Case. Part. 6. Any AC. Solutions. On one of the sides not sought, as AB,! let fall a perdendicular CD from the opposite angle; let ABC be the greater of the two angles A and ABC; the perpenIdicular falls either within or without the triangle, according as the angles A and 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 proportional to Cot. ACB T. 2 ABC+A 2 and in the latter, an angle, whose cotangent is a fourth proportional Ito T. ACB T. 2 ABC+A and T. 2 ABC-A 2 ; 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 tri-f angle ADC, right angled at D, the an-j gles A and ACD being given, AC may be found by prob. 1 solutions Sph. Tr. gles. the an-side. CAB, JABC Solutions. Otherwise. Let DEF, see fig to prop. 13 Sph. Tr., be the supplemental triangle to the triangle ABC; the arch DE is the complement of the angle ACB, EF of the 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 jopposite 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. ELEMENTS OF 439 NATURAL PHILOSOPHY, 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 knowledge. 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 appearances, 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; which are 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. The rules of reasoning in Philosophy, which have been formed after mature deliberation, are as follow: Rule 1. That more causes of natural things ought not to be admitted, than 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 things. Rule 2. Therefore of natural effects of the same kind, the same causes are, as far as possible, to be assigned. 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 what ever. 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 hypotheses. 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 whence motions so accurate and beneficial are produced; it seems proper to mentiou previously, some of the principal laws of the planetary motions, discovered by that eminent astronomer, John Kepler, from actual observations, according to the Copernican hypothesis, among which are the following: 1st. The areas, which the planets, which revolve round the sun, describe by right lines drawn to it, are proportional to the times. 2nd. The orbits, which they describe, are not circles, as was before generally supposed, but ellipses, the sun being in one of the focuses. 3rd. The cubes of their mean distances from the sun are to each other, as the squares of their periodick times. The two first laws being applicable to the moon's motion round the earth, and all three to the motion of Jupiter and Saturn's satellites round their primaries. It remained for the great Newton to deduce these and other laws of the system of the world, from the laws of motion, by mathematical reasoning. Some of his principal discoveries on this subject are delivered in the following elements, DEFINITIONS. 1. The quantity of matter, is a measure thereof, arising from its density and magnitude jointly. The air, for instance, its density being doubled, in a double space is four-fold, in a triple, six-fold. This quantity may be ascertained by its weight, especially in an exhausted receiver. 2. The quantity of motion, is a measure thereof, arising from the velocity and quantity of matter jointly. The motion of the whole, is the sum of the motions of all the parts, and therefore in a body of double the quantity of matter, with an equal velocity, is double, and with a double velocity, four-fold. And ever so small a power may be made to move ever so great a weight; namely, by making the velocity of the power compared with that of the weight such, that the product of the quantity of matter of the power multiplied by its velocity, may be greater than the product of the quantity of matter of the weight by its velocity, and so much greater as to overcome such resistance as may arise from friction, &c. |