VARIATION AND DIFFERENTIATION IN
CERATOFHYLLUM.

BY

RAYMOND PEARL

WITH THE ASSISTANCE OF

OLIVE M. PEPPER and FLORENCE J. HAGLE

WASHINGTON, D. C. :

Published by the Carnegie Institution of Washington

February, 1907

Q.'^'

CARNEGIE INSTITUTION OF WASHINGTON
Publication No. 58

-3 ^/ / 7

CORNMAN PRINTING CO.,
CARI.IST.E, PA.

CONTENTS.

Page

Introduction 5

Material and methods 9

Variation in Ceratophyllum — general results 16

Variation in different portions of the plant 27

Variation in whorls on the main stem 28

Variation in whorls on primary branches 36

Variation in whorls on secondary branches 40

Variation in whorls on tertiary and quaternary branches 45

The relative size of the different portions of the plant, and the variation in

this character 48

Summary of results 54

Relation between the number of leaves in the whorl and position on the plant 57

Position regression in different portions of the plant — the first law of growth

in Ceratophyllum 57

Summary 88

The correlation between differently situated whorls in respect to leaf-number ... 93
The variability of successively formed whorls— the second law of growth in

Ceratophyllum 95

Summary 106

The relation of the presence of branches to the number of leaves in the whorl.... 109

The position of branches 116

Summary... 124

General discussion of results 125

General summary 133

Literature cited 135

3

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

By Raymond Pearl,

With the assistance of Olive M. Pepper and Florencb J. Haqle.

INTRODUCTION.

The purpose with which this investigation was undertaken was to
attempt to work out as exactly and completely as possible for a particular
organism the laws according to which post-embryonic differentiation and
growth occur. In higher plants and animals the phenomena of growth
are accompanied by phenomena of differentiation. The two processes
go along together until finally the adult condition is reached, in which
the organism has attained not only a certain size, but also a complex
condition of differentiation of parts or organs. In studies of variation
it has been the usual— though by no means invariable— custom to take a
particular character or set of characters found in the adult as something
given, and then proceed, by processes now becoming well known, to
investigate the nature and degree of the variation exhibited in these
characters. But since the condition of the adult organism is, with
respect to every character, the result of a process of gradual develop-
ment and growth, it is clear that in order to gain anything approaching
a satisfactory analysis from the biological standpoint, we can not take
the adult structure as something given, but must investigate the laws
according to which the morphogenetic processes concerned in its pro-
duction operate. There is no doubt that the problem presented by the
phenomena of morphogenesis is one of the most fundamental in biology.
Just at present we are witnessing a period of remarkable activity in the
investigation of the problems of heredity, and we are told that here
lies the way to follow when in search of biological truth. No one can
doubt either the immense importance of a determination of the laws of
inheritance, or the value of the contributions to a knowledge of these
laws which have been made during the last few years, both by the
biometricians and by those working along Mendelian lines. Yet one
ventures to think that it is of equal importance that our knowledge of
the laws of morphogenesis be extended. In the zeal for the new thing
represented by Mendelian inheritance and the phenomena of mutation
there is a tendency to overlook the fundamental significance for our
whole outlook on the broader problems of biology, of the results which

6 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

have been obtained in the field of ' ' Entwickelungsmechanik, * ' and to for-
get that there is still a very great deal to be done in this field. Indeed,
hardly more than a good start has been made towards the analysis of the
factors concerned in form production.

It is the belief of the writer that in the methods of biometry we
have an analytical tool capable of rendering great aid in the investiga-
tion of the problems of morphogenesis, and in just the direction where
aid is most needed. Observation and experiment yield results which,
whether they are quantitative or qualitative, certainly demand quanti-
tative analysis if we are to get at their full meaning. In the physical
sciences not only has the necessity for quantitative (i. e., mathematical)
analysis of observational data long been recognized; but further, the
greatest generalizations of those sciences have come as the result of
such analysis. Unless one is prepared to maintain that the phenomena
of the inorganic world are fundamentally different in kind from those in
the organic, I can see no reason why a method of investigation which
has proven so valuable in the physical sciences should not, with proper
development, prove equally fruitful in the biological.

We may now consider briefly the precise nature of the present study.
As has been stated above, our problem was to determine and formulate
so far as possible the laws according to which differentiation with growth
occurs in Ceratophyllum. If we take an individual Ceratophyllum plant,
we find it to include a number of whorls of leaves all generally like each
other but differing in detail. For example, some whorls have a larger,
some a smaller number of leaves. A very little study suffices to show
that whorls with different numbers of leaves distribute themselves
about a typical condition in a characteristic way. Whorls with one par-
ticular number of leaves occur in a different proportion than do those
with either more or fewer leaves. In this way, if we take into account
all the whorls on the plant, we get a characteristic frequency distribu-
tion for the different numbers of leaves, such as is shown, for example,
in fig. 5 (p. 25, infra). Further, as we shall see, the character of the
frequency distribution is fundamentally the same, whatever may be the
absolute size or source of the plant. Our problem is to determine so
far as possible the biological factors that result in the production of this
characteristic distribution. A whorl of leaves is the product of a definite
morphogenetic process in the growing bud, and it seems not unreason-
able to suppose that there is a definite set of factors (internal and
external) which determine the number of leaves which each particular
whorl shall bear. Moreover, we can be reasonably certain that since the
nature of the distribution is the same for Ceratophyllum plants gener-
ally, some of these factors at least are constant in their action. Through

INTRODUCTION. 7

them, in some way, it is determined that each definite kind of differ-
entiated whorl shall occur among all whorls in a particular proportion.
What are these constant factors and according to what laws do they
operate? This is our problem.

We have, then, by analysis of the gross frequency distribution for
the plant as a whole, to investigate the biological laws which lead to
the production in this particular organism of the characteristic distribu-
tion observed. In biometrical terminology our problem is one of intra-
individual variability.

Specifically it seemed very desirable to study in detail such questions
as the following:

(1) The relation between the form of a given part and position in the
organism as a whole. Does the number of leaves in a given whorl of a
Ceratophyllum plant bear any definite relation to the position of that
whorl? Such a relationship between position and differentiation has
been found in a number of cases. (Cf . Pearson ( :05) , illustrations A and
D; also Shull (:05) and Tammes (:03), for example.)

(2) The relation of such a positional differentiation to the variation
and correlation of the differentiated parts. Does the variation exhibited
among all the whorls occupying the same position on Ceratophyllum
plants bear any definite relation to the position on the plant?

(3) The effect of environmental influences on positional differen-
tiation.

(4) The effect of environmental influences on the growth of the
organism as a whole.

(5) The relationship between intra-individual, intra-racial, and inter-
racial variation and correlation.

These statements will suffice to indicate the general standpoint from
which the work was done, and consequently it will not be necessary to
enumerate further at this point the specific questions investigated.

Ceratophyllum is in many respects a very favorable form for the
study of such problems as those outlined. It usually occurs in great
abundance wherever it is found at all, and, being a widely distributed
plant, can easily be obtained from a variety of habitats. Furthermore,
the individual plant attains to a large size, which is very important for
work on intra-individual variability. The plant is comparatively simple
in structure and presents characters easily capable of quantitative deter-
mination. It would, in fact, be almost ideal for an investigation of this
kind were it not for the fact that, as will be shown later in the paper,
the differences produced in the form of the plant by different environ-
ments are not very marked. Ceratophyllum appears to be a much less
plastic form than many of the land plants.

8 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

At the time when the work was begun the writer had not yet seen
the brilHant memoir by Professor Pearson on ' ' Homotyposis in the veg-
etable kingdom" (Pearson, :01). When later this memoir had been
read it was apparent that the problems which had been set for investi-
gation in Ceratophyllum were fundamentally similar to those which
Pearson had before him. There were, however, certain differences in
standpoint which seemed to make it desirable to go on with this work.
Pearson was concerned mainly with the determination in a wide series
of forms of the amount of the homotypic correlation. To quote his own
words (loc. cit., p. 294), the principle he was investigating was "the
principle that homotypes are correlated, i. e. , that the variation within
the individual is less than that of the race, or that undifferentiated like
organs have a certain degree of resemblance." From this standpoint
he very naturally dealt, so far as possible, with undifferentiated or
slightly differentiated like parts, not concerning himself at that time
with any special investigation of the factors which produce differentia-
tion in the repeated parts of plants. As will have been apparent
from what has gone before, it is with this latter problem that the present
work has to do. Our present aim is to examine as many as possible of
the factors concerned in producing differentiation of homologous parts,
and determine their effect on the intra-individual variability of the
differentiated parts. For such a study it seems best for practical
reasons to use at first only one organism, and make the investigation of
that as thorough and as detailed as possible. It will thus be seen that
this work, while not concerned with the determination of the degree of
homotyposis in a particular plant, yet deals with one of the fundamental
problems of homotyposis, and in so far may be considered supplementary
to Pearson's memoir on that subject.

This work was begun early in 1902 by Miss Olive M. Pepper, at that
time a student in the Zoological Laboratory of the University of Michigan.
During the summer of 1902 Miss Pepper collected and nearly finished
the counting of the plants in Series I, II, and III (cf. p. 13, infra).
During the first half of the academic year 1902-03 she continued the
work, sorting the data into frequency distributions and making some
start on the computing. During this year she determined and recorded
the data for Series IV. As it was impossible for Miss Pepper to go on
further with the work, it was then carried forward as opportunity
offered by the writer, by whom practically the whole of the raw material
was reduced. When in the spring of 1905 it became possible, through a
grant from the Carnegie Institution of Washington, to get some aid for
the biometric work in hand, an assistant, Miss Florence J. Hagle, was

INTRODUCTION. 9

put on this work. To her are due the determinations of the raw material
of Series V and VI, and an independent verification of the computations
for Series I, II, and III. The rest of the work and the arrangement of
the material for publication is due to the writer.

To the Carnegie Institution I am greatly indebted for a grant in aid
of this and other biometric work now in progress. Without this aid it
would have been quite impossible to have brought the work to comple-
tion at this time. To Prof. Karl Pearson I am very grateful for valuable
suggestions and advice, especially on the mathematical side of the work.

A word should be said regarding the arrangement of the paper. On
account of the number of topics dealt with and the consequent length
of the paper as a whole, it has seemed best to include a brief summary
and discussion of the results of a particular section in that section itself
rather than reserve all discussion for the end. This I believe will con-
duce to clearness.

MATERIAL AND METHODS.

The plant on which this work is based, Ceratophyllum demersum,^ is
a submerged aquatic which has a wide distribution. It is usually found
in quiet rather than running water, and under favorable conditions
forms great masses of vegetation. It commonly occurs in shallow water,
frequently extending to the very edge of the pool where it is growing.
The plant consists of a main axis, which may attain a length of 5 or 6
feet, from which spring a varying number of lateral branches. At more
or less regular intervals along the main axis and the branches are whorls
of leaves. These leaves are elongate and very narrow, being reduced
practically to the form of rods. This shape of leaf is undoubtedly to be
regarded as an adaptation to aquatic conditions. (Cf. Schenk, '86, and
Henslow, '95.) At their outer ends the leaves divide, leading to the
form of whorl shown in fig. 1. This division of the leaves at the
outer end is the usual condition for Ceratophyllum. The plant has no
root in the strict sense, but, as has been shown by the excellent physio-
logical study of Pond (:05), probably absorbs all its nutrition directly
from the water. It thus exhibits a practically perfect adaptation to
aquatic conditions of existence. The lower end of the main axis of the
plant is usually embedded for a distance of several inches in the layer of
soft mud and plant debris which forms the substratum of a Ceratophyl-
lum bed. This portion of the axis which is embedded is more or less etio-
lated and usually bears only the broken remnants of leaf -whorls. There

ij follow Pieters (:01) in designating the species of Ceratophyllum found about
Ann Arbor demersum. I have not myself been able to get plants in flower.

10 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

is absolutely no indication of root-hairs on this etiolated portion of the
stem. At the ends of the branches and of the main axis of normal,
healthy plants are found growing buds which at intervals form new
whorls. At the tips of the branches the internodes become progressively
shorter as we go towards the end, thus giving rise to the compact mass of
leaves at the tip of the branch which is so characteristic of Ceratophyllum.
The plant is monoecious, both staminate and pistillate flowers being
borne on the same individual. A detailed morphological study of the
development of the flower has been published recently by Strasburger
( :02) . Reproduction is not entirely by seed, but a form of vegetative
multiplication occurs very frequently (cf. Schenck, '86). At the ap-
proach of winter the tips of branches bearing growing buds break off
and sink to the bottom. Since the bud is protected by the thickly
matted leaves of the whorls near the tip it is able to winter over in this
condition. With the advent of spring, growth begins again and by its
continuance the bud gives rise to a new plant. In the early spring
young plants which have been formed in this way may be found in
various stages of growth. When the conditions are not too severe the
plant as a whole winters over, renewing its growth in the spring like a
perennial. This fact was first noted by Irmisch ('53, p. 528), who says
regarding the "Dauer der Ceratophyllum-Arten:"

Sie sind, wie ich mich iiberzeugt habe, bestimmt perennierend. Viele Examplare
fand ich im Friihjahr an der Spitze der deu Winter iiber frisch gebliebenen Zweige, die
sich nicht weiter verandert batten, weiterwachsen; in anderen Fallen waren die Blatter
der Zweigspitzen bogig iibereinander gekriimmt und die altern Internodien des Zweiges
waren abgestorben, so dass sie kleine lochere, isolirte BoUen darstellten. Auch diese
wuchsen im Friihjahr weiter.

Certain of the plants (in Series V and VI) used in the present study
were collected in the spring after having passed the winter in this way
without dying or being broken up.

• The characters of the plant with which the present study has prin-
cipally to do are the following:

(1) The number of leaves in the whorl. For verbal convenience this
character will be referred to throughout as "leaf-number."

(2) The position of the whorl on the plant, relative to the main axis.

(3) The size of the various divisions of the plant, as measured by
the number of whorls they bear.

(4) The position of the branches, relative to the proximal end of the
main axis.

With reference to the determination and recording of these char-
acters the following should be said: The character "number of leaves
to the whorl " was easily determined for the majority of the whorls, but

MATERIAL AND METHODS. 11

became difficult near the ends of the branches. There the whorls are
very small and closely packed together, and in order to make an accurate
count it was necessary to use a lens, and in some cases even the com-
pound microscope. For the sake of uniformity it was necessary to adopt
some arbitrary rule with reference to divided leaves. After examining
a considerable number it was decided to count only the proximal ends of
leaves. Thus the whorl shown in fig. 1, would be recorded as having 8
leaves.

The number of leaves in every unbroken whorl on
the plant was determined for each plant studied. In
case a w^horl was so mutilated as to make the deter-
mination of the number of leaves doubtful, it was so
recorded. Unfortunately Ceratophyllum is rather
liable to mutilation because of the fact that its tissues
are brittle. In comparison with the total number of fig. i.-Diagram of
whorls on the plant, however, the number so mutila- a whon, showing
ted as to be undeterminable was very small. form of leaves.

In recording the position of the whorls the plan followed was to give
each whorl on a particular portion of the plant a consecutive number,
beginning with the proximal. In doing this the different axial divisions
of the plant were treated separately. The first of these is the main
axis, or, as it will be called throughout the paper, the 'main stem."
Using the usual notation for branching, we have designated the lateral
branches arising from this main stem "primary branches;" those
arising from primaries, "secondary branches;" those arising from
secondaries, " tertiary branches, " and so on. "Quaternary branches"
were the highest lateral-branch elements found in any of our plants.
Of these divisions the main stem alone presented any practical difficulty
in making the records. It often happens in Ceratophyllum that the
main stem branches dichotomously at a distance from its proximal end
which varies in different cases. Are the two new axes to be considered
as continuations of this stem, or is one to be regarded as a continuation
of the main stem and the other as a large primary branch to be included
in the records with the lateral branches? If the axes really arise by
dichotomy of the main stem it is clear that they ought to be recorded as
parts of it. The only difficulty is the practical one of being certain in a
particular instance that we are not dealing with a case of unusually vig-
orous growth of a lateral branch which comes to rival the main stem in
size. As a matter of fact it was found very early in the work that
the first few whorls of lateral branches are so clearly differentiated
that it is always possible in a given case of branching to tell whether

12

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

Fig. 2. — Diagram of Ceratophyllum
plantjShowing plan of making record.
A, main stem; A',A"y secondary main
stem; B, B, primary branches; B', B',
secondary branches; B", tertiary
branch; B'", quaternary branch, The
numbers indicate the manner in
which position is recorded.

or not dichotomy of the main stem has
occurred. Consequently each case of this
kind was decided by a careful examina-
tion of the parts, and if it was found that
each of the two axes possessed the charac-
teristics of main stems they were recorded
as ' ' secondary main stems. ' '

The whorls on each of these divisions
of the plant (main stem, secondary main
stem, primary branch, secondary branch,
etc.) were numbered consecutively, be-
ginning with the most proximal as 1. In
this way the position of every whorl on
the plant was recorded. These position
records also gave the size, as measured
by number of whorls borne, for each di-
vision of a plant. Furthermore, in record-
ing a whorl from the axil of which one
or more branches arose, a note was made
of the origin of branches at that point.

The whole system of records will be
made plainer by reference to a diagram of
an imaginary Ceratophyllum plant. Such
a diagram is given in fig. 2, in which the
whorls are represented by short cross
lines. It will, of course, be understood
that in an actual plant the branching
would usually be much richer than is in-
dicated by the diagram. The diagram
shows, however, that our records were
such that we could reconstruct from
them the entire plant as it existed. Thus
the record on our data slips of a whorl
like that marked with an x in the diagram
would, when translated, signify that this
whorl was the fourth whorl on a secon-
dary branch from which no tertiaries
arise and that this particular secondary
branch sprung from the axil of the second
whorl of a primary branch, which in turn
arose at the ninth whorl of a secondary
main stem beginning just beyond the 19th
whorl of the original main stem. Since
our records are of this kind for every

MATERIAL AND METHODS. 13

plant studied, it will be seen that they give more complete data for
the study of positional differentiation than probably has hitherto been
available for any plant.

Our material came from three localities and is representative of four
distinct habitats. It is comprised in six series, an account of which
follows.

Series I, II, and III. —The plants composing these series were collected
in the summer of 1902, from Carp Lake, near Grand Traverse Bay, in
the northern part of Michigan. Carp Lake is a long and narrow body
of water extending about 18 miles in a generally north and south direc-
tion, with an outlet into Lake Michigan at its northern end. At its
narrowest part it is only about a half mile in width. At this point there
is a neck of marshy land extending out into the lake from the east side
and reaching more than halfway across it. On the north side of this
strip of land (known locally as Fountain Point), the plants of Series I,
II, and III were collected. They all came from a small open space sur-
rounded by an abundant growth of cat-tails and reeds (Juncus sp.).
The situation in which Ceratophyllum was growing was well protected
from the wind, and consequently the water about the plants was rarely
much disturbed. The plants were directly exposed to the sunlight, there
being no overhanging trees or bushes. The bottom was similar to that
usually found under Ceratophyllum beds. The proximal ends of the
plants were buried in soft, loose, black mud, containing much decayed
plant debris, which came mostly from the Ceratophyllum itself. The
growth of the plant at this place was very abundant. The individual
plants were so matted together in the main mass composing the bed
that it was only with difficulty that a single one could be disentangled
unbroken.

The dates of collection for the three series were as follows:

Series I: Collected July 22, 1902. At this time the water at the
point of collection was 2 feet deep.

Series II: Collected August 18, 1902. Water at point of collection
approximately 15 inches deep. The level of the whole lake had lowered
since the first collection. The water surrounding the plants was very
muddy.

Series III: Collected August 25, 1902. Conditions were the same
as when the Series II collection was made.

Series I included 5 plants. Series II, 2 plants, and Series III, 1 plant.
The reason why Series II and III did not include larger numbers was
the great difficulty of collecting unbroken specimens after the water
had become so low. The matting together of the plants in the bed was
so close as a result of this lowering that to disentangle a large plant

14 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

without breaking off many of its branches was practically impossible.
We had, then, to be content with few specimens. The individuals in
these series were fairly large, and considering the dates of collection
may be taken as representative of the "adult" condition of the plant.
In other words, it is not probable that if these plants had been left
undisturbed much further growth would have taken place that year.

To sum up: Series I, II, and III, include 8 complete Ceratophyllum
plants, coming from quiet water near the shore of Carp Lake in north-
ern Michigan. These plants were collected in mid-summer, the three
collections covering a period of time of approximately one month, and
may be considered to represent the condition at the height of the grow-
ing season.

Series IV. —This series was collected at Ann Arbor, Michigan, on
October 15, 1902. The plants were taken from a very extensive and
flourishing bed of Ceratophyllum which grew at that time' near shore
in the shallow back-water below the dam across the Huron River.
The water was comparatively quiet, but there was always a slow-
current, which after heavy rains became rapid. So far as this factor
in the environment is concerned the Series IV plants represent dis-
tinctly different conditions than do those of Series I, II, and III.
The character of the bottom was also somewhat different. In the river
habitat there was very little of the mass of plant debris which in ponds
and lakes accumulates under Ceratophyllum beds. Apparently a con-
siderable part of this material was carried away by the current in
the river. The plants were attached; that is, the proximal ends were
embedded in the soft mud of the bottom. The water at the point where
the collection was made was approximately 18 inches deep. The bed
was shaded by overhanging trees and bushes on its east side, but was
fully exposed on its west side. As will be noted, the collection was
made late in the season, so that the plants may be regarded as certainly
full grown.

The series includes two plants. One of these was a very large one,
bearing over 900 whorls and over 100 branches. In some respects this
was the most satisfactory series of all. There were comparatively few
broken whorls and the large size of the individuals gave an excellent
collection of data.

Series V and F/.— The plants in these series all came from a small
pond formed as a cut-off of the Huron River. It is situated some Si-
miles below Ann Arbor, just west of the upper end of the Geddes mill-
pond. The pond still retains its connection with the river through a

^Later this bed was completely washed away by the breaking of a dam in a spring
freshet.

MATERIAL AND METHODS. 15

narrow (6 to 8 feet) channel. It lies on the south side of the river.
The bottom of this pond is covered with a dense mass of aquatic vegeta-
tion, principally consisting of Chara, but with a good deal of Cerato-
phyllum about the edges of the Chara banks. On the north side of this
pond, which is about a hundred yards wide, there is a thick growth of
willows and alders, which overhang the edge and give a considerable
amount of shade to the Ceratophyllum beneath. On the south side, on
the contrary, there is no shade whatever, the aquatic plants being
exposed to the full glare of the sun during the whole day. On the south
side the Ceratophyllum was closely associated with Chara, forming a
fringe about the Chara bed. On the north side, however, the Cera-
tophyllum was growing alone. In other respects the conditions were
the same on the two sides of the pond. Series V, including 7 plants,
was collected from the north side of this pond, and Series VI, including
6 plants, from the south side. The date of collection was May 21,
1905. The plants thus represent the early spring conditions. Nearly
all the plants were small. The water where the different plants were
taken varied between 6 inches and 2 feet in depth. The bottom was
physically very like what has been described above for the Carp Lake
habitat— soft black mud, with much plant debris. All the plants were
attached. The water is very quiet, as the pond is well protected from
the wind, and there is no appreciable current. The plants of these two
series were of special interest because of the fact that they included
representatives of each of the methods of passing the winter which
have been mentioned above (p. 10) . Plants 1, 2, 6, and 7 of Series V,
and plants 1, 2, 4, 5, and 6 of Series VI had wintered over unbroken
and had begun new growth in the spring without having lost their
individuality of the previous year. The other plants of these two series
had started in the early spring, either from seeds or from separated
winter buds.

Putting all the series together, it will be seen that we have plants
from the beginning, the height, and the end of the growing season,
representing lake, pond, and river habitats. Our material must, then,
be regarded as fairly comprehensive.

Finally, a word may be said regarding the calculations. I have
followed throughout the plan of calculating the standard deviation from
the unmodified second moment, wherever it was to be used simply as
an index of variation. In calculating the moments for fitting curves to
the observations I have tried Sheppard's corrections wherever there
was any approach to high contact at the ends of the range, but usually
with not very satisfactory results (cf . p. 23, infra) . In each case a
statement is made as to whether the moments were modified or not.

16 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

VARIATION IN CERATOPHYLLUM-GENERAL RESULTS.

Before analyzing our material in detail it is desirable for purposes of
orientation to examine in a general way the nature and amount of the
variation exhibited by the plant. To this end the present section is
devoted.

In this section I shall only consider the variation in the number of
leaves to the whorl, when all the unmutilated whorls on the plant are
taken together without reference to their position, leaving until later
other characters studied. The frequency distributions for this char-
acter for each plant separately and for each series are given in table 1.

We see at once from this table that the range of variation in leaf-
number is rather limited in Ceratophyllum. Whorls with 4 leaves and
whorls with 11 leaves include practically the whole of the variation in
this character, though whorls with 12 and 3 leaves do occur very rarely.
The great bulk of the whorls have either 8, 9, or 10 leaves. Consider-
ing the first four series, it is seen that whorls with 9 leaves occur more
frequently than do any other single class, thus making 9 the observation
mode. From mere inspection of the frequency distributions it is quite
clear that while Series I, II, III, and IV are all very closely alike in
respect to the character under consideration, on the other hand the two
series taken at the beginning of the growing season (V and VI) are in
some respects quite diiferent. The plants of these latter series are
generally small, and the distributions are irregular. The reason for this
irregularity will be discussed later, but for the present it needs merely
to be kept in mind that the distributions of Series V and VI represent
the action of special factors which do not influence the other series.

In order to bring more vividly to the reader's mind the nature of the
variation in this character, the diagrams shown in fig. 3 have been pre-
pared. These give frequency polygons for the totals of each of the
series (distributions No. 44, 61, 62, 97, 139, and 176). These are so
plotted as to have equal areas; that is, the frequencies are reduced for
each series to percentages and then plotted to the same base unit.

From these diagrams the essential similarity in the frequency distri-
butions for Series I, II, III, and IV comes out in a very striking way.
Series IV shows a higher frequency of whorls with 9 leaves and a lower
frequency of those with 6 leaves than do Series I, II, and III, but other-
wise differs very little from these. All the distributions show a marked
degree of asymmetry about the mode. The range below 9 whorls is
roughly twice as great as the range above that number. One point to
which I wish to call attention here, as it is shown very clearly by the

VARIATION— GENERAL RESULTS.

17

diagrams, is that if we compare Series I, II, and III with each other it
is noticeable that, of the three, Series I has the lowest frequency of
whorls with 10 leaves. Series II has about 2 per cent more 10-leaved

Table 1.— Frequency distribution for variation in number of leaves per whorl for
all unmutilated whorls, without reference to position on the plant.

Series.

Plant.

Distri-
bution
num-
ber.*

Leaves per whorl.

3

4

5

6

7

8

9

10

11

12

Total.

r
I-

r

m...

f

r
v<

I

r
1

1

VI-
I,II,III

1

1
2
3
4
5

1
1
1

12
11

17

17

4

25
18
17
35
4

45
32
38
71
16

65
56
56
77
22

78
38
31
58
13

2
4
3
5
2

"i
1

228
161
164
263
61

2

3

4

5

All plants..
1

44

3

61

99

202

276

218

16

2

877

45
46

2

1

3

48
16

64

72
37

109

126
66

192

191
96

161

87

15
10

... 615
... 313

2

All plants..
1

61

287 ,248

25

928

62

44

50

101

170 151

7

... 523

1

73
74

"i "6

"i

3
16

54
118

140
196

228 182
326 261

3
3

620
922

2

All plants..
1

97

1 0

1

19

172

336

554

443

6

1,532

98
99
100
101
102
103
104

... j 4

30

i

30

"4
12

85
2
1
2
6
6

34

68
2
7
5
7
10
33

93

2
5
6
6
25
50

72
2
9

"5
16
16

3

"i

385
8

22
13
24
61

147

2

3

4

5

6

7

All plants..
1

139

4

31

46

136

132

187

120

4

...

660

140
141
142
143
144
145

t
... 1

"'3
1

"'5
2

11

7
4
5
3
6
2

27

7
15

5
12

15
23
10
11

27

95

1

79
76
46
41
216
236

2

20 11

"2
1

15

"i

3

15

9

60

43

174

8

5

74

130

251

4

5

26 ^P' 1

6

All plants..
All plants..

13

78

30
134

176

.-

18

1

694

177

6

169

258 495 733 617

48

2

2,328

«The "distribution numbers" are merely the serial numbers by which the distributions are
designated in our notes. They have been retained in the paper for convenience in referring to
particular distributions.

whorls, while Series III has 4 per cent more of such whorls than does
Series I. In other words, there is a progressive increase in the pro-
portion of 10-leaved whorls as we go from Series I to Series III. The

18

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

explanation of this fact will be taken up later when certain other results
are in hand. In anticipation of these results it may be said that the
matter is related to the fact that Series II and III were collected at later
dates than Series I, and the change is a growth phenomenon.

