VOL. XXXVIII.j PHILOSOPHICAL TRANSACTIONS. 67 I 



for a centre to different circles, which are not concentric. It is certain that 

 every arch on tlie limb may have a circle that will pass through the centre, and 

 be a locus or geometrical place for the angle made by that arch to fall on : but 

 then every arch has a different one from all others; as in fig. 10, pi. l6. Let 

 ABC be the quadrant, and ab, ef, gh be taken as arches of it : circles drawn 

 through each two of these respectively, and through the centre c as a tliird 

 point, will manifestly be such loci or places : for every pair of these points 

 stands in a segment of their own circle, as well as on a segment of the quadrant; 

 and therefore by the cited 21, 3d elem. the angles standing on these first seg- 

 ments will every where be equal at the periphery of their respective circles, and 

 their radius will always be equal to half the secant of half the arch on the 

 quadrant. For in the circle cedf, for instance, the angle ced is right, be- 

 cause in a semicircle, ce is the radius of the quadrant, ed the tangent of the 

 angle dce = ^ the arch ef, and cd is the secant of the same = the diameter 

 of the circle cedf, and therefore its radius is half that secant. 



Now from the figure it is plain, that in very small arches the radius of their 

 circular locus will be half the radius of the quadrant, that is, putting this 

 radius = 10, the other will be 5. And the radius for the arch of go, the 

 highest to be used on the quadrant, will be the square root of half the square 

 of the radius = sine of 43 degrees = 7.07 I, and the arches at the centre 

 drawn by these two radii are the extremes, the medium of which is 6.0355. 

 And if a circular arch be drawn with this radius -jL part of the length of it, that 

 is, in an instrument of 20 inches radius, the length of one inch on each side 

 of the centre affording 2 inches in the whole, to catch the coincidence of the 

 rays on, which must be owned is abundantly sufficient, the error at the greatest 

 variation of the arches, and at the extremity of these 2 inches, will not much 

 exceed one minute. 



But in fixing the curvature or radius of this central arch, something further 

 than a medium between the extremes in the radius, is to be considered : for in 

 small arches, the variation is very small, but in greater it equally increases, as 

 in the figure, where it appears, the difference between the angles abc and adc, 

 fig. 11, is much greater than the difference between ebc and edc, though both 

 are subtended by the same line bd : for their difi^erences are the angles bad and 

 BED. Therefore this inequality was likewise to be considered ; and compound- 

 ing both together, Tho. Godfrey pitched on the ratio of 7 to I 1, for the radius 

 of the curve to the radius of the instrument, which is 6.3636 to 10. But on 

 further consideration, he now concludes on 6-5*5-; and a curve of this radius of 

 an inch on each side of the centre, to an instrument of 20 inches radius, or of 

 ' of the radius, whatever it be, will in no case, as he has himself carefully 

 computed it, produce an error of above 57 seconds ; and it is well known that 



