RAMSDEN'S MACHINE for Dividing Mathematical Instruments, is an invention by which these divisions can be performed with exceeding great accuracy, such as would formerly have been deemed incredible. On discovering the method of constructing this machine, its inventor, Mr Ramden of Piccadilly, received £615l. from the commissioners of longitude; engaging himself to instruct a certain number of persons, not exceeding ten, in the method of making and using this machine from the 28th October 1775 to 28th October 1777: also binding himself to divide all octants and sextants by the same engine, at the rate of three fillings for each octant, and six shillings for each brass sextant, with Nonius's divisions to half minutes, for as long time as the commissioners should think proper to let the engine remain in his possession. Of this sum of £615l. paid to Mr Ramden, £300l. was given him as a reward for the improvement made by him in discovering the engine, and the remaining £315l. for his giving up the property of it to the commissioners. The following description of the engine, is that given upon oath by Mr Ramden himself.

"This engine consists of a large wheel of bell-metal, supported on a mahogany stand, having three legs which are strongly connected together by braces, so as to make it perfectly steady. On each leg of the stand is placed a conical friction pulley, whereon the dividing-wheel rests: to prevent the wheel from sliding off the friction-pulleys, the bell-metal centre under it turns in a socket on the top of the stand.

"The circumference of the wheel is ratched or cut (by a method which will be described hereafter) into 2160 teeth, in which an endless screw acts. Six revolutions of the screw will move the wheel a space equal to one degree.

"Now a circle of brass being fixed on the screw arbor, having its circumference divided into 60 parts, each division will consequently answer to a motion of the wheel of 10 seconds; five of them will be equal to a minute, &c.

"Several different arbors of tempered steel are truly ground into the socket in the centre of the wheel. The upper parts of the arbors that stand above the plane are turned of various sizes, to suit the centres of different pieces of work to be divided.

"When any instrument is to be divided, the centre of it is very exactly fitted on one of these arbors; and the instrument is fixed down to the plane of the dividing wheel, by means of screws, which fit into holes made in the radii of the wheel for that purpose.

"The instrument being thus fitted on the plane of the wheel, the frame which carries the dividing-point is connected at one end by finger-screws with the frame which carries the endless screw; while the other end embraces that part of the steel arbor, which stands above the instrument to be divided, by an angular notch in a piece of hardened steel; by this means both ends of the frame are kept perfectly steady and free from any shake.

"The frame carrying the dividing-point, or tracer, is made to slide on the frame which carries the endless screw to any distance from the centre of the wheel, as the radius of the instrument to be divided may require, and may be there fastened by tightening two clamps; and the dividing-point or tracer being connected with the clamps by the double-jointed frame, admits a free and easy motion towards or from the centre for cutting the divisions, without any lateral shake.

"From what has been said, it appears, that an instrument thus fitted on the dividing-wheel may be moved to any angle by the screw and divided circle on its arbor, and that this angle may be marked on the limb of the instrument with the greatest exactness by the dividing-point or tracer, which can only move in a direct line tending to the centre, and is altogether freed from those inconveniences that attend cutting by means of a straight edge. This method of drawing lines will also prevent any error that might arise from an expansion or contraction of the metal during the time of dividing.

"The screw-frame is fixed on the top of a conical pillar, which turns freely round its axis, and also moves freely towards or from the centre of the wheel, so that the screw-frame may be entirely guided by the frame which connects it with the centre: by this means any eccentricity of the wheel and arbor would not produce any error in the dividing; and, by a particular contrivance (which will be described hereafter), the screw when pressed against the teeth of the wheel always moves parallel to itself; so that a line joining the centre of the arbor and the tracer continued, will always make equal angles with the screw.

"Figure 1. represents a perspective view of the engine.

"Fig. 2. is a plan, of which fig. 3. represents a section on the line II.A.

"The large wheel A is 45 inches in diameter, and has..." Homiden's has ten radii, each being supported by edge-bars, as represented in fig. 3. These bars and radii are connected by the circular ring B, 24 inches in diameter and three deep; and for greater strength, the whole is cast in one piece in bell-metal.

