CHAPTER IV.

Geometrical Memoranda. - Paper Protractor-Inaccessible Distances.- Computation of Areas. - Triangles, Trapeziums, and Reduction of Figures.

A straight line lays evenly between its extreme points, as A B, Fig. 11; if we wish to draw as traight line on paper from one point to another, we lay a straight-edge close by the side of these, so that the point of a pencil may be carried from one to the other, and during the whole transit be in close contact with the straightedge. To do this on the ground, provided the points be not too far apart, we strain a line or a chain from point to point, and to lay down works, a pick traces the line by the side of the cord instead of a pencil. But if the points be too far apart for the cord to reach from one to the other, we must find intermediate points on the line sufficiently near to each other for the line to reach from each to each. To find these points set up ranging rods, so that standing a short distance from one or the other of the extreme points, one rod will cover the whole of those set up; the intermediate points are thus found; in doing this great care is required that the poles be all perfectly upright, which in very many cases can only be tested by the plumb line. By the same method a straight line may be produced, but here even greater caution is required as the length of the line is increased, for if there be a bend two lines containing an angle will have been laid out instead of a straight line.

Straight lines must either be so disposed with regard to each other as to be equidistant from each other throughout their lengths, when they are called parallels, or they must lie in such position that they meet and form an angle, or would do so if sufficiently produced. See Fig. 12.

When two straight lines meet each other, as in Fig. 13, they must do so either in such manner that the one of them does not lean to the other more on one side than on the other, as CD or A B, and C D is then perpendicular or at right angles to A B, and the angles are equal, and the numerical measure of each angle is 90°, or half 180°, or a fourth of 360°; or if the adjacent angles be not equal, then the one line E D meeting the

GEOMETRICAL MEMORANDA.

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other leans more on one side than on the other, and the angles are unequal, and the one measures more than 90°, and the other less; ED is then said to be oblique or askew to A B.

And any number of lines may meet a given line at a given point, and on the same side of the given line, but the angles contained by the given line and the concurrent lines will be equal to 180°, as in the last figure, where the angles A D F, FDC, CD E, and EDB, are all made up of, and equal to the two right angles A D C and C D B 180°. One straight line, therefore, meeting another straight line makes the adjacent angles equal to two right angles.*

An angle less than a right angle, as CDE or ED B, is an acute angle, and an angle greater than a right angle, as FDB, which is made up of the right angle CDB plus FDC, is obtuse. Also what any angle wants of 90° is called its complement; thus CDE is the complement of EDB; and what an angle wants of 180° is called its supplement; thus ADF is the supplement of FDB, or FDB is the supplement of AD F. Therefore the adjacent angles formed by one straight line meeting another are the supplements each to each.

The above circumstance is often of considerable use in surveying, as we will endeavour to explain, having still recourse to the same Figure. Let A DB be supposed a straight base of any length, and let A D be supposed to have been run over a range of hills and through broken ground, such as insures difficulty in tracing a straight line along the ground. It is very possible that the line may have become distorted before reaching D, which is supposed to be in a hollow, from which the greater part of the line A D is invisible, but DB visible; now the position of the latter enables us to check the former, for the angles FDB + FDA will together equal 180° if AD be straight. Again, suppose the position of A D to have been lost on the ground, and in the course of the survey wanted again, then the angle FDB being known, its supplement, or what it wants of 180°, enables us to set out D A with correctness.

Any number of angles formed by straight lines intersecting each other at one point are together equal to four right angles or 360°. Produce FD to G; then A DG and GDB together are equal to two right angles in the same manner as ADF, and FÚ B.

When two straight lines intersect each other, the opposite angles are equal to each other. In Fig. 13 let A B and D F

In mentioning an angle by three given letters, as ADC, the letter at the augle is always the middle one, and when there is no likelihood of confusion, an angle is often named by such letter alone.

intersect at C; then the angle ACE is equal to the angle DCB, and the angle ACD is equal to the angle ECB. For ACD and D C B are equal to two right angles; and DCA and ACE are also equal to two right angles; from each of these equal quantities subtract the angle ACD; then the remainders ACE and DCB will also be equal. In the same manner it may be shown that A CD is equal to EC B.

The circle is a plane surface circumscribed by a curved line, called circumference, any number of points on which are all at an equal distance from the point C called centre, Fig. 14; and this distance is called the radius; in the circle, therefore, any number of radii are all equal to each other.

The circle is divided into 360°; each degree into sixty minutes marked thus', and each minute is subdivided into seconds marked thus ".

A chord is any line drawn from any one point to any other point on the circumference, as A B, or B D or EA, and the greatest of chords is that which passes through the centre of the circle and is known as the diameter, which divides the circle in two equal parts, equal therefore each to 180°. The diameters of a circle can only intersect in the centre, as it is a condition of such a line to pass through it.

An arc is the portion of the circumference cut off by the chord.

A segment is any portion of a circle cut off between a chord and that portion of the circumference which the chord intersects, as BDEB, or A Ba A, or ADEB.

