2 REMARKS ON THE FIGURES OF PLATE X., 
tebral forms of spinal series, and we have drawn around it 
the ideal figure of its own archetype quantity. But we 
have taken care to locate it in the centrums of all the 
archetype vertebra, for it is no other elemental part of 
such archetype than a centrum. If we called E’”” the 
spinous process of a vertebra, we then would evidently 
have misnamed the form, since it does not hold series with 
spinous processes. 
As the part, therefore, speaks of the whole, as the last 
caudal bone speaks of the vertebral form, so does the 
whole series of spinal forms, from occiput to caudex, con- 
jure up in the ideas a serial line of vertebral archetypes. 
A series of plus and minus quantities, such as those 
which fashion the spinal axis, cannot be considered as 
equals or homologues; but, nevertheless, such is the order 
of their creation, that they generate the ideas of uni- 
formity when once we admit that it is only degradation or 
subtraction of quantity which gives rise to the difference 
between them. We can readily fancy how any plus figure 
of series may be so degraded or metamorphosed as that it 
shall equal even the smallest quantity of the same series ; 
and supposing this to have taken place, then a serial line 
of equal parts, such as the last caudal quantity, would 
range the whole length from occiput to the caudex. Now 
the admission of this possibility gives rise to another, viz., 
that as plus may be subtracted from and rendered minus, 
so minus may be added to and thus equal plus, however 
great this latter quantity may be. If this latter were the 
case, then a serial line of equal or plus parts, such as a 
vertebra, would range through the animal centre. How- 
ever, in the created form as it presents itself, we see the 
fitness of a series of various proportionals, we acknow- 
ledge the design as such, but at the same time we under- 
stand that the proportionals must be the minus quantities 
of an uniform plus series. 
It is possible for all the serial vertebre to suffer meta- 
morphosis to quantities equal to fig. EH”, and in this idea 
is, as it were, mirrored the accompanying one, that fig. 
E’” may have been so metamorphosed from a plus verte- 
bra equal to figs. D’C” B” or A”. If the fact of referring 
fig. H’” to fig. D” be too abrupt a comparison between 
plus and minus, then let us refer it to fig. E”’, the penul- 
timate caudal centrum next above it, and on perceiving 
that the differential quantity is but very slight between 
both these forms, there can be no reasonable objection to 
our rising through increasing series from fig. EK” to fig. 
E”; hence to fig. E”, hence to fig. HK’, and hence to fig. 
D’, or any other plus form in series. If there can be no 
objection to the comparison between the two slightly 
differenced quantities of figs. H’” and EH’, neither can 
there be objection to the comparison between figs. EH” 
and EK”, or E’ or D”. If we can readily grant that it is 
possible with nature to have metamorphosed the quantity 
of fig. E’”” from such a quantity as fig. HE”, what can 
there be improbable in the assertion that fig. E’”” may be 
a proportional of such another quantity as fig. D”? If we 
here venture to assert this latter reading, and that it is 
possible for fig. E’” to have been metamorphosed * from 
the ossa or figs. D’C’B’ A”, just as it is possible with 
Nature to “ degrade Mount Ossa to a wart,” who shall be 
able to find a mathematical rule in contradiction of the 
same assertion ? 
Proportions and progressions are known to be a sub- 
ject furnishing many theorems of practical utility in the 
sciences. Anatomical science teems with the beautiful 
subject, and the application of it to anatomy will no doubt 
yield some understanding of the law of form. 
A series of quantities is said to be in arithmetic propor- 
tion when every succeeding pair have the same difference 
as the preceding ; thus 1, 3, 4, 6, 9, 11, &c., constitute a 
series in arithmetic proportion; because 1 and 38, 4 and 6, 
9 and 11, have the same common difference, scil. 2. In 
the same way a, a+d, b,b+d,c, c+d, &e., or a, a— 
d, b, b—d, c, e—d, are series in arithmetic proportion ; 
because the first pair of terms in each have the same dif- 
ference, d, as every succeeding pair. If we apply these 
facts to the interpretation of the serial proportioning of 
figs. D’H’E” E” and E””, taken as quantities of osseous 
structure decreasing from D” to E””, or increasing from 
E”” to D”, we shall find that osteology and the law of 
skeleton formation will bear a much clearer interpretation 
than if we continue to draw analogy between fig. E”” and 
the xoxkvé. 
Again, a series of quantities is said to present in arith- 
metic progression, or arithmetic continued proportion, 
when they individually increase or diminish by a common 
difference. Thus the digital numbers 1, 2, 3, 4,5, 6,7, 8,9, 
form a series in arithmetic progression, because the terms 
increase by the common difference 1. Thus also a, a + d, 
a+2d, a+ 3d, &ce., or a,a—d, a—2d, a—3d, &e., are a 
series in arithmetic progression ; the terms of the former 
increasing, those of the latter diminishing, by the common 
difference d. Even this definition, if applied to the inter- 
pretation of the series of quantities ranging from fig. D” 
through figs. K’E”’E” and E”” in decreasing order, or 
from fig. EH” through the others to fig. D” in increasing 
order, will yield a clearer understanding of the ways of 
Nature than may be had from the following definition of 
her as “Principium et causa motus et ejus in quo est 
primo per se et non per accidens,” according to Aristotle, 
or else by viewing nature as a blind and mevitable course 
of necessity, according to the Stoics. 
* In the works of Vicq d’Azyr,. we find the dominant idea of a graduated or proportioning degradation of form: he writes, “ qu’on 
observe partout ces deux caractéres que la Nature semble avoir imprimés 4 tous les étres, celui de la constance dans le type et celui de la variéte 
dans les modifications.” —See Mémoire sur le Parallele des Extrémités, &c. 