Examining the frequency polygons for Series V and VI, we see that
these are essentially different both from each other and from all the
other series. Series V, while it has as the most frequent whorls those

Leaf number

Fig. 3.— Frequency polygons for variation in leaf number. Totals for series. The abscisses give the

number of leaves per whorl and the ordinates frequencies percent. Series I, • ; Series

II, o ; Series III, m ; Series IV <t ; Series V, a ; Series

VI, X.0— -o—o.

with 9 leaves, yet differs from the other series in having a compara-
tively high frequency of whorls with 7 leaves. The polygon for Series
VI is quite smooth and regular, yet differs from all the others in having,
as the most frequent class, 10-leaved whorls.

With these polygons before us the nature of our task becomes perhaps
plainer. We find that when we determine the number of leaves in
every whorl of each plant and plot up the results we get these very

VARIATION— GENERAL RESULTS. 19

smooth and regular polygons. But these polygons represent composites.
While they show the condition of the plants as a whole with respect to
variation, they tell us nothing about how this condition comes to be.
Our purpose is to analyze, piece by piece, these gross frequency polygons
and try to find out the different factors which determine the proportionate
frequencies of whorls of different types. In other words, we must by
statistical analysis dissect these polygons, and in that way, if possible,
gain some idea of the physiological factors to which they fundamentally
owe their origin. By proceeding in this way it should be possible to
obtain results of real significance.

In the continuation of our preliminary survey we may next examine
the values of the chief physical constants, means, standard deviations,
and coefficients of variation of the distributions given in table 1. These
constants are collected in table 2.

This table brings out a number of points of interest. In the first
place, we note again that in respect to all the constants Series I, II, and
III are very much alike, whether we compare the series as wholes or
take single plants. Series IV is significantly but not widely different
from Series I, II, and III. It has a somewhat higher mean number of
leaves to the whorl, and a lower variability in this character. Series V
differs from the others in having a lower mean and higher variability.

If, in order to get a general idea of the values of the constants for
variation in the Ceratophyllum of this region, we take the means of the
constants for the six series, weighting them in each case with the
number of whorls they include, we have the following results:

Mean of means 8.655

Mean of standard deviations 1.206

Mean of coefficients of variation 13.966

It may, then, be fairly concluded that the Ceratophyllum of this
region has an average number of leaves to the whorl of about 8.7, a
standard deviation of about li leaves, and a relative variability of approx-
imately 14 per cent.

It is of some interest to compare these results with the statements
of botanists who have investigated Ceratophyllum from other stand-
points. The first thoroughly scientific investigation of the biology of
Ceratophyllum was that of Schleiden ('37). Considering its date this
memoir is a most remarkable one, both for the wealth and the accuracy
of its observations. Regarding the number of leaves to the whorl
Schleiden says (loc. cit., p. 516): "Der Stengel der Pflanze ist stielrund
mit vollstandigen Knoten versehen, an denen sich die Blatter in 6-10
theiligen Wirteln befinden. Smith giebt falschlich 8 als Kegel (cf . Eng-

20

VARIATION AND DIFFERENTATION IN CERATOPHYLLUM.

lish Flora, I)." From this one must conclude that Schleiden either did
not count any considerable number of whorls, or else that he was dealing
with a highly differentiated local race. Judged by our data "Smith"
did not come far from the ''Kegel " by placing it at 8. Klercker ('85),
in his frequently quoted paper on the anatomy and embryology of

Table 2. — Constants of variation in number of leaves per whorl for all unmutilated
whorls, without reference to position on the plant.

Series.

Plant.

Distri-
bution
number.

Mean
(unit=l leaf).

Standard devia-
tion (unit=lleaf).

Coefficient of
variation.

f
I-

..i

I
III...

IV<

ay.

VI<

1
1

1.
2.
3.

4.
5

1.
2

1.

1.
2.

1.

7.

1.
2.
3.
4.
5.
6.

1
2
3

4
5

8.768±0.054
8.652± .068
8.470± .069
8.529± .050
8.689± .101

1.208±0.038
1.277± .048
1.309± .049
1.208± .036
1.167± .071

13.777±0.443
14.759± .567
15.451± .589
14.160± .425
13.432± .835

All plants ..

44

8.613±

.027

1.201±

.019

13.938± .229

45
46

8.624±
8.728±

.035
.047

1.283±
1.236±

.025
.033

14.876± .292
14.157± .389

All plants..

61

8.659±

.028

1.268±

.020

14.645± .234

62

8.679±

.037

1.254±

.026

14.446± .307

73

74

8.887±
8.758±

.026
.024

.966±
1.084±

.019
.017

10.870± .212
12.375± .197

All plants-

97

8.809±

.018

1.040±

.013

11.810± .146

98
104

139

7.987±
8.063±

.054
.067

1.568±
1.195±

.038
.047

19.636± .495
14.644± .588

All plants ..

8.155±

.038

1.430±

.027

17.540± .480

140
141
142
143
144
145

8.658±
8.132±
8.413±
8.098±
8.718±
9.394±

.093
.097
.140
.129
.058
.049

1.231±
1.260±
1.408±
1.226±
1.254±
1.117±

.066
.069
.099
.091
.041
.035

14.221± .778
15.497± .868
16.732±1.209
15.138±1.153
14.387± .476
11.888± .374

All plants-

176

8.820±

.032

1.243±

.022

14.091± .260

»In Series V the constants for the very small plants (plants 2, 3, 4, 5, and 6) are not separately
tabulated.

Ceratophyllum, has the following to say regarding the number of leaves
in the whorl (loc. cit., p. 8): "Le nombre [i. e., of leaves] dans chaque
verticelle en est en general constant sur le meme axe, mais varie avec
les axes. J'en ai trouve 7-12. ' ' This range agrees fairly well with what
we have found, though starting higher. It is in all probability, however,
a general estimate rather than the result of any extensive counting.

VARIATION— GENERAL RESULTS.

21

Schenk ('86) , in his well-known " Biologie der Wassergewaechse, " gives
the number of leaves to the whorl as between 6 and 10, but no special
weight is to be given to this statement, as it is clear from internal evi-
dence that his account of Ceratophyllum is very largely taken directly
from Schleiden. Strasburger (:02), in his recent memoir on Cerato-
phyllum, says (p. 486), regarding the number of leaves to the whorl:
''In den mir zur Verjiigung stehenden Exemplaren von Ceratophyllum
submersum war die Zahl der Blatter in den Quirlen sehr haufig zehn."
Further references to the literature regarding this matter are unneces-
sary. The point I wish to bring out is merely that so far as we can
judge from such very slender evidence, the condition of Ceratophyllum
with respect to number of leaves in the whorl in European countries
(Germany and Sweden) is apparently not very widely different from
what it is in America.

Table 3. — Coefficients of variation in plants.

Authority.

Plant.

Character.

Coefficient of
variation

Pearson (:03)

Ficaria ranunculoides
do

Number ofsepals

3.24 to 22.02

5.34 to 14.69

12.12 to 18.77

22.32 to 27.89

11.62 to 27.00

11.43 to 27.49

9.66 to 27.41

15.7

15.0

. 13.60

12.62

13.49

45.43

Do

Number of petals

Do

do

Number of stamens

Number of pistils

Do

do

Shull(:02and :04)...
Do

Aster prenanthoides ..
do

Number of bracts

Number of rays

Do

do

Number of disk florets-
Number of branches

....do

Shull (:05a)

Onagra lamarckiana ..
Onagra rubrinervis ...

Do

Do

Onagra lamarckiana..
Onagra rubrinervis ...

Length of leaf

Do

do

Pearson (:05)

Asperula odorata

Equisetum arvense ...

Branches per whorl

do

Do

The relative degree of variation in this character of Ceratophyllum
as indicated by a coefficient of variation of about 14 per cent is of the
same order of magnitude as that which Pearson (:01) has found for a
variety of characters in different plants. In his table xxxiv (loc. cit. ,
p. 361) are given the coefficients of variation for 26 cases, covering a
wide range of plant forms and characters. These coefficients of varia-
tion range between 7.80 and 41.96, with a mean of 19.97. Our value for
Ceratophyllum leaves comes very close to the coefficient measuring
variation in the stigmatic bands of Shirley poppies, for example, or to
that for the veins in the leaves of Spanish chestnuts. In order that a
comparison may be made between other well-known biometric results on
variation in plants and what we have found for Ceratophyllum, a table
of coefficients of variation (table 3) has been prepared.

22 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

It is apparent from inspection of fig. 3, or of the frequency distri-
butions in table 1, that the variation in Ceratophyllum in respect to leaf-
number is very skew. In order to bring out more precisely the amount
of this skewness, as well as other important facts regarding the nature
of the distributions, I have analyzed a portion of the data by the methods
given by Pearson, in his fundamental memoir on "Skew Variation"
(Pearson, '95) and its supplement (:01). I have dealt in this way with
two of the distributions made up of all the whorls on each plant. The
first of these distributions (No. 177, table 1) is that resulting if Series
I, II, and III are combined, that is, it is what is obtained by adding
together distributions 44, 61, and 62. It represents the variation of
the character under consideration in the Carp Lake population of Cera-
tophyllum, taken as a whole. As has been pointed out in the preceding
section, all the Carp Lake plants came from the same spot, and only
differed in the time of collection. It is clearly evident from the diagrams
in fig. 3 and the constants in table 2 that these three collections do not
differ from each other significantly. The differences between the con-
stants for the totals of these three series, as given in table 2, are small
in comparison with their probable errors, which 'are themselves small,
as we are dealing with relatively large numbers. Hence the series may
be considered as three random samples of a homogeneous population and
may be combined if it is desirable in order to bring out a particular
point. The other distribution, which I have graduated for comparison
with this, is that of plant 2, Series IV (No. 74, table 1). This plant was
the largest single plant in our collections and for that reason was chosen
for analysis. Others would have given essentially the same results,
but with larger probable errors.

The problem of how best to treat analytically distributions of the
sort given by Ceratophyllum is, as has been emphasized by Pearson
(:01, p. 456), a very difficult one. We have, in the first place, in these
distributions discrete variation, the observations proceeding by unit
steps, while the mathematical function which we take to represent them
is continuous; further, the distributions are markedly skew; the range
is very small, giving less than ten observations to determine the moments
from; and finally there is no approach to high contact at one end of the
range. Hence, the determination of the true moments is by no means
a simple matter. Proper corrective terms for use in such cases have
not yet been worked out. Sheppard's corrections are theoretically not
applicable. In the present instance by actual trial I obtained much
worse results with Sheppard's corrections than with the raw moments,
and consequently for lack of anything better the latter have been used
in graduating these two distributions.

VARIATION— GENERAL RESULTS.

23

The chief constants, both physical and algebraical, of the distribu-
tions are given in table 4. The unit is 1 leaf.

Table 4. —Analytical constants for variation in leaf-number.

Constant.

Total frequency

/*2

Ms

Mi

i3i

VPi

^2

/3,-3

Series I,
II, and
III com-
bined
(Distribu-
tiou No.
177).

2328
1.5888
-1.0503
6.5040
.2793
.5285
2.6028
- .3972
-1.6323

Plant 2 of
Series IV
(Distribu-
tion No.71),

922
1.1747

- .8331
4.4627

.4281

.6543

3.2341

.2341

— .8163

Constant.

Skewness

Modal divergence
Standard deviation

Mean

Mode

Modal frequency. .

Range

Lower end of range
Upper end of range

Series I,
II, and
III com-
bined
(Distribu-
tion No.
177.)

—0.6332

— .7961

1.2573

8.6465

9.4425

704.58

6.5028

4.2547

10.7575

Plant 2 of
Series IV.
(Distribu-
tion No. 74)

—0.4432

— .4804

1.0838

8.7581

9.2385

344.90

9.8186

1.1408

10.9594

The fact that the criterion «, {= 2/3, — 3A — 6) is in both cases nega-
tive indicates that curves of Type I are demanded. The equations to
the curves for the two distributions are as follows:

Series I, II, and III:

2y-704.5833
Series IV, plant 2:

V^^ 5.1878/ V^ 1.3150/

.5794

1.9760

.7209/

The histograms and their fitted curves are shown in figs. 4 and 5.

From both the constants of table 4 and the curves themselves it is
clear that the distributions deviate very far from normality. There is
no doubt that both the kurtosis (cf . Pearson, :05) and the asymmetry of
the curves are significant. In both cases the skewness is negative, the
mode lying above the mean. The amount of the modal divergence is
considerably greater in the case of the first curve (Series I, II, and III
combined) amounting there to more than three-fourths of a leaf. Since
frequency curves of Type I are of limited range, we have given for these
two curves the theoretical range of variation. It is of interest to com-
pare this with the observed range. In the Carp Lake race the observed
range is between 5 and 12 leaves, inclusive. The theoretical range is
between 4.25 and 10.76 leaves, a total of 6.51, as against the observed 7.
Thus the total theoretical range is only about a half leaf different from

24

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

the observed. It is not, however, distributed on the two sides of the mode
in the same manner. The theoretical range overestimates the observed
below and underestimates it above the mode. Thus the two whorls
with 12 leaves are theoretically excluded. It is altogether probable
that with proper corrective terms for the moments we should get a still
better estimate of the range. In the case of plant 2, Series IV, the
observed range is between 3 and 11 leaves, inclusive, or a total of 8.
The theoretical curve has a range between 1.14 and 10.96, or a total of
9.82 leaves. No better result could be desired for the upper end, but

eco

700

/

/

/-

\

^

600

500
0

c

3 1-00

CT

(0

li:

\

\

/

^

/

t

^
•i

1

/

/

/

100

/

/

y

/

Leaf number
Fig. 4— Frequency histogram and fitted curve for variation in Ceratophyllum. All
unmutilated whorls on all plants of Series I, II, and III.

the lower end is considerably overestimated. Thus, the occurrence of
2-leaved whorls is theoretically possible. There are three things which
should be kept in mind, however, in passing judgment on this result.
In the first place, the undue extension of the lower end of the theoretical
curve is in a large measure due to the single observation of a 3-leaved
whorl. Now, this whorl was almost certainly to be regarded as an
abnormality. It was a very small whorl in respect to size as well as
number of leaves. It presented the appearance, to borrow zoological
terminology, of being an intercalated, abnormal whorl between two
normal ones. What its origin was it is impossible to say, but it cer-

VARIATION— GENERAL RESULTS.

25

tainly presented the appearance of being- a teratological formation. In
the second place, it should be noted that while the end of the range is
theoretically at 1.14, the proportionate frequencies as given by the curve
for ordinates below that at 4.5 are excessively minute. Thus the abso-
lute frequency according to the theoretical curve for whorls of two
leaves is 0.00004.^ Or, in other words, the expectation of whorls with

ibO

/

\

\

3Z0
Z80

A —

/

/

\

\

Z-+0

/

\

c

0

3

cr

/

\

ll.

/

/

1

1

\

/

/

80
40

/

/

\

y

/•

1

'' '

^

y

Leaf number

Fig. 5.— Frequency histogram and fitted curve for variation in' leaf-number in Ceratophjilum.
All unmutilated whorls on plant 2, Series IV.

two leaves given by the theoretical curve is that there will be 5 such
whorls in every 100, 000, 000! Finally, we must remember that the range
is subject to a very large probable error, both in respect to its absolute

iThis is of course the area included in the range between 1.5 and 2.5, of which the
mid-ordinate is at 2, and not the ordinate at 2. To pass from ordinates to areas the
following formula, given by Pearson ( :04) for a curve of Type I was used.

On strip of base h:

h^ (mi + m; ) {x^ (mi + m; — 1)

Area = /i X ^ 1 +

['

^hcoT

24 d/rf/

where h is the base element having y as its mid-ordinate at a distance x from the
mode, and d^ and rfj are the distances of y from the ends of the range.

26 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

size and to its position. This is perhaps the most important point of
all, and it is one which is usually overlooked in criticisms of Type I
curves in regard to the estimation of the range.

On the whole, I think we may conclude that the curves give good
fits and lead to entirely reasonable values for the physical constants.

A point of interest brought out by these curves is the very close
similarity between the results for the combined Carp Lake plants, which
may be considered representative of the local race, and the single plant.
The curves are of very similar form, and the values of the constants
are of the same order.

SUMMARY.

In this section the general facts regarding variation in the number
of leaves to the whorl in Ceratophyllum have been brought out. It has
been shown (1) that the observed range of variation in this character
is from 3 leaves to 12 leaves, inclusive; (2) that the average number
of leaves to the whorl is about 8.7, with a standard deviation of about
1.2 leaves, leading to a coefficient of variability of about 14 per cent;
(3) that the variation is markedly skew in the negative direction, the
modal whorl having a higher number of leaves (about 9.3) than the
mean whorl; (4) that there are but comparatively small differences in
respect to this character between series collected at different times
and places; (5) that we reach essentially the same conclusions with
reference to these general facts of variation, whether we take a single
large plant or a number of plants from the same locality.

We have seen in this section, in a birdseye view, the facts presented
by the plants when we study the individual as a whole, taking into
account every whorl on the plant. The attempt may now be made to
analyze these facts.

VARIATION IN DIFFERENT PORTIONS OF PLANT.

27

VARIATION IN DIFFERENT PORTIONS OF THE PLANT.
In this section of the paper the material will be treated according
to the following scheme:

Variation in whorls on the main stem.

Variation in whorls on primary branches.

Variation in whorls on secondary branches.

Variation in whorls on tertiary and quaternary branches.

Relative size of the different divisions of the plant and variation in this character.

It will be understood throughout that when the whorls on a particular
division of the plant are referred to, all unmutilated whorls on that
division are included.

Table 5. — Frequency distributions for variation in leaf-number in main-stem whorls.

Series.

Plant.

Distri-
bution
num-
ber.

Leaves per whorl.

5

6

7

8

9

10

11

12

Total.

I<

i
I

II <
III...
IV-

V

v.

VI-

1

I

I,II,III

1

178
7

179

9

10

1

"i

3

1

1

4

5
3

8

2

19

5

16

12

5

43
3

19

16

3

1

"\
4
1

"i

70
13

2

3

1 50 1

4 ■

32
15

5

All plants..
1

24

5

7

18

57

84

7

2

180

47
48

1

2
3

13
16

35
19

12
4

63
42

2

All plants..
1

49

1

5

29

54

16

105

67

1

3

22

57

6

89

1

180
181

"i

4
3

6
17

22
29

32
50

2

All plants..
1

82

1

7

23

51

82

105
106
107
108
109
110
111

3

2
2

9
1

"i
2
3

8
2
1
2
1
2
3

11
2
2
2
5
2

18

13
2

7

"5
6
5

1
"l

...

47
7
10
4
12
14
30

2

3

4

5

6

7

All plants..
1

112

3

4

16

19

42

38

2

124

146
147
148
149
150
151

"i

1

2
'2

■'5

"5

1

1
8
3
3
5
3

3

12

5

4

13

10

1
8
6
3

22
21

"i

1

"7

7
34
16
18
41
41

2

3

4

5

6

All plants ..
All plants

152

2

4

11

23

47

61

9

157

180

5

9

26

108

195

29

2

374

28 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

VARIATION IN WHORLS ON THE MAIN STEM.

The frequency distributions for main-stem whorls of the different
plants and series are shown in table 5 (page 27) . In this table all the
divisions of the main stem are combined.

In order to bring out more clearly the form of these distributions,
and to facilitate comparison, the frequency polygons for the totals of
each of the six series have been plotted and are shown in fig. 6. All of

/it

? \:

/ ■'■

4p

/J

f

r' MM'

1 /\ y,-

/ \ \-

X \ «

f

ific

*** '■ \

i

\v\

%

1:

■.\l".\

8

/ /

---•^

-:-'Z'-'''

_ ../S-

- — /\

VJ

%:>.
K

\

Leaf number
Fig. 6.— Frequency polygons for variation in leaf-number in wliorls on the main stem.

Series I, »— — ~— — ; Series II, o — ^ ; Series III, e ; Series IV,

^ —.; Series V, o — ; Series VI, x-o----0"--o. The ordinates give percent-
age frequencies.

these polygons are so plotted as to have the same area. The scale of
the drawing is approximately the same as that of fig. 3 (p. 18) so that
direct comparisons may be made.

From the distributions and polygons it is seen that on the main
stem the most frequently occurring whorls are generally those having
10 leaves. Series V is the only exception to the rule, and in that case
whorls with 9 or 10 leaves are within the error from random sampling
equally frequent. There is distinctly less "scatter" in the main-stem

VARIATION IN DIFFERENT PORTIONS OF PLANT.

29

as compared with the "total" polygons given in fig. 3, indicating, of
course, that main-stem whorls are less variable. A more precise notion
of the variablity of the whorls on this portion of the plant may be gained
from table 6, which gives the means, standard deviations, and coefficients
of variation for the larger of the main-stem distributions.

Table 6. — Constants for variation in leaf-number in main-stem whorls.

■Series.

Plant.«

Distri-
bution
number.

Mean
(unit=lleaf).

standard devia-
tion (unit=l leaf).

Coefficient of
variation.

I^
III...

IV...
V.

r

v..

1

3

178
179

9.500±0.066
9.480± .099

0.824±0.047
1.034± .070

8.671±0.486
10.909± .743

All plants ..
1

24

9.317±

.053

1.062±

.038

11.397± .410

47
48

9.873±
9.571±

.069
.079

.807±
.760±

.048
.056

8.169± .494
7.945± .588

2

All plants ..
1

49

9.752±

.053

.802±

.037

8.225± .385

67

9.719±

.049

.687±

.035

7.066± .359

All plants

1

82

9.512±

.074

.703±

.037

7.387± .391

105
111

8.383±
8.933±

.150
.110

1.524±

.892±

.106
.078

18.175±1.305
9.984± .878

7

All plants ..
2

112

8.734±

.078

1.283±

.055

14.694± .643

147
150
151

8.588±
9.366±
9.780±

.135
.083
.086

1.166±

.789±
.812±

.095
.059
.060

13.578±1.130
8.422± .632
8.301± .623

5

6

All plants ..

152

9.089±

.065

1.212±

.046

i3.338± .517

"■Tlie constants were not separately calculated for plants in which there were only a few main-
stem whorls.

On account of the small number of the main-stem whorls the results
are not as regular as could be desired. Even in very large Ceratophyl-
lum plants there are usually a comparatively small number of main-stem
whorls, and of those originally present a part are usually so mutilated
as to be uncountable. As the main stem is the oldest portion of the
plant, it suffers most from accidental injuries, the attacks of aquatic
animals (e. g., insects, snails, planarians) and from other injurious
factors. Consequently, when one deals with comparatively few indi-
vidual plants the results are bound to be irregular, because the propor-
tion of unmutilated whorls in different regions of the main stem will
differ from plant to plant. In spite of this irregularity in the constants
for different individual plants, however, we are able to reach certain
definite results.

In the first place, it is quite clear, whether we compare single plants
or series, that the mean number of leaves to the whorl is higher for the

30

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

whorls on the main stem than for the plant as a whole. Further, we
see that both absolutely and relatively the main-stem whorls are on the
whole distinctly less variable than are those from the entire plant taken
together. But obviously the fair comparison is not between main-stem
whorls and all whorls in the plant, but between main-stem whorls and
all whorls on branches. Accordingly we may examine the distributions
and their constants for the whorls on all branches of the plant. In
table 7 are given the frequency distributions for variation in such whorls
for the totals for the six series. It is not necessary to tabulate these
distributions for each plant separately, for reasons which will appear
as we go on.

Table 7. — Frequency distributions for variation in leaf-number in whorls on all
branches. Totals for series.

Plant.

Distri-
bution
nuinbei".

Leaves per whorl.

3

4

5

6

7

8

9

10

11

12

Total.

I

All plants ..

43

3

56

92

184

219

134

9

697

II

do

60

3

64

108

187

258

194

9

823

Ill

do

72

44

49

98

148

94

1

...

434

IV

do

96

i ...

i

19

171

329

531

392

6

1,450

V

do

138

... 4

28

42

120

113

145

82

2

536

VI

do

175

9

23

67

111

127

190

9

i

537

I, II, III,

combined

do

181

6

164

249

469

625

422

19

1,954

The difference between these distributions and the corresponding
ones for the main stem given in table 5 is apparent. The "all-branch"
distributions resemble closely those for the plant as a whole. The most
frequently occurring whorls are those with 9 leaves (except in Series
VI) as against 10 in the case of the main-stem whorls. The constants
for the "all-branch" distributions are given in table 8.

Table

-Constants for variation in leaf-number in whorls on all branches.
Totals for series.

Distri- !

Mean

standard deviation

Coefficient of

Series.

Plant.

bution
number.

{unit=.lleaf).

(unit=l leaf).

variation.

I

All plants ..

43

8.432±0.031

1.227±0.022

14.550±0.268

II

do

60

8.520± .029

1.249± .021

14.657± .249

III

do

72

8.465± .040

1.237± .028

14.611± .342

IV

do

96

8.770± .018

1.042± .013

11.885± .151

V

do

138

8.021± .042

1.429± .024

17.819± .447

VI

do

175

8.741± .038

1.315± .027

15.047± .317

It is at once apparent that the whorls on the main stem have on the
average a higher number of leaves than do whorls on branches. Further,

VARIATION IN DIFFERENT PORTIONS OF PLANT.

31

main-stem whorls are distinctly less variable, both absolutely and rela-
tively. That these changes are significant is shown in table 9, which may
be called a "difference table." It gives for any designated series the
difference (and its probable error) between the value of a particular
constant for the whorls on all branches of that series as given in table
8, and the value of the same constant for the main-stem whorls of the
same series. The absolute differences, expressed as per cent of the
constant for the "all-branch" series, are tabulated as "relative differ-
ences." The sign of a difference is taken positive when the main-stem
constant is the greater.

Table 9. — Difference table, comparing main-stem whorls with whorls on all branches.

Totals for series.

Series.

I

II

III

IV]
V^
VI

Difterences.

Absolute difference

Relative difference..per cent.

Absolute difference

Relative diff erence..per cent.

Absolute difference

Relative difference. .per cent.

Absolute difference

Relative difference.-per cent,

Absolute difference

Relative difference, .per cent.

Absolute difference

Relative difference.. per cent.

Between means
(unit=lleaf).

0.885±0.061

10.5
1.232± .060

14.5

1.254± .063

14.8

.742± .076

8.5
.713± .089

8 9

.348± .075

4.0

Between standard

deviations

(unit=l leaf).

-0.165±0.044

— 13.4

— .447± .043

— 35.6

— .550± .045

— 44.5

— .339± .039

— 32.5

— .146± .060

— 10.2

— .103± .053

— 7.6

Between coeffi-
cients of vari-
ation.

-3.153±0.490

— 21.7

— 6.432± .459

— 43.9

— 7.545± .496

— 51.7

— 4.498± .419

— 37.8

— 3.125± .783

— 17.5
-1.709± .606

— 11.4

There can be no doubt that these differences are significant in compar-
ison with their probable errors. The only cases in which the differences
are not more than three times the probable errors are for the standard
deviations of Series V, and the standard deviations and the coefficients
of variation of Series VI. Even in these cases the differences are more
than twice the probable errors. There is a very substantial differentia-
tion between main-stem whorls and whorls borne on branches. Main-
stem whorls have on the average roughly 1 leaf more than do whorls on
branches, and are from 10 to 50 per cent less variable, both absolutely
and relatively. The very low variability of main-stem whorls both
absolutely and as compared with whorls on other portions of the plant
may be brought out by a totally different sort of consideration. We
say with regard to any organ or character that it varies very little
if the chance of finding any other than the typical condition is small.
If the chance of the occurrence of deviations from the typical con-
dition is large, we say that there is great variation. Now, in the case

32

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

of whorls on the main stems of Ceratophyllum plants, the chance that
a whorl chosen at random will have 10 leaves is, on the basis of the
combined Series I, II, and III, tVoV- Or, in other words, the chance
in favor of the 10-leaved whorl is greater than the combined chance of
all other whorls together. This means, of course, very low variability.
In order to gain a more precise idea of the nature of this differentia-
tion I propose to treat the distributions analytically. In doing this I
shall, as before, combine the frequency distributions for Series I, II,
and III. These series are so slightly differentiated from one another
that the error made by combining them is altogether negligible. In
fact it is practically certain that we shall come nearer to the true facts
by using the combined data for these three series from the same locality
than by taking any one of them separately. For the same reasons as
before I have used the raw moments in determining the constants.
The analytical results for 'main-stem" and "all-branch" whorls are
given in table 10. The combined distributions are, for the main-stem
whorls, No. 180 in table 5, and for the whorls on branches. No. 181 of
table 7.

Table 10. — Analytical constants for variation in leaf-number. Series I, II, and III

combined.

CJonstant.

