The watchmakers' hand book

PART I.

ARITHMETIC, GEOMETRY, DRAWING, ETC.

ARITHMETIC.

1. We often hear the theory advanced that in this country, at the present day, it is not necessary to have a knowledge of arithmetic, geometry, drawing, etc., because our interchangeable system of manufacturing watches, makes all knowledge in these lines superfluous, and that without any knowledge of arithmetic or geometry a man may become a thorough master of watchmaking. This is a mistake that too many of our young men make. The fact that the leading watch factories of the United States have adopted the interchangeable system, of course lessens the number of parts which the repairer will have to make and fit, but it by no means alters the situation as regards the repairing of foreign-made watches, nor even the changing of American watches from key to stem-winders. Without a thorough knowledge of arithmetic and, at least, an insight into the principles of geometry, no young man can hope to become a first-class watchmaker in the true sense of the word. Without these accomplishments he will be deprived of the pleasure of reading understandingly the best literature of the day, the works of those who are best fitted to impart knowledge to the members of the trade.

With a knowledge of geometry he will be able to comprehend the works of the best authors, to ascertain the dimensions of solid bodies, and be in a position to apply the rules that form the basis of linear drawing. Every watchmaker, worthy of the name, should be able to make and understand the drawing of any machine, or of any horological instrument. Many inventors, and even ordinary workmen, would avoid a large amount of hard work, often useless, and occupying much time, if, instead of at once putting an idea into practice with brass and steel, they were able as a preliminary, to make for themselves a correct design drawn to scale.

2. It is taken for granted that the reader is familiar with the rules of arithmetic at least, and we will touch upon some points in algebra and geometry that it will be well to mention. Should the reader have no knowledge of arithmetic, algebra and geometry, we would advise him to take up these studies during his leisure hours, using some of the standard text books for that purpose.[1] Besides possessing a knowledge of prime numbers, numbers which have no divisors but unity and themselves, the watchmaker should be able to determine the greatest common measure of several numbers, a rule which is of great importance in calculating a train of wheels that is complicated.

We shall confine our attention to the methods of extracting square roots and proportions, the rules for which may have been forgotten, owing to their being less frequently employed than the more common rules of arithmetic; they are of frequent use in horology.

SOME SIGNS EMPLOYED IN CALCULATIONS.

3. The sign of addition is an erect cross, +, called plus, and when placed between two quantities it indicates that the second is to be added to the first. Thus, 5 + 3 equals 8.

The sign of subtraction is a short horizontal line, -, called minus, and when placed between two quantities it indicates that the second is to be subtracted from the first. Thus, 8 - 3 equals 5.

The double sign, ±, is sometimes written before a quantity to indicate that in certain cases it is to be added and in others it is to be subtracted. Thus 5 ± 3 is read 5 plus or minus 3.

The sign of multiplication, ×, when placed between two quantities indicates that the first is to be multiplied by the second. Thus, 3 × 5 equals 15.

The sign of division is a short horizontal line with a point above and one below, ÷, and when placed between two numbers or quantities it indicates that the first is to be divided by the second. Thus, 6 ÷ 2 equals 3.

The sign of equality is two short, horizontal, parallel lines, =, representing the words equal to. Thus, 6 ÷ 2 = 3.

Inequality is denoted by the angle >, the opening always being toward the larger number or quantity; thus, in 12 + 7 > 14, the sign >, indicates that the sum of 12 and 7 is greater than 14, and the whole expression is read, 12 plus 7 is greater than 14. The expression 9 < 4 + 7 is read, 9 is less than 4 plus 7.

A parenthesis, (), denotes that the several numbers or quantities included within it are to be considered together, and subjected to the same operation. Thus, (10 + 4) × 3 indicates that both 10 and 4, or their sum, are to be multiplied by 3.

A horizontal vinculum, ———, placed over the numbers or quantities, is frequently used instead of the parenthesis. Thus, 2 + 4 + 6 × 7 is equivalent to (2 + 4 + 6) × 7.

Division is more usually indicated by a line between the two figures, the dividend being written above and the divisor below the line. Thus, ¹⁶⁄₈ indicates division, the same as 16 ÷ 8.

Algebra is that branch of mathematics in which the operations are indicated by signs or symbols, and the quantities are represented by letters.

The sign of ratio consists of two points like the colon, :, placed between the quantities compared. Thus, the ratio of a to b is written a : b.

