PRACTICAL GEOMETRY.

INTRODUCTION.

THE word Geometry is composed of two Greek words, signifying "the earth," and "measure;" and if the statements of Herodotus be true, that Geometry took its birth from the annual overflowing of the Nile, and the consequent re-measurement of land, the term applied to this process was literally correct. That Geometry in some shape existed in the earliest ages of the world, can easily be conceived, as it would have been impossible for the ancients to have constructed edifices, except upon geometrical principles; besides which, on some of the most ancient relics, astronomical emblems have been traced, which prove that Geometry at an early period was applied to astronomy; and as it is quite clear that most departments of the exact sciences are up to this day entirely dependent upon Geometry for their truth and usefulness, it is likely that it, from the first, advanced as rapidly as the limited facilities of the ancients would admit; till at the time of Euclid, who lived about 277 years B. C., it had arrived at so great a state of perfection, that upon his "data" are based all the great scientific truths of the present day.

Geometry is the science of magnitude, in its threefold properties; extension of length-extension of surfaceand extension of solidity: and the forms of objects, superficial and solid, can be so represented by its aid, as to convey the most correct ideas of their localities, bulk, and proportions; and upon it depend, to a much greater

extent than is at first sight perceived, the sources of enjoyment of civilized life. Every parent ought therefore to make it a special branch of his children's education; and whilst it will materially aid the acquisition of general knowledge, it will afford the highest gratification to the student's mind, and enable him to reason more acutely, more correctly, and with more moral effect, than the adoption of any other merely human system ever discovered.

Geometry is divided into two parts, called Theoretical, and Practical: the first comprises the principles of the science; the second, the application of those principles. Books on theoretical Geometry have proceeded from many authors, but not one is used more than Euclid's Elements, of which many excellent editions are published, as Potts', Colenso's, etc.

Both the theory and practice of Geometry will mutually assist each other, yet they may be studied separately with great advantage. The present work has been undertaken chiefly to guide practical men, who have not much time to devote to the theory of the subject. Geometrical terms are illustrated, and problems selected. These are followed by examples for exercise, in various useful forms, so that the mechanic may, by the substitution of his "line" and "chalk" for compasses and pencil, apply them to his daily occupation.

The first problem is explained at full length, in order that the student may know how to use his instruments; after which, it is presumed that he will be able to understand the construction of others with greater ease: but should he meet with difficulty, it may stimulate his efforts when told, that anything in science acquired through perseverance, will amply recompense him for all his toil.

DEFINITIONS.

IN every branch of science things occur which must have names assigned to them: these names require to be explained, or the novice cannot learn the science to which they apply. The explanations ought further to be such as to give the meaning of each thing referred to clearly and concisely: such explanations of names are called definitions. For example, if a pupil were asked, "What is a line?" it is probable the question would be proposed thus, "Define a line ;" and the proper reply to this demand would be, "A line is length without breadth."

Note.-Such definitions as are marked with a star are copied from the first book of Euclid's Elements.

1*. A Point is that which hath no parts, or which hath no magnitude.

A point has also been defined as having position, but no magnitude.

2*. A Line is length without breadth.

3*. The extremities of a Line are Points.

Lines are either Straight, Curved, or Mixed.

4*. A Straight (or Right) Line is that which lies evenly between its extreme points.

5. A Curved Line continually changes its direction between its extreme points.

6. A Mixed Line is formed by a right line joining a curved line.

When a line is mentioned simply, it means a straight or right line.

A Straight Line (as it appears to the eye) must be either Horizontal, Vertical, or Oblique.

7. A Horizontal Line is level.

It takes its name from the Horizon, which is the most distant edge of the sea, as seen on a clear calm day. The true Horizon partakes of the convex form of the earth, and in some scientific calculations this must be taken into the account; but in drawings of Machinery, and in perspective, the horizon is always represented by a straight line, which in a rectangular drawing is parallel to the base of the paper. In many good drawings, the horizon is not drawn parallel to the base, because positions are given to objects which would require more paper than could be spared to assign to the horizon its true position; but this is never forgotten either by the experienced draughtsman, or the judicious connoisseur.

8. A Vertical Line is that which presents the greatest contrast possible to a horizontal line, being always perpendicular to the horizon.

Thus, if a weight be freely suspended by a fine hair, when the oscillation has ceased, the hair will present a near approximation to a vertical line, since the weight will, (by the attraction of gravitation,) invariably tend towards the centre of the earth.

9. An Oblique Line is any straight line which is neither vertical nor horizontal.

10. A Tangent is a line that touches a circle or other curve without cutting

it.

A line or circle is tangential, or is a tangent to a circle, or other curve, when it touches it, without cutting, although both be produced.

The place where they touch is called the point of contact. When a tangent is mentioned simply, it means a straight line touching a curved line.

11. A Secant is a line which cuts a circle, lying partly within, and partly without it.

12*. A Superficies is that which hath only length and breadth.

13*. The extremities of a superficies are lines.

Superficies are either Plane, Concave, or Convex.

14*. A Plane Superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

Every plane superficies is completely flat or even, and is either a Horizontal, Vertical, or an Inclined Plane.

15. A Concave Superficies is that which is curvilineally hollowed.

16. A Convex Superficies is that which is globular.

17*. A Plane Angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.

Plane Angles are either Rectilineal, Curvilineal, or Mixtilineal.

18*. A Plane Rectilineal Angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

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N. B.-"When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the other line: Thus, the angle which is contained by the straight lines A B, C B, is named the angle ABC, or CBA; E that which is contained by AB, DB, is named the angle ABD, or DBA; and that which is contained by DB, CB, is called the angle DBC, or CBD; but, if there be only one angle at a point, it may be expressed by a letter placed at that point."

B

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Two adjoining angles, as ABE, ABC, having one line, as BA, common to both, are called Adjacent angles, provided the other lines EB and BC be in the same right line, but if not, they are called Contiguous angles, as ABE, ABD.

The word common in Geometry, signifies belonging to.