60. DAY, MONTH, AND YEAR.--If the day, the month, and the year are to be used concurrently, it is necessary to determine how many days are contained in the month and year, and when this has been done by the astronomer the numbers are found to be very awkward and inconvenient for daily use; and much of the history of chronology consists in an account of the various devices by which ingenious men have sought to use integral numbers to replace the c.u.mbrous decimal fractions which follow.

According to Professor Harkness, for the epoch 1900 A. D.--

One tropical year = 365.242197 mean solar days.

" " " = 365d. 5h. 48m. 45.8s.

One lunation = 29.530588 mean solar days.

" " = 29d. 12h. 44m. 2.8s.

The word _lunation_ means the average interval from one new moon to the next one--i. e., the time required by the moon to go from conjunction with the sun round to conjunction again.

A very ancient device was to call a year equal to 365 days, and to have months alternately of 29 and 30 days in length, but this was unsatisfactory in more than one way. At the end of four years this artificial calendar would be about one day ahead of the true one, at the end of forty years ten days in error, and within a single lifetime the seasons would have appreciably changed their position in the year, April weather being due in March, according to the calendar. So, too, the year under this arrangement did not consist of any integral number of months, 12 months of the average length of 29.5 days being 354 days, and 13 months 383.5 days, thus making any particular month change its position from the beginning to the middle and the end of the year within a comparatively short time. Some peoples gave up the astronomical year as an independent unit and adopted a conventional year of 12 lunar months, 354 days, which is now in use in certain Mohammedan countries, where it is known as the wandering year, with reference to the changing positions of the seasons in such a year. Others held to the astronomical year and adopted a system of conventional months, such that twelve of them would just make up a year, as is done to this day in our own calendar, whose months of arbitrary length we are compelled to remember by some such jingle as the following:

"Thirty days hath September, April, June, and November; All the rest have thirty-one Save February, Which alone hath twenty-eight, Till leap year gives it twenty-nine."

61. THE CALENDAR.--The foundations of our calendar may fairly be ascribed to Julius Caesar, who, under the advice of the Egyptian astronomer Sosigines, adopted the old Egyptian device of a leap year, whereby every fourth year was to consist of 366 days, while ordinary years were only 365 days long. He also placed the beginning of the year at the first of January, instead of in March, where it had formerly been, and gave his own name, Julius, to the month which we now call July. August was afterward named in honor of his successor, Augustus.

The names of the earlier months of the year are drawn from Roman mythology; those of the later months, September, October, etc., meaning seventh month, eighth month, represent the places of these months in the year, before Caesar's reformation, and also their places in some of the subsequent calendars, for the widest diversity of practice existed during mediaeval times with regard to the day on which the new year should begin, Christmas, Easter, March 25th, and others having been employed at different times and places.

The system of leap years introduced by Caesar makes the average length of a year 365.25 days, which differs by about eleven minutes from the true length of the tropical year, a difference so small that for ordinary purposes no better approximation to the true length of the year need be desired. But _any_ deviation from the true length, however small, must in the course of time shift the seasons, the vernal and autumnal equinox, to another part of the year, and the ecclesiastical authorities of mediaeval Europe found here ground for objection to Caesar's calendar, since the great Church festival of Easter has its date determined with reference to the vernal equinox, and with the lapse of centuries Easter became more and more displaced in the calendar, until Pope Gregory XIII, late in the sixteenth century, decreed another reformation, whereby ten days were dropped from the calendar, the day after March 11th being called March 21st, to bring back the vernal equinox to the date on which it fell in A. D. 325, the time of the Council of Nicaea, which Gregory adopted as the fundamental epoch of his calendar.

