is fastened to the cover.  The silver plate is not soldered, in order that the thread may be replaced when it chances to break.  On the inner part of the cover are marked in the first place the horary lines, traversed by curves that are symmetrical with respect to the vertical and having the aspect of arcs of hyperbolas.  At the extremity of these lines are marked the signs of the zodiac.  At the top, a pretty banderole, which appears at first sight to form a part of the ensemble of the curves, completes the design.  Such is this wonderful little instrument, in which everything is arranged in harmonious lines that delight the eye and easily detract one’s attention from a scientific examination of it.  Let us enter upon this drier part of our subject; we shall still have room to wonder, and let us take up first the higher question.

[Illustration:  FIG. 3.—­DIAGRAM EXPLANATORY OF THE MANDOLIN SUN DIAL.]

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.