The whole of the shadow of the moon is so small that only a few places on the earth’s surface can be simultaneously immersed in it, and when we come to discuss the conditions of an eclipse with regard to a particular observer, the problem becomes a complicated one. At some places the eclipse may be total, at others it will be only partial, while at others no eclipse will occur at all. These differences are due to the fact that the sun is scarcely appreciably displaced by the change of locality, while the apparent position of the moon may be affected to the extent of nearly a degree. Again, the observer situated on the earth’s surface has a movement of his own, produced by the earth’s rotation, and his rate of motion depends upon the latitude in which he is situated. The effect of this movement upon the conditions of the eclipse are very pronounced. Suppose for a moment that the sun, moon, and earth, are fixed along the same straight line S M E in Fig. 32, a terrestrial observer at a on the earth’s Equator would see an eclipse at noon; if he were not in rotation, and the three bodies remained at rest, the eclipse would be a perpetual one. He is, however, carried onward by the earth’s rotation, and even if the moon were at rest, it would appear to him to pass over the sun in the reverse direction. This retardation of the moon will be less in amount for observers away from the Equator, and also for observers to whom the sun is not on the meridian when eclipsed. The effect of rotation on an observer at b (Fig. 32), for example, is to move him almost in the direction of the line joining the moon and sun, and the backward tendency of the moon due to rotation is very slight. On account of the earth’s rotation, then, the duration of a solar eclipse is lengthened, the greatest increase occurring at those places where the sun is on the meridian at the time of eclipse.

There is another source of gain of duration of an eclipse to the observer who sees the phenomenon about noon. The moon’s apparent diameter is then augmented by a greater amount than at other places, because the observer is then nearest to the moon; while the sun’s apparent diameter is not appreciably affected. The greater the difference in the apparent diameters of the sun and moon, the longer will totality last.

These and other circumstances have all to be taken into account in computing the conditions under which an eclipse will be seen at any given place.

According to an eminent authority, Professor Young, the greatest possible diameter of the moon’s shadow, where it strikes the earth, is 167 miles. It may, however, cover a larger space on the earth’s surface, because the latter does not pass perpendicularly through the shadow. To all persons within the shadow, the eclipse will be total, but to those on its outer boundary the duration of totality will be for an instant only. The penumbral shadow has a cross section about 4,500 miles in diameter, covering sometimes a space on the earth’s surface 6,000 miles across. To all persons within this area, but not in the central shadow, the eclipse will be partial. The shadow spot travels over the earth’s surface, because of the moon’s movement, but its track and speed are greatly modified by the earth’s rotation. The movement of the shadow, as affected by the earth’s rotation, would be along a parallel of latitude; but its ultimate direction of movement, though trending eastwards, depends upon this, combined with the direction of the moon’s movement at the time of the eclipse. Thus, a portion of the track of the total eclipse of April 16, 1893, is as that shown in Fig. 33.

These considerations will suffice to explain the necessity for very precise calculations as to the position of the central line of an eclipse, when observers are sent out for the purpose of recording the phenomena.

Under the most favourable combination of conditions, that is, when the eclipse occurs at noon at a place on the Equator, an eclipse cannot be total for more than 7 minutes 58 seconds, nor be annular for a longer time than 12 minutes 24 seconds. From first to last contact may occupy as much as 2 hours, when all the circumstances are similarly favourable. (Loomis.)

The Solar Ecliptic Limit.—In order that an eclipse of the sun may occur, the moon must be so near the ecliptic that it can be seen projected on the sun, either wholly or partially, from some point on the earth. It must therefore not be very far from the node, and the distance it may be from the node, while still being seen upon the sun, is called the solar ecliptic limit. As in the case of lunar eclipses, this distance is determined by the inclination of the moon’s orbit, and the distances of the moon and sun from the earth. The latter being variable quantities, the limit is not always the same. It is calculated without much difficulty that an eclipse must occur if the new moon happens when it is within 15° 21′ of the node, and may occur within 18° 31′. These are called the minor and major ecliptic limits respectively. For total or annular eclipses, the limits are respectively 9° 55′ and 11° 50′.