The calendar having thus been brought back into agreement with that of old time, Gregory purposed to keep it in such agreement for the future by modifying Caesar's leap-year rule so that it should run: Every year whose number is divisible by 4 shall be a leap year except those years whose numbers are divisible by 100 but not divisible by 400. These latter years--e. g., 1900--are counted as common years. The calendar thus altered is called Gregorian to distinguish it from the older, Julian calendar, and it found speedy acceptance in those civilized countries whose Church adhered to Rome; but the Protestant powers were slow to adopt it, and it was introduced into England and her American colonies by act of Parliament in the year 1752, nearly two centuries after Gregory's time. In Russia the Julian calendar has remained in common use to our own day, but in commercial affairs it is there customary to write the date according to both calendars--e. g., July 4/16, and at the present time strenuous exertions are making in that country for the adoption of the Gregorian calendar to the complete exclusion of the Julian one.

The Julian and Gregorian calendars are frequently represented by the abbreviations O. S. and N. S., old style, new style, and as the older historical dates are usually expressed in O. S., it is sometimes convenient to transform a date from the one calendar to the other. This is readily done by the formula

G = J + (N - 2) - N/4,

where _G_ and _J_ are the respective dates, _N_ is the number of the century, and the remainder is to be neglected in the division by 4. For September 3, 1752, O. S., we have

J = Sept. 3 N - 2 = + 15 - N/4 = - 4 ------------------ G = Sept. 14

and September 14 is the date fixed by act of Parliament to correspond to September 3, 1752, O. S. Columbus discovered America on October 12, 1492, O. S. What is the corresponding date in the Gregorian calendar?

62. THE DAY OF THE WEEK.--A problem similar to the above but more complicated consists in finding the day of the week on which any given date of the Gregorian calendar falls--e. g., October 21, 1492.

The formula for this case is

7q + r = Y + D + (Y - 1)/4 - (Y - 1)/100 + (Y - 1)/400

where _Y_ denotes the given year, _D_ the number of the day (date) in that year, and _q_ and _r_ are respectively the quotient and the remainder obtained by dividing the second member of the equation by 7.

If _r_ = 1 the date falls on Sunday, etc., and if _r_ = 0 the day is Sat.u.r.day. For the example suggested above we have

Jan. 31 Feb. 29 Mch. 31 April 30 May 31 June 30 July 31 Aug. 31 Sept. 30 Oct. 21 --- D = 295

Y = 1492 + D = + 295 + (Y - 1) 4 = + 372 - (Y - 1) 100 = - 14 + (Y - 1) 400 = + 3 ------- 7) 2148

_q_ = 306 _r_ = 6 = Friday.

Find from some history the day of the week on which Columbus first saw America, and compare this with the above.

On what day of the week did last Christmas fall? On what day of the week were you born? In the formula for the day of the week why does _q_ have the coefficient 7? What principles in the calendar give rise to the divisors 4, 100, 400?

For much curious and interesting information about methods of reckoning the lapse of time the student may consult the articles Calendar and Chronology in any good encyclopaedia.

[Ill.u.s.tration: THE YERKES OBSERVATORY, WILLIAMS BAY, WIS.]

CHAPTER VII

ECLIPSES

63. THE NATURE OF ECLIPSES.--Every planet has a shadow which travels with the planet along its...o...b..t, always pointing directly away from the sun, and cutting off from a certain region of s.p.a.ce the sunlight which otherwise would fill it. For the most part these shadows are invisible, but occasionally one of them falls upon a planet or some other body which shines by reflected sunlight, and, cutting off its supply of light, produces the striking phenomenon which we call an eclipse. The satellites of Jupiter, Saturn, and Mars are eclipsed whenever they plunge into the shadows cast by their respective planets, and Jupiter himself is partially eclipsed when one of his own satellites pa.s.ses between him and the sun, and casts upon his broad surface a shadow too small to cover more than a fraction of it.

But the eclipses of most interest to us are those of the sun and moon, called respectively solar and lunar eclipses. In Fig. 33 the full moon, _M'_, is shown immersed in the shadow cast by the earth, and therefore eclipsed, and in the same figure the new moon, _M_, is shown as casting its shadow upon the earth and producing an eclipse of the sun. From a mere inspection of the figure we may learn that an eclipse of the sun can occur only at new moon--i. e., when the moon is on line between the earth and sun--and an eclipse of the moon can occur only at full moon.

