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°, 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°.
Let us carry, at H and E, the angles, O H E = H E O = 43° = 90°-47°; the angle at 0° will be equal to 180-86 = 94°. 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°. 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°; P H H¹ = 18°; P E E¹ = 65°. The first shows that the instrument has been constructed for a place on the parallel of 50°, and the others show that, at the solstices, the height of the sun was respectively 18° and 65°, decompounded as follows:
| 18° = | polar height of the place | -23½°. |
| 65° = | " | +23½°. |
The polar height of the place where the object was to be observed would therefore be 41½°, that is to say, its latitude would be 48½°.
Minor views of construction and measurement and the deformations that the instrument has undergone sufficiently explain the divergence of 1½° between the two results, which comprise between them the latitude of Paris.
After doing all the reasoning that we have just given at length, we have finally found the means by which the hypothetic bead was to be put in place. A little beyond the curves, a very small but perfectly conspicuous dot is engraved—the intersection of two lines of construction that it was doubtless desired to efface, but the scarcely visible trace of which subsists. Upon measuring with the compasses the distance between the insertion of the thread and this dot, we find exactly the distance, N P, of our diagram. Therefore there is no doubt that this dot served as a datum point. The existence of the bead upon the thread and the use of it as a rude calendar therefore appears to be certain.
The compass is to furnish us new indications. After dismounting it—an operation that the quite primitive enchasing of the face plate renders very easy—we took a copy of it, which we measured with care. The arrow forms with the line O C-O R an angle of 90° + 8°. The compass was therefore constructed in view of an eastern declination of 8°.
Now, here is what we know with most certainty as to the magnetic declination of Paris at the epoch in question: