1830 .] 
Description of a Repeating Circle. 
9 
reduced within narrow limits, hy confining our dependence on the steadiness of the 
rate to the shortest possible period. With' an uncertainty, as above supposed, of 
one fourth of a second on the daily rate, the above distance, which may be travelled 
in two days, might be determined to within one furlong, a proportion of 1 in 1600. 
36. There would still, however, be this doubt affecting the result. The rate of the 
watch may have changed ; there is no absolute proof it has not ; the result xnay 
then be erroneous. Against this we have only to alledge the probability of no such 
thing having taken place. This probability has its value, and can be expressed in 
figures. If indefinitely increased it becomes the next thing to certainty. It is evident 
that using two watches instead of one, if they both agree in giving the same result, 
the probability of the result being correct will be increased two fold. If three, three 
fold, &c. As to the original value of the probability, it will be expi*essed by a fraction, 
the numerator of which is 1, and the denominator the number of good watches, divid- 
ed by the number of instances in which their rate has changed within two days. Let us 
suppose this is to be that is, that I watch in 10 will be found to fail us in a period 
of two days. If we use two, the chances we are right (they both agreeing) will be 
20 to 1, if three, 30 to 1, and so on. By using more watches, we then increase the 
probability so much, that we can have every necessary degree of confidence in the 
result. The employment of a greater number of chronometers is calculated also to 
reduce the inevitable errors of observations, which were fixed at ‘ of a second, and to 
estimate this reduction in the most moderate manner, we may safely suppose it one 
half. I would say then, that 10 chronometers would determine the whole longitude 
of India, within two or three seconds of time, half or three quarters of a mile. This 
is one part in 3000 nearly 1 4 . 
37. Thus we see that operating with proper means, we may determine the longi- 
tudes with an accuracy of £ a mile, however great the distance, and the latitudes to 
a sixth of a mile, or even less ; no distance need then be erroneous to so much as one 
mile 15 , however great it may be. I say need be , — as indicating the necessity of 
the work being properly performed. 
38. Having thus briefly described the several methods which have been, or may be 
employed for mapping a given portion of the earth’s surface \ I shall, as this commu- 
nication lias already swelled to such an extent, defer my observations on the com- 
parative economy and expedition of each method, as well as my ideas of the circum- 
stances and conditions under which each is applicable, to a future occasion. It 
was necessary to establish the preceding detail, to refer to in the course of what I 
have further to urge ; as I know of no work to which I could appeal for authority, 
with regard to many of the statements I shall have to bring forward. D. 
II . — Description of an improved Repeating Reflecting Circle, by Mr. 
DoLloud, with observations by Captain Sabine, extracted from his 
work on the Pendulum , 
The reflecting circle, being constructed upon the same principles as the sextant, 
requires the same process for making the observations. The circle having the ad- 
vantage of enabling the observer to take the angle on, what is usually termed, the on 
arc ; that is, lie may first take the angle precisely in the same manner as with the 
sextant ; and then, by reversing the face of the instrument, take the angle on the off 
arc, or the other side of the zero. By this method, he will correct any error that 
may arise from a want of parallelism in the glasses, or in the adjustment of the 
14 . Dr. Tiarks determined the difference of longitude between Falmouth and Madeira 
to be, by seventeen chronometers, O/i. 47/«. 28s. 2, the extreme difference was 20 s, nine 
of them gave a mean Oh. 47 m. 28s. 2, and of these the extreme difference was only 3s, 7 . 
Vide Phil. Trans, for 18*24, p. 365. 
1 5 . In the case of a difference of longitude, the error on the distance will be equal to 
the error on the longitude. In the case of the latitude, equal to that of the latitude, and 
intermediately will be between these extremes. The maximum will be when m=p, in 
which case the error on the distance will be 1, 4 times that of the latitude -f- 1,4 times 
that of the longitudes a little less thau a mile. 
