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 must have proper regard to the direction in which we estimate the inclination or opening between the two straight lines which contain the angle; i.e. to the direction in which one of the straight lines must be supposed to revolve from coincidence with the other in order to pass over the angular space in question.

For instance, the straight lines AB, AC, according to Euclid, would only bound one right angle, but in accordance with the more extended definition of an angle, they may also be considered as containing an angle whose magnitude is three right angles, the line AB in this case being supposed to revolve from right to left in order to move into coincidence with AC.

A trigonometrical angle then must be regarded not merely as the opening between two straight lines, but as the angular space swept over by a revolving line, which starts from coincidence with one of the bounding lines of the angle and moves into coincidence with the other. Moreover, in order to effect this, the revolving line may be supposed to have made any number of complete revolutions, so that under this supposition we can have angular space of any magnitude whatever.

For instance, the minute hand of a watch at a quarter past four o'clock will since twelve o'clock have revolved through an angle the magnitude of which is 17 right angles.

3. Angular units.

In order to measure angles some particular angle must be chosen as a standard or unit. This selection is of course quite arbitrary, and is influenced only by considerations of convenience.

4. ''Degrees, minutes, seconds. Sexagesimal division of the right angle.''

The ninetieth part of a right angle is called a degree, the sixtieth part of a degree a minute, and the sixtieth