The watchmakers' hand book

When the rule and vernier are placed as shown in fig. 12, so that the o on both scales coincide, the successive divisions on the rule (marked with Roman numerals for distinction) will be progressively more and more in advance of the corresponding divisions on the vernier in the following proportion:—

The marks I and 1 are 1-10th apart; the marks II and 2, 2-10ths; III and 3, 3-10ths; and so on, the mark X being 10-10ths, or one complete division in advance of 10, this division being a unit on the scale.

Thus if the vernier is caused to slide along the edge of the rule, when 1 coincides with I the vernier has advanced 1-10th; when 2 coincides with II, it has advanced 2-10ths; and so on.

Let it be required to determine the distance P d, fig. 13. The division 6 on the vernier coincides with a division of the scale; hence it follows that the extremity d of the vernier is at a distance of 6-10ths millimeters from III, the next division of the scale to the left. The distance between P and d is thus 3.6 millimeters.

With a vernier showing tenths, if two consecutive divisions of the vernier fall between two divisions on the rule, and there does not appear to be a tendency towards one side more than towards another, even when observed with a strong glass, it is possible to take an approximate reading to the twentieth.

In measuring circular arcs a curved vernier is used in place of a straight one, and its graduations are made to correspond with those on the circle as shown in fig. 14.

30. Micrometer screw. By employing a micrometer screw it is possible to measure infinitesimal amounts, but the screw must be perfectly accurate, and must work without appreciable backlash or loss of time.

Assume V, fig. 15, to be such a screw, having a pitch of 1 millimeter. It will advance by this amount with each complete rotation.

To the head of the screw is attached a disc of such a size that its rim can be divided into a number of equal parts, say a hundred. These graduations may be marks on the edge or notches cut in it when an index is required to stop in them; but the index is less frequently met with than a simple divided straight-edge almost in contact with the disc. The divisions round the disc are numbered in ascending order as the points c and a separate, so that zero comes under the index or rule when these points are in contact. Readings of the numbers will thus afford a measure of the displacement of the point of the screw.

When the disc is rotated the point a will move towards or from c by 1-100th of a millimeter for each division passing under the straight-edge, and one millimeter for each complete rotation. It is thus possible to obtain the dimensions of an object when it enters without play between the two jaws to within an error of about 1-100th of a millimeter if the instrument is accurately made.

If, instead of passing the object between the two jaws, it is gripped by them, the measurement will be less exact, as no account is taken of the pressure exerted and of the elasticity. (44.)

GEOMETRICAL DRAWINGS.

31. Sketches. It is advisable from an early age to accustom oneself to make rapid freehand sketches of objects as they present themselves to the eye. Such a sketch will help in the preparation of a more exact drawing, which involves a knowledge of the several geometrical methods given below.

A drawing may be transferred, reduced or enlarged as follows:

Draw across the original picture a number of equidistant vertical and horizontal lines, forming perfect squares, and number the two sets of lines in succession. Then draw a similar series of lines on a clean sheet of paper, setting the lines at an equal, less or greater distance apart, and copy in succession the parts of the figure that are enclosed within the several squares.

As it is not always possible to draw lines across a figure, they may be replaced by a frame carrying fine threads or wires stretched in the two directions. The frame is laid over the original drawing, which can then be copied, as above explained, on a sheet of paper divided into squares (fig. 16).

The frame may, moreover, afford assistance in the drawing of solid objects. Having placed it above or in front of the object and in contact with it, copy on to the sectional paper the contents of each corresponding square, taking care to look at the object perpendicularly. With a little practice, and by placing the eye in the correct position and always at the same distance from the frame (a distance which may be regulated by a glass), a sketch in fair proportion may easily be obtained.

32. To erect a perpendicular on a straight line. Either the compass or a set-square can be employed; the use of the latter instrument is so simple that no further reference need be made to it. Assume a, fig. 17, to be the point in the line n m at which a perpendicular is to be drawn. On either side of a measure off equal distances a n, a m; from n and m, with a radius about equal to the distance n m, draw two circular arcs cutting one another. If their point of intersection b be joined to a, the line a b will be the required perpendicular.

33. To erect a perpendicular at the extremity of a line. From the extremity c, fig. 18, mark off four equal parts towards s. From s, with a radius equal to five such parts, describe a circular arc, and from c, with a radius of three such parts, describe another arc cutting the first at d. The line joining c and d will be perpendicular to s c.

For the square of 5 is 25, and this is equal to the square of 4 or 16 plus the square of 3 or 9. Thus the triangle s c d must be right-angled (15).