The sign of proportion consists of a combination of the signs of ratio. Thus, : :: : . The first two and last two dots are read is to, while the four in the middle are read as. Thus, if a, b, c, and d, are four quantities which are proportional to each other, we say a is to b as c is to d, and this is expressed by writing them thus:

a : b :: c : d.

POWERS AND ROOTS.

4. The power of a number is the product formed by successive multiplication of the same number by itself. Thus,

An exponent is a number written above a quantity, at the right-hand, to indicate how many times the quantity is to be taken as a factor, as 6³ = 6 × 6 × 6.

The root of a quantity is a factor which, multiplied by itself a certain number of times, will produce the given quantity. Thus, in the above examples 2 is the square root of 4 and the cube root of 8.

The radical sign, √, indicates that the root of the quantity placed under it is to be taken, and the index of the root is expressed by a little figure placed outside of the bend. If a square root the index figure is usually omitted.

√4 = 2 or ²√4 = 2 and ³√8 = 2.

5. Extracting the square root of whole numbers. [2]I. Point the given number off into periods of two figures each, counting from the units place to the left. For example, we wish to find the square root of 399424, we point it off thus: 39,94,24.

II. Find the greatest perfect square in the left-hand period, and write its root for the first figure in the required root; subtract the square of this figure from the first period, and to the remainder bring down the next period for a dividend. Thus:

III. Double the root already found, and write the result on the left for a divisor; find how many times this divisor is contained in the dividend, exclusive of the right-hand figure, and place the result in the root and at the right of the divisor. Thus:

IV. Multiply the divisor thus completed by the last figure of the root; subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. Thus:

V. Double the right-hand figure of the last complete divisor for a new divisor, and continue the operation as before. Thus:

PROPORTION.

6. It is often convenient to express the relations of qualities in the form of a proportion and from the proportion derive an equation.

Ratio is the quotient of one number divided by another. Thus the ratio of 30 to 6 is ³⁰⁄₆.

7. Proportion is the equality of ratios: Thus if ³⁰⁄₆ = 5 and ⁴⁰⁄₈ = 5 then we may state that ³⁰⁄₆ = ⁴⁰⁄₈ or the proportionality is usually expressed (3) thus:

30 : 6 :: 40 : 8

and this constitutes what is called a geometrical proportion, and 30 and 8 are called the extremes and 6 and 40 the means.

8. The product of the extremes is always equal to the product of the means. Thus: 30 × 8 = 6 × 40 = 240. Hence it follows that if we only know three terms we can always determine the fourth, or unknown term, which is usually represented by the letter x. Thus in the proportion

12 : 3 :: 16 : x

we find the product of the means, or 3 × 16 = 48; this product divided by 12, the known extreme, gives us the value of x, or the unknown extreme, as equalling 4.

9. If we know the two extremes and only one of the means the same rule is applied. Thus in the proportion

20 : 5 :: x : 25,

we have: 20 × 25 = 500. ⁵⁰⁰⁄₅ = 100, the value of x.

ELEMENTS OF PRACTICAL GEOMETRY.

10. The object of geometry is to measure the extent of bodies. A body has three dimensions, length, breadth and thickness, and one of these latter is sometimes termed weight or depth.

Either dimension taken by itself is measured by a straight line.

When the extent of a body is expressed by combining any two dimensions, it is termed area or surface, and when three are employed we obtain the solid measure or volume.

Plane geometry only takes cognizance of figures situated in one plane or surface, and therefore only possessing two dimensions; solid geometry, however, regards bodies as having all three dimensions.

Two lines are parallel when their distance apart is the same at all points. The same is also applicable to parallel planes.

Two lines or planes meeting each other will form an angle. The point at which they meet or intersect is termed the apex or summit of the angle.

A straight line is perpendicular or at right angles to another straight line, or to a plane, when all the angles which it makes with that line or plane are equal.

A circumference of a circle is a curved line l c d f fig. 1,) such that all its points are equally distant from an internal point, o, termed the center. The circle is the space enclosed by the circumference.

It will be noticed that in geometry these two words are distinguished, although they are frequently referred to as identical. Thus, the rim of a wheel or balance is generally termed a circle.

Two circles (l c d f and b r a, fig. 1,) described from the same center are said to be concentric. When their centers do not coincide they are called excentric with regard to one another.

