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Given. Sought.



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All Any an

From one of the angles not sought, as the

gle. ACB, let fall the perpendicular CD on sides. A. the opposite side AB ; let AC be the AB,

Igreater of the two sides AC and CB ; and BC & if the perpendicular fall within the trianAC.

gle, AB is the sum, if without, the differ. ence, of the segments AD and DB; in either case, find an arch, whose tangent

is a fourth proportional to T

and T.


; AB be the sum of AD and DB, the arch so found is half their difference, if AB be their difference, that arch is half their sum (28 Sph. Tr. and 16. 6 Eu.); in either case, the sum of that arch and the half of AB, is equal to the greater of the two AD and DB, and their difference, to the less (7 Pl. Tr.); and AD and DB being found; in the triangle ADC, right angled at D, AD and AC being given, the angle A may be found by Prob. 1. solutions Sph. Tr.

Otherwise. The rectangle under the sines of AB and AC : the rectangle under the sines of the arches AB+BC+AC—AB and


AB+BC+AC-AC : : the square of


radias : the square of the sine of half the angle A (30 Sph. Tr. and 20.6 Eu).

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All) Any On one of the sides not sought, as AB, the an-side. let fall a perdendicular CD from the opgles. AC. posite angle ; let ABC be the greater of

the two angles A and ABC; the perpenABC

idicular falls either within or without the and

triangle, according as the angles A ad ACB.

ABC are of the same or different affections (20 Sph. Tr.); in the former case, the angle ACB is the sum, in the latter, the difference, of the vertical angles ACD and BCD ; in the former case, find an angle, whose tangent is a fourth propor

ACB ABCA tional to Cot.


and ABC-A T.

in the latter, an angle, whose cotangent is a fourth proportional

to T.

and T. ABC-A

; the angle so found is, in the former case, half the difference, and in latter, half the sum of ACD and BCD (29 Sph Tr, and 16. 6 Eu.); in either case, the sum of the angle so found and the half of ACB, is equal to the greater of the two ACD and BCD, and their difference to the less (7 Pl. Tr.); and ACD and BCD being found ; in the triangle ADC, right angled at D, the angles A and ACD being given, AC may be found by probá 1 solutions Sph. Tr.




Case. Part.

Given. Sought.


6. All Any

Otherwise. the an- side.

Let DEF, see fig to prop. 13 Sph. gles. AC. CAB,

Tr., be the supplemental triangle to the ABC

triangle ABC ; the arch DE is the com

plement of the angle ACB, EF of the land АСВ.

angle BAC, and DF of the angle ABC, to semicircles ; the sides of the triangle DEF are therefore given ; from which, by 'ease 5, find the angle DEF which is opposite the sought side AC; which side may of course be found, being the complement of the measure of the angle

DEF to a semicircle (13 Sph. Tr). In the preceding solutions of the several cases of oblique angled spherical triangles, the rules are given for ascertaining the affections of the arches or angles sought, and removing ambiguities, where it could be conveniently done. For farther remarks on this subject, and particularly on the first solutious of the fifth and sixth cases, deduced from prop. 28 and 29 Sph, Tr. see note on Problem 2 Spherical Trigonometry.




As far as it relates to Astronomy, according to the Newtonian System.

Philosophy, which signifies a knowledge of things, is a word of Greek origin, and in that language means, a love of know. ledge. It is divided into Moral and Natural. Moral Philosophy, which is also called Etticks, and by some Metaphysicks, treats of the duties and conduct of man, considered as a rational being. Natural Philosophy, called also Physicks, treats of the properties of natural things, the causes of the different phenomena or ap? pearances, and the laws, by which the various operations, which we observe in natural things, are regulated; and of such natural laws, as may be applied to various useful purposes.

The assemblage of natural bodies or things, is called the Universe.

Though it is by no means the intention of this little tract to enter into the business of Natural Philosophy, farther than may be necessary to explain the motions of the heavenly bodies, and the laws by which these motions are regulated, deduced from the laws of motion ; yet it seems not unimportant, previously to mention some of the principal axioms of philosophy, which have been deduced from common and constant experience ; so evident, and so generally known, that a recital of a few of them will be sufficient.

1. Nothing has no property. Hence,

2. No substance or being can be produced from nothing by any created being

3. Matter cannot naturally be annihilated, or reduced to nothing ; and though things may appear to be utterly destroyed, as, for instance, by the action of fire, by evaporation, &c., yet in such cases the substances are not annihilated, but they are only dispersed, or divided into particles, so minute as to elude our senses.

4. Every effect has, or is produced by, a cause, and is proportionate to it.

which are Rule 1.

The rules of reasoning in Philosophy, which have been formed after m ture deliberation, are as follow:

That more causes of natural things ought not to be admitted, thun are both true, and sufficient to explain their appearances.

Philosophers say, Nature does nothing in vain , and that is done in vain by more causes, which can be done by fewer.For nature is simple, and abounds not in superfluous causes of thinre.

Rule 2. Therefore of natural effects of the same kind, the same causes are, as far as possible, to be assigneil.

As of respiration in a man, and in a beast ; of the descent of stones in Europe and in America ; of the light of a culinary fire and of the sun ; of the reflection of light in the earth and in the planets.

Rule 3. The qualities of bodies which can neither be increased or diminished, and which are found in all bodies on which we can make experiments, are to be reputed qualities of all bodies whatever.

Such as the extension, hardiness, impenetrability, mobility and vis inertiæ of matter. And if it appear from experiments and astronomical observations, that all bodies about the earth gravitate towards the earth, and that, in proportion to the quantity of matter in each ; that the moon, according to its quantity of matter, gravitates towards the earth, and our sea towards the moon; and all the planets and comets towards each other and the sun; we must by this rule affirm, that all bodies whatever gravitate towards each other. Indeed the argument from the appearances, for the universal gravitation of bodies, is stronger than for their impenetrability, of which we can have no experiment or observation in the celestial bodies.

Rule 4. In experimental philosophy, we should consider propositions collected by general induction from phenomena, as accurately or very nearly true, notwithstanding any contrary hypotheses which may be imagined, till other phenomena occur, by which they may be made more accurate, or liable to exceptions.

This rule should be followed, that the argument of induction may not be evaded by bypotheses.

These rules are evidently formed, in order that in our enquiries about the nature of bodies, we may be rather directed by experiment, than by hypotheses not founded on experiment, as appears to have been often done, to the evident danger of being led into errors ; and as the object of research in these elements, is the system of the world, and to investigate the causes, from

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