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land-surveying, angles are the chief subject of measurements. The direct measurements of length are only rarely possible with any accuracy; rivers, houses, forests, mountains, and general irregularities of ground all get in the way. The survey of a whole country will depend only on one or two direct measurements of length, made with the greatest elaboration in selected places like Salisbury Plain. The main work of a survey is the measurement of angles. For example, $A$, $B$, and $C$ will be conspicuous points in the district 23 surveyed, say the tops of church towers. These points are visible each from the others. Then it is a very simple matter at $A$ to measure the angle $BAC$, and at $B$ to measure the angle $ABC$, and at $C$ to measure the angle $BCA$. Theoretically, it is only necessary to measure two of these angles; for, by a well-known proposition in geometry, the sum of the three angles of a triangle amounts to two

right-angles, so that when two of the angles are known, the third can be deduced. It is better, however, in practice to measure all three, and then any small errors of observation can be checked. In the process of map-making a country is completely covered with triangles in this way. This process is called triangulation, and is the fundamental process

in a survey.

Now, when all the angles of a triangle are

known, the shape of the triangle is known–-that is, the shape as distinguished from the size. We here come upon the great principle of geometrical similarity. The idea is very familiar to us in its practical applications. We are all familiar with the idea of a plan drawn to scale. Thus if the scale of a plan be an inch to a yard, a length of three inches in the plan means a length of three yards in the original. Also the shapes depicted in the plan are the shapes in the original, so that a right-angle in the original appears as a right-angle in the plan. Similarly in a map, which is only a plan of a country, the proportions of the lengths in the map are the proportions of the distances between the places indicated, and the directions in the map are the directions in the country. For example, if in the map one place is north-north-west of the other, so it is in reality; that is to say, in a map the angles are the same as in reality.

Geometrical similarity may be defined thus: Two figures are similar (i) if to any point in one figure a point in the other figure corresponds, so that to every line there is a corresponding line, and to every angle a corresponding angle, and (ii) if the lengths of corresponding lines are in a fixed proportion, and the magnitudes of corresponding angles are the same. The fixed proportion of the lengths of corresponding lines in a map (or plan) and in the original is called the scale of the map. The scale should always be indicated on the margin of every map and plan. It has already been pointed out that two triangles whose angles are respectively equal are similar. Thus, if the two triangles 24 $ABC$ and $DEF$ have the angles at $A$ and $D$ equal, and those at $B$ and $E$, and those at $C$ and $F$, then $DE$ is to $AB$ in the same proportion

as $EF$ is to $BC$, and as $FD$ is to $CA$. But it is not true of other figures that similarity is guaranteed by the mere equality of angles. Take for example, the familiar cases of a rectangle and a square. Let $ABCD$ be a square, and $ABEF$ be a rectangle. Then all the corresponding angles are equal. But 25 whereas the side $AB$ of the square is equal to the side $AB$ of the rectangle, the side $BC$ of the square is about half the size of the side $BE$ of the rectangle. Hence it is not true that the square $ABCD$ is similar to the rectangle $ABEF$. This peculiar property of the triangle, which is not shared by other rectilinear figures, makes it the fundamental figure in the theory of similarity. Hence in surveys, triangulation is the fundamental process; and hence also arises the word "trigonometry,"