Graduation is the action of receiving or conferring an academic degree or the ceremony that is sometimes associated, where students become Graduates. Before the graduation, candidates are referred to as Graduands. The date of graduation is often called degree day. The graduation itself is also called commencement, convocation or invocation. At the University of Cambridge, the occasion on which most graduands receive their BA degree is known as general admission. After degree completion, graduates can be referred to by their graduating year. In the United States and Canada, it is also used to refer to the advancement from a primary or secondary school level.
When ceremonies are associated, they usually include a procession of the academic staff and candidates. Beginning at the secondary school level in the United States, the candidates will almost always wear academic dress, and increasingly faculty will do the same. At the college and university level, the faculty will usually wear academic dress at the formal ceremonies, as will the trustees and degree candidates. "Graduation" at the college and university level occurs when the presiding officer confers degrees upon candidates, either individually or en masse, even if graduates physically receive their diploma later at a smaller college or departmental ceremony.
Graduation ceremonies are in March followed by entrance ceremonies in April. This coincides with the Japanese school calender and the Japanese government fiscal year. Graduation is also related to the changing of the seasons and is often reflected in the ceremony decor and related advertising (i.e. new uniforms, backpacks, school supplies...).
Trends in a Japanese graduation ceremony remain similar throughout the nation. It takes place in a school auditorium, agora, or gymnasium. Wide, red and white, striped banners are hung to cover the walls and doors. Three large flags hang on the stage. Flag of Japan, city, and the school. A bonsai tree (usually rented)is next to the podium. Chairs are reserved for parents, as well as local officials, special guests including teachers from previous years. The underclassman bring in there classroom chairs and are seated in the middle, in front of the graduating class. Elementary school students wear their new Junior High uniforms. Secondary schools wear their current uniform. Sometimes high school students must buy a graduation uniform.
Underclassmen, parents, and teachers are seated first. The vice principal leads in the local officials and special guests. The teachers all stand and bow to them as the enter. When the graduates enter, they are led by their homeroom teacher. The walk in straight lines and make 90 degree turns if necessary. They walk very slow and deliberate. The students line up next to their chairs and wait for the home room teacher to signal they are all here and they all sit at once. The homeroom teacher joins the rest of the teachers.
Someone leading the ceremony asks everyone to stand and bow towards the Japanese flag. This is coordinated with three chords on the piano. 1. prepare 2. bow 3. return upright. This is followed by singing. Singing plays a large part in Japanese schools and graduation is no different. The first song is Japan's national anthem, followed by the city song and the school song. The latter two are usually printed in the program. This is the last time the graduating class will sing the school song together. This is followed by the portion of the program for giving out diplomas.
Elementary and Junior High School (the order of these events can change)
After singing the principle, wearing a Japanese style tuxedo, goes to the podium. Usually a female teacher wearing a kimono brings out diplomas on a large tray. The home room teacher announces the students names, who queue up walking in straight lines, 90 degree turns, at a deliberately slow pace. At some Elementary schools the students give a short speech about what they want to do at Junior High before receiving their diplomas. The principle reads the diploma out loud to everyone before handing it to the first student. When the students name is called, they walk to the podium exchange bows with the principal, and step forward. The principal turns the open diploma around to face the student and offers it to the student. The student receives the diploma by using their left hand first, and then their right hand before pulling it towards them. The student steps back and exchanges bows with the principle. The student then slowly closes the diploma and folds it under their left hand before turning and walking away. On the way back to their seats, the students top and bow to the special guests.
After all the students are seated again the principle makes a speech. The speeches are written vertically from top to bottom, right to left on a fan folded piece of paper. The principles speech is followed by the head of the PTA. At junior high school an underclassmen may give a speech thanking the graduating students for things like being good senpai. And this is followed by a student speech from the student president.
After speeches students sing to each other, lower class to upper class, and upper class to lower class. And then everyone sings together. Some of the possible songs are: Tabidachi no hi ni, Sayonara, "Until the World is One" by Ya-ya-yah and Sakura . Students may give set group speeches, as if a dialogue between the lower class and upper class. Students take turns yelling out parts of the dialogue sometimes being accompanied by everyone or a few other students in unison. This might happen before, in-between and/or after the songs.
