Nomenclature, 22.

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Notation, ib.

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-

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, 27. - Problems of maxima and minima,
ib.. Example, ib. Quadrature of curves, 28. Example:
Parabola, ib. Rectification of curves, ib.
cular arcs, ib. Measurement of solids, 29.
sphere, and cylinder, ib. Finding radius of curvature, ib.
Example: Parabola, 30. Addition of constant quantity in

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integration, ib.

Newton, ib.

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Example: Cir-

Example: Cone,

Method of investigation used by Sir Isaac

Subjects of the Three Books, 31.

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(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, 37.
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.

rin's, ib. - J. Bernouilli's, ib.

Prop. VI. B. I., Principia, 47. Keill's imperfect acquaintance

with this subject, ib. — Herrman's mistake, 48. Formulæ

exemplified in the case of the parabola, 49. Ellipse and

hyperbola, ib. - Centrifugal forces. - Formulæ 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.

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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-

59.

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-

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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 times 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,
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, 61. 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 observations 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, 67.- Comets of 1680, 1665, 1682,

1683, ib. General remarks on the importance of these sections,

68.

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iii. Motion (1) in given conic sections, (2) in straight lines,

ascending or descending.

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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.

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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. —

iv.

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.

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(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.

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-

-

-

-

-

- (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,

89. 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. Misstate-

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, 102.-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

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-

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, 107. 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

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to the earth and moon, 108.

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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, 116. 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

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-

--

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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.

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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.

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Attraction under two heads, i. that of spherical; ii. that of non-

spherical bodies.

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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, 139.- Five general theo-

rems, ib.
Corollary comparing corpuscular attraction with cen-

tripetal forces, 140. Peculiarity of the actual law of gravi-

tation, ib.-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,

147.- Attraction according to any power of the distance in any

symmetrical solids, ib. General solution, 148. Laplace's

formula for attraction, 149.

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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

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