Any portion of a circumference, such as f n d, is termed an arc of the circumference, or, more commonly an arc of a circle.

A chord is a straight line, f d, which unites the two extremities of an arc. When the chord passes through the center of a circle it is termed a diameter.

The radius of a circle or circumference is a straight line drawn from the center to the circumference; and all the radii that can be thus drawn are equal. A diameter is, then, always double the radius, and, conversely, the radius, is always half the diameter.

A tangent is a straight line that only touches a circumference at one point, as g l (fig. 1); whereas a secant cuts the circle, as i j.

A circumference is assumed to be divided into 360 equal parts, termed degrees. The degree is subdivided into sixty equal parts, or minutes, and the minute into 60 seconds. These are respectively symbolized by the marks ° ′ ″ placed at the right-hand top corner of the figure.

Such an expression as 18° 30′ 15.5″ would, then, be read 18 degrees, 30 minutes, and 15.5 seconds.

11. Ratio of the circumference to the diameter. The diameter of a circle is to the circumference as 7: 22; or, employing decimal fractions, as 1: 3.14159 (a number which, in algebra, is always represented by the Greek letter π.)

Knowing a diameter (D), the circumference, x, can be ascertained from the proportion:—

1: 3.14159:: D: x.

Knowing a circumference (C), the diameter, x, can be determined from the proportion:—

3.14159: 1:: C: x.

The latter proportion will also give the value of the radius, which is half a diameter.

12. The superficial area of a circle is equal to the circumference multiplied by half the radius, or to the square of the radius multiplied by 3.14159.

A sector is the circle enclosed between an arc and two radii bounding it, as b O r k (fig. 1.)

The area or surface of a sector can be ascertained by multiplying the length of the arc by half the radius.

A segment of a circle is the portion intercepted between an arc and its chord, as f d n (fig. 1.)

The surface of a segment, as b k r s, can be obtained by subtracting from the area of the sector O b k r, the area of the triangle, b r O (15).

13. Ring. To determine the surface of a flat ring, the area of the inner circle must be subtracted from that of the outer circle; in other words, take the difference between the areas of the two circles that fix the inner and outer diameters of the ring.

The area of a flat ring can also be calculated by adding together the internal and external diameters; then multiply the number so obtained by their difference and by the decimal fraction 0.7854 (that is, 3.1415/4). The product will be the required area.

14. Angles and their measurement. When two lines meet one another, they form an angle, as we have already seen. If we take the apex as the center of a circle, the number of degrees intercepted between the two straight lines gives a measure of this angle.

The angle measured by a quarter of a circumference, or 90°, is termed a right angle.

An obtuse angle is greater than a right angle, and an angle that is less is termed an acute angle.

15. Triangles, squares, etc., and their measurement. The triangle or plane area enclosed within three straight lines joined two and two together (A, B, C, fig. 2,) is said to be rectangular when one of its angles is a right angle; it is equilateral when the three sides are equal, under which circumstances the three angles are also equal; and isosceles when only two sides are of equal length.

The sum of the three angles of a triangle is always equal to two right angles. If only two of the angles are known, it is thus easy to determine the third.

Similar triangles are characterized by the fact that their homologous sides (that is, the sides opposite to equal angles) are proportional.

Peculiarity of the right-angled triangle. The square described on the longest side, termed the hypothenuse (B, fig. 2,) is equal to the sum of the squares described on the two other sides. Hence, it follows that, if the lengths of the two shorter sides are known, that of the hypothenuse can be ascertained by extracting the square root of the number formed by adding together the squares formed on these two sides (5).

If the hypothenuse is known and one of the shorter sides, the third can be determined by extracting the square root of the number formed by subtracting the square of the known side from the square of the hypothenuse.

The surface of a triangle is determined by multiplying one of the sides by half the perpendicular height of the angle opposite to this side.

16. The surface of a square or of an oblong or rectangle (a b c d, fig. 3) is equal to the product of the base multiplied by the height.

The sum of the squares described on the four sides is equal to twice the square described on a diagonal. This diagonal divides the rectangle into two equal rectangular triangles.

17. The surface of a parallelogram or lozenge, a plane figure with four sides, opposite pairs of which are parallel (c f g d and c i j d, fig. 3,) is equal to the product of one side multiplied by the perpendicular height of the figure.

