Everyone knows what an angle is, and you say at once it is the inclination of two lines that meet each other. These lines by their branching off form an opening more or less wide. This opening is measured by the aid of an instrument called a _protractor_ made of brass or horn, which finds its place in nearly every box of mathematical instruments. It represents a semi-circumference, divided into 180 equal parts, called degrees, written thus: 180°. Each degree is divided into 60 minutes, expressed thus: 60 min.; and finally the minutes are divided again in 60 parts, called seconds, indicated thus: 60 sec. There are therefore in a whole circumference, 360 deg., 2,160 min., and 12,960 sec. One degree, therefore, is the 360th part of a circumference, and thus we have a measure independent of all dimensions. For example, on a round table of 36 yards in circumference, one degree will be marked by one tenth of a yard; on a pond of 360 yards in circumference, one degree will be equal to one yard. The degree, therefore, may be more or less, but it is always the 360th part of the circumference of a circle. Let it be quite understood that, whether an angle is to be on a sheet of paper, or in the skies, the divisions do not change. [Illustration] This must be well grasped, it is of the utmost importance for the explanations which follow. It is therefore settled: the measure of the angles has nothing to do whatever with a measure of length. We have shown how to measure an angle. Let us examine now what is a triangle, without pondering too much over this geometrical figure, which every one knows. The essential property of this three-cornered figure is that the sum of its three angles is always equal to 180 degrees. In other words, the protractor placed successively at each angle will give three numbers, which, added, make up 180 degrees. Keep this property well in mind, as it will serve us hereafter. Now, to what distance does a degree correspond? For example, take a yardstick, and with the _graphometer_ (an instrument by which angles are measured), in readiness, carry it from the latter instrument to a certain distance, till the two extremities of the yardstick measure one degree; this yard is then said to subtend an angle of one degree. Now, measure the distance which divides the yardstick from the instrument, and you will find it to be 57 yards. Therefore, one degree corresponds to an object being at a distance of 57 times its height. A man two yards high at a distance of 57 times his height, or 114 yards will measure one degree. One minute will be represented by a piece of cardboard of a hundreth part of a yard long seen from a distance of 34 yards; and finally, a second will be given by a card a hundreth part of a yard seen from a distance of 2062 yards. A hair seen at 20 yards about represents a second. This perhaps, you think to be too small to be seen by the naked eye.