  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 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. 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 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. Suppose that you to measure the distance of a church situated on a height, and from which you are separated by a river (see fig.) Choose on the river’s bank two spots from which the steeple C can be seen, say A and B. At B plant a surveying-staff, and with the graphometer, go to A and find the angle formed by B A C. Suppose for example, it reads 84 degrees. Repeating the operation at B for the measure of the angle C B A, suppose it to be 95 degrees. Measure the distance from A to B and let it be 80 yards. Now here is the statement of our problem: How to resolve a triangle of which the base is known to be 10 yards, and two of its angles. Well, we have said We know that an angle of one degree corresponds to a distance of 57 yards. Multiply the base of our triangle by 57 yards and you obtain a distance of the church from the points A and B, 10 by 57, equal to 570 yards. Nothing is more simple than this. The smaller the measured angle the further off the object will be. As seen in our figure, the upright lines, m o, m’ o’, m, o,, do not vary, but according to their distances from point C, they form various angles, ac, a’c’, a,c,, becoming smaller and smaller. A graphometer is not always to be had. When approximate distances only are required, the following contrivance may be used. Trace on a cardboard of large size a semi-circumference which one divides first into 180 equal parts, then each of these is divided again in 2, 3, 4 divisions, etc., according to the size given to the circumference, which constitutes a large protractor. To measure an angle place the cardboard upright in an horizontal position, supporting it by the center of the semi-circumference by means of a screw fixed on a stick. Then proceed as stated above. From a pin stuck in the center mark the spot where the visual ray passes, go to A and to B, and you get approximately the desired result. |
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