238 Mr. Lax on a Method of examining 
maining arc of 8° against the second, we shall get the value 
of the first 4 0 , which added to the arc of 7 0 , before determined, 
will give us the length of the first arc of 1 1°. The first 2 0 of 
the remaining arc of 4 0 must then be measured against the 
second, and we shall get the value of the first 2 0 , and by add- 
ing this arc to the arc of 1 1°, we shall obtain the value of the arc 
of 13 0 . By taking away the first arc of i° from the arc of 15 0 , 
we get the remaining arc of 14 0 , and then having determined 
the length of the first 7 0 of this arc, by measuring them 
against the second, we must add it to the arc of i°, and we 
shall obtain the arc of 8°. The length of the first 4 0 of this 
arc will then be easily known, by measuring them against the 
second, as will afterwards that of the first 2 0 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 the first arcs 
of 10, 20, 30, 40, and 30 minutes contained in each degree, 
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 10' being 
measured against the second of the arc of 20', in which it is 
included, and also against the two arcs of io' contained in 
the last arc of 20', its own value, and that of the last io' in the 
degree will be determined from a comparison with the arcs 
of 20', in which they are respectively comprehended. The 
length of this last arc of io' being taken from that of the 
whole degree, will give us the length of the first 50', and 
complete the operation. 