Senior High School
The homeroom teacher for each class calls out the names of his or her students in the usual gender-split alphabetical order. This means that boys are called out in alphabetical order first, then the girls. Upon hearing their names, the students say はい (Hai) or "Yes" and remain at attention until all students have been called. Recently some schools have discontinued splitting the class by gender. Both the national anthem and school song are sung by everyone. The head of the student council reads a short congratulatory address to the graduates. This is different from a valedictorian speech. Unlike a valedictorian's speech, it is somewhat pre-set and heavily edited by the teachers responsible for the ceremony. Afterwards, the principal launches into a long-winded speech as is the tradition in most schools. Perseverance, hard work and patience are the most common themes brought up on the occasion.
The principal might wear black tie, complete with handkerchief and white gloves. The student’s ID number and name are read out loud, the diploma is handed over in full size (not rolled-up). The student receives it with both hands, raises it up in the air and bows to the principal before leaving the stage. There can be background music playing in the meantime, either from tape or CD, or provided by the school's brass band. Common songs include "Aogeba tōtoshi" and "Hotaru no hikari" (Sung to the tune of Auld Lang Syne)
Once the diplomas have been all handed out, a few more announcements and speeches are made, by PTA (parent-teacher association) representatives or someone from the municipal or local government, depending on the school’s status. To the sound of another march, the students leave the auditorium and go back to their class for a final address by their homeroom teacher. During that time, the rest of the school, teachers and students alike, proceed to undress the auditorium, put the chairs away and clean up. A few moments later, the graduates are free to roam around the school, in and out of the teachers’ office, saying their goodbyes to their favorite teachers and reminiscing the good times. Although some tears can be shed at the time, and genuine smiles are seen on all faces, the whole process remains stiff by Western standard. There are no handshakes or hugs to be seen, but instead a lot of bowing and sniffling.
The school calendar does not end with graduation. The next business day after the ceremony 1st and 2nd year students continue classes. This is anywhere from a few days to a few weeks.
In Junior High and High School, the graduation ceremony doesn't get as much importance, only in a few private schools. However, in college, particularly in the National Autonomous University of Mexico and National Polytechnic Institute, the graduation ceremony takes place in a very similar way to the USA.
Graduations from elementary, middle, and preparatory schools (6th,9th and 12th grades respectively) are usually accompanied by a school-organized dance, "ball" -a rehearsed dance routine for parents and family members- and a parent-organized mass, usually in a local Roman Catholic Church.
At the University of the Witwatersrand (WITS), the graduation ceremonies are formal affairs, which include an academic procession by the faculty staff. The WITS choir is always present, and as a fun twist once the academic procession has left the hall, the song "I Got You (I Feel Good)" by James Brown is played over the loudspeakers.
In the United Kingdom, unlike the United States, students do not usually 'graduate' from school below university level. They will normally leave secondary school, high school or sixth form college (if applicable) with specific qualifications, often GCSEs and A-levels respectively (Standard Grades and Higher National Courses in Scotland). However, these are not diplomas and are not necessarily presented in a formal ceremony.
Many university graduation ceremonies in the United Kingdom begin with a procession of academics, wearing academic dress. This procession is accompanied by music, and a ceremonial mace is often carried. After this, an official reads out the names of the graduates one by one, organized by class of degree or by subject. When their names are called, the graduates walk across the stage to shake hands with a senior official, often the university's Chancellor or the vice-chancellor. Graduands wear the academic dress of the degree they are receiving. At Oxford, however, they wear the dress of their status before graduating (their previous degree, or undergraduate academic dress), afterwards changing into the dress of the degree they have just received. Serving members of the armed forces may wear their military uniform underneath. Member institutions of the University of Wales hold their graduation ceremonies almost entirely in the Welsh language. Some of the older universities may hold their graduation ceremonies in Latin, even though few students understand this language. The Latin section of the ceremony may include a rendition of an anthem, sometimes called the unofficial anthem of all universities, the De Brevitate Vitae, also known as The Gaudeamus.
At the University of Cambridge, each graduation is a separate act of the university's governing body, the Regent House, and must be voted on as with any other act. A formal meeting of the Regent House, known as a Congregation, is held for this purpose.