The sum of the squares described on the four sides of a parallelogram is equal to the sum of the squares described on the two diagonals.

18. Measures of various solid bodies. The volume of a cube of parallelopiped (that is, a body bounded by six four-sided figures, every opposite two of which are parallel) is obtained by multiplying the surface of the base by the height.

The volume of a straight cylinder is the product of the surface of the circle which forms its base into the height of the cylinder.

The area of the curved surface of a cylinder is obtained by multiplying the circumference of the circle forming its base by the height.

The volume of a tube or cylindrical ring of rectangular section, such as the arbor-nut of a barrel, or the rim of a circular balance, etc., is equal to the product of the plane surface of its base (13) into its height.

The volume of a right cone or of a regular pyramid is the product of the base into a third of the height.

The surface of a sphere may be determined by multiplying the square of the diameter by 3.1416 (11).

The volume of a sphere is equal to this surface multiplied by a third of the radius.

GEOMETRICAL DRAWING.

19. An elementary knowledge of the art of drawing, an ability to represent the outlines of objects by simple lines, is of the first importance to the watchmaker.

Such a design is obtained by projecting on to one plane all the visible points of the object represented.

Projection on a vertical plane gives an elevation; the object is looked at from one side.

Projection on to a horizontal plane produces a plan; the object is observed from above, thus giving a bird’s-eye view.

The projection of a point on a vertical or horizontal plane is the foot of the perpendicular, from the given point on to the plane. Assume the line n m (fig. 4), to be fixed in space; its horizontal projection will give c d, and its vertical projection, r s.

Miscellaneous details. When one portion of the object to be represented is found to pass behind other pieces so that it cannot be seen, the continuation is frequently indicated by dotted lines.

Surfaces that are situated in planes one behind the other are shaded, the more deeply according as they are farther back. This shading is produced by a number of parallel lines which may be vertical or horizontal.

Parts that are in relief are indicated by projected shadows, or by increasing the thickness of a line that would cast a shadow.

In order to distinguish the several shadings or to emphasize the lines by which they are separated, it is a very usual, though not invariable practice to assume the light to be coming from the left-hand upper corner.

When drawing a square in relief, such as a b c d (fig. 3), the lines c d b d, will be made darkest; but if it is a recess, the lines a b, a c should be brought into prominence by means of dark lines.

These several directions will be found useful when a hole, any cavity, a pin, a round object etc., has to be depicted, as in fig. 5. As a general rule, the thick lines should indicate the position at which a shadow would form, the light being assumed to fall on the drawing in the manner indicated above.

A section shows a body as it would appear if cut in two, and one portion removed in order to expose the interior, as in fig. 6. A section is indicated by a series of parallel lines drawn close together and at an inclination of about 45° to the vertical.

In order to leave more room for important details, or to show objects that are situated behind, a piece is often broken off by an irregular line, as shown in that drawing.

Lines formed by a series of detached points sometimes serve as a means of associating several figures representing the same object looked as in different directions.

20. Tracing and transfering. These two operations are resorted to when it is required to obtain one or more copies of a picture or design already drawn.

Tracing consists in laying a piece of tissue or other translucent paper over the drawing and copying it by following over the lines that are visible with a pencil. Or ordinary paper can be used for the purpose, providing it is not too thick, if the picture be placed against the pane of a window or, what is more convenient, on a sheet of glass used as a desk and illuminated from below. When either sheet of paper is too thick to allow sufficient light to pass, one or other of the methods of transfering indicated below must be resorted to.

21. This operation consists in reproducing a tracing on a separate sheet of paper or on metal that is to be engraved. Either of the following methods may be adopted:

1. The picture to be transferred is fixed to a table or drawing board if tracing paper is to be used, or to a sheet of glass if only ordinary paper is available. The lines are then traced with a black lead pencil that must not be too hard. When this is finished it is laid, face downwards, on a sheet of white paper, taking care that both sheets are so fixed that they shall not slip. Apply pressure to the upper surface by tapping with a small pad made on purpose and, at the same time, gently rubbing. Experience will very soon show how hard the pad should be. Now remove the tracing, still taking care to avoid any slipping, and a faint reproduction of the design will be found on the lower sheet of paper. It is only necessary to follow over the lines with India ink. The figures will be reversed but a transfer with it in the original direction may be obtained by inverting the picture and laying it on glass so as to make a reverse tracing.

