748, pp. 421, 434.), but no one APPENDIX. 427 can read that paper without feeling that the acknowledg- ment was too coldly made, after he had gone so far as to suppose that the whole Newtonian doctrine was over- thrown, and to propose a new law of -^ + - T , the whole of this arising from his own error. It is to be remarked, however, that the investigation of 1745 was in all respects most accurately conducted, and must have led to the same result as in 1748 but for the supposition that certain quan- tities might safely be neglected. Even in 1745 Clairaut, upon Newton's assumption of the excentricity of M being nothing, comes to his conclusion that the proportion of the axes is as 69 to 70. 4. Legendre treats the subject very fully, as far as re- gards two centres, and also confining himself to the forces being inversely as the square of the distance (Exercises de Calcul. Integral, part iv. sect. 2.). He deduces from his analysis several theorems, two of which he regards as very remarkable. The first apparently strikes him in this light, because it shows the same orbit to be produced by the combined action of the two forces directed towards two foci, as either force would produce acting on the body, and directed to one of the foci. If V is the velocity at the vertex of the ellipse which would make the body de- scribe that curve when acted upon by the force directed to one focus, v the velocity at the same point which would make the body describe the ellipse when acted upon by the other force directed to the other focus ; then if the two forces act together upon the body, and I is the initial velocity, or velocity of projection, it will describe the same ellipse, provided I 2 = V 2 + v" 2 . 5. The other theorem follows from his integration which gives the expression for the time. It is that if two equal forces act upon the body directed to the two foci, and the masses of the attracting bodies consequently are equal, the revolving body will describe the ellipse in a shorter peri- 428 APPENDIX. odlc time, will move more swiftly, than if the whole mass were placed in one focus and acted from thence upon the revolving body. He takes the example of the tangent of the orbit with the axis making an angle of 30, and finds the periodic time shorter in the proportion of nearly 78 to 100 when the attracting mass is divided into two, one acting in each focus, than when both combined act from one focus, 6. What renders this problem of more centres than one so difficult, is that the resultants of the forces pass through different points, and that they vary by a law which differs in each case, as the locus of their extremities is a different curve. Take the least complicated, but still full of diffi- culty, that of two fixed points as the centres of force, and take the instances in nature of the forces being inversely as the square of the distance ; the radius vector to one point being r, the force -% ; to the other point the radius vector q, the force ^-. Now the force which acts on the body being the resultant of these two, and these forces not being as r and q, the diagonal does not pass through the middle point of the line joining the two centres; except in the single point of the orbit where r=q, and even then it may not reach that line, for it is 2 ' m . At every other point it runs in a different direction. Let S and S' be the two fixed points ; S P = r, and S' P = q. Then P a being taken = 2 -, and Pa' = , the resultant at that point P bisects a a! and is P c, and produced, P M cutting the axis. From hence may be seen how complicated would be the analysis, how next to impossible the geometrical con- struction of the locus of P, by referring the lines PM to APPENDIX. 429 S S' as an axis. We know indeed that one of the forces -^ or -3- acting towards S or S', the locus of P is an ellipse ; but it would not follow that if both forces acted the same curve would be the locus. That the force would be differ- ent is certain, because it would be as P c, and not as either Pa or P of. But it may be said that the curve also would be different. Let us, however, suppose the case of the curve, whatever it be, cutting the axis S S' produced at S and S', points equally distant from S and S', so that SS = S' S' ; also that the angle and the initial velocity of projection from S and S' is the same, and further that the 430 APPENDIX. attraction as the mass is the same from S and S', or that the mass of the body in S and in S' is the same ; then it seems impossible to avoid the conclusion that an ellipse, and the same ellipse, must be described ; because one of the forces alone acting from S, as 2 , would give the ellipse passing through 5 and S'; and the other force alone acting from S, as ^ would give the ellipse passing through the same points "2, and 2'; and the initial velocity * and angle of projection would prevent any difference in the length of the conjugate axis; and in the middle point answering to the centre C, the equality of r and q and of Pa and Pa' would make the diagonal PC coincide with the conjugate axis. But a further combination of forces may be sup- posed in this case ; two forces acting towards the points S and S' and in the proportion of r and q, or and . How will this addition affect the locus of P ? It should seem, for a reason similar to that before given, that the curve would remain the same ; for the two new forces and , acting in r or q or PS and PS' respectively, their resultant must, if there were none other acting, pass through the middle point C, between S and S'; and as we know that a force acting from that point, and in proportion to the distance from that point, causes the body to move in an ellipse whose centre is that point, and r + q being con- stant, the ellipse must have the same axis and coincide * The condition of Legendre (mentioned in page 427), that I 2 = V 2 4- ? is supposed to hold; for otherwise the centrifugal force would not be suffi- cient to balance the centripetal. APPENDIX. 