GRADUATION. 



GRADUATION. 



u ihown by B, should = the true error of the dot under B, which it 

 bo known from the table of error*. Any sensible discrepancy in 

 these two value* will (how faultineM in the centre. Again, if the 

 artUt hare, M he ought to hare, a convenient pier and a lufflcienoy of 

 micrometer microscopes, he may ascertain the erron of hU four or 

 eight primary dote, by placing four or eight equidistant microscopes 

 round hU circle, and reading off at each quarter or one-eighth of * 

 revolution. It would be prudent to repeat this several times and on 

 several days, selecting the times when the temperature had been and 

 promised to be steady. This prim-iple might be carried on farther in 

 examining a graduated circle by changing the number of microscopes. 

 Indeed, if the centre were very irregular, ami therefore the curve 

 traced out by iU revolution, a better division into equal ports might 

 be got after a preliminary division by tlrppiitg than by Troughton's 

 method unaltered. As the + and signs used in forming the tables 

 of apparent and true errors may cause a little confusion, it is usual to 

 cut a few slight divisions, which may easily be rubbed out, and thus 

 to lot the accuracy of the table of errors before commencing the actual 

 division. 



It is easy to see how Bird's method of dividing a scale may be 

 pursued, adopting microscopes instead of the beam-compass. The 

 examination of such a scale is completely exemplified in Mr. Daily's 

 memoir referred to in the sequel. 



In Hamsden's dividing-engine the circle carrying the instrument to 

 bo graduated was made to turn round by means of a perpetual screw, 

 the teeth of which worked in corresponding notches cut in the edge of 

 the circle : the screw was caused to turn on its axis by a cat-gut band 

 passing several times round a cylinder on the some axis, and made fast 

 In-low to a treadle which was pressed down by the foot, and allowed to 

 rise, when the foot was removed, by the unbending of a spring con- 

 nected with the cylinder. In the machine now to be described, which 

 was the invention of Mr. Alexander Ross, and of which an account is 

 given in the ' Transactions of the Society of Arts ' for 1881, the circle' 

 has a more steady motion, and it possesses the means of being adjusted, 

 so that the inequalities produced by the wear of the ports may be 

 obviated. 



A rectangular cast-iron frame, of which A B is a part, carries the 

 circle c, which turns on a vertical axis at its centre ; this axis is in the 



form of two frustums of cones, and its lower extremity rests in a cup 

 which is at one end of a lever ; at the other end is applied a counter- 

 poise which, causing a pressure upwards on the axis, diminishes the 

 pressure of the circle on the frame which supports it. Hound the 

 circumference of tho circle are 48 projections, a, a, a, fas.; and at one 

 i-nd (A) of the rectangular frame, a sliding apparatus carries a cylinder 

 i>, which turns on a horizontal axis, and has on its convex surface a 

 -piml projection ('/ M like a rectangular thread of a screw ; this enters 

 between the projections a, a, ftc., and the distance between the two 

 turns of the spiral is rather greater than the breadth of one of those 

 projection*. 



On the rim of the circle c, and under the projections a a, &c., is a 

 groove in which enters an endless cat-gut bond, a B ; on each side of 

 the frame, at the end A of the latter, this band passes over a fixed 

 pulley, e, and under a moveable pulley, from which is suspended a 

 weight, K : it then passes through a notch at d, and above the frame, 

 in the direction A d. 



r pillars, of which X are two, rising from the rectangular 

 frame A n, support ajitage which carries the tool for cutting tin divi- 

 sions on the instrument to I* graduated : the npparatiiH by which the 

 cutting-tool is connected with the stage is formed with joints so as to 

 allow the tool to be moved up or down, or in the direction of a radius 

 of tnc circle c, but neither to the right nor left of that radius ; and 



there is a contrivance for determining the length of the lines of 

 ..-.-.. 