Or the following method may be adopted: With any center i and radius i g (fig. 19), as large as possible, describe a circumference passing through g. From the point p, where the circle cuts the line, draw the diameter p i h. If the point h be joined to g, it is the required perpendicular; for, by a property of the semicircle, the angle h g p is a right angle.

34. To let fall a perpendicular on a straight line. In order to let fall a perpendicular from the point a on to the line b c (fig. 20), describe from a as a center, a circular arc sufficiently large, cutting the straight line in two points, b and c. From these two points, with the same opening of the compass, draw on the under side of the line two arcs that intersect. The point of intersection o joined to a gives the required perpendicular.

35. To draw parallel straight lines. Having fixed a good straight-edge over the drawing, as many parallel lines as are required may be drawn with the aid of a set-square which is caused to slide along the rule. They will be vertical, horizontal or inclined, according to the position of the rule, which must be set exactly perpendicular to the direction in which the parallel lines are to be drawn (fig. 21).

To draw, from a given point, a line parallel to a given line. Let d be the given point, and a b the given line (fig. 22). From d draw the circular arc a c, and from a where it cuts a b, with the same radius describe the arc d b. From a set off on a c, a distance equal to d b. The line joining d and c is the required parallel.

36. To subdivide a line into equal parts. Let it be required to divide the line p v (fig. 23), into five equal parts. Draw a line p q inclined at any angle, and mark off on this line five equal parts of any length; join q, the extremity of the five lengths, and v, and through the points a, b, c, d, draw lines parallel to q v. In virtue of a property of similar triangles these lines will divide p v into equal parts. It is advisable that the lines p v and p q should not differ very considerably in length, as, otherwise the inclination of the parallel lines to p q will render it difficult to observe the exact point of intersection.

To divide a line into proportional parts. The proposition can be solved in a similar manner. Let it be required to divide a line t r (fig. 24), into two sections that are to one another in the proportion of 5 to 3. On t s mark off a series of equal parts given by the addition of these numbers together, that is 8; and join the last point s to r. Then draw through c, the fifth division, a line parallel to s r. This line c d will cut t r into two parts, which are to one another in the proportion of 5 to 3.

By an analogous construction a fourth proportional can be graphically obtained, as already indicated in articles 25-27.

37. To construct an angle equal to a given angle. The angle may be measured by means of the protractor (24), which then enables us to draw a similar angle; but greater accuracy is obtainable by using the compass.

Let it be required to construct at m on the line m p (fig. 25), an angle equal to b a d. With as large a radius as possible, draw from the points a and m the arcs b d and n p. Measure the distance d b and mark it off with the compass from p on the arc p n. A line drawn through m to the intersection of the two arcs will give the required angle equal to b a d.

38. To subdivide an angle into 2, 4, or 8 equal parts. In addition to the use of the protractor, the following graphic method is often given in works on geometry.

An angle e f g being given (fig. 26), from its apex f as a center describe the arc e g, and from its two points of intersection with the sides, with a radius greater than half their distance apart, draw two short arcs cutting each other at s. A line drawn from f through the intersection s will divide the angle into two equal parts.

If four divisions are needed, repeat the process on the two angles s f e, s f g, and so on for a further sub-division.

The line that divides the angle into two equal parts will also bisect or divide into two equal parts the chord and the arc e g.

39. To find the center of a circle or of a circular arc. Take on the circumference, or on the arc, three points b c r (fig. 27). Join b to c and c to r. At the middle point of each of these lines[3] erect a perpendicular. The point of intersection of these perpendiculars is the required center.

A similar method should be resorted to when it is desired to describe a circle passing through three given points.

40. To connect up or associate lines. In order to join up a straight line, such as i j (fig. 28), with the curve l p, erect a perpendicular at j, and through the middle point of a chord, l p, draw a second perpendicular cutting the first in k. This point will be the center from which the curve uniting the two lines should be struck.

To unite a curve a b (fig. 29), with another curve, c x or c z, at the point c, first find o, the center of the curve a b, draw the line a o, continuing it beyond the center; join a and c, and erect a perpendicular at the middle point of this chord. The intersection of this perpendicular with a o, produced if necessary, should be taken as the center for a curve uniting b a with c.

To join up two lines inclined to each other or parallel lines of unequal length, such as a r, b s, fig. 30, draw midway between the two another line, z d; join the two extremities r and s, and from these points let fall perpendiculars r i and s c; then from d draw a line perpendicular to s r. The point o thus obtained will be the center of the arc r d, and c will be the center for d s.

