III. Common way of making globes. BOOK following Books, it would be premature to treat here more at length of the rules to be observed in choosing the best, and in transferring them to the globe with the greatest exactness. We shall only remark, that the method of delineating geographical outlines immediately on a ball of copper, wood, or any other matter, is only employed by lovers of the science who wish to join instruction with amusement, or by geographers who are particularly employed by some great nobleman. The globe-makers use a method less tedious, less expensive, which allows them to multiply copies. They have a general map of the world drawn and engraven, and distributed into slips, that is, spherical segments, with which they cover the ball destined for a terrestrial globe. The way of tracing these slips will be indicated in its place. Uses of the globe. Distance The first use that is made of the globe, is to determine the distance from one place to another. The shortest of places. distance of two points on the sphere, is measured by the arc of the great circle which joins them; and as all great circles are equal, the degrees of any of them contain the same number of itinerary measures as those of the meridian. We take therefore with a compass the opening of the arc comprised between the points proposed, and carry it to the meridian or the equator, which are graduated. If, for example, the arc comprised between two places marked on the globe, and brought to the meridian, contain 10° 45', we shall have the shortest distance between these points in itinerary measures, by converting the degrees and minutes into marine leagues of 20 to a degree. We first obtain 200 leagues for the 10°, and each minute being equivalent to a third of a league, or a nautical mile, the 45' will give 15 leagues: thus the total result will be 215 marine leagues. In delicate operations, however, it is better to use calculation, which gives a more precise result. Let us consider, for example, the spherical triangle APL, Fig. 6, formed by the meridians AP and PL of the places A III. and L, whose distance we require, and by the arc of the BOOK great circle AL, which joins them. In this triangle we know the sides AP and PL, which are the distances of the points A and L from the pole P, or the complement of their latitudes, and the angle APL, which is measured by their difference of longitude; the rules of spherical trigonometry will give us, in degrees and parts of degrees, the side AL, which we can convert into itinerary measures. In the case where the places A and L are in different hemispheres, one of their distances from the pole will be greater by 90° than the latitude of the place. itself.* If the places of which we wish to know the distance have the same meridian, it is only necessary to take the difference of their latitudes, and to convert it into itinerary measures. A difference of some minutes in longitude has no sensible effect on the result; thus, we should hardly mistake more than a league in measuring the distance from Some of our readers will be pleased, perhaps, with an example of this sort of calculation. The distance from Paris to Philadelphia is required. Longitude of Philadelphia, 77° 36' 0" W. Long. of Paris, 0° 0' 0". Difference of Long. A = 77° 36′ 0′′. Lat. N. of Paris, 48° 50′ 15′′; therefore the complement B = 41° 9' 45". Lat. N. of Philadelphia, 39° 56' 57"; therefore the complement C=50° 3′ 3′′. Multiply the tangent B by the cosine A, you will have a tangent which we shall call x. It must be subtracted from C if A is below 90°, and added if A is above. There results the quantity we shall call y. Now, as the cosine is to the cosine B, so the cosine y is to the cosine of the distance required D. The calculation is made by means of the Tables of Sines. See the trigonometry, and the general formulas in Puissant, Traité de Geo desie, art. 89. Comp. art. 30. BOOK Paris to Algiers on the meridian of Paris, though it is 41' more to the west than that of Algiers. III. Remarks on the measures of distances. It would be a great error to take the difference of longitude in degrees of two places, situated on the same parallel, for the measure of their distance; this can only be done when the places are situated on the equator, which is a great circle; but its parallels being small circles, the radius of which diminishes as we approach the poles, it follows from the principle stated above, that the absolute length of their arcs does not give the true measure of the shortest distance from the extremities of those arcs; this distance can only be measured by a great circle passing through the two extreme points. For as the radius of the parallel is shorter than that of the great circle, the arc of the parallel must necessarily have a greater curvature than that of the great circle comprised between the same points, and is consequently longer. Here is a striking example: Petersburgh is almost under the same latitude as the isle of Kodiak, in Russian America; the difference of longitude is about 180°, equivalent under this parallel to 1800 marine leagues; but the shortest distance between these two places is, counting on a meridian