AB, say a', draw xy, distances draw bb', cc,

parallel to line EF, and at any

dd', &c., parallel to AB, and draw

the lines 1.0, 2 p, 3 q, &c., parallel to xy.

It will be then obvious that at distance a'o the line CD approaches to line EF by the portion a' 1, of the distance between them; at distance a' p, by the portion a' 2, and so on, and consequently they must at length meet; and further, by a simple proportion the exact length of the distance to which each line must be continued from its point of intersection with the line AB may be calculated.

Thus for line EF continued, a'l: lo a'a distance required; and for line CD continued, a'1: ao:: a'a: distance required to point of intersection.

It must also be on the side where the angles are less than two right angles, as on the other side the distance between the lines will continually increase.

DIVISION B.

CONSTRUCTIONALS.

1. As shown in my "Practical Geometry" and in the Programme of a new Text Book on Geometry submitted to the "Society for the Improvement of Geometrical Teaching," that in order to have geometrical instruction made thoroughly useful as well as interesting, the common-sense course of first teaching how to do a thing, and then showing why it is done, and subsequently when it should be done (as adopted in teaching a business or a profession), must be adopted; and when adopted will be found the most effectual means to readily attain, retain, and apply knowledge.

I intend therefore in this work to follow out the principle adopted in my "Practical Geometry," and show the mechanical results, before showing the geometrical principles on which the operations are based.

The course I adopt is attended with the very great advantage that a student, without any strain on his mind, learns to perform many operations, the utility and interest of which must be obvious to him, and instead of being driven, as it were, to know how it is that the operations he performs are attended with the results attained, he desires to obtain such information, and thus enters with spirit and interest into a study which as generally taught is far from attractive.

2. Before showing how the work of constructionals is performed, it may be of use to make some remarks on the requisite tools.

On that subject, whilst drawing the figures with a fair degree of accuracy will be found, not only very easy, but of great use in the understanding the different operations, as compared with the common practice of having figures of scarcely any shape to represent circles, squares, &c., which should appear before the eye of the student in the different operations, the minute accuracy and finish required in the office of the engineer or architect is in no way required, and consequently more simple tools will be amply sufficient.

3. For geometrical drawing, a pair of compasses, with pen and pencil points, and a smaller pair for minute circles, a straight-edge or ruler and ruling pen or ordinary pen for lines, a pencil and piece of indiarubber, and a pair of set squares, will be sufficient.

4. Practically most even of these simple tools may be dispensed with, though useful and inexpensive; as for actual use as regards lines, an ordinary ruler, pen, pencil, and indiarubber will suffice, and for circles a piece of card with a pin-hole to represent centre, and another, or hole through which the point of a pencil may pass to represent radius, any circle may be drawn by a pencil; and for set squares, to draw perpendiculars or parallels, an ordinary canvassing card cut diagonally will give a most useful pair of set squares.

5. For fine lines, the ruling pen is far preferable to an ordinary pen, and the regular set squares, which with the ruling pen may be purchased for about 2s., will be found very useful.

6. The ordinary set squares as sold, consist of one having a right angle and other adjacent sides equal, and consequently giving each an angle of 45 degrees, and the other with a right angle and one of the adjacent sides double the length of the other, and consequently giving angles of 60 and 30 degrees.

7. A large parallel ruler is useful for drawing long lines, but is seldom required in mere geometrical work, though very useful in navigation, and in general the set squares are more useful than small parallel rulers.

8. In geometrical operations it is seldom necessary to draw figures to scale, but where desirable to do so, Fig. F 37 will give a scale to inches and hundredths of an inch.

If desirable to prepare a scale of feet and inches, the same may be made by doubling down a sheet of paper a foot long, and with a halfpenny marking in twelve operations the successive inches, or by a penny marking in ten operations the twelve inches in the foot, the halfpenny exactly giving an inch, and the penny an inch and one-fifth.

The division should be marked by a pin or point of a knife, a pencil not touching with sufficient accuracy.

It may not be useless to remark that three pence and five halfpence weigh an ounce.

9. In many geometrical operations, and especially in the very common and important matter of finding a mean proportional or side of a square equivalent with a given rectangle, the operations may be performed by simply using a thin strip of paper (see Div. F, Sect. 15, Fig. 4, post).

10. As to pencils, where an extremely fine line is required, the pencil should not be sharpened to a point, but to the form of a wedge, by two long and broad and flat slices being cut on opposite sides, to get the full width of the lead, and two narrower slices on the other opposite sides.

In drawing lines, the pen or pencil should be kept as perpendicular as possible, and at the same degree of closeness to the ruler or set square, for the whole length of the line drawn.

11. In a large proportion of the operations in the subsequent sections, in which arcs of circles are used in Euclid, the more simple and more neat mode of obtaining the results of parallels is adopted; not merely from the direct advantages incidental to such use, but, as before remarked, to limit the number of tools as much as possible, and thus by using only a few tools to acquire the greatest readiness in the use of them.

The next Section will commence the practical work of Constructionals.

12. TO DRAW PERPENDICULAR LINES.

(a) Geometrically. The best mode of drawing a perpendicular line from a given point in a given line, or letting fall a perpendicular from a given point on a given line, will be that shown in Fig. B 1, and which is based on the angles in a semicircle being all right angles (Div. E, Sect. 36). Application 1. From point a in line xy to raise a perpendicular line.