FIG. 244. S=s, C=c, and make y=1; hence, Ss + Cc=S2 + C2 = 1; therefore the great Cc C12 =cos. BЛe. Hence, in V. When the moon is in the equator, S= 1, C=0, and the altitude of the tide =s❜m, which therefore varies as the square of the cosine of latitude. this case, at the pole there is no tide. VI. The height of the tide when the moon passes the meridian = Ss + Cc2 × m and when the moon is at the opposite meridian the height is Ss+ Cc2 × m. Hence, when the moon is in the equator CO, and the height of both tides are equal. To a place on the north of the equator, when the moon has south declination, C becomes negative, and the latter tides are the greatest; but when the moon has north declination, C is positive, and the former is the greatest. Hence, to us in this case, the high tide is greater when the moon is above the horizon than when below. In all cases, e is nearer to or further from C, acCc cording as y - is positive or negative. The difference of the two tides S's VII. The height of the two tides when the moon passes the meridian being Ss+Ccx m and Ss+Ccx m, the mean height is 5's+Ccx m. VIII. Hence, the same north and south declination of the moon give the same mean altitude. This is confirmed by observations. IX. In latitude 45°, SC; hence, the mean altitude = 1/2 × s2 + c2 × m =m; therefore whatever be the declination of the moon, the mean altitude is, in this latitude, always the same. Hence, in our latitude, the mean altitude will vary but very little. X. Under the equator, the mean height = S'm, which therefore varies as the square of the cosine of the moon's declination. 1207. As the tides rise from the collecting of the waters on the whole surface of the main sea, if there be any quantities of water separated from it, the variation must be proportionally smaller. For if ro be a small surface of water detached from the rest, its surface will put on the figure as similar to ď't, consequently the variation ar from the mean altitude must be very small. Hence, there have never been any tides observed in the Caspian sea; for from the dimensions of that sea, the greatest altitude will not be above 1 inches at the eastern and western extremities, according to M. de la LANDE, who has corrected an error made by M. D. BERNOULLI in his computation; and it is manifest, that the middle of the sea will never be affected. Very small tides have been observed in the Black sea, which, from its connection with the Mediterranean sea only by a very small passage, may be considered as a de tached sea. The Mediterranean sea is connected to the main sea only by a narrow passage at Gibraltar, so that only a small quantity of tide from the open sea can flow in, and the sea itself is not large enough to produce any very sensible tides; accordingly we find the tides there to be but very small. The best observations are those which have been made by M. le Chevalier d'ANGOS at Toulon, from which it appears, that the tides produce a variation of about one foot in the height of the water. There were frequently greater variations, but those appeared to arise from high winds. 1208. The general phænomena of the tides from observation agree very well with the conclusions deduced from the theory of gravity; indeed much more accurately than could have been expected, when we consider how many circumstances there are which take place, and which cannot be reduced to computation. The theory supposes the whole surface of the earth to be covered with deep waters-that there is no inertia of the waters--that the major axis of the spheroid is constantly directed to the moon, and that there is an equilibrium of all the parts. But the inertia of the waters will make them continue to rise after they have passed the moon, although the action of the moon begins to decrease, and they come to their greatest altitude in the open seas about three hours after, at which time there is not a general equilibrium, but the waters rise and fall by a reciprocation; hence, the longest axis is not directed to the moon, nor is the figure a perfect spheroid-The waters have not a free motion on account of the shallow places, rocks, islands and continents, the force of currents and winds; also, as the waters approach the equator where the earth has a greater velocity about its axis, they must necessarily be left behind and obstruct the regular motion of the water when it moves from west to east, but conspire with that from east to west. All these circumstances must affect the measures of the phænomena as deduced from theory; it may however in many cases give the relative measures without any great error, so that by accurate observations once made on their absolute quantity in some one particular case, the measures, in all other cases, may be ascertained to a considerable degree of accuracy. 1209. If a place communicate with two seas, or has two inlets to the same sea, two tides may arrive at that place at different times, and produce various phænomena. An instance of this kind takes place at Batsha, a port in the kingdom of Tunquin in the East Indies, in 20°. 50′ north latitude. The day in which the moon passes the equator the water stagnates; as the moon recedes from the equator towards the north, the water begins to rise and fall once a day; and it is high water at the setting of the moon, and low water at its rising. This daily tide increases for six or eight days, and then decreases for the same time by the same degrees, and the motion ceases when the moon has returned to the equator. When it has passed the equator, and approaches the south pole, the water rises and falls as before, but it is now high water at the rising, and low water at the setting of the moon. Sir I. NEWTON thus accounts for this phænomenon. There are two inlets to this port, one from the Chinese ocean between the continent and the Manillas, the other from the Indian ocean between the continent and Borneo; and he supposes that a tide may arrive at Batsha, through one inlet, at the third hour