Number of Eclipses in a Year.—If the moon’s nodes were fixed, the sun would pass through the line of nodes twice a year. At such times an eclipse of the sun must necessarily occur if the moon were within 15° 21′ of the node on either side. The sun requires more than a month to traverse this space of 30° 42′, and the moon must therefore pass through each node at least once while the sun is traversing these limits. It follows, then, that there must be at least two eclipses of the sun in a year. Since the line of nodes of the moon’s orbit revolves backwards in a period of about nineteen years, the sun returns to the same node after an interval of 346·6 days, and there must accordingly be two solar eclipses in this interval. If, then, there be an eclipse early in January, there will be another about the middle of the year, and another at the end of the year, so that on this ground alone there is a possibility of three solar eclipses in a year.

Again, while the sun is passing through the ecliptic limits, it may happen that an eclipse occurs on its entrance, and then another will occur before it gets beyond on the other side of the line of nodes. In this way two eclipses may occur in the region of each node passage, and if the first of the series occurs early in January, five eclipses of the sun may occur in a single year.

The sun, however, is not a month in traversing the lunar ecliptic limit. Consequently, a whole year may elapse without the moon being sufficiently near the node to pass within the earth’s shadow, and in many years there are accordingly no eclipses of the moon. Only one full moon can occur within the lunar ecliptic limits when the sun passes the node, but if there be an eclipse at one node, there may also be one six months later at the other node. As in the case of the solar eclipses, the “eclipse year” is one of 346·6 days, so that if there be an eclipse of the moon early in January, there may possibly be three altogether in the course of the year, but there could not be three lunar eclipses if the extra solar eclipse were possible. Altogether, then, there may be seven eclipses in the course of a year—five of the sun and two of the moon. Usually there are four or five, some particulars of which are furnished by all respectable almanacs. It will be observed that the number of solar eclipses is much larger than that of lunar ones, but as the latter are visible at all places having the moon above the horizon, while the former are restricted to small parts of the earth’s surface, more lunar than solar eclipses are visible at any specified place.

Recurrence of Eclipses.—We have seen that the sun requires only 346·6 days to travel from one of the moon’s nodes back to the same node again, in consequence of the regression of the nodes, while the moon requires 27·2 days. Suppose, then, that the moon and sun are at a node, and there is an eclipse at new moon; after 346·6 days the sun will return to the same node, but the moon will not be at the node, nor will it be exactly new. It will not be until the sun has returned nineteen times to the node that the moon is also very nearly new at the same node again. Nineteen returns of the sun to the moon’s nodes occupy a period of 6,585·78 days; 223 intervals between successive new moons (synodic months) cover 6,585·32 days, while 242 node passages of the moon require 6,585·357 days. In this period of 18 years 11⅓ days (or 10⅓ days if there are five, and 12⅓ if there are three leap years in the interval), the sun and moon thus return to nearly the same conditions as affecting the possibility of eclipses. This period was called the Saros by the Chaldeans, by whom it was employed in the prediction of eclipses. The adjustment of periods, however, is not quite precise, so that predictions based upon the Saros are only approximations, which serve as a guide for more accurate computations.

This eclipse period is still more remarkable from the fact that it almost exactly represents 239 passages of the moon through perigee, so that after the lapse of 18 years 11⅓ days the moon is almost at the same distance from the earth, as well as nearly at the same phase and the same distance from a node.

As the Saros includes a fraction of a day, an eclipse is not necessarily repeated at the same place after the lapse of 18 years 11⅓ days, for the reason that the eclipse will not occur at the same time of day, and the sun may be below the horizon. After three Saroses, however, the eclipse will be repeated nearly at the same hour, but even then it will not be seen under the same conditions, because the track of the shadow will be in different latitudes, for the reason that the moon does not return exactly to the node in the interval between 223 new or full moons, and eclipses can only occur when the moon is new or full.

Beginning as a partial eclipse, an eclipse of the moon will gradually become of greater magnitude at successive intervals of 18 years 11 days, until it becomes a total eclipse, and will again gradually become of smaller magnitude, until it ceases to be reproduced at all. Altogether, it would be repeated once in every 223 months for 865 years.

Since the solar ecliptic limit is greater than the lunar, a solar eclipse is repeated at similar intervals of 18 years for about 1200 years. Most of these eclipses would be partial, 27 would be annular, and 18 total. During this period, the track of the central eclipse would shift northwards if the moon were at a descending node, and southwards if at an ascending node, until finally it passed altogether clear of the earth.