Total frequency

M2

^3

/*4

^i-^-

l//3i

^2

^2—3

Main-stem

whorls (Dis

tribution

No. 180).

374

.8798

— .9013
4.0691
1.1928
1.0921
5.2568
2.2568

+ .9354

+1.2455

Whorls on

all branches

(Distribution

No. 181).

1,954

1.5350

— .8739
5.8052

.2112

.4595

2.4637

— .5363
—1.7060

Constant.

Skewness —0.4270

Modal divergence — .4005

Standard devia-
tion 9380

Mean 9.5348

Mode 9.9353

Range

Lower end of
range

Upper end of \
range 13.2096

Main-stem
whorls (Dis-
tribution
No. 180).

Whorls on

all branches

(Distribution

No. 181).

—0.6119

— .7581

1.2390
8.4765
9.2345
6.0426

4.5435

10.5861

This table brings out clearly the essential differences between ' 'main-
stem" and "all-branch" whorls in respect to variation in leaf-number.
We note at once that the main-stem distribution is markedly less skew
than is the branch distribution, the distance from mean to mode being
in the former case only a little more than half what it is in the latter.
The direction of the skewness is the same however, in both cases, namely
negative, or the mode is larger than the mean. The range for the
"all-branch" distribution is, as was to be expected, less than what it is
for the plant as a whole (cf . table 4) .

VARIATION IN DIFFERENT PORTIONS OF PLANT. 33

For the "all-branch" distribution the criterion *<! (=2/?.,— 3^—6) is
negative, indicating that a curve of Type I is demanded for graduation.
In the case of the main-stem distribution, however, '<i is positive, whence
at first thought it would be concluded that a curve of Type IV was
demanded. But it has been shown by Pearson (:01) that ^ positive,
while a necessary condition, is not a sufficient condition for Type IV.

In addition,

A (/3.+3)^
"■' 4 (4A-3/8i) (2)8-3/8-6)"

must be > 0 and < 1. We see at once that the main-stem distribution
does not fulfill this latter condition, since «, = 1.2455, or is > 1. Clearly,
then, one of the transition curves of Type V or Type VI is wanted.
The condition for Type V is that <, = 1, while for Type VI we must
have ^2 > 1 and < oo . Strictly speaking, then, our main-stem distribu-
tion should be graduated by a curve of Type VI, but insomuch as k,
differs from 1 by only a small amount we shall probably get sufficiently
good results with Type V. I have accordingly fitted the main-stem
distribution with a Type V curve and the "all-branch" distribution with
a Type I curve. The equations to the curves are:
Main-stem distribution (Type V) :'

Log ?/=19. 6849467-18. 34909 log 0^-26.092355 (-)

Origin at 13.2096; x positive towards small whorls.
All branch distribution (Type I) :

j/=581.6454 (l+^:69ro) (l-lTMs)

Origin at mode = 9.2345.

The histograms and their fitted curves are shown in fig. 7. The
frequencies for both curves were reduced to percentages before plotting,
so that, since the base elements are the same, both curves have the
same area in the diagram. This method brings out most clearly the
points of difference between the two distributions.

^For obvious reasons I have put the curve in the logarithmic form. The equation
to a curve of this type is of the form

~P — y/*

where y^, p, and y are constants.

34

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

It is evident that the graduations are as good as could reasonably be
expected, considering the small number of classes from which the
moments had to be calculated.

I wish now to consider more fully a point that was raised earlier in
the paper. In the section dealing with the methods used in collecting
the material it was pointed out (p. 11) that in some cases the main stem
of a plant divided, forming what we have called secondary main stems,
and in some cases even a secondary main-stem was divided to form

bb

A

\

1

\

o

C
"Hi-

/

\

1

/-"-

'\

1

«)
oo
ro
-£24

8

I.

16
8

■■/

[/

/

1

--V-

A'

/

k

i

1

\

^

— -«—

1
^4'

/

\

\

/

j^^

1

1

Leaf number

Fig. 7,— Frequency histograms and fitted curves for variation in leaf-number in main-stem
and "all-branch" whorls of Series I, II, and III combined. Main-stem distribution and
curve in solid lines; "all-branch" distribution and curve in broken lines.

"tertiary" stems. The statement was made that there was no difficulty
in practice in distinguishing such main stem divisions from primary
lateral branches. I wish now to present statistical evidence to indicate
that the variation in leaf-number in these portions of the plants shows
that they are parts of the main stem rather than branches.

In table 11 are given the frequency distributions for all the secondary
main-stem whorls which were found in the course of the work, and in
table 12 are given the distributions for tertiary main-stem whorls.

VARIATION IN DIFFERENT PORTIONS OF PLANT.

35

It is evident from these tables that division of the main stem occurred
relatively infrequently in our material. All plants of Series II, V, and
VI and and three out of five plants of Series I had the main stem undi-
vided. Plant 1, Series III and both plants of Series IV had long
secondary main stems. In these plants the main stem proper was very
short, bearing only a few whorls. In the case of plant 2, Series IV,
there were no unmutilated whorls on the main stem below the point of

Table 11. — Frequeiicy distributions for variation in leaf-number of whorls on
secondary main stems.

Series.

III..,

IV

Plant.

All plants,

All plants.

Distri-
bution
number.

Leaves per whorl.

7

8

9

10

11

Total.

182
183

"i

1
2

3

8

10
13

"i

14
25

16
66

1

3
1

11
11

23
51

1

6

39
69

184
185

4

1

6
2

18
2

28
5

80

5

8

20

33

Table 12.— Frequency distributions for variation in leaf-number oftvhorls on

tertiary main stems.

Series.

Plant.

Distri-
bution
number.

Leaves per whorl.

6

7

8

9

10

Total.

I

IV...

1

186

187

'l

1

2

2
3

9

4

13
2

25
12

3

All plants..
2

23
81

1

3
1

5
2

13
15

15
27

37
45

division. The distributions exhibit even on casual inspection a marked
difference from what has been shown to be characteristic for the varia-
tion of whorls on branches. - These differences are brought out in a still
more striking way by the constants of the distributions, which are given
in table 13.

Now, I think it is perfectly obvious if we compare the values given
in this table with those for the "all-branch" distributions given in table
8, or with those for the primary, secondary, or tertiary branches given
in tables 15, 18, and 23, infra, that the whorls on those divisions of the

36

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

plant which we have called "secondary" and "tertiary" main stems are
sensibly differentiated from whorls borne on any branches, of whatever
order. These whorls have a higher mean number of leaves and are
markedly less variable (absolutely and relatively) than are whorls on
branches. That is, they have those characters which we have just seen
hold for main-stem whorls generally. There can be, I believe, but one
conclusion, namely, that these portions of the plant are parts of the main
stem rather than primary branches.

The results of this section of the paper may be summarized as follows:
It has been shown that (1) whorls borne on the main stem of the plant
have a significantly higher mean and modal number of leaves than do
whorls borne on branches; (2) main-stem whorls are markedly less vari-
able, both absolutely and relatively, than are whorls borne on branches;
(3) the distribution of variation in leaf -number is distinctly less skew

Table 13. — Constants for variation in leaf-number in whorls on secondary and

tertiary main stems.

Series.

Plant and
portion.

Distri-
bution
number.

Mean (unit =
1 leaf).

Standard devia-
tion (unit=l leaf.)

Coefficient of
variation.

1 and 3, second-
ary main stem-

16

9.513±0.084

0.780±0.060

8.202±0.631

land 3, tertiary
main stem

23

9.027± .117

1.052± .083

11.657± .926

III...

1, secondary
main stem

66

9.899± .044

.542± .031

5.478± .315

IV-

1 and 2, second-
ary main stem..

80

9.455± .087

.742± .062

7.851± .655

2, tertiary main
stem

81

9.511± .069

.687± .049

7.224± .515

in the case of the main-stem whorls than in whorls borne on branches;
(4) the divisions of the main stem have the same characteristics in
respect to variation in leaf-number that the main stem proper does.

From these facts we conclude that ivhorls borne on the main stem of
the plant are, as a class, clearly differentiated from ivhorls borne on
branches.

VARIATION IN WHORLS ON PRIMARY BRANCHES.

Anyone familiar with Ceratophyllum will recognize that it is the
primary branches which make up the greater portion of the plant.
They usually contribute a considerably larger number of whorls to the
total than does any other single division. This fact will be apparent by
comparing the data of table 14, which gives the frequency distributions
for primary branches, with those in the tables for other divisions of
the plant.

VARIATION IN DIFFERENT PORTIONS OF PLANT.

37

Comparing these distributions with those given in table 1, it is at
once evident that there is a great similarity. The most frequently-
occurring whorls in both cases are those with 9 leaves. The range
of variation is practically the same for the primary branches as it is for

Table 14. — Fi-eqiiency distributions for variation in leaf-number of whorls on

primary branches.

Series.

Plant.

distri-
bution
number.

Leaf-number per whorL

3

4

5

6

7

8

9

10

11

12

Total.

I<

r
II.

III..

IV.

1

I

V.

L

VI.

i,ii,iii

1

25
26

27
28
29

1
1
1

9

4

11

7
1

18
10

9
22

3

29
25
26

47
14

43

27
37
53
17

34

27
12
42
10

1
4
2

1
1

135

98

98

172

46

2

3

4

5

All plants..
1

30

3

32

62

141

177

125

9

549

50
51

"i

19
10

46
25

79
47

114
64

102
63

2
6

362
216

2

All plants ..
1

52

1

29

71

126

178

165

8

578

68

13

25

59

76

59

1

233

1

83
84

"i

"i

3

7

19

46

62
82

136

187

127
139

2
2

349
465

2

All plants..
1

85

1

1

10

65

144

323

266

4

814

113
114
115
116
117
118
119

2

9

17

'2

7

33
1
1
2
5
4

24

10

6
3
6
8
15

54

"3

4

1

23

22

32
"2

16

7

2

159
1

12
9
12
47
75

2

3

4

5

6

7

All plants..
1

120

2

9

26

70

48

107

51

2

315

153
154
155
156
157
158

"2

2
2

2
3

5
1
2
2

"8
5
7

18
9

7

11

7

8

24

16

10
8

10
5

23

24

9
3
2
2
31
85

"i

"6

...

"i

28
35
30
23
100
145

2

3

4

5

6

All plants..
All plants

159

...

6

15

47

73

80

132

7

1

361

188

4

74

158

326

431

349

18

1,360

the whole plant. In fact, these distributions make it clear that the
characteristic features of the variation of the plant as a whole with
respect to leaf-number are very largely determined by the primary
branches. The same essential similarity as before is observed when

38

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

we compare the first four series. Tliis is so evident from the distribu-
tions themselves and from the values of the constants which follow
that it does not appear to be necessary to figure separately the polygons
of variation in the primary branches. The general character of the
distributions is shown graphically in fig. 8, p. 44, in which primary and
secondary branches are compared in respect to variation in leaf-number.
From distribution 84 it is seen that the single 3-leaved whorl observed
in the course of the work was situated on a primary branch. In Series
VI, the single 12-leaved whorl of that series was on a primary branch.
Thus it would appear that the whole range of variation in leaf-number
in Ceratophyllum may be shown by primary-branch whorls.

Table 15. — Constants f(

?r varxat

wn %n leaf-numb(

?r m whorls

onp

rimary orancnes.

Series.

Plant.

Distri-
bution
Qumber.

Mean(unit=l

leaf).

standard devia-
tion (unit = 1
leaf) .

CoefHeieut of
variation.

I-

II-
III...
IV-

V

V-
VI-

1

25
26
27
28
29

8.556±0.072

8.735± .085
8.337± .085
8.605± .058
8.761± .100

1.245±0.051
1.250zb .060
1.253± .060
1.124:fc .041
1.004± .071

14.555±0.610
14.310± .703
15. 031 ± .740
13.057± .483
11.460dr .816

2

3

4

5

All plants..
1

30

8.581± .035

1.199±

.024

13.976i= .290

50
51

8.663± .042
8.741± .056

1.179±
1.228d=

.030
.040

13.605+ .347
14.044± .465

2

All plants..
1

52

8.692± .034

1.198±

.024

13.780± .278

68

8.627± .051

1.147±

.036

13. 296 i .423

1

83
84

85

9.063± .033
8.867± .033

.925±
1.057±

.024
.023

10. 206 ± .263
11. 923 ± .267

2

All plants..
1

8.951± .024

1.007±

.017

11.254± .190

113
119

120

8.138± .086
7.973± .091

1.608±
1.166±

.061
.064

19.754± .776
14. 622 t .822

7

All plants..
2

8.184± .053

1.386±

.037

16.938=b .468

154
157
158

7.829±: .145
8.570=h .085
9.359+: .066

1.276±
1.259±
1.178±

.103
.060
.047

16.297±1.348
14. 691 ± .715
12. 592 ± .506

5

6

All plants..

159

8.759± .047

1.332±

.033

15.202± .390

The chief physical constants for the distributions of table 14 are

given in table 15.

As we should expect, the values of the constants for primary branches
throughout are of the same order as those for the "all-branch" distri-

VARIATION IN DIFFERENT PORTIONS OF PLANT.

39

butions given in table 8. We note, however, that primary-branch
whorls by themselves are somewhat less variable, both absolutely and
relatively, than are the "all-branch" whorls. The means are also
slightly higher in the primary-branch than in the "all-branch" distri-
butions. This definite system of differences would appear to indicate
that whorls on primary branches are as a class differentiated from
whorls on other branch divisions of the plant. It will be seen later that
this is the case. It should be noted how slight the differences are be-
tween the constants for Series I, II, and III, for the portion of the plant
under consideration. The means for the three series differ by 0.1 leaf
or less, and the standard deviations by < 0.06 leaf. It is clear that
these differences are only what would be expected to arise from random
sampling, and that there is no sensible, real differentiation between
these series. As has been the case in the other portions of the plant so
far examined, there is a definite, though not large, divergence between
the constants for Series IV and those for Series I, II, and III. Series
V and VI fall in a class by themselves.

Table 16. — Difference table comparing pHmary -branch whorls with main-stem
whorls. Totals for seiries.

Series.

DiflFerences.

Between means
(unit =1 leaf)

II
III
IV

V
VI

Absolute difference — 0.736±0.064

Relative difference. . per cent. . . 7.9

Absolute difference — 1 . 060^r . 063

Relative difference. . per cent. . . 10.9

Absolute difference ;-1.092± .071

Relative difference. . per cent. . . 1 11.2

Absolute difference.

Relative difference.. per cent...

Absolute difference

Relative difference. . per cent...

Absolute difference

Relative difference.. per cent...

.561± .078

5.9
.550± .093

fi ^
— .330± .080

3.6

Between stand- Between coeffi-
ard deviations cients of vari-
(unit = 1 leaf). ation.

0.137±0.045

12.9

.396± .044

49.4
.460± .050

67.0
.304±: .041

43.2
.103zt .066

8.0

.120± .057

9.9

2.579±0.502

22.6
5.555± .475

67.5
6.230± .555

88 2
3.867zfc .435

52.3
2.244± .795

15.3
1.864± .648

14.0

The main-stem whorls form a definite base with which the primary-
branch whorls may be compared. This comparison is made in table 16,
which is a "difference table" corresponding in plan to table 9 above.
In this case the differences are given the positive sign when the primary-
branch constant is the greater. The relative difference is in each case
the percentage which the absolute difference is of the main-stem con-
stant. Only the totals for the series are compared.

Considering the first four series, we see that both absolute and relative
differences are large. There can be no doubt that they are significant.
In Series V and VI the differences are smaller, but still probably in all
cases significant. We conclude, then, that primary-branch whorls as

40

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

compared with main-stem whorls have on the average nearly one less
leaf to the whorl, and are relatively 20 to 80 per cent more variable.

The analytical consideration of the variation in primary-branch whorls
will be taken up later in connection with the secondary-branch distri-
butions.

It is shown in this section that ivhorls on primary branches are as a
class differentiated from whorls borne on the main stem. As we pass
from main-stem whorls to primary-branch ivhorls the mean number of
leaves in the whorl decreases and the variability increases.

VARIATION IN WHORLS ON SECONDARY BRANCHES.

The frequency distributions for the variation in leaf-number in this
portion of the plant are given in table 17.

Table 17. — Frequency distributions for variation in leaf-number in whorls on

secondary branches.

Series.

Plant.

Distri-
bution
number.

Leaves per whorl.

4

5

6

7

8

9

10

11

Total.

l<

II-
III..

\v<
v'

YU
I,II,III

1

31

32
33
34

2

5

5

10

6

5

4

12

11

4

4

24

3
24

3
12

1

8

23

46
16
58

2

3

4

All plants..

1

2

35

22

27

43

42

9

143

53
54

55

2

26
6

21
12

40
16

63
16

24

4

1

177

54

All plants..
1

2

32

33

56

79

28

1

231

69

24

19

34

60

28

165

1

86
87

"9

34
64

73
105

86
119

33
92

1

1

227
390

2

All plants..
1

88

9

98

178

205

125

2

617

121
122

2

7

1

4
3

22
5

14

8

15

7

12

4

76
28

7

All plants..
1

123

2

8

7

27

22

22

16

104

160
161
162
163

2

3
1
3

6
2

4

4

6

4

9

11

12

13
9

12

21
24

"2

39

7
52
50

2

5

6

All plants..
All plants

164

2

7

16

30~

34

57

2

148

189

2

78

79

133

181

65

1

539

It is apparent from the totals of this table that secondary branches
(in the present material at least) contribute less than half as many

VARIATION IN DIFFERENT PORTIONS OF PLANT.

41

whorls to the total number borne by Ceratophyllum plants than do primary
branches. Also, one notes that the character of the distributions is
changing. While the most frequently occurring whorls on the second-
ary branches are, as in the case of the primaries, those having 9 leaves,
yet whorls with fewer than 9 leaves occur proportionately much more
frequently on secondary than on primary branches. We should expect,
then, that the average number of leaves to the whorl would be less
for the secondaries than for the primaries. That this is in fact the
case is shown in table 18. Series V and VI still maintain their peculiar
character. In Series V the actually most frequent whorls are those
having 7 leaves, though 8 and 9 leaved whorls are nearly as frequent.
Series VI has the greatest frequency at 10, as in the case of the pri-
maries.

Table IS.— Constants for variation in leaf-number in whorls on secondary branches.

Series.

Plant.

Distri-
bution
number.

Mean (unit
leaf).

= 1

standard devia-
tion (unit = l
leaf).

Coefficient of
variation.

I

"{

III...

V j
VI <

1.

2.
4.

1.
2.

1.

1
2.

1.

5.
6.

31
32
34

35

7.783d=0.131
8.543± .120
7.655± .088

0.930 ±0.093

1.211± .085

.992± .062

11.956±1.206
14.170±1.016
12. 962 ± .825

All

plants..

7.923±

.065

1.159±

.046

14. 631 ± .596

53

54

8.198±
8.000±

.066
.103

1.311±
1.122±

.047
.073

15.989± .587
14.027± .928

All

plants-

55

8.152=b

.056

1.272±

.040

15.603± .501

69

8.297±

.067

1.285±

.048

15.487± .589

86

87

8.533±

8.574±

.042
.037

.930±
1.095±

.029
.026

10.904± .349
12.767± .313

All

plants..

88

8.559±

.028

1.037±

.020

12. 121 ± .236

121

7.737±

.123

1.584±

.087

20.476±1.166

All

plants..

123

7.817±

.101

1.524±

.071

19.494± .946

162
163

8.750±
9.180±

.131
.102

1.399±
1.071±

.093
.072

15.987±1.084
11. 669 ± .798

All

plants..

164

8.797±

.072

1.294±

.051

14.712± .589

We see in this case as before a close similarity between the results
for Series I, II, and III. Series V shows the highest variabilities for
this portion of the plant, and Series VI the highest means. Comparing
the results with those for primary branches, it appears that secondary-
branch whorls have fewer leaves to the whorl and are more variable
than those on primaries. The nature and the extent of the differences

42

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

are shown in table 19. In this ' 'difference table' ' the differences are taken
as positive when the primary-branch constant is the greater and nega-
tive when the constant for the secondaries is the greater. The absolute
differences are taken as percentages of the primary-branch constant, in
each case, to obtain the relative differences.

Table 19. — Difference table comparing secondary with primary -branch whorls.

Totals for series.

Series.

n!
m{

IV {

VI j

Diflferences.

Absolute
Relative
Absolute
Relative
Absolute
Relative
Absolute
Relative
Absolute
Relative
Absolute
Relative

difference .
difference..

difference
difference..

difference
difference..

difference
difference..

difference
difference..

difference
difference

per cent,
per cent.

per cent,
per cent.

per cent,
per cent.

Between means
(unit = 1 leaf).

0.658±0.074:

7.67

.540± .066

6.21
.330± .084

3.83
.392± .037

4.36

.367± .114

4.48

- .038± .086

.43

Between standard

deviations

(unit = l leaf).

0.040±0.052

3.33

— .074±

047

6.17

— .138±

060

12.03

— .030±

026

2.97

— .138±

080

9.95

.038±

061

2.85

Between

coefflcients of

variation.

-0.655±0.663

4.7
-1.823± .573

13.2
-2.191± .725

16.5
- .867± .303

7.7

-2.556dt:1.055

15.1

.490± .706

3.2

We see that generally the differences are positive in the "mean"
column and negative in the two variability columns, thus indicating
what was pointed out above, that whorls on secondary branches have
fewer leaves to the whorl and tend to be more variable than those on
primary branches. In respect to variation in leaf-number, then, the
secondary branches stand in much the same relation to the primaries as
the latter do to the main stem. In one case. Series VI, the order of the
differences is reversed, the mean being higher and the variability lower
for the secondary-branch whorls than for the primary, but no special
stress can be laid on this apparent exception to the rule, because the
differences are altogether insignificant in comparison with their proba-
ble errors. The differences between the means are, with this single
exception, significant in comparison with their probable errors. In the
variability columns the individual differences when taken singly are in
most cases not certainly significant. Due weight must, however, be
given to the fact that all tend in the same direction (except of course Series
VI) . I think we may safely conclude that, in general, whorls on sec-
ondary branches are as a class more variable in respect to leaf -number
than are whorls on primary branches.

We may now consider analytically the variation in primary and sec-
ondary branch whorls, in comparison with each other and with whorls
on the main stem. As in other cases I have used the combined distri-

VARIATION IN DIFFERENT PORTIONS OF PLANT.

43

butions for Series I, II, and III in the analytical treatment. The justi-
fication for combining these three sets of data will have been apparent to
anyone who has inspected the values of the constants which have been
given in the preceding tables. The ''raw" moments were used in this
as in the other cases. The values of the chief physical and algebraical
constants of distributions 188 and 189 of tables 14 and 17 are given in
table 20.

Table 20. — Analytical constants for variation in leaf-number. Series I, II,

and III combined.

Constant.

Primary-
branch

whorls( Dis-
tribution
No. 188).

Total frequency

^2

f^i

Ml

8,-3.

1360

1.4183

— .8671
5.3199

.2636

.5134

2.6448

— .3552
—1.5011

Secondary-
branch

whorls (Dis-
tribution

No. 189).

539

1.5753

— .6964
5.3816

.1240

.3522

2.1685

— .8315
—2.0351

Constant.

Skewness

Modal divergence...
Standard deviation

Mean

Mode

Range

Lower end of range
Upper end of range

Primary-

Secondary-

branch

branch

whorls(Dis-

whorls (Dis-

bution

tribution

No. 188).

No. 189).

—0.5484

—0.9996

— .6530

—1.2547

1.1909

1.2551

8.6360

8-1354

9-2891

9.3901

6.5021

5.1929

4.2354

5.1923

10.7375

10.3852

From this table we note that:

(a) The distribution for the secondary branches is markedly more
skew than is that for the primaries. Consequently, since the standard
deviation is also greater in the case of the secondary branches, we find

that—

(6) The distance from the mean to the mode is very nearly twice as
great in the secondary branches as it is in the primaries.

(c) Secondary branches have a lower mean number of leaves to the
whorl, but a higher modal number than the primaries. Too much stress
must not, however, be laid on the fact that the secondaries show the
higher modal number, because, as has been pointed out, the values of
the moments from which the mode has to be calculated have not been
in any way corrected.

id) The theoretical range of variation is smaller for secondary
branches than it is for primaries by more than one leaf. The second-
ary curve starts at a higher and ends at a lower value than does the
primary.

(e) The skewness is negative for both curves.

(/) The kurtosis {v =(^2 — 3) is negative in both curves, but has a
considerably higher value in the case of the secondary branches, thus
indicating that the secondary-branch distribution is the more flat-topped.

44 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

(g) Both curves are of Type I, the equations being as follows:
Distribution No. 188, primaries:

2.7363

!,= 433.4048(1 , ^^^^
Distribution No. 189, secondaries:

""•"^ 5^0536) \^ ~ 1.4484/

.784

.8726

y = 151.m2{l + ^y'^{l--^)

.2063

The curves are shown graphically in fig. 8. The frequencies are
reduced to percentages, so that both curves have the same area in the
diagram. The greater skewness of the secondary distribution is very
evident from the diagram.

40

30

20

4>
•00
ID

./

'

/ ,

/

• *"

7---

/

/ .

1 ^

1 1

!?

.__.

._±_

\

/

/

fw m,-!^

=^

/

/

la)

/

/

',

l4_.

—4

10

li

\2.

4 5 6 7 8 9

Leaf number
Fig, 8. — Frequency histogiams and fitted curves for variation in primary and secondary branch
whorls. Series I, II, and III combined. primaries; secondaries.

It is interesting to note that the analytical results give a greater
range of variation for primary than for secondary branches. This is
what we should expect on general principles to find, because the prim.ary
branches are so much longer, and consequently bear so many more
whorls, than do the secondaries.

Comparing the results for primary branches with those for the main
stem given in table 10 (p. 32) we note that (a) the variation is dis-

VARIATION IN DIFFERENT PORTIONS OF PLANT.

45

tinctly more skew in the whorls on primary branches than in those in
the main stem; (6) the primary-branch whorls are the more variable;
(c) the mean number of leaves per whorl is 0.8988 higher in the main-
stem than in the primary-branch whorls, while the modal number of
leaves per whorl is only 0.6462 higher; (d) the sign of the kurtosis
changes in passing from main-stem to primary branches.

From the results presented in this section we may conclude that
whorls borne on secondary branches are as a class differentiated from
those borne on either the main stem or primary branches. The changes
in the constants as we pass from primary to secondary branches are
similar in kind to those which we find in passing from main-stem to pri-
mary-branch whorls. The most important of these changes are (1) the
lowering of the mean number of leaves to the whorl, and (2) the increase
in the variation in this character.

VARIATION IN WHORLS ON TERTIARY AND QUATERNARY BRANCHES.

As the number of whorls is comparatively small in these portions of
the plant we may conveniently consider tertiary and quaternary branches
together. It will be understood that by "tertiary branches" are meant
those which arise from secondary branches, and by "quaternary" those
which arise from tertiaries.

Table 21. — Frequency distributions for variation in leaf-number in whorls on

tertiary branches.

Series.

Plant.

Distri-
bution
number.

Leaves per whorl.

5

6

7

8

9

10

Total.

I...

II...

III...

IV<

vJ

Vli

2 and 4

36
56

70

•••■

2
3

7

3

4
5

"5
5

"i

12

5
14
36

1

1

1

89
90

1
7

1
6

"3

2

17

2

All plants-

1

7

91

8

7

3

19

124
125

7

5

1

12
1

18
4

11
1

11

64

7

All plants..
1

. 126

7

6

13

22

12

71

165
166

"i

"i

1
3

1

7

2
11

5
23

5

All plants.

167

1

1

4

8

13

28

The frequency distributions for tertiary and quaternary branches
are given in tables 21 and 22, respectively.

The totals here are so small that much regTilarity in the distribu-
tions can not be expected, nor can very accurate comparisons between

46

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

the constants be made, because of the large probable errors. It is of
interest to note that the number of tertiary whorls is greater in Series
III, than in Series II or I. The relative increase is greater than is indi-
cated by the figures in table 21, since the total of Series III is smaller
than that of either I or II. This increase in tertiary whorls is a rough
index of the change which took place in the Carp Lake population, due to
growth in the interval between the collection of Series I and Series III.

Table 22.— Frequency distributions for variation in leaf-number in whorls on

quaternary branches.

Series.

Plant.

Distri-
bution
number.

Leaves per whorl.

5

6

2
1

7

8

9

10

Total

f

1
I

1

127
128

4

9

1

18
3

2
2

4

39

7

7

All plants-

129

4

3

10

21

4

4

46

In order that some judgment may be formed of the characteristics
of tertiary-branch whorls I have prepared table 23, which gives, in addi-
tion to the constants for the distributions of table 21, the absolute
differences between these and the constants for secondary-branch
whorls. Only totals for series are tabled, because of the paucity of
material. The difference is taken as positive when the secondary-branch
constant is the greater.