41. To describe an ellipse. Let a b (fig. 31), be the major axis of the ellipse; divide it into three equal parts, and from the two points, c and i, at which it is divided, with a radius equal to i c, draw (in pencil) two circles, intersecting in the points x and z. Through these points draw the lines x i g, x c h, z i f, z c d.

With the center z describe the arc d f, and from x draw h g; the ellipse will be completed by the two arcs, f b g, d a h, of the primitive circles.

If it be required to describe an ellipse that shall have a shorter minor axis, divide the major axis into four equal parts, thus obtaining three points of sub-division. With each point as a center and with a radius equal to one of the spaces describe circles. Those to the right and left will determine the extremities of the ellipse, and the central circle will intersect the minor axis in two points which must be taken as centers for describing the top and bottom portions of the figure.

When the length of the long and short axes are given, proceed as follows (fig. 32): From the center a, where they intersect at right angles, mark off the distances a n, a o, equal to the difference in the length of two semi-axes. Join n o, and add one half of n o to a o measured in the direction of a v, thus obtaining the point k; with the radius a k describe a circle. On this circumference will lie the four centers; k for the arc r u s, m for the arc p v q, t and i for the short arcs q j s, p e r.

The figures obtained by the methods here given closely resemble the ellipse, but are not of the strict mathematical form. It is well to acquire some facility in drawing ellipses, for the projection of a circle on a plane, when the two are neither parallel nor perpendicular, is an ellipse, and one often has occasion to describe it.

42. The following may be added as a mode of describing an ellipse:

The major axis and the two foci (points in this axis) being known, fix two pins in these foci. Then tie a piece of string into a loop and place it over the pins; stretch it with a pencil, the point of which is on the paper, and on moving this around in a circular direction, the string being maintained stretched, an ellipse will be described. When the string is so stretched that it lies along the major axis, the length should be such that the pencil is exactly at its end.

43. To draw a spiral curve. Draw four lines forming a small square (fig. 33). The point o is taken as the center of the first arc, i j; s is the center of j k; u of k l; i of l n. Then, to continue the curve, o is again taken as the center for n p, and so on. This method produces a volute in which the coils are at a considerable distance apart, such as has no special applicability to horology.

As the balance-spring of a watch is partially concealed by other pieces, it is generally sufficient to represent the parts that show themselves by concentric circular arcs, or arcs described from two centers. If a more accurate representation be required, the following method may be resorted to: when working on a small scale it involves the use of the eyeglass, for the figure (fig. 34,) here given is exaggerated in order to avoid confusion in the lines, numbers, letters, etc.

A small circle having been described, it is divided into an even number of equal parts, say four; a less number than this should never be adopted. From the same center describe another circle as small as possible, which will be cut by the two diameters drawn between opposite points of division numbered 1, 2, 3, 4.

Assuming a to represent the starting-point of the curve, from the center 1 with radius 1 a draw the arc a b; from 2 with radius 2 b draw the arc b c; from 3 with 3 c draw c d; from 4 with 4 d draw d s; then recommencing with 1 and the radius 1 s draw s f, and so on.

The less the radius of the small circle and the greater its number of divisions, the closer will the successive coils be together. To secure accuracy when working on a small scale, it is advisable that the center and the several points be in a thin brass or horn plate, which is maintained in position by steady pins.

THE MICROMETRICAL DIVIDING TABLE.

44. This instrument is no more than a simple application of the screw to dividing straight lines, but it will suffice to enable the reader to understand the principles on which the more complicated instruments are based.

A plate, P, fig. 35, supports a bracket a, in which a screw, similar to the one described in paragraph 30, is engaged by means of a collet; it rotates, being supported between this bracket and the small bearing b, that receives the pivot at the end of the screw.

The screw is fitted carefully into a nut n, which is rigidly attached to the small plate h; this carries a fine marker, movable on an axis, and terminating with a chisel edge or a fine diamond point, according as the instrument is to be used for engraving metal or glass; or it may be provided with a fine pencil if the object is merely to make subdivisions on a drawing.

This being understood, it will be evident that, if a rule or rod of any form be fixed by screws or otherwise between f and g, it can be graduated by means of the marker, the screw being made to advance; the millimeter screw can be used for dividing into millimeters and fractions direct, or, with a little calculation, into fractions of an inch. Each complete rotation of the head means a displacement of the marker by a millimeter; a half turn will be half a millimeter, etc.

OTHER METHODS OF DIVIDING INTO EQUAL PARTS.

