Let us consider a horizontal plane (Fig. 3, No. 2)--a plane perpendicular to the meridian, and a right line parallel with the axis of the world. Let P be a point upon this line. As we have seen, such point is the summit of a very wide cone described in one day by the solar rays. At the equinox this cone is converted into a plane, which, in a vertical plane, intersects the straight line A B. Between the vernal and autumnal equinoxes the sun is situated above this plane, and, consequently, the shadow of P describes the lower curves at A B. During winter, on the contrary, it is the upper curves that are described. It is easily seen that the curves traced by the shadow of the point P are hyperbolas whose convexity is turned toward A B. It therefore appears evident to us that the thread of our sun dial carried a knot or bead whose shadow was followed upon the curves. This shadow showed at every hour of the day the approximate date of the day of observation. The sun dial therefore served as a calendar. But how was the position of the bead found? Here we are obliged to enter into new details. Let us project the figure upon a vertical plane (Fig. 3, No. 1) and designate by H E the summits of the hyperbolas corresponding to the winter and summer solstices. If P be the position of the bead, the angles, P H H¹, P E E¹, will give the height of the sun above the horizon at noon, at the two solstices. Between these angles there should exist an angle of 47 deg., double the obliquity of the ecliptic, that is to say, the excursion of the sun in declination: now P E E¹-P H H¹ = E P H = 47 deg.. Let us carry, at H and E, the angles, O H E = H E O = 43 deg. = 90 deg.-47 deg.; the angle at 0 deg. will be equal to 180-86 = 94 deg.. If we trace the circumference having O for a center, and passing through E and H, each point, Q, of such circumference will possess the same property as the angle, H Q E = 47 deg.. The intersection, P, of the circumference with the straight line, N, therefore gives the position of the bead. Let us return to our instrument. We have traced upon a diagram the distance of the points of attachment of the thread, at the intersection of the planes of projection. We have thus obtained the position of the line, N S. Then, operating as has just been said, we have marked the point, P. Now, accurately measuring all the angles, we have found: N S R = 50 deg.; P H H¹ = 18 deg.; P E E¹ = 65 deg.. The first shows that the instrument has been constructed for a place on the parallel of 50 deg., and the others show that, at the solstices, the height of the sun was respectively 18 deg. and 65 deg., decompounded as follows: 18 deg. = polar height of the place -231/2 deg.. 65 deg. = " " " " +231/2 deg.. The polar height of the place where the object was to be observed would therefore be 411/2 deg., that is to say, its latitude would be 481/2 deg..