168 PROCEEDINGS OF THE CALIFORNIA 
Hydrographic surveying was reduced to a real, a practical entity, by the 
discovery of the three-point problem, by Pothenot. This problem being wide 
in its application, accurate in its determinations, and yet most simple in its 
graphic solutions, has, from the first, stood the grand central truth of prac- 
tical hydrography. But to fix a position by this problem is required, on 
three known points, two connected angles observed simultaneously. And 
with only two known signals in sight it utterly fails to fix a position. Now 
it is to remedy these defects, to fill up these gaps left open by the three-point 
problem, and thus enable the hydrographer to determine his position under 
a wider range of contingencies, that I propose the application of the two- 
point problem to hydrographic surveying. 
In determining positions of the sounding boat, equalin accuracy, and sec- 
ond only in point of usefulness to the three-point problem, is the two-point 
problem, which, with its many varied phases and fewer known points, 
greatly increases the hydrographer’s capability of ascertaining his position 
under every contingency. This problem determines any two points on an 
unknown range (or inter-range) if at each of these points are measured the 
(two) angles contained by this range and two known signals. The boat’s 
path may either coincide with the range or inter-range (see Fig. 1), or cross 
it at two or more points (as shown in Fig. 2). In the first case we can fix 
the position of the boat at any two or more instants by ‘‘ angling ”’ at those 
instants on two known signals (‘‘A’’ and ‘‘ B’’), and the undetermined range. 
When the boat only crosses the range at two or more points, its position 
can be fixed by this problem only at these points, and that, of course, by 
‘‘angling’’ at the very instants of crossing the range. The better condi- 
tioned the quadrilateral, formed by the two known points and the two places 
of observation, the better will these places be determined; and will be wholly 
undetermined when the right line, through the places of observation, pro- 
longed, traverses either one or both of the observed signals. 
Where ranges are ready prepared for us—as when adjacent to cities with 
their flag-staffs, chimneys and spires, or where the country rises into high- 
lands and mountain peaks back from the shore—the determination of a boat’s 
position by this problem is alike easy and expeditious. And even where nat- 
ure does not offer such ready prepared facilities, we can readily supply them, 
where the water is comparatively shallow, by dropping temporary spar buoys 
(a pole with rock to one end). One buoy will furnish a stern range, if we 
have another visible stationary object directly astern, or an inter-range if di- 
rectly ahead. But if there is no such stationary object visible, then continue 
the line of soundings, and drop further along a second buoy, and at the same 
instant measure the angles contained by two known signals, and the first 
buoy, and from another point on the range of these buoys, catch the angles 
between this range and any two known signals, and the soundings are deter- 
mined. By cutting on a third point on shore, from two or more of these de- 
termined positions of the boat, it can be fixed in position without visiting the 
shore or even stopping the sounding boat; other signals may thus readily be 
substituted for those swept away by storms, etc. 
The buoys thus dropped being determined fixed points, may serve as sig- 
nals for carrying a hydrographic triangulation further on out off shore. This 
