the first given line GL, from which the distance is to be obtained; place the ruler (with, in this case, index 40) diagonally, so that its edge coincides with the hypothenuse AC of the triangle, the arrow in its centre coinciding also with the zero O on the scale; then, whilst keeping the ruler steady with the left hand, slide the triangle with the right till the arrow is transferred and coincides with whatever may be the measurement required on the artificial scale—in this case 16. The perpendicular Bb will be found equal to one third of the space passed over by the arrow of the hypothenuse, and equal to that which would be obtained with the compass on the natural scale.

40

The following practice is recommended to those who wish to become familiar with the use of these scales :

Draw two lines seven inches in length parallel and one inch apart; divide the space between these lines into 14 compartments each an inch in breadth (vide Fig. 14), and subdivide by Marquois each of these compartments of 1 inch high by .5 inch broad into given numbers of equal parts, as already given in the explanation of plain scales.

The scale of 55 being wanting, the division of the inch into 11 equal parts is omitted, as well as that of 13, 17, and 19, which would be difficult and exposed to inaccuracy.

Dimensions can be obtained on the natural scales by means of the compass, precisely in the same manner as on the plain decimal scales affixed to the protractor, from which they do not differ in principle, but only in length.

Should we require to obtain through these scales the length of a line decimally given, say 3.5 inches, it is evident that that distance multiplied by the index of any scale will change its denomination, therefore let us assume the scale of 20, then, 3.5 × 20=18, the distance required. Take 5.83 by the scale of 30, then 5.83 × 30=

130.5
45

Take 2.9 by 45, then 2.9 × 45= &c.

174.9
30

In practice, the student will often discover some new means of using these scales with advantage, especially in the construction of diagonal and other scales, and in their application as substitutes to the triangle and T square.

THE PROPOSITIONS OF MOST FREQUENT
APPLICATION.

Those on which depend the construction of most Geometrical diagrams, the solution of most questions likely to be asked, are chiefly founded on the following Theorems :

EUCLID, BOOK I.

Prop. 15.-Theorem. If two straight lines cut one another, the vertical or opposite angles shall be equal.

Prop. 29.-Theorem. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.

Prop. 32.-Theorem. If one side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of every triangle are equal to two right angles.

Prop. 34.-Theorem. The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them, i.e., divides them into two equal parts.

Prop. 35.--Theorem. Parallelograms upon the same base and between the same parallels are equal to one another.

Prop. 37.-Theorem. Triangles upon the same base and between the same parallels are equal to one another. Prop. 47.-Theorem. In any right angled triangle, the square which is described upon the side subtending

C

the right angle, is equal to the squares described upon the sides which contain the right angle.

EUCLID, BOOK III.

Prop. 3.-Theorem. If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it shall cut it at right angles, and if it cuts it at right angles it shall bisect it.

Prop. 31.-Theorem. In a circle, the angle in a semicircle is a right angle, but the angle in a segment greater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle.

EUCLID, BOOK VI.

Prop. 2.-Theorem. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides or those produced proportionally, and if the side or the sides produced be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.

Prop. 33.-Theorem. In equal circles, angles whether at the centres or circumferences have the same ratio which the circumferences on which they stand have to one another, and so have the sectors.

Many of the following problems can be worked in several ways, but to avoid confusion, I will as much as possible confine myself to that single solution which I consider the best and most simple, indicating the propositions of Euclid demonstrating them.

All the problems and illustrations given in this work should be constructed at least twice or three times larger than those exemplified in the diagrams.

PRACTICAL GEOMETRY.

1. At a given point C upon a given line AB to erect a perpendicular. (Euclid 4 and 8, I.)

From Cas a centre and with any radius describe the semicircle DE. From the points D and E as centres, and with any radius greater than DC or EC, describe arcs intersecting each other in F, join CF, the perpendicular required.

2. At the extremity B, of a given line AB, to erect a perpendicular. (Euc. 31, III.)

Any where above AB determine the point C.

From C as a centre, with radius CB, describe the arc DB, intersecting AB in E. Through C draw the line.

ECD, and join DB, the perpendicular required.

Another method, exemplified at the extremity A of the given line, is also often useful.

From A as a centre, and with any radius, describe the arc FG. From F, with radius FA, describe the arc AH; and from H, with the same radius, describe the arc AK. From H and K, as centres, describe two arcs intersecting in L, and draw LA, the perpendicular required.

3. From a point in space C, nearly opposite the extremity of a given line AB, to drop a perpendicular. (Euc. 31, III.)

Draw any line CD, cutting AB in D, and bisect it in E. From E as a centre, with radius EC, describe the arc CBD, and draw the perpendicular C B.

4. From a point in space G, over a given line AB, to drop a perpendicular.. (Euc. 4 and 8, I.)

From G as a centre, with any radius greater than its distance from the given line, describe an arc cutting AB in H and K.

From H and K as centres, describe two arcs intersecting in L, join GL, meeting AB in M.

5. Through a given point C, to draw a line parallel to a given line AB. (Euc. 27, I. and 27, III.)

From C as a centre, with any radius, describe the arc DE.

From D as a centre, with the same radius, describe the arc CG.

Make the chord DE equal to the chord CG, and draw the line CE, which will be the parallel required.

6. To bisect a given line AB. (Euc. 4 and 8, I.)

From the extremities of the line A and B, and with any radius greater than half the whole length of the line, describe arcs intersecting above and below it, in C and D. The line CD being drawn, will bisect AB.

7. To divide a given line parts (say nine.)

AB into any number of equal (Euc. 2, VI.)

From A draw the line AC of any convenient length, and at any angle with AB (an angle from 45° to 60° is usually found the most convenient) with any convenient distance on the compass step on AC nine equal parts.

From the ninth, or last space C, draw a line joining CB, and from each of the other points draw parallels to CB.

N.B.-It is evident that this problem, which is of most frequent occurrence in Geometrical drawing, useful in the construction of scales, and on which the construction of the sectoral lines depends, will also enable to divide given lines according to any given ratio.