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perpendicular D C, which make 3-draw A C and CB; then A B C is the triangle required.
To describe a square, whose side shall be of a given length.
Let the given line AB be three chains. At the end B of the given line erect the perpendicular B C, (by Prob. IV. 2.) which make A B :—with A and C as centres, and radius A B, describe arcs cutting each other in D: draw A D, DC, and the square will be completed.
To describe a rectangled parallelogram having a given length and breadth.
Let the length AB
and, the breadth B C = 2. At B erect the perpendicular B C, and make it 2-with the centre A and raB dius BC describe an arc; and with
centre C and radius AB, describe another arc, cutting the former in D:-join A D, D C to complete the rectangle.
The base and two perpendiculars being given to construct a trapezoid.
Let the base AB 6, and the perpendiculars AD and BC, 2 and 3 chains respectively. Draw the perpendiculars A D, DC, as given above, and join DC, thus completing the trapezoid.
To make a triangle equal to a given trapezium ABCD.
Draw the diagonal D B, and CE parallel to it, meeting AB prolonged in E-join DE; then shall the triangle ADE be equal to the trapezium ABCD.
To make a triangle equal to the figure ABCDEA.
Draw the diagonals DA, DB, and the lines EF, CG,
To make a square equal to a given rectangle ABC D. Produce one side A B till BE be equal to BC-bisect A E in o; on which as a centre, with radius A o, describe a semicircle, and prolong BC to meet it in F:-on B F describe the square BFG H, and it will be equal to the rectangle A B C D, as required.
To set off an angle to contain a given number of degrees. Let the angle be required to contain 41 degrees. Open the compasses to the extent of 60° upon the line of chords, and, setting one foot upon A, with this extent, describe an arc cut
ting A B in B; then taking the
extent of 41° from the same line of chords, set it off from B to C; join A C; then B A C is the angle required.
To measure an angle contained by two straight lines. (See last figure.)
Let AB, AC contain the angle to be measured. Open the compasses to the extent of 60°, as before, on the line of chords, and with this radius describe the arc B C, cutting A B, AC produced, if necessary, in B and C; then extend the compasses from B to C, and this extent, applied to the line of chords, will reach to 41°, the required measure of the angle B A C.
A right angle, or perpendicular, may be laid off by extending the arc B C, and setting off the extent of 90° thereon. Also an angle greater than 90° may be laid off, by still further extending the arc, and laying the excess of the arc above 90°, from the end of the 90th degree.
NOTE. Angles are more correctly and expeditiously laid off and measured by an instrument called the protractor, to be hereafter described.
A right line EF, cutting two parallel right lines AB, CD, makes the alternate angle equal, &c:-thus the angles A G H, GHD are equal; also the exterior angle E G B is equal to the interior and opposite G H D. (Euc. I. 29.)
The greatest side of every triangle is opposite the greatest angle (Euc. I. 18.)
Let the side AB of the triangle ABC be produced to D, the exterior angle CBD is equal to the interior angles at A and C; also the three interior angles of the triangle are equal to two right angles. (Euc. I. 32.) angles of a triangle being given the third
(See figure to Definition 11.)
Let A B C be a right angled triangle, having a right angle at B; then, the square on the side AC is equal to the sum of the square on the sides AB, B C. (Euc. I. 47.) Whence any two sides of a right angled triangle being given the third becomes known.
Let ABC, AED be similar triangles; then, the triangle ABC is to the triangle A ED as the square AB is to the square of AE: that is, similar triangles are to one another in the duplicate ratio of their homologous sides. (Euc. VI. 19.)
All similar figures are to one another as the squares of their homologous, or like, sides. (Euc. III. 20.)
All similar solids are to one another as the cubes of their like linear dimensions. (Euc. VI. 24.)
DESCRIPTION OF INSTRUMENTS USED FOR MEASURING AND PLANNING SMALL SURVEYS.
THE chain, usually called Gunter's chain, is almost generally used in the British dominions, for measuring the distances required in a survey. It is 66 feet, or 4 poles, in length, and is divided into 100 links, which are joined by rings. The length of each link, together with half the rings connecting it with the adjoining links, is consequently: of a foot, or
66 × 12
= 7.92 inches. At every tenth link from each end is attached a piece of brass with notches; that at the tenth link has one notch, that at the 20th two notches, that at the 30th three, that at the 40th four, the middle of the chain, or the 50th link being marked with a large round piece of brass; hence, any distance on the chain may be readily counted. Part of the first link, at each end, is formed into a large ring for the purpose of holding it with the hand.
The chain acquires extension by much use, it should, therefore, be frequently examined, and adjusted to the proper length by taking out some of the rings between the links: for this purpose, chains having three rings between each link are to be preferred to those having only two.
THE OFFSET STAFF.
The offset staff is used to measure short distances, called offsets; hence its name. It is usually ten links in length, the links being numbered thereon with the figs. 1, 2, 3, &c. It is usually pointed with iron at one end, for the purpose of fixing it in the ground, as an object for ranging lines, for marking stations, &c.
The cross is an instrument used by surveyors to erect perpendiculars. It is usually a round piece of sycamore, box, or
mahogany, about four inches in diameter, with two folding sights at right angles to each other, or more commonly with two fine grooves sawed at right angles to each other, which answer the purpose of sights. It is sometimes fixed on a staff, of convenient length for use, pointed with iron at the bottom, that it may be fixed firmly in the ground: but it is found more commodious in practice to have a small pocket cross, which may be readily fitted to the offset-staff, either by an iron spike on the cross being inserted in a hole made in the offset-staff, or the offset-staff being passed through a hole made in the cross, to about the eighth link from the piked end, at which place the staff must be shouldered, that the cross may rest firmly.
DIRECTIONS FOR MEASURING LINES ON THE GROUND.
Besides the instruments already described, ten arrows must be provided, about 12 inches long, pointed at the end, so as to be readily pressed into the ground, and turned at the other end, so as to form a ring to serve for a handle.
In using the chain, marks are to be set up at the extremities of the line to be measured, as well as at its intermediate points, if its extremities cannot be seen from one another, on account of hills, woods, hedges, or other obstructions. Two persons are then required by the surveyor to perform the measurement. The chain leader starts with the ten arrows in his left hand, and one end of the chain in his right; while the follower remains at the starting point, who, looking at the staff or staves, that mark the line to be measured, directs the leader to extend the chain in the direction of the staff or staves. The leader then puts down one of his arrows, and proceeds a second chain's length in the same direction, while the follower comes up to the arrow first put down. A second arrow being now put down by the leader, the first is taken up by the follower; and the same operation is repeated till the leader has expended all his arrows. Ten chains, or 1000 links, having now been measured and noted in the field book, the follower returns the ten arrows to the leader, and the same operation is repeated as often as necessary. When the leader arrives at the end of the line, the number of arrows in the follower's hand shews the number of chains measured since the last exchange of arrows noted in the field book, and the number of links extending from the last arrow to the mark or staff at the extremity of the line, being also added, gives the entire measurement of the line. Thus, if the arrows have been exchanged seven times, and