The following easy Problems, fully worked out, will serve to direct the student in the practical application of the preceding propositions and principles.

PROB. 1. Two equal rafters are to be framed together, at right angles, to span a building of given width; find the length of each rafter.

AB or BC=5×1·4142...... ft.-7.071...... ft.

AC

C

The object of multiplying by √2, as has been

√2

done here, is to avoid having to divide by the interminable divisor, or 1·4142...... For though it is very easy to multiply by such a number, yet to divide by it

Thus, if AB=7 feet, and CD-4 feet, the triangle ABC=7×4=14 square feet.

To measure, therefore, a proposed triangle, measure any one of its sides, and the distance of that side from the vertex of the opposite angle; then half the product of these two lengths will be the area required.

But, observe, by distance of a side from the vertex of the opposite angle is meant the shortest distance, that is, the perpendicular let fall from the vertex to the side. And this perpendicular will in certain cases fall not upon the side itself, but the side produced. Thus in the annexed fig. the area of the triangle ABC is equal to × ABX CD,

C

226. To measure a given rectilineal surface of any number of sides.

D

Let ABCDE be the proposed surface to be measured (the process is the same whatever be the number of the sides). From the vertex of any one of the angles, as A, draw the diagonals AC, AD, so as to divide the whole surface into the triangles ABC, ACD, ADE. Then the surface ABCDE is manifestly equal to the sum of the three

C

b

B

triangles, each of which may be measured separately by (225).

Thus, drawing the perpendiculars Bb on AC, Dd on AC, and Ee on AD, by Art. 225,

Area of triangle ABC-AC- Bb,

ACD-AC- Dd,

ADE=ADx Ee;

therefore area of

ABCDE=AC × Bb + 1 AC × Dd+ 1⁄2 AD × Ee;

and by measuring the lines AC, AD, Bb, Dd, and Ee, the measure of the surface required is known.

[This Problem will be more fully discussed in the section on SURVEYING.]

227. To measure the perimeter and area of a given regular Polygon of any number of sides.

This is obviously only a particular case of the preceding problem; but as it furnishes a general result applicable to all regular polygons whatever, it may be fitly inserted here.

Let AB be one of the sides of a regular polygon; O the centre of the circumscribing circle (162, Part 11.); draw OC perpendicular to AB, and join Ол, ов.

(1) Then, to obtain the measure of the perimeter, it is plain that we have only to measure AB, and multiply it by the number of sides in the polygon.

A

C

B

(2) Also, the area of the polygon will be equal to the area of the triangle OAB multiplied by the number of sides in the polygon, that is, AB OC× number of sides, or OC× perimeter.

OBS. It is to be observed that the proposition proved in (43, Part 1.), viz. that in any right-angled triangle the square of the hypothenuse is equal to the sum of the squares of the sides bounding the right angle, is of continual application in Mensuration, and enables us to measure squares, rectangles, and triangles, without the precise data supposed in the preceding Articles. Thus,

228. To measure a square when the diagonal only is given.

Let ABCD be the square, of which the diagonal AC is known: then, by (48),

AC AB+BC

=twice AB, AB=BC,

D

C

.. ABAC;

or ABCD= half the

square of AC.

A

B

line, or length, to be measured, and the tape is then unwound, until, being tightly stretched, there is sufficient of it to cover the line in its whole extent. The figures marked on the tape, where it coincides with the other end of the line, express the length of the line in feet and inches.

Or, if the length to be measured be greater than the whole length of the tape, it is only necessary to repeat the operation by successive measurements, as in the first

case.

The method of measuring still longer lines, or lengths, on the Earth's surface, as adopted by surveyors, will be given hereafter.

219. To measure a given crooked line or length.

(1) If the crooked line consist of two or more straight lines joined together, it is obvious that the whole line may be measured

by adding together the measures of the

several lines taken A

separately (determin

ed as in the last Art.),

which make up the whole.

B

D

Thus, it is evident that the measure of the crooked line ABCD will be found by adding together the measures of AB, BC, and CD.

Or, with the tape, it may often be done in one single measurement. For, if ABCD be the boundary of a rigid body, or if pegs be fixed at B and C, the tape may be tightly stretched so as to coincide with AB, BC, and CD, and thus shew at once the measure of ABCD.

(2) If the line, or length, to be measured be a curved line, its measure may be found by carefully laying a string upon it throughout its whole extent, and then applying the foot-rule, or other standard, to find the length of the string stretched out into a straight line.

Or, by means of a tape, a curved line, or length, may sometimes be measured at one step, since the tape combines in itself both the flexible string and the graduated measure. Thus, the woodman finds the girth of a tree,

or the tailor the circumference of a man's body, in a moment of time.

II. OF SUPERFICIES, SURFACES, OR AREAS.

220. In the same manner as the Arithmetical Measure of a line is the ratio which that line bears to another line, taken as the unit, or standard-so the Arithmetical Measure of a superficies, surface, or area (all which mean the same thing), is the ratio which that surface bears to another surface, taken as the unit, or standard, of superficial measure. And as the lineal foot was stated to be often the most convenient unit of length for measuring lines, so the square foot (that is, the square* of which each side is a lineal foot) is a common and convenient unit, or standard, of superficial measurement.

Hence, taking this unit, the measure of a surface, or area, is the number of times which that surface, or area, contains a square foot; and that number will be sometimes a whole number, and sometimes fractional.

For example, suppose the annexed diagram, ABCD,

to be a miniature representation of a rectangle, of which the side AB is 4 feet, and the side AD is 1 foot; then, dividing AB into four equal parts (168, Part 11.) in E, F, G, and drawing Ee, Ff, Gg, parallel to AD, or BC, it is obvious that we have divided the rectangle into 4 equal squares, each of which is a square foot; therefore the rectangle ABCD is plainly equal to 4 square feet, that is, the measure of the rectangle is 4, when the unit is a square foot.

221. But as it often happens, that a given superficies, surface, or area, which it is proposed to measure, does not contain an exact integral number of square

* It must be borne in mind that a square is not four straight lines of equal length and at right angles to each other, but the plane area included within those lines.