Graduates receiving an undergraduate degree wear the academical dress that they were entitled to before graduating: for example, most students becoming Bachelors of Arts wear undergraduate gowns and not BA gowns. Graduates receiving a postgraduate degree (e.g. PhD or Master's) wear the academical dress that they were entitled to before graduating, only if their first degree was also from the University of Cambridge; if their first degree is from another university, they wear the academical dress of the degree that they are about to receive, the BA gown without the strings if they are under 24 years of age, or the MA gown without strings if they are 24 and over.
At Durham University the graduation ceremony is known as Congregation and takes place in Durham Cathedral, with graduands processing across Palace Green from the Great Hall of University College.
Due to the large number and geographical dispersion of students, unlike most UK universities, degree ceremonies at the Open University are not the occasion on which degrees are formally conferred. This happens in absentia at a joint meeting of the University's Council and Senate ahead of the ceremony. The University's ceremonies – or "Presentations of Graduates" – occur during the long summer throughout Britain and Ireland, as well as one ceremony in Versailles.
In the United States, besides "commencement," the term "graduation" is also used in schools below university level such as the high school, middle school and even kindergarten and preschool ceremonies.
The American Council on Education is the authority on academic regalia in the US, and has developed an Academic Ceremony Guide that is generally followed by most institutions of higher learning. The ceremony guide and the related Academic Costume Code provide the core of academic ceremony traditions in the US. High school graduation regalia, however, is generally left to the discretion of the school and often cap and gown colors vary from school to school.
At many large US institutions, where many hundreds of degrees are being granted at once, the main ceremony (commencement) involving all graduates in a sports stadium, amphitheater, parade ground or lawn, or other large – often outdoor – venue is usually followed, but sometimes preceded, by smaller ceremonies (diploma ceremony) at sites on or around campus where deans and faculty of each academic organization (college, academic department, program, etc.) distribute diplomas to their graduates. Another means of handling very large numbers of graduates is to have several ceremonies, divided by field of study, at a central site over the course of a weekend instead of one single ceremony. At large institutions the great number of family members and guests that each graduating student wishes to attend may exceed the capacity of organizers to accommodate. Universities try to manage this by allocating a specified number of graduation tickets to each student that will be graduating.
It is also common for graduates not to receive their actual diploma at the ceremony but instead a certificate indicating that they participated in the ceremony or a portfolio to hold the diploma in. At the high school level, this allows academic administrators to withhold diplomas from students who are unruly during the ceremony; at the college level, this allows students who need an additional quarter or semester to satisfy their academic requirements to nevertheless participate in the official ceremony with their cohort before receiving their degree. In addition, with large numbers of students receiving diplomas and often no specific order they walk in, it is impossible for their actual diplomas to be given to them at the ceremony, thus them receiving simply a blank diploma to be filled later.
At most colleges and universities in the US, a faculty member or dean will ceremoniously recommend that each class of candidates (often by college but sometimes by program/major) be awarded the proper degree, which is then formally and officially conferred by the president or other institutional official. Typically, this is accomplished by a pair of short set speeches by a senior academic official and a senior institutional official:
For students receiving an advanced degree, many colleges include a Hooding Ceremony in their commencement program, in which the students get to wear a hood. A hood is a symbolic garment, which is worn draped around the neck and over the shoulders, displayed down the back with the lining exposed. The hood’s length signifies the degree; with the institution's colors in the lining and a velvet trim in a color that signifies the scholar’s field. The hood is a part of traditional academic dress whose origins date back many centuries. Today, the hood is considered by some to be the most expressive component of the academic costume. Today’s hoods have evolved from a practical garment to a symbolic one. At Fordham University, graduates of a college put on the hood by themselves en mass after the university president confers the degree upon them from the podium. This is called 'to self-hood'. Doctorates are hooded upon the stage.
A graduation or commencement speech, in the U.S., is a public speech given by a student or by alumnus of a university to a graduating class and their guests. Common themes of the graduation speech include wishing the graduates well in the "real world", cautioning that the world of academe is a special place where they were taught to think (a common variation contradicts this view). Most recently, the trend has been to find a celebrity (often one with no apparent connection to the specific institution or education in general) or a politician to deliver the speech. Notable exceptions are Columbia University, Davidson College, and Belmont University, where the tradition has been that only the current university president gives the commencement address. Though there is only one commencement, individual colleges and schools of Columbia often invite a speaker at separate graduation ceremonies held earlier or on another day, however.
