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the altitude required. If the object be inaccessible, but still such that the difference of the lengths of its shadows, taken at two different times, can be ascertained, the altitude may be found nearly the same as in the last example. Thus, make ab= a, ac = b, and the unknown length of the shadow = a ; let the second shadow of the rod = b, and the second shadow of the tower = a + d, the height of the tower = y; then, by the preceding proportion, 1st operation b : a = a + 2d operation b : : a = r + d : y
the altitude of the object required. When the object is inaccessible, that is, when the distance CA cannot be measured, two such operations as that above must be employed. Thus, Let ED = a, DC = d, and the unknown distance CA = a ; also let d' represent the analogous distance D/C in the second operation, and a + c the second distance CA, that is, making c = the distances between the two situations of the mirror; and let the required height of the object = y. Then substituting the above letters in the preceding proportion, we have, 1st operation d : a = r : y 2d operation d' : a = x + c : y Whence, by - - - ...}} dard: a = c : y ac and consequently, y= d//
To measure an accessible or inaccessible Object, by the Geometrical Square. (Plate I. fig. 4.)
Having fixed the instrument at any place C, turn the square about the centre of motion D, till the top of the object B is perceived in
9 the direction of the sights placed
on the side of the square DE, and
to which adding the height of the
tude sought. In the case of an inaccessible
object, two observations must be made similar to that above, in which the only variable lines will be GF and CA. Let, therefore, the side of the square = s, and the variable part GF = a, in the first observation, and as in the second ; also put the unknown distance = r in the first case, and a + c, in the second, so that c will be the distance of the observer’s two stations, and make the required height of the object = y. Then on the same principle as those above,
1st operation s : a = a, : y 2d operation s : a) = a + c : y Whence, by equality,
To measure an accessible Object Trigonometrically. (Plate I. fig. 5.) Let AB be the object of which the altitude is required.
maticians, soon after the experi
At any convenient station C. with a quadrant, theodolite, or other graduated instrument, measure the angle of elevation ACB ; then, having also measured the distance CA, we have, from the elementary principles of trigonometry, as
12°3686132 Log. radius . . . . . 10:0000000
Log. BA = 23367 . . 2.3686132 To measure an inaccessible Object by two Stations. (Plate I. fig. 6.)
Let AB be the object of which the altitude is required.
Take the angles of elevation at the two stations D and C, and mea sure the distance between them. The angle DBC = the difference of the two measured angles RCA and BDA, (Euc. prop. 16, book 1.), which angle therefore becomes known. And by trigonometry, as
sin. DBC : P. = sin. CDB :
DC X sin. CDB
sin. DBC CB.
Again, as radius: CB = sin. ACB :
CB X sin. ACB BA
Rad. -- a--a,
the altitude required.
Or substituting, in the second expression, the value of CB, we
BA Rad. sin. DBC. Or by Logarithms.--From the sum of the logarithms of the terms in the numerator subtract the sum of those in the denominator, and the remainder will be the logarithm of the required altitude B.A. To measure. Altitudes by the Barometer.—The application of this instrument to the measuring of al titudes, suggested itself to mathe
ment of the Puy de Domme, which was executed in order to confirm the Torricellian experiment. Let EAQ, (Plate I. fig. 7.) represent the surface of the earth, and A, B, C, D, &c., a column of the atmosphere. Conceive this to be divided into a number of equal and indefinitely small parts, AB, BC, CD, &c. in each of which we may suppose the density to be uniform, because they are indefinitely small. Now since the density o the air is always directly as the compressing force; and the compressing force being in this case the pressure of the superior strata, we shall have the density of the air in any of those parts, as the weight of the column of atmosphere above; that is, if P represent generally the pressure, and D the density at any place, and Pl and D' the pressure and density at any other place whatever, we shall
P : D = P1 : DI; and, consequently, the pressure is to the density in a constant ratio, which may be represented by n : 1; that is, P : D = Pl: D = n : 1;
Then if d to represent the density of the atmosphere at the surface of the earth, we shall have