The probable errors are very large, as was to be expected, and the
results are not as smooth as could be desired. Still, I think, we may
safely draw the following conclusions:

(a) The mean number of leaves to the whorl is distinctly lower for
tertiary than for secondary branch whorls. In only one case out of the
five is the difference between the means negative, and then it is sensibly
zero, in comparison with its probable error. In Series II, IV, and VI,
the difference is 0.6 to 0.7 leaf, a very considerable amount, and cer-
tainly significant in comparison with the probable error. It is of the
same order as the difference between main-stem and primary-branch
whorls (cf. table 16), and is distinctly greater than the difference
between primary and secondary branches in respect to mean number
of leaves per whorl.

(6) The tertiary-branch whorls tend on the whole to be less rather
than more variable than those borne on secondaries. The differences in
the variability columns are all very small, and, taken singly, quite insig-
nificant in comparison with their probable errors. Due weight, however,
must be given to the fact that the sign of the differences is uniformly

VARIATION IN DIFFERENT PORTIONS OF PLANT.

47

plus. In fact, Series III forms the only exception to this rule. This
exception arises from the fact shown in table 28, p. 52 that the tertiary
branches tend to be more variable in length than is the case in the other
series. It will be shown in later sections of the paper that the positional
differentiation of whorls within an axial division of the plant is such that
the variation exhibited by groups of whorls may be greatly influenced by
the length of the branches on which they are borne.

Table 23. — Constants for variation in whorls on tertiary branches. Totals for series.

Series.

Portion of plant.

Distri-
bution
number.

Mean.

standard
deviation.

Coefficient
of variation.

Il|

mi

IV i

v]

VI<

Secondary branches
Tertiary branches ..

Difference

Secondary branches
Tertiary branches ..

Difference

Secondary branches
Tertiary branches ..

Difference

Secondary branches
Tertiary branches ..

Difference

Secondary branches
Tertiary branches..

Difference

55
56

69
70

88
91

123
126

164
167

8.152±0.056
7.500± .202

1.272±0.040
l.llSi .142

15.603±0.501
14.907±1.942

-+- .652±

.210

+ .154±

.148

+ .696±2.006

8.297±
8.194±

.067
.159

1.285±
1.411±

.048
.112

15.487± .589
17.215±1.408

+ .103±

.173

— .126±

.122

— 1.728d=1.526

8.559±

7.842±

.028
.135

1.037±

.874±

.020
.096

12. 121 ± .236
11.150±1.235

+ .717d=

.138

+ .163±

.098

+ .971±1.257

7.817±
7.831±

.101
.119

1.524±
1.473±

.071
.083

19. 494 ± .946
18.805±1.101

- .014±

.156

+ .051±

.109

+ .689±1.452

8.797±
8.021i

.072
.140

1.294±
1.102±

.051
.099

14.712± .589
13.735±1.261

+ .776±

.157

+ .192±

.111

+ .977±1.392

(c) The tertiary-branch whorls thus appear to be differentiated as a
class from those borne on other portions of the plant.

Turning to the whorls on quaternary branches, we have, unfortu-
nately, only a single series which gives any data at all. In our experience
the extreme condition of lateral growth implied in quaternary and higher
order branches is very rare. It seems possible that this may be due
merely to the restriction imposed by the shortness of the growing season.
The finding of quaternary branches in the plants of Series V lends
support to this view, for the reason that, as has been pointed out above
(p. 15) the large plants in this series had wintered over and were grow-
ing like perennials. An examination of Ceratophyllum from more
southern latitudes would be interesting in this connection.

48

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

The constants for the quaternary-branch whorls of Series V are given
in table 24, together with their differences when compared with tertiary-
branch whorls.

Table 24. — Constants for variation in whorls on quaternary branches. Series V.

Distri-
bution
number.

Portion of plant.

Mean.

standard
deviation.

Coefficient
of variation.

126
129

Tprtinrv Viranfhps

7.831-^0.119

1.473-t-0.083

18.805^1.101

Quaternary branches

7.652± .125

1.255± .088 ! 16.939±1.225

Difference

+ .179± .173

+ .218± .121

+1.866±1.647

We see that the mean number of leaves in the quaternary-branch
whorls is less than in those on tertiaries. The difference, however, is
small, being of the same order as its probable error. The quaternary-
branch whorls are also slightly the less variable. In the case of the
standard deviations the difference is nearly twice its probable error.
We may conclude, provisionally at least, that the same kind of change
occurs in the mean when we pass from tertiary to quaternary branches
as when we pass from secondaries to tertiaries. The differences between
the constants for whorls on these higher-order branches in any case
would probably be small in absolute amount, for reasons which will
appear as we go on.

THE RELATIVE SIZE OF THE DIFFERENT DIVISIONS OF THE PLANT
AND THE VARIATION IN THIS CHARACTER.

It will readily be admitted that if we are to get any light on the
question of the factors which produce the characteristic frequency dis-
tributions of variation for the whorls of a particular division of the
plant, by a study of the laws of growth and differentiation of the whorls
themselves, it will be necessary to know in addition something about
what proportions of the total whorls on a plant fall within particular
divisions, and to what extent these proportions are constant for different
plants. Also, it is desirable to have definite information regarding the
average length of the different axial divisions of the plant and the
variation in this character. It is with these matters that the present
section has to do.

Taking first the question of the proportion of the total number of
whorls found on different parts of the plants, we have the results set
forth in table 25. The figures in the vertical columns give the percent-
ages which whorls borne on the specified part of the plant are of the
total.

RELATIVE SIZE OF DIFFERENT DIVISIONS OF PLANT. 49

Table 25. — Proportionate numbers of whorls in different axial divisions of the plant.

Series.

Plant.

Percentage of total whorls on—

(a)
Main stem

and
divisions.

Pi'imary
branches.

(c)
Second-
ary
branches.

(d)

Tertiary
branches.

Quater-
nary
branches.

Total
number
of whorls

l<

"i

III...

.vj

V<
VI<

1
2
3

4
5

1
2

1

1
2

1
2
3

4
5
6

7

1
2
3
4
5
6

30.7
8.1
30.5
12.2
24.6

59.2
60.9
59.8
65.4
75.4

10.1

28.6

9.7

22.0

"2'. 4
""'.'4

228
161
164
263
61

10.2
13.4

58.8
69.3

28.8
17.3

2.2

615
313

17.0

44.6

31.5

6.9

523

5.3

5.4

57.2
50.4

37.2

42.3

.3
1.9

610
922

12.2

87.5
45.5
30.8
50.0
23.0
20.4

41.3
12.5
54.5
69.2
50.0
77.0
51.0

19.8

ig'.b

16.6

"4'. 8

10.1

"4.8

385
8
22
13
24
61

147

8.9
44.7
34.8
43.9
19.0
17.4

35.4
46.1
65.2
56.1
46.3
61.4

49.4
9.2

24.1
21.2

6.3
16.6

79
76
46
41
216
236

These results are reasonably uniform, if, as is obviously fair, we
leave out of consideration the very small plants, such as, for example,
plants 2, 3, 4, and 5 of Series V. When the plant is so small as to have
only a main stem and a few primary branches it is clear that the per-
centage of whorls on these two portions will not agree with what is
found on large plants with many higher-order branches. Making due
allowance for these irregularities, the results set forth in the table show
that—

(a) In well-developed plants on the average more than half the total
number of whorls are borne on primary branches. The actual v/eighted
mean percentage given by the values in column (b) of the table is 54.7
per cent. As it is obviously unfair that a plant like No. 2 in Series V
should have the same weight as No. 2 in Series IV, I have weighted
each entry in determining this and the following means with the abso-
lute size (i. e., in number of whorls) of the plant with which it is con-
cerned.

50

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

Table 26. — Frequency distributions for size of primary branches.

Series I.

Series II.

Series
III.

Series IV.

Whorls per

A

c<i cc 1 ■fl' , US 1

N

.

rt"

^

c^-

branch.

...3

■^

♦i

■u

♦^ c8

.4

.*J

f^

■4-1

♦a

.^

d

a

a

a

a

0 -S

a

a

a

a

d

cS

cj

ea

zi

cS

o

eS

d

c3

sS

d

0

H

H

Oh 1 Oh 1

Ph

Sh

&<

(U

Q-

Qu

Ph

P^

1

4

2
3
2

2

1

1

i

"i
1

2

1
i

2

3

2
3
4
4

"i

2

6

1

'2
2

i

2
1
2
1

3
5

1

i

14
8

12
9
9
2
5
3
4
5
3
6
0

2
7
11
2
3
3
3
2
3
4
2
1
2

3
4

1

5
3
3
2
2
3

2

2
10
15
3
8
6
6
4
5
7
2
3
2

1

5

8
1
6
3
6
2
2
2

1

3
2
2
2
2
4
5
7
4
5
3

4
1
5
3
4
8
4
7
5
2
5

5
4
7
5
6

10
8

12

12
«

10
4
1

2

3

4

5

6

7 . ..

8

2

1
2

9

10

11

12... .

2

13

14

'2

2

2

4

1
3
0

8

2

3
2
0
0

1

3

1
1
1

15

16. ..

17

18

1
1

1

2

1
0
0

2

3
1

1
1

2
0
1
0

19

20

21

...! ..

22

0

0

2

2

23

0

0

0

24

1

1

0

0
0

1
0

25

26

0

1

1

0

27

0

1

0

28

0

0

0

29

1

1

0

0

30

1

1

0

0

31

0

0

i

0

Total

23

14

21

23

13

94

55

31

86

40

44

58

102

Whorls per

Series V.

Series VI.

branch.

■^

'

<— (

ci

w

•

t^ f

■♦i

.^

+j <->

«i

c3

•«^

■iJ

■u -u

*=

.«j

a c

c

HI

a c

a

-

a

C

fl d

«5

d

2

cS 0

3 c9

rf,

c3 (S

ci

St

d

« 03

CS

es

0

H

H

Ph fl

4 Oh

dl

S 1 S

PLI

^

s

S 1 E

Oh 1

P.I

1

2
4
4

4 .
2 .
1 .

I ..

'. i
1
1

3
1

i

2

1

2
5
2
3
1

4
6
14

8
7
5
0

i

'2
1

i

i

1
3
2

"i

1

'2

2
2
2

1
1

2
2
7
3
2

0
4
5
12
13
4
5

2

3

4

5

6

7

8

1 .

3

1

2
2

6
3

1

i

"2

3
4

9

10

2 .
1 .

2
1
0

1

1
1

i

2

1
1

11

12

13

1 .

1

0

0
0

14

15

1
1 .

1
1
0

1

1
1
0

16

17

18

0

1

1

19

0

1

1

20

0

0

21

u

0

22

0

1

1

23

0

0

24

0

0

25

0

0

26

0

0

27.

0

0

28

0

0

29

1

1

0

0
0

30

31

u

0

Tot

al

28

1 3

2

4

7

18

60

4

7

8

4

16

21

59

1

RELATIVE SIZE OF DIFFERENT DIVISIONS OF PLANT.

51

(6) Of the remaining whorls usually somewhat more are borne on
secondary branches than on the main stem. This of course only applies
to plants which have secondary branches. Taking all the plants together
as they stand in the table, the weighted mean percentage contribution
of the main stem to the total number of whorls is 14.1 per cent. Reck-
oned in the same way the mean percentage contribution of the secondary
branches is 27.0 per cent, or very nearly twice that of the main stem.

(c) Tertiary and quaternary branches contribute, on the whole, a
very small proportion of the total number of whorls. Calculating the
mean percentage contribution from table 25, I find that it is 3.3 per cent
for tertiary branches.

Table 27. — Frequency distributions for size of secondary branches.

Whorls

per
branch.

Series 1.

Series 11.

Series
III.

Series IV.

Series V.

Series VI.

a

c4

C

03

a

a

1

a

a

a!

a

a
E

t

p4

a

a

d

i

a
5

a

a

«3

a

03

3
0

H

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

ToUl..

5
8
2
2

1

1

2
1

"i
1

i
"i

\

4
4

2

.

5
5
3

2
3

1

15
17
9
3
3
4
2
0
0
1
0
0
0
0
0

1

0
0
0

5

8
8
5
5
4
4
2
1

i

4
8
3
1
2

9
16
11

6
7
4
4
3
1
0
0
0

1

0
0
0
0
0
0

12
10
2
4
3
2
2
1
3
2
1
1
0
0
0
0
0
0
0

6
8
8
5
7
6
4
2
2
1
1

'i
1

3
6
K
12
4
5
3
3
4
2
1
2
1
2
1
2
1
1
1

9
14
14
17
11
11
7
5
6
3
2
2
2
3
1
2
1
1
1

9
4

1
6

1
1
1
1
1

"i

'2
1
1

i

9
5

1

8
2
2
1
1
1
1
0
0
0
0
0
0
0
0
0

'2
2
1

1
1

"i

'2
1

i
1
3
1

4

i

■3

2
1

1
2

0
1
5
9
4
6
2
2
1
0
0

1

0
0
0
0
0
0
0

18

8

10

19

55

48

19

62

43

52

60

112

25

6

31

8

3

11

9

31

The plants of Series IV differ in their percentage constitution very
distinctly from the others, in the direction of having a lower proportion
of main-stem whorls, and a higher proportion of secondary-branch
whorls. The most striking exception in the table is afforded by plant
1 of Series VI, in which there are 14 per cent more whorls on secondary
than on primary branches. But this is a small plant and clearly the
probable errors of the percentages are high. As a matter of fact this
plant had only three primary branches, while there were eight secondaries
and one tertiary. It is not surprising, since the plant as a whole was so
small, that the eight secondary branches should contribute a larger
number of whorls to the total than the three primaries.

I wish to call attention at this point to a matter which otherwise
might be overlooked. If the values given in table 25 be studied in con-
nection with the constants given in table 2, it will be seen that there is

52

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

in general marked agreement between the variation constants of plants
which have similar percentage constitutions in respect to number of
whorls. Thus, to take only a single example, we see from table 25 that
plant 2 of Series I, and plant 1 of Series II, have very nearly the same
percentage constitutions, although they are very different in abso-
lute size. Table 2 shows that their variation constants are likewise in
remarkably close agreement. This is of course what we should expect
from our result that whorls on different portions of the plant form
differentiated classes, and the fact that experience accords so closely
with expectation is strong corroborative evidence, if any be needed,
of the truth of our previous conclusions.

Table 28. — Frequency distributions for size of tertiary branches.

Whorls

Series I.

Series
II.

Series
III.

Series IV.

Series V.

Series VI.

per
branch.

Plant

Total.

Plant 1

Plant 1

Plant.

Total.

Plant.

Total.

Plant.

Total.

2

4

1

2

7

1

5

1...

2...
3...
4...
5...
6. .
7...
8...
9-15
16..

3

1

4
0
0
0
0
0
0
0
0
0

3
4
2
0
0
0
0
0
0
0

2
2
1
0
3
2
0
0
0
0

1
1

3
2
3
1

4
3
3
1
0
0
0
0
0
0

i

1

1
0
2
4
1
2
0
2
0
1

"2
2

1

0
0
0
2
3
1
0
0
0
0

T(

)ta

1...

3

1

4

9

1(1

2

9

11

11

2

13

1

5

6

We may turn now to the question of variation in the size of the
different axial divisions of the plant. In dealing with this phase of the
subject, the number of whorls on any division of the plant will be taken
as the measure of its size. For our present purpose this is a more
satisfactory measure than absolute length would be. What we wish to
determine is the mira-plant variability in respect to size of branch.
In the analysis of the morphogenetic processes which will be taken up
in a later section of the paper, it will be necessary to have some quan-
titative appreciation of how the branches of different orders compare in
size, and what degree of variation in size they exhibit. We may turn
at once to the data. The data respecting size of main stem have already
been given in tables 5, 11, and 12. In table 26 (p. 50) is given for each
plant, the frequency of primary branches of different sizes (i. e,, con-
taining different numbers of whorls). In tables 27 (p. 51) and 28 are
given in the same way the frequencies of secondary and tertiary branches
of different sizes.

As was to be expected, the range of variation in the number of whorls
to the branch is very large for the primary branches, and distinctly less

RELATIVE SIZE OF DIFFFERENT DIVISIONS OF PLANT.

53

for the secondaries. In order that a more definite idea may be obtained
of the comparative distributions for the different orders of branches, I
have prepared the diagrams given in fig. 9. These were made by plot-
ting to the same base-unit the percentage frequencies of primary,
secondary, and tertiary branches of different sizes, taking all the plants
of Series I, II, and III together. In order to smooth out some of the
irregularities in the distributions which arise from the relative smallness
of our samples, I have doubled the size of the classes given in the tables.

J530

(

; 1
. 1
■ 1
', 1

\ \
\ ;
\ ;
\ •

—

; 1
.'1
; 1
1 1

.'1

\

;l

;l

k *

[

■' /

■ \

'.

1

<

■

'\^

"v

1/

'.

'v^

'--<

\>

>~~.

1 t

, <

r~ — 1

^. -<

•■ ' I i^ '^ ~ -- ' I — >V- ^ *^— '»' ■

I-H2 3t+ 5+6 7+8 9+10 11 + 12 13+14 15+16 17+18 19+20 2I+Z2 Z3+Z4 Z5+Z6 27+28 29+30

Number of whorls per branch

Fig. 9.— Frequency distributions for length of primary, secondary, and tertiary
branches. The ordinates give percentage frequencies and tlie abscissas tlie num-
ber of whorls per branch. Primary branches, » ; secondary branches,
o ; tertiary branches,® .

From these diagrams we see clearly the general nature of the varia-
tion in the size of branches. The smoothness of the primary and
secondary branch curves is remarkable. The most frequently occurring
branches, whatever their order, are those having from 1 to 4 whorls.
The great range of secondary branches in length is rather surprising.
As was to be expected, the proportionate number of branches with 1 or
2 whorls increases enormously as we pass from primary to tertiary
branches. The diagrams show that the proportionate number of

54

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

branches having from 3 to 6 whorls is approximately the same for pri-
mary, secondary, and tertiary branches.

In table 29 are given the constants for the variation in the size of
primary and secondary branches for the totals of the series. On
account of the relatively small numbers in most instances it is not worth
while to give separately the values for each plant, nor for the tertiary
branches.

Table 2^.— Constants for variation in size of primary and secondary branches.

Series.

Order of branch.

I

II

III

IV

V

VI

Primary

Secondary.,
Primary —
Secondary..

Primary

Secondary.

Primary

Secondary.

Primary

Secondary.

Primary

Secondary.

Mean
(unit = 1 whorl).

7.085±0

3.036±

7.756±

3.726±

6.650±

4.023±

8.324±

5.857±

5.567±

3.548±

6.559±

5.161±

.427
.238
.411
.208
.622
.335
.313
.263
.392
.301
.367
.252

standard devia-

Coefficient

tion(unlt=l whorl)

of variation

6.073±0.302

85.7

2.621± .169

86.3

5.657± .291

72.9

2.424± .147

65.1

5.829d= .440

87.7

3.253± .236

80.8

4.691± .222

56.4

4.127± .186

70.5

4.503± .277

80.9

2.487± .213

70.1

4.175± .259

63.7

2.081± .178

40.3

From this table we see that:

(a) There is surprisingly little difference between the different series
in respect to mean size of branches.

(6) The mean size of primary branches is roughly twice that of
secondaries.

(c) As was to be expected, the mean size of branches is the greatest
in the series which was collected latest in the growing season (Series IV) .

{d) Both absolutely and relatively to the mean size the variation is
greater in primary than in secondary branches, with the single excep-
tion of Series IV, where, probably as a result of the great number and
relatively large size of secondary branches, we get a higher proportion-
ate variability in this group.

SUMMARY OF RESULTS.

Before taking up the discussion of the positional differentiation in
detail, it is desirable to put together in a connected way the results
which we have so far gained regarding the variation in leaf-number.
By taking separately the whorls borne on different axial elements of
the plant body we have found that there is a very sensible differentia-
tion of whorls in respect to their leaf -number when they are grouped
according to their general location on the plant. The nature of this

SUMMARY— VARIATION RESULTS.

55

differentiation is such that a progressive change occurs in the charac-
ter of the variation constants as one passes from central to peripheral

10

1.6
1.4
I.Z
1.0
0.8
0.6
20

16
12

8
4

0.6
0.2

-..^_

>,.,:;----^^^|r^;v^^^

■ ->

^ rTTTr-r»C;53^

i

' )

.,J

:- —

f..

•)

- ---^j

'

■'X

^^.

*^

'

1

"^'-K

<

i-^"

,

< _^

'" >

<

<

^^r..— ri

r'

}

(

.-<^'''

(

>

<

^^

(

^

I 2

to 0)

c O

O (Q

0) JQ

i_ in

u c
0--°

Fig. 10.— Diagram showing changes in the constants for variation in leaf-number
in different portions of plant. Series I, II, and III combined » ;

Series IV, o ; Series V, X- "X. Series VI, »• .

portions of the plant until tertiary branches are reached. Here the
order of events in part changes, and where previously certain of the

56 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

constants had been increasing, they now begin to decrease, and vice
versa. The nature of these changes in the character of the leaf-
number variation in different portions of the plant may best be grasped
as a whole from a graphical representation, such as is given in fig. 10.
This diagram shows for four groups (viz, Series I, II, and III combined,
Series IV, Series V, and Series VI) the changes as we pass from central
to peripheral divisions in the following constants— mean, standard devi-
ation, coefficient of variation, and skewness. The different constants
are plotted to different vertical scales, which are in each case given at
the side of the diagram.

This diagram makes plain at once the law which the variation in leaf-
number in different sets of whorls follows. It may be stated in the
following way: The mean number of leaves per whorl is highest in the
whorls on the most central division of the plant (the main stem) , and
decreases regularly in the peripheral divisions. The whorls on the main
stem are the least variable, and the variation increases regularly in the
more peripheral divisions, till a maximum is reached in secondary-branch
whorls. The variation then tends to diminish in the whorls on higher-
order branches. Now, from the method of growth of Ceratophyllum, it
is clear that as a class the main-stem whorls are the oldest, the primary-
branch whorls as a class stand next in age, secondary and tertiary
branch whorls next, while quaternary-branch whorls will on the whole
be youngest. Of course these distinctions are not absolute for every
whorl; there may, for example, be individual whorls on the main stem
which are younger than individual whorls on any branch, but on the
average it is evident that the more peripheral parts will be the younger.
So, then, we find that as a general rule the older the portion of the plant
the greater will he the average number of leaves to the whorl. Further,
the variation in leaf-number is least in the oldest portion of the plant and
increases in the younger portions, but reaches a maximum one or tiuo divi-
sions short of the youngest.

Besides the changes in type and variability, there are marked differ-
ences in other respects between whorls in different parts of the plant.
Thus for example, the skewness appears to be greater in the variation
of whorls on the younger portions of the plant, though on account of
paucity of material we can not go farther than secondary branches,
with the analytical constants.

It is perfectly clear from the results which have been presented that
in a general way at least the number of leaves in a whorl and the posi-
tion of the whorl on the plant are related. But how close is the relation?
Does it hold within an axial division (say primary branches) that the

POSITION REGRESSION— PRIMARY BRANCHES. 57

size of the whorl is a function of its position on the axis? So far we
have demonstrated positional differentiation of broad classes of whorls.
It now remains to determine whether this extends to the individual
whorls, and in general to find out the laws of growth which result in
main-stem or primary-branch, or secondary-branch, or other whorls,
showing the particular characteristics in their variation which they do.
Cases of positional differentiation in like parts are well known in a
number of plants, ' but a complete analysis of the phenomenon which
takes into account the whole series of repeated parts or characters for
a large and richly branching plant like Ceratophyllum has not hitherto
been undertaken.

THE RELATION BETWEEN THE NUMBER OF LEAVES IN THE WHORL
AND POSITION ON THE PLANT.

We come now to the more direct investigation of the morphogenetic
laws concerned in the formation of the leaf -whorls. From the method of
growth in lateral branches of Ceratophyllum it is clear that we have
an almost ideal form for such a study. The whorls on a branch present
a linear series of parts in which we know the order of formation in time
(cf. p. 10, supra). Such a system suggests at once a number of very
interesting problems in morphogenesis. It will conduce to clearness,
if we base the discussion of our results on such individual problems,
taking up the different ones in order.

POSITION REGRESSION IN DIFFERENT PORTIONS OF THE PLANT— THE
FIRST LAW OF GROWTH IN CERATOPHYLLUM.

The first problem which logically presents itself may be stated in
this way: In a series of like parts produced in a regular ordinal succes-
sion what relation exists between the form of a particular member of
the series and its ordinal position? If there is a change in the character
of successively formed parts, what law governs this change? It is
obvious that this problem is a very fundamental one, because of the fact
that one of the most frequently occurring plans of structure which we
know in the organic world is the metameric. In this type of structure
the organism is built up of a series of primitively similar units arranged
in linear order. Our problem is to find out, if possible, in a very simple
case the laws of differentiation in such a system. The biometrical solu-

^Many examples are given by Miss Tammes (:03) and the subject has been inves-
tigated biometrically by Pearson ( :05) in Asperula odorata and Equisetum arvense
and by Pearson and Radford ( :04) in beech leaves.

58 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

tion of the problem is theoretically clear. What we want to know are
the correlations and regressions between any character of the members
of such linear series and their position. In this particular case we have
to determine the correlation and regression between the number of
leaves in the whorl and position.

It is evident that the different axial divisions of the plant must be
treated separately. Logically the main stem should be considered first,
but for reasons which will appear as we go on, it is desirable to take up
the primary and secondary branches before the main stem. The cor-
relation tables showing for each series the relation between number of
leaves in the whorl and position on primary branches follow. In the
case of Series V and VI the tables have been cut off at the tenth whorl
because the very small number of entries beyond that point did not
make it worth while to keep them. In the other four series every
possible whorl has been included. Furthermore, the tables for Series V
and VI differ from the others in that in these two series all branches
(primary, secondary, tertiary, etc.) have been clubbed together.
Otherwise the numbers would have been too small to get any results at
all for these series. No confusion need result from this procedure, as
the whole discussion of positional differentiation will be based on the
first four series, the last two (V and VI) being used merely to illustrate
and confirm the conclusions reached from the others. The method of
designating position has been fully explained and illustrated in an earlier
section of the paper (p. 12 and fig. 2, supra) and need not be repeated
here in detail. It is merely necessary to recall that the whorls are
numbered in order, beginning at the proximal end of the branch; the
"first whorl" (1) thus being in the case of a primary branch the whorl
nearest the main stem, in the case of a secondary branch the whorl
nearest the primary, etc.

Mere inspection of these tables shows us at once that the character
of the whorls in respect to leaf-number changes as we go out on the
branch. It is perfectly clear in a general way that the farther a pri-
mary-branch whorl is from the main stem, the more leaves it is likely
to have. Or, in other words, we see that there is a clear positional
differentiation within an axial division of the plant. The character of
successively formed parts changes with the order of their formation.
We may at once proceed, then, to the analysis of the laws of this change.
As a first step it is necessary to know the exact degree of the correlation
between leaf-number and position, and to test whether the regression
is linear or not. The test for linearity of regression has been given by
Pearson ( :05) . It consists in determining for any system of correlated

POSITION REGRESSION— PRIMARY BRANCHES. 59

Table 30.— Correlation between leaf-number and position of primary-branch whorls.

SERIES I,

Number of

Position of whorl on branch.

whorl. ^ 2

3 4 5 6 7

8 9

10

11 12 13

14

15

16 17

18 19 20 21 22

23 24 25 26

27

28

29

30

Total.

5 3

3
32
62
141
177
125
9

549

6 2li 5

i i i i i

7 6 3 11
26 17 15 7 9
22 18 9 17 13

2 8 4 7 7
11

58 50 42 34 32

■'2 'i
2 6
12 10
11 9
1 2

28 28

1
1

1
12
11

26

7 2116

1 .
3
8
9

211

. 1
4 1

4 3
fi 7

i

3

9

1

g . .19 26

1 .. 1 .. ..

1

1

i

2

'i
1

i
1

"i
1

9" "[[[['.'.['.. ...... 6 16

3

8

3 2

4 4

2 1.. 11
2 3 2 12
1 ..

.. 1 .. ..
3.. .. 1
.. 1 .. ..

3 2 0 2

]0 1

11

Total 70 61

1 1

5 13

13

11

8 6

5 4 3 3 3

1

SERIES II.

Number of

Position of whorl on branch.

whorl. J 2

3 4 5 6 7

8 9

10 11 1

2 13

14

15

1617

18 19 20 21 22

23 24 25 26

27

28

29

30

Total.

5 1

1

29
71
126
178
165
8

6 19 2

1 2 .. ..