As the whole weight of the wheel A rests on its ring B, the edge-bars are deepest where they join it; and from thence their depth diminishes, both towards the centre and the circumference, as represented in fig. 3.

The surface of the wheel A was worked very even and flat, and its circumference turned true. The ring C, of fine brass, was fitted very exactly on the circumference of the wheel; and was fastened thereon with screws, which, after being screwed as tight as possible, were well rivetted. The face of a large chuck being turned very true and flat in the lathe, the flattened surface A of the wheel was fastened against it with holdfasts; and the two surfaces and circumferences of the ring C, a hole through the centre and the plane part round (b) it, and the lower edge of the ring B, were turned at the same time.

D is a piece of hard bell-metal, having the hole, which receives the steel arbor d, made very straight and true. This bell-metal was turned very true on an arbor; and the face, which rests on the wheel at b, was turned very flat, so that the steel arbor d might stand perpendicular to the plane of the wheel; this bell metal was fastened to the wheel by six steel screws l.

A brass socket Z is fastened on the centre of the mahogany stand, and receives the lower part of the bell-metal piece D, being made to touch the bell-metal in a narrow part near the mouth, to prevent any obliquity of the wheel from bending the arbor; good fitting is by no means necessary here; since any shake in this socket will produce no bad effect, as will appear hereafter when we describe the cutting frame.

The wheel was then put on its stand, the lower edge of the ring B resting on the circumference of three conical friction-pulleys W, to facilitate its motion round its centre. The axis of one of these pulleys is in a line joining the centre of the wheel and the middle of the endless screw, and the other two placed so as to be at equal distances from each other.

F is a block of wood strongly fastened to one of the legs of the stand; the piece g is screwed to the upper side of the block, and has half holes, in which the transverse axis h turns: the half holes are kept together by the screws i.

The lower extremity of the conical pillar P terminates in a cylindrical steel-pin k, which passes through and turns in the transverse axis h, and is confined by a cheek and screw.

To the upper end of the conical pillar is fastened the frame G, in which the endless screw turns: the pivots of the screw are formed in the manner of two frustums of cones joined by a cylinder, as represented at X. These pivots are confined between half poles, which press only on the conical parts, and do not touch the cylindric parts: the half holes are kept together by screws a, which may be tightened at any time, to prevent the screw from shaking in the frame.

On the screw-arbor is a small wheel of brass K, having its outside edge divided into 60 parts, and numbered at every 6th division with 1, 2, &c., to 60. The motion of this wheel is shown by the index y on the forew-frame G.

H represents a part of the stand, having a parallel slit in the direction towards the centre of the wheel, large enough to receive the upper part of the conical brass pillar P, which carries the screw and its frame; and as the resistance, when the wheel is moved by the endless screw, is against that side of the slit H which is towards the left hand, that side of the slit is faced with brass, and the pillar is pressed against it by a steel spring on the opposite side: by this means the pillar is strongly supported laterally, and yet the screw may be easily pressed from or against the circumference of the wheel, and the pillar will turn freely on its axis to take any direction given it by the frame L.

At each corner of the piece I are screws n of tempered steel, having polished conical points: two of them turn in conical holes in the screw-frame near o, and the points of the other two screws turn in holes in the piece Q; the screws p are of steel, which being tightened, prevent the conical pointed screws from unturning when the frame is moved.

L is a brass frame, which serves to connect the endless screw, its frame, &c., with the centre of the wheel: each arm of this frame is terminated by a steel screw, that may be passed through any of the holes q in the piece Q, as the thickness of work to be divided on the wheel may require, and are fastened by the finger-nuts r.

At the other end of this frame is a flat piece of tempered steel b, wherein is an angular notch: when the endless screw is pressed against the teeth on the circumference of the wheel, which may be done by turning the finger-screw s, to press against the spring t; this notch embraces and presses against the steel arbor d. This end of the frame may be raised or depressed by moving the prismatic slide u, which may be fixed at any height by the four steel screws v.