It must be remarked, however, that, in the period corresponding to a single Saros, about 28 eclipses of the moon, and 43 of the sun, usually appear, so that altogether about 71 series of eclipses are in progress. Of the solar eclipses which occur during a period of 18 years, about 12 are total at some places upon the earth.

Occultations of Stars and Planets by the Moon.—In its monthly round, the moon is constantly passing in front of some of the stars which lie in its apparent path, and these luminaries will, therefore, at times, be hidden temporarily by the moons disc. Occasionally a planet may appear in the same line of vision as the moon, and that also will pass from view until subsequent motion again removes the intercepting body. These disappearances are closely allied to the phenomena of eclipses, and receive the name of occultations. On account of the moon’s eastward movement, it is evident that the disappearance of stars or planets when occulted will take place on the eastern edge of the moon; but since the moon trends north or south in some parts of its orbit, the disappearance near the northern and southern edges may occur slightly on the western side of the north or south point of the moons limb. Similarly, the reappearance generally occurs on the western side of the moon, but occasionally may occur on the eastern side—that is, when the northern or southern edge of the moon does not much more than appear to graze the stars.

The calculation of the circumstances of an occultation is very similar to that involved in the computation of eclipses. (A simple graphical method for working out the conditions of an occultation is described by Major Grant, R.E., in the Geographical Journal for June, 1896.)

Eclipses and Occultations of Satellites by Planets.—Just as we find the moon eclipsed by passing through the earth’s shadow, we find the satellites of other planets to be at times invisible for a similar reason. We thus observe eclipses of the satellites. The satellites may also be invisible to us for the reason that they are behind the planet, and they are then said to be occulted. These satellite phenomena are especially remarked in the case of Jupiter, and their observation is one of great interest. When a satellite passes between the sun and the planet it throws a shadow on the surface of the planet similar to that of the moon upon the earth. This is visible to us as a dark spot, and from the centre of that dusky patch an inhabitant of Jupiter would undoubtedly see a total eclipse of the sun. To us on the earth the passage of such a shadow across the planet’s disc is but a “transit of the shadow” with its “ingress” and “egress.”

The times of all these appearances are computed from a knowledge of the movements of the satellites.

CHAPTER IX.
HOW TO FIND OUR SITUATION ON THE EARTH.

Determination of Latitude.—In order that we may precisely define our situation upon the terrestrial sphere, we have seen that two measurements are necessary, namely, latitude and longitude. The first of these indicates the angular distance from the Equator, and the latter the angular distance east or west of an arbitrary initial meridian. It is necessary for us then to learn something of how these important co-ordinates can be determined.

In considering the apparent movements of the heavenly bodies in different latitudes, we have already seen that at places on the earth’s Equator the north celestial pole is on the horizon, while at the North Pole it is in the zenith, and in other latitudes is elevated at different angles. If one sails from England to the Cape, for example, the Pole Star is seen to gradually get lower and lower in the sky, until, on crossing the Equator, it descends below the northern horizon and is no longer visible. Sailing northward, as to Norway, the Pole Star is seen to get higher in the sky.

Now, although the Pole Star is not exactly at the north celestial pole, it is a convenient guide to the eye as to the location of that very important mathematical point, and what we learn from its behaviour as our latitude is changed is that the altitude of the Pole above the horizon is equal to the latitude of the place of observation.

One of the methods employed for finding the latitude of a place is accordingly to determine the altitude of the Pole. This can be obtained by an instrumental measurement of the altitude of the Pole Star, from which, if the time of observation be known, the altitude of the true Pole, which occupies the centre of the small diurnal circle traversed by the star, can be computed. Tables which save an immense amount of labour in the calculations involved are given in the “Nautical Almanac,” and in “Whitaker’s Almanac.”

Another method of finding the elevation of the Pole is to take advantage of the fact, that at intervals of twelve sidereal hours the Pole Star passes the meridian alternately above and below the Pole. If, then, one finds the altitudes at the upper and lower transits, and corrects them for refraction, the average of the readings is a measure of the altitude of the true Pole, and therefore of the latitude. Other stars which are circumpolar may be employed for the same purpose, and this method has the great advantage that a knowledge of the correct time, or of the exact position of the star observed, is superfluous. The disadvantage is that the correction for refraction, especially in low latitudes, cannot be made with the necessary degree of accuracy. It must be remembered that an error of only 1′ in latitude implies a mistake of a mile measured on the earth’s surface.

Other methods, however, are available. As we go southwards, not only does the Pole Star become lower in the sky, other stars in the southern part of the sky become higher at the same rate that the Pole Star descends. Other stars can therefore be utilised, and in order that refraction may affect the observations as little as possible, stars of known declination near the zenith are observed. Suppose an observer, situated at O (Fig. 34) on the earth’s surface, observing a star S on his meridian, O Z will represent his zenith, and O E, parallel to the Equator, will be the direction in which he will see the celestial equator where it crosses his meridian. The declination of the star, represented by the angle S O E, has been previously determined with great accuracy, and the angle S O Z, the zenith distance of the star, is the angle which he measures. In the case illustrated by the diagram, the difference between the declination and the zenith distance will give the angle Z O E, which is evidently equal to the latitude O C Q. To get rid of the ever troublesome refraction of our atmosphere, stars which pass as nearly as possible through the zenith are selected for observation, and stars both to north and south are observed.

Another way of determining the latitude, which is very commonly employed, is known as Talcott’s method. The observations are made with the aid of a zenith telescope. The latitude being approximately known, two stars are selected which transit nearly at the same time and nearly at the same distance from the zenith, one to the north and the other to the south. That which transits first is brought to the centre of the field of view, which is marked by a spider thread. The instrument is then reversed in its bearings so that it points at the same angle on the opposite side of the zenith. When the second star comes into the field, the telescope is kept fixed, and a moveable spider thread is made to coincide with the star passing through the field. The distance between the spider threads furnishes a measure of the difference in zenith distances. Half the sum of the declinations added to half the difference of zenith distances gives the latitude when this method is employed.

Various other methods have been devised for the precise determination of latitude, but the foregoing will sufficiently serve to illustrate the processes followed when the observations are made on land.

Before the invention of astronomical instruments, latitude was approximately measured by the lengths of shadows. At the summer solstice, at noon, the shadow of a vertical stick is at its shortest, while at the winter solstice it is longest. By measuring these lengths, a diagram can be made showing the altitude of the sun at noon on each occasion. Midway between these will be the altitude of the celestial equator where it crosses the meridian. Since the altitude of the Pole is equal to the latitude, the altitude of the Equator, subtracted from 90°, thus gives the latitude.

It will be noted that this gnomon experiment also furnishes a measure of the obliquity of the ecliptic. The gnomon was in use by the ancient Chinese, and it is also believed that the Egyptian obelisks which are now embellishing various cities were originally erected for the same purpose.

Determination of Longitude.—As we have imagined an observer travelling in a north or south direction in connection with the measurement of latitude, let us consider what will happen to an observer who travels only in longitude—that is, east or west. At the starting-point, he will see the Pole at a certain altitude, and the stars will perform their diurnal revolutions at a certain inclination to the horizon depending upon his latitude. If he travels towards the east, the Pole will remain at the same angle above the horizon, and he will detect no difference in the apparent movements of the stars. What then is there to indicate that he has changed his place at all? The answer is simple; he will find that the sun and stars cross the meridian earlier, and if he be 15° east of his first station they will transit an hour sooner, because it takes the earth an hour to turn through that angle. If he travel westward in the same way, the earth must turn through a greater angle to bring him back to the same star, so that the stars will appear to cross the meridian later.

The determination of longitude is accordingly based upon a measurement of the difference in the times of transit of sun or stars at the place of observation, and the place from which longitude is reckoned.

Let us take Greenwich as the start-point for our longitudes, and suppose we are in Dublin. The sun, or a star, will cross the meridian of Dublin at a certain interval after it has passed that of Greenwich, and if we measure this interval, the angle turned through by the earth in that time will determine the longitude. With a transit instrument one can readily tell the exact moment when the star crosses the meridian of Dublin, but how is one to know the exact moment at which the star crossed the meridian of Greenwich without going there?

Looking at the question in another way, let us remember that the clocks in Dublin register local time, that is time reckoned from the passage of the sun over the meridian of Dublin, while the Greenwich clock indicates times based on the transit of the sun over the Greenwich meridian. Evidently the difference of these times is the difference of longitude, and our question becomes, how to find the time at Greenwich when stationed at the observatory in Dublin.

In all modern work, the telegraph is employed whenever it is available, the two stations being directly connected. An observer at Greenwich is thus enabled to transmit a signal to the observer in Dublin at the exact moment a star passes through the centre of his transit instrument, and the latter observer then notes the interval which elapses before the same star passes the central line of his own instrument. If the signals were transmitted instantaneously, the interval elapsed from the reception of the signal to the observed transit of the same star would give the longitude as reckoned in time.

Practically, what is done is for each observer to determine his local sidereal time very accurately, with the aid of his transit instrument, and in this way to find the error of his clock. It is then only necessary to compare the two clocks, and this is done in the following way: the clock at Greenwich has an attachment by which an electrical contact is made every second, and this is switched in to the telegraphic circuit, so that the Dublin observer receives a signal every second so long as the clock is connected. These signals are automatically recorded by a chronograph, together with similar signals from the Dublin clock, and the times to which each of them corresponds is easily identified. Immediately afterwards the Dublin clock is switched into the circuit, and records its beats on the chronograph sheet at Greenwich, alongside those sent by the Greenwich clock. In this way the differences between the clocks can be very accurately measured, and the longitude can then be reckoned in degrees and minutes by allowing 15° for each hour. Before the invention of the telegraph, less accurate methods were of necessity employed. Among others the entrance of the moon into the earth’s shadow during an eclipse was noted by an observer desiring to know his longitude. As we have already seen, this occurrence is independent of the observer’s position on the earth, so that if he records the local time of the observation and compares with the calculated Greenwich time of the commencement of the eclipse, he can find his longitude. Similarly, the eclipses of the satellites of Jupiter may be utilised to signal Greenwich time to an observer situated elsewhere. Unfortunately, the shadows are too ill-defined at the edges to permit very accurate determinations in this way.

Methods Employed at Sea.—One of the most important applications of astronomy to the needs of everyday life is in enabling the navigator on the open ocean to determine the situation of his ship. Without the help supplied by astronomical predictions the sea would be truly trackless, and commerce by sea would be almost impossible.

A sextant and two or three good chronometers, together with a copy of the current “Nautical Almanac,” furnish the means of ascertaining the geographical position of a ship. With the aid of the sextant, the sun’s greatest angular distance above the sea horizon—that is, its meridian altitude—is measured, and from the known declination of the sun at the time, the latitude is deduced in exactly the same way as in the case of an observation of a star (p. 124).

The sextant also enables the observer, by measuring the sun’s altitude in the early morning or evening, to determine the local time, as already explained (p. 83). Greenwich time is kept by the chronometers, and the difference between this and the local time is a measure of the longitude. More than one chronometer is carried by a ship, for fear that a single one might fail, through accident or other causes, to give correct readings. The rate of each has been previously very accurately gauged, and by taking the average indications, Greenwich time is known with considerable accuracy.

Should the chronometers fail, or any doubt be thrown upon their accuracy, there is another method by which the Greenwich time, and thence the longitude, can be ascertained. This is the lunar method, in which the heavens become the equivalent of the dial of a clock, while the moon, with its rapid easterly movement, plays the part of the hands.

In the words of Dr. Lardner, this is “a chronometer of unerring precision; a chronometer which can never go down, nor fall into disrepair; a chronometer which is exempt from the accidents of the deep; which is undisturbed by the agitation of the vessel; which will at all times be present and available to him wherever he may wander over the trackless and unexplored regions of the ocean.”

From the known movements of the moon, its position with regard to the sun, planets, or conspicuous stars, at definite Greenwich times, can be calculated in advance, and “lunar distances” are accordingly tabulated in our nautical almanacs. We find, for instance, that the apparent distances of the moon from the star Regulus, as they would appear from the earth’s centre, were as follows on Jan. 1, 1896:—

To utilise these predictions for the purpose in hand, the observer would measure with the sextant the apparent distance of the moon from Regulus at a known local time, and he would then compute what the apparent distance would have been if his observation had been made from the earth’s centre. From the tabulated distances, he would then be able to find the Greenwich time at which his observation was made; and, as we have seen, the difference between this and local time is a measure of the longitude.

