the Division^ of astronomical Inslrumenis. 283 



arc of 15% we shall obtain the first arq of 14°. ThQ first 

 7° of this arc being measured against the second, we as- 

 certain the value of the first 7"; and then, by measuring 

 the first 4'' of the remaining arc of 8^ against ihe second, 

 we shall get the value of the first 4°, which added to the 

 arc of 7°, before determined, will give us the length oF the 

 first arc of 11°. The first -2° of the remaining arc of 4^ 

 must then be measured against the second, and we shall 

 get the value of the first 2", and by adding this arc to the 

 arc of 11°, we shall obtain the value of the arc of 13^. 

 By taking away the first arc of 1° from the arc of 15°, we 

 get the remaining arc oF 1 1^; and then having determined 

 the length of the first 7^ of this arc, by measuring them 

 aeainst the second, we must add it to the arc of 1°, and we 

 shall obtain the arc oF 8°. The length of the first 4^ of this 

 arc will then be easily known, by measuring them against 

 the second, as will aFterwards that of the first 2° in the arc 

 of 4° itself, by measuring them against the second in the 

 same arc. 



We have still to ascertain the lengths of all ihe first arcs 

 of 10, 20, 30, 40, and 50 minutes contained in each de- 

 gree, for I shall only consider the case in which the circle 

 is divided into parts oF 10 minutes. Now the length of 

 the first arc of 30' will be obtained by measuring it against 

 the second, and the lengths of the first and second arcs of 20' 

 (whose sum will give the arc of 40^) by measuring the first 

 against each of the remaining arcs. The length of the 

 third arc of 20' must likewise be put down, and then the 

 first arc of lO' being measured against the second of the 

 arc of 20*, in which it is included, and also against the 

 two arcs of lO' contained in the last arc of 20', its own 

 value, and that of the last lo' in the de2:ree will be deter- 

 mined from a comparison with the arcs of 20', in which 

 they are respectively comprehended. The length of this 

 last arc of lO' being taken from that of the whole degree, 

 will give us the length of the first 50', and complete the 

 operation. 



In order to ascertain the greatest possible error to which 

 we are liable in the examination, let s denote in parts of a 

 second the greatest that can be committed in bisecting any 

 point upon the limb ; then, since this error may occur at 

 each end of the arc, it is evident that e in the expression 



deduced above O^—^ x '2pe\ will become 2e, and the ex- 

 pression itself -^^ — X 4/)£. Hence the possible error will 



be 



