457 



GRADUATION. 



GRADUATION. 



458 



space between them, which can be most accurately divided with a 

 pointer by the hand, aided by a magnifying lens. In the mural quad- 

 rant which Graham erected at Greenwich, he carried this principle 

 into full operation. The beam-compass, which was used for drawing 

 the divisional arc of the quadrant, was used for setting off the chord of 

 60 ; this was bisected, and the radius being again set forward from 

 30, he had the quadrant exact. The arc of 60 was divided by con- 

 tinual bisections into 64 (or 2) equal parts, and the arc of 30 in like 

 manner into 32 (or 2 5 ) parts. The subdivisions were, on the same 

 principle, into 16 parts each. This division of the quadrant into 96 

 parts was continued as long as quadrants remained in use, but the 

 trouble of reducing every observation into ordinary degrees, minutes, 

 and seconds, was a considerable increase of labour to the observer. 

 The quadrants were generally divided to 90 as well as into 96 parts, 

 and in Bradley's and Maskelyne's ' Greenwich Observations,' the zenith 

 distances are recorded in both divisions, with a recommendation to 

 trust to the 96 division in cases of discrepancy. The improvements 

 introduced by Bird chiefly apply to the division into 90. He made a 

 long scale of equal parts by stepping three times with a beam compass 

 51 "2 inches, and subdivided each of these parts by continual bisections 

 into tenths. For further subdivision, a space of 25,856 inches was 

 taken and divided into 256 parts by perpetual bisection ; hence, as 

 each of the new spaces was O'lOl inches, he had, with his vernier, a 

 scale of equal parts to O'OOl of an inch. From such a scale of equal 

 parts as this and the proper tables, all the lines of Gunter's and other 

 scales are laid off upon a standard. In dividing the mural quadrant 

 Bird made great use of his scale, chiefly to obtain the arc of 85 20', 

 which admits of perpetual bisection, being 1024 x 5'. The chorda of 

 the several arcs were computed and beam-compasses prepared, which 

 were adjusted by the scale to be the chorda 30, of 15, of 10 20', and 

 of 4 40', to the proposed radius. The scale, quadrant, and beam- 

 compasses were then left all night to come to the same temperature, 

 and before sunrise were examined, and readjusted if incorrect. The 

 arc being struck, the radius marked off the chord of 60, which was 

 bisected with the beam-compass containing the chord of 30, and the 

 radius protracted from 30 gave the quadrantal arc exactly as in 

 Graham's mode of dividing. The arc between 60 and 90 was then 

 bisected in 75 by its proper beam-compass, the chord of 15, and then 

 the chord of 10 20' was carried forward from 75, and the chord of 

 4 40' was carried backwards from 90. The exact joining of these 

 two chords in the same point proved the accuracy of the operation. 

 The fifth beam-compass had been set to the chord of 42 40' and with 

 this the arc of 85 20' was bisected. When this part had been sub- 

 divided, the chord of 64 subdivisions, or of 5 20', was taken from the 

 divided portion and carried forward from 85 20', and then perpetually 

 bisected. Bird remarks " that the points at 30, 60, 75", and 90 fell 

 in without any sensible inequality." Bird's manual skill and the great 

 care he took to avoid errors arising from the partial expansion of the 

 quadrant or tools during the operation gained him great and merited 

 reputation, but we are inclined to doubt whether in engineering or 

 theoretical accuracy of division he made any step beyond Graham. 

 The careful division into 90 is a retrograde step. 



The divisions of Graham, Bird, Ramsden, and the elder Troughton, 

 were all performed by the beam-compass, and in a great measure by 

 touch; magnifying lenses were indeed applied to the points of the 

 beam-compasses, but when an erroneous point is once made, the beam- 

 compass naturally falls into it, and there is considerable trouble in 

 rectifying the error. A French nobleman, the Due de Chaulnes, after 

 perfecting the micrometer microscope for reading off the divisions of 

 astronomical instruments [CIRCLE], first showed how it might be used 

 in actual dividing. He did not however follow Graham's rule and 

 proceed by perpetual bisections ; hence his method was neglected, 

 although Kamsden saw the advantage of the micrometer microscope, 

 and used it for reading off the divisions of his circles and theodolites. 

 In this state the art was taken up by Edward Troughton, who by a 

 happy combination of the principle of Graham, the Due de Chaulnes' 

 BOde of reading off, and his own ingenuity, brought the division of 

 astronomical instruments to its present state of perfection. We will 

 suppose that a circle is to be divided originally. After the edge of the 

 circle is very carefully turned upon its own centre, a small circular 

 roller, 16 revolutions of which carry it exactly round the circle, is pre- 

 pared and so fitted to the circle by a radial frame joining the two 

 centres, that on turning the frame round the roller is also turned in 

 an opposite direction by friction between the edges of the roller and 

 circle. The roller in divided into 16 parts, and a microscope placed 

 over the divisions, and as each division comes under the microscope, 

 a fine round dot is marked upon the circle, which thus receives an 

 approximate division into 256 (or 2") equal parts. Troughton expected 

 that as the roller could be carried round the circle by any number of 

 times without over or under lapping, it would also mark out equal 

 portions at each revolution, but he found himself mistaken, and he 

 therefore proceeded to examine the dots optically. Two microscopes 

 A and B are fixed above the circle, nearly in a diameter, and the dot? 

 which are to determine the divisions and 180 are bisected by them. 

