5 ON DIVIDING INSTRUMENTS. 



viding bisec the means of dividing bisectionally without one. His me- 

 put'alLTJ.^'^ thod I will brlffly state as follows, in the manner in which 

 it would apt)ly to dividing a mural quadrant. The arcs of 

 60° and SO"" give the total arc as before; and let the last arc 

 of 30° be bisected, also the last arc of ]5", and again the 

 last arc of 7° 30': the two marks next 90* will now be 82* 

 30' and 86° 15 , consequently the point sought lies between 

 them. Bisections will serve us no longer; but if we divide 

 this space equally into three parts, the most forward of the 

 two intermediate marks will give us 85°, and if we divide 

 the portion of the arc between this mark and 86° 15' also 

 into three, the most backward of the two marks will denote 

 85° 25'. Lastly, if we divide any one of these last spaces 

 into five, and set off one of these fifth parts backwards from 

 85° 25', we shall have the desired point at 1024 divisions 

 upon the arc from 0*. All the rest of the divisions which 

 have been made in this operation, which 1 have called 

 marks because they should be made as faint as possible, 

 must be erased ; for my brother would not suffer a mark 

 to remain upon the arc to interfere with his future bisec- 

 tions. 



Snie3ton's pre- Mr. Smeaton, in a paper to be more particularly noticed 

 fert'nceofdivi- ,. • 1 i .1 . /. ■,,••, 



«ion by the presently, justly remarks the want 01 a unity or principle 



computed jn Mr. Bird's method ; for he proceeds partly on the ground 

 of the protracted radius, and partly upon that of the com- 

 puted chord ; which, as Smeaton observes, may or may not 

 agree. Bird, without doubt, used the radius and its parts 

 in order to secure an exact quadrant; but Smeaton, treat- 

 ing exactness in the total arc as of little value to astronomy, 

 would, in order to secure the more essential property of 

 equality of division, reject the radius altogether, and proceed 

 entirely upon the simple principle of the computed chord. 

 Advantages of The means pursued by my brother, to reach the point which 

 Mr. Jodn ^ terminates the great bii^ectional arc, is the only part in 

 /aeih(?d. which it differs from Bird's method ; and I think it is with* 



out prejudice, that 1 give n the preference. It is obvious, 

 that it is as well calculated to procure equality of division, 

 as the means suggested by Smeaton; at the same time that 

 it is equal to Bird's in securing the precise measure of the 

 *o!fil arc. It proceeds entirely upon the principle of the pro- 



t I acted 