431 with the ellipse produced by the combination of the forces m , m -r and ? . 7. This had appeared to be a necessary consequence of the conditions stated, but not as at all proving the ve- locity to be the same in the ellipse, when described by one force % or ^, or when described by the combined action of both, or when described by the combined p r 3 <7 m m r , q , action of - - and . or of -^, ^, , and ; be- m m r 2 q* m m cause in all those cases the velocity will be different, and particularly the action of = H with - + will occa- r 2 m q* m sion adifferent velocity in each point from that occasioned by 3- + 3. Thus to take the velocity at one point answering to C. If IT and ITa' be taken as -^ and -^, the diagonal He' is the force of ^ and -y combined, IlCisthe resultant of and combined (supposing m = l). There- fore the velocity in IT will be as lie' + IIC, when all the forces act, and only as Etc', when the two former act alone, and as IIC when the two latter act alone.* But the curves appear to be the same in each case. 8. These consequences seeming to follow from a con- * The difference in velocity is easily obtained, in comparing the effect of one force and of the combined forces, from the equation u 2 =5 (/x half chord osculating circle, the chord being = 2 p , p = perpendicular to the tan- gent, and R = radius of curvature. 432 APPENDIX. sideration of the conditions stated, but without a full and rigorous investigation, it was very satisfactory to find that Lagrange had arrived at the same conclusion in one case of his solution of the problem of two fixed centres (Mcc. Anal. pt. ii. sect. 7. ch. 3.) That solution is marked throughout with the stamp of his great genius. Euler had, in the Berlin Memoirs for 1760, treated the case of the inverse square of the distance and the centres and orbit being in the same plane. Lagrange's solution is general for the force being as any function of the distance, and of x, y, z, being the coordinates. Pressed by the great difficulties of the problem, and the impossibility of a ge- neral solution, he first confines himself to the inverse square of the distance (p. 97.), and a general integration being still impossible, even after obtaining a differential equation with the variables separated, he makes a supposi- tion which enables him to obtain two particular integrals (p. 99.), and this gives for the orbit an ellipse in the one case and an hyperbola in the other, with the foci in the two centres offeree; and it follows, he observes, from the investigation, that the same conic section which is described in virtue of a force to one focus, acting inversely as the square of the distance, or to the centre and acting in the direct ratio of the distance, may be still described in vir- tue of three such forces (" trois forces pareilles*"), tending to the two foci and to the "centre." He adds: "ce qui est tres remarquable" (p. 101.). It having appeared to many persons that a portion of the demonstration was not so rigorous as might be desired, M. Serret has very ably and satisfactorily supplied the defect (Mec. An. torn. ii. note iii. p. 329. ed. 1855), but he arrives at the same * It is plain that "pareilles "does not mean | of the same kind as -L and v ; for he resolves the force to the centre into two acting to the foci, and calls the whole forces ^- + 2 7 r and -~ + 2 7 q. APPENDIX. 433 result. There is also given a very important generalisa- tion of Lagrange's solution, and of Legendre's theorem already mentioned, by M. Ossian Bonnet. (Ibid, note iv.) 9. The same reason already given proves that if, instead of two points not in the trajectory we take two in it, as 2 and 2', and refer the forces to those two, and make the forces -5- and g- in 211' and 2TI' respectively, and the angle of projection and initial force the same, the same circle will be described by the body; and that if two other forces also act on it, as 2n' and 2'H' (or and - ) the same \ m m) circle will be described by the joint action of the forces. This is even a more remarkable consequence than the other ; because the forces acting to the centre would of course give a uniform motion, and those acting to the points in the circumference an accelerated motion, and the forces combined will give an accelerated motion. At the middle point IT, the velocity will be, if only the forces ~ and . act, as ^ ^ if the forces and also act, it r 1 q & a mm will be as A/-r~H -. It must, however, be added, V 4 a 4 m that Lagrange's solution does not contain this case, of the circle and two points in the circumference, and there is very great difficulty in applying to it his analysis. Indeed, it appears that if the problem be worked upon the datum /) of E, = ~ + 2 y r, and Q = ~ + 2y q, there is no possi- bility of obtaining an expression freed from the integral sign ( r\ in the same way as Lagrange does from his equa- g tion, founded upon the datum R = ^ + 2 y r and Q = 2 F F 434 APPENDIX. -H 2yy, m = 2, and consequently m -f 2 = seems necessary to his process. There seems reason to suppose that the kind of reasoning on which we have relied as to the identity of the trajectories had influenced Legendre in confining his investigation to the case of curves which have not infinite branches. He expressly says (Ex. de Calc. Int. 11. 