CHAPTER X.
THE EXACT SIZE AND SHAPE OF THE EARTH.

Geodesy.—We have already seen that the earth is a sphere, or of some form which differs but little from a sphere, and a rough method of determining its size, on this supposition, has been indicated. Now we have to inquire more minutely into the size and shape of our planet, for, as we shall see presently, a knowledge of these facts is essential to the adequate explanation of the various movements of the heavenly bodies, besides forming the basis of all our knowledge of the distances which separate us from the other bodies which people space. As an illustration of the importance of an exact knowledge of the size of the earth, it may be remarked that Newton’s grand law of gravitation was kept from the world for ten years, owing to an error in the generally accepted value of the earth’s radius, which was afterwards rectified by the labours of a French astronomer, Picard.

A great amount of labour has been expended in the endeavour to arrive at the true size and shape of the earth, and the name geodesy is given to the science which deals with these operations. As a secondary object, geodesy is concerned with the measurement and description of tracts of country.

An Arc of Meridian.—The measurement of the size of the earth is accomplished by first measuring relatively small parts of its surface, and then applying geometrical principles, in order to determine the whole circumference. If the earth were a true sphere, and we could measure the exact distance in miles between two places on the same meridian, a subsequent determination of the difference of latitudes of the two places would enable us to find the length of a degree, measured on the earth’s circumference. As there are 360° in a circle, the circumference would be 360 times the length of a degree, and the diameter of the earth would be the length of the circumference divided by 3·14159, this number expressing the constant ratio which exists between the circumference and diameter of a circle of any size whatsoever.

The determination of the size and shape of the earth thus involves two distinct sets of operations; first, measures of distances; and second, astronomical observations to determine the angular measurements of the arcs on the earth’s surface comprised between stations separated by known distances. When two such stations lie on the same meridian, the arc measured in this way is called an arc of meridian. We have already seen what means are available for finding the latitudes and longitudes of places on the earth, and it now remains for us to apply a yard measure, or its equivalent, to the precise measurement of the distance between places which are many miles apart.

The Base Line.—In the first instance a line of unimpeachable straightness is measured with scrupulous accuracy. The measuring-rod which has been most successfully employed is one consisting of a combination of brass and steel bars, which automatically corrects itself for changes of temperature in very much the same way that the balance-wheel of a chronometer, or of a good watch, corrects itself so as to perform its swing in equal periods at all temperatures. Several of these compensated rods are used, and they are enclosed in wooden boxes which are provided with levels and sights. When in use the outer boxes rest on adjustable trestles, and instead of putting the rods end to end they are placed a certain definite distance apart by the use of microscopes, which are themselves mounted on compensating bars. The first rod is put in position and levelled, and the others are successively placed in line with it by means of the sights. As the ground ceases to be perfectly flat it becomes necessary to raise the level of succeeding bars, but they are kept in the same vertical plane. Six bars are frequently employed in laying out a base line, and in order to protect them from extremes of temperature they are usually kept covered with long tents. In this way a distance of several miles can be measured with no greater probable error than a couple of inches, and the ends of such a measured base line are marked on metal plugs built in columns of masonry. The chief base lines measured in connection with British map construction were on the sandy shores of Lough Foyle in Ireland, 41,614 feet in length, and on Salisbury Plain, 36,578 feet long.

Triangulation.—When a base line has been accurately measured in this way, a distant object which is clearly visible from both ends is observed with the aid of an instrument called the theodolite, and the angles between the base line and the lines joining its ends with the object are very carefully determined. Thus if A B in Fig. 36 represent the base line, and C a conspicuous object several miles away, the angles C A B and C B A are measured, and then it becomes easy to determine the distances A C and B C by trigonometrical calculations. A check on the accuracy of the observations is obtained by transferring the theodolite to C and measuring the angle A C B. The sides of the triangle may then be employed as new base lines for the measurement of other distances. With the theodolite at C, another object, D, is sighted, and the angle D C A is measured; similarly, with the theodolite at A, the angle C A D is determined, and from these observations the distances of D from the points A and C are easily computed. These distances again become available for base lines, and so the triangulation can be extended indefinitely.

