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Again, pointing out that every section of a sphere must be a circle, and that knowing the circumference they can find the diameter-or the line which would reach from any one point to the one differing in longitude 180° from it-also the area of the section or slice of the earth which the plane of a parallel of latitude makes. · The following questions may interest a teacher who has a tolerable knowledge of the subject, and suggest others.' ' . (1) The length of a degree of longitude in our latitude is 37.76 geographical miles : compare the velocity of a point on the earth's surface here arising from the motion of rotation, with the velocity of a point on the equator.
(2) If the earth's diameter were only one half what it is, what proportion would the mass, the surface, and the different land divisions of this new globe bear to those of the present one, and what would be the size of each in square miles ?
The teacher should work this question out numerically to its final results; it only requires a knowledge of the properties of a circle and of a globe, that the circumferences of circles vary as their diameters, the areas as the squares; and that the solid contents of spheres are as the cubes of their diameters.
Archimedes* more than two thousand years ago discovered that the superficies of a sphere is equal to the convex surface of the circumscribing cylinder, or to the area of four of its great circles; and that the solidity of the sphere is to that of its circumscribing cylinder as 2 to 3. He was so pleased with this discovery, that he ordered a sphere inscribed in a cylinder to be placed on his tomb, and the numbers which express the ratio of these solids.
As a means of giving correct ideas of the apparent motions of the heavenly bodies, a celestial globe will be necessary. This, to an unpractised eye, seems a mass of confusion, but by confining the attention at first to a few particular stars, particularly those near the pole, and by degrees extending it to others, it will be found very simple.
It is essential to make them understand bow the elevation of the pole, or the apparent place of the pole-star, variesthat at the equator the poles are in the horizon, and at the poles directly over head.
* See Appendix B, Table 1.
Having elevated the pole according to the latitude, and otherwise regulated it for any particular day and hour in the year, they may conceive the equinoctial and ecliptic as the corresponding lines of the terrestrial globe swollen out to the blue vault of the sky, the teacher would point out, for instance, the constellation of the Great Bear, and how to find the pole-star from it; others, as Capella in Auriga, &c., which never get below the horizon—that the stars near the polestar appear to move in circles round it from east to westthat this is in consequence of their own motion with the surface of the globe from west to east-that the farther a star is from the pole-star, the greater the circle it describes, until you get to those which rise due east-that such a star would describe a greater circle than one rising either to the north or south of east, and that stars rising further to the south will appear to describe smaller and smaller arcs in the heavens, until you get to those which only just make their appearance on the horizon—such as a star of the first magnitude (Fomalthaut) in Piscis Australis—those further south not rising to us at all, but describing circles round the south pole, in the same way as the stars in the Great Bear and others do round the north.
Then by degrees to call the attention to others, such as a star (Vega) of the first magnitude in Lyra-Arcturus, Regulus, Antares in the Scorpion, &c., marking those in and near the ecliptic-point out also the direction of the Milky Way, and the particular stars near it on each side, east or west of it.
Then turning the globe from west to east, show the rising, &c., or particular parts of the heavens where the more remark. able stars are to be found, at hours when they may themselves observe them—where they will be at eight, nine o'clock, &c., near the horizon in the east-or that they must turn their faces to the south, the west, &c., to see them; as also their apparent distance from the pole-star; and they will have the greatest pleasure in hunting them out and watching their motions.
When a right conception of the apparent motion of a few of the more important stars is formed, that of the rest scattered among them becomes an easy matter of reasoning which is soon filled up, always bearing in mind their apparent distances from the pole-star-watching those which never set in their highest and lowest points, beginning with them in the east;
conceive how the observers must turn in order to see them in the different parts of the circle they appear to describe, until they come to the same point again.
That if they can observe any of those stars to change its position with respect to any star which they know to be fixedif they find its angular distance from a fixed point increase or decrease—that this is called a planet—that the planets move in orhits inclined to the plane of the ecliptic, but that their path is never far from that of the sun—some nearer the sun some farther from it than the earth—the difference this must cause in the quantity of heat and light falling upon themthat in one it would melt iron and lead—that they would not be known as solids, water only as an elastic vapour—while in another, perhaps, quicksilver, water, &c., would be solid substances, capable of being quarried out in blocks like Aberdeen granite-gases would become solid, &c.
Then to point out their respective distances from the suntheir periods of revolution in their orbits-their satellites, &c.the exactness with which astronomers are able to make all these calculations—changes of the moon and her different phases.
