the Divisions of astronomical Instruments. 235 
temperature of the instrument. The rest may be performed 
at our leisure. 
You will immediately perceive the object of this kind of 
measurement. It enables us to determine, with any degree 
of accuracy that may be required, the proportion which the 
first and every succeeding arc of the circle, contained betwixt 
the micrometer and the microscope, bears to the whole circle, 
and of course the absolute length of the arcs themselves. Let 
a denote the real length of the first of these, and + a',± a",±a'", 
&c. the difference betwixt the first and second, the first and 
third, &c. respectively; let A represent any other arc whose 
length is known, and which is a multiple of a, as marked upon 
the instrument, and let this multiple be expressed by n. 
i hen will a -j- ( a -f- a' ) -J- ( a -j- a!' ) -f- ( a -f- a!" ) -J- &c. . . . 
is evident, that if there is no error committed in the measure- 
ment of any of these arcs, we shall have the value of a, and 
consequently of a -f- a! , a -J- a", a -{- a"', &c. and of any arc, 
comprehending any number of these, accurately determined. 
But if there be an error of e in the measurement of the first, 
of e ', e" , e'" , &c. in the measurement of the second, third, &c. 
respectively, then we shall have the following equation for 
determining a , viz. a -f (a -f a' -f e -f e') -f- (a -f- a" + e 
-f e") + &c. ... (a + a" r ' n ~ l + e + = A, and 
( a -f a!""' n 1 ) == A, and a 
. Hence it 
n 
consequently a will appear to be equal to — - — -- 
a" a' 
n 
, which differs from its true value by 
n 
n— 1 e + e + e " -f 
,u . -w • • . • n — 1 
MDCCCIX. 
n 
. Hence it follows, that the value of 
li 
