32 MATHEMATICS. 
tions with those drawn from the other extremity of the line, determine the 
position of these points on the reduced plane of the table. 
If a great surface is to be measured—an entire country, for instance—a 
trigonometrical net-work must be constructed, as already mentioned under 
the head of Trigonometry. This consists in dividing the part of the earth’s 
surface in question, into a great number of connected triangles, whose 
corners form stations visible one from the other. In these triangles, only 
one side, rarely over 15 mile long, needs to be measured ; in addition to 
which, the angles must be measured with a theodolite. Care must be taken 
that the angles of these triangles be neither too acute nor too obtuse; those 
most nearly equilateral are most convenient. The net must be so arranged 
that each sheet of the plane table contains at least two of the corners of the 
trigonometrical net. In pl. 4, fig. 10, AB represents the base ; from this 
the points D and C are determined, by measuring the angles BAC, DAB, 
and ABC, DBA, and two triangles thus obtained, whose sides, AC, BC, AD, 
BD, may be calculated trigonometrically. AD may now be taken as base, 
and the point E determined; as also K, from the base DE, &c. In this 
manner the network, ABCDEKH, is produced. It will add greatly to the 
accuracy of the work to determine each point from several stations if 
possible. This serves to control the various measurements. Suppose the 
point K to be determined from DH, and likewise from DE, if it should fall 
towards L, some error must have occurred, which must be detected either by 
repeated measurements or by special calculations to which we have not 
time here to refer. 
A very important problem, and one of frequent occurrence, is to determine 
the point on the plane table corresponding to the one where it was originally 
set up; this, knowing the positions, a, 8, y, of the three points of the field, 
A, B, C (fig. 64-68). If the triangles, a, 8, y, can be brought into a position 
perfectly parallel to the field triangle ABC, then the point required will be 
determined by applying the sight ruler at a, 8, y, and sighting towards 
A, B, C; the intersections of these three lines will determine the point. It 
is very difficult, however, to attain this parallel position. If the two triangles 
are not parallel, the three sight lines will form a triangle, by means of which 
the desired point may be attained. We cannot here go into the minute 
details of the operation. 
To determine the area of a rectilineal figure, all that is necessary is to 
divide it by diagonals into triangles, whose individual areas are to be com- 
puted from their ascertained bases and altitudes, and added together. The 
figure may also be divided into trapezia and triangles, which method is some- 
times preferable. When the figure to be calculated is curvilineal, the latter 
method may sometimes be employed (as in fig. 13), if the parallel lines are 
drawn so closely to each other that the included parts of the circumference 
may, without material error, be considered as rectilineal. In the case 
represented in fig. 12, the two triangles, ABC, BCD, are first calculated, 
then the mixed lined parts by which the triangles exceed the curvilineal 
fizure. This latter is effected by dividing them into trapezia and triangles, 
by perpendiculars erected, and subtracting their sum from that of the 
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