That if the plane of the orbit in which the moon moves were extended, it does not lie in the plane of the ecliptic, but inclined to it, at an angle of about 5°; that at new moon, the sun, moon, and earth are in a straight line, and that side of the moon which receives light from the sun is turned entirely from us, so that none of her reflected light can reach the earththat by her motion in her orbit she separates herself, moving to the east, about 13° daily from the sun—that a day or two after the change we see a small crescent of light, concave towards the east; this goes on increasing daily with her angular distance from the sun, until she appears in the part of the heavens directly opposite to him, when it is full moon—the whole enlightened surface of the moon being turned towards the earth. She now goes on decreasing, rising later on successive evenings, the waning side being convex towards the west.
Call attention also to the points of the horizon on which she rises—when due south-the arc described in the heavensber varying distance from particular stars—and why the difference in time between successive risings of what is called
the Harvest Moon, is less than at any other tinie of the year, That the orbit in which the earth moves is not a circle, but an oval or ellipse with the sun in one of the foci-show how an ellipse may be described that the sun is nearer the earth in winter than in summer-how the point of the horizon on which he rises varies, being farthest to the south in winter, and to the north of east in summer-how his altitude when on the meridian varies, being much greater in summer than in winter; the effect of this, so far as heat is concerned—that the length of time between sunrise and sunset varies, as you leave the equator, all the way up to the pole--the duration of twilight short at the equator, longer at other places as the latitude increases, and why? The sun not getting so bigh in the heavens in winter as in summer, the rays fall in a more slanting direction on the earth's surface, and on this account at this season, as well as from his not being so long above the horizon, less warmth is communicated to the earth than in summer. On fields with an aspect to the north, the rays will fall still more slantingly than on those turned to the south or on a horizontal plain, and in such situations less warmth will be given to the soil or to any substances upon it; hence vegetation in the spring is not so forward in a northern as in a southern aspect-the hoar frost in autumn remains up till noon, or even the whole day, in aspects turned to the north, but vanishes early in those to the south—the same of snow remaining on the north side of hills-other reasons also, such as cold winds from the north. What must be the inclination towards the north, on any given day, that the rays may fall parallel to the surface? What the inclination to the north beyond which the surface would be entirely in the shade? What the aspect to the south, that the rays of the sun may fall perpendicularly to the surface on any given day?
Light travels from the sun to the earth in 7 minutes, at the rate of 192,500 miles in a second of time.
It moves through a space equal to the circumference of the earth in th part of a second -a space which would take the swiftest bird three weeks to fly over.
Again, point out the difference between sidereal and solar time-day-year: how a solar day is not always of the same length - clocks regulated by mean solar time, &c. : how the period of time we call a year does not consist of an exact
number of days, as 365 ; and hence the difficulty in regulating the calendar. *
That the sidereal day, or the time between any meridian leaving a particular star, and coming to it again, is always the same, the star not having moved in the interval --- that this is not the case with the sun that in the interval between any two successive passages of the same meridian under him, he has moved on towards the east, and this daily motion being unequal, causes the length of a solar day to vary. A clock tells mean time, and is therefore sometimes before, and sometimes behind solar time. - That the time of the earth's making a complete revolution in its orbit is 365 days 5 hours and 48 minutes; so that if leap-year is made to occur every four years, this would be too often, and require correction.
On the subject of Eclipses. There is no phenomenon connected with the appearances and motions of the heavenly bodies which creates so much astonishment among those who have never thought on the subject, as an eclipse of the sun and moon; and that the time of their having happened, or of their happening for the future, can be so exactly computed, is a subject of no less wonder. ** It is familiar to every one, that an opaque body of sufficient size may be so placed between a luminous body and the eye of an observer, as to stop all the light proceeding from it, and in this case the luminous body becomes invisible.
* “ Hipparchus, the most celebrated astronomer of antiquity, and who lived about a century and a half before Christ, first paid great attention to the rising and setting of the stars ; he discovered that the period of 365 days 6 hours, which had been considered as the true length of the solar year, was too great by about five minutes, and observed that the four parts, into which the year is divided by the solstices and equinoxes, are by no means equal, the sun occupying 944 days in passing from the vernal equinox to the summer solstice, and only 92 from the same solstice to the autumnal equinox, and that therefore the sun remained 187 days in that part of the ecliptic which lies north of the equator, and only 178 in the other part."Encycl. Brit.
Laplace concludes that the mean heat of the earth cannot be altered by lo of Reaumen since the time of Hipparchus, inasmuch as the dimensions of the globe would be thereby changed in a small amount, its angular velocity increased or diminished, and a sensible difference be made in the length of the day-and this is found not to be the case.