338 



GRADUATION 



Fig. 1. 



therefore be considered first in reference to original 

 graduation ; and afterwards the reproduction, by 

 hand or machine, of originally graduated lines or 

 arcs may be dealt with. 



The most elementary process in original gradua- 

 tion is the operation of dividing a line into a given 

 number of equal parts. Let AB ( fig. 1 ) be a line : 



it is required to 

 divide it into, 

 say, seven equal 

 parts. From A 

 draw AC, mak-~ 

 ing any conveni- 

 ent angle with- 

 AB, and on AC 

 lay off with com- 

 passes or a scale 

 seven equal dis- 

 tances from A. 

 Join BD, sup- 

 posing D to be 

 at the end of the seventh part on AC, and 

 through the other points of division of AC draw 

 lines parallel to BC. These will cut AB into 

 seven equal parts. In practice, however, this 

 method is not very accurate and is not often em- 

 ployed. In another method, that known as con- 

 tinual bisection, the length of half the line is laid 

 off, by means of the beam-compass, from both ends; 

 these lengths from the two ends should agree in one 

 point as being the middle point of the line ; if they 

 do not, then the point midway between them is 

 taken as being the middle point, and is found by 

 means of a pair of fine compasses and a lens. Each 

 half of the line is, by the same means, halved 

 again, and so on until the required number of 

 divisions is obtained. Similar division of a straight 

 line may be obtained by laying off, by a pair of 

 spring-dividers, one after another, from one end of 

 the line, the smallest part required. Obviously, if 

 there is any error in the first distance thus laid 

 down, it will be multiplied in the last in proportion 

 to the whole number of divisions. This method is 

 known as stepping. 



The original graduation of circles or of circular 

 arcs is a matter of some difficulty, as it requires on 

 the part of the operator such skill, patience, and 

 care as is possessed by few. In tins connection 

 the names of Graham, Bird, Trough ton, Ramsden, 

 and Simms may be mentioned as those whose 

 work has been of high value to the astronomer 

 and physicist. The first method which may be 

 described by which a circle can be divided is practi- 

 cally the same as that of bisection in the case of 

 the straight line. Since the chord of an arc of 

 60 is equal to the radius of the circle ( the chord 

 and two radii to its extremities forming an equi- 

 lateral triangle ), if this length be laid off from any 

 point on the circle an angle of 60 is thereby deter- 

 mined. The half of this angle may be obtained, 

 and when added to 60 forms the quadrant or 

 90. Continual bisection of 60 gives the smaller 

 divisions of degrees and fractions of a degree. 

 Troughton's method depends on an entirely different 

 principle. A cylindrical roller is employed, whose 

 dimensions are such that in rolling on its axis once 

 round the outside edge of the circle it revolves six- 

 teen times. The edge of the roller is itself divided 

 into sixteen equal parts by repeated bisection. It 

 is then held firmly oy a frame against the edge of 

 the circle, so that, on being moved always tangenti- 

 ally to the circle, it revolves on its axis round the 

 edge of the circle. In doing so marks are made 

 on the edge of the circle corresponding to the 

 divisions on the roller ; of these there will be 256, 

 each interval being equal to 84| minutes of arc. 

 The further division of these intervals into degrees 

 and minutes is effected by means of a subdividing 



sector, placed concentrically, and rolling with the 

 roller. For the details of the manipulation of this 

 sector reference may be made to Troughton's paper 

 in the Philosophical Transactions, 1809. 



The reproduction, or copying as it is termed, of 

 graduated straight scales, circles, or circular arcs, 

 by copying them from patterns originally graduated 

 with great accuracy, may be done by hand or by 

 mechanical contrivances. In copying a straight, 

 scale the 'work' i.e. the piece of metal or other 

 material whose division is required is laid parallel 

 to and flat with the pattern whose graduation is 

 copied. A straight-edge is laid across both, so as 

 to coincide with one of the divisions in the pattern, 

 and the dividing- knife is drawn carefully along the 

 edge, and across the work. In copying circles the 

 work is screwed firmly down on and concentric 

 with the pattern ; the dividing-knife is then used 

 in the same manner as in copying straight scales, 

 being guided by an index steel bar, the edge of 

 which is exactly coincident with a radius of the 

 circle. 



Copying is now more usually effected by instru- 

 mental means, the machines for this purpose being 

 the linear and circular dividing-engines. In the 

 linear dividing-engine the principal part is a care- 

 fully turned screw, which revolves in bearings in 

 two supports. Connected with the screw is a crank 

 handle and a disc whose plane is perpendicular to 

 the axis of the screw, and whose flat edge is divided 

 into a number, usually 400 or 500, of equal parts. 

 As the handle is turned the screw rotates, but does, 

 not move in the direction of its length (it may, 

 therefore, for distinction be referred to as the fixed 

 screw). At the same time the disc also revolves, 

 and each division on its edge passes an index line- 

 on a part of the support close to it ; the number of 

 complete turns and fractions of a turn of the screw 

 may thus be easily counted. If, now, the fixed 

 screw pass through a hollow travelling nut or 

 screw, the latter will move backwards and forwards 

 according as the fixed screw is rotated one way or 

 the other. Thus, e.g., if the ' pitch ' of the fixed 

 screw be one millimetre, and the handle be turned 

 ten times and a little more, corresponding to fifty- 

 six divisions (of which let there oe 500) on the 

 graduated edge of the disc, the travelling-screw, 

 and anything there may be in connection there- 

 with, will advance through 10^% mm. i.e. 10'112 

 mm. The handle is so connected with a ratchet- 

 wheel that the fixed screw can only be rotated in 

 one direction, so that the travelling-screw can only 

 travel in one direction, usually from left to right. 

 Attached to the travelling-screw is the dividing- 

 apparatus, which is a light frame supporting a. 

 vertically-placed steel needle, with a fine, hard 

 point, and capable of a to-and-fro motion in a hori- 

 zontal line at right angles to the fixed screw. This- 

 needle serves as a marker whereby divisions may 

 be made on any object whose graduation is desired. 

 For instance, to divide a given length into a certain 

 number of equal parts, the travelling-screw is 

 allowed to advance, by turning to the requisite 

 amount, so that the point of the needle, starting 

 from one end of the line, moves through a distance 

 equal to one of the equal parts. A mark is then 

 made with the needle ; the travelling-screw is again 

 advanced through precisely the same distance, and 

 another mark is made ; this process is continued 

 until the whole length is divided. The length of 

 mark made by the steel point may be adjusted, 

 within certain limits, by increasing or decreasing, 

 by screws, the range of the marking-point. In 

 addition to this, however, it is desirable in some 

 cases to make every tenth mark longer than the 

 others, excepting the fifth, which may be inter- 

 mediate. This is effected by a wheel whose cir- 

 cumference is cut up by rectangular notches, into- 



