Chap. VIII.

GEOMETRY.

803

thrown aside, and the entire attention devoted to minutuc: in taking the levels of avalley through which a stream discharges itself into the sea, and where there are many millweirs and their fall can be ascertained, it will be of considerable service previously todecide the level of the river’s mouth, its entrance into the sea, and also the slope of its bed,which may be calculated by adding the several falls together, and taking an average in-clination per mile of the stream : although this is not a very accurate way of proceeding, itwill serve as a check to gross errors.

Inland districts are not necessarily higher than the level of the ocean, and in the fens ofLincolnshire and elsewhere, the slope of the streams is so inconsiderable as to be hardlyperceptible, the fall being frequently less than 2 inches to the mile; the Thames fromLechdale to London bridge is 146j miles, and in this distance the rise from low-watermark is 248 feet, consequently its fall averages about 20 inches per mile, though for a partof its course the slope of the bed is not more than a foot. Surveys made through a countrywhere the falls of the rivers are known may befrequently rendered more accurate by com-paring the levels as taken with the instru-ment with those observed in the manner al-luded to.

Trigonometry teaches the method of measuringall kinds of distances as well as heights bytriangles; it enables the engineer to ascertainbow many feet or yards there arc in a rightline from one place to another ; to measure thebreadth of a river ; the length of a line of for-tification ; the opening of a breach; the distanceof a fort even when water intervenes, or thesurrounding country is inaccessible: it alsoenables him to measure the heights of hills,mountains, and buildings of every kind withgreat precision: formerly these two branchesof trigonometry were called longimetry andaltimetry.

By the first was understood the method ofmeasuring in a right line from one place toanother, as to find the width of a river, or thedistance of one building from another, as thedistance of the castle A from the church B :it is evident from the stations at G and H,two angles may be measured ; that by computa-tion afterwards the distance may be knownaccurately.

By the second the height of the tower at Cfrom the point D may be found from the sta-tion points where the instrument is placed.

For suppose the circumference of a circledivided into 360 equal parts or degrees, theseeach into minutes, and these into seconds, it willbe easy to measure the angle taken by thedemicircle with the points C and D; its mag-nitude may be expressed by degrees, minutes,and seconds; this division of the circle, calledusually the sexagesimal, was that adopted bythe ancients. Supposing the point occupied by

the demicircle to be marked by the letter B, the ratio that C D bearsto C B is called the sine of the angle, and the ratio of B D to B C thecosine of the angle, or, as they are usually written,

DC . * BD A

B C = Sm A ’ BC = ° 0S A

The ratio which the sine of an angle bears to the cosine is called thetangent of the angle ; the inverse of the ratio the cotangent; the ratio ofunity to the cosine of an angle the secant, and that of unity to the sinethe cosecant. The difference between unity and the cosine is termedthe versed sine, and the difference between unity and the sine of an an-gle the coversed sine.

The depth of all places may also be found, as the depth and widthof all ditches and cavities, as that of E F, and the breadth at thebottom.

Fig. 930.

Fig. 940.

Fig. 911.

3 F 2