(81) General Rule. When the given figure has many sides, with angles sometimes salient and sometimes re-entering, the operations of reduction are very liable to errors, if the draftsman attempts to reason out each step. All difficulties, however, will be removed by the following General Rule:

1. Produce one side of the figure, and call it a base. Call one of the angles at the base the first angle, and number the rest in regular succession around the figure.

2. Draw a line from the 1st angle to the 3d angle. Draw a parallel to it from the 2d angle. Call the intersections of this parallel with the base the 1st mark.

3. Draw a line from the 1st mark to the 4th angle. Draw a parallel to it from the 3d angle. Its intersection with the base is the 2d mark.

4. Draw a line from the 2d mark to the 5th angle. Draw a parallel to it from the 4th angle. Its intersection with the base is the 3d mark.

5. In general terms, which apply to every step after the first, draw a line from the last mark obtained to the angle whose number is greater by three than the number of the mark. Draw a parallel to it through the angle whose number is greater by two than that of the mark. Its intersection with the base will be a mark whose number is greater by one than that of the preceding mark.*

6. Repeat this process for each angle, till you get a mark whose number is such that the angle having a number greater by three is the last angle of the figure, i. e. the angle at the other end of the base. Then join the last mark to the angle which precedes the last angle in the figure, and the triangle thus formed will be the equivalent triangle required.

In practice it is unnecessary to actually draw the lines joining the successive angles and marks, but the parallel ruler is merely laid on so as to pass through them, and the points where the parallels cut the base are alone marked.

(82) It is generally more convenient, for the reasons given at

the end of Art. (80), to reduce

Fig. 44.

half of the figure on one side and

secutively all the way around, but, after the numbers have gone around as far as the angle where it is intended to have the vertex

of the final triangle, the numbers should be continued from the other angle of the base, as is shown in Fig. 45. In it only the intersections are marked.*

(83) It is sometimes more convenient, not to produce one of the sides of the figure, but to draw at one end of it, as at the point 1 in Fig. 46, an indefinite line, usually a perpendicular to a line

joining two distant angles of the figure, and make this line the base of the equivalent triangle desired. The operation is shown by the dotted lines in the figure. The same General Rule applies to it, as to the previous figures.

(84) Special Instruments. A variety of instruments have been invented for the purpose of determining areas rapidly and correctly. One of the simplest is the "Computing Scale," which is on the same principles as the Method of Art. (75). It is represented in Fig. 47, given on the following page. It consists of a scale divided for its whole length from the zero point into divisions, each representing 24 chains to the scale of the plat. The scale carries a slider, which moves along it, and has a wire drawn across its centre at right angles to the edges of the scale. On each side of this wire, a portion of the slider equal in length to one of the primary, or 24 chain, divisions of the scale, is laid off and divided into 40 equal parts.

This instrument is used in connection with a sheet of transparent paper, ruled with parallel lines at distances apart each equal to one chain on the scale of the plat. It is plain, that when the

instrument is laid on this paper, with its edge on one of the
parallel lines, and the slider is moved over one of the divi-
sions of 24 chains, that one rood, or a quarter of an acre,
has been measured between two of the parallel lines on the
paper (since 10 square chains make one acre); and that
one of the smaller divisions measures one perch between
the same parallels. Four of the larger divisions give
one acre.
The scale is generally made long enough to
measure at once five acres.

To apply this to the plat of a field, or farm, lay the transparent paper over it in such a position that two of the ruled lines shall touch two of the exterior points of

2

3

2

4

to make the spaces cand d about equal. Hold the scale firm, and move the slider till the wire cuts the right hand oblique line in such a way as to equalize the spaces e and f. Without changing the slide, move the scale down the width of a space, and to the left hand end of the next space; begin there again, and procced as before.

So go on, till the whole length of the scale is run out, (five acres having been measured), and then begin at the right hand side and work backwards to the left, reading the lower divisions, which run up to 10 acres. By continuing this process, the content of plats of any size can be obtained.

A still simpler substitute for this is a scale similarly divided, but without an attached slide. In place of it there is used a piece of 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.

(86) 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."