2. Lay the picture on a desk or drawing board and trace it with ink on a very transparent sheet of paper. When the ink is dry, invert the tracing and blacken the back with a No. 2 pencil. Now lay the tracing, with the ink side uppermost, on a sheet of clean paper, taking care to avoid slipping, and go over the several lines with an agate or metal style, avoiding excessive pressure on account of the risk of tearing the paper. On removing the upper sheet an impression will be found not reversed. Go over all the lines with India ink and clean the paper with India-rubber or stale bread.

Observations. The choice of paper and pencil is not a matter of indifference. All kinds of paper do not receive an impression equally well, neither do all pencils transfer with equal facility. The Faber pencil No. 3, will generally be found best suited to such work.

In preparing drawings which you desire to preserve, as drawings of escapements, etc., a good quality of light weight bristol board will be found more desirable than the best drawing papers. White wedding bristol, about two-ply in thickness, answers admirably, and India ink lines drawn upon it will not spread as they often do on drawing papers. The prepared liquid India inks now on the market are superior to any you can prepare by grinding the sticks.

22. To transfer an engraved design. This method is available when it is desired to obtain an impression, for example, of the engraved surface of a watch case.

Procure some of the inks used by copper-plate engravers, or, in its absence, ordinary stencil ink may be used. Taking a small quantity on the end of the finger, tap it on the surface of a glass plate, in order that the ink may be distributed, leaving only a small quantity evenly spread over the finger: tap with the finger thus prepared over the watch case long enough to make sure that all the surface in relief has received some ink; take a piece of writing paper and, after slightly moistening it, spread it over this surface. Lay above this a piece of paper folded in four and pass over it in all directions any round body, such as a small tool handle, and with some pressure; then raise the papers without allowing them to slide.

If the operation has been carefully performed a very clear impression will thus be obtained of the engraved surface. The relief will be black and the hollows white, but, of course, the figure is reversed like that in a looking-glass. If required in the right direction it must be traced through to the other side of the paper.

DRAWING INSTRUMENTS.

23. It is needless here to describe the rule, set-square, T-square, bow-compass, etc., as every one knows them.

To verify the accuracy of a rule. On a perfectly flat smooth surface carefully draw, with the rule in question, a fine straight line. Then turn the rule over, hinging it as it were on the line just drawn; if quite straight the edge of the rule will exactly coincide with the line, in this new position, throughout its entire length. Each edge should be thus examined.

To verify the accuracy of a set-square. Having fixed an accurate rule on a smooth surface, place one edge of the set-square against it, and draw a line along the edge perpendicular to the rule; then, having turned the set-square, hinging it on the line just drawn, bring it against the rule and along the line. If the square is true the edge and line will coincide throughout their length.

24. The Protractor. Fig. 7 represents a common form of this instrument. It is made of horn, or, if of metal, the inner portion is cut away, leaving only a base and a semicircular arc, which is divided into 180 equal parts or degrees; a complete circle would therefore consist of 360 such degree. The point indicating the center of the arc, should be very small in order to facilitate the exact setting of it at the apex of an angle.

When an angle has to be drawn with accuracy, the protractor is unsuitable; it will be better to adopt one of the methods described at paragraph 37, or trigonometrical methods.

25. Drawing scales. When an object is represented by a drawing, if the dimensions are the same as those of the object itself, or, rather, as they would project on to a horizontal or vertical plane, the drawing is said to be full size; but the object is generally represented either on an increased or diminished scale, which is defined, the proportions between all the parts being still, however, maintained the same.

With a view to avoid the many calculations that such a change would involve, it is usual to employ drawing scales. The following notes will sufficiently explain their construction and use.

Let it be required to reproduce a large drawing on a small scale, in such a proportion that the dimensions are reduced in the ratio of 10 to 1.

Take a straight line of indefinite length, a b, fig. 8, and mark out on it spaces equal to a 1, which represents any measurement taken on the original object; at c, the 10th division, draw a perpendicular, and on it measure c g equal to a 1, or one-tenth of a c and join a g.

Through the points indicating the divisions into tenths draw lines parallel to c g, and you will thus have a series of triangles, d a 1, d′ a 2, d″ a 3, etc., similar to the triangle g a c. In virtue of a well-known property of such triangles (15), d 1 will be one-tenth of a 1; d′ 2 one-tenth of a 2; and so on.