 The circle is then turned half round, and dot 180 brought under A ; 

 if at the same time falls exactly under B, the points are diame- 

 trically opposite, if not, the quantity and direction of the error is 

 measured by the micrometer of B, half of which is evidently the error 



of dot 180. The microscope B is then shifted and fixed over dot 90, 

 while A bisects 0. By a quarter revolution of the circle 90 is 

 brought under A, and 1SU under B, and the error, if any, measured and 

 uoted. In like manner the error of dot 270 is detected, after which 

 the microscope B is again shifted, and fixed over dot 45, when the 

 errors of dots 45, 135, 225, and 315, are determined. It is easily 

 seen that this is exactly Graham's principle of perpetual bisection, only 

 using an optical beam-compass instead of one without points, and 

 registering the errors of the dots instead of cutting actual divisions. 

 In this way Troughton proceeded by continual bisections to note 

 the relative or apparent errors of the 256 dots. His next step was to 

 compute the actual or true error of each dot, and to form a table. 

 Suppose that in examining the 180 dot, he found that the arc from 

 O 3 to 180 was less than the arc from 180 to 360 by 20", he would 

 conclude that the dot 180 was 10" behind its true place. Again, let 

 arc from dot to dot 90 exceed the arc from dot 90 to dot 180 by 

 30". If 180 were right, 90 would be too forward by 15" ; but as 

 180 is 10" behind its true ptace, 90 will on this account alone be 5" 

 behind, and therefore on the whole will be 10" too forward.* The 

 true errors of each of the 256 dots being thus computed, Troughton 

 returned to the roller, and by help of a small sector which revolved 

 with it and gave him an enlarged scale, enabling him at the same time 

 to reduce the division into 256 parts into 360 mechanically, proceeded 

 to cut the actual divisions on the circle. This method was com- 

 municated to the Royal Society, and printed in the ' Phil. Trans, for 

 1809,' p. 105. The Copley medal was granted for this very valuable 

 and original memoir. Several circles have since been divided after 

 Troughton's method, by his successor Mr. Sinims, and by Mr. Thomas 

 Jones, and it has been thus proved that the merit of Troughton's 

 dividing depended, as he asserted, on the excellency of his processes, 

 and not on his manual dexterity. It is not worth while to divide a 

 circle originally which is less than two feet in diameter. 



There is a caution with respect to this mode of dividing which will 

 be sufficiently obvious when pointed out, namely, that very great 

 care indeed must be taken that the pivots on which the circle turns 

 shall be perfectly true and round. The circular line to be divided, and 

 the run on which the roller moves, are respectively drawn and turned 

 from these pivots, and the figure of neither is a circle unless the pivots 

 be so. The large collar of the mural circle on which the rim is turned 

 is of steel, and several inches in diameter. It often happens that hard 

 knots occur in the steel which ordinary tools will not touch, and it 

 would be prudent in the artist to perform the finishing part himself 

 with a diamond. 



We should also recommend the following extension of Troughton's 

 mode of examining his primary dots : After determining the errors of 

 dots 0" and 180, we should leave the microscopes A and B undisturbed, 

 and fix two new microscopes, c and D, at 90 and270. Then, having 

 adjusted c and D in 90 and 270, and having ascertained their errors, 

 as has been already described, the circle should be turned round till 

 and 180 are bisected by B and A, when 90 is under D and 270 under 

 c. The errors are then to be again determined exactly as before. 

 Now, if the circle turn round a mathematical point, the two results 

 must, of course, be identical ; but if not, it may happen that the 

 observations will give two errors for dot 90 and two for dot 270, 

 which should, however, have the same difference. The final error, or 

 that which the artist should adopt, for each dot, is the mean of the 

 two determinations, which will give the position of a line at right 

 angles to the diameter from to 180, and the nearest possible to the 

 variable centre. It is also clear when this discrepancy is found that 

 the centre has not been properly turned. By extending the above 

 process to the dots which bisect the quadrants, that is, shifting the 

 microscopes c and D to 45 and 225, and trying the dots, as in 

 Troughton's method, and also after a half-revolution, a series of bisect- 

 ing diameters may be found which will cut the diameters already 

 determined at angles of 45, and pass as near the variable centre as 

 may be. This process should be continued one or two steps more, and 

 then Troughton's method may be considered sufficient for the rest. If 

 the above system of examination should appear too troublesome, it 

 would be at least advisable, when Troughton's subdivision has been 

 carried to 16 or 32 parts, that the table of true errors should be 

 checked by opposite readings. This is easily done by bringing each 

 dot in its turn under microscope A, and reading off the apparent error 

 of the opposite dot by microscope B. As the true error of the dot 

 under A is known from the table of errors, this, + the apparent error 



* Let i and i' be tho errors of any two dots, a and b, -\- when too forward, 

 and when behind their true place, and the distance to the bisecting dot r bo 



k 

 from a = m and from 6 = m-fJk.- then the apparent error of c is -; for it 



k 



should be at a distance '-}:, from a. But the dot a is wrong i, and the dot 



6 is wrong ' ; therefore there is a further correction of . , and the whole 



i+" * 

 error of c is (attention being paid to the signs) r~ T ; from which expression 



the mode of forming the table of real errors is very evident, cure being taken 

 not to confound the signs, and also to pass from the arcs to the half arcs in 

 succession. 