In a mountainous country, the sides of the triangles are often as much as 100 miles in length. Signals on the Wicklow Mountains in Ireland have been observed from Ben Lomond in Scotland and from Scafell in Cumberland. The stations are chosen so that none of the angles to be measured are very small, and in this way the chances of error are greatly reduced. Hence the triangles in the immediate neighbourhood of the base line are comparatively small, but the sides are gradually extended as the survey proceeds.

The process of triangulation forms the basis of the construction of accurate maps, and for this purpose the great triangles are subdivided by a secondary triangulation, so that the exact situations of a very great number of places are determined. These, again, serve for another set of still smaller triangles, with sides perhaps a mile in length; and finally the details are filled in by local chain surveys and draughtsmanship.

There is another point of some importance in connection with these triangulations when on a large scale. The larger triangles must be corrected for the curvature of the earth’s surface. The construction of the theodolite is such that two adjacent sides of any triangle, measured from their intersection, are referred to the same horizon; but when the instrument is transferred to another corner of the triangle, the adjacent sides are referred to a new horizon. The sum of the three angles of a triangle in these geodetical surveys thus exceed two right angles, whereas in plane triangles they are always equal to two right angles; the difference is called the spherical excess, and in the computations the observed angles have to be corrected on this account.

Thus, after an extremely laborious survey, it becomes possible to determine with great accuracy the distance between any two places whatever, and so the number of miles between two places at the extremities of an arc of meridian is ascertained. An arc of meridian extending nearly 18° has been measured in India, and another over 25° long extends from Hammerfest in Norway to the mouth of the Danube.

Exact Shape and Size of the Earth.—From the facts which have been gleaned by the measurements of arcs of meridian in different parts of the world, it is found that the length of a degree of latitude as measured on the earth’s circumference increases towards the Poles. In latitude 66° N. a degree is about 3,000 feet longer than a degree near the Equator. This means that the curvature of a meridional arc is greatest at the Equator, whence it is concluded that the earth is flattened at the Poles. The figure which best accords with the observations is the ellipse, and thus it becomes possible to calculate the polar diameter, although no arcs have been measured in the immediate neighbourhood of the Poles.

Arcs of longitude, extending between two places which have the same latitude, have also been measured and applied to the determination of the figure of the earth, and, indeed, any arcs between two places of known latitude and longitude can be utilised.

When all the facts are brought together it is found that the earth’s polar diameter is about 26 miles shorter than the average equatorial diameter, while an equatorial section of the earth is also elliptical, the diameter passing through longitude 14° E, being two miles longer than the one at right angles to it. According to the calculations of Colonel Clarke, R.E., we have the following principal dimensions:

A solid which has a shape like that of the earth, with three axes of unequal lengths, is called an ellipsoid.

A very important consequence of the ellipsoidal form of the earth is that lines which are vertical—that is, perpendicular to the surface of water—do not pass through the centre of the earth, unless they are at the Poles or at certain points on the Equator.

There is every reason to suppose that at one time the earth was in a molten condition, and in response to physical laws, such a mass of matter could not retain a spherical form when set in rotation, although the sphere would be its natural shape if at rest. This has been demonstrated by a variety of experiments.

Thus, taking it generally, the shape of the earth is very intimately associated with its rotation, and it will subsequently appear that the same holds good for the sun and planets. Those bodies which have the most rapid rotation show the greatest flattening in the direction of the polar diameter.

In addition to direct measurements of the earth, there are other ways of studying the shape of our planet. One of these depends upon observations of the swing of a pendulum at different parts of the earth’s surface; as the time of oscillation of a pendulum depends upon the force of gravity, which itself varies with the distance from the earth’s centre, it is evident that this method is a practicable one. It is true that the matter is complicated in various ways, but after everything has been taken into account, these pendulum observations indicate, not only that the earth is flattened at the Poles, but they show further that the amount of polar compression deduced from geodetical work is in all probably very near the truth.

Again, the movement of the moon around the earth is found to be subject to certain irregularities which would not exist if the earth were a perfect sphere. These inequalities being deduced from observations of the moon’s position, the amount of polar flattening necessary to produce them can be calculated, and this is found to agree very closely with the value derived from the measurements of arcs of meridian.

Different Kinds of Latitude.—If the earth were a smooth spherical body, the latitude of a place would be simply equal to the angle made by a line joining it to the earth’s centre with the plane of the Equator. Owing to the bulging out of the earth in its equatorial part, however, it becomes necessary to distinguish between different kinds of latitude. If we adopt the definition given above, the name of geocentric latitude is given to the angular measurement. Taking the earth as a smooth geometrical spheroid, and assuming it to have certain dimensions, the angle which a line perpendicular to the surface makes with the plane of the Equator determines the geographical latitude. As the line perpendicular to the surface does not pass quite through the centre of the earth, the geographical and geocentric latitude differ by as much as 11′ in mid-latitudes, although nearly agreeing at the Poles and on the Equator.

As there are no direct means of finding the direction of a line passing through the earth’s centre, or of one perpendicular to the imaginary standard spheroid, geocentric and geographical latitudes must be calculated from the astronomical latitude, which is determined by observations of the elevation of the Pole, or its equivalent. The astronomical latitude is the angle between the direction of gravity and the Equator, and is therefore to a small extent dependent upon local irregularities of the earth’s surface.

A knowledge of geocentric latitude is chiefly of use in making corrections for parallax, in order that the data calculated for the earth’s centre may be precisely corrected for the place of observation, or vice versâ, as in the case of a lunar distance measured for the determination of longitude, or in the calculation of a solar eclipse.

Variation of Latitude.—For some years past a widespread interest has been taken in the question of a possible change in the position of the earth’s axis with regard to its surface. The subject is by no means a new one, for as far back as two thousand years ago, such variations were suspected. Changes amounting to several degrees were then believed to have occurred, but it is now certain that the supposed variation was due solely to the imperfection of the observations. As astronomical science became more and more precise, even before the discovery of aberration, it became evident that if any changes of latitude were taking place at all, they must be very minute.

In its geological aspect, the possibility of great changes of latitude having occurred in the past history of our globe is evidently well worth serious investigation. Granted a sufficient change in the position of the earth’s axis, the climate of London might become Arctic, or that of Greenland tropical. From this point of view the subject has been mathematically investigated by Professor G. H. Darwin, and it appears that if only the varying distribution of land and sea indicated by the geological records be taken into account, past changes of more than about three degrees are very improbable. Admitting that at any time during the life-history of our globe the earth was sufficiently plastic to be deformed by earthquakes or other disturbances, it is possible that changes amounting to 10° or 15° may have occurred.

Opinion is perhaps best reserved as to what has happened in the past. We are on surer ground when we consider the variations of latitude which are now going on.

Many competent observers have investigated the present movements of the Pole, and it has been conclusively demonstrated that changes in the position of the earth’s axis do really occur. Dr. Küstner, of Berlin, commenced a series of observations for a different purpose in 1884, and found that some anomalous results could only be explained by supposing that the latitude of Berlin was from 0″·2 to 0″·3 greater from August to November, 1884, than from March to May in 1884 and 1885. Great interest was excited by this striking result, and steps were at once taken to test its truth. Old observations were re-discussed and compared, and new observations were made, with the final result that the movement of the earth’s axis of rotation was placed beyond dispute. It was not until Dr. Chandler attacked the problem, however, in 1891, that the nature of the changes became clear. His masterly analysis indicated that the observed variations in latitude arise from two periodic fluctuations superposed upon each other; one of these has a period of 427 days, and a semi-amplitude of 0″·12, while the other is an annual change which has ranged between 0″·04 and 0″·20 during the last fifty years. The resultant of the two movements produces changes which are seemingly very irregular in amount and of varying period, but a cycle is completed about every seven years. When the two sources of difference are at their maximum at the same time, the total range reaches about two-thirds of a second of arc. In consequence of the inequality of the annual part of the change, the apparent average period between 1840 and 1855 approximated to 380 or 390 days; widely fluctuated from 1855 to 1865; from 1865 to about 1885 was very nearly 427 days, afterwards increased to near 440 days, and very recently fell to somewhat below 400 days.