The bottom of this slide has a notch k, whose plane is parallel to the endless screw; and by the point of the arbor d resting in this notch, this end of the frame is prevented from tilting. The screw S is prevented from unturning, by tightening the finger-nut w.

The teeth on the circumference of the wheel were cut by the following method:

Having considered what number of teeth on the circumference would be most convenient, which in this engine is 2160, or 360 multiplied by 6, I made two screws of the same dimensions, of tempered steel, in the manner hereafter described, the interval between the threads being such as I knew by calculation would come within the limits of what might be turned off the circumference of the wheel: one of the screws, which was intended for ratching or cutting the teeth, was notched across the threads, so that the screw, when pressed against the edge of the wheel and turned round, cut in the manner of a saw. Then having a segment of a circle a little greater than 60 degrees, of about the same radius with the wheel, and the circumference made true, from a very fine centre, I described an arch near the edge, and set off the chord of 60 degrees on this arch. This segment was put in the place of the wheel, the edge of it was ratched, and the number of revolutions... Ramden's revolutions and parts of the screw contained between Machine. the interval of the 60 degrees were counted. The radius was corrected in the proportion of 360 revolutions, which ought to have been in 60 degrees, to the number actually found; and the radius, so corrected, was taken in a pair of beam-compasses: while the wheel was on the lathe, one foot of the compasses was put in the centre, and with the other a circle was described on the ring; then half the depth of the threads of the screw being taken in dividers, was set from this circle outwards, and another circle was described cutting this point; a hollow was then turned on the edge of the wheel of the same curvature as that of the screw at the bottom of the threads: the bottom of this hollow was turned to the same radius or distance from the centre of the wheel, as the outward of the two circles before mentioned.

"The wheel was now taken off the lathe; and the bell-metal piece D was screwed on as before directed, which after this ought not to be removed.

"From a very exact centre a circle was described fig. 1, 2, 3, on the ring C, about four-tenths of an inch within where the bottom of the teeth would come. This circle was divided with the greatest exactness I was capable of, first into five parts, and each of these into three. These parts were then bifected four times: (that is to say) supposing the whole circumference of the wheel to contain 2160 teeth, this being divided into five parts, each would contain 432 teeth; which being divided into three parts, each of them would contain 144; and this space bifected four times would give 72, 36, 18, and 9: therefore each of the last divisions would contain nine teeth. But, as I was apprehensive some error might arise from quintisection and trisection; in order to examine the accuracy of the divisions, I described another circle on the ring C, one-tenth of an inch within the former, and divided it by continual bifections, as 2160, 1080, 540, 270, 135, 67½, and 33¾; and as the fixed wire (to be described presently) crossed both the circles, I could examine their agreement at every 135 revolutions; (after ratcheting, could examine it at every 33¾): but, not finding any sensible difference between the two sets of divisions, I, for ratching, made choice of the former; and, as the coincidence of the fixed wire with an intersection could be more exactly determined than with a dot or division, I therefore made use of intersection in both circles before described.

