PRACTICE.

1. By scale of chords, determine angles of 32°, 47°, 76°, 92° and 117°, and 153°, and prove those angles by the graduated edge of the protractor.

2. Construct a triangle having two of its sides 4.62 and 3.47 inches long respectively and the included angle 48°: figure the remaining angles and the length of the third side.

3. Construct a triangle ABC, of which the sides AB, BC and CA are respectively 1.72, 2.9 and 2.68 inches : determine the number of degrees contained in each of its angles.

USE OF SCALES.

In order to understand the use of scales, it is desirable that we should be acquainted with their construction. The scales generally in use are either the plain scale or the diagonal scale.

Plain scales contain primary and secondary divisions. The primary or larger divisions represent whole numbers or multiples of 10 units.

The secondary divisions are subdivided into as many units or fractional parts as are contained in a primary division.

To draw a plain scale, say of 10 units to 1 inch and 60 units in length. (Vide Fig. 5.)

Draw a line 6 inches long and divide it into 6 equal parts, (each of these parts will equal 10 units).

Subdivide the first left hand division into 10 equal parts. (Euclid 2, VI. book.) (Vide Practical Geometry, problem 7.)

Draw a second line (thick) at th of an inch below and parallel to the first, and number it as in the given example. So far the scale of 10 units to one inch will be completed. All plain scales are constructed according to the same

principle, and vary from each other only in their proportions, which are determined by the index accompanying them, thus the scales preceded by the numbers 60, 50, 40, 30, 20, &c. signify that the space of one inch is subdivided in the ratios of 60, 50, 40, 30, 20 equal parts, &c. The mode of division remaining the same in every one.

PRACTICE.

1. Draw a line A-B 6 inches long, and upon it, from A as a common starting point, determine by the plane scale and number the distances of

147 88 176 81 and 55
공공

60

40

It is clear that these scales will also enable us to obtain vulgar fractional parts of the inch-as, should we require

The scale of 18=2 of 10=3 of 18=4 of 50=5 of £0=6

of 35=7 of 40=8 of 45=9 of 59 or

=

10 of 55=11

of 60=12

of 35=14 of 2015, of 40=16 of 45=18, &c.

PRACTICE.

1. Find 1. and 24 inches on your scale.

Now 14 and 21= = 13.6

= =

2. Find 12.3, 3, and 19 of an inch.

The diagonal scales accompanying the protractors are constructed according to the following principles.

Required a scale, say of 100 units to 1 inch and of 600 units in length. (Vide Fig. 6.) Draw eleven lines (to inclose 10 spaces) 6 inches long, parallel and equidistant (any space between these lines will do, but of an inch is usually found a convenient distance, the size of the scale being however the best guide to follow.) Divide these lines into 6 equal parts, forming 6 spaces or primary

divisions separated by vertical lines. Subdivide the lower line of the left hand primary division into 10 equal and secondary parts-from the first left hand subdivision of which draw a diagonal line to the upper intersection of the highest line with the left hand bounding extremity of the scale from each of the other secondary divisions draw lines parallel to this and number them as in example.

The primary divisions will represent 100ths of units, the secondary divisions will represent 10ths, and the tertiary divisions each of the secondary or one single unit. (Euclid 2, VI. book.)

All diagonal scales are constructed on similar principles-but, like the plain scales, they differ in their rate of division.

On most protractors the diagonal scales bear the ratios of 200 and 400 to 1 inch.

On a scale of 100 to 1 inch, required distances of 2.5, 3.27 and 4.665. (Vide Fig. 6.) The dark line drawn from the second primary division 2 to the fifth secondary division on the base line will show the first distance. The dark line drawn from the point at which the seventh parallel line from the base is crossed by the second diagonal to the third primary division will give the second division.

And, in the same manner, that drawn midway between the two 6th and 7th parallels, will point the way of obtaining distances requiring 3 places of decimals.

PRACTICE.

On the scale whose method of construction has been given of 100 to 1 inch, mark off with a red line, or with a dark pencil, spaces of 1.7, 3.45, 4.78, 5.27, 5.655 and 5.785.

Should we require to obtain these distances on the scale of 200 to 1 inch, we would either multiply the distances by 2, or obtain them in half inches and double them on the

paper-the same operation by 4 will enable us to use similarly the scale of 400 to 1 inch.

This last method has the advantage of enabling us to obtain lengths of greater magnitude than those offered by the scale.

Required a line 9.92 inches long.

Take that distance on the inch scale and double it.
Required a space of 16.765 inches long.

Take very carefully that distance on the inch scale and step it four times.

One moment of ocular demonstration will show the convenience of the plotting scales to obtain equidistant or proportional spaces, &c. and in many instances to dispense advantageously with the compass.

SECTOR.

A sector is an instrument either of boxwood or of ivory, consisting of two moveable limbs each containing a series of lines acting in pairs, and divided so as to obtain certain dimensions according to given ratios. (Euclid 2, VI. book.)

The three principal pairs of lines contained by the sector

are,

The lines marked L, or lines of lines, which are divided up to 100 equal parts and are principally used for obtaining proportionals to given lines; to divide lines into fractional parts, to construct scales, &c.

The lines marked C, or lines of chords, by means of which angles can be accurately measured to degrees and half degrees, and by which other fractional parts of degrees can be approximatively obtained. They are graduated up to 60 degrees.

The lines of Polygons marked Pol, which are used for the construction of polygons, whether in given circles or on given bases, and from 4 to 12 sides.

N.B.-It is is to be observed, that the dimensions on the sectorial lines are to be obtained from the lines, at the extremity of which there is a small brass nail.

There are other pairs of sectorial lines, of which two, marked T, are used for tangents, and are graduated up to 45° and 75°. One S for sines, graduated to 90, and one pair for secants s, and graduated to 75°.

The sector contains likewise logarithmic scales of sines, tangents and numbers.

When open to its whole length, the sector is 12 inches long, and its back is divided into 100 equal parts.

On one of its edges, the same distance is divided into 12 inches, each decimally subdivided into 10ths of inches, or in 120 equal parts to one foot.

The spaces marked longitudinally on the sectorial lengths are termed lateral distances.

The dimensions taken on the open sector, reaching to similar indices of a pair of lines having the same denomi-. nation, are termed "transverse distances."

LINE OF LINES (L).

The lines of lines are divided into 10 equal parts, termed primary divisions, each of which is subdivided into 10 secondary subdivisions, the value of each of which may optionally equal .1, 1, 10, 100, 1000, &c. Application, (vide fig. 7.)

1. To divide a line 2 inches long into 10 equal parts. Take the given length (2 inches) in the compass, and adapt the sector to it at the denomination required; then open the sector so that the transverse distance 10-10, equals two inches. It is evident (Euclid 2, VI. book) that the transverse distances, 9-9, 8-8, 7-7, &c. will equal .9, .8, .7 of 2 inches.

2. Show .73 of a line 2 inches long?