44. The size of the earth.—The student is presumed to have learned, in his study of geography, that the earth is a globe about 8,000 miles in diameter and, without dwelling upon the "proofs" which are commonly given for these statements, we proceed to consider the principles upon which the measurement of the earth's size and shape are based.

In Fig.25 the circle represents a meridian section of the earth; PP' is the axis about which it rotates, and the dotted lines represent a beam of light coming from a star in the plane of the meridian, and so distant that the dotted lines are all practically parallel to each other. The several radii drawn through the points 1, 2, 3, represent the direction of the vertical at these points, and the angles which these radii produced, make with the rays of starlight are each equal to the angular distance of the star from the zenith of the place at the moment the star crosses the meridian. We have already seen, in ChapterII, how these angles may be measured, and it is apparent from the figure that the difference between any two of these angles—e.g., the angles at 1 and 2—is equal to the angle at the center, O, between the points 1 and 2. By measuring these angular distances of the star from the zenith, the astronomer finds the angles at the center of the earth between the stations 1, 2, 3, etc., at which his observations are made. If the meridian were a perfect circle the change of zenith distance of the star, as one traveled along a meridian from the equator to the pole, would be perfectly uniform—the same number of degrees for each hundred miles traveled—and observations made in many parts of the earth show that this is very nearly true, but that, on the whole, as we approach the pole it is necessary to travel a little greater distance than is required for a given change in the angle at the equator. The earth is, in fact, flattened at the poles to the amount of about 27 miles in the length of its diameter, and by this amount, as well as by smaller variations due to mountains and valleys, the shape of the earth differs from a perfect sphere. These astronomical measurements of the curvature of the earth's surface furnish by far the most satisfactory proof that it is very approximately a sphere, and furnish as its equatorial diameter 7,926 miles.

Neglecting the compression, as it is called, i.e., the 27 miles by which the equatorial diameter exceeds the polar, the size of the earth may easily be found by measuring the distance 1-2 along the surface and by combining with this the angle 1O2 obtained through measuring the meridian altitudes of any star as seen from 1 and 2. Draw on paper an angle equal to the measured difference of altitude and find how far you must go from its vertex in order to have the distance between the sides, measured along an arc of a circle, equal to the measured distance between 1 and 2. This distance from the vertex will be the earth's radius.Exercise 19.—Measure the diameter of the earth by the method given above. In order that this may be done satisfactorily, the two stations at which observations are made must be separated by a considerable distance—i.e., 200 miles. They need not be on the same meridian, but if they are on different meridians in place of the actual distance between them, there must be used the projection of that distance upon the meridian—i.e., the north and south part of the distance.

By co-operation between schools in the Northern and Southern States, using a good map to obtain the required distances, the diameter of the earth may be measured with the plumb-line apparatus described in ChapterII and determined within a small percentage of its true value.45. The mass of the earth.—We have seen in ChapterIV the possibility of determining the masses of the planets as fractional parts of the sun's mass, but nothing was there shown, or could be shown, about measuring these masses after the common fashion in kilogrammes or tons. To do this we must first get the mass of the earth in tons or kilogrammes, and while the principles involved in this determination are simple enough, their actual application is delicate and difficult.

In Fig.26 we suppose a long plumb line to be suspended above the surface of the earth and to be attracted toward the center of the earth, C, by a force whose intensity is (ChapterIV)

F = k mE/R²,

where E denotes the mass of the earth, which is to be determined by experiment, and R is the radius of the earth, 3,963 miles. If there is no disturbing influence present, the plumb line will point directly downward, but if a massive ball of lead or other heavy substance is placed at one side, 1, it will attract the plumb line with a force equal to

f = k mB/r²,

where r is the distance of its center from the plumb bob and B is its mass which we may suppose, for illustration, to be a ton. In consequence of this attraction the plumb line will be pulled a little to one side, as shown by the dotted line, and if we represent by l the length of the plumb line and by d the distance between the original and the disturbed positions of the plumb bob we may write the proportion

F : f :: l : d;

and introducing the values of F and f given above, and solving for E the proportion thus transformed, we find

E = B · l/d · (R/r)².

