Fig. 274. (1) The following method is useful and easy when we have a portion of a single whorl, such as to shew both its inner and its outer edge. A broken whorl of an Ammonite, a curved shell such as Dentalium, or a horn of similar form to the latter, will fall under this head. We have merely to draw a tangent, GEH, to the outer whorl at any point E; then draw to the inner whorl a tangent parallel to GEH, touching the curve in some point F. The straight line joining the points of contact, EF, must evidently pass through the pole: and, accordingly, the angle GEF is the angle required. In shells which bear longitudinal striae or other ornaments, any pair of these will suffice for our purpose, instead of the actual boundaries of the whorl. But it is obvious that this method will be apt to fail us when the angle öÝ is very small; and when, consequently, the points E and F are very remote.
Fig. 275. An Ammonite, to shew corrugated surface-pattern. Fig. 276. dctr01 (2) In shells (or horns) shewing rings, or other transverse ornamentation, we may take it that these ornaments are set at a constant angle to the spire, and therefore to the radii. The angle (ö¡) between two of them, as AC, BD, is therefore equal to the angle ö¡ between the polar radii from A and B, or from C and D; and therefore BDã₤ãã₤AC =ã₤eÿ£¢ö¡ã₤cotã₤öÝã₤, which gives us the angle öÝ in terms of known quantities. {538}
(3) If only the outer edge be available, we have the ordinary geometrical problem,ãgiven an arc of an equiangular spiral, to find its pole and spiral angle. The methods we may employ depend (1) on determining directly the position of the pole, and (2) on determining the radius of curvature.
Fig. 274. (1) The following method is useful and easy when we have a portion of a single whorl, such as to shew both its inner and its outer edge. A broken whorl of an Ammonite, a curved shell such as Dentalium, or a horn of similar form to the latter, will fall under this head. We have merely to draw a tangent, GEH, to the outer whorl at any point E; then draw to the inner whorl a tangent parallel to GEH, touching the curve in some point F. The straight line joining the points of contact, EF, must evidently pass through the pole: and, accordingly, the angle GEF is the angle required. In shells which bear longitudinal striae or other ornaments, any pair of these will suffice for our purpose, instead of the actual boundaries of the whorl. But it is obvious that this method will be apt to fail us when the angle öÝ is very small; and when, consequently, the points E and F are very remote.
Fig. 275. An Ammonite, to shew corrugated surface-pattern. Fig. 276. dctr01 (2) In shells (or horns) shewing rings, or other transverse ornamentation, we may take it that these ornaments are set at a constant angle to the spire, and therefore to the radii. The angle (ö¡) between two of them, as AC, BD, is therefore equal to the angle ö¡ between the polar radii from A and B, or from C and D; and therefore BDã₤ãã₤AC =ã₤eÿ£¢ö¡ã₤cotã₤öÝã₤, which gives us the angle öÝ in terms of known quantities. {538}