Thus, if a measurement taken on the object, or on a large drawing, is equal to a x, it will only be needful to turn the compass on the point x as a center, and to observe accurately the perpendicular height, x z, to ascertain the corresponding measurement on the reduced scale.

Such a scale can be employed to measure meters and decimeters, or feet and inches (but in this latter case, it would have been necessary to mark off 12 instead of 10 divisions from a). Since a 1 might be made to represent one metre; a 2, two meters, etc., in virtue of the principle of the triangle already referred to, d 1 will be the tenth of a 1, and will therefore represent a decimeter; d′ 2 will represent 2 decimeters, etc. The length required to represent, say 5.3 will be ascertained by taking the distance a 5, to which the distance d″ 3 is added. Similarly 6 feet 2 inches would be given by a 6, to which d′ 2 is added on a 12-division scale.

26. The following description of one of these decimal scales, which is engraved on metal or ivory, and often included in cases of drawing instruments, will suffice to enable any one to construct a scale on this principle, that goes to a still further degree of accuracy, measuring, for example, meters, decimeters and millimeters, or yards, feet and inches.

Let A B, fig. 9, be a flat rectangular rule, divided throughout its length by parallel equidistant lines into ten strips. At right angles to these are the lines o o′, a a′, etc., separated from one another by a distance of one centimeter (doubled in the drawing in order to make the details more clear.) The first centimeter is subdivided along the two edges, A o, n′ o, into 10 equal parts or millimeters, and the division, o, on the upper edge is joined by an oblique line with the 1 on the lower edge, and the others by parallel oblique lines as shown in the figure. Thus c i will be one-tenth of a millimeter, s j two-tenths, and so on.

If the compass is opened so as to reach from x to z, it will be seen that it covers a space of 16 millimeters and 2-10ths of a millimeter, for there are one large division (or 10 mm.), 6 smaller divisions (or millimeters) plus a fraction of a millimeter equal to s j or 2-10ths of a millimeter.

27. Sector. When it is required to reduce the scale of a drawing, subject to the condition that the dimensions shall be all diminished in the ratio of two given lines, we may state the problem thus:

The longer of the two given lines is to the shorter, as any given dimensions of the old drawing is to x. The value of x thus determined will be the corresponding dimension of the new figure.

Such a rule of three proposition would involve a considerable amount of work, and the required result can be arrived at with greater facility by the geometrical methods which forms the basis of the scale just described, or, better still, by using the sector shown in fig. 10. It consists of two brass or ivory legs hinged about a center m which is at the apex of the angle n m c formed by two straight lines similarly divided into equal parts.

It is employed as follows: Let us assume that a drawing has to be reduced in the ratio of the line A to the line B; set off the length A along m n, and suppose its extremity to be at s, where division number 5 occurs. Open a compass to a distance equal to B, and placing one point on s, open the two legs of the scale until the second point coincides exactly with the corresponding division t, that is, with the 5 on the other leg, m c. Maintaining the scale open to this amount, it is only needful, after measuring a distance on the original drawing or object, to set it off along m n, and to measure the distance between its extremity and the corresponding point on the other leg; this distance will be the dimension on the reduced scale.

28. Proportional compass. This consists of two equal stems terminating with points, fig. 11. They are cut through for a portion of their length, and provided with a slide forming a hinge, that can be clamped by a screw a in any position. Graduations on the two slots and a mark on the slide indicate in what position of the slide a, the length a b (equal to a g) is equal to ½, ⅛, ¼, etc., of a d; and thus show what is the ratio of g b to c d, a ratio which is independent of the extent to which the arms are opened.

29. The vernier. The vernier consists of a small graduated slide which is adapted to a graduated rule or circular arc with a view to ascertain the value of small fractional parts of the divisions marked on the rule or arc.

Let A B, in fig. 12, be a rule divided into millimeters (the proportions are enlarged in the drawing so as to avoid confusion among the lines), and let it be required to determine a length to within the tenth of a millimeter.

As the measurement is required to the tenth, take ten less one or nine of the divisions of the scale; they will extend from O to IX, and this represents the acting length of the vernier.

Subdivide the vernier into ten equal parts; it is manifest that each graduation of the vernier differs from the original subdivisions of the rule by 1-10th of a graduation of the latter. In other words, unity on the vernier is equal to 9-10ths of unity on the rule.