By moving one end of a lever at the opposite extremity of the frame 

 A B, a bar, //', which is attached at one end to that lever, gives n 

 to a catch, or click, so as to cause a ratchet-wheel to turn on its axis, 

 .nl thus give the requisite movement to the cutting-tool : the other 

 extremity,/', of the bar, by turning a lever on a horizontal axle con- 

 taining within it the axle of the cylinder D, allows a catch, or click, to 

 turn a ratchet-wheel on the latter axle, and, with it, the cylinder itself; 

 and thus a side of the spiral pn>j. .-t:..n I, I, is removed a little wn\ 

 its position when in contact with one side of a projection n. Moving 

 then the first-mentioned lever Kick again, a spring, which had pre- 

 viously pressed against the cat-gut band on the circumference of c, is 

 drawn off; and thus one of the weights, K, by drawing down the band 

 and pulley above it, gives a small movement to the circle r, and conse- 

 quently brings the side of a projection a again in contact with the side 

 of the spiral. According to the place at which the bar//' is applied 

 to the lever on the axle of D, the catch may pass over any n> 

 number of the teeth of the ratchet-wheel ; and thus the circle c may 

 l>e turned through any angle consistently with the values of the 

 divisions intended to be cut. 



Through each of the projections a a, Ac., posses a steel, screw, whose 

 ends appear in the above cut ; and through the spiral thread b b pass a 

 number of steel screws, whose ends also appear. The right-hand 

 extremity (in the diagram) of each of the former screws is ground 

 quite flat and perpendicular to the length of the screw ; while t ! 

 band extremities of the screws in bb are turned in the form of hemi- 

 spheres; and, after every movement of the circle c and of the spiral b b, 

 the machine is held in a state of rest by the abutment of the 

 hemispherical end of one screw against the flat end of the other. 



The original divisions on the circle c are made on silver studs let 

 into its surface ; and the screws passing through a, a, ftc., ore so 

 adjusted, by means of their capstan heads, that if the end of a screw in 

 4 6 is in contact with the screw in any one of the projections a, one 

 complete revolution of the spiral 4 4 may bring the end of the screw in 

 the next projection a in contact with the eame screw in the spiral. 

 The corresponding movement in the circle is equivalent to the ii 

 between two of the original graduations ; and the movement is verified 

 by the primitive divisions coming successively to a wire in a 

 microscope. 



It may be useful, before we give a list of references for the history 

 and practice of graduation, to point out the effect of excentrieitv in 

 engine-divided circles, which is almost always mistaken for bad <li 

 by unskilful persons. If the centre of the divided circle is not 

 concentric with the axis of rotation (we here suppose the circle to 

 revolve, and the verniers or microscopes to be fixed), it is clear that 

 the centre of the divisions will describe a very small circle about the 

 axis. Now, suppose we set out from the position when the centre of 

 the divisions is in a line with and between the axis and the reading 

 apparatus. On turning the circle a little round, it will be seen that 

 the angle through which the instrument rrally moves is less than the 

 angle read off at that vernier or microscope; and on drawing tho 

 figure it will be seen that the angle read off is the exterior angle of a 

 triangle of which the true angle is the interior and opposite ; and also 

 that the error or the difference between these two angles is the other 

 interior angle of the triangle, the measuring arc of which to the same 

 radius is, as to sense, equal to a perpendicular from the centre of 

 divisions on the primary line.* Hence the error caused in any vernier 

 by ejventricity is an arc equal to the distance between the two < 

 x sine of the angle from the position we have set out from. From 

 this consideration it will be evident to one who knows the nature of 

 trigonometrical lines, that two, three, or any number of equidistant 

 readings will cure excentricity, as the + errors must always equal 

 the errors. Now, if the possessor of a circular instrument baring 



* Let A be the axis of rotation, c r the circle described by tho centre of tho 

 divided circle v r v the place of the reading. In the position first mentioned in 

 the text, t coincide* wi:h c, and r with v. When the instrument has been 

 moved round the angle v A r, the angle read of Is v t at v < v, nnd the error is 

 r x _ A c 



the angle AVC, which is 



C V 



x ln VAC. 



A V 