372.), that lie confines himself to curves where the orbit is restricted to a definite space. Certain it is, that the reasons applied to the identity in the case of curves returning into them- selves is wholly inapplicable to curves having infinite branches. 10. The extreme complication of the problem arising from the resultants passing through innumerable points in the axis has been above noted, as regards the case of two forces only ^ and . When we add the other two and the complication is not considered by Lagrange m m to be increased (p. 99.), and probably it is not as regards the analytical investigation. But it certainly is increased as regards the geometrical construction ; for we then have to take the resultant of P c with P C (which is the re- sultant of r and g}, and this will carry the ultimate dia- gonal representing the whole force applied to P beyond the axis S S'. Lagrange indeed does not take P C into his analysis, because he supposes the forces r and q to act in the same line of the radii vectores with the forces -y and -3-. But this would cause these radii vectores to be produced, and make their resultant also fall below the axis. It can hardly be doubted that these considera- tions weighed with Sir Isaac Newton, in disinclining him to the investigation of a problem which could afford no hope of a geometrical, or of any synthetical solution. That he had deeply considered the subject of attraction to various APPENDIX. 435 centres, in the more difficult case of moveable centres is certain. The justly celebrated LXVIth proposition of the First Book affords ample proof of it; and indeed the LXIVth proposition comes so near the subject of this note, that it may be correctly said to contain the grounds both of Clairaut's and Legendre's more full investigation. 11. In connection with this subject Lagrange expresses great admiration of a theorem of Lambert, which no doubt is remarkable, that in ellipses (the central force being as 3! the time taken to describe any arc depends only on the transverse axis, the chord of the arc, and the sum of the radii vectores at its extremities. We may observe, in passing, that the vanishing of the expression for the conjugate axis in some fundamental formuhe con- nected with the ellipse, for example, the subtangent, gives rise to other curious properties of the curve similar to the one noted in this theorem, which is itself related to that peculiarity. (See a porism arising from this circum- stance; Life of Simson, p. 154.) The same theorem had occurred to Lagrange himself, in examining the problem of deflecting forces to two centres ; it is indeed derivable immediately from the case of that problem when one force vanishes and the centre connected with it is in an arc of the ellipse ; for then the radius vector belonging to that centre becomes the chord. But Euler, long before either of them, in 1744, had given the theorem for parabolic arcs, which they only extended to elliptic arcs, arid had published it in the Berlin Mem. 1760. Yet when Lam- bert claimed it as his own in 1771, and Lagrange gave him the honour of it in 1780, Euler, though he lived three years after, never thought of reminding them of his prior claims. It was thus, too, with the first of analysts, respecting the extension of the Differential Calculus to that of Partial Differences (Life of VAlembert, p. 466.), by far the greatest step in mathematical science which FF2 436 APPENDIX. has been made since the age of Newton and Leibnitz, if it have not a rival in the calculus of variations, the honour of which also is shared by him with Lagrange. 12. It must be observed that when in 1771 (Berlin Mem.} Lambert extended the theorem to elliptic arcs, he was ignorant of Euler having anticipated him as to para- bolic arcs. But Lagrange truly states (Mec, Anal. ii. 28., ed. 1855), what shows that all of them had been antici- pated by Newton. For in the IV. and V. Lemmas of the Third Book he had very distinctly given the whole materials of the proposition as far as parabolic arcs are concerned. Lagrange notes the uses of the theorem, and observes upon the remarkable circumstance of the time not depend- ing at all on the form of the ellipse, provided the trans- verse axis remains the same. This must have frequently recurred to his recollection, when engaged in those great investigations which show the connection that the trans- verse axis remaining unchanged, has with the permanency of the system. 13. He further remarks- upon another consequence of the conjugate axis ? or the form of the orbit, not affecting the time; namely, that the conjugate wholly disap- pearing, and the orbit becoming rectilinear, the theorem applies to the time of falling to the centre, on the centri- fugal force or that of projection ceasing to act. (Berlin Mem. 1778.) But Newton's Vlth Lemma, to which he does not refer, in some degree anticipated this also. 14. The great difficulty of the problem of several cen- tres, has been stated. Euler was clearly of this opinion, and he was the first that undertook the solution. After speaking of the general problem (Berlin Mem. 1760, p. 228.) as alike important and difficult, he confines him- self to the case of two bodies in fixed positions, acting upon a third, which moves in the plane of those disturbing bodies ; in a word, to the motion of a body drawn towards two fixed centres. He says that, whoever undertakes the APPENDIX. 