# Encyclopædia Britannica, Ninth Edition/Graduation

GRADUATION is the name given to the art of dividing straight scales, circular arcs, or whole circumferences into any required number of equal parts. It is the most im portant and difficult part of the work of the mathematical instrument maker, and is required in the construction of most physical, astronomical, nautical, and surveying instru ments, such as thermometer scales, linear measuring instru ments, quadrants, sextants, mural circles, theodolites, &c. The art was, undoubtedly, first practised by clockmakers for cutting the teeth of their wheels at regular intervals ; but so long as it was confined to them, no particular delicacy or accurate nicety in its performance was required. This only arose when astronomy began to be seriously studied, and the exact position of the heavenly bodies to be deter mined, which created the necessity for strictly accurate means of measuring linear and angular magnitude. Then graduation began to be looked upon as an art which required special talents and training, and hence we find that all the best artists have spent their best efforts on the perfecting of astronomical instruments. Of these may be named Abraham Sharp, Bird, Smeaton, Ramsden, the Due de C haulnes, John and Edward Troughton, Simms, and Ross. It is obvious that the first graduated instrument must have been done by the hand and eye alone, whether it was in the form of a straight-edge with equal divisions, or a screw, or a divided plate ; but, once in the possession of one such divided instrument, it was a comparatively easy matter to employ it as a standard, and copy its divisions on any other article that might be desired. Hence graduation naturally divides itself into two distinct branches, original graduation and copying, which latter may be done either by the hand or by a machine called a dividing engine. We may thus speak of graduation under the three heads of original graduation, copying, and machine graduation

*Original Graduation*.—This is by far the most difficult
part of the art so difficult, indeed, and requiring such
accuracy of hand and eye, that but few in any generation
have been completely competent for the task. The earlier
astronomers graduated their own instruments, and, from
the examples that have come down to us, it must have been
very roughly done as compared with modern work.
in regard to the graduation of straight scales, we have,
by elementary geometry, the means, theoretically, of divid
ing a straight line into any number of equal parts ; but the
practical carrying out of the geometrical construction is so
beset with difficulties as to render the method untrustworthy.
This method, which employs the common diagonal scale,
was used in dividing a quadrant of 3 feet radius, which
belonged to Napier of Merchiston, and which only read to
minutes a result, say Thomson and Tait (Nat. 7Yu7.),
" giving no greater accuracy than is now attainable by the
pocket sextants of Troughton and Simms, the radius of
whose arc is little more than an inch."
The original graduation of a straight line is, in practice,
clone either by the method of continual bisection or by
stepping. In continual bisection the entire length of the
line is first laid down. Then, as nearly as possible, half
that distance is taken in the beam-cotnpass and marked off
by faint arcs from each end of the line. Should these marks
coincide the exact middle point of the line is obtained. If
not, as will almost always be the case, the distance between
the marks is carefully bisected by hand with the aid of a
magnifying gbss. The same process is again applied to the
halves thus obtained, and so on in succession, dividing the
line into parts represented by 2, 4, 8, 16, I.Y.C., till the desired
divisions are reached. In the method of stepping the
smallest division required is first taken, as accurately as
possible, by spring dividers, and that distance is then laid
off, by successive steps, from one end of the line. It is
evident that, in this method, any error at starting will be

multiplied at each division by tlie number of that division. Errors so made are usually adjusted by the dots being put either back or forward a little by means of the dividing punch guided by a magnifying glass. This is an extremely tedious process, as the dots, when so altered several times, are apt to get insufferably large and shapeless. The division of circular arcs is essentially the same in principle as the graduation of straight lines, and of this we shall now give some examples. The first example of note is the 8-feet mural circle which was graduated by Graham for the National Observatory in 1725. In this two concentric arcs of radii 96 85 and 95 - 8 inches respectively were first described by the beam-compass. Ou the inner of these the arc of 90 was to be divided into degrees and 12th parts of a degree, while the same on the outer was to be divided into 96 equal parts, and these again into 16th parts. The reason for adopting the latter was that, 96 and 16 being both powers of 2, the divisions could be got at by continual bisection alone, which, in Graham s opinion, who first employed it, is the only accurate method, and would thus serve as a check upon the accuracy 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 60 was at once determined. With the points 0" and 60 as centres successively, and a distance on the beam-compass very nearly bisecting the arc of 60, 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 gave the point 30. The chord of 60 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 these quinquesected gave degrees, the 12th parts of which 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 indi cating the required divisions were obtained, either straight strokes all directed towards the centre were drawn through them by the dividing knife, or sometimes small arcs were drawn through them by the beaui-compass having its fixed point somewhere on the line which was a tangent to the quadrantal arc at the point where a division was to be marked. The next important example of graduation was done by Bird in 1767. His quadrant, which was also 8 feet radius, was divided Into degrees and 12th parts of a degree. He employed the method of continual bisection aided by chords taken from an exact scale of equal parts, which could read to 001 of an inch, and which he had previously graduated by continual bisections. AVith the beam- compass an arc of radius 95 938 inches was first drawn. From this radius the chords of 30 , 15 , 10 20 , 4 40 , and 42 40 were com puted, and each of them by means of the scale of equal parts laid off on a separate beam-compass to be ready. The radius laid off from D gave the point 60 ; by the chord of 30 the arc of 60 was bisected ; from the point 30 the radius laid off gave the point 90 ; the chord of 15 laid off backwards from 90 gave the point 75 ; from 75 was laid off forwards the chord of 10 20 ; and from 90 was laid off backwards the chord of 4 40 ; and these were found to coincide in the point 85 20 . Now 85 20 being = 5 x 1024 = 5 x 2 10 , the final divisions of 85 20 were found by continual bisections. For the remainder of the quadrant beyond 85 20 , containing 56 divi sions of 5 each, the chord of 64 such divisions was laid off from the point 85 40 , and the corresponding arc divided by continual bisections as before. There was thus a severe check upon the accuracy of the points already found, viz., 15, 30, 60, 75, 90, which, however, were found to coincide with the corresponding points obtained by continual bisections. The short lines through the dots were drawn in the way already mentioned. The next eminent artists in original graduation are the brothers John and Edward Troughton. The former was the first to devise a means. of graduating the quadrant by continual bisection without the aid of such a scale of equal parts as was used by Bird. His method was as follows : The radius of the quadrant laid off from gave the point 60. This arc bisected and the half laid off from 60 gave the point 90. The arc between 60 and 90 bisected gave 75; the arc between 75 and 90 bisected gave the point 82 30 , and the arc between 82 30 and 90 bisected g,ve the point 86 15 . Further, the arc between 82 30 and 86" 15 trisected, and two- thirds of it taken beyond 82 30 , gave the point 85, while the arc between 85 and 86 15 also trisected, and one-third part laid off beyond 85, gave the point 85 25 lastly, the arc between 85 and 85 25 being quinquesected, and four-fifths taken beyond 85, gave 85 20 , which as before is = 5 x 2 10 , and so can be finally divided by continual bisection.

*Edward Troughton's Method*.—The method of original graduation
discovered by Edward Troughton is fully described in the Philo
sophical Transactions for 1809, as employed by himself to divide a
meridian circle of 4 feet radius. The circle was first accurately
turned both on its face and its inner and outer edges. A roller
was next provided, of such diameter that it revolved 16 times on
its own axis while made to roll once round the outer edge of the
circle. This roller, made movable on pivots, was attached to a
frame-work, which could be slid freely, yet tightly, along the circle,
the roller meanwhile revolving, by means of fiictioual contact, on
the outer edge. The roller was also, after having been properly
adjusted as to size, divided as accurately as possible into 16 equal
parts by lines parallel to its axis. AVhile the frame carrying the
roller was moved once round along the circle, the points of contact
of the roller-divisions with the circle were accurately observed by
two microscopes attached to the frame, one of which (which we
shall call H) commanded the ring on the circle near its edge,
which was to receive the divisions, and the other viewed the roller-
divisions. The exact points of contact thus ascertained were marked
with faint dots, and the meridian circle thereby divided into 256
very nearly equal parts.
The next part of the operation was to find out and tabulate the
errors of these dots, which are called apparent errors, in conse
quence of the error of each dot being ascertained on the supposition
of its neighbours being all correct. For this purpose two micro
scopes (which we shall call A and B) were taken, with cross wires
and micrometer adjustments, consisting of a screw and head
divided into 100 divisions, 50 of which read in the one and 50 in
the opposite direction. These microscopes, A and B, were fixed so
that their cross-wires respectively bisected the dots and 128,
which were supposed to be diametrically opposite. The circle was
now turned half-way round on its axis, so that dot 128 coincided
with the wire of A, and, should dot be found to coincide with B,
then the two dots were sure to be 180 apart. If not, the cross
wire of B was moved till it coincided with dot 0, and the number
of divisions of the micrometer head noted. Half this number gave
clearly the error of dot 128, and it was tabulated -for -according
as the arcual distance between and 128 was found to exceed or
fall short of the remaining part of the circumference. The micro
scope B was now shifted, A remaining opposite dot as before, till
its wire bisected dot 64, and, by giving the circle one quarter of a
turn on its axis, the difference of the arcs between dots and 64
and between 64 and 128 was obtained. The half of this difference
gave the apparent error of dot 64, which was tabulated with its
proper sign. AVith the microscope A .still in the same position the
error of dot 192 was obtained, and in the same way by shifting B to
dot 32 the errors of dots 32, 96, 160, and 224 were successively
ascertained. By proceeding in this way the apparent errors of all
the 256 dots were tabulated.
From this table of apparent errors a table of real errors was drawn
up by employing the following formula:
^(x a + x e ) + z = te real error of dot b,
where x a is the real error of dot a, x c the real error of dot c, and *
the apparent error of dot b midway between a and c. Thus having
got the real errors of any two dots, the table of apparent errors gives
the means of finding the real errors of all the other dots.
The above formula is easily, derived as follows :
Let a and c be the number of micrometer divisions from to dots
a and c respectively; and let b be the number of similar divisions
from to the point b, supposed to be midway between a and c.
Also let x a and x c be the real errors of dots a and c, and z the
apparent error of b. Then we have
z=b-a- ^(c- a) = b- ! 2 (
Now the real positions of a and c are a ,r and c a c respectively.
Therefore the real position of b, the point midway between a and
and c, is
= a - x a + i { c - y c - (a - x a ) ]
= 4( + e)-i(a!a + !,,).
Therefore the real error of b
= b - real position of b
Having obtained the 256 dots and their real errors, the third and
last part of Trough ton s process was to employ them to cut the
final divisions of the circle, which were to be spaces of f/ each.
