REMARKS 
ON THE FIGURES OF PLATE XXXI. 
SSS Ss 
THE COSTO-VERTEBRAL FIGURE HAS A BILATERAL BUT NOT AN ANTERO-POSTERIOR SYMMETRY. THE DUPLI- 
CATION OF THE COSTO-VERTEBRAL FIGURE GIVES BOTH CONDITIONS OF SYMMETRICAL CLEAVAGE. 
Ny BOhs quantities of form are duplex. The circle is a duplicate of the semicircle, and that whole 
figure which consists of two tangent or secant circles is a duplicate of the circle. Any right 
line which shall pass through the centre of the circle will divide it symmetrically. The circle is a 
symbol of complete unity, symmetry is its attribute, and homology is its chnmdaneine ; its halves are 
homologous, and so likewise are all its segments, provided they be cut of equal bases. When we draw 
two tangent circles, and view both as constituting an entire form, we find. that this figure is also 
capable of symmetrical cleavage by two lines intersecting each other at right angles. The tangent 
circles are cleft bilaterally symmetrical by that perpendicular line which passes through the centres 
of both circles. And this figure of tangent circles is likewise cleavable into antero-posterior symmetrical 
halves by that transverse line which passes through their secant points. Now, when we compare the 
costo-vertebral figure with that of the secant circles, it will be seen that the former differs from the 
latter by the want of antero-posterior symmetry, whereas both forms retain the common feature of 
bilateral symmetry. The costo-vertebral form is rendered bipartite by that line of cleavage which 
passes through its spinous process behind, and its sternal piece in front; but the line of transverse 
cleavage carried through its vertebral centrum will prove that the dorsal and ventral structures are 
unequal quantities. The neural or dorsal arch is minus, while the costal or ventral arch is plus. 
In the opposite figures we have drawn comparison 
between the dorsal and ventral aspects of the several 
classes of vertebral quantities, and by the comparison to 
. prove this fact, viz., that in the mammalian skeleton axis 
the vertebral form is prone to describe plus variety by 
producing the autogenous pieces through a ventral 
circle. The neural arches at dorsum enclose space all 
through the spinal series excepting at the caudal region, 
where minus proportioning is at its lowest extreme. But 
the costal or ventral arches of the thoracic units enclose 
space which is not found to be similarly embraced by the 
units of a cervix, loins, or sacrum, excepting in those few 
cases where the cervical or lumbar vertebra developes 
anomalous costz. When those cervical or lumbar cost 
are produced, then they imitate the thoracic quantities in 
enclosing thoracic space ; but when the lumbar or cervical 
units do not develope costz like a thorax, then they may 
be said to be minus the costal structures, by which we are 
given to understand that these structures have been sub- 
tracted from them. 
The cervical vertebra fig. A is symmetrically cleavable 
by the line 2, but not by the transverse Ime 1; and even 
when fig. A describes the ventral circle, still the line 2 
cleaves the whole figure into equal sides, though the line 1 
would still divide the form transversely into unequal quan- 
tities. Still we must confess that when the vertebra per- 
forms the ventral arch, it approaches in some degree to the 
equation of dorsal and ventral form. Fig. B shows how 
two circles placed one above the other would form an 
entire figure, capable of being cleft symmetrically by a line 
passing either transversely or perpendicularly through the 
centre 3; and in fig. C we find that if the autogenous 
elements described the ventral.or costal circle, whilst the 
neural arches suffered expansion to the dorsal circle, then 
- the vertebra would simulate the form of double circles, and 
be capable, like this form, of both modes of symmetrical 
cleavage, either by the transverse line | or by the perpen- 
dicular line 2; and upon this the question may be asked, 
is there a vertebral quantity in any skeleton axis, fossil or 
recent, produced in such condition that it could be cleft 
from back to front and from side to side symmetrically ? 
If there be in all creation such an archetype quantity, 
then fig. A is a proportional of such another quantity. 
And as figs. D and G are equal to fig. A, so must they - 
