We all know that an object appears smaller, in proposition with its distance from us. This diminution is not a matter of chance. It is geometric, and proportional to the distance. Every object removed to a distance of 57 times its diameter measures an angle of 1 degree, whatever its real dimensions. Thus a sphere 1 meter in diameter measures exactly 1 degree, if we see it at a distance of 57 meters. A statue measuring 1.80 meters (about 5 ft. 8 in.) will be equal to an angle of 1 degree, if distant 57 times its height, that is to say, at 102.60 meters. A sheet of paper, size 1 decimeter, seen at 5.70 meters, represents the same magnitude.
In length, a degree is the 57th part of the radius of a circle, i.e., from the circumference to the center.
The measurement of an angle is expressed in parts of the circumference. Now, what is an angle of a degree? It is the 360th part of any circumference. On a table 3.60 meters round, an angle of one degree is a centimeter, seen from the center of the table. Trace on a sheet of paper a circle 0.360 meters round—an angle of 1 degree is a millimeter.
Fig. 80.—Measurement of Angles.
If the circumference of a circus measuring 180 meters be divided into 360 places, each measuring 0.50 meters in width, then when the circus is full a person placed at the center will see each spectator occupying an angle of 1 degree. The angle does not alter with the distance, and whether it be measured at 1 meter, 10 meters, 100 kilometers, or in the infinite spaces of Heaven, it is always the same angle. Whether a degree be represented by a meter or a kilometer, it always remains a degree. As angles measuring less than a degree often have to be calculated, this angle has been subdivided into 60 parts, to which the name of minutes has been given, and each minute into 60 parts or seconds. Written short, the degree is indicated by a little zero (°) placed above the figure; the minute by an apostrophe (′), and the second by two (″). These minutes and seconds of arc have no relation with the same terms as employed for the division of the duration of time. These latter ought never to be written with the signs of abbreviation just indicated, though journalists nowadays set a somewhat pedantic example, by writing, e.g., for an automobile race, 4h. 18′ 30″, instead of 4h. 18m. 30s.
This makes clear the distinction between the relative measure of an angle and the absolute measures, such, for instance, as the meter. Thus, a degree may be measured on this page, while a second (the 3,600th part of a degree) measured in the sky may correspond to millions of kilometers.
Now the measure of the Moon's diameter gives us an angle of a little more than half a degree. If it were exactly half a degree, we should know by that that it was 114 times the breadth of its disk away from us. But it is a little less, since we have more than half a degree (31′), and the geometric ratio tells us that the distance of our satellite is 110 times its diameter.
Hence we have very simply obtained a first idea of the distance of the Moon by the measure of its diameter. Nothing could be simpler than this method. The first step is made. Let us continue.
This approximation tells us nothing as yet of the real distance of the orb of night. In order to know this distance in miles, we need to know the width in miles of the lunar disk.