D parallel to them, meeting the C F B G PROBLEM XIII. To make a square equal to a given rectangle A B C D. Produce one side A B till B E be F G equal to BC:-bisect AE in 0; on which as a centre, with radius A 0, describe a semicircle, and prolong BC Ć D to meet it in F:-on BF describe the square B F G H, and it will be equal to the rectangle A B C D, as required. E B Η Α PROBLEM XIV. 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 cutting A B in B; then taking the A 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. PROBLEM XV. (See last figure.) Let A B, 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 AB, 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. B E. GEOMETRICAL THEOREMS. TAEOREM I. Angles vertically opposite are equal: -В 6 thus the angle A GE angle H GB, C. -D and EGB = AGH. (Euc. I. 15.) H THEOREM II. (See last figure. A right line EF, cutting two parallel right lines AB, CD, makes the alternate angle equal, 8c. :-thus the angles AGH, GHD are equal; also the exterior angle E GB is equal to the interior and opposite G HD. (Euc. I. 29.) THEOREM III. The greatest side of every triangle as opposite the greatest ungle. (Euc. I. 18.) THEOREM IV. Let the side A B of the triangle A BC be produced to D), the exterior angle CBD is equal to the interior angles at A and C; B also the three interior angles of the triangle D are equal to two right angles. (Euc. I. 32.) Whence any two angles of a triangle being given the third becomes known. THEOREM V. (See figure to Definition 11.) Let ABC be a right angled triangle, having a right angle at B; then, the square on the side A C is equal to the sum of the square on the sides A B, BC. (Euc. I. 47.) Whence any two sides of a right angled triangle being given the third becomes known. THEOREM VI. In any triangle A B C, let D E he drawn parallel to one of its sides, CB; then, A B is to A E as B C is to DE; E and the triangles are said to be similar. (Euc. VI. 2.) THEOREM VII. (See last figure.) Let ABC, A ED be similar triangles; then, the triangle A B C is to the triangle A ED as the square of A B 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.) D с B THEOREM VIII. All similar figures are to one another as the squares of their homologous, or like, sides. (Euc. VI. 20.) THEOREM IX. All similar solids are to one another as the cubes of their liko linear dimensions. CHAPTER II. DESCRIPTION OF INSTRUMENTS USED FOR MEASURING AND PLANNING SMALL SURVEYS. THE CHAIN. 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 66 66 x 12 adjoining links, is consequently of a foot, or =7.92 100 100 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 ot' fixing it in the ground, as an object for ranging lines, for marking stations, &c. THE CROSS. The cross is an instrument used by surveyors to erect perpendiculars. It is usually a round piece of sycamore, box, oi a 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 cach 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, 80 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 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 shew the number of chains measured since the last exchange of arrows noted in the field book, and the number of 7000 links extending from the last arrow to the mark or 600 staff at the extremity of the line, being also added, 83 gives the entire measurement of the line. Thus, if the arrows bave been exchanged seven times, and 7683 arrows. if the follower have six arrows, and from the arrow last put down to the end of the line be 83 links, the whole measurement will be 7683 links, or 76 chains 83 links, which is usually written thus-76-83 chains, the two last figures being decimals of a chain. In using the chain, care must be taken to stretch it always with the same tension, as it will extend by much use, and will therefore require to be examined occasionally, and shortened, if necessary. But a good chain may be used several days, on tolerably smooth ground, without any material extension. The surveyor must mark, or cause, to be marked, every station on the line, while it is being measured, with a staff or cross on the ground, entering its distance in the field book. When a survey is made for a finished plan, all remarkable objects should be noted down; as buildings, roads, rivers, ponds, footpaths, gates, &c. The boundary of the estate measured ought to be carefully observed. If the ditch be outside the boundary fence, it usually belongs to the estate, and vice versa ; although this is not universally the case; therefore, inquiry ought to be made with respect to the real boundary. In some places five links from the hedge-posts or roots of the quickwood are allowed for the breadth of the ditch, but this breadth varies to as far as even ten links, especially in swampy countries. All ditches and fences must be measured with the fields to which they belong, when the full quantity on the plan is required: but when the growing crops only are to be measured, only so much as is occupied by the crops. INSTRUMENTS FOR LAYING DOWN OR PLOTTING SURVEYS. THE COMMON DRAWING COMPASSES, This instrument consists of two legs moveable about a joint, 80 that the points at the extremities of the legs may be set at any required distance from one another: it is used to transfer and measure distances, and to describe arcs and circles. THE HAIR COMPASSES. The hair compasses ought to be used where greater accuracy is required in transferring distances, than can be obtained by the set of the joint of the common compasses. In the hair compasses the upper part of one of the steel points is formed into a bent spring, which, being fastened at one extremity to |