ACADEMY OF SCIENCES. 7 
yet approaching on account of the progress of the boat; then with the second 
index glass he makes the direct and reflected images of the middle and right 
hand objects coincident, and keeps them coincident with tangent screw 
until the first two objects become coincident, then clamps, and he has the two 
angles observed at the same instant—and also has them set off on the proper 
limbs of the instrument simultaneously with, and by the same effort that 
measured the angles. And hence after measuring two connected angles with 
this instrument, we have only to lay it down on the ‘‘ Field Sheet’’ (which 
should always be spread on a board before the observer in the boat), and 
shift it until the fiducial edges of the three protractor arms traverse the three 
points (representing the signals observed upon), and the center of the in- 
strument will then occupy the relative place of the observer; now dot the cen- 
ter, and the position is plotted, without any of those tedious transfers of 
angles from the limbs of sextants to the limbs of the protractor, which are un- 
avoidably incident to the execution of practical hydrography with the forms 
of sextants and protractors now in general use. 
However, with the hydrographer, it is necessary to read the angles off of 
the instrument and record them for future reference and closer plotting on 
the ‘‘ Office Sheet.”’ 
The angles observed with the Protracting Sextant, or any other reflecting 
instrument, are measured in the plane of the objects. If this plane be in- 
clined to the horizon and a result rigorously accurate be sought, the angles 
of elevation of each station above the horizon should at the same time be 
observed to afford data for reducing the hypotheneusal to the horizontal 
angle. But this reduction may be neglected in all cases where the difference 
of elevation of the objects does not exceed two or three degrees, and when 
the observed angle is larger than (the minimum angles allowed in determin- 
ing a boat’s position by observations from the boat), twenty or twenty-five 
degrees—for the reduction to the horizon would, in such cases, deal with 
quantities more minute than the amount of error to which the measures of 
all angles observed at an unstable station are liable. When the difference of 
the objects is considerable, an ideal vertical line may be drawn from the 
highest object downward to an elevation corresponding to that of the lower 
object, and the angle measured between this vertical line and the lower object 
—this with some experience and correctness of eye, will give results sufficiently 
near the truth, z.e., within the limit of the errors of plotting. Objects very 
close should not be observed on account of the parallax of the instrument. 
The Protracting Sextant should have supplementary attachments (such as 
were described by the writer before the Academy, February 16th, 1874), so 
that angles between one hundred and forty and one hundred and eighty 
degrees may be measured with equal facility with those of smaller magnitude. 
But these larger angles cannot be plotted in the usual way, for they are too 
great to be set off at the same time on the limbs of the instrument because of 
the jamming of the movable protractor arms; now, under this contingency, 
if we have no tracing paper, and don’t wish to sweep the circles of position, 
then we may use the following easy and accurate method of plotting by 
supplementary angles, viz.: Suppose A, B and C, the left, middle and right 
hand objects on which are measured two angles, too large to be set off on the 
