APPLIED GEOMETRY. 29 
The distances, aa’, bb’, cc’, &c., from a, b, c, &e., are next to be measured at 
right angles to the base (which may be done by the eye, or more accurately 
with a string divided as the numbers 3, 4, 5, of a right angle). The 
measured distances of both kinds, the abscissas, Aa, Ab, Ac, &c., as also 
the ordinates, aa’, bb’, &c., are traced on paper on a reduced scale, and the 
points, a, b, c, &c., united. The accuracy of the outline will be evidently 
in proportion to the number of abscissas and ordinates measured. The 
outline may sometimes be such as to render it advisable to stake off 
two lines, as in fig. 1, whose relation to each other must be known. 
In the second place, the distance between two points may often be deter- 
mined even when no direct measurement is possible. Three principal 
cases are here to be distinguished: 1. When the distance between two 
points cannot be measured directly, but only that from a third point to 
each of these two (fig. 2). In this case, we measure the distances, CA, CB ; 
continue the prolongations of these lines beyond C, towards D and E; take 
CE=CA, and CD =CB, or the reverse ; the measured distance from D to E 
will be the same as that from A to B. It is much more convenient, when the 
prolongations of the lines, AC and BC, cannot be made equal to them, to take 
a certain part of the distance, as one fourth Cd=4Cb, Ce=4Ca; then 
de will be the same fraction of AB, or de=+AB, or AB=4de. 2. When 
we can reach only one of the two points whose distance from each other is 
desired, as in pl. 4, fig. 3. Here we assume any point, C, at pleasure, from 
which B may be reached in a straight line, measure CB, and continue the 
prolongation of this line to D, so that CD =CB, and then in the direction 
DE, making the angle CDE=CBA. (To effect this take on BA and BC, 
any portions, Ba, Bb—five feet, for instance—measure the distance ab, make 
Dd= Bb, and with a beam compass, from d as centre, with ab as radius, 
describe an arc, intersecting another arc from D as centre, with aB as 
radius. Stake off the line DE through e, and we shall then have the direc- 
tion of the required angle.) We then continue in the direction DE or De 
until we reach a point, E, which lies in the same straight line with C and A, 
as ascertained by two staves. The distance DE will then equal AB. Fig. 
4 represents another method of attaining the same result: Take on AB any 
point, C, between A and B, and then a point, D, whose distance from B and 
C may be directly measured; continue the lines CD and BD beyond D, 
making DF=CD, DE=DB. Finally, draw EF, and continue it to a point, 
G, in the same straight line with Dand A; EG will be the distance required. 
We may here also, instead of the whole line, BD, CD, take a fractional part 
of these prolongations; thus, if we make De =}DB, and Df=1CD, then, 
if g lie in a straight line with AD, as well as with ef, eg will= AB. 
3. When we can reach neither of the points, A, B (fig. 5). In this case, 
many methods may be employed; the one represented is perhaps the 
simplest: lay off the line CD approximately parallel to AB, and on it 
take cD equal to an aliquot part, as + of CD; make Dca=DCA, and 
Dch = DCB, taking the distance ca so that a may be in the line AD, and Dd 
so that b may be in the line BD, then abd will, in our figure, be ¢ of AB (the 
line cb is not represented). 
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