In this equation the mass of the ball, B, the length of the plumb line, l, the distance between the center of the ball and the center of the plumb bob, r, and the radius of the earth, R, can all be measured directly, and d, the amount by which the plumb bob is pulled to one side by the ball, is readily found by shifting the ball over to the other side, at 2, and measuring with a microscope how far the plumb bob moves. This distance will, of course, be equal to 2d.

By methods involving these principles, but applied in a manner more complicated as well as more precise, the mass of the earth is found to be, in tons, 6,642×10¹⁸—i.e., 6,642 followed by 18 ciphers, or in kilogrammes 60,258×10²⁰. The earth's atmosphere makes up about a millionth part of this mass.

If the length of the plumb line were 100 feet, the weight of the ball a ton, and the distance between the two positions of the ball, 1 and 2, six feet, how many inches, d, would the plumb bob be pulled out of place?

Find from the mass of the earth and the data of §40 the mass of the sun in tons. Find also the mass of Mars. The computation can be very greatly abridged by the use of logarithms.46. Precession.—That the earth is isolated in space and has no support upon which to rest, is sufficiently shown by the fact that the stars are visible upon every side of it, and no support can be seen stretching out toward them. We must then consider the earth to be a globe traveling freely about the sun in a circuit which it completes once every year, and rotating once in every twenty-four hours about an axis which remains at all seasons directed very nearly toward the star Polaris. The student should be able to show from his own observations of the sun that, with reference to the stars, the direction of the sun from the earth changes about a degree a day. Does this prove that the earth revolves about the sun?

But it is only in appearance that the pole maintains its fixed position among the stars. If photographs are taken year after year, after the manner of Exercise7, it will be found that slowly the pole is moving (nearly) toward Polaris, and making this star describe a smaller and smaller circle in its diurnal path, while stars on the other side of the pole (in right ascension 12h.) become more distant from it and describe larger circles in their diurnal motion; but the process takes place so slowly that the space of a lifetime is required for the motion of the pole to equal the angular diameter of the full moon.

Spin a top and note how its rapid whirl about its axis corresponds to the earth's diurnal rotation. When the axis about which the top spins is truly vertical the top "sleeps"; but if the axis is tipped ever so little away from the vertical it begins to wobble, so that if we imagine the axis prolonged out to the sky and provided with a pencil point as a marker, this would trace a circle around the zenith, along which the pole of the top would move, and a little observation will show that the more the top is tipped from the vertical the larger does this circle become and the more rapidly does the wobbling take place. Were it not for the spinning of the top about its axis, it would promptly fall over when tipped from the vertical position, but the spin combines with the force which pulls the top over and produces the wobbling motion. Spin the top in opposite directions, with the hands of a watch and contrary to the hands of a watch, and note the effect which is produced upon the wobbling.

The earth presents many points of resemblance to the top. Its diurnal rotation is the spin about the axis. This axis is tipped 23.5° away from the perpendicular to its orbit (obliquity of the ecliptic) just as the axis of the top is tipped away from the vertical line. In consequence of its rapid spin, the body of the earth bulges out at the equator (27 miles), and the sun and moon, by virtue of their attraction (see ChapterIV), lay hold of this protuberance and pull it down toward the plane of the earth's orbit, so that if it were not for the spin this force would straighten the axis up and set it perpendicular to the orbit plane. But here, as in the case of the top, the spin and the tipping force combine to produce a wobble which is called precession, and whose effect we recognize in the shifting position of the pole among the stars. The motion of precession is very much slower than the wobbling of the top, since the tipping force for the earth is relatively very small, and a period of nearly 26,000 years is required for a complete circuit of the pole about its center of motion. Friction ultimately stops both the spin and the wobble of the top, but this influence seems wholly absent in the case of the earth, and both rotation and precession go on unchanged from century to century, save for certain minor forces which for a time change the direction or rate of the precessional motion, first in one way and then in another, without in the long run producing any results of consequence.

The center of motion, about which the pole travels in a small circle having an angular radius of 23.5°, is at that point of the heavens toward which a perpendicular to the plane of the earth's orbit points, and may be found on the star map in right ascension 18h. 0m. and declination 66.5°.Exercise 20.—Find this point on the map, and draw as well as you can the path of the pole about it. The motion of the pole along its path is toward the constellation Cepheus. Mark the position of the pole along this path at intervals of 1,000 years, and refer to these positions in dealing with some of the following questions:

Does the wobbling of the top occur in the same direction as the motion of precession? Do the tipping forces applied to the earth and top act in the same direction? What will be the polar star 12,000 years hence? The Great Pyramid of Egypt is thought to have been used as an observatory when Alpha Draconis was the bright star nearest the pole. How long ago was that?