Now the mean interval between any two dots is _ = 5 x 16|,
256
and hence, in the final division, this interval must be divided into
16| equal parts. In order to accomplish this a small instrument,
called a subdividing sector, was provided. It was formed of thin
brass and had a radius about four times that of the roller, but made
adjustable as to length. The sector was placed concentrically on
the axis, and rested on the upper end of the roller. It turned by
frictional adhesion along with the roller, but at the same time was
sufficiently loose to allow of its being moved back by hand to any
position without affecting the roller. Now it is evident that, while
the roller passes over an angular space equal to the mean interval between two dots, any point of the sector must pass over 16
times that interval, that is to say, over an angle represented by
?^L x 16 = 22 30 . This interval was therefore divided by 16g, and
256
a space equal to 16 of the parts taken. This was laid off on the arc
of the sector and carefully divided into 16 equal parts, each equal
to 1 20 ; and, in order to provide for the necessary gths of a division,
there was laid off at each end of the sector, and beyond the 16
equal parts, two of these parts each subdivided into 8 equal parts,
A microscope with cross wires, which we shall call I was placed on
the main frame, so as to command a view of the sector divisions,
just as the microscope H viewed the final divisions of the circle,
Before the first or zero mark was cut, the zero of the sector was
brought under I and then the division cut at the point on the circle
indicated by H, which also coincided with the dot 0. The frame
Avas then slipped along the circle by the slow screw motion provided
for the purpose, till the first sector-division, by the action of the
roller, was brought under I. The second mark was then cut on the
circle at the point indicated by H. That the marks thus obtained
are 5 apart, as they should be, is evident when we reflect that the
distance between them must be ^th of a division on the section
which by construction is 1 20 . By proceeding in this Avay the
first 16 divisions were cut; but before cutting the 17th it was
necessary to adjust the micrometer wires of H to the real error of
dot 1, as indicated by the table, and bring back the sector, not to
zero, but to th short of zero. Starting from this position the
divisions between dots 1 and 2 were next filled in, and then H was
adjusted to the real error of dot 2, and the sector brought back to
its pi oper division before commencing the third course. By pro
ceeding in this manner through the whole circle, the microscope H
was finally found with its Avire at zero, and the sector Avith its 16th
division under its microscope indicating that the circle had been
accurately divided.

*Copying*.—In graduation by copying the first requisite is
a pattern, which must be either an accurately divided
straight scale, or an accurately divided circle, commonly
called a dividing plate.
In copying a straight scale the pattern and scale to be
divided, usually called the vork, are first fixed side by side,
with their upper faces in the same plane. The dividing
square, which closely resembles an ordinary joiner s square,
is then laid across both, and the point of the dividing kuife
dropped into the zero division of the pattern. The square
is now moved up close to the point of the knife; and, while
it is held firmly in this position by the left hand, the first
division on the work is made by drawing the knife along
the edge of the square with the right hand. Great care
must bs taken that the knife is held exactly in the same
position in cutting the division and in setting the square.
It frequently happens that the divisions required on a
scale are either greater or lass than those on the pattern.
To meet this case, and still use the same pattern, the work
must be fixed at a certain angle of inclination with the
pattern. This angle is easily found in the following way.
Take the exact ratio of a division on the pattern to the re
quired division on the scale. Call this ratio a. Then, if
the required divisions are longer than those of the pattern,
the angle is cos~ l a, but, if shorter, the angle is sec~ J a. In
the former case two operations are required before the divi
sions are cut : first, the square is laid on the pattern, and
the corresponding divisions merely notched very faintly on
the edge of the vork; and, secondly, the square is applied to
the work and the final divisions drawn opposite each faint
notch. In the second case, that is, when the angle is sec^a,
the dividing square is applied to the work, and the divisions
cut when the edge of the square coincides with the end of
each division on the pattern.
In copying circles use is made of the dividing plate.
This is a circular plate of brass, of 36 inches or more in
diameter, carefully graduated near its outer edge. It is
turned quite flat, and has a steel pin fixed exactly in its
centre, and at right angles to its plane. For guiding the
dividing knife an instrument called an index is employed,
This consists of a straight bar of thin steel of length equal to
the radius of the plate. A piece of metal, having a V notch
with its angle a right angle, is riveted to one end of the bar
in such a position that the vertex of the notch is exactly in a
line with the edge of the steel bar. In this way, when the
index is laid on the plate, with the notch grasping the cen
tral pin, the straight edge of the steel bar lies exactly along
a radius. The Avork to be graduated is laid flat on the divid
ing plate, and fixed by two clamps in a position exactly con
centric with it. The index is now laid on, with its edge
coinciding with any required division on the dividing plate,
and the corresponding division on the work is cut by draw
ing the dividing knife along the straight edge of the index.

*Machine Graduation*.—The first dividing engine was
probably that of Henry Hindley of York, constructed in.
1740, and used for the most part by him for cutting the
teeth of clock wheels. This was followed shortly after by
an engine devised by the Due de Chaulnes ; but the first
engine which obtained distinct notoriety was that made
by Ramsden, of which an account was published by the
Board of Longitude in 1777. He was rewarded by that
board with a sum of 300, and a further sum of 315 was
given to him on condition that he would divide, at a certain
fixed rate, the instruments of other makers. The essential
principles of Ramsden s machine have been repeated in
almost all succeeding engines for dividing circles, and it
Avill be well, therefore, to give a brief description of it.
It consisted of a large brass plate 45 inches in diameter, carefully
turned, and movable on a vertical axis. The edge of the plate was
ratched with 2160 teeth, into which a tangent screw Avorked, by
means of Avhich the plate could be made to turn through any
required angle. Thus six turns of the screw moved the plate
through 1, and ^th O f a turn through -^-^th of a degree. On the
axis of the tangent screAV was placed a cylinder having a spiral
groove cut on its surface. A ratchet-wheel containing 60 teeth
Avas attached to this cylinder, and Avas so arranged that, when the
cylinder moved in one direction, it carried the tangent screAV with
it, and so turned the plate, but Avhen it moved in the opposite
direction, it left the tangent screw, and with it the plate,
stationary. Round the spiral groove of the cylinder a catgut band
was wound, one end of A T hich Avas attached to a treadle and the
other to a counterpoise Aveight. When the treadle Avas depressed
the tangent screw turned round, and when the pressure was
removed it returned, in obedience to the Aveight, to its former posi
tion without affecting the screw. Provision Avas also made Avhereby
certain stops could be placed in the Avay of the screAV, which only
allowed it the requisite amount of turning according to the gradua
tion required. The Avork to be divided Avas firmly fixed on the
plate, and made concentric Avith it. The divisions were cut, while
the screw Avas stationary, by means of a dividing knife attached to
a swing frame, Avhich alloAved it to have only a radial motion. In
this Avay the artist could divide very rapidly by alternately depress
ing the treadle and working the dividing knife.
Ramsden also constructed a linear dividing engine on
essentially the same principle. If we imagine the rim of
the circular plate with its notches stretched out into a
straight line and made movable in a straight slot, the
screw, treadle, &c., remaining as before, we shall get a very
good idea of the linear engine.
In 1793 Edward Troughton finished a circular dividing
engine, of which the plate was smaller than in Ramsden s, and
which differed considerably otherwise in simplifying matters
of detail. The plate was originally divided by Troughton s
own ingenious method, already described, and the divisions
so obtained were employed to ratch the edge of the plate
for receiving the tangent screw with great accuracy.
In the Transactions of the /Society of Arts for 1830-31
there is a full description, with illustrative figures, of a
dividing engine, constructed by Andrew Ross, which differs
considerably from those of Ramsden and Troughton.
The essential point of difference is that, in Eoss s engine, the
tangent screw does not turn the engine plate ; that is done by an
independent apparatus, and the function of the tangent screw is
only to stop the plate after it has passed through the required
angular interval betAveentAvo divisions on the work to be graduated.
Round the circumference of the plate are fixed 48 projections which
just look as if the circumference had been divided into as many-
deep and somewhat peculiarly shaped notches or teeth. Through
each of these teeth a hole is bored parallel to the plane of the plate and also to a tangent to its circumference. Into these holes are
screwed steel screws with capstan heads and flat ends. The tangent
screw consists only of a single turn of a large square thread which
works in the teeth or notches of the plate. This thread is pierced
by 90 equally distant holes, all parallel to the axis of the screw,
and at the same distance from it. Into each of these holes is
inserted a steel screw exactly similar to those in the teeth, but with
its end rounded. It is the rounded and flat ends of these sets of
screws coming together that stop the engine plate at the desired
position, and the exact point can be nicely adjusted by suitably
turning the screws.
