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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 comland
plement of the angle ACB, EF of the ACB.
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; 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 form-ed after m ture deliberation, are as follow: Rule 1.
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 things.
Rule 2. Therefore of natural effects of the same kind, the same causes are, as far
as possible, to be assignel. 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 inay 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,
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 ascertained by its weight, especially in an exhausted receiver.
2. The quantity of motion, is a measure thereof, arising from
2 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.
3. The force of inertness, or tis inertiæ, or vis insita of matter, is the power of resisting, by which every body, as much as is in it, perseveres in its state of rest, or of uniform motion in a right line.
This force is proportional to the quantity of matter.
4. An impressed force, is an action exercised on a body, te change its state of rest, or uniform motion in a right line.
This forco consists in the action alone, nor does it remain in the body after the action. For the body perseveres in every new state by its force of inertness alone. But the impressed force is of different origins, as from a stroke, a pressure, a centripetal or centrifugal force.
5. A centripetal force, is that, by which bodies are drawn, impelled, or any how tend towards any point as a centre.
Of this kind is gravity, by which bodies tend to the centre of the earth ; magnetism, by which iron is attracted towards a magnet ; and that force, whatever it be, by which the planets are perpetually drawn from rectilineal motions, and caused to be revolved in curve lines. A stone, whirled about in a sling, endeavours to recede from the hand which turns it; and by that endeavour, distends the sling, and with so much the greater force, as it is revolved with the greater velocity and as soon as it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because it is directed towards the hand as the centre of the orbit, may be called the centripetal force. And the same thing is to be understood of all bodies revolved in any orbits. They all endeavour to recede from the centres of their orbits, and were it not for the opposition of a contrary force, by which they are retained in their orbits, and which may therefore be called centripetal, would go off in right lines with a uniform motion. A projectile, if it were not for the force of gravity, would not deviate towards the earth, but would go off in a right line, and with a uniform motion, if the resistance of the air were taken away. By its gravity it is perpetually drawn aside from its rectilineal course, and made to deviate towards the earth more or less, according to the force of its gravity, and the velocity of its motion.-. By how much the less the force of gravity is, and the greater the velocity, with which it is projected, by so much the less it will deviate from a rectilineal course, and the farther it will go. If a leaden ball, projected from the top of a mountain, by the force of gun-powder, with a given velocity, in a horizontal direction, be carried to the distance of two miles before it falls to the