Why? Also, the eclipsed moon, _M'_, will present substantially the same appearance from every part of the earth where it is at all visible--the same from North America as from South America--but the eclipsed sun will present very different aspects from different parts of the earth.

Thus, at _L_, within the moon's shadow, the sunlight will be entirely cut off, producing what is called a total eclipse. At points of the earth's surface near _J_ and _K_ there will be no interference whatever with the sunlight, and no eclipse, since the moon is quite off the line joining these regions to any part of the sun. At places between _J_ and _L_ or _K_ and _L_ the moon will cut off a part of the sun's light, but not all of it, and will produce what is called a partial eclipse, which, as seen from the northern parts of the earth, will be an eclipse of the lower (southern) part of the sun, and as seen from the southern hemisphere will be an eclipse of the northern part of the sun.

[Ill.u.s.tration: FIG. 33.--Different kinds of eclipse.]

The moon revolves around the earth in a plane, which, in the figure, we suppose to be perpendicular to the surface of the paper, and to pa.s.s through the sun along the line _M' M_ produced. But it frequently happens that this plane is turned to one side of the sun, along some such line as _P Q_, and in this case the full moon would cut through the edge of the earth's shadow without being at any time wholly immersed in it, giving a partial eclipse of the moon, as is shown in the figure.

In what parts of the earth would this eclipse be visible? What kinds of solar eclipse would be produced by the new moon at _Q_? In what parts of the earth would they be visible?

64. THE SHADOW CONE.--The shape and position of the earth's shadow are indicated in Fig. 33 by the lines drawn tangent to the circles which represent the sun and earth, since it is only between these lines that the earth interferes with the free radiation of sunlight, and since both sun and earth are spheres, and the earth is much the smaller of the two, it is evident that the earth's shadow must be, in geometrical language, a cone whose base is at the earth, and whose vertex lies far to the right of the figure--in other words, the earth's shadow, although very long, tapers off finally to a point and ends. So, too, the shadow of the moon is a cone, having its base at the moon and its vertex turned away from the sun, and, as shown in the figure, just about long enough to reach the earth.

It is easily shown, by the theorem of similar triangles in connection with the known size of the earth and sun, that the distance from the center of the earth to the vertex of its shadow is always equal to the distance of the earth from the sun divided by 108, and, similarly, that the length of the moon's shadow is equal to the distance of the moon from the sun divided by 400, the moon's shadow being the smaller and shorter of the two, because the moon is smaller than the earth. The radius of the moon's...o...b..t is just about 1/400th part of the radius of the earth's...o...b..t--i. e., the distance of the moon from the earth is 1/400th part of the distance of the earth from the sun, and it is this "chance" agreement between the length of the moon's shadow and the distance of the moon from the earth which makes the tip of the moon's shadow fall very near the earth at the time of solar eclipses. Indeed, the elliptical shape of the moon's...o...b..t produces considerable variations in the distance of the moon from the earth, and in consequence of these variations the vertex of the shadow sometimes falls short of reaching the earth, and sometimes even projects considerably beyond its farther side. When the moon's distance is too great for the shadow to bridge the s.p.a.ce between earth and moon there can be no total eclipse of the sun, for there is no shadow which can fall upon the earth, even though the moon does come directly between earth and sun.

But there is then produced a peculiar kind of partial eclipse called _annular_, or ring-shaped, because the moon, although eclipsing the central parts of the sun, is not large enough to cover the whole of it, but leaves the sun's edge visible as a ring of light, which completely surrounds the moon. Although, strictly speaking, this is only a partial eclipse, it is customary to put total and annular eclipses together in one cla.s.s, which is called central eclipses, since in these eclipses the line of centers of sun and moon strikes the earth, while in ordinary partial eclipses it pa.s.ses to one side of the earth without striking it.

