APPLIED GEOMETRY. 33 
triangles. In fig. 14, the computations are to be made in the same way, and 
‘the sum of the mixed lined portions added to that of the triangles. In 
jig. 15, the parts BCL, DEK, &c., may be determined in the same 
manner. 
Levelling forms a particularly important branch of Geodesy. This 
consists in ascertaining the difference in height of two points of the earth’s 
surface, by direct measurement, and not by projection or calculation. The 
object of levelling may be thus expressed generally: to ascertain how much 
further one point of the earth’s surface lies from its centre, than another 
point. As a general rule, it is not great elevations that are here in question, 
put simply the gradual rise and fall of the ground. The instruments neces- 
sary for this purpose will be described hereafter; the operation itself is 
explained by pl. 4, figs. 16-19. In general, two methods for determining 
the difference of height of two points, may be distinguished—either to set up 
at one of the two points ( fig. 16), or between the two (fig. 17): the latter 
is, perhaps, preferable. ‘The distance between the two points whose 
difference in height is to be ascertained, must not be very great (from 1— 
2000 feet). At a greater distance, intermediate stations must be assumed, 
which the nature of the surface sometimes renders necessary for slight 
distances. Thus, in fig. 18, the difference of elevation between A and J is 
to be ascertained by means of the intermediate stations, C, IX, G, assumed 
in the lines A, J; four levellings are here required. In fig. 19, this difference 
between A and D is determined by means of the two intermediate stations, 
B and C. 
It will be necessary to add, in conclusion, a few words with respect to 
topographical drawing. ‘This consists in representing portions of the earth 
upon paper, in their natural appearance. A topographical drawing differs 
from a chart in its much larger scale, which admits of the insertion of more 
details. While, for ordinary maps or charts, the scale rarely exceeds 
suse Of the natural size, in special plans for economical or military purposes, 
20000 
it may amount to z;4,;, so that one line, or ;2, of a foot in the drawing, 
2500? 
would represent 25 feet of ground. 
A topographical drawing represents not only streams, roads, houses, 
forests, &c., but also mountains and valleys; and this in such a manner 
that from the drawing the steepness of the declivities may be ascertained. 
This is done, according to the almost universally adopted method of the 
engineer Lehman, by means of rectangular pen strokes, made side by side, 
in such a manner that the amount of black is to the amount of white, as the 
given angle of inclination to 45° minus the same angle; consequently, a 
horizontal surface appears entirely white; that inclined at an angle of 45°, 
entirely black; at 5, 10, 15, 20° of slope, the breadth of each black space is 
respectively 4, 2, 1, 4, of the white interval succeeding; while at 40, 35, 30, 
25 degrees in succession, the reverse order takes place. This method is 
not calculated for slopes of from 45° to 90°, for the simple reason that they 
seldom occur, are always much broken in their declivities, and are entirely 
impracticable for military purposes, which the inventor had chiefly in view. 
Figs. 58-60, on pl. 5, are intended to elucidate the preceding remarks. 
ICONOGRAPHIC ENCYCLOPADIA.—VOL. I. 3 33 