437 solution of this less difficult problem " will find difficulties almost as insurmountable as in the great fundamental problem of astronomy ; " and adds that, after making many fruitless attempts, he had at last been led to a solution by the accident of an error into which he had fallen in his in- vestigation. What he proposes is to find the cases in which the curve is algebraical ; there being, according to the con- ditions, an infinite variety, most of them transcendental. He considers, however, that if this case of two bodies in fixed centres, and in the same plane with the body at-, tracted, should be incapable of solution, the general pro- blem must prove still more so. Nothing can exceed the clearness of his investigation ; and the ingenious subtlety of the contrivances by which he facilitates the reduction of his differential equations to those of a lower degree. Of this Lagrange expresses great admiration, who, in giving a solution of the case in some respects more ex- tended, but in others less, became fully sensible of the difficulties of the process, and whose investigation is less luminous than his great predecessor's. Euler reduces his investigation to the integration of the equation !. d x v dy and obtaining the relation between the angles made by the two radii vectores with the axis. It is clear that La- grange's solution is obtained by another course altogether. No. V. LEIBNITZ'S DYNAMICAL TRACTS. EARLY in 1689, about a year and a half after the pub- lication of the Principia, there appeared in the Ada Eru- ditorum, of Leipsic, two papers of Leibnitz, entitled, " G. Gr. L. Schediasma de Resistentia Medii et Motu pro- jectorum gravium in medio resistente," and " Tentamen 438 APPENDIX. de Motuum Coelestium Causis." As these tracts coincide in their subjects and in many of their doctrines with the propositions of Sir I. Newton, it has been held by some, and apparently by Sir Isaac himself, that their author had obtained from those propositions the substance at least of his own. This is certainly a grave charge against Leibnitz; inasmuch as he affirms that he had not seen the Principia when he wrote his two papers, admits that he had seen an abridgment of it in general terms, and expresses regret at not having seen the work itself. He further states his inability to comprehend by what me- thods the most important of Newton's discoveries, as de- scribed in the abridgment, the elliptical motion of the heavenly bodies, is demonstrated mathematically from the data obtained by observation, and the law of attrac- tion governing that motion ; a proposition which Leibnitz gives as deduced by himself with the aid only of the differential calculus. The whole statement, though not in direct terms, yet by manifest implication, represents that he had made his investigation, and arrived at the result, without any further knowledge of the Principia than the fact of Sir I. Newton having obtained the same result by his process, whatever it might be. It cannot be denied, therefore, that the strongest proof is required to authorise the belief of his having seen the Principia, and borrowed his propositions from thence. But instead of proofs there appear to exist only certain sus- picious circumstances. His statement (Acf. Erud. 1689, January, p. 36.) is perhaps too particular in describing his absence from home on his official journey. Newton says that a copy of the Principia had been given imme- diately after its publication, to Facio Duiller, a young mathematician, a friend of Huygens and of his own, for the purpose of its being sent to Leibnitz ; but there is no evidence that it reached him. (Macclcsfield MS. Paper in Newton's handwriting. Rigaud's Historical Essay, App. Nos. XIX. XX.) The same paper of Newton APPENDIX. 439 charges Leibnitz with having endeavoured to appropriate Mouton's discovery by denying that he had seen his work before he made the discovery himself.* It must unfortunately be added, that Leibnitz's con- duct in the controversy relating to the invention of the calculus, leaves an unfavourable impression. There was great disingenuousness, to give it no harsher name, in John Bernouilli's proceedings, by the confession of his own family ; and Leibnitz, who had encouraged him, betrayed the secret confided to him of his authorship, for the mere purpose of grasping at an advantage, by means of the autho- rity which Bernouilli's great name in the mathematical world gave to his decision against Newton, whom he had opposed by anonymous writings to please his patron. All this, however, we must admit, only affords ground for en- tertaining suspicions; and the proof required must be sought for in the internal evidence of the works compared together. The fact of the abstract having appeared in the same work the June preceding is admitted by him, as is his having read it. The account, however, which it contains of the Principia is exceedingly general ; none of the inves- tigations are given of the propositions which it states that the woi'k enunciates. We can only consider it as showing that the truths of which it gives a concise summary, are proved by the application of mathematical reasoning to the known phenomena ; and a person so learned in this science as Leibnitz could not have read that the Principia treated of the descriptions of various trajectories, particularly the conic sections, according to various data (juxta varia data, p. 308.), without perceiving at once that this must refer to dynamical considerations. But especially must he have per- ceived in what manner Newton had conducted his investi- gations, when he found it stated (p. 310.) that the heavenly * This was probably Mouton's method of interpolation for places between those calculated, instead of proportional parts. 