GE, and ED, each part will be an arc of of 90°, or 30°; and if AG be divided into three equal parts AK, KI, and IG, each part will be an arc of of 30°, or 10°, and so on to any other subdivisions.

The theodolite is

A

14. It will be useful for the student to know, at this stage of his instruction, something about the way in which angles are taken by the surveyor. the most useful instrument for measuring the angle contained by lines drawn from a point to two distant objects. The principle upon which this instrument is constructed may be readily understood. Let DGE be a circle divided into degrees like the protractor; upon the centre B of this circle

G

E

D

F

let a tube or spy-glass DE turn upon a pivot; direct the tube first towards the object A, and then turn it until it comes to the position FG, in a line with the other object C; then the angle formed by the lines AB and CB will be measured by the number of degrees in the arc EG.

Problems.

1. In order to find the distance of A from the station B, without actually measuring it, the surveyor finds the ABC (=40°); he then measures the distance BC (= 300 yds.), and placing his theodolite at c finds the ▲ BCA (= 70°).

Here, by construction, from the scale of equal parts, Art. 8., take off BC=300; with the protractor* draw a line BA, making the ABC= 40°, and CA, making the ▲ ACB=70°; then these lines will intersect each other in a point a; with the compasses take off BA, and apply it to the scale of equal

* Or with the scale of chords, hereafter described.

parts, and the units on the scale will be the units of yards in the required distance = 205 yards.

2. To find the distance of two towers A and B, inaccessible from each other. With

the theodolite take the ▲ c(=110°), measure A C (=32 yds.), and with the theodolite at A take the

angle ▲ (= 30°), which

A

B

the tower B makes with a staff placed at c; then we find by construction AB = 46 yards.

3. To find the height of a tower DC. measures from the bottom D of the tower the horizontal line DB = 400 ft.; he then places his theodolite at B, and takes the angle CBD (= 40°), which the top of the tower makes with the level line BD. Required the height DC of the tower.

From the scale of equal

parts take BD 400;

B

The surveyor

with the protractor draw BC, making the B=40°, and with the square, Art. 10., draw DC perpendicular to BD; then these lines will intersect each other in a point c. With the compasses take off the height DC, and apply it to the scale of equal parts, and the units upon the scale will give the units of feet in the height of the tower = 335 ft. nearly. 4. To find the height of a steeple BC, whose base B is inaccessible. The surveyor measured a base line AD = he then took the CDB = 52°, and the Required the height C B of the steeple.

=76 ft.; CAB=27°

From a scale of equal parts mark off AD=76; with the protractor draw DC, making the D= 52°, and draw AC.

they will give the units of feet in the height, as required, =64 ft.

5. To find the height of a tower CB standing upon a hill

Draw any line AB, and from a scale of equal parts take off BD = 50, and DA=

the

75;

with the protractor draw DC, making D=41°, and AC, making the ▲ A = 24°; then these two lines will intersect each other in a point c; join c and B, and CB measured off the scale of equal parts will give the height required = 69 ft.

Explanation of Terms and Processes.

15. A definition, in geometry, explains the peculiar way in which a geometrical line or figure is formed. Thus, when we say that an isosceles triangle is a triangle that has two of

It

its sides equal to each other, we define this sort of triangles. The definitions of figures form the premises, the properties admitted, upon which our conclusions, or properties to be demonstrated, are based. Thus from the definition of an isosceles triangle we can prove that the angles opposite to the equal sides are also equal. The property to be proved in any figure is called a theorem. Too much importance can scarcely be attached to the clearness of our definitions; and care should be taken to distinguish between the properties which we assume and those which we must prove. matters not, so far as our conclusions are considered, whether or not the figures, which we define, be actually described, provided we conceive them to be so according to the form of the definition; nor is it necessary, as in the course of demonstration, that we actually place one figure upon another, provided we conceive the thing to be done in the manner described. The fundamental method of geometrical demonstration is that of superposition, or the placing of one figure upon another.

The converse form of a theorem consists in stating the terms of the theorem in a reverse order. Such theorems are usually proved by the method called reductio ad absurdum, or the reducing of all other suppositions to an absurdity. In most cases the converse form of a theorem is really involved in the very proof we give of the theorem itself.

An axiom is an intuitive or self-evident truth, that is, a truth which is admitted the moment the terms in which it is expressed are understood. If equals be added to equals, the wholes are equal; if equals be taken from equals, the remainders are equal; things which are equal to the same thing, are equal; &c., are examples of axiomatic truths. The student will have no difficulty in observing, in the course of any demonstration, what principles are taken as axioms. It must be conceded that some theorems are as simple, and are as readily admitted, as the axioms upon

which they are made to depend. In this class may be ranked the properties of parallel lines. Such propositions, in a first course of geometry, may be taken as axioms without materially infringing upon the logical strictness of the geometrical course.

In a problem there is something required to be done from certain things that are given, which are called the data of the problem. Theorems, as well as problems, may be regarded as propositions, inasmuch as in both cases there is something proposed to be done.

In postulates it is assumed that it is possible to draw figures exactly as they are described in our definitions. Thus, although we may not actually draw a parallelogram, yet it is obvious that it is possible to construct this figure in accordance with its definition.

A corollary is an obvious deduction from one or more propositions.

An hypothesis is something which we suppose to be true, either in the enunciation of a theorem or in the course of its demonstration.

The sign of equality is =; equal to B. When two or

thus AB, signifies that a is more quantities are to be added, the signor plus is used; thus AB=AC+CB, for here the whole line is made up of the two

parts AC and CB. When one quantity is to be taken from another,

A

B

the sign or minus, is used; thus AB CB AC, for when the part CB is taken away from AB, the part AC must be left. In the case of angles we have, ≤ ABD= /ABC + ▲ CBD, for the whole angle is made up of these two parts; and also ABD-CBD = ABC, for when one of the parts is taken away from the whole angle, the other part must be left.

B

D

A

The symbol.. signifies therefore; indeed, every symbol used in algebra admits of the same interpretation in geometry.