The motion of the pole of course carries the equator and the equinoxes with it, and thus slowly changes the right ascensions and declinations of all the stars. On this account it is frequently called the precession of the equinoxes, and this motion of the equinox, slow though it is, is a matter of some consequence in connection with chronology and the length of the year.

Will the precession ever bring back the right ascensions and declinations to be again what they now are?

In what direction is the pole moving with respect to the Big Dipper? Will its motion ever bring it exactly to Polaris? How far away from Polaris will the precession carry the pole? What other bright stars will be brought near the pole by the precession?47. The warming of the earth.—Winter and summer alike the day is on the average warmer than the night, and it is easy to see that this surplus of heat comes from the sun by day and is lost by night through radiation into the void which surrounds the earth; just as the heat contained in a mass of molten iron is radiated away and the iron cooled when it is taken out from the furnace and placed amid colder surroundings. The earth's loss of heat by radiation goes on ceaselessly day and night, and were it not for the influx of solar heat this radiation would steadily diminish the temperature toward what is called the "absolute zero"—i.e., a state in which all heat has been taken away and beyond which there can be no greater degree of cold. This must not be confounded with the zero temperatures shown by our thermometers, since it lies nearly 500° below the zero of the Fahrenheit scale (-273° Centigrade), a temperature which by comparison makes the coldest winter weather seem warm, although the ordinary thermometer may register many degrees below its zero. The heat radiated by the sun into the surrounding space on every side of it is another example of the same cooling process, a hot body giving up its heat to the colder space about it, and it is the minute fraction of this heat poured out by the sun, and in small part intercepted by the earth, which warms the latter and produces what we call weather, climate, the seasons, etc.

Observe the fluctuations, the ebb and flow, which are inherent in this process. From sunset to sunrise there is nothing to compensate the steady outflow of heat, and air and ground grow steadily colder, but with the sunrise there comes an influx of solar heat, feeble at first because it strikes the earth's surface very obliquely, but becoming more and more efficient as the sun rises higher in the sky. But as the air and the ground grow warm during the morning hours they part more and more readily and rapidly with their store of heat, just as a steam pipe or a cup of coffee radiates heat more rapidly when very hot. The warmest hour of the day is reached when these opposing tendencies of income and expenditure of heat are just balanced; and barring such disturbing factors as wind and clouds, the gain in temperature usually extends to the time—an hour or two beyond noon—at which the diminishing altitude of the sun renders his rays less efficient, when radiation gains the upper hand and the temperature becomes for a short time stationary, and then commences to fall steadily until the next sunrise.

We have here an example of what is called a periodic change—i.e., one which, within a definite and uniform period (24 hours), oscillates from a minimum up to a maximum temperature and then back again to a minimum, repeating substantially the same variation day after day. But it must be understood that minor causes not taken into account above, such as winds, water, etc., produce other fluctuations from day to day which sometimes obscure or even obliterate the diurnal variation of temperature caused by the sun.

Expose the back of your hand to the sun, holding the hand in such a position that the sunlight strikes perpendicularly upon it; then turn the hand so that the light falls quite obliquely upon it and note how much more vigorous is the warming effect of the sun in the first position than in the second. It is chiefly this difference of angle that makes the sun's warmth more effective when he is high up in the sky than when he is near the horizon, and more effective in summer than in winter.

We have seen in ChapterIII that the sun's motion among the stars takes place along a path which carries it alternately north and south of the equator to a distance of 23.5°, and the stars show by their earlier risings and later settings, as we pass from the equator toward the north pole of the heavens, that as the sun moves northward from the equator, each day in the northern hemisphere will become a little longer, each night a little shorter, and every day the sun will rise higher toward the zenith until this process culminates toward the end of June, when the sun begins to move southward, bringing shorter days and smaller altitudes until the Christmas season, when again it is reversed and the sun moves northward. We have here another periodic variation, which runs its complete course in a period of a year, and it is easy to see that this variation must have a marked effect on the warming of the earth, the long days and great altitudes of summer producing the greater warmth of that season, while the shorter days and lower altitudes of December, by diminishing the daily supply of solar heat, bring on the winter's cold. The succession of the seasons, winter following summer and summer winter, is caused by the varying altitude of the sun, and this in turn is due to the obliquity of the ecliptic, or, what is the same thing, the amount by which the axis of the earth is tipped from being perpendicular to the plane of its orbit, and the seasons are simply a periodic change in the warming of the earth, quite comparable with the diurnal change but of longer period.

