MODERN CLASSIFICATION. 
229 
Mr. Naumann to call it a ' concho-spiral.' Taking, however, the former curve as an 
approximation to the form produced, we know that this depends for its shape entirely on 
the angle at which it is bent ; and this depends on the law of the growth of the shell. 
Now, since this law is the same for all the parts of the shell, it follows that the curve 
described by every point in the same plane is the same, only representing earlier or later 
portions of one and the same curve. If, therefore, in an Ammonite which is coiled on 
one plane it is necessary to go back an exact revolution of 360° to reach the part of the 
curve corresponding to the inner edge of the whorl, it is plain that this will be just in 
contact with the outside of the preceding whorl, which is an exact revolution behind. 
If we must go back further the inner edge ,of the last whorl will lie within the outer edge 
of the preceding, or the shell will be more or less involute ; if, on the contrary, we have 
to go back less, it will be uncoiled. The shape, then, of any shell coiled in one plane 
depends on the magnitude of two angles, viz. the angle of the spiral and the angle of 
retardation of the inner edge." 
Fiftldy. — The absence or presence of the Aptychus is the fifth character in the 
anatomy of the shell which requires our consideration. In several extensive groups the 
Aptychus is supposed to be absent, as in Phi/Hoceras, Lytoceras, Arcestes, Tracliyceras ; at 
all events it has not yet been found in these. It is present as a horny undivided 
body in Ai'ietites, Aegoceras, and Amaltlieus. It is divided or bivalved and calcareous in 
Harpoceras, Oppelia, and Haploceras. It is thin, bivalved, and granulated externally in 
Stephana cer as, PerispJiinctes, and Cosmoceras. It is thick, smooth, and punctated exter- 
nally in Simoceras and Aspidoceras, and greatly resembles the large flat grinding teeth 
of certain genera of fishes from the Carboniferous Limestone ; these thick calcareous 
Aptychi consist of two layers, an inner and outer. The inner alone exhibits lines of growth 
concentric with the angle of each plate, which is situated on that side of its broad end 
which is applied to its fellow of the opposite side. The outer layer is composed of many 
laminae, which in some forms have a porous structure and a punctated and polished 
surface ; there is a remarkable contrast between the thick porous Aptycld of the 
Kimmeridgian Clay, and Upper Jurassic rocks of Germany, and the thin shelly Aptycld 
of the Ammonites from the Inferior Oolite, or the horny Anaptyc/d of the Lias 
formation. When found undisturbed they occupy the lateral wall about the middle 
of the body-chamber, having their base in general turned towards the mouth, as 
seen in figs. 97 and 99. The Solenhofen Lithographic Limestones contain several 
species of Ammonites with the body-chamber, mouth border and lateral processes of 
the shell all complete, and in these specimens the relative position of the Aptychi agrees 
with what I have stated. 
Having described in detail the five characters upon which palaeontologists have based 
the diagnosis of genera, I now proceed to give the classification proposed by Dr. W. 
Waagen, as it exhibits, in an analytical form, the results of modern investigations on 
the different groups of the Amraonitidse. 
