REMARKS ON THE FIGURES OF PLATE LIV. 3 
traction of quantity to have taken place on fig. G, it is 
evident that all the serial proportional parts which would 
remain persistent within the lines d%, would one and all, 
even to the smallest fractional quantity within those lines 
at unit 61, be the remains of archetype plus quantities 
such as constitute now the entire serial skeleton axis. 
The same oblique lines d 2, of figs. I and H describe that 
the like subtraction of quantity from plus archetypes has 
yielded the lumbo-sacro-caudal spine for those skeleton 
The last caudal nodule 56 of fig. I, is a proportional 
of such a plus quantity as we find in its own thoracic region, 
and also of such a quantity as that marked 56 in fig. G. 
Consequently this caudal ossicle must be granted to be a 
proportional of any serial plus quantity in the skeleton axis 
fig. G, forasmuch as all units of this axis are uniform with 
each other. 
Now the horizontal line h, divides fig. G through the 
38rd costo-vertebral unit of series, and shews that all the 
units above this line are homologues of the units now 
happening below it. It must follow, therefore, that even 
when subtraction of quantity pursues the oblique lines 
di, through all the serial units ranging between unit 20 
and unit 61, and shall leave those units as proportional 
quantities, still we must regard them as the parts of full 
archetypes such as we now see them; and hence we must 
say that the line h, which divides the actual serial arche- 
type of fig. G through unit 33, divides also the ideal serial 
archetype of fig. H through unit 33; forasmuch as fig. H 
is a proportional creation of such as fig. G. Consequently 
we conclude. that the line h, passing through unit 38 of fig. 
I, diviges the ideal serial archetype, of which fig. I as well 
as fig. H, are minus proportionals. 
In fig. G we see that homologues happen above and 
below the line #. In figs. I and H we discover that 
unequal proportionals of such homologues happen above 
the line 2, and below it. How then stands the law of 
unity in variety between these three figures? How can 
it be interpreted truthfully, otherwise than by saying that 
the archetype plus unity is fig. G, while variety marks fig. 
I or H, which are the proportional quantities of such an 
archetype as fig. G? - 
In figures A, B, E, and F, we see homologous forms 
such as the geometrician would draw them, but these forms 
“are not more absolutely homologous to each other both 
as to figure and quantity, than are the serial costo-vertebral 
units of fig. G, such as Nature herself draws them. 
The geometrician means by the term homologous, 
corresponding. Rectilineal figures are said to be similar 
when the angles of the one are respectively equal to the 
angles of the other, and the sides about the equal angles 
proportional : hence that in similar figures the correspond- 
ing sides and angles are homologous. 
In a series of four proportionals, the antecedents are 
homologous to each other, and also the consequents. Fig. 
A is a triangle homologous to fig. B, both in form and 
quantity. Now, if the angles of one triangle be respec- 
tively equal to the angles of another, the three sides of 
one triangle have, to the corresponding sides of the other, 
Thus, let abc—def, be 
aXes. 
respectively, the same ratio. 
two triangles drawn in fig. B, the one having the angle 
abc, equal to the other angle def, the angle ac é, equal 
to the angle d fe, and, consequently, the angle bac, 
equal to the angle edf. Then, as ac, is to df, so is a J, 
to de, and as ac, is to df, so is bc, to ef; and also, as 
ab, is to de, so is be, to ef. 
Full quantities are homologues: fig. A is homologous 
to fig..B. Equal proportionals of such full quantities are 
also homologous, just as fig. C is homologue of fig. D. 
Now, it is evident that the only difference existing 
between figs. C D, and A B, is that occurring by the 
subtraction of quantity ; for we see that fig. B, minus the 
parallelogram g,e,d,b, would equal fig. D, or C, and so we 
have equalled figs. C and D, with figs. A and B, by filling 
up the parallelogram quantity for C and D, as seen in figs. 
E and F. 
Fig. A contains a proportional quantity equal to fig. 
C; and fig. B shews within itself that proportional 
quantity which is equal to fig. D; and hence, as we 
know that fig. B, minus the parallelogram, g,e,b,d, would 
equal fig. D, so we reasonably infer that fig. D is now a 
speciality, owing to that quantity, g,e,6,d, which has been 
subtracted from it. 
Fig. A, therefore, may be accounted archetype of fig. C, 
and the equation of both these quantities is shown in fig. E. 
The presence or absence of the parallelogram g,e,b,d, 
is that which renders them proportionately different: plus 
quantity has been subtracted from, and minus quantity 
persists ; and so it is that we interpret the vertebral quan- 
tity as aminus figure or species compared with the thoracic 
costo-vertebral archetype. The archetype has been sub- 
tracted from, and rendered minus or vertebral. A cervix 
or a lumbar spine of the skeleton axis, fig. I, is in minus 
condition compared with the plus thoracic series of fig. 
I, and also minus compared with all the serial quantities 
of costo-vertebral form seen in fig. G; therefore fig. I, as 
it presents to us, still refers to the archetype unity from 
which it has been proportioned, and to the law of design 
which has fashioned it as a special form, when we know 
it to be a proportional of the archetype uniform series, 
which fig. G now manifests. 
The persistence of archetype quantity through all regions 
of series renders the skeleton axis thoracic through all 
regions, such as is shown in fig. G; whereas, the persistence 
of archetype quantity in some one region of series, and 
the metamorphosis or subtraction of quantity from the 
like archetypes in other regions, establishes the thoracic 
series in one region of fig. I, and the cervical, lumbar, and 
caudal proportionals, in other proper localities of the 
same serial axis. 
The skeleton serial axis fig. G, is that condition of 
formation to which our comparisons have conducted us, 
and with it we shall for the present terminate our obser- 
vations upon the laws which preside over the development 
of skeleton quantity. It will be necessary, however, to 
recapitulate briefly those facts which, under comparison, 
have led us here to recognise fig. G as the integer or 
archetype form of all those serial skeleton designs, whether 
of “normal” or “abnormal” cast, which we have figured 
