The reader who is curious in such matters will find directions for constructing "a dial, for any latitude, on which the shadow shall retrograde or move backwards," in Ozanam's "Recreations in Science and Natural Philosophy," Riddle's edition, page 529. Professor Ball in his "Mathematical Recreations," page 214, gives a very clear explanation of the phenomenon. The subject is somewhat too technical for these pages.
HOW A WATCH MAY BE USED AS A COMPASS
everal years ago a correspondent of "Truth" (London) gave the following simple directions for finding the points of the compass by means of the ordinary pocket watch: "Point the hour hand to the sun, and south is exactly half way between the hour hand and twelve on the watch, counting forward up to noon, but backward after the sun has passed the meridian."
Professor Ball, in his "Mathematical Recreations and Problems," gives more complete directions and explanations. He says:
"The position of the sun relative to the points of the compass determines the solar time. Conversely, if we take the time given by a watch as being the solar time (and it will differ from it only by a few minutes at the most), and we observe the position of the sun, we can find the points of the compass. To do this it is sufficient to point the hour-hand to the sun and then the direction which bisects the angle between the hour and the figure XII will point due south. For instance, if it is four o'clock in the afternoon, it is sufficient to point the hour-hand (which is then at the figure IIII) to the sun, and the figure II on the watch will indicate the direction of south. Again, if it is eight o'clock in the morning, we must point the hour-hand (which is then at the figure VIII) to the sun, and the figure X on the watch gives the south point of the compass.
"Between the hours of six in the morning and six in the evening the angle between the hour and XII, which must be bisected is less than 180 degrees, but at other times the angle to be bisected is greater than 180 degrees; or perhaps it is simpler to say that at other times the rule gives the north point and not the south point.
"The reason is as follows: At noon the sun is due south, and it makes one complete circuit round the points of the compass in 24 hours. The hour-hand of a watch also makes one complete circuit in 12 hours. Hence, if the watch is held with its face in the plane of the ecliptic, and the figure XII on the dial is pointed to the south, both the hour-hand and the sun will be in that direction at noon. Both move round in the same direction, but the angular velocity of the hour-hand is twice as great as that of the sun. Hence the rule. The greatest error due to the neglect of the equation of time is less than 2 degrees. Of course, in practice, most people would hold the face of the watch horizontal, and in our latitude (that of London) no serious error would thus be introduced.