176 CAPTAIN A. E. CLAEKB ON THE COMPARISON OE ENGLISH AND FOREIGN 
sufficiently direct, and judging by the expressed probable errors, should be very accu- 
rate. Unhappily, however, there is no information as to the precision of the compa- 
risons made by M. Arago between either F s or F s and the toise of Peru. But it is to 
be remarked that, if we eliminate N and F s between the equations (9), (11), and (13), 
we get 
T 10 =C— 0'-00086, 
as the result of M. Struve’s comparisons ; while from General Baever’s, (10) with (14), 
T 10 =C— 04)0099, 
the difference between these two entirely independent values of the Prussian toise is 
only CF00013, or less than the six millionth part of a toise. This shows that Arago’s 
assigned lengths of F s F 4 are at any rate admirably consistent. 
We must now explain how the toises a bouts have been compared with the toise a 
traits. Suppose for a moment a cube of steel, one-eighth of an inch side, its faces 
polished, and a fine dot engraved on one of the faces at about one-hundredth of an inch 
from one of the edges and exactly opposite the middle point of that edge. Suppose the 
toise lying horizontally, and consequently its terminal planes in a vertical position, and 
let a cube as described above be applied against each end of the toise, the face carrying 
the dot being uppermost and horizontal ; then the distance T + <r between the dots when 
so held is about two-hundredths of an inch greater than the toise. Next let the cubes 
be placed in contact under the microscope, and the distance s between the dots measured ; 
we shall then, by subtracting this quantity, know the exact length of the toise. But the 
mechanical difficulties to be overcome in this theoretically simple arrangement are found 
to be very great. After numerous experiments in different ways, the following modifica- 
tion was adopted : suppose a sphere of steel, three-quarters of an inch in diameter, to be 
cut by two parallel planes, one-eighth of an inch apart, on opposite sides of and equi- 
distant from the centre. Taking the central segment, let it be laid on a horizontal 
plane, and cut in two along a diameter, leaving two semicircles ; next let these two 
pieces, without removing either of them from the horizontal plane, be placed so that 
their curved surfaces shall come in contact, while their bases or semidiameters are 
parallel and at the maximum distance apart ; then the common tangent plane at the 
point of contact will be a vertical plane, even if there should have been any error in the 
cutting of the sphere, so that one of the planes was nearer the centre than the other. 
Next suppose each of these semicircles to be placed An and fastened to a carefully 
planed rectangular plate of steel, say 4 inches long, the diameter of the semicircle being 
perpendicular to the length of the rectangle, and the curved surface projecting slightly 
beyond the end of the plate : suppose we have the means of levelling this plate, of 
raising or lowering it small quantities, of giving it a small motion in the direction of its 
length, and also in the direction perpendicular to its length, and lastly of giving it an 
azimuthal movement ; then it is clear that we have absolute command as to position 
over the semicircular pieces. On the upper surface of each semicircle suppose a fine 
line drawn parallel to the base (or perpendicular to the length of the plate), and as near 
