This method is of use in cases where the actual height in inches is not fixed. AB is divided into ten parts, and they are numbered from each end as shown. Then at each point set up an ordinate representing the product of the two numbers at the point; either vertically, as in the figure, or all parallel in any direction.

The curve then passes through the tops of the ordinates.

The property which leads to this method of construction is, in words

The ordinate is proportional to the rectangle contained by the segments into which it divides the base. For the area we have

RULE.-The area of a segment of a parabola is of that of the circumscribing parallelogram; i.e. of the product of base and perpendicular height.

The circumscribing parallelogram means the parallelogram AEFB in Figs. 9 or 10.

The Hyperbola.-This curve together with the two preceding constitute the conic sections, which are treated in Analytical Geometry. For our purposes we select only those properties which are directly useful; and we define the curve by those properties, and not as it is defined in geometry.

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For our purposes then we define the curve thus:The hyperbola is the curve traced by a point which moves so that the product of its distance from two rectangular axes is constant.

This is strictly a special case, and is the rectangular hyperbola. It is, however, the only one we concern ourselves with.

Construction of the hyperbola.

Let OX, OY be the axes, and A one known position of the moving point. Through A draw lines AN and AM, parallel to OX and OY respectively. Produce AN as far as necessary. (The curve has no limits.)

Mark off along MX distances M1, 12, etc., equal or

not, so far as the curve is required to extend, and draw the ordinates II, 22, etc. Join now the top points 1, 2, etc., to O, cutting MA in 1', 2', etc.

Through ' draw 'I parallel to OX, cutting II in I; through 2', 2'II cutting 22 in II; and so on for III, IV, V, etc.

Then I, II, etc., are points on the required curve, and we draw it through them.

By Euclid we easily prove that the rectangles, OI, OII, etc., are equal to OA.

Area. The particular area required is that bounded by the curve, two ordinates, and OX; such as ABCD in Fig. 13.

The log. is the Napierian or natural

log., or from its present property the

hyperbolic log., and is 2.3026 times the ordinary tabular log. The number 2.3026 being known as the modulus.

Any Curve.

To find the area between the irregular curve CD, the ordinates AC and BD, and the base AB; we proceed as follows:

Divide AB into an even number of small parts, and

draw ordinates.

The sum of the extremes, four times the even, and twice the remaining odd; all multiplied by one-third of the common interval.

For example in Fig. 14 the lengths are given in inches, the interval being ". Then

Extremes

Remaining odd 1.125 +1.075+.975+1.225=4.4.

Evens

1.1+1.225=2.325.

1.075+1.15+.95+ 1.1 + 1.325=5.6.

(2.325+2 × 4.4+4×5.6)†

.. Area=

sq. ins. 2.8 sq. ins.

3

Peripheries. These are not of importance; and can

not in most cases be accurately found, hence we omit them except for the circle, which is given on page 4.

We come next to solid figures. These we treat briefly. Square-edged Plate.

RULE.—The volume of a piece of plate of any shape is equal to the product of the area of its face and its thickness.

Volumes are required chiefly for the determination of weights. To determine this we must know the weights of unit volumes of the particular materials, which will for the principal ones be found in a table in chap. xxi. The Sphere.

RULE. The volume of a sphere of radius r is πr3.
RULE. The surface of a sphere of radius r is 4ær2.
The Cylinder.

Fig. 15 shows two views of a cylinder. Side view and end view or cross-section. This is a right circular cylinder, the two ends being perpendicular to the axis.

HO FO

Fig. 16 shows a circular cylinder, the two ends not being perpendicular to the axis but parallel to each other. The two cross-sections are the same. Then for each case we have

RULE. The volume of a cylinder is obtained by multiplying the area of its cross-section by its length measured along the axis.

Thus in each case

Volume 2h.

RULE.-The surface of a cylinder is obtained by multiplying the periphery of its cross-section by its length along the axis.

In each case then

Surface 2πrh.

If the cylinder be hollow and thin, and cut through along ab, then it can be flattened into a plate; the area of which is evidently as above.

The Parallelopipedon. This is a solid, whose cross-section is not circular, but

h

Fig. 17.

is the same at all points of its length; as for example a wrought or cast-iron beam (Fig. 17).

The rules for the cylinder apply exactly.

Lastly, though of minor importance, we will take— The Cone.

RULE.—The volume of a cone is that of a cylinder on the same base, and of the same height.

.. Volume={πr2h.

The curved surface can be found by supposing the cone hollow, cutting it down a line as ab, and flattening it out as shown in Fig. 19, forming a section of a large circle of radius 1.