Crinoidea, Pentacrininae. 29 
diameter that corresponds with the columnar perradius; if it bisects that diameter it is 
central; but it may be supra-central or infra-central, according as it iS nearer 
the proximal or distal edge of the columnal. The fulcrum is a ridge parallel to the trans- 
verse axis, and above, below, or coincident with it. The fulcrum may be above, below, or 
on the level of the lumen, and in the last case it may be either interrupted by the 
lumen or continuous around it, The margin of the facet may be raised in a rim, 
which, however, is not at the extreme edge, and therefore is visible even when the facet is 
indented, The semicircular areas between the rim and the fulcrum are muscle-fossae, 
In transverse section the proximal cirrals usually have an outline identical with, or 
approximating to, that of the cirrus-facet. Their joint-faces may differ in detail from the 
cirrus-facet, but they have the same essential elements of the bifascial articulation. 
Mode of measurement, For purposes of comparison it is not enough to give 
measurements ; one must also state how these have been taken, Since no rules have hitherto 
been formulated for the measurement of crinoid-stems, it is necessary to explain the method 
here followed. 
Text-figure 4. The diameters of a cylindrical and of a stellate columnal contrasted. The exact ratios 
for a pentagon are given in the text. 
The diameter of a columnal passes through the lumen at right angles to two 
parallel planes, which touch the periphery, but nowhere cut it, and ends where it meets 
those planes (text-figure 4). When the section of the columnal is circular, subpentagonal, or 
pentagonal, the diameter lies wholly within the columnal. But when the section is stellate or 
quinquelobate, the diameter passes outside it along a radius, and the plane that cuts it at this 
end touches the periphery (if the section be a mathematically regular figure) in two 
points. If the distance from the centre of the lumen to the periphery along the interradius 
be JR, and if the distance from centre to periphery along the opposite radius be 7, then 
‘ina cylindrical or basaltiform columnal, Diameter —JR-+r; but in stellate or quinque- 
lobate columnals, Diameter —= JR + r+ a_(i. e. depth of re-entrant radial angle). Only in 
a mathematically circular section does Diameter = 2 IR. For purposes of comparison it is 
advisable to reduce all measurements to a common standard. The diameter of the columnal 
is adopted as this standard and assigned a value of 100. 
Thus, the length TR = 50 in a cylindrical columnal; but in a pentagonal or stellate 
columnal JR = 55.28, while r + 2 = 44.72. If TR—50 were taken as the constant, 
yr +a would be 40.45, But it is usually easier to measure the diameter in actual practice. 