that is almost common to them, 60 degrees of latitude, equivalent to 1200 leagues. It is true that, to take advantage of this, it would be necessary to cross the eternal ice of the pole. Thus, in geography as in politics, the straight road is not always the most advantageous. It is necessary, therefore, in many cases, to measure the distances on the parallels, and, consequently, to know exactly the value of the degrees of longitude marked on the Law of the parallel circles. The globe renders the diminution of these the degrees degrees towards the poles sensible to the eye; our tables decrease of of longi tude. indicate it in detail:* But we should know the mathematical principles of it. The length of the degrees marked on the parallels is proportionate to the radii of the circles ; but the radii of the equator, and of its parallels, are per See the Tables annexed to this volume. pendiculars let fall from the different points of the meridian on the diameter of each of these circles, as in Fig. 6. the lines EC and HK. Consequently, if we take the radius EC for the length of the degree of the equator, and if we divide it into twenty parts, representing marine leagues, the number of these parts which the radius HK of the parallel M may contain, will indicate the value of the degree of this parallel in leagues. Hence it results, that, to determine the length of the degrees on each parallel, we have only to describe on a line EC, which represents the length of the degree of the meridian, or of the equator, a quarter of a circle EP, divide it into degrees, and let perpendiculars fall from each point of division on the radius CP; these lines will mark the respective lengths of the degree of the parallel for each latitude. As the line HK is the sine of the arc PH, and the cosine of the arc EH, one of which indicates the distance from the parallel HM to the pole, and the other the latitude of that parallel, it is evident that, taking for unity the degree of the equator, that of any parallel whatever will be the cosine of the latitude given by the trigonometrical tables. For example, the latitude of Paris is 48° 50', and the cosine of this angle 0.658 of the radius; multiplying this number by 20 marine leagues, we have for the value of the degree of the parallel 13 leagues. In the latitude of Petersburgh, or 60°, the degree of longitude is reduced to 10 leagues, because the cosine of 60° is the half of the radius. BOOK III. the points C rizon. We have mentioned what is to be understood by north Relation of and south, east and west; it is by studying the globe atten- places to tively that we come to understand perfectly the value of of the hothose terms. Two terrestrial points, situated under the same meridian, are directly north and south of each other, and all the intermediate points, that is to say, all the points of the line of distance, are equally north and south of each other, and all reciprocally on the same point of the compass. In like manner, any two points whatever, taken under the terrestrial equator, are directly east and west of each other, BOOK and all the intermediate points are equally so, and are reciprocally on the same point of the compass. If we take two places which are neither under the same meridian, nor under the equator, whatever be otherwise their relative position, none of the intermediate places will be, with respect to the other places, on the same point of the compass. For the arc of a great circle which measures the distances, is an arc of a vertical circle which passes by the zenith of the two places in question; but every vertical circle which is itself neither a meridian, nor perpendicular to the terrestrial meridian, (like the equator,) will cut all the intermediate meridians under angles unequal among each other. But it is these angles of position which determine the point of the compass on which a place is relatively to another. Therefore, as all the intermediate places between the two places in question will offer angles of position unequal in degrees, each of them will be on another point of the following place from what the preceding place was from it. Thus, in following the shortest route, between two places situated out of the equator, and under different meridians, the point of the compass would change at every step. This is demonstrated by Fig. 17, where P Ep represents a meridian, EGI i the equator, HLQ a parallel, and HIK i the great circle perpendicular to the meridian in H. We perceive, also, that all the great circles perpendicular to the same meridian, meet in two opposite points I and i, which are the poles of that meridian. These great circles must, therefore, continually approach each other; and it is only in a very small space, on each side of the meridian PE p, that the circles IE i and IH i can be considered as parallel with each other; and hence, too, it can only be in a small extent that the lines east and west, or the perpendiculars to the meridian, can be and west. considered as parallel.* Lines east As the great circle IHK, perpendicular to the meridian p' EP, cuts the other meridians under angles different for See afterwards Projection of the Maps of Cassini. |