28 MATHEMATICS. 
A plane passed through one of the three axes forms an ellipse by its inter- 
section with the surface. When two axes are equal, the spheroid becomes 
elliptical, being generated by the revolution of an ellipse around one of its 
two axes (forming an ellipsoid). A paraboloid is generated by the revolu- 
tion of a parabola about its axis, and a hyperboloid by that of a hyperbola. 
V. APPLIED GEOMETRY. 
1. GEODESY, OR SURVEYING. 
Practical geometry, which is itself only a part of applied mathematics, 
embraces, in a restricted sense, 1, the greater and lesser arts of surveying, 
or geodesy; 2, descriptive geometry, or theory of proportion. In a 
restricted sense, we understand by practical geometry only the first of 
these divisions, which proposes to itself the problem, accurately to deter- 
mine the size, shape, and position of a larger or smaller part of the earth’s 
surface, and to represent it pictorially on a reduced scale. 
We distinguish, as above mentioned, a lower geodesy or field surveying, 
which has to deal only with small parts of the earth’s surface, as a field, or 
estate, and a higher geodesy, having reference to whole countries. 
Under geodesy are also reckoned, generally, levelling and surveying of 
mines. 
The first problem in field surveying is to mark off a straight line. This is 
done by means of straight cylindrical staves of wood, from 6 to 8 feet high, 
and 1 to 1} inches thick, with iron points at the lower end for more conve- 
nient insertion into the ground; together with a number of stakes, called 
also arrows, pickets, &c. Of these staves, A and B are placed perpendicu- 
larly in the ground about 100 feet apart, and a third, C, still further forward 
in the same straight line. In order to place these staves in the same 
straight line, we may have them so adjusted, that, standing behind A, the 
others, B and C, shall both be covered by it; or A and B may be covered by 
C in the line of sight. This latter method is, perhaps, preferable. We 
must proceed in the same way to extend the line of these staves. 
The second problem is to measure a line which has already been staked off- 
This is done either by means of the measuring chain, which is most generally 
employed, or by measuring tapes or threads, which are commendable for 
their cheapness and convenience, but do not afford accurate results; or, 
finally, by measuring staves, which give by far the ‘most correct measure- 
ments. 
With stakes and a chain, or some other means of measuring a given line, 
quite a number of the more difficult problems may be solved, without any 
other apparatus. We can, in the first place, survey any irregularly curved 
line on the surface of the ground, as, for instance, the outline of a field or 
plain ( pl. 4, fig. 1). To this end, a straight line, AB, is staked off, and on 
this as many successive distances, Aa, ab, bc, &c., as possible, measured. 
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