440 APPENDIX. motions were explained by the propositions in the three first sections of the First Book, and still more particularly (p. 311.), that from the phenomena and the things contained in that First Book, the demonstration was given of the elliptical orbits, the principle of gravitation, and the law of the inverse square of the distances, in the case of all planets and their satellites. It is manifest, therefore, that, without having seen the Principia, Leibnitz may have been so far enlightened by the view given of Newton's labours, as to be set upon applying the differential calcu- lus to dynamical investigations, and to the motions of the heavenly bodies as the most important of all. He found, from the account of Newton's work, that he had succeeded in solving the great problem by mathematical investiga- tion. He never had made any such attempt before ; he now made it when he found Newton had successfully made it ; and to a certain extent he himself succeeded. A great difficulty ai-ises in examining these propositions of Leibnitz and comparing them with Newton's, from the singular manner of using the letters in the diagrams and referring to them, as well as from the inaccurate printing. It however appears clearly enough that there are incon- sistencies between different parts of his investigations ; and Newton, when he breaks off with the words " Newtoniana tantum descripsit suo more,ac describendo nonnumquam " after, in partly the same terms, having charged him with imitating fluxions and then erring in his imitation from not well understanding that method, appears to have in- tended making a similar remark upon his copying the dynamical propositions. (Rigaud, App. XIX. XX.) No doubt, if taken literally, and using the words centrifugal force in the sense in which Newton and indeed all others use it, there seems the greatest inaccuracy in the position that it varies inversely as the cube of the distance or of the radius vector, this being only true if the curve is a circle. But when we find that by conatus centrifugus he means what would be the centrifugal force in a circle APPENDIX. 441 whose radius is the radius vector at the given point of the curve, there is no error, and indeed he obtains his result in the same way as Newton does, nor could it well be obtained otherwise. His XVth proposition equally coincides with the equation to the centripetal force de- duced from the Vlth proposition of Book I. of the Prin- cipia ; and the resolving the forces into two, one of which is in the direction of the radius vector, is according to that proposition.* By harmonical motion he means motion whereby equal curves are described by the radius vector in equal times ; of which he gives no demonstration. Though we may be surprised with several other coinci- dences, it must be remembered that Leibnitz had the great benefit of Huygens' theorems on centrifugal forces ; and if it be alleged that he threw his propositions into the form of dealing with centrifugal forces, the circumstance just adverted to will account for it without the suspicion that he did so to distinguish his investigations from those of the Principia. Still less have we a right to suggest that the attempt at reducing the whole within the scope of the hypothesis of vortices was made to conceal his knowledge of the Principia. It without doubt originated in the favour still entertained generally in that day for the Car- tesian philosophy, of which not only Huygens was a zealous supporter f, but Euler himself a disciple, half a century latent Upon the whole, we may affirm that the internal evi- dence is insufficient to support the charge. * The exact agreement of his XVth prop, with our equation ~ - r =. centripetal force, being the angle of the radius vector with the axis, is to be noted among other coincidences. j- Letter to Leibnitz against the principle of gravitation, 1690. J Mem. Acad. Paris, on the Tides, 1740. It should seem that M. Biot has, though with some hesitation, arrived at the same conclusion, from several passages of his most learned and valu- able papers in the Journal des Savants, 1852; a paper which deserves to be 442 APPENDIX. diligently studied by all who take an interest in the history of the mathe- matical parts of the Newtonian philosophy. It was occasioned by the truly admirable publication of Mr. Eddleston ( The Newton and Cotes Cor- respondence); and it throws important light upon some points that had not been sufficiently examined. No more profound admiration can be cherished for Newton than this eminent mathematician constantly and warmly expresses; nor can any disciple find the least reason to complain, unless, perhaps, he might dissent from the observation (p. 275.) on the inferiority of the Newtonian notation and supposition of generation compared with the Leibnitz plan, as if on the latter alone the calculus of partial differences could have been invented. The investigations from pp. 458. to 476., and from 522. to 535. are of great importance ; they strikingly illustrate the facilities afforded by analytical process in the solution of dynamical pro- blems; they, indeed, show how the propositions of Sections II. and III. (Lib. L), are easily deduced from one general analytical formula, and may afford new ground for the supposition of those who think that Newton investigated algebraically, and demonstrated geometrically. M. Biot inclines strongly to the belief that the abstract