45. First method. Having fixed a sheet of drawing paper on a smooth board, draw the line M N, fig. 36, longer than the rule which is required to be divided. Then, with a compass or graduated scale, mark off a series of equidistant points, commencing at N, equal in number to the required series on the rule, and let M be the last division. With the center N and radius N M describe the circular arc p v, and with M as a center and the same radius, describe a second arc r s, intersecting the first at O. Join O with M and N. Assume a c to be the rule that is to be divided into equal parts; slide it on the paper parallel to M N until the extremities, a and c, coincide with the lines O M, O N, and are equidistant from O. This position can be easily found by the aid of a compass with one of its centers at O. Now fix the rule in position with sealing-wax, or by some other means, and, with a firm upright pin, center the brass rule R at O, so that it can rotate round this center on the pin as a pivot. It now only remains to trace a series of lines O k, O b, O d, etc., with the rule, to the division points of the line M N. The line a c is thus divided into as many equal parts as the line M N. The graduations will be all the more exact according as the divisions of the line M N are longer.

46. Second method. By the side of the chuck of a wheel-cutting engine, arrange a horizontal slide y f fig. 37, that can travel easily in a direction perpendicular to a T. A watch fusee-chain, or a very flexible spring, is fixed by one end to the chuck, and by the other to the slide at d. The chain or spring is kept stretched by a weight which tends to draw the slide from f towards d.

The rule to be graduated, a, is now fixed on the slide, and an initial division is marked on it with a pointed rotating cutter in the position usually occupied by the wheel cutter, or else by striking a small pointed or flat-edged chisel arranged for the purpose, in such a manner as not to be liable to derangement.

Rotate the table through a definite distance; the rule a will advance through the same distance; mark the second division; then having moved the division-plate through a distance equal to its first displacement, mark the third graduation, and so on. Suppose, for example, that it be required to make 30 divisions on the rule between f and d; select on the plate the circle corresponding to twice or thrice this amount, so that the radius of the chuck may not be relatively too short, and that the chain or spring may not act at a disadvantage; take the number 60 for example:

The two marks at d and g on the spring indicate the length that corresponds to the straight line to be divided.

The chuck is placed in the lathe and reduced in diameter until the half circumference is exactly equal to the distance between these two marks on the spring, which thus fall on a diameter, i g, of the chuck.

The spring having been fixed by its two extremities, the slide with the rule attached is placed in position, so that the mark g is on the line a T; it will be evident from the figure that each displacement of the division-plate through one-sixtieth of its circumference will cause a to advance through one-thirtieth of the space between f and d.

Remarks.—Knowing the relation of a diameter to the circumference (as 1 : 3.1416), we can determine the diameter of the chuck at once by calculation.

Its form should be a true cylinder, and it is well to place guides that will prevent the spring or chain from assuming a helical position.

The side that carries the rule should be strictly perpendicular to a T; and the portion of the spring that is not coiled on the chuck should always be parallel to this slide.

The chuck and spring must be quite clean and smooth, and the latter should be very pliable. A greater weight will be needed to keep the spring stretched than will suffice for a chain, and it must be increased as the strength of springs is greater.

The slide, y f, may simply travel over a horizontal surface between pins planted in two parallel lines. But it would be preferable to adopt some other method, for instance, to make this piece (F, fig. 37), travel with a little friction along a perfectly true cylindrical rod.

47. Third method. This is merely an application of the arrangement mentioned in paragraph 44. The lathe can be employed for marking off a series of equidistant points in a straight line. Knowing the pitch of the slide-rest screw, determine the distance apart in, say, millimetres, of the required divisions, and fix the rule perfectly flat on the face-plate, which must be rendered immovable by any convenient means. Then mark the first point with the drill-stock. Advance the screw by the amount previously determined upon and mark the second point. After withdrawing the drill, again advance by the same amount and mark the third point, etc. Always be careful, before making the first mark, that the screw has already traveled some distance in the direction it will continue to move, so as to avoid backlash, or loss of time.

TO SUBDIVIDE A CIRCLE.

48. To divide the circumference into equal parts. After having drawn the circle, A, fig. 38, draw two diameters, d a, b c, at right angles to each other, dividing the circle into four equal parts. Join the points, c a, and divide the line, c a, accurately into nine equal parts.

Draw a series of circles concentric with the first, at distances apart equal to one of the divisions of c a, and to the number of one, two, three, etc., according as it is required to subdivide the circle, say for a pinion, into seven, eight, nine, etc., equal parts.

With a fine-pointed compass, measure off the radius of the initial circle A. Placing one point of the compass at c, the other point will give the position of the next leaf, and so on, all around the circumference. If the innermost circle A be selected for sub-division, six divisions will be obtained, and there will be one more division for each larger circle.

