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As institutions increase and progress, suitable books are required to meet the tastes and pursuits of the members, not one of which is more loudly called for than a treatise on Practical Geometry. The Author of the following pages does not presume to rank himself with writers on mathematical science, yet having been honored with many engagements at the Philosophical Theatres in London, &c., including that under His Royal Highness's patronage; he has been requested to give in as condensed, cheap, and simple a form as is practicable, the substance of his public lectures. To compile such a work has proved to be a more difficult task than he could have at first conceived, his custom having been to lecture wholly without notes, in addition to which, he was compelled, in consistency with the title of his work, to confine himself to a practical view of the subject; still, he hopes that the long experience of nearly thirty years as a draughtsman and teacher, has enabled him to treat his subject so as to remove difficulties which have hitherto discouraged the student; and whilst his work is intended to meet the demands of Mechanics' Institutions, he trusts that it will not be unworthy the perusal of the private gentleman, nor unadapted to the studies and tastes of ladies.
Woolwich, 27th of January, 1843.
THE word Geometry is composed of two Greek roots, 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 antique relics, astronomical emblems have been traced, which prove that geometry at an early period was applied, though crudely, 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 before Christ, it had arrived to 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 surface and 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, their application. Books on theoretical geometry have proceeded from many authors, but not one is patronized more than Euclid's Elements, an excellent edition of which, containing valuable explanatory notes, has recently been compiled by W. Rutherford, Esq., F. R. A.S., of the Royal Military Academy, Woolwich.
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 to guide practical men, who have not time enough 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 precisely 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.
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.”
All the following definitions, printed in large type, are copies of those in the best editions of Euclid's Elements; and having been approved by the most eminent mathematicians, they are here transcribed literally. Those persons who intend to study Euclid, should commit them to memory, so as to be able to quote them according to the order in which they stand, as I. II. III. &c. Besides these, others are added, but not being found in Euclid, they are numbered, 1, 2, 3, &c., so that they can easily be referred to when necessary. The notes and definitions, printed in small type, need only be read so that their sense may be clearly understood.
I. 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.
On reading the foregoing definition, the intelligent student will at once perceive that, although Practically, he must work with points which have "magnitude," yet these points should be made so small, that they can only just be visible to the naked eye. If this rule be not observed, there will be no approach to truth in geometrical drawings; it therefore follows that the legs of compasses with which points are to be stabbed upon paper, should be sharpened with great delicacy, and carefully preserved from injury.
II. A Line is length without breadth.
It may here also be repeated that, although the student must in Practice, work with lines which have "breadth," yet they should be drawn as thin as possible, as most of the results in line drawing depend upon the intersections of lines, in order to obtain points; if, therefore, the lines themselves are thick, the true points of intersection, cannot be decided upon. These remarks may at first sight appear to be premature; as the student in this stage of his work, is supposed not to have used his drawing instruments; but the cautions are necessary, because young students are frequently at a loss to know the practical meaning of the preceding definitions. Most of those here given, apply literally only to Theoretical Geometry, for the eye and hand of a draughtsman cannot avoid errors in practice; still, a very near approximation to practical accuracy may be obtained, if the student will fix in his mind a high standard of truth, and the first step to this attainment, is a clear comprehension of the theoretical and practical import of the above definitions.
III. The extremities of a line are points.
IV. A Straight Line is that which lies evenly between its extreme points.
Lines are either Straight, Curved, or Mixed.
1. A Curved Line continually
changes its direction between its
2. 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.
3. 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 con
vex form of the earth, and in 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 par