horn having a line drawn across it and rivetted to the end of a short scale of box-wood, divided like the former slide. It is used like the former, except that at starting, the zero of the short scale and not the line on the horn is made to coincide with the zero of the long scale. The slide is to be held fast to the instrument when this is moved.

The Pediometer is another less simple instrument used for the same object. It measures any quadrilateral directly.

(85) Some very complicated instruments for the same object have been devised. One of them, Sang's Planometer, determines the area of any figure, by merely moving a point around the outline of the surface. This causes motion in a train of wheel work, which registers the algebraic sum of the product of ordinates to every point in that perimeter, by the increment of their abscissas, and therefore measures the included space.

Instruments of this kind have been invented in Germany by Ernst, Hansen, and Wetli.

A purely mechanical means of determining the area of any surface by means of its weight, may be placed here. The plat is cut out of paper and weighed by a delicate balance. The weight of a rectangular piece of the same paper containing just one acre is also found; and the "Rule of Three" gives the content. A modification of this is to paste a tracing of the plat on thin sheet lead, cut out the lead to the proper lines and weigh it.

(87) FOURTH METHOD.-TRIGONOMETRICALLY. By calculating, from the observed angles of the boundaries of the piece of ground, the lengths of the lines needed for calculating the content. This method is employed for surveys made with angular instruments, as the compass, &c., in order to obtain the content of the land surveyed, without the necessity of previously making a plat, thus avoiding both that trouble and the inaccuracy of any calculations founded upon it. It is therefore the most accurate method; but will be more appropriately explained in Part III, Chapter VI, under the head of "Compass Surveying."

PART II.

CHAIN-SURVEYING;

By the First and Second Methods:

OR

DIAGONAL AND PERPENDICULAR SURVEYING.

(88) The chain alone is abundantly sufficient, without the aid of any other instrument, for making an accurate survey of any surface, whatever its shape or size, particularly in a district tolerably level and clear. Moreover, since a chain, or some substitute for it, formed of a rope, of leather driving reins, &c., can be obtained by any one in the most secluded place, this method of Surveying deserves more attention than has usually been given to it in this country. It will, therefore, be fully developed in the following chapters.

CHAPTER I.

SURVEYING BY DIAGONALS :

OR

By the First Method.

(89) Surveying by Diagonals is an application of the First Method of determining the position of a point, given in Art. (5,) to which the student should again refer. Each corner of the field or farm which is to be surveyed is "determined" by measuring its distances from two other points. The field is then "platted" by repeating this process on paper, for each corner, in a contrary order, and the "content" is obtained by some of the methods explained in Chapter IV of Part I.

The lines which are measured in order to determine the corners

of the field are usually diagonals of the irregular polygon which is to be surveyed. They therefore divide up this polygon into triangles; whence this mode of surveying is sometimes called "Chain Triangulation."

A few examples will make the principle and practice perfectly clear. Each will be seen to require the three operations of measur ing, platting, and calculating.

A

Fig. 49.

C

DE

B

(90) A three-sided field; as Fig. 49. Field-work. Measure the three sides, AB, BC, and CA. Measure also, as a proof line, the distance from one of the corners, as C, to some point in the opposite side, as D, at which a mark should have been left, when measuring from A to B, at a known distance from A. A stick or twig, with a slit in its top, to receive a piece of paper with the distance from A marked on it, is the most convenient mark.

Platting. Choose a suitable scale as directed in Art. (44). Then, by Arts. (42) and (49), draw a line equal in length, on the chosen scale, to one of the sides; AB for example. Take in the compasses the length of another side as AC, to the same scale, and with one leg in A as a centre, describe an arc of a circle. Take the length of the third side BC, and with B as a centre, describe another arc, intersecting the first arc in a point which will be the third corner C. Draw the lines AC and BC; and ABC will be the plat, or miniature copy - as explained in Art. (35) — of the field surveyed.

Instead of describing two arcs to get the point C, two pairs of compasses may be conveniently used. Open them to the lengths, respectively, of the last two sides. Put one foot of each at the ends of the first side, and bring their other feet together, and their point of meeting will mark the desired third point of the triangle. To "prove" the accuracy of the work, fix the point D, by setting off from A the proper distance, and measure the length of the line DC, by Art. (43). If its length on the plat corresponds to its measurement on the ground, the work is correct.*

Calculation. The content of the field may now be found as directed in Art. (65), either from the three sides, or more easily though not so accurately, by measuring on the plat, by Art. (43), the length of the perpendicular CE, let fall from any angle to the opposite side, and taking half the product of these two lines.

Example 1. Figure 49, is the plat, on a scale of two chains to one inch, of a field, of which the side AB is 200 links, BC is 100 links, and AC is 150 links. Its content by the rule of Art. (65), is 0.726 of a square chain, or OA. OR. 12P. If the perpendicular AD be accurately measured, it will be found to be 72 links. Half the product of this perpendicular by the base will be found to give the same content.

Ex. 2. The three sides of a triangular field are respectively 89.39, 54.08, and 45.98. Required its content.

sure also the other diagonal, or BD, for a "Proof line."

B

Platting. Draw a line, as AC, equal in length to the diagonal, to any scale, by Arts. (42) and (49). On each side of it, construct a triangle with the sides of the field, as directed in the preceding article.

To prove the accuracy of the work, measure on the plat the length of the "proof line," BD, by Art. (43), and if it agrees with the length of the same line measured on the ground, the field work and platting are both proved to be correct.

Calculation. Find the content of each triangle separately, as in the preceding case, and add them together; or, more briefly, multiply either diagonal (the longer one is preferable) by the sum of the two perpendiculars, and divide the product by two.

Otherwise: reduce the four-sided figure to one triangle as in Art. (78); or, use any of the methods of the preceding chapter. Example 3. In the field drawn in Fig. 50, on a scale of 3 chains to the inch, AB = 588 links, BC = 210, CD = 430, DA = 274, the diagonal AC = 626, and the proof diagonal BD = 500. The total content will be 1A. OR. 17P.

Ex. 4. The sides of a four-sided field are AB = 12.41, ВС = 5.86, CD = 8.25, DA = 4.24; the diagonal BD = 11.55, and the proof line AC = 11.04. Required the content.

Ans. 4A. 2R. 38P.

Ex. 5. The sides of a four-sided field are as follows: AB 8.95, BC = 5.33, CD = 10.10, DA = 6.54; the diagonal from A to C is 11.52; the proof diagonal from B to D is 10.92. Required the content.

Ans.

Ex. 6. In a four-sided field, AB = 7.68, BC = 4.09, CD = 10.64, DA = 7.24, AC = 10.32, BD = 10.74. Required the