Taking the circumference of the globe at 24856.148 Eng lish miles, the radius is 7911.964, or, making the circum ference 21600 geographical miles, the radius becomes 6875.499. By the former the area of the globe is 196660948 English square miles, and by the latter 148510778.4 geo graphical square miles. The area of a zone is found by multiplying these numbers by the fractional value of the zone. Thus the zone included between the parallels of and 57" is equal to 196660948 X.004815 =946922.46 English square miles,or 148510778.4x.004815=715079.39 geographical square miles. Thus also the area of Portu gal is equal to 196660948 x .000187= 36775.6 English square miles, or 148510778.4X.000187=27771.5 geogra phical square miles.
There is another problem connected with the figure and dimensions of the earth, which, though properly belonging to trigonometry, may, from its application to the present subject, and the facility with which it can be solved, be properly introduced here, viz. to find the most distant point of the globe visible to the eye at any elevation ; or to determine the extent of the visible horizon from any given point.
Let ABG, Plate CCLXV. (Fig. S.) represent the circum ference of the globe, and GB a diameter produced to E, a given elevation above B, it is required to find the most dis tant point visible to the eye at E, supposing the eminence BE to be in a level country, or on the sea coast.
Through E draw EF a tangent to ABG in D, then D is the limit of the horizon as seen from E ; the arch BD is the measure of the distance required in degrees, minutes, or seconds, and DE the tangent of that arch to the radius CB or CD. But in very small arches, as BD must always be, even though E were the summit of the highest moun tain on the globe, the tangent hardly differs from the arch itself ; therefore ED may, without any sensible error, be considered as the distance required. Now ED' =GE.BE (see BE is very small compared to GB) GB.BE nearly, and thereforeED =,./GB.11E. Hence, if d represent the diameter of the globe in English miles, f the given height of the eye in feet, that is the 5280 height in miles, and d' the distance required in miles, we have d'= d x 5280 5280 —= — XJ: If the-diameter of the globe be assumed = 7912 English miles, the formula becomes d'= Xf 1.224126 X V f. In 5280 every case, therefore, the square root of the height in feet multiplied by 1.224.126 will give the radius of the visible horizon in English miles, supposing the t ay of light to come from the verge of the horizon to the eve in a straight line, viz. in the direction of the tangent DE. This, bowel-el, is not exactly the case, the ray, by the refractive power of the atmosphere, being bent downwards, so as to meet BE at a point below E ; that is, the point 1) is visible to the eve at an elevation less than BE. From E, therefore, the hori zon extends to a greater distance than 1), and consequent ly the value old', as found by the preceding formula, is too little by a quantity corresponding to the refraction, and which is found to vary from 1- to of the whole distance, according to the state of the atmosphere with regard to weight and humidity. In a medium state, the refraction is about or .0714, which may therefore be considered as a near approximation to the truth in all ordinary cases. With this correction the preceding formula becomes d' = 1.224126x 1.0714x Vf =--1.3115Vf ; and reduced to the form of a rule, it may be stated thus : Multiply the square root of the height of the eye, in feet, by 1.3115, and the quotient quill be the radius of the visible horizon in English miles.
Example 1. Required the distance or radius of the visi ble horizon to the eye, situated six feet above the surface of the sea.
Here d'=-.1.3115 V 6 = 1.3115 X2.449 = 3.2118 English miles, the distance required.
Example 2. At what distance may a mountain 21440 feet in height be seen, the eye being on the surface of the sea.
In this case d' = 1.3115 V 21440 = 1.3115 x 146.42 = 192 English miles. Hence the summit of Chimbo razo, the highest of the Andes, ought to be seen at a dis tance of 192 miles, if its height be, as it is stated, 21,440 feet.
As soon as geographers had discovered the spherical figure and diurnal revolution of the earth, they would na turally be led to a very simple method of representing its motion and various positions, by means of an artificial sphere. We find, accordingly, that from a very early pe riod, the globe, with certain modifications, has been made use of for this purpose, and notwithstanding all the disco veries and improvements in the astronomical apparatus of modern times, it still continues to afford the simplest, and at the same time a correct illustration, of the principles of mathematical geography. We have already seen in vat way, and to what extent, the earth differs in figure from a true sphere, and how imperceptibly small the irregularities .of its surface become, when represented on a sphere six or seven feet in diameter. If the sphere be reduced to one third of this, which is more nearly the size of ordinary globes, these irregularities will totally disappear, and the difference between the polar and equatorial diameters, or between the meridian and equator, be itself inappreciable. The earth therefore, with all its inequalities, can alone be r4resented by a sphere ; and the only remaining question is, hew can the instrument be accurately constructed, and most extensively applied ? In constructing an artificial sphere or globe, the first operation is, to prepare a spherical body of wood, metal, ivory, or such other substance as may he found most con venient. The materials commonly employed, and per
haps upon the whole best adapted for this purpose, are pa per and plaster, prepared and combined by the following process : On a spherical block or mould of wood, some what less than the size or the intended globe, are laid suc cessive coverings of paper or pasteboard, attached to one another by glue or paste, till the whole is about the thick ness of Atli or moths of an inch. When completely dry, this covering is cut into two hemispheres, by which it is separated from the mould ; and the hemispheres being again placed on a wooden axis, previously prepared for the polar diameter of the globe, they are stitched together in the same position, as when attached to the block or mould. In the extremities of the wooden axis, are fixed pins of iron or other metal, which represent the poles, and by which the globe is suspended in a metallic semicircle, whose di ameter is exactly equal to that of the intended globe. In this state, a composition of whiting and glue is applied to the surface of the paper, the globe in the mean time being made to revolve, so that the interior edge of the semicir cle, which is prepared for the purpose, pares oil the super fluous plaster from the projecting parts of the surface. The whole being thus made perfectly smooth and spherical, and at the same time equally balanced on its axis, so as to re main in any position in which it may be placed while sus pended by the poles, it is set aside to dry and harden, when it is ready to receive the various circles which geographers have imagined to be described on the surface of the earth, and which we have already explained, viz. the equator, ecliptic, meridians, the tropics, polar circles, and other parallels of latitude. These circles being described by some of the methods afterwards to be explained, the va rious parts of the surface of the earth are then delineated, according to their actual situation, the position of every place being determined by the intersection of its meridian and parallel of latitude. The iron pins in the extremities of the axis, formerly used for fixing the globe in the me tallic semicircle, for the purpose of applying the plaster, are now employed to suspend it in a brass circle, of such a diameter that the globe may revolve easily without coming in contact with any part of its interior edge. This ring is called the universal meridian, because, by the revolution of the globe, it may be made to represent the meridian of any place. The frame in which the globe is placed is various ly constructed, according to the taste and fancy of the work man ; but its top or upper part always consists of a broad horizontal circle of wood or metal WNES (Plate CCLXV. Fig. 4 ) of which the interior diameter WE is equal to the interior diameter of the brazen meridian. The latter, with the globe suspended in it, passes through notches at N and S, and rests by its under edge in a groove in which it may be made to slide, so as to elevate or depress the pole at pleasure. In every position, however, one half of the globe is above, and another below the surface of WNES, which is therefore taken to represent the rational horizon. On the surface of this horizontal rim are described several concentric circles, variously divided, according to the pur poses which they are intended to serve. The largest, or that towards the outer edge, is named the calendar, being divided into 365 parts, representing the days of the year, classed under their respective months. The next repre sents the ecliptic divided into signs and degrees, and so ar ranged that each point of the ecliptic stands opposite to the day on which the sun is at that point. The names or cha racters of the different signs arc placed at the beginning, or opposite the middle of each. The innermost circle re presents the horizon divided into quadrants, two of these being reckoned from \V the west point, and the other two from E the east point, towards N and S, the not th and south points. This circle, or rather another concentric with it, but larger, is divided into 32 equal parts, representing the points or rhombs of the ma•iner's compass. The side of the brazen meridian facing the west is divided into de grees, or if the size of the circle will admit into dcgress and minutes, reckoning from the equator towards both poles on two quadrants, and from the poles towards the equator on the other two, each quadrant being numbered from 1 to 90. The circle representing the equator is also divided in to degrees in two directions, on each side of the first meri dian, which, on British globes, is that of Greenwich. The 15th, SOth, 45th, &c. degrees towards the west are marked I. II. III. &c. to facilitate the conversion of longitude, and difference of longitude into time, and the contrary. The ecliptic, which is generally made to intersect the equator and first meridian in the same point, is divided into 12 signs, each sign being again subdivided into 30 degrees, and reckoned from the first meridian eastward. The cha racters of the signs are placed at the beginning, or opposite the middle of each, as on the horizon.