the one station to the other, and the line BA from the station at B to the top of the mountain at A; in the same way the angle BCA is to be measured, when in the triangle BAC knowing the side BC, and the angles ABC and ACB, and of course the angle BAC, the remaining sides BA and CA may be found.

Again, supposing the imaginary line CD to represent the horizon, the angle DCA formed by the horizon and the line CA to the summit of the mountain, may be measured, and AD will represent the perpendicular altitude of the mountain, in which case the angle at D will be a right angle; hence in the right-angled triangle DAC, having the hypothenuse AC and the angle ACD, the perpendicular may be found (Trig. Prop. 5,) to which adding the height of the observer's eye above the surface, the total altitude of the mountain will be ascertained.

EXAMPLE IX. Fig. 9, Plate 4.

Let A, B, and C represent three steeples of a town whose positions and relative distances the one from the other are known, and let an observer at D measure the angles formed at his eye by these objects; it is desired to know how far he is from each of them.

Laying down upon paper, from the plan of the town, the positions of the three steeples, assume any point at pleasure for the observer's station, as at D, through which and the two objects A and B draw a circle: draw lines from D to A and B, and also to C, producing it until it meet the opposite circumference of the circle in E, and draw the lines AE and EB.

From the plan of the town the distances AB, BC, and CA are known, and, consequently, the angles at A, B, and C: the angles ADC ADE and CDB = EDB mea. sured by the observer at D are also known. In the triangle

BAE the side BA is given, and the angle EAB is known, for it is equal to the angle EDB, being angles in the same segment of a circle, (Geom. Prop. 10,) and standing on the same arch or chord EB; hence the two sides AE and EB may be calculated. Again, in the triangle CEB the sides BE and BC are known, as also the angle CBE, which is composed of the angles CBA which was given, and ABE now found, consequently the angle BCE may be calculated, and its supplement to 180 degrees will be the angle BCD, (Geom. Prop. 1.) Then in the triangle DBC the side BC was given, and the two angles BCD and BDC are known, consequently the angle CBD, and the remaining sides BD and CD may be found, which will give the distance of the observer at D from the two steeples at B and C.

Lastly, in the triangle EAC, the sides EA and AC are known, together with the angle EAC, which is composed of EAB already found, and BAC given in the proposition, consequently the angle ECA may be found, the supplement of which to 180 degress will be the angle ACD: then in the triangle ADC are known the angles ACD and CDA, together with the side AC; the side AD may therefore be found, which will give the distance from the observer at D to the third steeple at A.

In the preceding examples mention has often been made of the level or horizontal line: by such a line, in strictness, should be meant one at all parts equally distant from the centre of the earth; but in ordinary language, by a level or horizontal line, is understood a tangent to the earth's surface at any one place, which, on account of the magnitude of the earth, will, in short distances, have no sensible deviation from the true level or horizon.

Fig. 10 of Plate 4, represents a section of part of the earth, in which C is the centre of the globe, and the circular arch is the surface; CA is a radius, or semidiameter, drawn from the centre to the surface of the sea at the ship A; the dotted line AB is the apparent level or horizontal line of an observer in the ship, which touching the surface at the point A, and produced either towards B or in the opposite direction, falls without the circle, or above the surface; it is, consequently, a tangent at the point A, and at right angles, or perpendicular to the semidiameter, or to the axis of the earth. The point B is the summit of a ⚫ mountain raised above the true circular surface of the earth at D, and the space DB represents the elevation of the mountain above that surface, or, as it is usually termed, above the level of the sea.

Were the surface of the earth a perfect plain, the tangent at the point A would consequently coincide with every part of that plain, and be likewise a tangent at the point D; but experience shows that this is not the case; for an observer in the ship at A, on approaching the land, discoyers first the summit of the mountain at B, and as his distance from the mountain continues to decrease, the more of its elevation does he discover; whereas, had the earth been a plain, the whole elevation BD would have been perceptible at once. It is therefore evident, that as the direction of the tangents to the earth's surface at A and D must be very different, varying in proportion to the distance between the points of contact, the line of sight, or the apparent horizontal level, must also be continually changing its direction; hence the level of the point A would, if produced, extend not to the surface of the sea at the bottom of the mountain at D, but to the summit at B: and hence we have a method of ascertaining the elevation of remarkable mountains, provided we know their distance from the place of observation; as also of determining the magnitude

of

of the earth, supposing it to be a perfect globe or sphere, which in this case may readily be granted.

Let an observer at A observe the point B, the summit of the Peak of Tenerif just appearing along the surface of the sea, (the state of the atmosphere which renders this impossible is not here considered). The height of this mountain he knows to be about 15,400 feet, and that the diameter of the earth, supposing it a perfect sphere, is about 7,944 English miles, or 41,944,320 feet; consequently the radius or semidiameter will be 20,972,160 feet. In the triangle ABC are given the right angle at A, for BA is a tangent to the circle at that point, and consequently perpendicular to the radius or semidiameter AC, (Geom. Prop. 31;) also the side AC 20,972,160 feet; and the hypothenuse CB is likewise known, for CD being a radius of the same circle, must be equal to CA, to which adding the space DB, the height of the mountain 15,400 feet, the whole CB, will be 20,987,560 feet. Then knowing the two sides of a right-angled triangle, the remaining side AB may be found =729,040 feet, or about 140 English miles, which is the distance at which the summit of the Peak of Tenerif might be perceived just rising from or sinking into the sea, provided the atmosphere, would admit of vision along the surface at so great a distance, which is not the case.

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On the other hand, by knowing the distance AB, we could calculate the height of the mountain, BD, which is the difference between the hypothenuse BC and the semidiameter DC and by measuring the angle ABD, formed at the summit of the mountain by the line of vision BA with BD or BC, the vertical line, or that indicated by a plummet as tending to the centre of the earth, we might ascertain the quantity of the angle at the centre ACB, or of the arch AD, and, consequently, of the whole circumference of the globe, to which this arch bears a determined proportion.

From

From the inspection of the figure (10) it will be evident, that if two places, on the surface of the earth, are on the same horizontal line, as A and B, these must be at different distances from the centre, as AC and BC: and the method of ascertaining this difference is termed levelling, where in practice the short distances from one station to another, although in fact portions of the spherical surface of the earth, may safely be considered as straight lines, or tangents, to the curvature of the globe: then supposing the horizontal distance AB to be one English mile, or 5280 feet, the line DB which represents the difference between the distances of the points A and B from the centre of the earth will be about ,665 of a foot, or nearly 8 inches, supposing no allowance to be made for the effect produced in the apparent altitude of objects in the horizon by the state of the atmosphere: and if these calculations were continued,

would be found that the difference between the horizontal line AB and the true level AD, might be assumed as sufficiently correct in the proportion of the squares of the distances.

The following rule is near enough the truth, to be used in the practice of levelling. "Multiply the number of chains contained in the distance between the objects, whose difference of level is required, by itself, and this product by 124, a common multiplier in cases of this sort, on account of the curvature of the earth's surface; then divide this last product by 100,000, or cut off five figures from the right hand, when whatever stands on the left of the division will be inches, and the figures cut off will be decimal fractions of an inch.

The following Table of the Curvature of the Earth points out the quantity of depression of the true level below the apparent, calculated for every chain's length, or the 80th part of a mile.