2. From C, as centre, with any radius, describe the semi

circle DIE. 3. From Das centre, with radius DE, describe the semi

circle E2F. 4. From C as centre, with radius CF, describe the semicircle

F3G. 5. From D as centre, with radius DG, describe the semicircle

G4H, &c., &c. Note. In this manner, a common spiral may consist of any number of semicircles, the points C' and D being alternately the centres of the required semicircles.

Problem 210.

To construct a spiral of one revolution.

1. Divide the given circle A into any number of equal

parts (say in this case eight), and draw radii to each point of division 1, 2, 3, &c.

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2. Divide one of the radii, say A1, into the same number of

equal parts (Pr. 15), and number them from the circum

ference 1, 2, 3, &c. 3. From A, with radii A1, A2, A3, c., describe arcs on Al,

cutting the corresponding radii 8, 7, 6, &c., in B, C, D,

&c. 4. Through points B, C, D, &c., draw the required spiral


NOTE.—The above spiral is usually termed the Archimedes spiral of one revolution, in honour of Archimedes, one of the most celebrated mathematicians of antiquity, who flourished about 300 A.C.

Problem 211.

To construct the involute of a given circle A.

1. Divide the given circle A into any number of equal parts

(say in this case twelve), and draw radii to each point of division 1, 2, 3, &c.

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2. From the points of division 1, 2, 3, &c., draw tangents

(Pr. 54), all being produced in the same direction, 3. On the tangent drawn from point 1, mark off a space

equal to one-twelfth of the circumference.

4. On the tangents drawn from points 2, 3, 4, &c., mark off

spaces equal to two, three, four-twelfths, &c., of the cir. cumference; the tangent thus drawn from point 12 will

be equal to the circumference of the circle. 5. Through the outer extremities of the several tangents

draw the required involute.

Problem 212. To construct the spiral, known as the Ionic volute, its longest diameter AB being given.

1. Bisect AB in C (Pr. 1).
2. Divide BC into four equal parts (Pr. 15), and let CD be

one of those parts.
3. On CD as diameter, describe a circle, which is called the

eye of the volute.

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4. In this circle, inscribe a square, having two vertical

diameters. 5. Divide each of these diameters into sic equal parts, and

number the divisions, as shown in E, the eye enlarged.

6. Produce 1, 2, indefinitely beyond 2, and from centre 1,

with radius 1A, describe the arc AF. 7. Produce 2, 3, indefinitely beyond 3, and from centre 2,

with radius 2F, describe the arc FG. 8. Produce 3, 4, indefinitely beyond 4, and from centre 3,

with radius 3G, describe the arc GH. 9. By proceeding in this manner, the required volute is

completed at point C. The radius for each successive arc is obtained by producing a line from the preceding centre through the point next in advance.


In geometrical drawing, it is often required to make a copy of an ohject much smaller than the object itself. For this purpose, we must make use of a scale, so that the several portions of the object may be drawn proportionally.

Scales are of various kinds , e.g., plain, diagonal, and a scale of chords. The simplest form of scale is the plain scale, which consists of a line divided into equal (or unequal) portions of various lengths, each portion representing some fixed measurement. For example, let the given line AB, which is in reality about 3 in. in А.



length, represent an actual length of 3 yards ; then one-third of the given line ; i.e., AC, will represent 1 yard ; one-third of AC will represent 1 foot, &c. &c.

NOTE.-Such a scale is called a scale of 36, because the whole line AB is ze of the distance which it represents, i.e. 3 in.

3 of 3 yards, or 108 inches. In this case, the fraction (38) is called the representative fraction of the scale.

Moreover, we may make a line of any length correspond to a foot, e.g., 1 in. as in A, 3 in. as in B, 1 in. as in C.

A B с

By dividing each of these lines into twelve equal parts, each part will correspond to an inch. Such scales are called “ duodecimal scales.”

Note. Sometimes the line corresponding to a foot, as in the preceding, is divided into ten equal portions, e.g.


Such scales are called “ decimal scales."

Diagonal Scale.--A diagonal scale is a scale used for measuring more minute distances than can be done by an ordinary plain scale. It is usually divided into 100ths.

A 9 8 7 6 5 4 3 2 1 B 9

E 9 8 7 6 5 4 3 2 16

Its construction is as follows :- Any indefinite straight line is taken, from which a distance AB is set off according to the intended length of the scale ; repeat AB any number of times as BC, CD, &c. Draw EF parallel to AD at any convenient distance from it; and draw the perpendiculars AE, BG, CH, &c. Divide AB and AE each into ten equal parts, and through 1, 2, 3, &c., draw lines parallel to AD; and through 1, 2, &c. (on the line AB), draw 1G, 2 1,3 2, &c., as in the above figure.

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