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,

2 × 2 = 4, the second power or square of 2. 2 × 2 × 2 = 8, the third power or cube of 2. 2 × 2 × 2 × 2 = 16, the fourth power of 2, etc.

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 6^3 = 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:

39,94,24(6 36 -- 394

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:

39,94,24(63 36 -- 123 394

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:

39,94,24(63 36 -- 123 394 369 --- 2524

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

39,94,24(632 36 -- 123 394 369 --- 1262 2524 2524

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.)

[Illustration: _Fig. 1._]

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=).

[Illustration: _Fig. 2._]

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.

[Illustration: _Fig. 3._]

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_.

[Illustration: _Fig. 4._]

_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.

[Illustration: _Fig. 5._]

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.

[Illustration: _Fig. 6._]

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.

[Illustration: _Fig. 7._]

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.

[Illustration: _Fig. 8._]

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.

[Illustration: _Fig. 9._]

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.

[Illustration: _Fig. 10._]

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.

[Illustration: _Fig. 11._]

=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.

[Illustration: _Fig. 12._]

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.

[Illustration: _Fig. 13._]

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.

[Illustration: _Fig. 14._]

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.

[Illustration: _Fig. 15._]

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).

[Illustration: _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.

[Illustration: _Fig. 17._]

=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.

[Illustration: _Fig. 18._]

=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=).

[Illustration: _Fig. 19._]

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.

[Illustration: _Fig. 20._]

=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.

[Illustration: _Fig. 21._]

=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).

[Illustration: _Fig. 22._]

_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.

[Illustration: _Fig. 23._]

=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.

[Illustration: _Fig. 24._]

_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.

[Illustration: _Fig. 25._]

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.

[Illustration: _Fig. 26._]

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_.

[Illustration: _Fig. 27._]

=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.

[Illustration: _Fig. 28._]

=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.

[Illustration: _Fig. 29._]

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_.

[Illustration: _Fig. 30._]

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.

[Illustration: _Fig. 31._]

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.

[Illustration: _Fig. 32._]

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.

[Illustration: _Fig. 33._]

=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.

[Illustration: _Fig. 34._]

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.

[Illustration: _Fig. 35._]

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.

[Illustration: _Fig. 36._]

=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_.

[Illustration: _Fig. 37._]

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.

[Illustration: _Fig. 38._]

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.

[Illustration: _Fig. 39._]

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

1 degree of longitude equals 4 minutes of time. 1 minute ” ” ” 4 seconds ” ” 1 second ” ” ” ¹⁄₁₅ or .066 seconds of time.

=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.

FOOTNOTES:

[1] Loomis’ Treatise on Arithmetic.

Loomis’ Treatise on Algebra.

Loomis’ Elements of Geometry.

Robinson’s Algebra and Geometry.

[2] Adapted from Robinson’s Algebra.

[3] Determined in the manner explained for erecting a perpendicular in par. 32 except that intersecting arcs are described on both sides of the line (_n m_, fig. 17); the perpendicular will be a line joining these points of intersection.

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