VOL. IX.] PHILOSOPHICAL TRANSACTIONS. IQl 



1.00000 = r Qj^ 



"I = 316920 ) 319827 ( 1.00918 = a ^ l6 



I = 314013 ) 319827 ( 1.01852 = 6 ^, 17 



I = 311103 ) 319827 ( 1.02803 = c ^^^ ig 



I = 308198 ) 319827 ( 1.03773 = d ^^^ 19 



I = 305290 ) 319827 ( 1.047^2 = c ^^^ is 



302343 ) &c. ( 1.05769 =/ ^^^ 20 



29m 5 ) ( 1.06796=^ 1^27 2^ 



296567) (1.07843= A ^^^ 31 



293660 ) ( 1.08911 = i ^^^^ 21 



290752) (1.10000= A: 1089 



So far my memorandum. Where we may observe two cases; viz. when the 

 breadth of the limb is made the 5th part, and the ] 0th part, of the shorter 

 radius; and the angle to be divided lO'; so far accurate according as the com- 

 mon trigonometrical canon is so ; and indeed the last figure ambiguous, some- 

 times in excess, sometimes in defect. As to the radius, I make it 1, as com- 

 monly I do ; not 10000000, as most others do ; by which all the multiplications 

 and divisions by radius are saved. Therefore the sines will be in decimal parts; 

 to which therefore, when necessary, I prefix ciphers, which also occupy the 

 unit's place of integers. In like manner are we to proceed, mutatis mutandis 

 when the limb of the instrument is taken in any other proportion to the length 

 of the radius. 



But it will be more convenient, in order to avoid so often finding the pro- 

 portional part, that the angle p be of some convenient magnitude, to be deter- 

 mined by exact minutes, without the annexed seconds or thirds; and so finding 

 the length of the greatest radius z, in the same manner as the others a, b, c, &c. 



Suppose that, as before, there be taken r = 1, and the angle A = 10', and 

 the angle ^ be taken, not 178° lO' \" bf'% as before found, but rather 178"* 

 lO', whose supplement is 1° 50', the tabular sine of which is 0.03 1 9922 ; and 

 the same of the «, p, y, J", &c. The sines will be had like as before, but by one 

 division only ; and the length of that radius z being, not precisely 3.1 as before 

 but nearly so, as I.09996. Then, 



Sin. « = 1° 49' = 317015 ) 319922 

 Sin, /3 = 1 48 = 314108 ) 319922 

 Sin. y = 1 47 = 3112O0 ) 319922 

 &c. 308293 ) 319922 



305385 ) &c. 

 302478 ) 

 299570 ) 

 296662 ) 

 293755 ) 

 290847 ) 



1.00000 = r ^*' 



917 

 1.00917=. '^ 17 



1.01851 = 6 052 ^« 

 1.02803 = c ^^ 17 



m ^ 



588 '^ 



1007 ^^ 

 20 



20 



1.03772 = d 

 1.04760 = c 

 1.05767 —f 

 1.06794 =g 1027 



1.07841 = A 1047 ^^ 

 1.08908 = i 1007 ^^ 

 1.09996 = k 1088 



= z. 



