FIG. 62.-(PLATE VIIα.)

To construct a scale to show 70 yards, when 1 inch represents 9 yards.

Here we have 1 inch equal 9 × 12 × 3 = 324, or the representative fraction is; also, the scale has to be long enough to measure 70 yards, and will therefore assume a proportion as follows:9 yds. 1 in. :: 70 yds. : x = 7.77 inches. That is, if it takes 1 inch to represent 9 yards, it will take 7.77 to represent 70.

In the figure the scale is half the real size.

NOTE.-If the drawing of the scale be not accurate it is of no service. The student should therefore construct them with very great care, and should thoroughly test them before inking in.

Drawing Scales.-The general method of drawing scales is to draw two parallel lines about an eighth or three-sixteenths of an inch apart, and to make the bottom line a thick one, as is shewn on the drawings.

can be

Comparative Scales.-One scale is said to be comparative to another when the distances measured by the one measured by the other. As, for example, if we have a scale of miles, and we wish to measure the same distances on that scale in furlongs, then the scale of furlongs would be a comparative scale to that of the miles; consequently, in making any comparative scale there must of necessity be two units of comparisons, as an inch, a mile, a foot, a yard, &c.

FIG. 63.-(PLATE VIIα.)

To construct a comparative scale of yards so that 1 inch represents 4 feet, to measure 60 feet.

By proportion we have first

4 ft. 1 in. :: 3 ft. : x =

inch.

And to measure 60 ft. = 20 yrds. we have 20 ×

= 15 inches.

Draw, therefore, the scale 15 inches long, divide it into two primary divisions, the first primary division into 10 sub-divisions, to measure single yards.

Diagonal Scales are always divided decimally, and consequently (except when very large) completed to some power or multiple of 10.

FIG. 64. (PLATE VIIα.)

To construct a diagonal scale tenths and hundredths of one inch, the scale to be 4" long.

Draw a rectangle of 1" x 4", and through every inch of length draw lines parallel to the ends.

At one end A B divide A B into 10 equal divisions, and draw lines parallel to the sides.

Divide A C and B D into 10 equal divisions.

Join A with the ninth division on B D, and through every other division draw lines parallel.

The diagonal scale will be constructed as required.

To measure on the diagonal scale, 43, 3, 57, &c., place one leg of the dividers on the horizontal line E in the line C D, and stretch it until it meets the line drawn from 4 in B D, as shown by the dots.

For 3, we simply measure on B D.

For 57 place one leg of the dividers on the horizontal line 7 from A B at its point of intersection with C D, and stretch it until it meets the line drawn from 5 from B D, the distance will be what is required.

1. A line A B is 2.7" long. Divide it in the point C so that AB: BC as 7.5 4.

2. A B is the mean proportional between two lines 3" and 1.8" long. Find its true length.

3. Determine an equilateral triangle whose area equals the sum of three squares of 1", 1.5" and 2" sides.

4. Find by construction the value of the following, the unit being 1 square inch :—

5. Construct a nonagon whose diagonal shall be 4", and describe an equilateral triangle about it.

6. Construct a scale of yards 13. Show 60 yards.

7. Draw a diagonal scale of 1 foot to show cubits of an inch.

8. Construct a scale of fathoms whose representative fraction is 780.

9. Construct a scale of feet whose representative fraction is 18.

10. Draw a a rhombus (included angle 40°) of six square inches.

11. Draw a scale 6 feet to 1", the scale to be at least 50 feet long. What is the "fraction of the scale"?

12. Two circles have their centres 31" apart; the radius of one is ", that of the other is. Draw a straight line to touch both.

* The student is strongly advised to procure the work on Geometry in the Science Examination series of Answers to the Questions set at the Government Science Examination. Price, 9d. Elementary and Advanced separate. To be had from the publisher.

SOLID GEOMETRY.

CHAPTER I.

PROJECTION OF LINES.

Projection enables us to represent the true shape and dimensions of any object, by means of plans and elevations, so that by the aid of the drawing the objects themselves may be constructed.

Planes of Projection, which are usually conceived to be at right angles to each other, are employed. These planes are vertical and horizontal; and as it is of the greatest importance that the student should have a clear conception of their position and relation to one another, we shall proceed to explain their position and use.

All solid objects have three dimensions-length, breadth, and thickness; and for the construction of any object we require to know these three dimensions. If we take a box in our hands, and look straight down upon it, we have two dimensions, viz., length and breadth. Again, on looking direct at one of the sides we have two dimensions-height and length, but not breadth; so that two views of an object, called respectively the plan and elevation, are sufficient to determine its true dimensions, and are supposed to be obtained by straight lines perpendicular to the plane, going from every point in the object whose position is required until met by the plane. Although two planes are sufficient for the construction of any object from its projection, yet we frequently, for reason of obviating the necessity of measuring on two drawings, employ three planes, which can be well illustrated by supposing a book to be placed half open-that is, the leaves at right angles to each other-and a closed book

C

placed at right angles to its ends. The projection on the plane below the object, and which represents the view obtained when looking direct over the object is called the plan. The other projections are called respectively front and side elevations, as they represent the front or side.

Sections are formed by a plane cutting an object in any direction, and are employed when we require to represent some of the interior of a structure which is out of sight. When that section is parallel to the horizon, it is termed a horizontal section; when perpendicular to it, a vertical section; when forming any angle with the horizon, an oblique section; a section taken in the direction of the length of the object, a longitudinal section; when taken in that of its breadth, a transverse section; and when taken at right angles with two parallel sides of the object, so as to show the shortest distance between its several parts, a profile section.

The Ground Line is the line in which the planes intersect each other, and is called also the intersecting line, the base line, and the line of level, and is denoted by X Y.

Descriptive Geometry and Orthographic Projection. -In many works we have various names applied to the same subjects; as, for example, Orthographic Projection, Descriptive Geometry, &c. There is no distinct line of demarcation, however, between these divisions; those appearing being only accidental circumstances, or the omission of what is not required, but is otherwise supposed to be present. As, for example, in descriptive geometry the co-ordinate planes, or planes of projection, are supposed to extend indefinitely, forming four dihedral angles, and in problems in this subject we find objects frequently represented which are in any of the planes of projection. In orthographic projection only the vertical and horizontal planes are employed, they being supposed to terminate at the base or ground line. The planes which may be required for sections and the development of any section or solid are also generally formed by planes at right angles to the planes of projection, planes which are oblique to both being only occasionally used.

DEFINITIONS.

1. The intersections of a line or of the line produced with the planes of projection are called the traces of the line.

2. The traces are distinguished as vertical and horizontal, according as they are on the vertical or horizontal planes.

3. The traces of a plane are synonymous with those of a line, being the lines in which the plane meets the planes of projection. When the traces of a plane are given the plane itself is given.