THE RAMSDEN DIVIDING ENGINE. 729 



manner in whicli it was accoinplisbed is described suhstautially as 

 follows (see p. 332, Smith's Optics, 1738) : 



"Two concentric arcs of radii 1)G.85" and 95.8" respectively were 

 first described by the beam compass. On the inner of these arcs 90° was 

 to be divided into degrees and twelfth parts of a degree, while the same 

 on the onter was to be divided into 90 equal parts, and these again into 

 sixteenth parts. The reason for adopting the latter was that 90 and IG 

 both being powers of 2, the divisions will be got at by continual bisec- 

 tion alone, which, in Graham's opinion, who first employed it, is the 

 only accurate method, and would thus serve as a check upon the accu. 

 racy of the divisions of the outer arc. With the same distance on the 

 beam compass as was used to describe the inner arc, laid off from 0°, 

 the point 00° was at once determined. 



" With the points 0° and 00° as centers successively, and a distance 

 on the beam compass very nearly bisecting the arc of 00°, two slight 

 marks were made on the arc; the distance between these marks was 

 carefully divided by the hand, aided by a lens, and this gives the point 

 30°. The chord of 00° laid off from the point 30° gave the point 90°, 

 and the quadrant was now divided into three equal parts. Each of 

 these parts was similarly bisected, and the resulting divisions again 

 trisected, giving 18 parts of 5° each. Each of the qninquesected gave 

 degrees, the twelfth parts of which were were arrived at by bisecting 

 and trisecting as before. The outer arc was divided by continual 

 bisection alone, and. a table was constructed by which the readings of 

 the one arc could be converted into those of the other. After the dots 

 indicating the required divisions were obtained, either straight strokes, 

 all directed towards the center, were drawn through them by the divid- 

 ing knife, or sometimes small arcs were drawn through them by the 

 beam compass having its fixed i^oint somewhere on the line which was 

 a tangent to the quadrantal arc at the point where a division was to 

 be marked." 



In 1707 John Bird, an English mathematical-instrument maker, 

 graduated a quadrant of 8 feet radius. His method was that of con- 

 tinual bisection, and is described in a pamphlet published by order of 

 the commissioners of longitude, 1707, entitled " The Method of Divid- 

 ing Astronomical Instruments," by John Bird, mathematical-instrument 

 maker in the Strand. 



The exact radius which he used was 9oi^o-o% inches. The radius laid 

 off from the point 0° gave the point 00°. This arc of 00° was care- 

 fully bisected, giving the point 30°, from which the radius, that had 

 remained undisturbed on the original beam compasses, was laid off, 

 giving the point 90°. 



The chords of 30°, 15°, 10° 20' 4° 40', and 42° 40' were computed and 

 carefully laid oft", each on a se[)arate pair of beam compasses. Bird 

 used an exact scale of equal parts, which by the aid of a magnifying 

 glass he was able to read to one one-thousandth of an inch. 