GRADUATION (see also Graduate), the art of dividing straight scales, circular arcs or whole circumferences into any required number of equal parts. It is the most important and difficult part of the work of the mathematical instrument maker, and is required in the construction of most physical, astronomical, nautical and surveying instruments.
The art was 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 determined, which created the necessity for strictly accurate means of measuring linear and angular magnitudes. Then it was seen that graduation was an art which required special talents and training, and the best artists gave great attention to the perfecting of astronomical instruments. Of these may be named Abraham Sharp (1651-1742), John Bird (1709-1776), John Smeaton (1724-1792), Jesse Ramsden (1735-1800), John Troughton, Edward Troughton (1753-1835), William Simms (1793-1860) and Andrew Ross.
The first graduated instrument must have been done by the hand and eye alone, whether it was in the form of a straightedge 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. Hence graduation 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. Graduation may therefore be treated under the three heads of original graduation, copying and machine graduation. Original Graduation. - In regard to the graduation of straight scales elementary geometry provides the means of dividing a straight line into any number of equal parts by the method of continual bisection; but the practical realization of the geometrical construction is so difficult as to render the method untrustworthy. This method, which employs the common diagonal scale, was used in dividing a quadrant of 3 ft. radius, which belonged to Napier of Merchiston, and which only read to minutes - a result, according to Thomson and Tait (Nat. Phil.), " 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 done 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-compass 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 glass. 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, &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. In this method, any error at starting will be multiplied at each division by the 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.
The first example of note is the 8-ft. mural circle which was graduated by George Graham (1673-1751) for Greenwich Observatory in 1725. In this two concentric arcs of radii 96.85 and 95.8 in. respectively were first described by the beam-compass. On 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 beamcompass as was used to describe the inner arc, laid off from o°, the point 60° was at once determined. With the points o° 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 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 indicating 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 beam-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-ft. 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 ooi of an inch, and which he had previously graduated by continual bisections. With the beamcompass an arc of radius 95.938 in. was first drawn. From this radius the chords of 30°, 15°, 10° 20', 4 0 40' and 42° 40' were computed, 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 o° 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'X2 10, the final divisions of 85° 20' were found by continual bisections. For the remainder of the quadrant beyond 85° 20', containing 56 divisions 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 o° 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 gave the point 86° 15'. Further, the arc between 82° 30' and 86° 15' trisected, and twothirds 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' X2 10, and so can be finally divided by continual bisection.
The method of original graduation discovered by Edward Troughton is fully described in the Philosophical Transactions for 1809, as employed by himself to divide a meridian circle of 4 ft. 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 frictional 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. While 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 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 consequence of the error of each dot being ascertained on the supposition that its neighbours are all correct. For this purpose two microscopes (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 were fixed so that their cross-wires respectively bisected the dots o 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 o be found to coincide with B, then the two dots were 180° apart. If not, the cross wire of B was moved till it coincided with dot o, and the number of divisions of the micrometer head noted. Half this number gave clearly the error of dot 128, and it was tabulated or - according as the arcual distance between o and 128 was found to exceed or fall short of the remaining part of the circumference. The microscope B was now shifted, A remaining opposite dot o 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 o 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. With 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. 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: 2 (x a +x c) +z = the real error of dot b, where x a is the real error of dot a, x, the real error of dot c, and z the apparent error of dot b midway between a and c. 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 last part of Troughton's process was to employ them to cut the final divisions of the circle, which were to be spaces of 5' each. Now the mean interval between any two dots is 360 0 /256 = 5' X 161, and hence, in the final division, this interval must be divided into 161 equal parts. 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 was sufficiently loose to allow of its being moved back by hand to any position without affecting the roller. 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 360° X16/256 =22° 30'. This interval was therefore divided by 168, and a space equal to 16 of the parts taken. This was laid off on the arc of the sector and divided into 16 equal parts, each equal to 1° 20'; and, to provide for the necessary $ths 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 o. The frame was 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 is evident when we reflect that the distance between them must be 6th of a division on the section which by construction is 1° 20'. In this way 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 8th short of zero. Starting from this position the divisions between dots 1 and 2 were filled in, and then H was adjusted to the real error of dot 2, and the sector brought back to its proper division before commencing the third course. Proceeding in this manner through the whole circle, the microscope H was finally found with its wire at zero, and the sector with its 16th division under its microscope indicating that the circle had been accurately divided.
