THE FRENCH REPEATING CIRCLES. 
451 
of the two extreme reading:s at the commence- 
ment and termination of the observations.” 
Tliat this conclusion of M. Biot may be true, 
it is necessary that there be no, or at least an 
insensible, resistance in the centre work lo 
the action of the tangent screws, and that 
there is no imperfection in the tangent screws 
in producing motion, nor in the clamping 
screws in securing permanent positions. Now, 
it is clear that if there is the least 
defect in all or any of these, M. Biot’s con- 
clusion will be erroneous, and such must of 
necessity be the case to a certain degree, 
since it depends upon the materials of which 
the instrument is constructed, and cannot 
be removed by the abilities of the artist, or 
the perfection of the workmanship, however 
excellent it may be. Hence, it necessarily 
follows that a slight relative motion must take 
place between the verniers and the circle for 
each repetition, causing by that means a small 
error, which will be continually repeated, and 
which, therefore, the principle of repetition 
cannot cure. It is, I believe, owing to this 
cause that a constant error of about 6 , 
according to Baron Zach, m^y remain in 
some instruments in a series of many hundred 
observations made "with the repeating circle 
when the clamping irons are imperfect. M. 
Biot goes on to say, that the errors of the 
extreme readings at the commencement and 
termination are much diminished, because 
the circle has generally four verniers that are 
read separately, and of which, the mean 
marks the commencement and termination of 
the total with a great probability of accuracy. 
Finally, the small error which still remains, 
notwithstanding these precautions in the 
extreme readings, is distributed over the 
entire arc measured on the limb, and there- 
fore has an insensible influence on the simple 
value of one observation, when these obser- 
vations are sufficiently multiplied. The errors 
of the division, then, in the repeating circle 
itself are also thus diminished by repetition, 
and the compensation of errors is not the 
effect of probability, but of certainty. 
“ To estimate the extent of this compensa- 
tion, it may be remarked, that our (French) 
repeating circles are generally about 15 inches 
in diameter, and the error of division cannot 
exceed 15' . If the error would be reduced to 
half a second after thirty observations, what 
would it become aftei eighty or one hundred ? 
What does it become after, as has often been 
done, the series of different days are made to 
succeed one another, without interruption, 
upon the limb, so that the two errors of the 
extreme readings are extended upon a total 
arc, which contains the simple arc many 
thousand times ? I he errors of division, 
then, in this instrument become evanescent, 
and it is impossible they can be entirely de- 
stroyed in the largest instruments, if they 
are not repeaters. Never can the address of 
an artist equal a mathematical proceeding.” 
But there are other errors which are de- 
stroyed by the principle of probabilities in the 
use of the repeating eircle, that still remain 
in other instruments. Such are, the errors 
of the level, which were small in the original 
repeating circles, and in those later construct- 
ed still less, in which the divisions of the 
level give immediately fractions of a second. 
Such is also the case with the errors of 
pointing, or those arising from directing the 
inter>ection of the cross wires of the teles- 
copes to the object observed, which, though 
small of themselves, ate destroyed like those 
of the level by their fortuitous compensation 
in many thousands of observations. 'I hese 
errors exist also (though I may add in a less 
degree) in the observations made with large 
instruments, as the mural circles. For the 
error of pointing is still found, though 
diminished by the greater power of the teles- 
cope, and that of the level is represented by 
the error of the plumb line. But in this case 
the small number of observations does not 
admit of a compensation as exact as in the 
repeating circle. If we suppose that the ac- 
curacy of mean results is in the ratio com - 
pounded of the number of observations, and of 
the length of the rauius of the instrument, one 
hundred observations made with a repeating 
circle of two decimetres, or about eight Eng- 
lish inches radius, would be equivalent to one 
observation made with a mural circle of twen- 
ty metres radius, or about sixty-six English 
feet. “ Could we obtain such instruments,” 
says M. Biot, “ and, above all, could we em- 
ploy them in observations which require us to 
transport them from place to place ?” Now, 
though the repeating circle is, in the hands 
of an able observer, an instrument capable of 
great precision, yet we cannot assent to the 
extravagant eulogium thus betowed upon it 
by M. Biot in his Astronomic Physique, be- 
cause it rests on assumptions too gratuitous 
to be granted without qualification ; and, as 
we have already remarked, he has not alluded 
at all to the errors inseparable from its construe - 
tion, and the method of using it when executed 
by the best artists. 
However perfect the damping screws may 
be, yet still, by repeating the observations, 
repeated small relative motions by pressure 
must take place between the verniers and 
limbs, which remain as a constant error that 
no continuation of the process of repetition 
can remove, because it arises from that very 
principle. If, however, an equal number of 
observaiions at nearly equal zenith distances 
on opposite sides of the zenith be taken, then 
on the principles of probabilities, it may be 
expected that the errors from this cause will 
likewise have a tendency to destroy each 
other. Thus, by a judicious use of the re- 
peating circle, it may be employed to great 
advantage in all observations which require 
a moderately sized instrument capable of easy 
transportation. Still, however, the complex 
nature of its construction and the involved 
methods of observation are inherent disad- 
vantages, which render a commodious instru 
ment of similar dimensions but more simple 
in principle a desideratum to a numerous 
class of practical astronomers. 
The only other instruments, whose prices 
are moderate, and dimensions convenient, are 
Captain Rater’s circle somewhat enlarged, 
and Mr. 1 roughton’s portable altitude and 
