Proceedings of the Polytechnic Association. 847 



2. The liour lines, on -wliatcvcr surface, plane or not, are the inter- 

 sections of that surface by planes, all meeting at the shadow casting 

 edge, and making equal angles thereat. Consequently liavnig found 

 the twelve o'clock plane (which is the meridian), the eleven and one 

 one o'clock planes must be inclined thereto fifteen degrees, and the 

 ten and two o'clock thirty degrees, and so on ; their common inter- 

 section being the gnomon edge. 



Most persons probably know that the theoretic sun dial is not, 

 except on four days of the year, the true clock time of the place. 

 The minutes of the equation of time for any day arc generally the 

 same for every year, but not the seconds / and the four times of ;io 

 equation, or of maximum eqnation, may each fall, in different 

 years, on the former or the latter of two days (just as the equinoxes 

 may). The only reason the same table will not serve for two suc- 

 cessive years is the excess of the year over the exact number of days. 

 Thus 1868, being a leap year, was above eighteen hours longer than 

 a true year, while 1869 is to be nearly six hours less than a true one. 

 If the excess were an exact quarter of a day, this table would be the 

 same for any year as for the fourth before or after it ; and four tables 

 would serve by turns continually. But the defect of the fraction from 

 six hours prevents this, and makes another gradual accumulation of 

 differences, to the table of two dates ninety-six years apart differ 

 nearly as much as two successive years. This again, in our present 

 Gregorian style, is over corrected by one day omitted from the cen- 

 turial year ; and yet again under corrected by the retention of tlie 

 day in the 4:00th year ; and thus it happens that, practically, while 

 the world stands, each year must have its own table. In astronomi- 

 cal books, however, we see a practically permanent " equation of 

 time " table made by omitting days of the month and substituting 

 tlie sun's longitude. "VVe may also regard it as dependent on the 

 sun's declination only in this way. He crosses any given delineation 

 circle twice a year, and to each delineation belong no more than two 

 amounts of dial correction in going southward, the otlier when he 

 passes it in coming northward. To each tropic, however, and to one 

 intermediate circle, about ten degrees north there belongs only one 

 dial error. Hence the raediEevals got a simple and elegant way of 

 setting their clocks. In some Italian cathedrals the transept has a 

 meridian line marked on the pavement, and a lens in the south window, 

 casts an image of the sun, whose center, of course, crosses this line 

 every day at dial noon ; but farther north or south each day as his 