In graduation by copying the pattern 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 work, 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 knife 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.
It frequently happens that the divisions required on a scale are either greater or less 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 found in the following way. Take the exact ratio of a division on the'pattern to the required division on the scale. Call this ratio a. Then, if the required divisions are longer than those of the pattern, the angle is cosl a, but, if shorter, the angle is sec1 a. In the former case two operations are required before the divisions are cut: first, the square is laid on the pattern, and the corresponding divisions merely notched very faintly on the edge of the work; 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 l 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 in. or more in diameter, carefully graduated near its outer edge. It is turned quite flat, and has a steel pin fixed in its centre, and at right angles to its plane. For guiding the dividing knife an instrument called an index is employed. This is 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 central pin, the straight edge of the steel bar lies exactly along a radius. The work to be graduated is laid flat on the dividing plate, and fixed by two clamps in a position exactly concentric 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 drawing the dividing knife along the straight edge of the index.
The first dividing engine was probably that of Henry Hindley of York, constructed in 1740, and chiefly used by him for cutting the teeth of clock wheels. This was followed shortly after by an engine devised by the duc de Chaulnes;but the first notable engine 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.
Ramsden's machine consisted of a large brass plate 45 in. 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 worked, by means of which the plate could be made to turn through any required angle. Thus six turns of the screw moved the plate through I°, and nth of a turn through gi oth of a degree. On the axis of the tangent screw was placed a cylinder having a spiral groove cut on its surface. A ratchet-wheel containing 60 teeth was attached to this cylinder, and was so arranged that, when the cylinder moved in one direction, it carried the tangent screw with it, and so turned the plate, but when 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 which was attached to a treadle and the other to a counterpoise weight. When the treadle was depressed the tangent screw turned round, and when the pressure was removed it returned, in obedience to the weight, to its former position without affecting the screw. Provision was also made whereby certain stops could be placed in the way of the screw, which only allowed it the requisite amount of turning. The work to be divided was firmly fixed on the plate, and made concentric with it. The divisions were cut, while the screw was stationary, by means of a dividing knife attached to a swing frame, which allowed it to have only a radial motion. In this way the artist could divide very rapidly by alternately depressing the treadle and working the dividing knife.
Ramsden also constructed alinear 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 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 in simplifying matters of detail. The plate was originally divided by Troughton's own 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. Andrew Ross (Trans. Soc. Arts, 1830-1831) constructed a dividing machine which differs considerably from those of Ramsden and Troughton.
The essential point of difference is that, in Ross'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 between two 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.
A description is given of a dividing engine made by William Simms in the Memoirs of the Astronomical Society, 1843. Simms Dividing Engine.
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 more exact than original graduation. His machine involved essentially the same principle as Troughton's. The accompanying figure is taken by permission.
The plate A is 46 in. 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 gunmetal. 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 detached from the axis and other parts to which it has 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. H is the spiral barrel round which the catgut band is wound, one end of which is attached 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 0, 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 can be cut. 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 forward or backward upon these rails to suit circles of different diameters. The box N is movable upon the bar R, and the rod I is adjustable as to length by having a kind of telescope joint. The engine is self-acting, and can be driven either by hand or by a steam-engine or other motive power. It can be thrown in or out of gear at once by a handle seen at S.
Mention may be made of Donkin's linear dividing engine, in which a compensating arrangement is employed 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, Repsold and others in Germany, Gambey in Paris and by several other astronomical instrument-makers. A machine constructed by E. R. Watts & Son is described by G. T. McCaw, in the Monthly Not. R. A. S., January 1909.
Bird, Method of dividing Astronomical Instruments (London, 1767); Due de Chaulnes, Nouvelle Methode pour diviser les instruments de mathematique et d'astronomie (1768); Ramsden, Description of an Engine for dividing Mathematical Instruments (London, 1777) ; Troughton's memoir, Phil. Trans. (1809); Memoirs of the Royal Astronomical Society, v. 325, viii. 141, ix. 17, 35. See also J. E. Watkins, "On the Ramsden Machine," Smithsonian Rep. (1890), p. 721; and L. Ambronn, Astronomische Instrumentenkunde (1899). (J. Bi.)
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