"The arms of the frame L were connected by a thin piece of brass of three-fourths of an inch broad, having a hole in the middle of four-tenths of an inch in diameter; across this hole a silver wire was fixed exactly in a line to the centre of the wheel; the coincidence of this wire with the intersections was examined by a lens seventeenths of an inch focus, fixed in a tube which was attached to one of the arms L (A). Now a handle or winch being fixed on the end of the screw, the division marked on the end of the screw, the division marked 10 on the circle K was set to its index, and, by means of a clamp and adjusting screw for that purpose, the intersection marked 1 on the circle C was set exactly to coincide with the fixed wire; the screw was then carefully pressed against the circumference of the wheel, by turning the finger-screw S; then, removing the clamp, I turned the screw by its handle nine revolutions, till the intersection marked 240 came nearly to the wire; then, unturning the finger-screw S, I released the screw from the wheel, and turned the wheel back till the intersection marked 2 exactly coincided with the wire, and, by means of the clamp before mentioned, the division 10 on the circle being set to its index, the screw was pressed against the edge of the wheel by the finger-screw S; the clamps were removed, and the forew turned nine revolutions till the intersection marked 1 nearly coincided with the fixed wire; the screw was released from the wheel by unturning the finger-screw S as before, the wheel was turned back till the intersection 3 coincided with the fixed wire; the division 10 on the circle being set to its index, the screw was pressed against the wheel as before, and the screw was turned nine revolutions, till the intersection 2 nearly coincided with the fixed wire, and the screw was released; and I proceeded in this manner till the teeth were marked round the whole circumference of the wheel. This was repeated three times round, to make the impression of the screw deeper. I then ratched the wheel round continually in the same direction without ever disengaging the screw; and, in ratching the wheel about 300 times round, the teeth were finished.

"Now it is evident, if the circumference of the wheel was even one tooth or ten minutes greater than the screw would require, this error would in the first instance be reduced to \( \frac{1}{25} \) part of a revolution, or two seconds and a half; and these errors or inequalities of the teeth were equally distributed round the wheel at the distance of nine teeth from each other. Now, as the screw in ratching had continually hold of several teeth at the same time, and, these constantly changing, the above-mentioned inequalities soon corrected themselves, and the teeth were reduced to a perfect equality. The piece of brass which carries the wire was now taken away, and the cutting screw was also removed, and a plain one (hereafter described) put in its place: on one end of the screw is a small brass circle, having its edge divided into 60 equal parts, and numbered at every fifth division, as before mentioned. On the other end of the screw is a ratchet-wheel C, having 60 teeth, covered by the hollowed circle d, which carries two Fig. 5a clicks that catch upon the opposite sides of the ratchet when the screw is to be moved forwards. The cylinder S turns on a strong steel arbor F, which passes through and is firmly screwed to the piece Y: this piece, for greater firmness, is attached to the screw-frame G by the braces v; a spiral groove or thread Fig. 4a is cut on the outside of the cylinder S, which serves both for holding the string, and also giving motion to the lever J on its centre by means of a steel tooth n, that works between the threads of the spiral. To the lever is attached a strong steel pin m, on which a brass socket r turns: this socket passes through a slit in

(A) The intersections are marked for the sake of illustration, though properly invisible, because they lie under the brass plate. Ramden's in the piece \( p \), and may be tightened in any part of the slit by the finger-nut \( f \): this piece serves to regulate the number of revolutions of the screw for each tread of the treadle \( R \).

"T is a brafs box containing a spiral string; a strong gut is fastened and turned three or four times round the circumference of this box, the gut then passes several times round the cylinder \( S \), and from thence down to the treadle \( R \). Now, when the treadle is pressed down, the string pulls the cylinder \( S \) round its axis, and the clicks catching hold of the teeth on the ratchet carry the screw round with it, till, by the tooth \( n \) working in the spiral groove, the lever \( J \) is brought near the wheel \( d \), and the cylinder stopped by the screw-head \( x \) striking on the top of the lever \( J \); at the same time the spring is wound up by the other end of the gut passing round the box \( T \). Now, when the foot is taken off the treadle, the spring unbending itself pulls back the cylinder, the clicks leaving the ratchet and screw at rest till the piece \( s \) strikes on the end of the piece \( p \): the number of revolutions of the screw at each tread is limited by the number of revolutions the cylinder is allowed to turn back before the stop strikes on the piece \( p \).

"When the endless screw was moved round its axis with a considerable velocity, it would continue that motion a little after the cylinder \( S \) was stopped: to prevent this, the angular lever \( n \) was made; that when the lever \( J \) comes near to stop the screw \( x \), it, by a small chamfer, presses down the piece \( x \) of the angular lever; this brings the other end \( n \) of the same lever forwards, and stops the endless screw by the steel pin \( x \) striking upon the top of it; the foot of the lever is raised again by a small spring pressing on the brace \( v \).