It is evident that the period within which the succession of winter and summer is completed, the year, as we commonly call it, must equal the time required by the sun to go from the vernal equinox around to the vernal equinox again, since this furnishes a complete cycle of the sun's motions north and south from the equator. On account of the westward motion of the equinox (precession) this is not quite the same as the time required for a complete revolution of the earth in its orbit, but is a little shorter (20m. 23s.), since the equinox moves back to meet the sun.48. Relation of the sun to climate.—It is clear that both the northern and southern hemispheres of the earth must have substantially the same kind of seasons, since the motion of the sun north and south affects both alike; but when the sun is north of the equator and warming our hemisphere most effectively, his light falls more obliquely upon the other hemisphere, the days there are short and winter reigns at the time we are enjoying summer, while six months later the conditions are reversed.

In those parts of the earth near the equator—the torrid zone—there is no such marked change from cold to warm as we experience, because, as the sun never gets more than 23.5° away from the celestial equator, on every day of the year he mounts high in the tropic skies, always coming within 23.5° of the zenith, and usually closer than this, so that there is no such periodic change in the heat supply as is experienced in higher latitudes, and within the tropics the temperature is therefore both higher and more uniform than in our latitude.

In the frigid zones, on the contrary, the sun never rises high in the sky; at the poles his greatest altitude is only 23.5°, and during the winter season he does not rise at all, so that the temperature is here low the whole year round, and during the winter season, when for weeks or months at a time the supply of solar light is entirely cut off, the temperature falls to a degree unknown in more favored climes.

If the obliquity of the ecliptic were made 10° greater, what would be the effect upon the seasons in the temperate zones? What if it were made 10° less?

Does the precession of the equinoxes have any effect upon the seasons or upon the climate of different parts of the earth?

If the axis of the earth pointed toward Arcturus instead of Polaris, would the seasons be any different from what they are now?49. The atmosphere.—Although we live upon its surface, we are not outside the earth, but at the bottom of a sea of air which forms the earth's outermost layer and extends above our heads to a height of many miles. The study of most of the phenomena of the atmosphere belongs to that branch of physics called meteorology, but there are a few matters which fairly come within our consideration of the earth as a planet. We can not see the stars save as we look through this atmosphere, and the light which comes through it is bent and oftentimes distorted so as to present serious obstacles to any accurate telescopic study of the heavenly bodies. Frequently this disturbance is visible to the naked eye, and the stars are said to twinkle—i.e., to quiver and change color many times per second, solely in consequence of a disturbed condition of the air and not from anything which goes on in the star. This effect is more marked low down in the sky than near the zenith, and it is worth noting that the planets show very little of it because the light they send to the earth comes from a disk of sensible area, while a star, being much smaller and farther from the earth, has its disk reduced practically to a mere point whose light is more easily affected by local disturbances in the atmosphere than is the broader beam which comes from the planets' disk.50. Refraction.—At all times, whether the stars twinkle or not, their light is bent in its passage through the atmosphere, so that the stars appear to stand higher up in the sky than their true positions. This effect, which the astronomer calls refraction, must be allowed for in observations of the more precise class, although save at low altitudes its amount is a very small fraction of a degree, but near the horizon it is much exaggerated in amount and becomes easily visible to the naked eye by distorting the disks of the sun and moon from circles into ovals with their long diameters horizontal. The refraction lifts both upper and lower edge of the sun, but lifts the lower edge more than the upper, thus shortening the vertical diameter. See Fig.27, which shows not only this effect, but also the reflection of the sun from the curved surface of the sea, still further flattening the image. If the surface of the water were flat, the reflected image would have the same shape as the sun's disk, and its altered appearance is sometimes cited as a proof that the earth's surface is curved.