The operation will be facilitated by selecting the first circle, so that the line a c contains exactly nine divisions equal to those of some scale that is accessible. Such a circle can be easily found, by first drawing the two diameters, laying the scale in the direction c a, and determining by trial the radius for which the first and ninth divisions correspond to a and c respectively.

49. To divide a surface into rings of equal or proportional superficial area. The following solution is due to M. Brocot:

Let a d be the radius of a circle (fig. 39), that is required to be subdivided into four rings of equal area by concentric circles. Taking a d as a diameter, draw the semicircumference, a b d; accurately divide a d into four equal parts, and at each point so obtained, erect a perpendicular. Through the intersections of these perpendiculars with the semicircle, draw a series of concentric circles; they will trace out rings, 1, 2, 3, 4, that have equal superficial areas.

If it be required to divide the surface in a given proportion, divide the line a d, according to that proportion.

The right-hand side of fig. 39 gives a special application of this method to the division into two equal areas of the interior of a barrel exclusive of the space occupied by the arbor-nut. If the mainspring accurately covers i c when wound up, and i j when unwound, it will give the greatest possible number of turns.

TIME.

50. Solar time is taken from the revolutions of the earth, and the watchmaker can easily get the exact solar time of any point at which he may happen to be by a little calculation from known standards. These standards are: 1. The zenith. 2. The longitude of the point of observation. 3. The difference between noon at the point of observation and noon of a known meridian either east or west of the point of observation. The zenith is that point in the heavens where the rays of the sun are in a plane exactly perpendicular to the surface of the earth at the point of observation, and when the rays of sunlight are in this plane it is noon at that point. The circumference of the earth is divided into 360 degrees or meridians of longitude, so that as the earth revolves once every twenty-four hours, each of these meridians will pass the zenith, or fixed point, in that time. In twenty-four hours there are 24 × 60 = 1,440 minutes, so that the interval between the passage of one meridian and the next will be 1,440 ÷ 360 = 4 minutes. A degree of longitude is divided like an hour, into minutes and seconds, so that

51. Thus it happens that, at a town one degree east of a given point the sun will be visible four minutes sooner, and if to the west, four minutes later than at that point. The “local,” or solar time, therefore, will be four minutes earlier at the first town, and four minutes later at the second town, than it is at the point of observation.

52. It will be readily seen that, having any two of the three factors given above, the other can be readily found. Thus having the time of a given meridian and the local noon or meridian time, the longitude can be readily found; or, having the longitude (which can be readily obtained from a surveyor) and the time of a given meridian, “noon,” can be calculated, etc. The first method is followed in calculating distances at sea; the chronometer keeping Greenwich time, and the local noon giving the longitude.

When great accuracy is necessary, however, a fixed star is used as a means of observing the exact time when a revolution of the earth is completed, as the revolution of the sun in its orbit causes a slight variation during the year. For further information on this point the reader is referred to works on astronomy.

To obviate the constantly varying time in running east or west, the railroads use the time of a given meridian over each fifteen degrees of longitude, and as each degree of longitude equals four minutes of time, it follows that only the hour is changed in changing from one standard to another. In Europe the zero of longitude, or the time of the meridian of Greenwich is used. In the United States the time of the 75th degree, which passes through Philadelphia, is used from the 67th to the 80th degree, which comprises the territory from Princeton, Maine, to a line drawn north and south, passing through Erie and Pittsburg, Pa., and is called Eastern time. The time of the 90th meridian is used from the 80th to the 102d meridian, and is called Central time. The time of the 105th meridian is used from the 102d to the 114th meridian, and is called Mountain time. The time of the 120th meridian is used from the 114th meridian to the coast (which ends at about the 124th meridian) and is called Pacific time. The time of the various standards is telegraphed through their various territories at noon each day, and furnishes an accurate standard of comparison to all watchmakers.

53. In very many cities the actual or solar noon has been discarded and the railway standard adopted, thus making but one standard and removing the source of confusion and annoyance to many people. In others, however, the two standards are still used, and it becomes necessary for the watchmaker to be able to calculate both standards, in case of accident or irregularity in his regulator. Hence he should calculate his longitude within one second by means of the difference between railroad and local noon, and have the nearest surveyor correct his reckoning; then, by means of the accurate longitude and the railroad time, correct the solar time; then by means of the solar noon and the longitude calculate the railroad time. When all these calculations check each other perfectly, he possesses all the time data he needs for that place, and can correct his standard or regulator if at any time it should become irregular. The calculations are very simple, and can be easily performed from the data given above.