20 
Some allowance must be made for state of preservation, some for the 
measurements being taken from figures, and some for the method used in 
estimating diameter when the whorl is concealed, as, for instance, at the 
beginning of the last whorl. Unless the specimen be broken or purposely 
cut through the middle this method of estimating has to be employed in 
order to ascertain the proportions at the beginning of the last whorl: it is, 
therefore, advisable to describe it. In a specimen when the rate of coil 
appears to be regular the major diameter, from centre to end of whorl, is 
ascertained. This gives the percentage of the major radius to the dia- 
meter: in a very large number of Ammonites this percentage is 57. The 
nearer the specimen is to a circle — that is, the slower the rate of volutional 
increase or the more polygyral the Ammonite — the nearer the major radius 
will approximate to 50; in oligogyral forms, with rapid volutional increase 
the major diameter may rise much beyond 57 per cent. But, the major 
radius of a specimen having been ascertained, the diameter of the shell at 
the commencement of the last whorl can be found by measuring from 
centre to circumference and, on a slide-rule, which is the most useful 
instrument for the work, setting, say, 57 opposite the measure of the 
major radius: the position of 100 will then give the diameter. 
When the major radius is 57 per cent, the minor radius is 43. Slide 43 
to the position occupied by 57 on the rule : the position of 100 now indicates 
the diameter of the shell half a whorl farther on, allowing for the ratio of 
coiling being constant. Repeating the process gives the diameter one 
whole whorl farther. Reversing the process from the starting point gives 
the diameters half and a whole whorl previously: and so on. 
By such processes as these, diameters of partly concealed whorls can 
be estimated: also the diameters of fragments of whorls: the process has 
been employed for the estimation of measurements in Newton’s Fig. 3 of 
Am. ( Macrocephalites ) ishmae. 
The table shows that nearly all the species supposed to belong to 
Macrocephalitidae, including the so-called Cadoceras grewingki, show 
thickness constant or declining. A small exception, M . metastatus, is due 
to the presence of body-chamber — an inflation of body-chamber in a 
normally thinning species is usual. The other exception is in the mid 
stages of P . glabrescens — early stages not seen: the thickening in this case 
may be due to faulty preservation. The slight drop in I, M. ishmae, may 
be due to the same cause. 
On the other hand the rapid increase in thickness of the specimens 
and species of Cadoceras stands out in marked contrast. This would, in 
the case of C. sublaeve, be continued much farther at about the same rate. 
Comparison may be made with the line given for Cadoceras modiolare 
Nikitin {See data given above). Nikitin gives some remarkable figures of 
Cadoceras and Cadoceratoids in section: they are marred, however, for the 
graphic purpose by the fact that the amount of reduction of the figures is 
not given: if details be taken from a reduced figure as if it were natural 
size, because of lack of warning, it makes increase or decrease begin too 
early: this may be the difference between species and is misleading. 