The total amount of the refraction at the horizon is a little more than half a degree, and since the diameters of the sun and moon subtend an angle of about half a degree, we have the remarkable result that in reality the whole disk of either sun or moon is below the horizon at the instant that the lower edge appears to touch the horizon and sunset or moonset begins. The same effect exists at sunrise, and as a consequence the duration of sunshine or of moonshine is on the average about six minutes longer each day than it would be if there were no atmosphere and no refraction. A partial offset to this benefit is found in the fact that the atmosphere absorbs the light of the heavenly bodies, so that stars appear much less bright when near the horizon than when they are higher up in the sky, and by reason of this absorption the setting sun can be looked at with the naked eye without the discomfort which its dazzling luster causes at noon.

51. The twilight.—Another effect of the atmosphere, even more marked than the preceding, is the twilight. As at sunrise the mountain top catches the rays of the coming sun before they reach the lowland, and at sunset it keeps them after they have faded from the regions below, so the particles of dust and vapor, which always float in the atmosphere, catch the sunlight and reflect it to the surface of the earth while the sun is still below the horizon, giving at the beginning and end of day that vague and diffuse light which we call twilight.

Fig.28 shows a part of the earth surrounded by such a dust-laden atmosphere, which is illuminated on the left by the rays of the sun, but which, on the right of the figure, lies in the shadow cast by the earth. To an observer placed at 1 the sun is just setting, and all the atmosphere above him is illumined with its rays, which furnish a bright twilight. When, by the earth's rotation, this observer has been carried to 2, all the region to the east of his zenith lies in the shadow, while to the west there is a part of the atmosphere from which there still comes a twilight, but now comparatively faint, because the lower part of the atmosphere about our observer lies in the shadow, and it is mainly its upper regions from which the light comes, and here the dust and moisture are much less abundant than in the lower strata. Still later, when the observer has been carried by the earth's rotation to the point3, every vestige of twilight will have vanished from his sky, because all of the illuminated part of the atmosphere is now below his horizon, which is represented by the line 3L. In the figure the sun is represented to be 78° below this horizon line at the end of twilight, but this is a gross exaggeration, made for the sake of clearness in the drawing—in fact, twilight is usually said to end when the sun is 18° below the horizon.

Let the student redraw Fig.28 on a large scale, so that the points 1 and 3 shall be only 18° apart, as seen from the earth's center. He will find that the point L is brought down much closer to the surface of the earth, and measuring the length of the line 2L, he should find for the "height of the atmosphere" about one-eightieth part of the radius of the earth—i.e., a little less than 50 miles. This, however, is not the true height of the atmosphere. The air extends far beyond this, but the particles of dust and vapor which are capable of sending sunlight down to the earth seem all to lie below this limit.

The student should not fail to watch the eastern sky after sunset, and see the shadow of the earth rise up and fill it while the twilight arch retreats steadily toward the west.

Duration of twilight.—Since twilight ends when the sun is 18° below the horizon, any circumstance which makes the sun go down rapidly will shorten the duration of twilight, and anything which retards the downward motion of the sun will correspondingly prolong it. Chief among influences of this kind is the angle which the sun's course makes with the horizon. If it goes straight down, as at a, Fig.29, a much shorter time will suffice to carry it to a depression of 18° than is needed in the case shown at b in the same figure, where the motion is very oblique to the horizon. If we consider different latitudes and different seasons of the year, we shall find every possible variety of circumstance from a to b, and corresponding to these, the duration of twilight varies from an all-night duration in the summers of Scotland and more northern lands to an hour or less in the mountains of Peru. For the sake of graphical effect, the shortness of tropical twilight is somewhat exaggerated by Coleridge in the lines,

In the United States the longest twilights come at the end of June, and last for a little more than two hours, while the shortest ones are in March and September, amounting to a little more than an hour and a half; but at all times the last half hour of twilight is hardly to be distinguished from night, so small is the quantity of reflecting matter in the upper regions of the atmosphere. For practical convenience it is customary to assume in the courts of law that twilight ends an hour after sunset.

How long does twilight last at the north pole?

The Aurora.—One other phenomenon of the atmosphere may be mentioned, only to point out that it is not of an astronomical character. The Aurora, or northern lights, is as purely an affair of the earth as is a thunderstorm, and its explanation belongs to the subject of terrestrial magnetism.