ARITHMETIC, GEOMETRY, DRAWING, ETC

Page 5

**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 several geometrical methods given below.

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

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

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

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

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

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

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

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

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

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

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

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

*To divide a line into proportional parts.* The proposition can be solved in a similar manner. Let it be required to divide a line *t r* (fig. 24), into two sections that are to one another in the proportion of 5 to 3. On *t s* mark off a series of equal parts given by addition of these numbers together, that is 8; and join 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 in articles (**25-27**).

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

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

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

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

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

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

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

* 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

Assuming *a* to represent the starting-point of the curve, from the center 1 with radius 1 *a* draw the arc *a b*; from 2 with radius 2*b* draw the arc *b c*; from 3 with 3 *c* draw *c d*; from 4 with 4 *d* draw *d s*; then re-commencing 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.

Submitted by: Samuel Kirk (##)