are required turns in the spiral. Then subdivide one of these spaces into four equal parts. Produce the line BE to 4, making the extension E-4 equal to two of the subdivisions. At 1 draw the line 1-D, lay off 1-2 equal to one of the subdivisions. At 2 draw 2-A perpendicular to 1-D and at 3 in 2-A draw 3-C, etc. With center 1 and radius 1-B describe the arc BD with center 2 and radius 2-D describe the arc DA, with center 3 and radius 3-A, etc. until the spiral is completed. If carefully laid out the spiral should terminate at E as shown in the drawing.

MENSURATION.

Mensuration is that branch of arithmetic which is used in ascertaining the extension and solidity or capacity of bodies capable of being measured.

DEFINITIONS OF ARITHMETICAL SIGNS.

Sign of Equality, as 4+8=12.

+ Sign of Addition, as 6+612, the Sum.

Sign of Subtraction, as 6-3-3, the Remainder. × Sign of Multiplication, as 8×4 32, the Product. Sign of Division, as 24-6=4 24 24 = 4.

✓ Sign of Square Root, signifies Evolution or Extraction of Square Root.

2 Sign of to be Squared, thus 828×864.

3

Sign of to be Cubed, thus 33

3 × 3 × 3 = 27.

MENSURATION OF PLANE SURFACES.

To find the area of a circle-Fig. 52. Multiply the square of the diameter by .7854.

To find the circumference of a circle. Multiply the diameter by 3.1416.

To find the area of a semi-circle.-Fig. 52. Multiply the square of the diameter by .3927.

To find the circumference of a semi-circle. Multiply the diameter by 2.5708.

Semi-circle: Area = .3927D2

To find the area of an annular ring-Fig. 53. From the area of the outer circle subtract the area of the inner circle, the result will be the area of the annular ring.

To find the outer circumference of an annular ring. Multiply the outer diameter by an 3.1416.

To find the inner circumference of an annular ring. Multiply the inner diameter by 3.1416.

Annular ring: Area = .7854 (D2—H2)

To find the area of a flat-oval-Fig. 54. Multiply the length by the width and subtract .214 times the square of the width from the result.

To find the circumference of a flat-oval. The circumference of a flat-oval is equal to twice its length plus 1.142 times its width.

Flat-oval: Area D (H-0.214D)

Circ. 2 (HX0.571D)

=

To find the area of a parabola Fig. 55. Multiply the base by the height and by .667. Parabola: Area = .667 (DXH)

To find the area of a square-Fig. 56. Multiply the length by the width, or, in other words, the area is equal to square of the diameter.

To find the circumference of a square. The circumference of a square is equal to the sum of the lengths of the sides.

To find the area of a rectangle-Fig. 57. Multiply the length by the width, the result is the area of the rectangle.

To find the circumference of a rectangle. The circumference of a rectangle is equal to twice the sum of the length and width.

Rectangle: Area DX H

To find the area of a parallelogram Fig. 58. Multiply the base by the perpendicular height. Parallelogram: Area DXH

Fig. 59.

To find the area of a trapezoid-Fig. 59. Multiply half the sum of the two parallel sides by the perpendicular distance between the sides.