2 1

1

1

7 35 18

7 4 3.. 1
25 20 6 10 6
25 24 28 22 18
5 5 11 9 13
2 .. 1 .. 1

;J5 53 51 41 39

1 2
5 4
10 7
14 17

30 30

8 12 25

3 --

2 2

3 2
8 7

0,

1
1
9

2

7

.. 1
1 1
6 5

112....

6
12
1

24

1
13

151

1 1 .... 2
4 2 2 3..

10 4

2 12 2
.. 1 .. ..

1
1

1
1

-

-

11

Total 70 69

611

11

9

7 7

7 4 4 3 2

2 2 2 2

578

1 1

Position of whorl on branch.

whorl.

1 z

3 4 5 6 7

8 9

10111

213

14

15

1617

18 19 20 21 22

23 24 25 26

27

28

29

30

Total.

6 . 12 1

13
2>
59
76
59
1

7 10 8

2 'i i .. ..
810 6 2 3
151013 9 6
4 3 3 3 3

1 2

2 ..
5 2
2 3

8 12 15

i

3

i ■

3

i 'i

2 2
1

"i

2

'2

.. 1
1 ..
1 1

.. ..!..

Q 9

i

16 1

12 2 2 2

2 2 2 2

1

1

1

1

11

Total 34 34

29 24 23 14 12

10 7

4

4

4I

3

2

2 2

2 2 2 2 2

2 2 2 2

1

1

1

1

233

1

SERIES IV.

Number of

Position of whorl on branch.

whorl. ^ 2

3 4 5 6 7

8 9

10
1

111

L213

14

15

16 17

18 19 20 21 22

23 24 25 26 27

28

29

30

Total.

1
0

1

=; 1

6 9 1

10

7 41 g

4 2 5 3 i

:25:i4 7 5 a

5li5138 37 3E

8|15 27 26 2J

.. 1

88 83 77 71 6^

1

1
3

7
14

25

'

65
144
323
266
4

5 4
18 11

1
ifl

'

1 i

2 ]

7 (

.

9 8 38

3 4

121c

1 .

16 11

1
) £

'211..!
) 4 3 2 2 1

) 32 24 21

1.. 1 .

155 40 35

11...

11 1

Total 93 9£

ir

2 1(

)10

) 6 4 3 2 2

11...

■ —

—

814

J_

_

—

—

60

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

variables a constant, v, called the correlation ratio, and defined by the
expression

V

In other words, the correlation ratio is the ratio of the variability of
the means of the arrays of one correlated character to the total vari-
ability of that character, as exhibited by the sample as a whole. The
constant v has the same value as the coefficient of correlation r only
when the regression is perfectly linear. If the regression is not linear
r] will be greater than r. Then evidently -7 ~ r is a measure of the
approach of the regression to linearity. Recently Blakeman ( :05) has
given methods of obtaining the probable error of various functions of
7/ ~ r.

Table SI.— Correlation between leaf-number and position. Whorls on all branches.

Number of

leaves per

whorl.

Position of whorl oa
branch— Series V.

Number of

leaves per

whorl.

Position of whorl on
branch— Series VI.

I

2

2
h

6

29
28
20
5

95

3

i

10

i

19
6

79

4

'2
1
14
17
26
i

64

5

'i

9

8

22

6

46

6

17
10

37

7

"i

'2
3

13

8

27

8

i
4

'6
14

25

9

'i

3
4
4

12

10

i

Total.

1

7
19

S7

2

■4

16
33
26
12

3

'3

24
30
29

1

87

4

1

'4
12
32
2.5

1

75

5

"2
8
11
31

52

6

"i

6
6
19

1

33

7

i

1
5

16

1

24

8
1

"i

3
9

1

15

9

"i

2
10

1

14

10

i

"2
4

Total.

4

2

17
16
40
19

4
27
36
118
111

5

9

23
66
105
123
160
6

5

6

6

7

7

8

8

20
6
5

9

9

5

7

5I 1S7

10

10

3
9

67

1

11

11

1
107

Total

94

91

Total

501

7

492

Table 32. — Correlation between leaf-nutnber and position. Whorls on primary
branches. Series I, II, and III combined.

Number of

leaves per

whorl.

Position of whorl on branch.

1

2

3

4

5

6

'i

1

19
48
19

1

89

7

'i

2

18
37
23

2

83

8

'4

9

27
27

1

68

9
'5

10

3

1

11

"i

1
3
10

25
40

12

i

"6
8

16
4

35

13

i

3
6
16

1

27

14

"2

5

20

27

15

'5
17

22

16

i

'5
11

17

17

'2
3
10

15

18

i

"2

4

7

14

19

"i

2
7

10

20

"3
"6

"9

21

'i

6

1

8

22

'3
4

7

23

"7
7

24

i
3
2

6

25

"4
4

26

"i

■5

6

27

"l
■3

4

28

29

30

Total.

5

6

3

1

8

42

66

"2

16
59
62
11

2

152

'i
11

■3

7

■3

3

'2
2

2
2

4
74
168
326
431
349
18

66
43

8

47 27

10 4

9

10 44

52
16

12-^

60
18
1

116

19

29

2

65

19

26

1

54

10

11

Total

174

6
167

1,360

In table 33 are given the values of r and rj with their probable errors
deduced from the data given in tables 30 to 32. The probable errors
of r tabulated were computed from the formula

(1-^0

Vn

Probable error of >/ — ±0.67449

POSITION REGRESSION— PRIMARY BRANCHES.

61

Pearson (loc. cit. ) has shown that this expression is sufficiently ac-
curate for all ordinary purposes, and it is much easier to calculate than
the complete expression for the standard deviation of v.

Table 33. — Correlation between leaf-number and position of primary -branch whorls.

We see at once from table 33 that—

(1) There is a very considerable degree of correlation between the
number of leaves per whorl and the position of the whorl.

(2) The degree of correlation is very closely the same for all series.
We should expect, of course, that Series V and VI would give different
values for the coefficients, because in those cases we are dealing with
three and four different orders of branches together.

(3) There can be no doubt that the regressions are not linear. The
differences between v and r are so considerable that I have not thought
it necessary to work out the probable errors for every case. The series
which gives the smallest difference between rand ^ is V (>?— r=0.0197),
but the apparent approach to linearity here is due to putting different
orders of branches together. Considering the primary branches alone,
the minimum difference between v and r is given by Series I (17 — r =
0.1142). We may take these two instances as a sample:

It has been shown by Blakeman (loc. cit.) that if we let

an approximate formula for the probable error of ^, i. e., E^^ is

C

Vn

E^ 0.67449

il/f

\n+(i-v'y-(i-ry

Working from this formula we have for Series I,

and for Series V,

C = 0.1364 ±0.0193,
^ = 0.0191 ±0.0082.

62

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

In the first case the constant is, of course, certainly significant in com-
parison with its probable error, and in the second case very probably
significant.

It thus being clear that the relation between leaf-number and posi-
tion of whorl is not a simple linear one, it becomes necessary to find out
what it is. To do this we must first see what the exact form of the
observed regression is. In table 34 is given the mean leaf-number for
each array of whorls occurring in a given position, as indicated in tables
30 to 32.

Table 34. — Mean nurtiher of leaves in successive whorls on primary branches.

Position
of Whorl.

Mean n

umber of leaves.

Series
I.

Series
11.

Series
III.

Series
IV.

Series 1,
Il.andlll
combined

Series

V.a

Series
VI.»

1

7.057

7.014

7.000

7.441

7.029

6.981

7.149

2

7.875

8.043

8.029

8.453

7.970

7.642

8.286

3

8.293

8.492

8.724

8.685

8.461

7.873

9.011

4

8.520

8.566

8.625

8.988

8.559

8.188

9.040

5

8.524

8.902

8.783

9.130

8.741

8.500

9.365

6

8.912

8.976

9.071

9.211

8.966

8.838

9.394

7

8.844

9.179

9.000

9.313

9.024

8.889

9.625

8

9.250

9.233

8.800

9.491

9.176

9.120

9.333

9

9.179

9.300

8.857

9.550

9.200

8.916

9.786

10

9.192

9.208

9.750

9.438

9.241

9.222

9.286

11

9.300

9.000

9.750

9.360

9.425

12

9.267

9.438

10.000

9.875

9.429

13

9.462

9.455

9.667

9.714

9.481

14

9.615

9.727

9.667

9.750

9.667

15

9.727

9.778

10.000

9.900

9.773

16

9.250

9.857

9.500

9.600

9.529

17

9.667

9.571

9.000

9.444

9.533

18

9.200

9.000

9.500

9.667

9.143

19

9.750

9.250

10.000

9.750

9.600

20

9.333

9.000

10.000

9.667

9.333

21

10.000

10.000

10.000

10.000

10.000

22

9.667

9.000

10.000

9.500

9.571

23

10.000

10.000

10.000

10.000

10.000

24

10.000

, 10.500

10.000

10.000

10.167

25

1 10.000

10.000

10.000

26

9.000

10.000

10.000

9.667

27

9.000

, 10.000

10.000

9.500

28

10.000

10.000

10.000

10.000

29

10.000

10.000

10.000

30

10.000

10.000

10.000

"It will be remembered that in these two series all branches are clubbed together, while In the
case of the otlier series we are dealing here with primary-branch whorls only.

There is no doubt from the figures in this table that the form of the
regression line is essentially the same for all the series. In order to
show this graphically I have prepared fig. 11, in which the observed
regression lines for Series I, II, III, and IV are plotted. To prevent con-

POSITION REGRESSION-

fusion Series V and VI have not
been included, but examination of
the values in table 34 is sufficient
to show that they are not essen- "^
tially different from the others, i^
In the diagrams the ordinates give
the means as recorded in table 34
and the abscissas the position of "'
the whorls. c

These diagrams at once make
clear the following points:

(a) The regression is evidently '^
not linear. Starting with a low ^
value for the first whorls, the
curves all show a sharp rise,
amounting to almost exactly one -
leaf, to the second whorl. From -
the second to the third whorl there ^ _
is approximately half as great a | '^
rise as from the first to the second, i +
From the third whorl on, while °r:
the general trend of the lines is j _
upward, their slope becomes more - "^
and more gradual. The maximum ^
towards which the lines tend is -
clearly 10-leaved whorls, though
on account of the small number
of entries in the outlying parts of s
the correlation tables at the upper ts>
end, the regression lines become
very irregular on these high "^
values. 5

(6) While the form of the re- n
gression line is clearly the same in
Series IV as it is in Series 1, II, '^
and III, it differs in being practic- S
ally uniformly higher. It starts «
nearly a half leaf higher on the
first whorl, and m.aintains this °'
difference on the whole very even- s
ly out to the 15th whorl. From ^
that point it becomes more irregu-
lar on account of the paucity of
observations.

■PRIMARY BRANCHES.

Mean leaf number

63

"--

"^.

'^

^.

\

s

w

.A

v

• \

I

\

\\

4f

1?

\

v.

■V

\

1

'

:

X. -^

\:

9

1

>>

TT

\i

<

\

i)>

<?^

\

b
/

/I

>

i^<

/ !

i

%

C'

.<<

t

-

,.

,

*'

r
1

''''

{

^

Fig. 11.— Regression line, stiowing ctiangeol mean
leaf-number with position. Primary ^branches.

Series I, • ; Series II, o — ;

Series III,^.. ; Series IV, ©-..— - — —©.

64 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

(c) There is evidently a definite functional relation (in the mathe-
matical sense) between the number of leaves in a whorl and its position.
Biologically this means that the form of a particular whorl is in some
manner related to the number of whorls which the growing bud has
previously formed. Successively formed whorls show an increase in
the number of leaves.

The values in table 34 and the diagrams indicate that this increase
follows a definite law. We can in general formulate this by mere in-
spection of the data. There is an increase in the mean number of
leaves in successively formed whorls, but the increment in leaf -number
diminishes with each successive whorl. So much is clear, but we are
only thrown back to the discovery of the law according to which the
increment diminishes. So far as I can see, any kind of biological rea-
soning is powerless to help us further. Observation shows that there
is some sort of a functional relation between number of leaves in the
whorl and position, just as observation indicates to the physicist that
there is a functional relation between two things. But to determine
the nature of the functional relation, or in other words, to find out the
law which the phenomena follow, we are compelled, so far as I can see,
to resort to mathematical treatment of the data. The physicist has
done this, of course, for a very long time. If this is a logical, scientific
method in physics (and I presume no biologist has any doubt that it is) ,
why is it not equally logical and scientific when a precisely similar
problem is presented by biological phenomena?

To formulate the law of growth according to which the change in
mean leaf -number in the successively formed whorls of a Ceratophyllum
plant occurs we must turn to mathematics, as biological reasoning will
not help us further. I have dwelt at some length on this point in order
to show that in one particular class of biological problems at least we are
compelled to call mathematics to our aid, or else to be content to stop
considerably short of a goal within easy reach. The idea seems to
prevail among many biologists that the application of higher math-
ematical methods to biology is altogether idle and futile. It seems
possible that something may be done towards removing this unfortunate
prejudice if clear and definite statements telling just why it is neces-
sary to resort to mathematics if we are to advance on particular prob-
lems, are more frequently made in biometrical writings.

The mathematical problem before us is this: If we let y indicate the
mean number of leaves per whorl and x the position of the whorl on a
primary branch, then direct observation shows us that

y=<l>{x).

POSITION REGRESSION— PRIMARY BRANCHES, 65

We have to determine the form of the function ^. It is obvious that
the proper way to set about this is to find out what curve best graduates
the observed data, and the equation to this curve will be the expression
of the law of growth v/hich we are seeking. While the problem is thus
theoretically a simple one, practically it is an extremely difficult one,
because if the data give no clue at the start as to what the nature of
<^ (x) is, which unfortunately is usually the case, we have to resort to a
very laborious process of trial and error. Different curves must be
fitted one after another to the data until the right one is found. It is
the usual custom among physicists to assume a parabola and get the
best fit possible by increasing the number of constants. This method
is not, however, a theoretically justifiable one, because, as Pearson
(:02, p. 19) has pointed out:

There are often considerations, lying outside the actual data, which suffice to indi-
cate that trigonometrical, exponential, or other types of curves will give better results
than parabolas. A parabola which passes even through all the observations may
indeed be a most undesirable representation of the facts, for it has twisted and curled
to account for error as well as to give the general sweep of the observations.

The figures in table 34 and the diagrams in fig. 11 show clearly that the
nature of the regression curve is the same for all the series, and that
further, the absolute values for Series I, II, and III are, within the limits
of error from random sampling, identical. Hence on account of the
very laborious character of the arithmetic involved, these three series
(I, II, and III) were combined, and in what follows this combined mate-
rial will serve as a basis. By this proceeding a considerable gain in
smoothness of the figures to work from was made, without any loss
of accuracy. The means for the combined series are given in the fifth
column of table 34.

Having at the outstart no idea of the nature of </> (x) , it was decided
as a beginning to fit a series of parabolas to these data. That is to say,
in the expression

it was assumed that

<li{x) = c„ +A^ + c^x"" + Cgx' + . . . cX-

By taking more and more terms of this expansion we get successively
higher order parabolas. There is no theoretical limit to the process,
but obviously, if we are seeking a true graduation formula, it is idle
to go higher than the sixth-order term, for the reason mentioned
above. Furthermore, to get the sixth-order parabola, we require
moments up to and including the sixth, and as Pearson (:05 and else-
where) has shown, the probable errors of the higher moments rapidly

66 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

become very large indeed. Each entry in column 5 of table 34 was
taken as the ordinate (y) at a given position (x) . As there was no appa-
rent biological reason for weighting differently the whorls in different
positions, all whorls were given equal weight. The method of fitting
was that of moments given by Pearson ( :02) . Since it is necessary in
this method to have an odd number of whorls, the first 29 were dealt
with instead of the whole 30. The origin was taken at a; = 15. The
observations were considered to give a system of trapezia, and the proper
corrective terms for the moments of such a system as given by Pearson
( :02, p. 8) were used. The range is from 1 to 29, or we have

21 = 28, or l = U.
For the quantities \ = -yf^, where z^', is the s^^ moment about the ori-
gin and I is half the total range, the following values were obtained,

\ = 0.030132 K = 0.193246

A, = .326446 \= .015303

\,= .020219 \= .136507

Further, we have

Vo ^ 9.3454.

From these values the following series of parabolas was obtained,
in which x is measured from the mid-range (= 15) .

(I) y = 9.3454|l + 0.090397 ( j) \

(II) y = 9. 3454 |l. 025828 + 0.090397 (|) - 0.077483 (^ ) ^l

(III) ?/ = 9.3454|l.025828+0. 034226 (j) —0.077483 (~) %0.093608;(j) |

(IV) 2/=9. 3454 1 1.013268+0.034226 (^) +0.048110 (|) % 0.093608 (j) ^-0.146524 (|) H

^v) 2/=9.3454|l.013268+0.036131 {j ) + 0.048110 ( j) Vo.084725 (|) ^-0.146254 (|) +

0.008002 (^) ^1

(VI) y = 9. 3454! 1.029148+0. 036131 (y) + 0.285372( |) V 0.084725 (|) %

0.853919 [j) V 0.008002 (|) ^—0.733659 (|)

In order to show the nature of the results with these parabolas fig.
11 bis has been prepared. This gives the observations and three
of the higher-order parabolas (namely, the third, fifth, and sixth) . It

POSITION REGRESSION— PRIMARY BRANCHES.

Mean leaf number

67

seemed hardly worth while to plot
the lower-order curves, because
if the higher ones fail to represent
the data, it is reasonably certain
that the lower will, and further,
six superimposed curves on the
same data tend to make a rather
confused diagram. As will be
seen from equations iv and V
above , the fourth and fifth order
curves are sensibly identical, and
hence it is unnecessary to plot
both of them.

Examining these curves, we
see that they signally fail to give
as good a result as we want and
have reason to expect. The third
and fifth order curves do not give
at all the form of the curve at the
lower end of the range. This is
clearly the most important part of
the curve and a failure to give a
good graduation here is fatal and
at once throws out of court the
lower-order curves. The sixth-
order curve does somewhat better,
especially at the start of the range,
but it is evident that the fit is a
purely artificial one and does not
help us any towards the law of
growth we are seeking. In the
first place, this curve has alto-
gether too many points of inflexion
to correspond to the biological
facts. It is, in a word, fitting
itself to the "errors" rather than
to the general sweep of the obser-
vations. The artificial character
of the graduation is well brought
out at the upper end of the range,
where the curve takes a very
sharp and sudden turn downward,
corresponding to nothing in the
observations. We may be quite
sure that the correct expression

O

■t> cc

r) in C

>

' to

rJ <y

-

•<

"ij^

s

N

-^

^

OJ

■>

"* V-

•t>

\

.

01

^

>

CD

V.

w

SI

5

i'l

=

'i

;3

5

u

Ul

;l\

3

; 11/

o _

-+1 en

s

-1

■A

■ s

w

<

i
(1

Cd

i

S

^

'it.

K

i'r

b>

IV

f

:y

(2

1^

1
I
1

'\ 1

N

U1

1
1

0.

/

-

N

^

S

/

s

y

g

1

Fig. 11 6i.s.— Regression line and fitted parabolas,
showing change of mean leaf-number with po-
sition. Primary branches, Series I, II, and III
combined. Observations, » ; third

order parabola, — — — — — — ; fifth-order parabola,

^..>...».^.«^ ; sixth-order parabola,— •^.^•—•.

68 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

for the law of growth here will be such a one that it will give reasonable
results for extrapolation at the upper end of the range. That is to say,
the same curve which gives a good graduation for the first 29 whorls,
ought, if it is to have any biological validity, to give a reasonable
result when extended to the 30th and further whorls. It is clear that
none of these parabolas is at all satisfactory in this respect. The gen-
eral failure of parabolas to give good extrapolation results, though they
may be entirely satisfactory for interpolation, has been recently empha-
sized by Miss Perrin ( :04) . The results obtained in the present case
form an excellent confirmation of her general position. On the whole
these results show very clearly that to put

<^ (x) = Co + c^x + CiX^ + CsO;^ .... cX

is inadequate in the present case. The form of the function which
expresses the true relation between size of whorl and its position is
something different from this.

As has been mentioned above, inspection of the regression lines shows
that while the mean number of leaves per whorl increases the farther
out we go on the branch, yet at the same time the increment at each
successive whorl diminishes. This at once suggests another hypothesis
regarding the function. If we let dy denote a small change in mean
leaf -number and dx a small change in position, we may assume that

^=1 .const.

or in other words that the rate of increase in mean leaf -number varies
inversely as the position on the axis. This leads at once to a logarithmic
curve, which may be put in the following form.

2/ = A + C log a?,

where A and C are constants to be determined from the data, and, as
before, y and x measure respectively leaf-number per whorl and posi-
tion. In order to test the worth of our assumption it was decided to
try fitting a curve of this type.

The same data as before were used. Before beginning the actual
calculation of the constants 'for the fitting, however, a graphical ''first
smooth" of the observations given in column 5 of table 34 was made on
a large scale and the smoothed ordinates read off. The values so
obtained furnished the raw material for the actual fitting. As before,
all the ordinates were given equal weight. Since we may obviously
take the origin of y (i. e. , of the ordinates) anywhere we please, it was
for practical reasons taken at 7 leaves. So that in the actual work each

POSITION REGRESSION— PRIMARY BRANCHES. 69

ordinate was expressed as a deviation (in excess or defect) from 7.
The origin of x was taken at 0, i.e., at the proximal end of the branch.
A curve of the type

1) = A-\r C (log X)

was then fitted by the method of least squares.

It gave on the whole a very good graduation; so good as to indicate
clearly that one was on the right track. It signally failed, however, to
give a good result at the lower end of the range. It bent altogether too
gradually at the start to represent the facts. Weighting the ordinates
with their observed frequencies did not help matters at all. While in this
way a better fit at the start of the range was obtained, it was at the
expense of getting very improbable values for whorls beyond the 10th or
12th. The results clearly showed, however, that some logarithmic curve
offered the solution of the problem. It only remained to find the proper
logarithmic curve.

The next assumption to be tested was that the true law of growth
was such that

dy \ .

-r~ = const.

ax x—o.

or, in other words, that the rate of increase in mean leaf-number varied
inversely as the position measured from a fixed point (a) on the axis.
This leads to a curve of the form

y = A + C log (x — o).

In fitting this curve a value of 0.8 for the constant « was first found by
a rough method of approximation. Then putting this value of 0.8 for a
into the equation, the constants A and C were found by the method of
least squares. This gave a first approximation to the curve. The next
step was to proceed to get a better fit by modifying all the constants by
small amounts, the modifying terms being calculated by the method of
least squares. The final equation determined was

y = 0.9520 + 1.3608 log (x - 0.8015)

Remembering that y has been measured from 7, we have for the
final result

Y = 7.9520 + 1.3608 log {x - 0.8015) .... I

where Y denotes the actual mean number of leaves per whorl and x the
position of the whorl. Calculating the ordinates of this curve corre-
sponding to the successive ordinal positions of the whorls, we have the
series of values given in the second column of table 35, against which
are put for comparison the actually observed values.

70

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

The graduation is clearly a remarkably good one. A mean error of
0.0004 between observation and theory, when we are dealing with 30
ordinates is clearly as low as we could expect to get, considering the
probable errors to which the individual observations are subject. There
can be no doubt that we have found the mathematical expression of the
law according to which growth in the character under consideration
takes place.

Tab-le 35.— Comparison of observed and calculated mean leaf-number for successive
whorls of primary branches. Series I, II, and III combined. Logarithmic curve.

Position
of whorl.

Oljserved
mean leaf-
number.

Calculated
mean leaf-
number.

Ditter-
ence.

Pcsition
of whorl.

Observed

mean leaf-

uumber.

Calculated
mean leaf-
number.

Differ-
ence.

1

7.029

7.004

+0.025

16

9.529

9.560

—

.031

2

7.970

8.059

- .089

17

9.533

9.600

—

.067

3

8.461

8.418

+ .043

18

9.143

9.633

—

490

4

8.559

8.639

- .080

19

9.600

9.667

—

.067

5

8.741

8.800

- .059

20

9.333

9.698

—

.365

6

8.966

8.926

+ .040

21

10.000

9.728

+

.272

7

9.024

9.030

— .006

22

9.571

9.757

—

.186

8

9.176

9.119

+ .057

23

10.000

9.784

+

.216

9

9.200

9.195

+ .005

24

10.167

9.810

+

.357

10

9.241

9.264

— .023

25

10.000

9.835

+

.165

11

9.425

9.324

+ .101

26

9.667

9.859

—

.192

12

9.429

9.380

+ .049

27

9.500

9.882

—

.382

13

9.481

9.430

+ .051

28

10.000

9.904

+

.096

14';

9.667

9.477

+ .190

29

10.000

9.926

+

.074

15:

9.773

9.520

+ .253

30

10.000

9.946

+

+

.054
.011

Sum of d
Average

deviation p

er observa

-t-

.00037

The curve and the observations are shown in plate i.

This result I believe to be of very considerable interest and signifi-
cance from several points of view. In the first place, it gives us a pre-
cise and unique formulation of a fundamental law of growth and differ-
entiation in Ceratophyllum. We now know the nature of the change in
successively formed whorls on the growing branch. Before proceeding
to state this law in words it should be remembered that the differential
coefficient of our equation is

dy _ 1
dx X —

where the constant is, of course,

C

const.

log,, 10
Or the law of change in successively formed whorls on primary branches
in Ceratophyllum may be stated in the following way: The mean num-

POSITION REGRESSION— PRIMARY BRANCHES. 71

ber of leaves per whorl increases with each successive whorl, and in such
a luay that not only does the absolute increment diminish, but also the
rate of increase diminishes, as the ordinal number of the whorl meas-
ured from a fixed point increases. Or, put more briefly, the rate of
increase in leaf-number at any given point as we go out on the branch
varies inversely as the number of whorls which separate the given point
from a fixed point, the a. of our equation.

As a consequence of the fact that the rate of increase in leaf-number
constantly diminishes as we go out on the branch, it is evident that the
actual number of leaves per whorl observed on any branch becomes
practically almost constant after the first 10 to 15 whorls. Even though
there be a tendency to increase, the rate is so slow that in discrete variates
such as we have here it would only be detected w^ith very large numbers,
while with ordinary numbers likely to be met in practice it would appear
that there was a constant number of leaves. To show how slow the rate
of change is we may determine at what position on primary branches
the mean number of leaves would be 11, that is, one more than what
we find as the highest mean number on the plants here dealt with.
Put in another way, we may determine how many whorls would have to
be successively formed in accordance with the law of growth which
holds up to the 30th whorl before the mean number of leaves per whorl
would become 11. To do this we have merely to substitute 11 for Yin our
equation and solve for x. Doing this we have

log {x - 0. 8015) = ^-^5^5 = 2 . 23977

whence

a; =177.5

or in round numbers, the mean number of leaves per whorl will not
become 11 until the 177th or 178th whorl is reached! But the facts pre-
sented in the last section show that there is comparatively little diver-
gence in the number of whorls to the branch (i.e., the length of the
branch) in plants collected from different localities, provided they are
taken at roughly the same period of the growing season. All show
about the same number of whorls to the branch, and this number is very
far short of anything like 100, even. There is no evidence whatever
from our material that Ceratophyllum in a state of nature ever attains
to such size as to have 175 whorls, or anything approaching that number,
on primary branches. Our material includes several large plants, so
that it can not be maintained that we are dealing with smaller-sized
individuals than usually occur. In particular, plant 2 of Series IV
was one of the largest Ceratophyllum plants I have ever seen. Yet, as

72

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

has been seen in all our material, 30 whorls is the maximum for primary-
branch length. We must conclude, then, I think, that under natural
conditions the mean leaf-number per whorl becomes practically constant
after the formation of from 15 to 20 whorls. This number is 10. To
reach a mean leaf-number of 11 the branches must have more than 175
whorls, while to raise the mean to 12 would require the occurrence of
primary branches with more than 900 whorls each! Such an occur-
rence is of course far out of the bounds of reasonable expectation.

Table 36. — Observed and calculated mean leaf-number in successive whorls on
primary branches. Series IV.

Position
of wborl.

Observed
mean leaf-
number.

Calculated
mean leaf-
number.

Differ- 1
ence.

Position
of whorl.

Observed
mean leaf-
number.

Calculated
mean leaf-
number.

Differ-
ence.

