38 PHILOSOPHICAL TRANSACTIONS. [aNNO 1780. 



In my introduction to M. Roemer's method of division, I have shown, that 

 divisions laid off in succession, by the same opening of the compasses, either in 

 a right line, or in the arch of a circle, being in its idea geometrically true, and 

 in itself the most simple of all processes, it has the fairest chance of being 

 mechanically and practically exact, when cleared of the disturbing causes. The 

 objection therefore to his method is, the great number of repetitions, which de- 

 pending on each other in succession (requiring no less than 540 to a quadrant, 

 when subdivided to lO' each,) the smallest error in each, repeated 540 times, 

 without any thing to check it by the way, may arise to a very sensible and large 

 amount : but in the method I have hinted, this objection will not lie ; for, in the 

 first case, the assumed opening is laid off but 5 times ; and in the latter case but 

 4 times ; nor does this repetition arise out of the nature of the thing ; for, if 

 you like it better, you may, in the former case, at once compute the chord of 

 Oa° ; and in the latter that of 85° 20', and then proceed wholly by bisection ; 

 supplying what is wanted to make up the quadrant, from the bisected divisions, 

 as they arise. Mr. Bird prescribes this method himself, for the division of 

 Hadley's sextants and octants. 



I suppose he was the first who conceived the idea of laying off chords of 

 arches, whose subdivisions should be come at by continual bisection ; but why 

 he mixed with it divisions that were derived from a different origin, as prescribed 

 in his method of dividing, I do not well conceive. He says, that after he had 

 proceeded by the bisections, from the arc of 85° 20', the several points of 

 30^ 6o°, 75°, and 90°, all of which were laid down from the principle of the 

 chord of 60° being equal to radius, fell in without sensible inequality ; and so 

 indeed they might ; but yet it does not follow that they were equally true in 

 their places as if they had been, like the rest, laid down from the bisection from 

 85° 20', and therefore being the first made, whatever error was in them, would 

 be communicated to all connected with them, or taking their departure from 

 them. Every heterogeneous mixture should be avoided where equal divisions 

 are required. It is not the same thing, as every good artist will see, whether 

 you twice take a measure from a scale as nearly the same as you can, and lay 

 them off separately ; or lay off 2 openings of the compasses, in succession, 

 unaltered ; for though the same opening, carefully taken off from the same 

 scale a 2d time, will doubtless fall into the points made by the first, without 

 sensible error ; yet as the sloping sides of the conical cavities made by the first 

 point will conduct the points themselves to the centre, there may be an error 

 which, though insensible to the sight, would have been avoided by the more 

 simple process of laying off the opening twice, without ever altering the 

 compasses. 



The 96 arc was, I have no doubt, invented by Mr. Graham, from having 



