ACADEMY OF SCIENCES. 169 
problem will thus prove most useful in the survey of off-shore shoals and 
reefs, and in the location of buoys, where only two signals are in sight. And 
it may, moreover, be found especially serviceable in the surveys of those 
large shallow bodies of water which abound along the Atlantic and Gulf coasts 
of the Southern States, where the low shores and hazy atmosphere render it 
extremely difficult to keep three signals in sight. 
The geometrical construction of this problem is accomplished in the follow- 
ing obvious manner: On A B, in Fig. 3, describe segments containing re- 
spectively the angles A M Baud A P B, draw the chord B y, to cut off seg- 
ment, B A My, containing the angle, B My, and another chord A a, cutting 
off segment, A B P x, containing the angle, AP«. The points. x and y, will 
be in the same right line with M and P; join x y, which produce both ways 
till it cuts the circumferences in Mand P, which will be the required places 
of observation. 
The triogmetrical analysis furnishing the readiest means for computing 
this problem, is that known as the indirect. Thus, let any number, as 10 or 
100, represent mp, in Fig. No. 1. Then in the triangle, Amp, are known 
the angles, Amp, (=A MP), and Apm(=AP UM), with side, mp, from 
whence Am may be found. In the triangle, mp b, are’ known angles, m pb 
=MPB), andbmp (= MP), with mp, to find bm. Now, in the triangle, 
Amb, are known angle, Amb (=A MB), and the sides, Adm and bm, from 
which Ab may be found. And now, from the similarity of figures, 4b: AB:: 
mp: M P, and by like proportions any other of the required sides may be 
found. 
The two-point problem finds a ready graphic solution by laying off each 
set of observed angles on a separate piece of tracing paper, and shifting 
these two papers until the lines of sight traverse each its proper point, 
then prick the vertices of these angles on to the sheet, and they are (Zand P) 
the required points of observation. 
But a neater graphic solution, based upon very obvious geometrical consid- 
erations, is found in the three-arm protractor: with the angles measured at 
M (see Fig. 3) set off on the proper limbs of the protractoz, cause its left 
and middle arms to traverse A and B, and draw a line along its right arm. 
Shift center of protractor to some point, as m—taking care to keep A and B 
bisected by left and middle arms—and draw another line along right arm and 
y, the point of intersection of these lines, will be a point in the right line 
through the places of observation, M and P. Now, set off the angles ob- 
served at P on the corresponding limbs of protractor, bisect A and B with 
the fiducial edges of the middle and right arms. Draw line along left arm; 
shift center of protractor to some point as p, and with middle and right arms 
still bisectin's A and B, draw line along left arm, and 2, the point of intersec- 
tion of these {wo lines, will be a second point in right line through M and P. 
Draw an indefinite right line through randy. Now, with the angles observed 
at P on the protractor, cause its middle and right arms to traverse A and B 
while the true edge of its left arm coincides with line through z and y; dot 
its center, and we have P, one of the places of observation; and, in like 
manner, find-JZ, the other place of observation. 
Proc. Cau. AcapD. Sci., Vou. VI.—12, 
