« ΠροηγούμενηΣυνέχεια »
T0 0 UR READERS,
WE have both pleasure and satisfaction in looking back upon our labours during the past year for the diffusion of real
J, oxidoN, MAncil 26, 1853.
LESSONS IN ALGEBRA.
I. Section I., Introduction: Illustrations, Definitions,
Axioms ...................................... 354
LESSONS IN BOTANY.
XV. Classes: 18. Polydelphia, Dodecandria, Icosandria,
23. Polygamia, Monaecia, Diacia “.............. 41
W., WI. Outline Drawing from Simple Forms; Principle
Suffixes ....... ........................... 85
Conversations on English Grammar, No. II. 116
XXV. to XXVII. The Greek Element: Greek Stems,
XXVIII. Greek stems continued, &c. ................ 170
stances . . . . . . . . --------------------- ----- o
ditionals; Idioms.......................... 4
XXX. XXXI. Frequent Idioms; Dimension, Weight, 64, 78
the Subjunctive.................... ........ 98
XXXIX., XL. Idioms relating to Verbs............. .201,218
Present Participle, the Verbal Adjective.... 236
XIII. Bode's Law, Titius Law, Telescopic Planets, the
T------------- - -o-
Suppose that you were elevated in the heavens, or in the vast space in which roll all the stars, to a point millions of miles above the sun; and that you were furnished with a telescopic eye so powerful, that from that point you could observe the magnitudes, motions, and distances of all the bodies in the Solar System; that is, the bodies or planets which revolve round the sun in consequence of the laws of attraction and tangential impulse; you would perceive among them a highly-favoured planet called the Earth, accompanied by a satellite (an attendant) in its course, called the Moon. This Earth and her satellite, like all the other planets and their satellites which you would behold in this bird's-eye view, receive both their light and their heat from the sun; and the influences of these imponderable bodies are distributed to all the planets in the same ratio as the power of attraction which keeps them revolving in their orbits (tracks or paths); that is, in the inverse ratio of the squares of their distances; or, to express it more clearly, the power of the attraction, the light and the heat of the sun on one planet, is to that on another planet, as the square of the distance of the latter, is to the square of the distance of the former. In your elevated position, you would next perceive that the planets in their various revolutions, would at sometimes be nearer to the sun than at other times; and that if the orbit of each were traced by a white line in space, it would appear to your eye, if rightly placed, to have the form of an oval nearly, being in fact, what is called in the language of mathematics, an ellipse. In order that you may understand the nature of this curve, we shall explain it by means of a diagram.
ig. 1. Thus, in fig. 1, if you fix two pins on a board,
Fig. - . M at the points F and F', and fasten a string 2. FM F, of any convenient length, but greater /27 - than the distance between the two points, by A or b its extremities, at these points; and if you FTO r take a crayon or chalk pencil, and press it on the string horizontally at M, so as to keep it d always tense (i.e. stretched), and parallel to
the board, moving the pencil round and round at the same time, from one side to the other; you will describe the curve A c B p, which is called the ellipse. It is evident that the limits of the form of this curve are the circle and the straight line. If the two points F and F" are brought close together, the curve will be a circle; if they be separated as much as the string will allow, the curve will become a straight line. The two points F and F" are called the foci (the plural of focus) of the curve; the straight line A B drawn through them, and terminated both ways by the curve, is called the major aris; and the straight line cD drawn at right angles to this axis from its middle point o, and terminated both ways by the curve, is called its minor aris. If a straight line be drawn from F" to c, it will be equal to the straight line A or or half the major axis. The point o is called the centre of the ellipse, and the ratio of F o to Ao, that is, of the distance between the centre and the focus to half the major axis, is called the eccentricity of the ellipse. The distance from the focus f to any point M in the curve is called the radius vector of the ellipse; it is least at A, and greatest at B. With these explanations, while you are supposed to be looking at the orbit of a planet from your elevated position in space, you will now be able to comprehend the fundamental principles of Astronomy, viz. Kepler's Laws. The eminent German astronomer just mentioned, who flourished at the close of the 16th century and the beginning of the 17th, discovered, by laborious observations and calculations, the following remarkable laws, which were afterwards mathematically demonstrated by Sir Isaac Newton:–1. That the planets all revolve in elliptic orbits, situated in planes passing through the centre of the sun; the sun itself being placed in one of the foci of the ellipse. 2. That the radius vector or straight line drawn from the centre of
the sun to the centre of the planet passes over equal areas in equal times in every part of the orbit; that is, whether the planet be in its aphelion, or farthest from the sun, in its perihelion, or nearest to the sun, or at its mean distance from the sun. And 3. That the squares of the periodic times of the planets, that is, of the times of a complete revolution in their orbits, are proportional to the cubes of their mean distances from the sun; in other words, that the square of the periodic time of one planet is to the square of the periodic time of another planet, as the cube of the mean distance of the former from the sun is to the cube of the mean distance of the latter from it. Into the full explanation of these laws we cannot enter until we treat of astronomy; in the meantime it is necessary to give some explanation of the law which we have marked first, though it is generally accounted the second, in order to clear up some points connected with phenomena relating to the earth, and the circles drawn on the globe, which is the only true representation of the earth's surface. Supposing then the ellipse in fig. 1 to represent the earth's annual orbit round the sun, and