VOL. LXXVI.] PHILOSOPHICAL TRANSACTIONS. 35 



is evident that though this inequality may be very small, and even imperceptible 

 between neighbouring divisions, yet among distant ones, it may and will arise to 

 something considerable , which, in the mensuration of angles, will have the 

 same ill tendency as near ones. Now, as M. Roemer has given no means of 

 checking the distant divisions, in respect of each other, it is very probable that 

 no one has followed his steps, in cases where great accuracy was required, in a 

 considerable number of divisions. For in reality this method is likely to fall far 

 short of Dr. Hook's ; as Dr. Hook's divisions being cut in a similar successive 

 manner, by the rotation of the sharp edge of the threads of a screw against the 

 exterior edge of the limb of the instrument, a very slight degree of pressure will 

 bring a fine screw of 30 threads in an inch, which he prescribes, to touch 

 against an arch whose radius is 4 or 5 feet in more than 1, 2, 3, or 4 threads at 

 once ; so that the threads supporting each other, a small extuberance, or even a 

 small hard particle in the metal, will be cut through or removed by the grinding 

 or rather sawing motion of the screw ; and which, in regard to its contact, being 

 in reality an edge, will be much more effectual, that is, more firm, in its reten- 

 tion, than a mere simple point : and a repetition of the operation, from the 

 support of the threads to each other, will tend to mend the first traces ; whereas, 

 in Roemer's way, a repetition will make them worse ; for whatever drove forward 

 or backward the point on first entering, will, from the sloping of the point, be 

 confirmed and increased in driving it deeper. 



When Dr. Halley was chosen Astronomer Royal, Mr. Flamsteed's instru- 

 ments being taken away by his executors, Mr. Graham undertook to make a 

 new mural quadrant, about the year 1725; who, uniting all that appeared 

 valuable in the different methods of his predecessors, executed it with a degree 

 of contrivance, accuracy, and precision, before unknown : and he performed 

 the division with his own hand. The model of this quadrant, for strength, 

 easy management, and convenience, has been ever since pursued as the most 

 perfect. What I apprehend to be peculiar in it, was the application of the arch 

 of qQ° ; not only as a check on the arc of degrees and minutes, but as superior 

 to it, being derived from the more simple principle of continual bisection. To 

 make room for this, he has entirely rejected the subdivision by diagonals, and 

 has adopted the method of the vernier ; but the subdivision of the vernier divi- 

 sions he, as I apprehend for the first time, measured by the turns of the detached 

 adjusting screw, making it in fact a micrometer, by which the distance of the 

 set of the instrument was to be measured from the perfect coincidence of one of 

 the actual divisions of the limb with the next stroke of the vernier ; by which 

 means the observation could not only be read oflf with all the precision that the 

 division of the instrument was capable of, but the two sets of divisions could be 

 checked and compared with each other. Another thing that I apprehend to be 



p 2 



