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(such as the progression of the Lunar apogee) within Newton's system but not obeying its laws. Or he must be content to abide the trial of the other test and examine that series of confirmations (the agreement of the computed and observed phenomena) which have just before been spoken of. This second test is, indeed, to him who makes it, most formidable: for, the computations of the quantities and laws of the phenomena are conducted (as most of them must necessarily be) by the most refined and intricate processes of calculation.

There is one method, indeed, of eluding these difficulties. Newton's Theory may be brought to a test by mean of the Lunar Tables, which are partly constructed by it, and which are intended to serve for many years to come. If from these Tables we take the Moon's place, it is rarely found to differ from the observed place by more than fifteen seconds. Before Newton's discoveries, the error between the computed and observed places amounted to six minutes: although the coefficients and arguments of several of the principal equations had (from observation alone) been determined by the antient Astronomers and by Kepler and Tycho Brahé. The modern Tables contain many more equations than the antient, and, for that reason, are better. These equations have been deduced, (the forms of their arguments, at least), from the Theory of Gravity: and there will necessarily arise a very strong presumption for its truth, if the Tables so constructed, give year after year, and many years after their construction, the Moon's place to that exactness which has just been specified. This is a kind of test which may be resorted to by a person although he be not deeply versed in mathematical science : and, which, besides, may easily be resorted to by comparing Burg's Lunar Tables, and the Greenwich Observations: the former computed previously to the latter, and the latter not so made as purposely to uphold Newton's system.

The system of Newton is established on the Theory of Gravity; on its principle and law. The parts composing the system are phenomena of the same class, like effects, or results produced, on mechanical principles, from the same cause; the cause being no

occult quality but being always similarly expounded by some line or space (as was shewn in pages xxxv, xxxvi.) capable either of being algebraically expressed or arithmetically valued. The mode by which gravity causes its effects (such spaces as we have just spoken of) is beside the scope of the Physical Astronomer.

It is nevertheless a circumstance extremely curious that effects, such as are those of gravity, should be produced; that apparently so small a body as Mars, for instance, should be able sometimes to impede, and at other times to expedite the Earth in its course. The more we reflect on this matter the more mysterious it appears. It is truly wonderful that planetary influence should exist, and that the ingenuity of man should have detected it. Astronomy reveals things scarcely inferior, in interest, to the mysteries of Astrology. It does not indeed pretend to shew that the planets act on the fortunes of men, but it explains after what manner and according to what laws they act on each other.

The Author returns his thanks to the Syndics of the University Press for the liberal assistance afforded him in printing the present Volume.


P. x. 1. 24. for distance' read 'space'

P. xi. 1. 15. after results' dele comma; and line 26, after 'Astronomy' instead

of, put;

P. xiv. 1. 4. instead of a comma after processes' place it after 'being'

P. xxiv. 1. 3. from bottom, after 'of' place a comma.






Accelerating and Centripetal Forces; their Definitions: Differential Equations of Motion caused by their Action. Transformation of those Equations into others more convenient for Astronomical purposes. Three Equations necessary for determining the Length of the Radius Vector, the Latitude and Longitude of the Body.

Ir a body be supposed to be projected from the point A in the direction AC, or if it be merely supposed moving along the line AC with a certain velocity, then, according to the first law of motion, the body, if not compelled to change its state by any impressed force, will continue to move uniformly in the same direction AC.

If the body do not continue to move uniformly, or if, during any interval of time, its velocity suffer either increment or decrement, then such change in the uniform motion, or such increment or decrement of velocity, is said to originate from an accelerating or retarding force. Again, if the body do not continue to move in the same direction, or, if it be deflected or caused to deviate from the line LMC, then such deflection is said to arise from some force, which is variously denominated: it may be centripetal, or


repulsive, or disturbing. A force, in fact, is denominated according to the circumstances under which it acts.

For instance, if a body moving in the direction LC, be solicited besides by a force ƒ according to that same direction, then,

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such force ƒ produces no deflection from the line LC, but solely an acceleration of motion, and accordingly it is called an Accelerating force; and, if we consider the point C to be a centre towards which the body tends, it may also be called a Centripetal force.

If, however, the body moving in the direction of LC, be solicited at L by a force, acting from L towards T, then such force produces at once both deflection and acceleration. As a centripetal force, it solicits the body towards T as a centre, and deflects it from its right-lined course LC. As an accelerating force, it produces an acceleration of motion in the direction of LC, not proportional to its whole quantity, but to that part of it LM

which is expounded by

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or cos. TLC.

For, if we draw

TM perpendicular to LC, and consider LT to represent the entire force, LM alone produces acceleration in the direction LC; since TM has no tendency to move the body either from L towards M, or from L towards a similar point in the opposite direction.

The measure of an accelerating force is the increment of velocity generated by it during a given time. If the time be increased, the increment will be increased, and in the same proportion. Hence, if ƒ represent the accelerating force generating the increment of the velocity, or, more properly, the differential dv of the velocity, and dt be the corresponding differential of the time, we have, in symbols,

dv = f.dt.

If represent the velocity in the direction LC, and ƒ be the corresponding force tending from L to C, and, if V and F be the velocity and force in the direction LT, then we have these two equations,

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(since, by the action of the accelerating forces by which dv, d V are generated, p, q, are diminished). Hence, instead of the two former equations, we may use these two:

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If the body should be moving in the direction LC, and be solicited solely by a centripetal force (F) tending towards T, then

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