the focus F the place of the sun's centre ; then the point A will represent the . of the earth's centre at mid-winter, when it is nearest the sunloor in its perihelion; B will represent its position at mid-summer when it is farthest from the sun, or in its aphelion; c will represent its position at the spring or vernal equinox, when it is at its mean distance from the sun; and d its position at the harvest or autumnal equinox, when it is also at its mean distance from the sun. We think we hear some of our readers exclaiming, notwithstanding the elevated position in which we have supposed them to be placed, What! will you tellus that the sun is the cause of light and heat on the earth's surface, and yet you assert that the earth is nearer to the sun in winter than in summer ? How can this be : Paradoxical as this may seem, it is nevertheless true; and the reason we shall now give. As you are supposed to be looking from a great distance, and to be able to discern all the motions of the planets, if you look narrowly at the earth, you will perceive that besides its orbitual motion round the sun, it has a revolving or journal motion on its own axis. By aris here is meant that imaginary straight line passing through the globe of the earth, on which its rotation is supposed to take place, and which is aptly represented in artificial globes by the strong wire passing from one side to the other, at the points called the poles (that is, pivots), which are the extremities of the axis. This motion may be likened to the spinning of a top, a motion which continues while the top is driven forward in any direction from one place to another. In fact, the analogy would be so far complete, independently of the causes of the motion, if the top, while it is spinning or revolving as it were on its own axis, were made to run regularly round in an oval ring on the ground, under the lash of the whip. Thus, the earth has two motions; one on its own axis, performed once every twenty-four hours; and one in its orbit, performed once every 365 days 6 hours; we have stated these periods in round numbers, in order that they may be easily remembered; but the exact period of the earth's daily revolution on its axis, is 23 hours, 56 minutes, 4 seconds, and 9 hundredth parts of a second ; and the exact period of the earth's annual revolution in its orbit is 365 days, 5 hours, 48 minutes, 49 seconds. The analogy of the motions of the top, however, to the motions of the earth, as thus described, is incomplete in respect of the position of their axes. The axis of the spinning top is in general upright or perpendicular to the ground, which may be called the plane of its orbit, that is, of the oval ring in which it is supposed to move; but the axis of the earth in its daily motion is not perpendicular to the plane of its orbit, or the ellipse in which its annual motion is performed. In speaking of the plane of the earth's orbit our analogy fails, for there is nothing to represent the ground on which the motion of the
spinning top takes place. The mere attraction of the sun, coupled with the effect of an original impulse in the direction of a taugent to its orbit, is sufficient to preserve the earth in its orbitual motion in empty space. Hence, the sublimity and truth of the ancient passage in the book of Job : “He stretcheth out the north over the empty place, and hangeth the earth upon nothing.” (Job xxvi. 7.) This passage is singularly true in regard to the first sentence as well as to the second. For the axis of the earth is inclined to the plane of its orbit, at an angle of 66 degrees, 32 minutes; that is, rather more than two-thirds of a right angle, so that literally and truly “the north is stretched over the empty place,” and not over the body of the earth itself, in either of its motions, whether axial or orbitual. This inclination is preserved during the whole of its motion in its orbit, and is the cause of the variation of the seasons; the preservation of the inclination of this axis has been not unaptly called the parallelism of the earth's axis. Before explaining the effect of this parallelism and inclination of the earth's axis in producing the seasons, it will be proper to explain what is meant by tangential impulse. In fig. 2, let A con represent the orbit of the earth, which is nearly cir- ri cular; let D represent the place of the sun, and g-2. A the place of the earth at the moment when it began its revolution in its orbit. At this moment the force of the sun's attraction would be. gin to act on the earth in the direction a D, and had this alone been allowed to operate, would
Co F B A have drawnitrapidly towards the sun in a straight
line, until it had come finally in contact with the
sun itself; but at the same moment, an original impulse was, or is supposed to have been given to the earth in the direction a E, which is that of a tangent, or straight line touching the circle at the point A: so that the carth, which under the action of the former force would in a certain time have been found at some point in Ap, and under that of the latter force would, in the same time, have been found at the point y in A E, would, by the combined action of both forces, be found near the point c in the curvilinear orbit A ch. This original impulse, the effect of which remains to this day unaltered by the action of attraction, (seeing it has met with no resistance in empty space, and has been so balanced against the force of attraction as to retain the earth in its orbit,) is called the tangential impulse or force, which was imparted to it when it began its orbitual revolution. Young, in his “ Night Thoughts,” alluding to this tenet of the Newtonian philosophy, asks— “Who rounded in his palm those spacious orbs • Who bowl'd them flaming through the dark profound 2" Night IX. Let us now consider the effect of the inclination of the earth's axis to the plane of its orbit. In fig. 1, we have supposed the sun to be at the focus F, while the earth is at the point A in midwinter. Now, at this point, you would see from your supposed elevated position, that the northern half of the earth's axis is inclined to the major axis A B at an angle of 113 degrees 28 minutes, the supplement of its angle of inclination to the plane of the orbit; so that the North Pole, with the space on the earth's surface around it