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tion of such a book, which, to a certain extent, must necessarily be a compilation. Let the author follow the beaten path-repeat what has been explained before him —and he lays himself open to an accusation of plagiarism. On the other hand, let him emit new ideas, attempt new methods, and although he obtains the same correct results, he exposes his book to be rejected by those who, unaccustomed to the proposed alterations, often consider as " heresies”

any

deviations from their own familiar process. Disregarding these considerations, and without following invariably any particular method, I have selected what my experience induced me to consider as most useful. When I have found it necessary, I have not scrupled to consult works treating especially upon the branches having reference to the subjects under explanation. In most cases, however, I have endeavoured, when practicable, to simplify complicated methods. If in some of these I have failed, I am open to correction which will be thankfully received.

Some of my observations will perhaps appear scanty, because, wishing as much as possible to reduce the solution of individual subjects to general rules, I have seldom repeated the explanations given at the beginning of the chapter to which they refer, and which, to attentive and intelligent pupils should prove amply sufficient.

I beg, finally, to repeat that this small treatise is not in. tended to act as a substitute for, but merely as an adjunct to masters, and that its condensation was planned as a means of obtaining at the same time a cheap and useful text-book, acceptable both to tutors and pupils. That this may be the case, and meet with approval, is my sin

cere wish.

H. DE RHEIMS,

2, Bedford Terrace,

Plumstead Common. S.E.

CONTENTS.

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Problems relating to Points, Lines and Solids

Regular Polyhedrons

Plans, Elevations, Sections and Sectional Elevations of

Solids when standing on horizontal planes

Inclination of Planes to the Horizon

71

72

77

79

82

PRACTICAL LINES

IN

GEOMETRICAL DRAWING,

a

As in the following pages my intention is to be as concise as possible, and to avoid everything not absolutely necessary, I will confine myself to those parts of Geometrical drawing, to those descriptions of instruments, and to those observations which an experience of twenty-five years has taught me to be the most required by pupils.

Without entering into the description of those implements usually contained in a box of mathematical instruments, the use of which may almost be learnt intuitively, I will merely observe that “good work depends in a great measure on good tools,” and that too much care cannot be bestowed on our instruments, especially with respect to cleanliness, sharpness of points, smoothness of edges, and accuracy of scales, rulers, &c. The following rules, therefore, should always be borne in mind.

1. In geometrical drawing, all lines should at first be lightly drawn with a pencil moderately hard-HH.

2. The indian rubber should be used as sparingly as possible, and not until some time after the diagram has been inked in.

3. In inking in the different lines should, as much as possible, be drawn according to their order of construction, but when there are tangents to curves, begin by inking in the curves.

4. The beauty of a line consists in being throughout of an even and uniform thickness.

B

5. The indian ink should be carefully rubbed, very dark, and free from grit.

6. In drawing circles, the compass should be held very - lightly, and in a vertical position, yet so as to avoid making holes in the paper.

7. In geometrical constructions, the nature of the lines usually determine their meaning, thus :Thin lines,

are generally employed for data; thick lines,

for the representation of results, or quaesita, and dotted lines for lines of construction.

In intricate problems the nature of the lines of construction vary with the progress of the diagram, thus, in the first period, the lines are usually dotted, ........., or barred, --------; in the second period, alternately dot and bar .-.-.-.-.; in the third construction alternately two dots and bar, and so on.

Dotted lines are also used in solids to determine the position of lines unseen but necessary to the conception of the diagram, also to determine shadows, &c. As these lines are conventional, practice will teach what kind of lines should be used in any case.

8. When lines meet, the angle formed by them should be sharply defined.

9. For the sake of distinction original or given points are often surrounded by a small circle o to distinguish them from resulting points.

10. In determining points by the intersection of curves, the arcs should, if possible, cross nearly at right angles to each other, never at an angle less than 60°.

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The student should begin by drawing a series of pencil lines and then inking them in with lines of different thicknesses, and do the same with a series of concentric circles of different radii; before inking in, experience will teach

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