of the moon, and through the other inlet, six hours after; and supposing these tides to be equal, one flowing in whilst the other flows out, the water must stagnate. Now they are equal when the moon is in the equator; but as the moon begins to decline on the same side of the equator with Batsha, the diurnal tides exceed the nocturnal (as appears by the foregoing principles), so that two greater and two lesser tides must arrive at Batsha by turns. The difference of these will produce a motion of the water, which will rise to its greatest height at the mean time between the two greatest tides, and fall lowest at the mean time between the two lowest tides; so that it will be high water about the sixthi hour at the setting of the moon, and low water at its rising. When the moon has got to the other side of the equator, the nocturnal tide will exceed the diurnal; and therefore the high water will be at the rising, and low water at the setting of the moon. These principles will account for other extraordinary tides which are observed in those places whose situation exposes them to such irregularities. etwo CHAP. XXXIX. ON THE PRINCIPLES OF PROJECTION, AND THE CONSTRUCTION OF GEOGRA- Art. 1210. THE projection of an object is its representation upon a plane, and is formed by drawing lines from the eye to the plane through every point of the object; and according to different situations of the eye, the object and the plane, the representations will be different. The projection in order to be perfect, should be a perfect representation of the object, that is, the proportion and relative situation of all the parts of the figure should be the same as in the object; but in the construction of geographical maps, this is not practicable; it being impossible to give a true representation of a spherical surface upon a plane, retaining the true proportion of the figures, magnitudes and positions of the countries, with the relative degrees of latitude and longitude. We will first show the principles of the different projections, and then apply them to our present purpose. On the Orthographic Projection. 1211. If the eye be supposed to be at an indefinitely great distance, so that all the lines drawn from it to the object may be considered as parallel, and also perpendicular to the plane of projection, the projection is called Orthographic. pro 248. 1212. The figure of a straight line AB is a straight line in the projection. FIG. For draw AE, BD perpendicular to ay the plane of the projection, and join ED, and it will represent the intersection of the plane passing through EABD with the plane of projection; draw mn perpendicular to ED, and n is the jection of m; thus it appears that ED is the projection of AB. Draw AC parallel to ED, then ACDE is a parallelogram, and AC ED; AC may therefore represent the projection of AB. Hence, if we want to make the representation upon a plane at a distance from the body, it will be all the same if we suppose the plane to touch the body, the parallelism of the plane remaining the same. 249. 1213. The figure of the projection of a circle is an ellipse. For let ABC be FIG. a semicircle conceived to be inclined to the plane of the paper, which we will make the plane of projection; draw BE perpendicular to that plane, and ED, FIG. BD perpendicular to the diameter AC, then "ED is the projection of BD (1212). By the property of the circle, AD × DC=BD'; but as the angle BDE is constant, being the inclination of the circle to the plane of projection, BD is to DE in a constant ratio, namely, that of radius to the cosine of BDE therefore DE varies as BD, and consequently DE varies as BD"; hence, AD × DC varies as 'DE, which is the property of an ellipse the curve AEC is therefore an ellipse. And by the last Article it appears, that the projection will be the same, at whatever distance the circle is from the plane of projection. Let O be the center, and draw QP, OQ perpendicular to AC, then OQ is the minor axis; hence, to the radius of the circle, the minor axis is the cosine of the inclination of the circle to the plane of projection. If the circle be parallel to the plane of projection, the projection will be a circle equal to it. 1214. By the property of the ellipse 'and circle, the area ABC the area AEC:: BD: ED;: rąd. cos. BDE; hence, the area of the circle will be dis minished by this projection in the ratio of radius: cosine of the inclination of the plane of the body to the plane of projection. And this must manifestly be true whatever be the form of the body ABC (considered as a plane), because every line in the body is to its corresponding line in the projection in that ratio. Also, the projection is not similar to the body. Hence, equal parts upon the surface of a sphere will not be projected into parts either equal or similar. 1215. If ABC be perpendicular to the plane of projection, E and D coincide, and D is the projection of B; thus the circle ABC is projected into its diameter; the arc AB is projected into its versed sine AD, and BP is pro jected into DO, which is equal to the sine of BP, or the cosine of AB. If AB=60°, then BP-30°, and AD-DO; these two arcs therefore, one of which is double of the other, have their projections equal. This projection is used in the construction of solar eclipses, CHAP. XXIII. On the Stereographic Projection. 1216. Let an eye be situated any where upon the surface of a sphere, and from it draw a diameter, and perpendicular to this diameter draw a great circle; then if all the circles in the hemisphere opposite to the eye be projected upon that great circle by lines drawn to the eye, the projection is called Stereographic, and the point opposite to the eye is called the Pole. 1217. Let EQPR be a sphere whose center is O, E the place of the eye, 250. draw the diameter EOP, and QOR perpendicular to it, representing the plane of projection; draw ECA, EDB, and join BO. Now DO is the projection of PB; but DO is the tangent of the angle DEO BOP. Hence, the pro |