In this latter case we have to consider another cone called the _penumbra_--i. e., partial shadow--which is shown in Fig. 33 by the broken lines tangent to the sun and moon, and crossing at the point _V_, which is the vertex of this cone. This penumbral cone includes within its surface all that region of s.p.a.ce within which the moon cuts off any of the sunlight, and of course it includes the shadow cone which produces total eclipses. Wherever the penumbra falls there will be a solar eclipse of some kind, and the nearer the place is to the axis of the penumbra, the more nearly total will be the eclipse. Since the moon stands about midway between the earth and the vertex of the penumbra, the diameter of the penumbra where it strikes the earth will be about twice as great as the diameter of the moon, and the student should be able to show from this that the region of the earth's surface within which a partial solar eclipse is visible extends in a straight line about 2,100 miles on either side of the region where the eclipse is total. Measured along the curved surface of the earth, this distance is frequently much greater.

Is it true that if at any time the axis of the shadow cone comes within 2,100 miles of the earth's surface a partial eclipse will be visible in those parts of the earth nearest the axis of the shadow?

65. DIFFERENT CHARACTERISTICS OF LUNAR AND SOLAR ECLIPSES.--One marked difference between lunar and solar eclipses which has been already suggested, may be learned from Fig. 33. The full moon, _M'_, will be seen eclipsed from every part of the earth where it is visible at all at the time of the eclipse--that is, from the whole night side of the earth; while the eclipsed sun will be seen eclipsed only from those parts of the day side of the earth upon which the moon's shadow or penumbra falls. Since the point of the shadow at best but little more than reaches to the earth, the amount of s.p.a.ce upon the earth which it can cover at any one moment is very small, seldom more than 100 to 200 miles in length, and it is only within the s.p.a.ce thus actually covered by the shadow that the sun is at any given moment totally eclipsed, but within this region the sun disappears, absolutely, behind the solid body of the moon, leaving to view only such outlying parts and appendages as are too large for the moon to cover. At a lunar eclipse, on the other hand, the earth coming between sun and moon cuts off the light from the latter, but, curiously enough, does not cut it off so completely that the moon disappears altogether from sight even in mid-eclipse. The explanation of this continued visibility is furnished by the broken lines extending, in Fig. 33, from the earth through the moon. These represent sunlight, which, entering the earth's atmosphere near the edge of the earth (edge as seen from sun and moon), pa.s.ses through it and emerges in a changed direction, refracted, into the shadow cone and feebly illumines the moon's surface with a ruddy light like that often shown in our red sunsets. Eclipse and sunset alike show that when the sun's light shines through dense layers of air it is the red rays which come through most freely, and the attentive observer may often see at a clear sunset something which corresponds exactly to the bending of the sunlight into the shadow cone; just before the sun reaches the horizon its disk is distorted from a circle into an oval whose horizontal diameter is longer than the vertical one (see -- 50).

QUERY.--At a total lunar eclipse what would be the effect upon the appearance of the moon if the atmosphere around the edge of the earth were heavily laden with clouds?

66. THE TRACK OF THE SHADOW.--We may regard the moon's shadow cone as a huge pencil attached to the moon, moving with it along its...o...b..t in the direction of the arrowhead (Fig. 34), and as it moves drawing a black line across the face of the earth at the time of total eclipse. This black line is the path of the shadow and marks out those regions within which the eclipse will be total at some stage of its progress. If the point of the shadow just reaches the earth its trace will have no sensible width, while, if the moon is nearer, the point of the cone will be broken off, and, like a blunt pencil, it will draw a broad streak across the earth, and this under the most favorable circ.u.mstances may have a breadth of a little more than 160 miles and a length of 10,000 or 12,000 miles. The student should be able to show from the known distance of the moon (240,000 miles) and the known interval between consecutive new moons (29.5 days) that on the average the moon's shadow sweeps past the earth at the rate of 2,100 miles per hour, and that in a general way this motion is from west to east, since that is the direction of the moon's motion in its...o...b..t. The actual velocity with which the moon's shadow moves past a given station may, however, be considerably greater or less than this, since on the one hand when the shadow falls very obliquely, as when the eclipse occurs near sunrise or sunset, the shifting of the shadow will be very much greater than the actual motion of the moon which produces it, and on the other hand the earth in revolving upon its axis carries the spectator and the ground upon which he stands along the same direction in which the shadow is moving. At the equator, with the sun and moon overhead, this motion of the earth subtracts about 1,000 miles per hour from the velocity with which the shadow pa.s.ses by. It is chiefly on this account, the diminished velocity with which the shadow pa.s.ses by, that total solar eclipses last longer in the tropics than in higher lat.i.tudes, but even under the most favorable circ.u.mstances the duration of totality does not reach eight minutes at any one place, although it may take the shadow several hours to sweep the entire length of its path across the earth.