to a considerable extent, is prevented from receiving the rays of the sun, and consequently the heat of those rays; while the South Pole, with the space, around it to the same extent, is made to receive these rays and to enjoy their heat. Hence, while it is winter in the northern or arctic regions of the earth, it is summer in the southern or antarctic regions. While the earth is still in this position, the rays of the sun fall more obliquely upon the illuminated portions of the northern hemisphere, than they do upon the southern hemisphere, and thus have less power to produce heat then if they fell perpendicularly ; just as a person sitting at the side of a fire-place with a good fire in it, feels less heat than a person who sits exactly in the front of it. On the other hand, if you consider the earth from your elevated position, when it is at the point * in mid-summer, the reverse of all this takes place. The northern half of the earth's axis is inclined to the major axis (or line of apsides, as it is sometimes called; that is, the line of junction of the two opposite points A and B), at an angle of 66 degrees 32 minutes, which is its angle of inclination to the plane of its orbit; so that the North Pole, with the space on the earth's surface around it, above-mentioned, is made to receive the sun's rays, and consequently their heat; while the South Pole, with the similar space “ound it, is prevented from receiving those rays and enjoying * heat. Hence, while it is summer in the northern or arctic
regions, it is winter in the southern or antarcticizegions. While the earth remains in this position, the rays of the sun fall more directly upon the northern hemisphere than they do upon the southern hemisphere, and thus have more power to produce heat than if they fell obliquely, according to the illustration given above. Now, as we in this country are inhabitants of the northern hemisphere, and of that part which is within the circle of illumination all the year round, we experience the vicissitudes of the seasons just described as belonging to it, and we are consequently colder in winter than in summer, although the earth be actually nearer the sun in winter than in summer. But we must explain more fully what we mean by the circle of illumination. It is plain that the rays of light falling from the sun upon the opaque or dark body of the earth in straight lines, can never-illuminate more than-one-half of its surface at a time; as may be seen by the very simple experiment of making the light of a candle fall upon a ball at a-distance from it. Now, as the earth revolves on its axis once every 24 hours, it is evident that the illuminated half, and consequently the circle of illumination which is the boundary of that half, is perpetually changing, so that almost all parts of the globe receive-light for several hours in succession, and that they are also enveloped in darkness for several hours in the same manner. If the axis of the earth, instead of being inclined at a certain angle to the plane of its orbit, which we shall hereafter call the Eeliptic, were at:right angles to that plane, and preserved its parallelism, them the circle of illumination would continually extend from pole to pole, and all places on the earth'ssurface would enjoy light for 12 hours in succession, and would be enveloped in darkness for exactly the same period the whole year round. On the otherhand, if the axis of the earth were coincident with the plane, and preserved its parallelism, this would happen only twice a year; and each hemisphere would at opposite periods be in total darkness for a whole day, while the variations between these extremes would be both inconvenient and injurious. In the former case the seasons would-be all the same, that is, there would be perpetual sameness of season all the year-round; in the latter case, the seasons-instead of four would be innumerable, that is, there would be perpetual change. Here, then, creative wisdom shines unexpectedly forth. The inclination of the earth's axis is such as to produce the four seasons in a remarkable manner, and to permit sufficient time for-the-earth to bring her fruits to perfeetion, as well to letherlie fallow for a period that she may renew her fruitfulness. In fig. 1, when the earth is supposed to be at the point c, she is at her mean distance from the sun at the ternal equinor, which is the first time of the year when day and night are equal, which happens on or about the 21st of March. Now, at this point the inclination of the earth's axis to the minor axis of the ellipse is a right, angle, and as the focus F", in the case of the earth, nearly coincides with the centre-o, the rays of light proceeding from the sun nearly in the straight line oc, fall upon that axis nearly perpendicularly, and illuminate the globe-from pole to pole, so that the circle of illumination passes through the poles, and the days and nights are equal all over the globe, each consisting of 12 hours, while the earth is in this position. In the opposite position at D, the earth is again at her mean distance from the sun, at the autumnal equinor, which is the second time of the year when day and night are equal, which happens on- or about the 22nd of September. At this point-the-circumstances of the globe and the circle of illumination are exactly the same as we have just described. At these four points, A, c, B, and D, in the orbit of the earth, are found the middle points of the four seasons of the year, viz., at A, Mid-winter; at c, Mid-spring; at B, Mid-summer; and at D, Mid-autumn. At the point A, or Mid-winter, which is on or about the 21st of December, we have the shortest day-in the northern hemisphere and the longest day in the southern-hemisphere; and at the point B, or Mid-summer, which is on or about the 22nd of June, we have the longest day in the northern hemisphere and the shortest in the southern hemisphere. “Thus, is primeval prophecy fulfilled: While earth continues, and the ground is tilled; Spring time shall come, when seeds put in the soil Shall yield in harvest full reward for toil; Heat follow cold and fructify the ground, Winter and summer in alternate round; And night and day in close succession rise, While each is regulated by the skies. Supreme o'er all, at first, Jehovah stood, Aud, with creative voice, pronounced it good."