A quadrant, as EBC, is the quarter of a circle and the half of a semicircle; it is bounded by two halves of the two diameters intersecting each other at right angles, and by the quarter of the circumference cut off by them. This quarter of a circumference, which is equal to 90°, is the measure of the quadrant or angle ECB or BCA. An angle is thus measured by that portion of the circle intercepted between the two sides forming the angle, and subtending it. Thus each of the angles at C being equal by construction to right angles, is subtended by a fourth of the circle circumscribing the whole. An angle is also measured by the chord subtending it, as A B, which cuts off the arc Ba A, so that an angle is measured by the chord or by the arc, either of which will give the same measure.

A tangent, as FDG, is a line which touches a circle, as at D; a perpendicular to a tangent at the point of tangency will always pass through the centre of the circle if produced, as DB. An explanation of the trigonometrical signs will be found in another part of this work.

PAPER PROTRACTOR.

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The protractor is an instrument by means of which the mechanical operation of laying down angles on paper is performed. A variety of these are sold by the instrument-makers, and will be hereafter described; but a very useful instrument may be easily made, and which will answer a variety of general purposes: this is a paper protractor, constructed in the following manner :-With a radius of from four to six or seven inches describe a circle with your compasses; through this draw a diameter, and being careful that the legs of your compasses have not moved, and commencing at one end of your diameter, run the legs of your compasses right round the circumference. If you have done all this accurately, the third intersection of the circumference will be at one end of your diameter, and the sixth will bring you back exactly to your starting-point, and the circumference will have been divided into arcs of 60° each. Now take twice the natural sine of 15° and multiply it by the radius, which will give the chord of 30°, with which bisect each are of 60°, and next twice the natural sine of 10°, multiplied by the radius, which will give the chord of 20°, with which each are of 60° should be divided into three, whereby the circumference will be divided to 20°. Now, with twice the natural sign of 7° 30', divide the circumference to 5°. With twice the natural sine of 3° you may now subdivide to single degrees, and with twice the natural sine of 3° 45′ you may divide to half degrees, by working from the points you have already found. Twice the natural sine of 3° 20′, which is equal to the chord of 6° 40′, will enable you to divide the circumference to 10', if the size or radius of your protractor will admit of it.

This operation, which is rather a lengthy one, demands care and a nicely pointed pair of dividers and a good scale; the work may be checked first from one point and then from another, and if any trifling inaccuracy should be found, divide the error. Every ten degrees requires to be numbered from 0°, round to 360°, and the principal divisional lines should be longer than the rest, the lines for the single degrees should be somewhat less than these, and less again for the divisions of each degree.

Copies of this protractor may be obtained by pricking through the divisional points on a sheet of paper placed underneath. Should the reader be unacquainted with the use of trigonometrical tables, he will find instruction on the subject further on.

On one of the chords subtending any one of the quadrants of this protractor may be laid down "a line of chords," in the following manner :-Let A B, Fig. 15, be the chord to the quadrant or fourth part of the circle, and which is here only divided to every tenth degree. Set one foot of the compasses at B, and

describe arcs through the divisions on the arc on to the chord A B, which will then be a line of chords, by which any angle may be measured by making the two sides containing the angle equal to each other and equal to the radius of our protractor, and carrying the length of the third side in a pair of compasses on to the line of chords, which will give the measure of the angle required.

PROBLEM 1.

From a given point on a given line to set off a perpendicular.

Let AB, Fig. 16, be the given line, and C the given point. From C, and on each side of it, set off equal distances as at D and F, and from these points as centres, and with the same radius, describe two arcs intersecting at G; then a line from G to C will be the perpendicular required. Draw the lines G D and GF, which are equal by construction; G is therefore equidistant from D and F, as will also be any other point on the perpendicular G C. When not provided with any angular instruments, our means of setting off a perpendicular on the ground are generally very limited; so much so, that if such line be afterwards produced to any distance, it requires testing, which is readily afforded by the above means, since any point on the perpendicular must be equidistant from two points also equidistant from the point on the line from which the perpendicular has been set off. Let H be the point to which the perpendicular has been produced, and let I and K be the two points made equidistant from C; if CH be perpendicular to C, then IH and KH will be equal. Another means of testing this is merely to measure off C I and C K equal to each other, and produce D G and FG to intersect HI and H K at L and M, when, if H C be perpendicular to AB, H L will measure equal H M.

Of some of the Properties of Rectilineal Triangles.-Any side of any triangle may be taken as its base; the angle opposite to such base will then be the summit, and a perpendicular to the base from the summit will be the height or altitude of the triangle.

The sum of the interior angles of any triangle is equal to the sum of two right angles; therefore the sum of the three internal angles of any triangle is equal to the sum of the three internal angles of any other triangle; therefore, also, if two angles of a triangle are equal to two angles, each to each, of any other triangle, the remaining angles will be equal, and the triangles will be equiangular.