According to Whitmell the greatest possible duration of a total solar eclipse is 7m. 40s., and it can attain this limit only when the eclipse occurs near the beginning of July and is visible at a place 5 north of the equator.

The duration of a lunar eclipse depends mainly upon the position of the moon with respect to the earth's shadow. If it strikes the shadow centrally, as at _M'_, Fig. 33, a total eclipse may last for about two hours, with an additional hour at the beginning and end, during which the moon is entering and leaving the earth's shadow. If the moon meets the shadow at one side of the axis, as at _P_, the total phase of the eclipse may fail altogether, and between these extremes the duration of totality may be anything from two hours downward.

[Ill.u.s.tration: FIG. 34.--Relation of the lunar nodes to eclipses.]

67. RELATION OF THE LUNAR NODES TO ECLIPSES.--To show why the moon sometimes encounters the earth's shadow centrally and more frequently at full moon pa.s.ses by without touching it at all, we resort to Fig. 34, which represents a part of the orbit of the earth about the sun, with dates showing the time in each year at which the earth pa.s.ses the part of its...o...b..t thus marked. The orbit of the moon about the earth, _M M'_, is also shown, with the new moon, _M_, casting its shadow toward the earth and the full moon, _M'_, apparently immersed in the earth's shadow. But here appearances are deceptive, and the student who has made the observations set forth in Chapter III has learned for himself a fact of which careful account must now be taken. The apparent paths of the moon and sun among the stars are great circles which lie near each other, but are not exactly the same; and since these great circles are only the intersections of the sky with the planes of the earth's...o...b..t and the moon's...o...b..t, we see that these planes are slightly inclined to each other and must therefore intersect along some line pa.s.sing through the center of the earth. This line, _N' N''_, is shown in the figure, and if we suppose the surface of the paper to represent the plane of the earth's...o...b..t, we shall have to suppose the moon's...o...b..t to be tipped around this line, so that the left side of the orbit lies above and the right side below the surface of the paper. But since the earth's shadow lies in the plane of its...o...b..t--i. e., in the surface of the paper--the full moon of March, _M'_, must have pa.s.sed below the shadow, and the new moon, _M_, must have cast its shadow above the earth, so that neither a lunar nor a solar eclipse could occur in that month. But toward the end of May the earth and moon have reached a position where the line _N' N''_ points almost directly toward the sun, in line with the shadow cones which hide it. Note that the line _N' N''_ remains very nearly parallel to its original position, while the earth is moving along its...o...b..t. The full moon will now be very near this line and therefore very close to the plane of the earth's...o...b..t, if not actually in it, and must pa.s.s through the shadow of the earth and be eclipsed. So also the new moon will cast its shadow in the plane of the ecliptic, and this shadow, falling upon the earth, produced the total solar eclipse of May 28, 1900.

_N' N''_ is called the line of nodes of the moon's...o...b..t (-- 39), and the two positions of the earth in its...o...b..t, diametrically opposite each other, at which _N' N''_ points exactly toward the sun, we shall call the _nodes_ of the lunar orbit. Strictly speaking, the nodes are those points of the sky against which the moon's center is projected at the moment when in its...o...b..tal motion it cuts through the plane of the earth's...o...b..t. Bearing in mind these definitions, we may condense much of what precedes into the proposition: Eclipses of either sun or moon can occur only when the earth is at or near one of the nodes of the moon's...o...b..t. Corresponding to these positions of the earth there are in each year two seasons, about six months apart, at which times, and at these only, eclipses can occur. Thus in the year 1900 the earth pa.s.sed these two points on June 2d and November 24th respectively, and the following list of eclipses which occurred in that year shows that all of them were within a few days of one or the other of these dates: