EXAMPLES.

1. In the trapezium sketched above, the following measures were taken:

:

AC 40-18 feet, DF 14.32 feet, BE-12.86 feet; required

=

The method here illustrated is that usually employed for finding the area of a trapezium; but there is another way of proceeding, not given in books on Mensuration, which will frequently be more convenient in actual practice. It may be explained as follows:

D

Let ABDC be a trapezium, from the extremities of one of the sides of which let perpendiculars be drawn to the opposite side, which may be called the base: draw also the lines CF, DE. Then the trapezium consists of the triangle DEB, the triangle DEC, and the triangle CEA. But by Euclid

C

equal to the triangle DEB, the triangle CFE, and the triangle CEA; that is to the two triangles DEB, CFA, and thus we have a second rule as follows:

RULE II. From the extremities of any side from which perpendiculars can be drawn to the opposite side or base, let them be drawn and measured. Measure also the part of the base between one of its extremities and the remoter perpendicular, and multiply this measure by the nearer perpendicular: measure in like manner, the part of the base between the other extremity and the remote perpendicular, multiplying this measure by the

D

nearer perpendicular. Half the sum of these two products will be the area of the trapezium.

The following example will illustrate this rule:

2. Required the area of the trapezium sketched above, from the following measures:—

AF 745 links, BE-10001. CE=3521. DF=595 l.

=

In the above diagram, and in the foregoing expression of the rule, we have supposed the two perpendiculars to fall within the figure; because, in surveying, the

D

rule would scarcely be employed

in other circumstances;* but it is

C

prehends all cases; thus-Let the figure be as in the margin. Then the area of twice the trapezoid ED will be

EFXFD+EF × EC;

and twice the area of the exterior triangles will be

BFXFD+AEXEC;

so that twice the area of the trapezium will be

that is

(EF-BF) FD+ (EF-AE) XEC;

EBXFDAFXEC;

the measures along the base being taken as before, from each extremity up to the perpendicular nearest to the other extremity.

* In many other departments of Mensuration, the second case of the rule would however, be practically useful; as for instance, in measuring the sections of embankments, pits, &c. If the figure in the text be turned upside down, it may represent the transverse section of an embankment constructed upon sloping ground, CD being the sloping base of the section.

We may here remark that, when the side CD is parallel to AB (fig. 2), the rule becomes that already given for the trapezoid; for since, in this case, DF=CE, therefore AF CE+BE DF (AF+BE) CE= (AB+EF) CE= (AB+ CD) CE, the double area.

A

D

C

E

B

There are other modes of subdividing a trapezium in practice among surveyors: thus the figure is sometimes divided into two triangles, as in the annexed diagram, the diagonal and the two perpendiculars represented by dotted lines being measured in conjunction with the base. Sometimes also the diagonal and all four of the sides are measured, so that the figure then presents two triangles, in each of which the three sides are given. We have dwelt thus long on the trapezium, because it is a figure constantly occurring in the measurement of land, as the practical surveyor endeavours to cut up his fields into triangles and trapeziums for the better convenience of computation.

Fig. 4.

F

3. Required the area of a field ABCD (see fig. 1, at page 32), whose sides are AB, 600 links; BC, 480; CD, 320; and the diagonal AC, 760 links. Ans. 2 Acres, 27 Perches.

4. In the trapezium ABCD (fig. 1), the following dimensions were taken, namely,

AC 80.5, DF=30∙1, BE=24·5;

=

required the area. Ans. 2197 65.

5. What is the area of a field in the form of a trapezium of which the diagonal is 1660 links, and the perpendiculars upon it, 702 and 712 links respectively? Ans. 11 A. 2 R. 37·79 P.

6. One side AB, of a quadrangular field measures 11·1 chains, and the two perpendiculars upon it from the opposite corners, 3.52 and 5.95 chains respectively; also AE=1·1 chains, and AF=7-45: required the area (fig. 2). Ans. 4 Acres, 1 Rood, 5.79 Perches.

7. In measuring the field ABCD (fig. 4), the following dimensions were taken, namely:

AB 1621 5 links, DF-584, AD=1209, CE=305; required the area. Ans. 6 Acres, 2 Roods, 12·5 Perches.

8. How many square yards are there in a trapezium whose diagonal measures 126 feet 3 inches, and the perpendiculars upon it 58 feet 6 inches, and 65 feet 9 inches? Ans. 871-47569 Yards.

9. The south side of a quadrangular field is 27.4 chains, the east side 35.75 chains, the north side 37.55 chains, and the west side 41.05 chains; also the diagonal from south-west to north east, 48-35 chains; how many acres does the field contain? Ans. 123 Acres, 11.86 Perches.

10. In the quadrilateral ABCD (fig. 4), AB measures 3243 links; DF, 1168; AD, 2418; CE, 610: required its area. Ans. 26 Acres, 1 Rood, 10 Perches.

=

11. In the trapezium ABCD (fig. 1), the following measures are taken, namely:-AC=378 yards, AB=220 yards, DC 265 yards, AE=100 yards, CF=70 yards: required the area. Ans. 17 Acres, 2 Roods, 21-232 Perches.

12. In the irregular polygon represented in fig. 1, below, the following dimensions were taken in feet, namely:

AB-35.5, Cm=8

AD=36, Go=7·5, Cn—8·6
FD 34, Ep 10.4, Gr=9-2

Required the area of the polygon. Ans. 765 Feet.

13. In the above diagram (fig. 2), the following measures

were taken in links, namely:

AB-440, BE=70

BC145, CF=320

CD 405, MC=435

AH 315, HK=

= 350 IL=50

HI 295,
IM 410, DG=470

Ans. 3 Acres, 1 Rood, 16.12 Perches.

NOTE. The trapeziums FGMC and HMLK, may be computed like the trapezium in fig. 2, page 33.

14. In a five-sided field ABCDEA, which the learner may easily sketch, there are given the following measures in links, namely:

AB=2735, BC=3115, CD=2370, DE=2925, EA=2220; also these two diagonals, namely,

required the area of the field. Ans. 117 Acres, 2 Roods, 39-1345 Perches.

The plane figures now discussed, are the principal of those which ordinarily occur in actual practice; and as usual, the calculations have all been performed by the principles of common arithmetic. We have, however, already taken occasion to notice that some of these calculations may be considerably abridged by using what are called logarithms instead of common numbers; and as the stage at which the learner has now arrived, may be considered as the completion of one important division of his subject, it will be convenient, before entering upon new ground, to review a little what has been passed over under the somewhat different aspect which the employment of logarithms will give to it. The section, therefore, which immediately follows, is to be regarded as supplementary to that just completed; and in order that the learner may not be kept wholly in ignorance of any practical process which can fairly be reckoned among the fundamental operations of Mensuration, we shall present him in this supplementary section, with a glimpse of the subject of Trigonometry, attempting, however, only so much as may enable him to form some estimate of its value to practical men, and as may stimulate him to further study and inquiry.

SECTION III.

SUPPLEMENTARY.

ON THE PRINCIPLES OF LOGARITHMS AND PLANE
TRIGONOMETRY.

LOGARITHMS are a set of numbers, arranged in tables called Tables of Logarithms, and which were contrived by Lord Napier, early in the seventeenth century, to facilitate arithmetical computations. The original inventor explained his contrivance, and constructed his numbers upon principles which, at the present day, have given place to simpler and more expeditious methods. By aid of a few easy theorems in modern algebra, the doctrine may be satisfactorily developed in a very short compass, and the numbers themselves may be calculated with but a very small portion of the labour which Napier himself expended upon the undertaking. We shall here attempt to explain them to the learner by the help of these additional advantages, pre