of the Principia in the Acta Erudi- torum, referred to in the text, was the work of Newton himself. THE END. LONDON : F tinted by SPOTTISWOODE & Co. New-Street-Square. A CATALOGUE NEW WORKS IN GENERAL LITERATURE, rUBLlSHED BY LONGMAN, BROWN, GREEN, AND LONGMANS, 39, PATEBNOSTEB BOW, LONDON. CLASSIFIED INDEX. Agriculture and Rural Affairs. Pages. BavMon On valuing Rents, &c. - 4 Caird's Letters on Agriculture - 5 Cecil's Stud Farm - - - 6 Panes. Richardson's Art of Horsemanship 18 Riddle's Latin Dictionaries - lb&19 Roget's English Thesauius - - 19 Rowton's Debater - 19 Short Whist 20 Pages Rogers' Essays from Edinb. Review, 19 Roget's English Thesaurus - - 19 Russell's (Lady Rachel) Letter. - 19 " Life of Lord W. Russell 19 Loudon's Agriculture - 13 Thomson's Interest Tables - - 22 Schmiti's History of Greece" - 19 " Self-Instruction - - 13 Lew's Element* of Agriculture - 14 Webster's Domestic Economy - 24 West on Children's Diseases- - 24 Smith's Sacred Annals - - 20 Southey's Doctor - 21 Domesticated Animals - 14 Willich's Popular Tables - - 24 Stephen's Ecclesiastical Bioi .aphv 21 Wilmot's Blackstone - - - 24 " Lectures on French I: story 21 Arts, Manufactures, and Architecture. Botany and Gardening. Sydney Smith's Works - - - 20 J Select Work, - 23 " Lectures - - i.*0 Arnott on Ventilation - - - 3 Conversations on Botany C " Memoir. - - 20 Bourne On the Screw Propeller - 4 Hooker's British Flora - - 9 Taylor's Loyola - - - - 21 Brando's Dictionary of Science,&c. 4 " Organic Chemistry- - 4 Chtvreul on Colour - - - - 6 " Guide to Kew Gardens - 9 " " " Kew Museum - 9 Lindley's Introduction to Botany 13 " -Wesley - - - - 21 Thirlwall's History of Greece - 22 Townsend's State Trials - - 22 Cresj 's Civil Engineering - - C Eastlake On Oil Painting - - 7 Gwilt's Encvclo. of Architecture - 8 Jameson' Sacred & Legendary Art 10,11 " Commonplace Bok - 10 Theory of Horticulture - 13 Loudon's Hortus Britannicus - 13 " Amateur Gardener - 13 Trees and Shrubs - - 13 Turkey and Christendom - - 23 Turner's Anglo-Saxons - - 22 '': ' - Sacred Hist, of the World 22 Whitelocke's Swedish Embassy - 24 Konia'sPict" i;i ; Lit. of i.uther - 8 " Plants - - - 13 Woods' Crin.. ;, u C:.m].aiLn - - 24 Loudon's Rural Architecture - 13 Pereira's Materla Medica - - 17 Youug's Christ of History - - 24 Rivera's Rose Amateur's Guide - 19 Piesse^s ArfofPffrfulnery - - - 18 Richardson's Art of Horsemanship 18 Scrivenor on the Iron Trade- - 19 Wilson's British Mosse. - - 24 Chronology. Geography and Atlases. A rrow smith's Geogr. Diet, of Bible S Brewer's Historical Atlas - - 1 Stark's Printing - - - - 23 Steam Engine, hv the Artisan Club 4 Tate on Strength of Materials - 21 Blair's Chronological Table. - 4 Brewer's Historical Atlas - - - 4 Bunsen's Ancient Eeypt 5 Butler's Geography and Atlases - 6 Cabine* Gazetteer - ... 6 Cornwall, its Mines, &c. .S3 Ure's Dictionary of Arts, &c. - 22 Haydn's Beitson's Index - - 9 Durrieu's Morocco - - - 23 Biography. Jaquemet's Chronology - - 11 Johns* Nicolas' Calendar ofVictory.il Nicolas'. Chronology of History - 12 Hughes's Australian Colonies - 23 Johnston's General Gazetteer - 11 Lewis's English Rivers - - 1 1 M'Culloch's Geographical Dictionary 14 Bodensted't^and Wagner's Schaiml 23 Commerce and Mercantile " Russia and Turkey - 23 Buckingham's (J. S.) Memoirs - 5 Affairs. ' ' r Crimea ^ - - - - 18 Francis On Life Assurance - - 8 " Russia - - 15 Cockayne's Marshal Turenne - 23 Francis's Stock Exchange - - 8 Lorimer's Youne Master Mariner 13 Murrav's Encvclo. of Geography - 17 Sharp's Britis'h Gazetteer - - 19 Dennistoun's Strange & Lurmsden ^ Mac Leod's Banking - - - 14 M'Culloch'sC 'ommercf & Navigation 14 Wheeler's Geography of Herodotu. 24 Hobo rt d 's 8 Memo e irs fcld "^ - "' - 23 Holland's (Lord) Memoirs - - Scrivenor on Iron Trade - - 19 Thomson's" Interest Tables - - 22 Tooke's History of Piices - - 2J Juvenile Books. Amy Herbert 20 Cleve Hall 20 tMdner't Cabinet Cyclwedta T Maunder's Biographical Treasury- 15 Criticism, History, and Earl's Daughter (The) - - - 20 Experience of Life - - - 20 Memoirs. Gertrude - - - - 20 Me^aL^fMemoirs^'oTcfceTo^ - 15 Austin's Germany - - - - 3 Blair's Chron. and Histor. Table. - 4 Gilbarfs Logic for the Young - 8 Hewitt's Ben's < uuntrv i,,,l. - 10 Russell's M^" d \ v D Ru , seU ' \l Brewer's Historical Atlas - - - 4 Bunsen's Ancient Egypt - - 6 " (Mary) Children's Year - 9 Katharine Ashton Snuthev's Life of Wesley - - 21 Southey 1 . fe ^ ( . orr - espondence 21 Burton's Historv of Scotland - 6 Laneton I ^"'(""^versaMon. 1 I 15 Selert CmTi'S'i'ndcnce- n Stephen's Ecclesiastical Biography 21 Sydney Sn.ith's Memoirs - - 20 Convbeare and Howson's St. Paul 6 Eastlake's H istory of Oil Painting 7 Erskine's History of India - - 7 Margaret Percival - 20 Pyvroft's English Reading - - 16 T.y|orWla T - - - - W^^-^f&Essays 2! Wheeler's Life of Herodotus - 24 ! Books of General Utility.^ Francis's Annals of Life Assurance 8 Gleig's Leipsic Campaign - - 23 Gumev's Histor.cal Sketches - 8 Hamilton's Essays from the Edin- burgh Review - - - 8 Haydon's Autobiography, by Taylor b Holland's (Lord) Wl.ig Party - 9 Jeffrey's (Lord) Contributions - 11 Medicine and Surgery. Brodie's Psychological Inquiries - 4 B ^' 8 S 1 a n n% t cm M en t t h o e f r Chi,dr;n '- I Copland's Dictionary of Mtdicme - 6 Cust's Invalid's Own Book - - 6 BHcT'^etS on Brewing- - 4 Johns and Nicholas's Calendar of How to Nnrse"sick 'ch'ldrl'n - ' ' Cabinet Gazetteer - Kemble's Anglo-Saxons ^ - 11 Lat^am'on^iTeMes of ttw BMIt - 1 1 Pereira On Food and Diet - - 1 < Macaulay 8 J}- n * f England - 14 Pereira's Materia Medica - - 17 Ho D w ! to'Nurs?s C ic t kChild ; ren- - 9 Mackintosh^ sceUane'ous Work". 