In the Memoirs of the Astronomical Society, of date June
1843, a description is given of a dividing engine made by
William Simms. From experience he became convinced
that to copy upon smaller circles the divisions which had
been put upon a large plate with very great accuracy was not
only more expeditious but better than original graduation,
and hence he determined to construct a machine which would
do this work as perfectly as possible. That machine involved
essentially the same principle as Troughton s, and, with
some slight modifications, is at present to be seen at work
in the workshop of the eminent firm of Troughton &. Simms
at Charlton near London. The accompanying figure is
taken by permission from a photograph of that instrument.

[ *Dividing Engine.* ]

The plate A is 4t> inches in diameter, and is composed of gun-metal cast in one solid piece. It has two sets of 5 divisions one very faint on an inlaid ring of silver, and the other stronger on the gun metal. These were put on by original graduation, mainly on the plan of Edward Troughton. One very great improvement in this engine is that the axis B is tubular, as seen at C. The object of this hollow is to receive the axis of the circle to be divided, so that it can be fixed flat to the plate by the clamps E, without having first to be de tached from the axis and other parts to which it lias already been carefully fitted. This obviates the necessity for resetting, which can hardly be done without some error. D is the tangent screw, and F the frame carrying it, which turns on carefully polished steel pivots. The screw is pressed against the edge of the plate by a spiral spring acting under the end of the lever G, and by screwing the lever down the screw can be altogether removed from contact with the plate. The edge of the plate is ratched by 4320 teeth which were cut opposite the original division by a circular cutter attached to the screw frame. II is the spiral barrel round which the catgut band is wound, one end of which is attracted to the crank L on the end of the axis J and the other to a counterpoise weight not seen. On the other end of J is another crank inclined to L and carrying a band and counterpoise weight seen at K. The object of this weight is to balance the former and give steadiness to the motion. On the axis J is seen a pair of bevelled wheels which move the rod I, which, by another pair of bevelled wheels attached to the box N, gives motion to the axis M, on the end of which is an eccentric for moving the bent lever O, which actuates the bar carrying the cutter. Between the eccentric and the point of the screw P is an undulating plate by which long divisions, at required intervals, can be cut. It will be seen that the cutting apparatus is supported upon the two parallel rails which can be elevated or depressed at pleasure by the nuts Q. Also the cutting apparatus can be moved fons ardor backward upon these rails to suit circles of different diameters. The box N is mov able upon the bar K, and the rod I is also adjustable as to length by having a kind of telescope joint. The engine is quite self-acting, and can be driven either by hand or by a steam-engine or other motive power. When driven by the latter it can be thrown in or out of gear at once by a handle seen at S. II r Simms has also provided an ingenious arrangement whereby that is done automatically directly any piece of work was finished. Spaca permits of but the mere mention of Donkiu s linear dividing engine, in which the principle employed is a compensating arrangement whereby great accuracy is obtained notwithstanding the inequalities of the screw used to advance the cutting tool. Dividing engines have also been made by Reichenbach and others in Germany, and Gambey in Paris.

In addition to those already mentioned, the following references may be given : Bird, Method of dividing Astronomical Instru ments, London, 1767 ; Due de Chaulnes, Nouvclle Methodc pour diviser Ics Instruments de Matheinatiquc ct d Astronomic, 1768 ; Kamsden, Description of an Engine for dividing Mathematical Instruments, London, 1777; Troughton s memoir, 1 ldl. Trans., 1809 ; Memoirs of the Royal Astronomical Society, vol. v. p. 325, vol. viii. p. 141, vol. ix. pp. 17 and 35 ; Holtzapll el, Turning and Mechanical Manipulation, pp. 651-955.

(J. BL.)