"D, two clamps, connected by the piece \( z \), slide one on each arm of the frame \( L \), and may be fixed at pleasure by the four finger-screws \( t \), which press against steel springs to avoid spoiling the arms: the piece \( q \) is made to turn without shake between two conical pointed screws \( f \), which are prevented from unturning by tightening the finger-nuts \( N \).

"The piece \( M \) is made to turn on the piece \( q \) by the conical pointed screws \( f \) resting in the hollow centers \( e \).

"As there is frequent occasion to cut divisions on inclined planes, for that purpose the piece \( y \), in which the tracer is fixed, has a conical axis at each end, which turns in half holes: when the tracer is set to any inclination, it may be fixed there by tightening the steel screws \( g \).

**Description of the Engine by which the endless screw of the Dividing Engine was cut.**

"Fig. 9 represents this engine of its full dimensions seen from one side.

"Fig. 8. the upper side of the same as seen from above.

"A represents a triangular bar of steel, to which the triangular holes in the pieces \( B \) and \( C \) are accurately fitted, and may be fixed on any part of the bar by the screws \( D \).

"E is a piece of steel whereon the screw is intended to be cut; which, after being hardened and tempered, has its pivots turned in the form of two frustums of cones, as represented in the drawings of the dividing engine (fig. 5.). These pivots were exactly fitted to the half holes \( F \) and \( T \), which were kept together by the screw \( Z \).

"H represents a screw of untempered steel, having a pivot \( I \), which turns in the hole \( K \). At the other end of the screw is a hollow centre, which receives the hardened conical point of the steel pin \( M \). When this point is sufficiently pressed against the screw, to prevent its shaking, the steel pin may be fixed by tightening the screws \( Y \).

"N is a cylindric nut, moveable on the screw \( H \); which, to prevent any shake, may be tightened by the screws \( O \). This nut is connected with the saddle piece \( P \) by means of the intermediate universal joint \( W \), through which the arbor of the screw \( H \) passes. A front view of this piece, with the section across the screw arbor, is represented at \( X \). This joint is connected with the nut by means of two steel slips \( S \), which turn on pins between the cheeks \( T \) on the nut \( N \). The other ends of these slips \( S \) turn in like manner on pins \( a \). One axis of this joint turns in a hole in the cock \( b \), which is fixed to the saddle-piece; and the other turns in a hole \( d \), made for that purpose in the same piece on which the cock \( b \) is fixed. By this means, when the screw is turned round, the saddle-piece will slide uniformly along the triangular bar \( A \).

"K is a small triangular bar of well-tempered steel, which slides in a groove of the same form on the saddle-piece \( P \). The point of this bar or cutter is formed to the shape of the thread intended to be cut on the endless screw. When the cutter is set to take proper hold of the intended screw, it may be fixed by tightening the screw \( e \), which presses the two pieces of brass \( G \) upon it.

"Having measured the circumference of the dividing-wheel, I found it would require a screw about one thread in a hundred coarser than the guide-screw \( H \). The wheels on the guide-screw arbor \( H \), and that on the steel \( E \), on which the screw was to be cut, were proportioned to each other to produce that effect, by giving the wheel \( L \) 168 teeth, and the wheel \( Q \) 200. These wheels communicated with each other by means of the intermediate wheel \( R \), which also served to give the threads on the two screws the same direction.

"The saddle-piece \( P \) is confined on the bar \( A \) by means of the pieces \( g \), and may be made to slide with a proper degree of tightness by the screws \( n \).

For Ramden's equatorial or portable observatory, see Optics, No 89, and Astronomy, No 364. See also a long account of an equatorial instrument made by Mr Ramden by the direction of Sir George Shuckburgh in the Philosophical Transactions for 1793, art. x. p. 67. In this instrument the circle of declinations is four feet in diameter, and may be observed nearly to a second. The glass is placed between six pillars, which form the axis of the machine, and turn round by two pivots placed on two blocks of stone. See also Barometer.