1 J Reece's Medical Guide - - - 18 West on Diseases of Infancy - - 24 WB3SW : : Ktown*DoeSic Medicine - 11 Lardner'8 Cabinet Cyclopsedia - 1^ Maunder's Treasury of Knowledge 15 Historv of England - 14 M'Culloch'sGeographicalDictionary 14 Martineau's Church History- - 15 Maunder's Treasuryof History - IS Memoir of the Uuke of Wellington 23 Miscellaneous and General Carlisle's Lectures and Addresw. 23 I Sy^^Sy 1 " 7 ; J Merivale's History of Rome - - 15 " Roman Republic - - 15 Milner's Church Hist<>r> - - 6 Moore's 'Thomas M.-moirs.&C. - 17 Defence of C - - Piese' Art of Perfumery - - - 18 Piscator's Cookery of r i.h - Pocket and the Stud - Pvcroft's English Reading - - 18 Recce's Medical Guide - - - Mure's GreekJ-Herature _ - 17 Piddle'^ Sn%^c"onar'ie8 l0na [p & 19 Hassan on Adulteration of Food - Havan's Book of Dignities - - 9 Holland's Mental Pnyiology - 9 ^_Hooker'. Kew Guide. - - - 9 Rich's Comp. to T.atin Dictionary l 2 CLASSIFIED INDEX. Pages. Pages. Paces. Hewitt's Rural Life of England - 10 Laneton Parsonage - - - 20 Mann on Reproduction - - - 1 " Visitsto RemarkablePlaces 10 Jameson's Commonplace -Book - 10 Long's Inquiry concerning Religion, 13 Marcel's f Mrs.! Conversations - 11 Moseley'sEngineering&ArchiUcture 17 Jeffrey's (Lord) Contributions - 11 Last of the Old Squires - - 17 M^itiand^Church in Catacomb* - 14 Marsaret Perciva! - 20 Owen's Lectures on Comp. Anatomy 17 Our Coal Fields and our Coal Pits 23 Macaulay's Crit. and Hist. Essays 14 Speeches - - - 14 Maitineau's Christian Life - - 15 " Church History - 15 Pereira on Polari-ed Light - - 17 Peschel's Elements of Physics - IB Mackintosh'sMiscellaneous Works 14 Milner's Church of Christ - - 16 Phillips's Fo--il of Cornwall, Ac. 18 Memoirs of a Mailre d'Armes - 23 Maitland's Chnrchin the Catacombs 14 Martineau's Miscellanies - - 16 Pascal's Works, by Pearce - - 17 Montgomery Original Hymns - 16 Moore On the Use of the Body - 16 " Soul and Body - 16 " 's Man and his Motives - 16 " Mineralogy - - - 18 Guide to Geology - - 18 Portlock's G eology of Londonderry 18 Powell's Unity of Worlds - - 18 Printing: Its Origin, &c. - - 23 PycroiVs English Reading - - 18 Rich's Comp. to Latin Dictionary IS Riddle's Latin Dictionaries - lb A 19 Rowton's Debater - -13 Morn.onism - ... 23 Neale's Closing Scene - - - 17 " Resting Places of the Just 17 " Riches that Bring no Sorrow 17 " Risen from the Ranks - 17 Smee's Electro-Metallurgy - - 20 Steam Engine (The) - - - 1 Tate On S'trength of Materials - 21 Wilson's Electric Telegraph - - 23 Seaward 's Narrative of his Shipwreck 19 Sir Roger de Corerley ... 20 Smith's (Rev. Sydney) Works - 2O Southey's Common -place Book* - 21 Newman's J. H.) Discourses - 7 Ranke's ferdinand & Maximilian 23 Readings for Lent - - iO ConBrmation - - 20 Rural Sports. Baker's Rifle and Hound in Ce!on 3 Berkeley's Reminiscences - - 1 " The Doctor Ac. - - 21 Souvestre's Attic Philosopher - 23 Robins against the Roman Church, 19 Robinson's Lexicon to the Greek Elaine's Dictionart of Sports - 1 Cecil's Stable Practice - - - 6 " Confessions of a Working Man 23 Testament 19 " Records of the Chase - - 6 Spencer's Psychology - 21 Saints our Example - - - 19 " Stud Farm - 6 Stephen's Essays - - - - 21 Self Denial - - - - 19 The Cricket Field - - - - 6 Slew's Training System - - 21 Sermon in the Mount - - 19 Davys Piscatorial Colloquies- - 7 Tagmrt on Locke's Writings - - 21 Thomson's Laws of Thought - 22 Townsend's State Trials - - 22 Willich's Popular Tables - - 24 Yonge'8 English-Greek Lexicon - 24 " Latin Gradus - - 24 Sinclair's Journey of Life - - 20 Smith's (Sydney) Moral Philosophy 20 " (G.) Sacred Annals - - 20 Southey's Life of Wesley - - 21 Stephen's Ecclesiastical Biography 21 Tavler's 'J J.) Discourses - - 21 Ephemera On Angling - - - 7 " Book of the Salmon - 7 Hawker's Young Sportsman - - 8 The Hun ting- Field - - - 8 Idle's Hints on Shooting - - 10 Pocket and Ihe Slud - - - 8 Zumpfs Latin Grammar - - 21 Tavlor's Lovola - - - 21 '" Wesley - - - 21 Practical Horsemanship - 8 Richardson's Horsemanship - - 18 Natural History in general. Theologia Germanica - - - 5 Thomson on the Atonement - - 22 St John's Sporting Rambles - 19 Stable Talk and Tible Talk - - 8 Callow's Popular Conchology - 6 Ephemera and Young On the Salmon 7 Thumb Bible (The) - - 22 Turner's Sacred History - - - 22 Stonehenge On the Grevhound 21 The Stud, for Practical'Purposes - b Kemp's Natural Hist, of Creation 23 Kirbyand Spence's Entomology - 11 Lee's Elements of Natural History 11 Mann on Repi eduction - - 14 Wheeler's Popular ilfbl* Harmony 21 Poetry and the Drama. Veterinary Medicine, &c. Cecil's Stable Practice - 6 " Stud Farm - - - 6 Maunder's Natural History - - 15 Turton's Shells oftheBritishlslands 23 Waterton'sEssaysonNaturalHisU 22 Arnold's Poems - - - - 3 Aikin's(Dr.l British Poc-ts - - 3 Baillie's (Joanna) Poetical Works 3 Hunting Field (The) - - - 8 Miles's Horse-Shoeing - - - 16 Pocket and the Stud - 8 Yonatt's The Dog ... 