[19] Dr. O. Danckworth, _Vierteljahrschrift der Astronomischen Gesellschaft_, 16. Jahrgang 1881, p. 9. Dr. Stockwell’s results have been communicated to me by letter. Some, but not all, of Dr. Lockyer’s calculations appeared in _The Dawn of Astronomy_. CHAPTER XII ASTRONOMICAL HINTS FOR ARCHÆOLOGISTS--_Continued._ I next come to the sun observations. First we must consider the astronomical differences between the rising of a star and of the sun, by which we generally mean that small part of the sun’s limb first visible. It is frequently imagined that for determining the exact place of sunrise or sunset in connection with these ancient monuments we have to deal with the sun’s centre, as we should do with the sun half risen. As a matter of fact, we must consider that part of the sun’s limb which first makes its appearance above the horizon; the first glimpse of the upper limb of the sun is in question, say, when the visible limb is 2′ high; and we must carefully take the height of the hills over which it rises into account. The accompanying diagram will at once show the difference between the rising conditions we have now to consider. It deals with the summer solstice, as being the most precise case, in Lat. 59° N. At this time the position of the sun, _that is of the sun’s centre_, as given in the “Nautical Almanac,” is represented by the double circle on the sea horizon. [Illustration: FIG. 36.--The Conditions of “Sunrise” at the Summer Solstice in Lat. 59° N. ~Vertical axis from bottom: Altitudes SEA HORIZON, HILL ¹⁄₂° HIGH, HILL 1° HIGH, HILL 1¹⁄₂° HIGH.~ ~Horizontal axis from left: Azimuths N 37°-42° E.~] The azimuth of this position is N. 39° 16′ E. This is the equivalent of the declination of a star, but it will be seen that the real azimuths we want are very different. The dotted circles represent the actual position of the sun with regard to the horizon, the continuous circles the apparent positions caused by the lifting-up effect of refraction. We have the positions in azimuth of the apparent sun as it appears on a sea horizon, and when the horizon is formed by hills up to 1¹⁄₂° in vertical height. To make this quite clear I give a table which has been computed by Mr. Rolston, of the Solar Physics Observatory, showing azimuths with hills up to 1¹⁄₂° high for lat. 59° N., and 51° N. nearly the latitude of Stonehenge, of the sun’s upper limb for the summer solstice:-- Lat. 59° Lat. 51°