REMARKS ON THE FIGURES OF PLATE VIII. 
EQUAL AND UNEQUAL QUANTITIES ARE SUBTRACTED FROM EQUAL VERTEBRAL QUANTITIES, 
[’ from equal things we take away equal or unequal things respectively, the remainders will be equal or 
unequal, as the case may be. This observation applies to any known quantity and its equals, as well as to 
the vertebral figure and its homologues. The repetition of a quantity,-whether of form or number, is 
tantamount to the creation of homologues for the one, and equals for the other ; and so the words homology 
and equality may be applied to the repetitions of either a number or a form. Thus, the repetition of the 
integer 9 is 9 ad infinitum ; and the repetition of a cubic, a triangular, or a circular form, is a cube, a triangle, 
or a circle ad infinitum. In like manner, the repetition of a vertebra is, in the hands of Nature, the creation 
of vertebree all through the serial line, however long it may be. Now, it also happens, that the subtraction 
of an equal or unequal quantity, whether it be from an integral number, a cube, triangle, circle, or vertebra, 
will leave behind equals or unequals of any of those several forms or quantities. But it also occurs, that, in 
the very act of subtraction from a whole or integer which we already know, and in the act of rendering 
it unequal or various to itself by the loss of parts, we leave, within the memory, an idea of that quantity 
which was proper to the whole; and so, whether it be an equal or an unequal quantity which we now view 
remaining, we still associate it with the full quantity or the archetype. 
When we have once seen a whole 
quantity, then any of its parts, occurring separately, will invariably create the idea of that whole figure. 
The subtraction of elemental parts from archetype 
quantities is a process identical with the metamorphosis 
of such quantities, and both processes produce the same 
results. 
The dotted outlines of the opposite figures indicate those 
elemental parts of the vertebral archetypes which have 
been subtracted or metamorphosed. The persistent ele- 
ments of those vertebral archetypes are indicated by the 
firm outline. 
In fig. A’, the cervical vertebra, all parts are omitted 
except the centrum 1. Figs. B’ and C’and D’, which are 
the dorsal, lumbar, and sacral forms, have the like parts 
omitted, the centrums marked 1 alone remaining. 
Figs. A” B” C” and D” are cervical, dorsal, lumbar, 
and sacral vertebree, whose parts have all been metamor- 
phosed except the centrum 1, and the lateral autogenous 
pieces a. ; 
Figs. A” BY” C” and D” are cervical, dorsal, lumbar, 
and sacral vertebre, whose elemental parts have been all 
omitted except the lateral pieces 2 of the neural arch, and 
the spinous process 3. 
The object which we have had in view, when drawing 
those figures, was to prove that the elemental pieces of 
vertebree, when persistent, always appear in their own 
proper position, and thereby -clearly indicate the parts 
which have been subtracted. Also, that when we regard 
any one element of any of those vertebra, whether it be 
a centrum, a neural arch, a lateral process, or a spinous 
process, we invariably conjure up, in idea, the completed 
vertebral archetype. 
In fig. A’ the centrum 1, although the other parts are 
annihilated, mvariably names itself to be the centrum of 
a vertebree. 
In fig. A”, where we see the centrum | and the autoge- 
nous pieces a of the transverse processes to persist, whilst 
all other parts are wanting, still those persistent elements 
speak of the vertebral type. 
In fig. A” where the neural arch 2 with its exogenous 
processes and the spinous element 3 alone persist, still they 
indicate the archetype quantity of which they are the 
fractionals. The same remarks may be applied to all the 
other figures marked B, C, and D. 
The persistent elements a and 1 of the vertebral 
figure marked A” speak of the subtracted elements 2 and 
3 of that same vertebral quantity. The fractional refers 
to its integer, the part to the whole. 
In the figs. marked A’ A” A” let us suppose that the 
several elemental pieces indicated bythe dotted outline were 