21 Bode's Ballads from Herodotus - 1 Practical Horsemanship - - 8 " The Horse - - - 21 Calvert's Wife's Manual - - 5 Richar,, son's Horsemanship - 18 Flowers and their Kindred Thoughts 1 1 Stable Talk and Table Talk - 8 1-Volume Encyclopedias Goldsmith's Poems, illustrated - 8 Kippis's Hymns - - - - H Stud (The) - - - - 8 Youatt's The Dog - 24 and Dictionaries. L. E. L.'s Poetical Wor' * - - 13 Linwood's Antholcgia < xonie asis - 13 " The Horse - - - 21 Arrowsmith's Geogr. Diet, of Bible 3 Blame's Rural Sports - - - 1 Voyages and Travels. Brande's Science, Literature, A Art 1 Copland's Dictionary of Medicine - 6 Cresy's Civil Engineering - 6 Montgomery's Poetical Works - 16 Allen's Dead Sea - - - - 3 Baines's Vaudois of Piedmont - 23 Baker's Wanderines in Ceylon - 3 Gwilt's Architecture - - - 8 Moore's Poetical Works - - 16 Barrow's Continental Tour - - 23 Johnston's Geographical Dictionary 11 " LallaRookh - - - 16 Burton 's Medina and Mecca - - 5 London's Agriculture - 13 " Irish Melodies - 16 Carlisle's Turkey and Greece - 5 " Rural Architecture - 13 " Songs and Ballads - - 16 De Custine's Russia - - 23 " Gardening - - - 13 " Plants 5 - - 13 Shakspeare, by Bowdter - - 20 <> Sentiments A Similes 10 Duberly's Journal of the War - 7 Eothen 23 " Trees and Shinbs - - 13 M'CulIoch'sGeographicalDict,onary 11 Southey's Poetical Works - - 21 " British Poets - 21 Ferguson's Swiss Travels - - 23 r orester's Rambles in Norway - 23 " DictionaryofCommerce 14 Thomson's Seasons, illustrated - 21 Gironiere's Philippines - - - 23 Murray's Encyclo. of Geography - 17 Sharp's British Gazetteer - - 19 Ure's Dictionary of Arts, Ac. - - 22 Political Economy and Gregorovius's Corsica - - - 23 Hill's Trarrls in Siberia - - 9 Hope's Brillany and Ihe Bible - 23 1 Webster's Domestic Economy - 22 Statistics. " Chase in Brittany - - 23 Caird's Letters on Agriculture - 5 Hewitt's Art-Student in Munich - 9 Religions & Moral Works. Census of i>-ol - - - - 6 " (W.) Victoria - - - 10 1 Hue's Chinese Empire - - - 10 Amy Herbert - - 20 Arrowsmith's Getter. Diet, of Bible 3 Greg's^Es'says on Political and H uc and Gabet's Tartary A Thibet 23 Hughes 's Australian Colonies - 23 Bloomfield's Greek Testament - 1 Laird's Notes'of a Traveller - 11 A 23 Humboldt's Aspects ofNatur* - 10 " Annotations on do. - 1 Bode's Bampton Lectures - 1 Calrert's Wife's Manual - - 5 M'Culloch'sGeog. Statist. Ac. Diet. 11 " Dictionary of Commerce 14 tt London - - - 23 Ke^nTrd's Eastern Tour '- - 11 Jerrmann's St. Petersburg - - 23 CleveHall ----- 20 Conjbeare's Essays - 6 Conybeare and Howson's St. Paul 6 Dale's Domestic Liturgy - 7 Defence of Eclipse of Faith - - 7 Despres On the Apocalypse - 7 StatisticsofGl. Britain 11 Marcel's Political Economy - Rickards On Population & Capital Tegoborski's Russian Statistics - Willich's Popular Tables - Laing's Norway - - - - 23 " Notes of a Traveller 11 & 23 Marryafs California - - - 15 Mason's Zulus of Natal - - i3 Mavne's Ar.tic Discoveries - - 23 Miles's Rambles in Iceland - - 23 Earl's P Daughter (The) - - 20 Eclipse of Faith - 7 Englishman's Greek Concordance 7 The Sciences in Genera and Mathematics. Pfeiffer's Voyase round the World 23 " Second ditto - 18 Richardson's Arctic Boat Voyage 18 Englishman'sHeb.AChald.Concord. 7 Arago's Meteorological Essays Seaward's Narrative - - - 19 Experience of Life (The) - 20 Gertrude ----- 20 F Popular Astronomy - St. John's (H.) Indian Archipelago 19 Harrison's Light of the Forge - 6 Brande's Dictionary of ScFence, Ac. Sutherland's Arctic VovUe 8 - - 21 Hook's Lectureson Passion Week 9 " Lectures on OrsanicChemUtry Weld's United States and Canada- 21 Home's Introduction to Scripture* 9 " Abridgment of ditto - 9 Cresy's Civil Engineering - - DelaBeche'sGeologvofCornwall.Ac. Werne's African Wanrierings - 23 Wheeler's Travels of Herodotus - 21 " Communicant's Companiom 9 Jameson's Sacred Legends - - 10 " Geological Observer - De la Rive's Electricity Young 's Christ of History - - 21 Monastic Legends- - 10 " Legends <.f the Madonna 10 Faraday's Non Metallic Elements Herschel's Outlines of Astronomy Works of Fiction. " Sisters of Charity - 10 Holland's Mental Physiology - Arnold's OakfieM - 3 Jeremy Taylor's Works - - - 11 Kaltsch's Commentary on Exodus - 1 1 Humboldt's Aspects of Nature - 10 Lady Willonghbv's Diary - - 24 Katharine Ashton - - - 20 Hunt On Light 08 - - - 10 Sir^Roger de Covertey"*- '- - 20 Kippis's Hymns - - - - 11 Konig's Pictorial Life of Luther - 9 Kemp's Phasis of Matter - - 11 Southey's The Doctor Ac. - - 21 Lardner's Cabinet Cyclcpa-dia - 12 } Trollope's Warden - - - Z2 ALPHABETICAL CATALOGUE Of NEW WOBKS AMI NEW EDITIONS PUBLISHED BY Messrs. LONGMAN, BROWN, GREEN, and LONGMANS, PATERNOSTER ROW, LONDON. Modem Cookery, for Private Families, reduced to a System of Easy Practice in a Series of carefully-tested Eeceipts, in which the Principles of Baron Liebig and other emi- nent Writers have been as much as possible applied and explained. By ELIZA ACTON. "Newly revised and much enlarged Edition ; with 8 Plates, comprising 27 Figures, and 150 Woodcuts. Fcp. 8vo. price 7s. 6d. 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LANMAN's ADVENTURES in the WILDS of NORTH AMERICA 2/6 29. RUSSIA. By the MARQUIS DE CUST1NE 3/6 30. SELECTIONS from the Rev. SYDNEY SMITH'S WRITINGS, Vol. 1 2/6 j BODENSTEDT and WAGNER'S SCHAMYL; and) 81> \ M'CULLOCH'S RUSSIA and TURKEY J 2/6 32. LAING'S NOTES of a TRAVELLER, First Series 2/6 33. DURRIEU'S MOROCCO; and an ESSAY on MORMONISM 2/6 34. RAMBLES in ICELAND, by PLINY MILES 2/6 35. SELECTIONS from the Rev. SYDNEY SMITH'S WRITINGS, Vol. II 2/6 JHAYWARD's ESSAYS on CHESTERFIELD and SELWYN ; and) 2/6 ' | MISS MAYNE'S ARCTIC VOYAGES and DISCOVERIES J " 37. CORNWALL: its MINES, MINERS, and SCENERY 2/6 38. DE FOE and CHURCHILL. By JOHN FORSTER, Esq 2/6 39. GREGOROVIUS'S CORSICA, translated by RUSSELL MARTINEAU, M.A... 3/6 f FRANCIS ARAGO'S AUTOBIOGRAPHY, translated by the Rev. B. POWELL) ., \STARK'S PRINTING: Its ANTECEDENTS, ORIGIN, and RESULTS J ' 41. MASON'S LIFE with the ZULUS of NATAL, SOUTH AFRICA 2/6 42. 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