1

7.441
8.453
8.716
8.988
9.130
9.211
9.313
9.491
9.550

7.304
8.359
8.718
8.939
9.100
9.226
9.330
9.419
9.495

-f 0.137

+ .094

— .002
+ .048
+ .030

— .015

— .017
+ .072
+ .055

10

11

9.424
9.360
9.875
9.714
9.750
9.900

9.564
9.624
9.680
9.730
9.777
9.820

-0.140

- .264
+ .195

- .016

- .027
+ .080

2

3

12

4

13

5

14

6

15

7

8

Sum of d(
Average
vation

+ .230
+ .015

9

deviation per obser-

Another matter which demands consideration is this: We have
reached our generalized law of growth from a study of plants collected
at one place, namely. Carp Lake. Do plants from other localities follow
the same law of growth? I think it is evident on general grounds that
this is altogether likely to be the case, and table 34 and fig. 11 show
very clearly that it is so. The law of the change in mean leaf-number
with successive whorl formation must be the same to lead to such par-
allel results as we find on comparing any two series, as for example.
Series I with Series IV or Series VI, etc. I propose to show in another
way, however, that the law of growth which we have deduced is general
for Ceratophyllum. A glance at fig. 11 indicates that the most marked
difference between Series IV and Series I, II, or III in respect to the
regression of leaf-number on position is that for corresponding positions
the mean leaf -number is in general higher in Series IV than in the
other three series. Now, suppose that without in the slightest changing
the shape of our growth-curve, we simply change its position by moving
it up on the y axis only 0.3 leaf. This will of course be done by simply
increasing the value of the constant A by that amount. The result is
shown in table 36 and graphically in fig. 12. The first column in the
table and the zigzag line give the actually observed values for the mean

POSITION REGRESSION— PRIMARY BRANCHES.

73

leaf-number in the first 15 whorls on primary branches in Series IV,
as given in table 36. The data are not carried farther than the 15th
whorl, because from that point the material is too meager to yield very
reliable means. This will be evident by examining table 30. The contin-
uous curve is the graph of

r = 8.2520 + 1.3608 log {x — 0.8015)

the ordinates of which are given in the second column of the table.
That is, it is our first growth curve, changed only by a uniform addition
of 0.3 leaf in the absolute size of every whorl.

10.2

9.8

9.4-

9.0

u

E 8.6

D
C

ro

8.2

7.8

7.4

7.0

}

'--^. -

^

^^

"""^

.--■

/

__ _^ii

>^ '*

^

i^

r

y

^

'i'

jl

/•

jl

1
1

ll

7 8 9 10

Position of whorl

12

14

15

Fig. 12.— Regression of leaf-number on position and the fitted curve for Series IV. The ordinates
give mean leaf-number and the abscissas position. ^^— — observation; theory.

It will be seen that the agreement between the actual observations
and the values predicted by the growth curve is extraordinarily close.
Of course there is some irregularity in the observations after the
10th whorl, but this is only to be expected when we remember that the
means here are based upon comparatively few observations. Even
at the worst the maximum deviation of observation from prediction is
only a little over 0.2 leaf, while the average deviation is only 0.01 leaf.

This result seems to me to be of considerable importance, because,
in the determination of the shape of the growth curve the observatioroS which
we see it here expressing so very closely were in no wise taken into account.

74 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

Table 37. — Correlation between leaf-number and position of secondary-branch whorls.

SERIES I.

Number

of leaves

per whorl.

Position of whorl on branch.

Total.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

6

17

20
10

47

2

4
15

7

28

2
1
7
11
1

22

1
1

2

11

1

16

"5

5
1

11

"i
1
3

1

6

"i
1

1

3

"i
2

3

"i
1

2

"i

'"i
2

"i
1

"i

1

"1
1

1

22
27
43
42
9

7

8

9

10

Total

143

SERIES 11.

5

2

30

18

8

2

2
12
24
13

1

3

14

17

6

5

19
3

3
12

4

1
8
4

3

1
2
2

"2

"i

i

"i

"i

2
32
33
56

79

28

1

6

7

8

9

10

11

Total

61

52

40

27

19

13

9

5

2

1

1

1

1

231

6

24
8
8

8

15

6

1

3

13

4

1

4

11

3

"i

i'o
3

1
7
3

1

6
3

"4
4

2

1
4

"2
2

"i

"i

24
19
34
60
28

7

8

9

10

Total

40

29

21

19

14

11

10

8

7

4

1

1

165

SERIES IV.

6

7
61
34

5

2

21
43
31

1

4
37
33

8

7
20
41

5

3

18

28
8

1
8

20

10

1

1
4

20
10

6

10
10

2
11
10

2

2

14

"2

12

1

"i
11

1

1
9

"i

7

"5

"i

3

"2

1

9

98

178

205

125

2

7

8

9

10

11

Total

107

98

82

73

57

40

35

26

23

18

15

12

11

8

5

4

2

1

617

SERIES I, II, AND III COMBINED.

5

71
4€
2€

r

»

4
. 24
. 54

! 2e

I

t 5
\ I
. 24
> 41
11

1
2

11
41

7

1
27

1

S
IS

"2
13

7

2
g
6

2
2
7

1

2
3

1

3

"i

2

"2

2

78

79

133

181
65

1

6

7

8

9

10

11

Total

147 lOS

83

62

44

30

22

16

11

7

3

3

2

539

POSITION REGRESSION— SECONDARY BRANCHES.

75

That is to say, we find that a mathematical function deduced solely from
data given by Ceratophyllum plants collected at Carp Lake expresses
perfectly the method of growth of Ceratophyllum plants collected at Ann
Arbor. This seems to me to amount to a demonstration that our growth
curve is an expression of a real and fundamental morphogenetic law
in Ceratophyllum and not a mere chance result, or due to any skillful
figure juggling.

By changing the position of the curve slightly in the opposite direc-
tion we obtain a very fair graduation of the regression in Series V,
though on account of the fact that Series V and VI include spring
plants in which we have the results of a part of two seasons' growth, as
has been pointed out above (p. 15), the fit in these cases is not so good
as in the other series.

The result shown in fig. 12 enables us to reach a further interesting
conclusion. We see, namely, that in our fundamental growth equation

Y= A + Clog {x — a)

it is the constant A which is affected by environmental differences.
That is, the absolute size of the elements of the developing system given by
a Ceratophyllum branch is modified by environmental differences, but the
law which describes the proportionate differentiation of the elements is
independent of the environmental history of the plant. Thus we are able
by statistical analysis to separate clearly and definitely the effects of
external environmental factors and internal form-determining factors
in this case. The constant A takes different values in populations
living in different environments, while the portion Clog {x — «) remains
unaltered, or is only very slightly altered, and we may therefore look
upon A as the "environmental constant."

Table 38.— Correlation between leaf-number and position of secondary -branch whorls.

Series.

XL.
III.
IV

I, II, and III combined.

0.601 ±0.036
.688± .023
.692± .027
.650± .016
.671± .016

0.712±0.028*
.801± .016
.816± .018
.727± .013
.784d= .011

Table.

37
37
37
37
37

»As before, the probable error of the correlation ratio V is calculated from the short formula.

So far we have been dealing with primary-branch whorls in the
discussion of the regression of leaf-number on position. It remains to
determine in how far the same relations which we have found for
primary branches hold for other divisions of the plant. We may turn
first to the consideration of whorls on secondary branches.

76

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

The correlation tables showing the relation between number of leaves
in the whorl and position on the branch for this division of the plant
are given in table 37.

The correlation coefficients (r) and correlation ratios (v) deduced from
the preceding table are given in table 38. From this table the following
points are to be noted:

(a) As in the case of the primary branches there is a high correlation
between the number of leaves in the whorl and position on the branch.
Comparing the values of the correlation constants for secondary branches
with those for primary branches given in table 33 above, we see that

Table 39. — Mean number of leaves in successive whorls on secondary branches.

Position
of whorl.

Mean number of leaves.

Series I.

Series II.

Series III.

Series IV.

Series I, II,

and III
combined.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

6.851
7.964
8.364
8.625
8.636
8.667
9.000
8.667
9.500
9.000

10.000
9.000

10.000

6.633

7.981

8.650

8.926

9.053

9.231

9.333

9.200

10.000

11.000

10.000

10.000

10.000

6.600
7.931
8.842
8.071
9.182
9.200
9.200
9.500
9.286
9.500
10.000
10.000

7.346
8.082
8.549
8.603
8.719
9.050
9.114
9.154
9.348
9.667
9.800
9.917
9.727
9.875

10.000
9.750

10.000
8.000

6.694
7.963
8.651
8.823
8.954
9.100
9.227
9.250
9.455
9.571

10.000
9.667

10.000

they are without exception higher in the case of the secondaries. The
reason for this is found in part in the fact that the secondaries are
shorter than the primary branches. Consequently they do not have at the
distal ends of the long branches the long string of whorls with, as has been
shown above (p. 62) , a very nearly constant number of leaves. These
distal whorls, from about 15 on, operate in the case of the primaries to
lower the correlations. To get a fair test of the relative degree of the
positional correlations in the two orders of branches we must compare
branches of roughly the same length. From table 37 we see that the
secondary branches of Series I, II, and III run up to 13 whorls, while
in Series IV they go to 18 whorls. If, now, we calculate the coeffi-
cient of correlation between leaf -number and position, for the first 13

POSITION REGRESSION— SECONDARY BRANCHES.

77

Mean leaf number

^1 OD 00 00

b > to

<T>

whorls on primary
branches in Series I,
II, and III combined,
we find a value

r = 0.555.

In the case of Series
IV, since the primary-
branches (cf. table
30) only extend to 24
whorls, and the total
frequency beyond
the 18th whorl is
only 13, it is obvious
that for all practical
purposes the con-
stants given in table ^
33 for the primary- ^.
branch correlation©'
are directly compar- o
able with those for
the secondaries giv-
en in table 38. Com- —
paring the primary-
branch values with
the corresponding
ones for secondary
branches in table 38
it is clear that apart
from the reduction
of the correlation
caused by the great
length of primary
branches there is
still a higher corre-
lation between leaf-
number and ordinal
position of the whorl
in secondary branch- . . ^,

.... Fig. 13.— Regression of leaf-number on position in secondary-brancn
eStnan m primaries. whorls. series I, II, and in combined, o-. ; Series IV,

The significance of • •

|5

o

78

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

this fact will be pointed out later, when certain other results are in
hand.

(6) There is the same uniformity in the correlations for the different
series as was observed in the case of the primary branches.

(c) It is obvious from the values of r and v that the regressions are
not linear.

The precise character of the regression of the leaf-number on posi-
tion is given in table 39. In this we have for each of the first four series
and the combined Series I, II, and III, the mean number of leaves for
each position on the branch.

It is at once apparent that the secondaries start with a smaller mean
number of leaves to the whorl than do primaries, but that the increase in
the first few whorls from the origin is more rapid than in primary
branches. Unfortunately in the case of secondary branches the number
of observations on which the means are based gets small very quickly,
so that the results begin to be irregular even before we reach the 10th
whorl. The figures in the table show these points, but they are brought
out more clearly in the diagrams in fig. 13, which show graphically the
regression lines for Series I, II, III, and IV.

Table 40. — Comparison of observed and calculated mean leaf-num,ber for successive
whorls of secondary branches. Series I, II, and III combined.

Po.sltioa
of whorl.

Observed
mean leaf-
number.

Calculated
mean leaf-
number.

Difference.

Position
of whorl.

Observed
mean leaf-
number.

Calculated
mean leaf-
number.

Difference.

1

6.694
7.963
8.641
8.823
8.954
9.100
9.227
9.250

6.630

8.013
8.485
8.778
8.991
9.158
9.295
9. 412

+0.064

— .050
+ .166
+ .045

— .037

— .058

— .068 i

— .162

9

9.455
9.571

10.000
9.667

10.000

9.513

9.604
9.685
9.758
9.825

—0.058

- .033
+ .315

— .091
+ .175

2

10

11

12

13

3

4

5

R

7

Sum of deviations i + .208

Average deviation oer obser-!

8

vation..

+ .016

From these diagrams we see that the general features of the regres-
sion are essentially the same in secondary as in primary branches.
There is clearly a functional relation of very much the same sort between
the number of leaves to the whorl and the order of formation of the
whorl. To determine the nature of this functional relation, we may in
the light of our experience with primary branches assume at once that
it is of the form

y = A -\r Clog {x — o.)

and proceed to fit a curve of this type to the data by the method followed
in the former case. This was done, using as material the combined

POSITION REGRESSION— SECONDARY BRANCHES.

79

Series I, II, and III, as before. In this case, however, the origin of y
was taken at 6. 6 instead of at 7, as in the other case. The reason why
this was done will be obvious to the mathematical reader.
The curve obtained was

y = 1.2662 + 1.8024 log {x - 0.7939)
whence, taking the origin of ?/ at 0 we have finally

Y = 7.8662 + 1.8024 log {x — 0.7939) II

where Y indicates mean number of leaves to the whorl and x indicates
ordinal position on the branch. The observed and calculated ordinates
are given in table 40. The actual observations and the fitted curve are
shown in fig. 14. Considering the paucity of observations at the upper
end of the range, abetter graduation could not reasonably be expected.
We conclude, then, that the same type of curve expresses the regression
in secondary branches as in primaries. Or, in other words, the differen-
tiation of successively formed whorls follows the same general law in
both primary and secondary branches.

9.6
9.2

E

>♦-
« 8.0

C

a
V 7.6

7.2

6.8

I 2 3 •* 5 6 7 8 9 "0 II 12 13

Position of whorl

Fig. 14.— RogresHion of leaf-numbor on position In secondary branches, Kerios I, II,
and III combined. ObHervationn, • '— ; lltted curve,

Having seen that the growth curve is fundamentally the same for
the two orders of branches, we may next examine the differences in the
two cases, and see what they signify. We may first notice how the
increment in mean leaf-number per unit advance in position compares
in the case of secondaries with what it is in primaries, for corresponding

■'■

A

— -N

/'

--:3

^^■'

^"It

f^

/

y

/

//

f

//
//

y

80 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

whorls in the two cases. The value of the constant C in the case of the
secondaries (=1.8024) shows us at once that the increment in Yper
unit change in x will be greater in the secondary-branch whorls than
in the primary. That this in fact is the case is obvious enough to the
eye if we compare fig. 14, with plate I. The upward slope of the secondary
curve is much more rapid than is that of the primary curve. The fact
may be shown in another way. The successive increments in mean leaf-
number for the first 10 whorls of primary and secondary branches are
given in table 41.

Table 41. — Increment in leaf-number in successive whorls.

Increment in mean
leaf-number between-

On primary
branches.

On secondary
branches.

1.055
.359
.221
.161
.126
.104
.089
.076
.069

1.383
.472
.293
.213
.167
.137
.117
.101
.091

Second and third whorls

Third and fmirth whorls

Fourth and fifth whorls

Fifth and sixth whorls

Sixth and seventh whorls

Seventh and eighth whorls

Eighth and ninth whorls

Ninth and tenth whorls

It is evident that the increments are larger on the secondary branches.
The nature of the change in the increment is very clearly shown if we
plot the data from table 41 to the same scale for the two orders of
branches. This has been done with the result shown in fig. 15.

We see that though the secondary branches start lower, they attain
any designated mean leaf- number in a smaller number of whorls. Thus
while in the case of the primary branches a mean leaf-number per whorl
of 11 would only be reached after the formation of 175 successive
whorls, the secondaries would reach this mean after the formation of
only 56 whorls in round numbers (55.58 exactly). That is to say, if we
may speak in terms of analogy, the morphogenetic mechanism attains
its results more rapidly in the case of the secondary branches. This is
an expression of the fact noted above (p. 77) that the positional corre-
lation is higher in secondary than in primary branches. The law
of growth in the plant is, as we have seen, that the mean leaf-
number per whorl increases with successive whorl formation, the rate
of increase at any point varying inversely as the number of whorls
which have been formed up to that point. Clearly the production of a
maximum mean leaf-number may be regarded as the "end" towards
which the morphogenetic process here tends. But the secondaries
attain this "end" more quickly than do the primaries; that is, with the
formation of fewer successive whorls. If, merely as an analogy, we

^

^

.^

^

________

^

V

y-

X

/

^

^

— "

\

/

^

^

/

^

^

/

//

//

f

position of whor!

1 line and fitted logarithmic curve, showing the change in
Series I, II, and III combined. The ordinates give r

whorl formation in primary branches,
pbsitioii on the axis.

POSITION REGRESSION— SECONDARY BRANCHES.

81

may compare the developing branches of a Ceratophyllum plant with a
machine, of which the whoi-ls are the product, we may say that the
machine works more smoothly the longer it runs.

The fact that the increments in mean leaf -number with successive
whorl formation are greater in the case of secondary branches is shown
in the growth equation by the higher value of the constant C. This
indicates that C may be regarded as the constant which expresses the
action of the internal formative factors, in contradistinction to A, which,
as we have seen above (p. 75) in the case of primary branches, expresses
the effect of external environmental conditions. We should expect C,
then, to remain practically constant for the same division of the plant

C 0.8

E

u

<J 0.6

T

\
1 \
\ \
\ \

\ \

\ \
\ \
\ \

\ \
\ \

N

s. N

"-^^

^^""^

5-6

Whorls

9«-IO

Fig. 15.— Curves comparing growth of secondary and primary branch whorls. The ordinates
give the increments in mean leaf-number as we pass to successive whorls, the positions
being given by the abscissas. Primary-branch whorls ; secondary-branch whorls

(i.e., primary or secondary branches) without regard to external influ-
ences, while A changes with change of environmental conditions. In
other words, it would appear that external conditions affect chiefly the
absolute size of the elements of the plant while the internal morphoge-
netic factors which determine the way in which the proportionate
differentiation of the elements shall occur are practically independent
of environmental change. These relations hold for primary branches
very clearly, so far as the present material goes. We might test the
matter for the secondaries in the same manner as before by determining
whether with an appropriate change of the constant A in the secondary-
branch growth equation (II) which has been deduced from Carp Lake

82 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

data solely, we could get a reasonable graduation of Ann Arbor material.
Unfortunately the Series IV data for secondaries are so meager that the
results are rather irreg-ular. The number of plants is not large enough
to get rid of the disturbing influence of the individuality of a single
large specimen. This especially shows itself in the first whorls of sec-
ondary branches in this series, where we get an abnormally high mean,
due to the marked tendency for one of the plants in this series to have
exactly 7 leaves in these whorls. If, however, we change the posi-
tion of the growth-curve (equation II) by lowering it 0.1 of a leaf
on the y axis, it represents the data for Series IV, with the exception
of the first whorl, with very considerable accuracy. Thus, taking the
first 10 whorls (after this the observations are too few to give reliable
means) , I find the total deviation between observation and theory to be
+0.253, giving an average deviation per observation of 0.025. This is
a sufliciently close agreement to justify the contentions made that (a)
environmental differences affect chiefly the absolute size of the elements,
and (6) that the essential character of the differentiation with growth
is practically independent of environmental influences. We could of
course get a better graduation of the Series IV material by specially
determining all the constants of the curve anew from that data, but
what I have tried to show is that we get for all practical purposes a
sufficiently good fit by simply shifting slightly the position of the curve
deduced from the other series, this alteration being merely.to take into
account the difference in absolute size between the two sets of plants.

Summing up, then, we see that the same law of differentiation of
whorls with growth holds jor secondary as for primary branches. The
chief difference in the two cases is that the change in successively
formed whorls is, at any given point, greater in secondary than in
primary branches.

We have now to consider the question of whether the same law
which we have found for secondary and primary branches holds for the
other divisions of the plant (main stem and tertiary branches) . We
may take up first the main stem. It is of course to be expected that
the relations are not essentially different here. Unfortunately it is
impossible to test the matter directly in the way which has been used
for the other divisions, because of lack of material. A plant may have
50 or more primary branches, but of course it has only one main stem.
Therefore, to get satisfactory means for main-stem whorls in particular
positions it would be necessary to have a very considerable number of
plants. There is a further difliculty in the case of the main stem.
Ordinarily the plants have several inches of the lower (proximal) end
of the main stem embedded in the soft debris and mud of the bottom.

POSITION REGRESSION— MAIN STEM.

83

In almost every case one finds that the whorls which were originally
present in this region of the plant have been in whole or in part destroyed,
so that it becomes quite impossible to tell how many leaves they bore.
Only scars and short stumps of leaves remain to indicate the original
presence of whorls. It is probable that the destruction of the leaves in
this region is due in the main to the action of plant-feeding aquatic
animals. In any event it is impossible to get a determination of the
regression of leaf -number on position in the first-formed whorls of the
main stem. The extent to which this destruction of leaves has gone
varies very much in different plants. In some cases only a few whorls
are found to be mutilated, while in other plants a considerable number
are completely destroyed.

Table 42. — Correlation between leaf-number and position in main-stem ivhorls.
Series I, II, and III combined.

Leaves

per
whorl.

Position of whorl.

1-10

11-20

21-30

31-40

41-50

51-60

61-70

71-80

81-90

Total.

6

7

8

9

10

11

12

Total...

2
1

5
21
15

2
3
5
24
22
3
1

3

7
18
30
10

1

1

1

3

16

47
6

"i

3

15
35

6

"i

10
17

4

"i

2

17

2

8

4

5

9

26

108

195

29

2

44

60

69

74

60

33

20

10

4

374

r = 0.238±0.033.

In order to show the nature of our material, so far as it goes, in this
respect, table 42 has been prepared. This gives for Series I, II, and III
combined the actually observed frequencies of main-stem whorls with
various leaf-numbers, in different positions. Position is indicated in
the same way here as in the case of the branches; that is, the most
proximal whorl on the stem is numbered 1, the whorl next it 2, and so
on in regular succession. In the cases where the main stem has
dichotomized the whorls on each subdivision are numbered in order, both
beginning at the same point. For example, if the main stem divided
after the formation of the 25th whorl, the first whorl on each of the
subdivisions would be numbered 26, since both occupy the same position
relative to the main axis. Since the whole number of entries for main-
stem whorls is so small, I have not entered each whorl separately in the
table here shown, but have combined them in groups of 10. This, of
course, would not be legitimate if we expected to get very fine results,
but the material is too scanty to give close results in any event, and by
arranging it in this way we are able to see better the general trend.

84

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

It is seen at once by mere inspection of the table that there is clearly
a change in the mean number of leaves per whorl, with successive whorl
formation. The precise amount of this change is shown in table 43,
which gives the actual mean for each array of table 42.

Table A3.— Regression of leaf-number on position in main-stem whorls.

Position
of whorl.

1-10
11-20
21-30

Mean number

of leaves

per whorl.

9.045
9.233
9.580

Position
of whorl.

31-40
41-50
51-60

Mean number
of leaves
per whorl.

9.689
9.700
9.697

Position
of whorl.

61-70
71-80
81-90

Mean number
of leaves
per whorl.

9.800

9.800

10.000

This result shows that in main-stem whorls, just as in primary and
secondary branch whorls, there is a definite relation between the number of
leaves and the position on the axis. The later-formed whorls tend to have
a higher mean number of leaves than those formed earlier. The data of
table 43 are shown graphically in fig. 16.

The increase in mean leaf-number in main-stem whorls is clearly very
gradual. We have now to determine whether it is according to the same
law which holds for other divisions of the plant. On account of the paucity
of material we can not get at this directly as we have in the other cases, but
must resort to an indirect method. For reasons which have been stated
above, we know nothing about how the main-stem curve starts. The first
array in table 42 (whorls 1 to 10) is unfortunately largely composed of
whorls above the fifth, the lower ones being in most cases broken. The
consequence is that the mean for this group has an unduly high value.
Further, there can be no doubt that the main-stem curve actually starts at
a considerably higher level than does that for primary branches, so that
it would be obviously unfair to compare the absolute values predicted
from our primary-branch growth curve (equation I) with main-stem
means. The fairest way of approaching the problem seems to me to
compare the increment in mean leaf-number occurring between whorls in
different situations on the main stem with the increment between similarly
situated whorls on primary branches as predicted by equation I. If
there is reasonable agreement between the observed and predicted
increments we may fairly say that the growth and differentiation of
leaf -whorls follows the same law in the main stem and the branches.
The data for this comparison are given in table 44.

On the whole these results are quite satisfactory. Of course there
is wide divergence between observed and predicted results at the two
ends of the series. At the lower end (whorls 5 to 15) the observed incre-
ment is much too small, but, as has been pointed out, our first mean

POSITION REGRESSION— MAIN STEM.

85

gives no adequate idea of the actual condition at the proximal end of
the main stem. It is altogether too high, because of the destruction of
the earlier whorls in most cases. At the upper end (75 to 85 whorls) we

have only four whorls to base the
Mean leaf number mean on — an obviouslv insuffi-

OO tC lD P . ''

cient number. Leaving out of
account the upper and lower ends
of the series, we have quite as
good agreement between obser-
vation and theory as could be
expected. Or, we conclude that
the same latu of groivth describes
the change in mean leaf-number
in successively formed whorls,
whether the whorls are borne on
the main stem or on primary or
secondary lateral branches.

The discussion of main-stem
whorls has been based on the
data of Series I, II, and III. The
other series lead to exactly the
same result, and consequently
it does not seem necessary to
consider them in etail in this
connection.

We come finally to the consid-
eration of tertiary-branch whorls
from the standpoint of positional
differentiation. The same diffi-
culty is met here as in the case
of the main stem, namely, very
meager material. We have, how-
ever, sufficient data to get an idea
of the general conditions among
the whorls in this division of the
plant. Taking as before the
combined Series I, II, and III, we
have the observations set forth in
table 45. From this table I find
r = 0.603 ±: 0.059
V = 0.878 ± 0.021
We conclude, then, that just as in other divisions of the plant there
is a considerable degree of correlation between leaf-number and order

o o

O ui

J

izv:

Fig. 16— Regression of leaf-number on position in
main-stem whorls, Series I, II, and III com.
blned.

86

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

of formation of tertiary-branch whorls. We further note that the
correlation between these two characters is distinctly higher in tertiary-
branch whorls than in main-stem or primary-branch whorls. The cor-
relation coefficient is of about the same order of magnitude for tertiaries
as for secondaries, but the correlation ratio (which is the better measure
in these cases of skew correlation) is markedly higher for the tertiaries
than for secondaries (cf . table 38, supra) .

Table 44. Comparison of increments in mean leaf-number occurring between every

ten whorls on (a) the main stem, and (6) primary branches. Series I, II, and III
combined.

Increment in mean
leaf-number between—

Observed
on main

stem.

Predictedfrom

primary

branches

(equation I).

Difference.

^tVi anfl 1.5th whorl

0.188
.347
.109
.011
— .003
.103
.000
.200

0.720
.315
.203
.153
.121
.100
.085

+0.032

— .094

— .042

— .124
+ .003

— .085

Tith nnd 25th whorl

P'^th and 3,5th whorl

^'^tVi ^nH 4Bth whorl

Af>fV^ and Ftfiih vjhor]

f^f^tVi nnr] Pt^th whorl

fif^th nnri 7.^th whorl

I'^fh and 85th whorl

The change in mean leaf -number with advance in position is shown
in table 46, giving the means of the arrays of table 45. From this table
the following points are to be especially noted:

(a) The tertiary branches start at the first whorl with a lower mean
number of leaves than do either primaries or secondaries.

Table 45. — Correlation between leaf-number and position in whorls
on tertiary branches. Series I, II, and III combined.

Leaves per
whorl.

Position of whorl.

1

2

3

4

5

6

Total.

6

12
8
2

3

8
4

"i

"i

2

"i
1

"i

3

2

12
12
10
13
8

7

8

9

10

Total

22

15

7

5

4

2

55

(6) The rate of increase in mean leaf -number with successive whorl
formation, so far as we can judge from our present material, is much
more rapid in the case of the tertiaries than in the other branches. That
is, if we regard the production of a high mean leaf-number as the
"end" towards which the growth processes are tending, this "end" is

POSITION REGRESSION— TERTIARY BRANCHES.

87

apparently attained with the production of fewer whorls in the case
of tertiaries than in either of the lower-order branches.

Table 46. — Regression of leaf-number on position in
tertiary-branch whorls.

Position of
whorl.

Mean number
of leaves.

Position of
whorl.

Mean number
of leaves.

1
2
3

6.545
8.067
9.000

4
5
6

9.200

9.750

10.000

We have now to consider the question as to whether the differ-
entiation of successive whorls which exists here is in accord with the same
general type of growth curve which has been demonstrated for the other
branches. We can of course say a priori that it is altogether likely
that this is the case, but it is desirable to see just what result we get
from the actual data, even though they are small in point of numbers.
The data are too few to make it worth while to attempt to fit a special
curve for them, nor is it necessary to do this to bring out the point.
What we wish to find is whether in the case of the tertiaries the rate of
increase in mean leaf-number diminishes as the number of whorls formed
increases. It is clear that one simple test of this would be to determine
whether the ratio between the increments in leaf- number in successively
formed whorls diminishes. If it does, then clearly the rate of increase
is diminishing. Taking the first four whorls, we have the following
absolute increments in mean leaf -number:

Between first and second whorls 1.522

Between second and third whorls 933

Between third and fourth whorls 200

These yield the following ratios:

1:522 ~ ^-^^^

0:933 = ^-214

Or we conclude that, so far as we can form any judgment from the
present data, the rate of increase in mean leaf-number in tertiary-branch
whorls decreases as the number of successively formed whorls increases.
Or, in other words, it appears that the differentiation of the whorls on the
tertiary branches follows the same law which has been shown to hold for
the other axial divisions of the plant.

88 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

SUMMARY OF SECTION.

We may now put together in orderly form the chief results which
have been gained in this section of the paper. It has been shown that:

(a) There is a positive correlation between the number of leaves to
the whorl and position of the whorl on the plant, or, since in a plant
growing in the manner in which Ceratophyllum does, position on the
plant is determined by the order of the formation of the organ or
character under consideration, there is a correlation between the number
of leaves to the whorl and the ordinal rank in the process of successive
whorl formation.

(6) This correlation is considerable in amount. It is lowest for the
main-stem whorls, and increases steadily in the higher axial divisions
of the plant (primary, secondary, and tertiary branches).

(c) The regression of leaf -number on position is not linear, but log-
arithmic.

(d) The mean number of leaves to the whorl increases with succes-
sive whorl formation according to the equation

y = A -{- Clog {x — a)

where y denotes the mean number of leaves per whorl, x the position
of a whorl on an axial division of the plant, and^, C, and a'are constants.
Stated in words, the law of differentiation with growth in Ceratophyllum
is: The mean number of leaves per whorl increases with each successive
whorl, and in such a way that not only does the absolute increment in
each leaf-number diminish, but also the rate of increase diminishes as
the ordinal number\ofthe whorl, measured from a fixed point, increases.
This may be, for convenience, designated as i\\e first law of growth in
Ceratophyllum. It means, broadly speaking, that the form of any
particular whorl of a Ceratophyllum plant is a function (in the math-
ematical sense) of the number of whorls which have been produced
before it on the same axis.

(e) The same law of growth holds (with appropriate changes of the
constants) for all axial divisions of the plant (main stem, primary,
secondary, and tertiary branches) .

( f) The absolute size of the elements of the developing system given
by an axial division of a Ceratophyllum plant is modified by environ-
mental differences, but the proportional differentiation of the elements
is in accord with the same law in plants from different environments.

{g) The absolute increment in mean leaf-number between similarly
situated successive whorls is least on the main-stem, and increases
regularly in each of the higher axial divisions (primary, secondary, and

88

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SUMMARY OF SECTION. 89

tertiary branches) . That is to say, a given mean number of leaves per
whorl is produced with the formation of the least number of consecu-
tive whorls in the case of tertiary branches; more whorls are required
to attain the same result on secondaries, more still on primaries, and
most of all on the main stem.

Before going on to other subjects I wish to discuss certain points
brought out by the results of this section which could not well be taken
up till all the results for positional leaf differentiation were in hand.
In the first place, we note that from the correlation tables for the
characters, position, and leaf-number given in this section (tables 30-32,
37, 42, and 45) it is a simple matter to build up piece by piece the gross
frequency distributions for all whorls on the plant, such as are given
in an earlier section of the paper (e. g., table 1). In other words, the
arrays of the correlation tables form, so to speak, the dissected elements
of our earlier gross frequency distributions. It has been possible in the
present case to carry out this process of dissection of variation curves
with considerably greater completeness than has hitherto usually been
the case. The results of this procedure will, I believe, repay careful
study from several different points of view.

In order to bring out more clearly the general features of the analysis
of the frequency distributions, I have resorted to the'graphical method,
with the result shown in plate ii.

In plate li the whorls belonging to different divisions of the plant
are represented in different colors, main-stem whorls being given by
blue, primary-branch whorls by red, secondary-branch whorls by green,
and tertiary-branch whorls by yellow. The data are for Series I, II,
and III combined. The abscissas give number of leaves per whorl and
the ordinates, frequencies. Instead, however, of representing merely
the total height of the ordinates, the frequency of each size of leaf is
proportionately divided to show where whorls are located on the plant
as a whole and also on the individual axes. These proportional divisions
are indicated by the horizontal lines, and the areas they include
represent the frequency of whorls in the positions indicated by the
Arabic numerals. In order to avoid too great complexity in the diagram
the whorls have been to some extent grouped. In the case of the main
stem each 10 succeeding whorls were grouped together; in the primary
branches the 1st to the 10th whorls, inclusive, are given by single whorls,
and from that point to the end of the branch in groups of 5; in the
secondary branches the first 6 whorls are given singly, while the
remainder of the whorls (7th to 13th inclusive) are in a single group;
the first two tertiary-branch whorls are given singly, and the remainder
(3d to 6th, inclusive) together. The smooth curves are the graphs of
the equations given on pages 23 and 24.

90

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

The diagram brings out very sharply a number of points. In the
first place we see that, whatever the part of the plant, first whorls con-
tribute proportionately very largely towards the frequencies on the low
leaf -numbers. As we go out towards the distal end of the branch or
the main stem the proportionate frequency of 9 and 10 leaved whorls
increases. Thus if we compare the rectangle representing 7-leaved
whorls with that for 10-leaved whorls we see at once that the sub-
divisions of the first decrease for each division of the plant as we go
towards the top of the diagram, while the subdivisions of the latter
increase. This means of course that the skewness of the distributions
is changing in the manner schematically indicated in the following
diagram:

Small leaf-num-
ber per whorl
Proximal whorls,

") Positive )
■ j skewness. C

Leading
through var-
ious inter-
mediate stages,

Including

symmetry, to

extreme.

Positive

r High leaf-
„i,«^v,^.,o ■^ number per
skewness. I ^j^^^l_ Tistal

whorls.

A and C would represent the extreme conditions of proximal and
distal whorls, while B would represent one of a whole series of inter-
mediate stages passed through as we go from A to C. It will be under-
stood that this diagram is purely schematic, but fairly represents the
essential facts very closely. The same stages as those indicated may
indeed be seen in plate ll, or in the correlation tables (cf . supra) on
which it is based, while a careful examination of the data will leave no
doubt in the mind of the reader that the general trend of the frequency
distributions in respect to skewness is fundamentally that indicated.
We see, then, that homologous organs on the same individual plant run
through the whole gamut of skewness from positive (A), through perfect
symmetry* to negative (C) . The phenomenon of skew variation stands
forth in this case, free of doubtful interpretation through selection or
any similar factor, clearly and definitely as a phenomenon of growth. In

*Cf . for example the distribution for the third whorl on primary branches in Series
I, II, and III combined.

SUMMARY OF SECTION. 91

the face of facts of this kind it is difficult to understand how anyone
can be so firmly convinced of the Allgemeingilltigkeit of the normal or
Gaussian law, as some biologists still are [cf. for example, Ranke and
Greiner ( :04) ] . Skewness in variation is a very real biological phenom-
enon, which may be changed and modified, not only in degree, but in
direction, by various biological factors like growth, as, for example, in
the present case, or environmental influences,* etc.

The general features of our first law of growth are very well brought
out graphically by the diagram of plate li. Thus we see clearly that
for all divisions of the plant the pro'portionate frequency of whorls with
high leaf -numbers becomes greater the longer the plant grows, though
at a decreasing rate. This of course results in the increase in mean leaf-
number which has been discussed in this section of the paper.

It is also of some interest to see graphically the proportion of whorls
which the different axial regions contribute to the plant as a whole.
Primary branches manifestly outweigh any other part of the plant.

With a knowledge of the law of positional differentiation of whorls
we are able to interpret many of the facts regarding variation in Cera-
tophyllum which were before obscure. These interpretations will be
so obvious to anyone who has followed the results so far set forth that
it is not worth while to go over all of them in detail. A few cases
only need be cited as illustrations. Thus, on page 18 it was seen, when
the gross frequency distributions for Series I, II, and III were com-
pared, that "Series I has the lowest frequency of whorls with 10
leaves, Series II has about 3 per cent more 10-leaved whorls, while
Series III has 4 per cent more of such whorls than does Series I." The
explanation of this progressive difference is now clear. A reference to
page 13 will recall the fact that the dates of collection for Series I, II,
and III were respectively July 22, August 18, and August 25. The
proportionate frequency of 10-leaved whorls varies directly as the date
of collection, which is exactly what would be expected from our law of
growth, since as growth goes on there is an ever-increasing tendency
towards the addition of whorls with 10 or a higher number of leaves.

It has appeared at various points throughout the paper that the gross
frequency distributions for. Series V and VI were different in their
characteristics, both from all the other series considered and from each
other. Yet, as has been seen in this section and will be more fully
brought out later, these Series V and VI plants follow the same laws
of growth differentiation as do the others. The explanation of their
apparent abnormality is to be found principally in the fact that in the

*E. g., see Pearl (:06).

92 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

main they were plants which had wintered over without breaking up (cf .
p. 15, supra). When this happens and vigorous growth begins again
in the spring, we should expect to find the plants bearing branches* of
high average length and having an unusually large proportion of all
their whorls borne on primary branches. Reference to tables 25 and 26
shows exactly this to have been the case in Series VI. But on long
branches according to our law of growth there will be a great preponder-
ance of 10-leaved whorls. Hence we see the reason for the result
previously obtained, viz, ' 'The polygon for Series VI is not quite smooth
and regular, yet differs from all the others in having as the most frequent
class 10-leaved whorls. ' ' Series V differed from all the other series in its
gross frequency distributions in showing an unusually high proportionate
number of 7-leaved whorls. This arises from the fact that the two
large plants in this series (1 and 7) , when the spring growth began,
threw out an unusually large number of new lateral branches (second-
aries, tertiaries, and quaternaries). At the time the collections were
made these branches had not attained any considerable length. Hence
we have a preponderance of short branches in this series, and, as indi-
cated by the law of growth, we should in consequence expect a consid-
erable increase in the relative frequency of whorls with low leaf-number,
which is just what we find.

It hardly requires discussion to make it evident that the facts embodied
in our law of growth explain how the characteristic values and differ-
ences in the variation constants for different regions of the plant, as set
forth in an earlier section, arise. It was found that the older the portion
of the plant the higher was the mean number of whorls. But the reason
for this is now clear, since the older portions will have grown longer than
the younger and produced more whorls with high leaf-numbers. It will
be seen later that the differences in variation in different portions of
the plant are to be accounted for similarly.

We may turn now to another point which arises in connection with
positional differentiation.

*So far as I am able to judge from the present material there is apparently nothing
in Ceratophyllum when it grows in this way as a perennial corresponding to the
"localized stages" discussed by Jackson ('99), Cushman (:02, :03, and :04), Shull ( :05),
etc. The new growth in the spring appears to go on in precisely the same manner as
it would have if there had been no interruption. This is, of course, what would be
expected in a plant like Ceratophyllum. My material at present is not sufficiently
great to enable me to make a positive statement that no change whatever in the
growth curve takes place at points where new growth of this kind begins. So far,
though, I have seen no certain indication of such a change.

CORRELATION BETWEEN DIFFERENTLY SITUATED WHORLS. 93

THE CORRELATION BETWEEN DIFFERENTLY SITUATED WHORLS IN
RESPECT TO LEAF-NUMBER.

In the preceding section we have examined the correlation existing
between leaf-number and position of the whorl on the axis, and in that
way determined the law according to which the size of whorls changes
with different positions on the branch. We may now consider the
further problem of how the absolute sizes of different whorls are cor-
related together. Especially it is desirable to determine to what degree
the first, whorl on a branch is correlated with whorls farther out. In
this way we shall see whether a particular branch has as a whole a
definite tendency or *'set" towards a relatively large (or small) absolute
leaf-number for all its whorls. It will be seen that this problem is quite
distinct from that considered in the last section. We are not now con-
cerned with the law of change with growth, but rather with absolute
sizes. All the whorls on a given branch may be relatively large or rela-
tively small and still have been differentiated according to our logarithmic
law. The question, then, with which this section has to do may be
stated in this way: If the first whorl on a branch has more than the
average number of leaves, will the succeeding whorls on the same branch
also tend to have in each case more than the average number of leaves
for their respective positions? In order to test this question we have
to determine individually the correlation between the first and each
successive whorl on the branch. On account of lack of material I have
not been able to carry the correlations beyond the tenth whorl. For
the same reason it is necessary to confine the discussion to primary-
branch whorls. We may first examine the results from the data of
Series I, II, and III combined. In table 47, page 94, is given the raw
material. It may be noticed that the totals of these tables do not
agree with those for the corresponding whorls in table 32, because in
that table every unmutilated whorl is included, while here we have, of
course, been restricted to every unmutilated pair of whorls. Naturally,
a great many broken whorls pair with unmutilated ones which appear
in the earlier tables, but not in these.

The coefficient of correlation was determined for each of these tables
in the ordinary way. The values so obtained, together with their prob-
able errors, are given in table 48, page 95.

There is clearly a very distinct correlation between the first and
second and first and third whorls. As we go farther out on the branch
the correlation steadily decreases, till finally at the ninth whorl it
becomes sensibly zero. There is apparently a rise in the correlation at
the eighth whorl, but this is probably a purely accidental result arising
from the fact that the number of pairs is getting too small to depend

94

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

on for smooth results. The correlations are all positive, with the excep-
tion of that for the ninth whorl, which is negative, but so lov/ as to be
entirely insignificant in comparison with its probable error. The data
from Series IV lead to essentially the same results as the Series I, II,
and III material, and consequently it does not seem necessary to take
the space to reproduce the figures in detail.

Table 47. — Correlation between the first and each of the following nine whorls on
primary branches in respect to leaf-number. Series I, II, and III combined.

First
whorl.

A.— Second whorl.

B.— Third whorl

C— Fourth whorl.

5

6

7
1

17

16

5

8

18

20

14

2

9

10

Total.

6

7

8

9

10

11

Total.

7
"4

8

13
18

_9_

14

18

10

2

10

...

"3

10

2

Total.

31

43

26

5

5

6

"i

6
2

'4

16

14

5

"2
3
1

1

46

56

36

8

"i

5

1
21
20
10

"9

16
14

4

"2

7
1

"i
1

1

38

54

33

5

7

8

9

Total..

"i

6

1

8

39

54

39

6

147

1

13

52

53

10

2

131

9

37

44

15

105

First
whorl.

D.— Fifth whorl.

E.— Sixth whorl. j

F.— Seventh whorl.

6

7

8

9

10

11

Total.

8

9

10

11

Total.

6

"i

7

8

"4
6
2

9

16

12

5

2

10

"4
8
7
1

11

Total.

5

6

7

"i

1

1
1

ii
11

2

16

19

12

2

"2

5

i

1
30
37
21

3

"4
9

13

1

17
15
10

1

"i

8
6
2

"i

1
23

^9.

"1

"i

26

26

15

3

8

6

! Tfi

9

Total.

1 ...

3

2

2

24

49

14

1

92

44

17

1

75

1

12

29

20

2

64

First
whorl.

G.— Eighth whorl.

H.— Ninth whorl.

I.— Tenth whorl.

7
1

2

8

~3
2

1

9

9
5
5

10

1

13
6
3

11

Total.

7

8

9

10

11

Total.

6

7

8

9

10

11

Total.

6

"1

14
22

1

"i

3
2

4
6
6

1

5

10

6

1

... 12

1 19

... 12

3

"i

1
2

"i

1

1

4
7
5

6
5
5
2

...

11

14

11

2

7

8

9

Total..

3

6

19 23

1

52

1

5

17

22

1

46

1

16

18

38

We may safely conclude, then, that there is a distinct tendency for
branches which show an excess (or defect) from the average in the number
of leaves in the first whorl, to have the succeeding tvhorls greater (or less)
than their respective averages. The longer the brayich grows the weaker
this tendency becomes, till finally, when the ninth or tenth whorl is reached,
the number of leaves which it bears is entirely independent of the number
in the first tvhorl. Branches which start large maintain an excess over
the average for roughly 6 to 8 whorls. So far as our data go they show
the same kind of relation for the other axial divisions of the plant, viz,

VARIABILITY OF SUCCESSIVELY FORMED WHORLS.

95

a positive correlation between the first few successive whorls, diminish-
ing the farther out we go. The material for investigating the matter
in these cases is, however, so meager that it is not worth while to discuss
it in detail.

Table 48.— Coefficients of correlation from table ^7.

Correlation between—

r.

TJ'irQt nnrl cspmnfl whorl

0.456±0.044
.402± .049 1
.360± .057
.260± .066
.262± .072
.208± .077
.393± .079
— .007± .099
.002± .109

First and third whorl

First and fourth whorl

First and sixth whorl

First and seventh whorl

First and eighth whorl

First and ninth whorl

First and tenth whorl

THE VARIABILITY OF SUCCESSIVELY FORMED WHORLS.— THE SECOND
LAW OF GROWTH IN CERATOPHYLLUM.

We come now to the consideration of a matter of very considerable
interest and importance in connection with the general laws of morpho-
genesis in Ceratophyllum. It is as to whether whorls in different
positions on the plant show equal degrees of variability. We have seen
that the mean leaf-number of successively formed whorls changes in a
regular and orderly way. We have now the further problem: Does the
variability of whorls successively formed similarly show a tendency to
orderly change, and if so, what law does this change follow?

In discussing the subject we may adopt the notation and methods
used by Pearson (:05) in his memoir on "Skew Correlation." If we let
o■,^. denote the standard deviation of an a--array of a character B, and o-y
the total variability of the same character, then (Pearson, loc. cit.,
p. 10) : "A curve in which the ratio of o-», to the standard deviation o-^
is plotted to X may be termed a scedastic curve. "

Further, Pearson says (p. 22) :

I must remind the reader, however, that the form of the regression line does not
in any way limit the nature of the distribution of the array about its mean; the varia-
bility of an array, i. e., the standard deviation of an array, having for its mean value
ay^i — yj^^ may or may not be the same for all arrays. If it is the same, or all arrays
are equally scattered about their means, I shall speak of the system as a homoscedastic
system, otherwise it is a heteroscedastic system.

For every array of the correlation tables for position and leaf -number
given above (pp. 59 and 60) I have calculated the ratio -i^, with the re-

96

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

suits set forth in tables 49 and 50. Lest there should be any misunder-
standing as to just what these figures mean, it may be said that each
value gives the ratio which the variability of whorls in a designated
position on the branch in respect to leaf-number is to the variability
of all whorls taken together. We may consider first the primary
branches.

Table 49. — Variability of successively formed primary-branch whorls.

Before discussing these results we may examine those for the second-
ary branches, which are given in table 50.

Even the most superficial examination of the values given in these
tables shows us at once that the whorls in different positions on the axis
do not form a homoscedastic system. The variability is not the same
in whorls occupying different positions. We have now to consider the
further question of whether the change in variability as we consider
whorls in different positions is entirely irregular, or whether, on the
contrary, it follows some definite law, as we have seen to be the case

VARIABILITY OF SUCCESSIVELY FORMED WHORLS.

97

with the means. The matter is one of such considerable importance
that it will be considered in some detail.

Table 50. — Variability of successively formed secondary-branch whorls.

Position of
whorl.

Variability in leaf-number, measured by ratic

. '^J'^y

Series I.

Series II.

Series III.

Series IV.

Series, I, II,
III, com-
bined.

1

0.641
.710
.845
.800
.555
.813
.705
.406
.431
.863

0

0

0

0.684
.659
.644
.422
.473
.450
.368
.682

0

0

0.623
.538
.562
.579
.548
.447
.467
.389
.686
.389

0

0

0.647
.712
.708
.727
.739
.776
.682
.741
.610
.643
.721
.266
.594
.319

0
.417

0

0

0.655
.638
.693
.582
.536
.558
.476
.527
.623
.720

0
.376

0

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

Confining- our attention at first to the primary branches, I think it
reasonably clear that, disregarding minor fluctuations due to the relative
smallness of the numbers of observations, there is a tendency for the
variability of whorls in leaf-number [to decrease, the farther out on the
branch we go. The minor fluctuations are, however, rather disturbing,
and in order to get clear results we must resort to graphical representa-
tion and graduation. In that way we can get an idea of the general trend
of the variability, apart from its accidental fluctuations. In order that
the diagrams might not be too extended, and that at the same time we
might get a sort of "first smooth" of the observations, I have combined

the whorls into pairs and taken the weighted means of the ratio -^ for

a-y

each pair. Thus, whorls 1 and 2 have been combined and a weighted
mean taken; whorls 3 and 4 combined together, and so on for the whole
length of the branch. In taking the means, each single observation

was weighted with the frequency in the array from which— ^ was calcu-

a-y

lated. Proceeding in this way, we have for Series I, II, and III combined
the series of values given in table 51.

98

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

Table 51.— Weighted mean variability of successively formed primary-
branch whorls.

Position.

Mean of '^nJ'^y

Position.

Mean of '^nJ'^y

Whorls 1 and 2

0.788

Whorls 17 and 18

0.768

Whorls 3 and 4

.723

Whorls 19 and 20

.618

Whorls 5 and 6

.719

Whorls 21 and 22

.418

Whorls 7 and 8

.748

Whorls 23 and 24

.266

Whorls 9 and 10

.864

Whorls 25 and 26

.376

Whorls 11 and 12

.834

Whorls 27 and 28

.415

Whorls 13 and 14

.624

Whorls 29 and 30

•0

Whorls 15 and 16

.416

Even with this sHght smoothing it is clear from the figures them-
selves that the variability is tending to become smaller in the higher
whorls, till we finally reach a condition of no variability at the ends of
the branches, or in other words, till all the frequency falls on one type
of whorl. These data are shown graphically in fig. 17.

1.0

0.9

^

-\

^

}

■^

0.8

"

\^^^

^

-__

\

j

\

^

, ^

0.7
Xi^ 0.6

^

--^

r\

4

N

I

V

f

^

\

o

\

r

1

<v^^

v^

10 0.4.

\

A

\^

\

0.3

S

/

\

V

"^

Y

0.1

\

. —

l+Z 3+4 5+6 7+8 9+10

Whorls

Fig. 17.— Scedastic curve for primary-branch whorls of Series I, II, and III combined. The

abscissas give the position of the whorls, and the ordinates the ratio cnx/cy. o O^

observations. The fitted straight line and parabola are given by continuous lines.

From this diagram it is at once evident that there is a clearly marked
and definite tendency for the variability to decrease with the successive
formation of whorls. Whorls formed when the branch first begins its
growth are more variable than those formed later. In order to give a
more precise idea of the rate at which this decrease in variability goes
on I have fitted to the observations given in fig. 17 a straight line and a
parabola. The fitting was very easily done by the method of moments

VARIABILITY OF SUCCESSIVELY FORMED WHORLS.

99

(Pearson, :02) , using the corrective terms for the moments of trapezia.
The equations are as follows:

Straight line, y = 0.5845 1 1 — 0.5103 (y)

Parabola, y = 0.5845 ] 1.1084 — 0.5103 (^)- 0.3249 (q)'

where y is the ordinate (i. e. , the probable value of (^nj^^y) and I, the
half range, equals 7 units. In these and all the curves which follow in
this section of the paper the origin of x is taken at the central ordinate
of the group. Both line and parabola give very reasonable graduations,
and until we have much larger numbers than at present it is impossible
to say which one gives the truer representation of the scedastic curve for
Ceratophyllum. What I wish to bring out now is that when we smooth
the observations by a graduation formula there can be no doubt that
the general trend of the variability is to decrease as the position of the
whorl on the branch, or in other words, the order of its formation,
increases.

1.0

0.9

0.8

0.7

o

+-"

0.6

0.5

a3

0.2

0.1

^_-

\\

V

\
\

:: X

\
\

a

l+Z 3+4 5+6 7+8 9+10 M+12 13+14 15+16 17+18 19+20 El+ZZ 23+24

Whorls
Fig. 18.— Scedastic curves for primary-branch whorls of Serle.s IV.
Significance of lines as in fig. 17.

Do the other series show the same relation? Treating the data given
in table 49 for primary-branch whorls of Series IV in exactly the same

100

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

way as we have those for Series I, II, and III, we get the results shown
in fig. 18. The zigzag Hne gives the weighted means of the successive
pairs of observations given in table 49.

As before, I have fitted a straight line and a parabola to the data.
In order to have an odd number of points for the fitting of the constants,
the curves were calculated from the first 11 observations {I = 5) neglect-
ing the twelfth. Apart from the practical reason, this procedure was
theoretically justifiable because of the small number of whorls on which
this last observation is based. The equations to the curves are:

StraightHne, y = 0.6347 1 1 - 0.3292 (|) [

Parabola, y = 0.6347 ] 1.0701 - 0.3292 [fj - 0.2102 (|)

Again we see that the general trend of the curve is downwards as
we pass to the whorls farther out on the branches. The slope of the

curve is somewhat more
rapid than in the case of
Series I, II, and III, but
the diif erence in this re-
spect is not great. We
must conclude that here
as in the former case the
variability decreases with
successive whorl forma-
tion.

We may next consider
Series V and VI. Here it
will be recalled that we are
dealing not solely with
primary-branch whorls,
but with all the whorls
borne on branches on the
plant as a whole. Fur-
thermore, for reasons set
forth above (p. 58), we
have considered only the
first 10 whorls. Dealing with Series V in the same way that we have
with the other series (i. e., combining successive whorls in pairs and
taking the weighted mean for each pair) we have the result shown in
fig. 19.

\+z

3+4-

7+8

3+10

5+6
Whorls

Fig. 19.— Scedastic curve for whorls on branches. Series V.
Weighted means of successive pairs. Significance of lines
as in fig. 17.

VARIABILITY OF SUCCESSIVELY FORMED WHORLS.

101

The equations to the fitted curves, when I = 2, are
Straight Hne, y = 0.7812 ] 1 - 0.2297 (|) [

Parabola, y = 0. 7812 -{ 1. 0338 - 0. 2297 ( j) - 0. 1015 (|) [

The result is the same as before. There is very clearly a decrease
in the variability with successive whorl formation. Since the total
number of successive whorls dealt with in this case (only 10) is so small,
it is perhaps worth while to examine the results obtained when each
whorl is considered by itself instead of being- paired with the one
following. Fig. 20 shows the result in this case. Here each circle

represents the ratio — for the whorls in the particular position desig-

nated by the abscissal number.

\^'

.'^

0.9
0.8
0.7

x>

0.6

ID

^ 0.5
0.4
0.3

<k—

\
\
\

^-

\

o

IZ345&789I0

Whorls
Fig. 20.— Scedastic curve for whorls on all branches. Series V.

In fitting the straight Hne to the observations in this case I have used
only the first seven points. The last three are very irregular, on account
of the small number of whorls on which they are based. The equation
to the Hne, where now I = 3, is,

?/ = 0.8108 1 1 -0.1326(^)

We reach the same result as before, namely, that the variability
decreases as we go out on the branch. The smoothness of the scedastic
curve for the first five observations in this case is remarkable.

102

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

As shown by fig. 21, Series VI leads to the same conclusion as the
other series. In this case each observation is given separately. The
straight line was fitted to the first 7 points. As in the case of Series V,
the last three observations are based on a small number of whorls, and
are in consequence very irregular.

M

1.0
0.9

0.8
0.7
0.6

>0^

>0"

0.5

0.4

0.3

<

/

\

e

/
/

\
1

\

/
/

\

■ — ~.

N.

/

V

1

\

>1

V

/
1
1

\
\

i

I

\

/

N

X

H

1
1

I

\

\

1
1
1

1

1

f "^^

--^

I Z2.4-56789I0

Whorls

Fig, 21.— Scedastic curve for whorls on all branches. Series VI.

The equation to the straight line, where I = 3, is,

1/ = 0.7242] 1 — 0

,1348(f)

All our series agree, then, in showing that the variability in leaf-
number in successively formed whorls on primary branches changes
inversely as the order of formation of the whorl. The whorls first
formed are the most variable, and the degree of variability steadily
diminishes in the later whorls.

We may next consider the question of whether the same relation
holds for secondary-branch whorls. It is evident, I think, from an
examination of table 50 (p. 97) that such is the case. In order, however,
that there may be no doubt about it we may consider in detail the com-
bined data for Series I, II, and III. The data given in the last column
of table 50 are plotted in fig. 22. The irregularity in the last four obser-
vations is due to small numbers in the arrays here, as will be seen by
reference to table 37 (p. 74) .

VARIABILITY OF SUCCESSIVELY FORMED WHORLS.

103

The equation to the straight Hne is,

2/ = 0.5054 1 1 -0.4442(^) j-

where I = 6. The fit is a very reasonable one, and clearly a straight
line gives a sufficiently good graduation for our present purpose, which
is merely to show the general trend of the observations. Out as far as
the 9th whorl the scedastic curve is quite regular. I think that there
can be no doubt that the secondary-branch whorls follow the same law
as those on primary branches in respect to their variability. The varia-
tion in leaf-number decreases more and more as successive whorls are
formed.

0.9

03

ae

\i'

0.5

0.3

0.2

^

/<

>

(

-— <

h

N

y

r

I

V

>>^

k

k

N

.^

\
\

-^

>

/
/
/

\
\

-

/

/
1

\

\
\

1

1
1

\
\
\

c

^

I Z 3 4 5 6 7 8 9 10 II l^ 13

Whorls
Fig. 22.— Scedastic curve for secondary-branch whorls. Series I, II, and III combined.

We may fairly conclude that what is true of the variability of primary
and secondary branch whorls will also be true of whorls on other branch
divisions of the plant (tertiaries, etc. ) . On account of the relatively
small number of tertiary-branch whorls it is not possible to test the
matter directly there. Our results in the previous section of the paper
with reference to the mean number of leaves in successive whorls in
different divisions of the plant makes it very probable that a similar
uniformity prevails with reference to variability.

Fortunately, we can test the matter directly for main-stem whorls.
In table 42 (p. 83) are given, for the combined plants of Series I, II,

104 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

and III, the frequency of the various leaf-numbers for successive whorls,
proceeding by groups of 10 whorls. Calculating the standard deviation
{(Tn.) for each array of this table, and dividing it in each case by the
standard deviation M of all main-stem whorls, we have the results
shown in table 52.

Table 52.— Variability of successive main-stem whorls.

Position.

Variability.

Position.

Variability.

Whorls 1 to 10

Whorls 11 to 20

Whorls 21 to 30

Whorls 31 to 40

Whorls 41 to 50

1.041

1.190

1.110

.892

.843

Whorls 51 to 60

1 Whorls 61 to 70

Whorls 71 to 80

Whorls 81 to 90

0.808
.543
.426

0

The data in this table are shown graphically in fig. 23.

\

1.3

\

l.Z

^

^j

/

V

1.0

' — "t

\

\

0.9

I

N

\

0.8

^^ 0.7

^ 1^ 0.6
o

\

\ ^

V

A

^

+. 0.5

to

(t

\

0.'^

V

0.3

\~

\

0.2

\

O.i

.. .

_

\

> '

i-10 II-EO 21-30 31-40 41-50 51-60 61-70 71-80 81-90
Whorl
Fig. 32.— Scedastic curve for whorls on main stem. Seiles I,
II, and III combined,

VARIABILITY OF SUCCESSIVELY FORMED WHORLS. 105

I have fitted to the observations the following curves:
Straight line y = 0.7903 1 1 - 0.6428(^) [

Parabola y = 0.7903 ] 1.1096 - 0.6428(1^) - 0.3289(^) [

where ^ — 4.

There can be no doubt that the main-stem whorls show the same
decrease in variability which we have observed in the other divisions of
the plant. Indeed, the results for the main stem are more regular than
for any other of the groups considered. This is of course due to the
fact that we have a much larger range of position (1 to 80) and so get
a smoothing eif ect by taking the whorls in groups of 10. In this case
it is clear that the parabola gives a somewhat better graduation than
does the straight line, but the gain is mainly in the representation
of the first and last observations, on neither of which can much weight
be laid.

The difficulty with reference to the first observation is that so many
of the proximal main-stem whorls were mutilated and could not be
counted, thus giving disproportionate weight to the others (cf. p. 83).
This would of course operate to lower the variability for that group.
The last observation is based on too few whorls to be significant.

So far it has been shown that when the variability of the whorls
situated in a definite position on an axial division is measured by taking
the ratio of the standard deviation of such whorls to the standard
deviation of all whorls on the same axial division, the degree of varia-
bility decreases as the distance of the whorl from the proximal end of
the branch (or main stem) increases. Now, this measure of variation,
consisting as it does of the ratio of two standard deviations, gives no
idea of the degree of variability in proportion to the size of the thing
varying. It has been shown in a preceding section that the mean
leaf -number changes in a very regular and definite way in successive
whorls, the direction of this change being an increase in the mean with
every increase in the distance of the whorl from a fixed point on the
axis.

Since the means thus increase with successive whorl formation,
while, as we have seen, the absolute variability decreases, clearly there
must be a still more marked decrease in the relative variability in pro-
portion to the size.

If we take the percentage of the standard deviation to the mean
(coefficient of variation) for each array corresponding to a definite posi-

106

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

tion on the axis, we shall clearly have a measure of this relative varia-
bility. I have done this for all of the tables, but since it is evident, if
in an expression

100 (r

^ = ir

o- is decreasing while M is increasing in value, that v must also decrease,
it does not seem worth while reproducing these coefficients in detail.
In order to afford some idea of how the actual values run, however,
there are given in table 53 the coefficients of variation for the first four
whorls on primary branches in Series I, II, and III and IV, and on all
branches in Series V and VI.

Table 55. — Coefficients of variation for successive whorls of primary branches.

Position
of whorl.

Coefficients of variation.

Series I.

Series II.

Series III.

Series IV.

Series I, II,
III com-
bined.

Series V, all
branches.

SeriesVI,all
branches.

1

14.74

11.77

9.50

10.16

11.67

12.23

11.15

8.93

12.00

10.67

8.95

8.73

11.57
9.68
7.92

7.82

13.07

11.92

10.31

9.91

20.54
16.87
14.90
13.29

16.70

12.56

9.88

10.91

2

3

4

We see in all the series that while the branches start with a relatively
very variable first whorl, the variability rapidly decreases in the suc-
ceeding ones. The same thing is shown if we calculate the coefficients
of variation for secondary-branch whorls.

SUMMARY OF SECTION.

In this section of the paper it has been shown that both absolutely
and in proportion to the size the variability of successive whorls in
respect to leaf-number diminishes as we pass from the most proximal
to the most distal whorl on an axial division. This relation has been
demonstrated for the three most important axial divisions (main stem,
primary and secondary branches), and it can not reasonably be doubted
that it also holds in the same way for the other divisions (tertiary and
quaternary branches) . But from the method of growth of Ceratophyl-
lum we know that succession of whorls in position on the branch
denotes succession in order of formation or differentiation from the
growing bud. Our result, then, means that as whorls are successively
produced by a growing bud, they are formed with ever-increasing constancy
to their type, the ultimate limit towards which the process is tending being
absolute constancy. This may be designated as the "Second law of
growth" in Ceratophyllum.

SUMMARY OF SECTION. 107

The operation of this law is to be seen in other of the morphogenetic
activities of the plant than those which liave been discussed in this
section. In a later section data are given showing that it holds in branch
production. It seems to me that a number of results obtained earlier
in the paper can best be interpreted as due to the operation of this law.
Thus, in examining the data on which our first law of growth was based
it was found that the slope of the positional regression line was steepest
for secondary-branch whorls, less steep for whorls on primaries, and
had the least slope in the case of main-stem whorls. It was pointed
out in the discussion that this meant that a given type was attained
with the production of fewer whorls in secondary than in primary
branches, etc. Similarly it appears probable that the diminution in
variability with successive whorl formation goes on more rapidly the
farther distad on the plant we go. That is, the same rule appears to
hold between different axial divisions of the plant as holds for the organs
within a given division. In other words, as we go towards the periphery
of the plant the variability of repeated characters diminishes in such a
manner as to give the impression that in some way there is stored up
in the protoplasm, as it were, the results of previous morphogenetic
experience. An axillary bud on a primary branch goes through the
same series of events when it develops into a branch as does an axillary
bud on the main stem. It produces whorls whose type changes accord-
ing to a logarithmic law, and whose variability diminishes with successive
formations. But the rate at which it attains any given result in this
series is greatly accelerated over that at which affairs went on in the
case of the bud on the main stem. The formative activities of each
bud on the plant appear to be influenced in some very direct way by the
sum total of previous morphogenetic history of the portion of the plant
proximal to the bud. That the two things are objectively related is a
fact clearly demonstrated by the results which have been presented in
this and earlier sections of the paper. No one who will take the trouble
to study carefully these results can fail to be impressed with the reality
of the fact, I think. How such a relation as that of which we are
speaking is to be interpreted or explained is another question to which
we shall return in the concluding section of this paper.

The relation of this second law of growth to our earlier results on
the variation in whorls on different axial divisions of the plant is so
obvious as hardly to need special mention. In connection with the first
law of growth it enables us to interpret very clearly and completely the
results regarding variation obtained from grouped material. Thus, for
example, it was found that main-stem whorls as a class are least varia-

108 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

ble; primary-branch whorls are more variable; a maximum of variability
is reached in secondary-branch whorls as a class; and in tertiary and
quaternary whorls the variability tends to decline. It is obvious that
the variation shown by any composite group of whorls will depend on
two factors: (a) the real variability of the whorls in a given position,
and (6) the mixing of whorls of differentiated types (i. e., whorls which
occur at different positions) . Now, in the case of main-stem whorls as a
class the effects of both a and b are at the minimum as a result respectively
of the operation of our second and first growth laws. In primary-branch
whorls the effect of the factor b is greater than in the main stem, but
since many primary branches are very long the influence of the law of
diminishing variability comes in through a to keep down the variation ex-
hibited by the group as a whole. When we come to secondary-branch
whorls this lowering of the variation through the presence of long
branches with many whorls of low variability no longer occurs, because
there are few or no long branches. Finally, the drop in the variability
shown by tertiary-branch whorls as a class is clearly due to the fact
that owing to the extreme shortness of these branches factor b contrib-
utes very little to the sum total, but instead we have the expression
practically of a alone.

In following sections of the paper the operation of the law of dimin-
ishing variability in other phenomena than whorl production will be dis-
cussed. A discussion of its theoretical bearings and interpretation will
also be undertaken later in the paper. It need only be pointed out here
that it is merely a special case of a much more general biological law
applying to other phenomena besides those of growth. Jennings ( :05) in a
recent paper has enunciated what is essentially the same law in the field
of behavior in the following terms : ' 'The resolution of one physiological
state into another becomes easier and more rapid after it has taken
place one or more times."

BRANCHES AND LEAF-NUMBER.

109

THE RELATION OF THE PRESENCE OF BRANCHES TO THE NUMBER
OF LEAVES IN THE WHORL.

The lateral branches in Ceratophyllum always originate of course at
nodes, and hence in each case a branch comes into very close relation
with the whorl at that node. It seemed desirable to determine whether
the presence of a branch at a node in any way influenced the number of
leaves in the whorl belonging to that node. To this end frequency
distributions were formed for each plant and series, giving the leaf-
number in every whorl at which a branch originated. These distribu-
tions are given in table 54.

Table 54. — Frequency distributions for leaf-number in whorls at which branches

originate.

Series.

Plant.

Distri-
bution
number.

Leaves ]

9er whorl.

4

5

6

7

8

9

10

11

12

Total.

I

r
II.

III..

IV

v<

VI <

1

37
38
39
40
41

"i
"3

"i

1
3

1

2
5
8
16
4

13
12
13

27
8

29
13
14
25

7

"2
2
2

"i

44
33
39
73
25

2

3

4

5

All plants ..
1

42

5

6

35

73

88

6

1

214

57
58

1

8
5

27
13

65
38

88
53

14
5

203
114

2

All plants ..
1

59

1

13

40

103

141

19

317

71

4

12

47

109

113

6

291

1

92
93

"2

3
11

22
39

88
140

115
193

2

230
385

2

All plants..
1

94

2

14

61

228

308

2

615

130
131
132
133
134
135
136

2 1 8

4

"2
2

10

1

i
2

7

12
'1

"5
3

19
1
2
6
5
7

18

26
2

7

"'5
3

8

3

84
4
10
6
11
19
38

2

3

4

5

6

7

All plants ..
1

137

2

8

8

21

21

58

51

3

172

168
169
170
171
172
173

"i

1

1

"i

4
1

"4

4
2

2
5
3
1
4
5

8
10

2

4
15

4

7
5
4
5
42
29

'1
"6

22
22
9
15
67
46

2

3

4

5

6

All plants-

174

...

1

3

15

20

43

92

7

181

110

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

We note at once from this table that the most frequently occurring
whorls at the nodes where branches originate are those having 10 leaves
(with the single exception of Series V, in which the 9 and 10 leaved
whorls are approximately equally frequent) . Further, it appears that
generally whorls having a small number of leaves do not occur at the
origin of branches, though there are individual exceptions.

The constants for the largest of these distributions are given in
table 55.

Table 55. —Constants for variation in leaf-number in whorls at which branches

originate.
[Unit = 1 leaf.]

Series.

Plant.

rjistrl-

butlon

number.

Mean.

standard
deviation.

Coefficient of
variation.

I"

r
II.

III..

r

IV^
V.

VI

1.
2.
3.

4.
5.

1.
2.

1.

1.
2.

1.

7.

2.
5.
6.

37
38
39
40
41

9.614±:0.058
9.000± .129
9.282± .109
9.096± .072
8.840± .189

0.573±0.041
l.lOli .091
1.011± .077
.909± .051
1.405± .134

5.957±0.430
12.233±1.031
10.897± .842

9.994d= .563
15.895±1.554

All plants ..
All plants ..

42

9.192±

.046

1.007±

.033

10.959± .362

57
58

9.345±
9.351±

.046
.057

.962±

.898±

.032
.040

10.294± .348
9.605± .433

59

9.347±

.036

.940±

.025

10.052± .272

71

9.144±

.038

.949±

.026

10. 375 ± .293

92
93

9.396±
9.327±

.032
.028

.725±
.810±

.023
.020

7.720± .244
8.687± .213

All plants ..

94

9.353±

.021

.780±

.015

8.343± .162

130
136

137

8.357±
8.605±

.132
.127

1.790±
1.159±

.093
.090

21.424±1.165
13.473±1.061

All plants..

8.581zt

.077

1.502±

.055

17.499± .655

169
172
173

8.773±

9.343±

, 9.696±

.143
.090
.097

.997±

1.086±

.975±

.101
.063
.069

11.364=bl.l70
11.630± .687
10.057ifc .714

All plants ..

174

9.238±

•057

1.134±

.040

12.277± .442

From this table the following points are to be noted:
(a) The mean number of leaves in the whorls where branches start
is relatively high. If we compare the values given in this table with
those for the different axial divisions of the plant presented earlier in
the paper (tables 6, 13, 15, 18, 23, and 24) we see that these whorls
agree more closely in type with those borne on the main stem than with
any others.

BRANCHES AND LEAF-NUMBER.

Ill

(6) The variabilities, both absolute and relative, shown in this table
are comparatively low. Again they agree most closely with what has
been found for main-stem whorls.

We get the same results, only still more marked, if we consider the
whorls at nodes Vv^here two branches originate. This "double" branch-
ing at one node does not occur very commonly in Ceratophyllum ac-
cording to our experience, but in one plant (No. 2, Series IV) such cases
were especially abundant. Table 56 shows the condition, in respect to
leaf-number, of these whorls on the plant mentioned.

Table 56. — Frequency distribution for variation in leaf-number of whorls
where two branches originate. Plant 2, Series IV.

Leaves per whorl.

7
1

8

9
41

10

Total.

Frequency

120

164

The constants for this distribution are as follows:

Mean = 9. 707 ±0.027

Standard deviation = .518± .019
Coefficient of variation = 5.333 ± .199

Comparing these with the values in table 55 for the same plant we
see that there is a still further raising of the mean and lowering of the
variability in the whorls at which two branches start.

Now the question arises, can we consider the relatively high mean
and low variability of these whorls at which branches originate to be a
specific result of the presence of one or more branches at the node to
which the whorl belongs? Clearly, without further evidence, we can
not, because of the fact that among these whorls some are borne on the
main stem, others on primary branches, and still others on secondary
branches, and it has been shown that whorls are differentiated in respect
to leaf-number according to the part of the plant on which they are
borne. Further, according to the two laws of growth which have been
set forth above, it is clear that the number of leaves in whorls from
which branches originate will depend, in part at least, on the position
which such whorls occupy on their axes. Before we can reach conclu-
sions as to whether the presence of a branch influences the number of
leaves in the whorl we must determine the probable condition of such
whorls as a result merely of the operation of the usual laws of growth.
If, then, it be found that these v/horls differ considerably from the condi-
tion of "whorls in general" occupying the same relative position on the

112

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

plant, it may safely be concluded that the branching exercises a specific
influence.

In order to test this matter it is necessary to know first of all some-
thing about the positions at which branches originate. Is the chance
of a branch occurring the same for all nodes? Or is branching more
abundant in some positions on the axes of the plant than in others?
To answer these questions the frequency of occurrence of branches at
the different nodes must be determined. This has been done for the
secondary branches of the first four series, with the results shown in
table 57. The way in which the table was made was to enter for each
plant the number of secondary branches which originated at specified
nodes on primary branches, the nodes being numbered in order, begin-
ning with 1 as the most proximal. In case two secondaries arose at the
same node each was entered separately.

Table 57. — Frequency distributions showing position of secondary branches.

Frequency of secondary branches.

Series I.

Series II.

Series
III.

SerJesI
II, III.

Series IV.

Position.

Plant

Plant.

All
plants.

Plant.

Tot 1.

Total.

Plant 1

Total.

1

2

3

4

5

I

2

1

2

1

1

3

2

3

9

13

3

16

9

34

7

13

20

2

1

2

3

6

7

6

13

13

32

5

11

16

3

1

2

5

"4

12

15

6

21

23

56

14

29

43

4

1

"4

2

3

10

12

6

18

15

43

9

33

42

5

1

1

2

"i

5

11

2

13

13

31

12

27

39

6

2

1

2

1

6

12

4

16

12

34

12

31

43

7

1

"i

2

4

2

10

6

4

10

9

29

12

28

40

8

1

2

3

4

4

8

4

15

9

26

35

9

"i

"i

1

1

4

4 4

8

3

15

8

22

30

10

1

3

1

6

*i

12

5 1 3

8

3

23

7

20

27

11

3

1

3

7

2 i 3

5

4

16

4

19

23

12

1

"i

2

2

2

4

3

9

5

15

20

13

2

1

"2

"i

6

2

2

4

1

11

5

13

18

14

3

1

4

2

4

6

3

13

4

10

14

15

4

4

1

5

5

9

14

16

2

"i

"3

i

4

5

2

10

6

7

13

17

2

2

"i

5

1

2

3

2

10

6

4

10

18

1

1

2

1

3

4

1

7

5

5

10

19

"i

1

1

1

2

3

6

6

6

20

"i

1

2

1

1

2

3

7

4

"2

6

21

1

1

1

1

2

1

4

2

2

4

22

"i

1

2

2

1

3

3

8

1

2

3

23

1

1

1

2

3

1

5

2

2

24

1

1

2

2

4

1

6

25

1

1

2

1

3

26

"2

"2

1

1

2

5

27

1

1

1

2

28
Total

21

20

1

1

1

2

148

330

16

45

16

118

109

76

185

138

441

478

BRANCHES AND LEAF-NUMBER.

113

Calculating from this table the mean and median positions of second-
ary branches, we get the results set forth in table 58.

Table 58. — Constants for position of secondary branches. Totals for series.

Series.

Mean.

Median.

I

9.24
8.39

7.75
8.42
8.42

8.33
6.72
5.69
6.72
7.90

II

Ill

I, II, and III combined
IV

From these data we see that:

(1) The average position of origin of secondary branches is at roughly
the 7th to 9th node from the proximal end of primary branches.

(2) Fifty per cent of all secondary branches originate from the
seventh {ca. ) or more distal nodes of primaries.

(3) There is very close agreement between the different series in
respect to the point of origin of secondaries. The mean for the combined
Carp Lake material agrees to the third place of figures with that for
the Ann Arbor material (Series IV).

From these results it is clear that branches (a) do not occur with
equal frequency at all nodes, nor (6) as will be seen by comparison with
tables 26 and 27 (pp. 50, 51), can the positional distributions of branch
origins be regarded as random samples of the positional distributions of
"whorls in general." Instead it is found that the branches are so dis-
tributed in their points of origin that the great majority occur beyond
the first few proximal nodes. But it has been seen in what has gone
before that (a) taking all whorls on branches together there are in the
total more first whorls than second, more second than third, and so on
(cf . tables 30, 31, and 32, pp. 59 and 60) ; and (6) that the mean number
of leaves per whorl is lowest for first whorls and increases as we go in a
distal direction. That is to say, the frequency distributions for ' whorls
in general" contain a higher proportion of whorls which, owing to the
operation of our first law of growth, have a small number of leaves than do
the distributions for whorls at which branches originate. Naturally, then,
we should expect the means to be higher for the latter than for the former
distributions. Again, we have seen that distal whorls are less variable
than proximal, and that further, the whorls at which branches originate
include a higher proportion of distal ones than do the distributions for
the entire plants. So we should expect them to be less variable. While
it is thus clear that the ordinary growth factors account for a consider-
able part of the results which have been found for the "branch-origin"

114 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

whorls, we have yet to determine whether the whole effect is to be
explained in this way. To test this we have to ansv/er the following
question: Is there a sensible difference between the means and varia-
bilities of (a) whorls at the nodes where branches originate, and (6) an
equally large sample of similarly situated whorls taken without refer-
ence to the presence of branches? An approximate answer to this
question can be obtained by the use of our growth equation (i) and the
data provided in table 58. If in the equation

Y = 7.9520 + 1.3608 log (x — 0.8015)

we substitute for x the values for the mean position of "branch origin"
whorls in the different series given in table 58, and solve for Y, we shall
get the probable mean leaf- number in a group of primary-branch whorls
situated in the same average position on the axis as are the whorls
where secondary branches originate. But in the distributions for
"branch-origin" whorls all parts of the plant have been included, and
not merely primary branches. Consequently the predicted means from
the equation will not be strictly comparable with the observed means in
table 55. It is to be expected, however, that if the means for whorls
where branches originate were calculated for primary branches alone
they would not differ greatly from the values given in table 55. This
seems probable from the fact that the two other portions of the plant
besides primary branches which contribute most largely to the table 55
means are the main stem and secondary branches. But since, as has
been shown above (p. 31), main-stem whorls have a relatively high leaf-
number, while on the other hand secondary-branch whorls have a low
leaf-number (cf. p. 41), owing to the high proportion of proximal whorls,
it may fairly be supposed that the effect of these two portions of the
plant will about balance each other in the means given in table 55.
That this supposition is in fact a reasonable one is shown by the results
which follow.

In table 59 are given in parallel columns the observed mean leaf-
number in whorls at nodes where branches originate and the pre-
dicted leaf-number from equation I, where x takes successively the
mean values given in table 58 for Series I, II, III, and IV.

The agreement between observation and prediction is closer than
probably would have been expected. The table shows that at the outside
not more than 0.2 leaf in the excess of the means for "branch-origin"
whorls over "whorls in general" can be due to the combined effect of
(a) any hypothetical influence of the presence of a branch at the node
to which a whorl belongs, and (6) the inclusion in our observed means
of main-stem and secondary-branch whorls. In Series I the predicted

BRANCHES AND LEAF-NUMBER.

115

value is actually slightly in excess of the observed. In the other three
series the small excess of the observed mean is entirely due, I believe,
to the fact that in their influence the main-stem and secondary-branch
whorls do not exactly balance each other, but on the contrary the main
stem preponderates. It has been shown above (table 25, p. 49) that the
number of primary branches greatly exceeds the number of tertiaries.

Table 59. — Observed and predicted leaf-number in "branch-origin" whorls.

Series.

Observed
mean.

Predicted
mean.

Difference.

I

II
III
IV

9.192
9.347
9.144
9.353

9.212
9.150
9.098
9.152

+0.020

— .197

— .046

— .201

Every primary branch means, of course, an entry of one main-stem
whorl in the frequency distributions of table 54, while every tertiary
branch means the entry of one secondary-branch whorl (excepting,
naturally, in both cases mutilated whorls). Hence if primary branches
exceed tertiaries in number we should expect a preponderant effect of
main-stem whorls (with high leaf-number) over secondary-branch whorls
(with low leaf -number) in the means of table 55.

Taking all these points into account I think we may safely conclude
that the presence of a branch originating a.t a particular node is luithout
any influence on the number of leaves in the whorl belonging to that node.
This conclusion is confirmed by the data for whorls at which two
branches originate, but it hardly seems worth while reproducing the
evidence in detail. Strasburger ( :02) states that the same thing is true
with regard to the influence of flowers on the leaf -number in whorls.
He says (p. 486) : ' 'Die in einem Wirtel vertretenen Bliithen beeinflussen
nicht die Zahl der Blatter. ' '

116 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

THE POSITION OF BRANCHES.

In the last section the ordinal position with reference to the proxi-
mal end of the axis of the nodes at which lateral branches originate
was studied in order to get light on another question. I wish now to
consider on its own account some other phases of the problem of branch
production.

The first question which we may consider is as to where, on any given
axis of the plant, branches begin. That is, at what node, counting
from the proximal end of an axis, does the most proximal branch on
that axis appear. And what degree of variation is there in this mat-
ter? The frequency distributions giving data on the point for Series
I to IV are exhibited in table 60. The data are for the position of the
most proximal secondary branches.

Table 60.— Frequency distribution for positions offirsf branches (secondaries).

Node of primary branch
at which the first
secondary branch
develops.

9

10

11

12

13-18.
19

Total.

Frequency of '

'first" secondary branches.

Series I, II

Series I.

Series II.

Series III.

and III Series IV.

combiued.

9

12

8

29

21

2

9

7

18

7

4

10

7

21

20

6

4

2

12

12

2

2

6

3

1

4

3

2

i

6

i

1

2

3

i

2

2

1

1

"i

1

34

40

25

99

68

'Throughout the section of the paper the most proximal branch on any axis of the
plant will be designated as a "first" branch, tlic next branch distad of this as the "sec
ond" branch, and so on. (t;f. tig. 25, infra.)

Calculating from this table I find for the means, medians, and stand-
ard deviations the following values:

Table Q1.— Constants for variation in position of "first" secondary branches.

Series.

Mean.

Median.

standard
deviation.

I

4.71

2.85

2.32
3.35±0.20
2.81± .15

4.33
2.89
2.64
3.12
3.30

3.94

2.08

1.35
2.92dz0.14
1.84± .11

li

Ill

I, II, and III combined
IV

POSITION OF BRANCHES.

117

Table 62. — Frequency distributions for position of the first five secondary branches.
Series I, II, and III combined, and Series IV.

SERIES I, II, AND III COMBINED.

1 First
Node of primary branch at whicli \ secon-
secondary originates. dary

branch

9

10

11

12

13

14

15 and 16 .

17

18

19

20

21

22

23

24

25

26

29

18

21

12

2

4

6

Total.

Second
secon-
dary

branch,

14
16
12
6
7
2
2
1
5
1

Third
secon-
dary
branch.

12

11

10

5

4

1

2
2

Fourth

secon

dary

branch.

Fifth
secon-
dary
branch.

I 1

99

68

51

SERIES IV.

1

2

3

4

5

6

7

8

9

10

11

12

13

Total

7
11
7
5
2
1
1
2
2

41

30

21

7

9

20

■ 11

7

12

11

9

4

6

7

8

7

3

7

10

8

6

4

4

9

7

i

1

1

4

3
6

i

"i

"i

"i

1

68

51

40

33

26

118

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

Table 63. — Constants for position of secondary branches.

[Unit = 1 node.]

Series.

I, II and 1

III com- <

bined.

Branch.

Mean.

standard
deviation

1 Series.

1

Branch. | Mean.

standard
deviation

First

3.35
4.93
6.02
7.37
8.50

2.92
4.04
4.05
3.37
4.03

ivi

First 2.81

Second ! 4.29

Third 5.12

Fourth 6.27

Fifth 7.23

1.84
1.87
1.73
1.69
1.42

Second

Third

Fourth

Fifth

From tables 60 and 61 the following points may be noted:

(a) The first secondary branch does not usually occur at the first

node of the primary branch. The average position of first branches is

at about the third node.

(6) Roughly speaking, about 50 per cent of first secondary branches

arise at or below (i. e. , proximad of) the third node of primaries, and 50

per cent at or above (distad of) this node.

(c) There is a close agreement between all the series in respect to
the position of origin of secondary branches.

(d) There is a wider range and higher degree of variation in this
character than would have probably been predicted. Series I is very
remarkable in this respect, there being one first branch occurring on
the 19th node of the primary axis. All the series had first branches
occurring as far out as the 9th or 10th node.

(e) In table 60 it is to be noticed that there is apparently a tendency
for first branches to occur more frequently at odd than at even nodes,
at least at the beginning of the primary axis. In all the series (except
III, where the numbers are small) there is a high frequency at the
first node, a decided drop at the second node, and a more or less consid-
erable rise again at the third. Beyond this point the numbers are so
small that one can not make a safe judgment. So far as the data go it
of course looks as if we might possibly have the start of a Fibonacci
series here, but there is no evidence whatever of a relatively high
frequency at the 5th and 8th whorls as compared with the 4th, 6th, and
7th, which we should expect to find if we were really dealing with such
a series.

Having seen how the position of origin of first branches is distributed
on the axis from which they spring, we may examine into the same
matter in the case of the succeeding branches (second, third, fourth,
and fifth). It is not worth while to go beyond the fifth on account of
the small number of observations. The data are given in table 62.

POSITION OF BRANCHES.

119

The means and standard deviations for these distributions are given in
table 63. It is seen from these tables that:

(a) In mean position successive branches are roughly one node apart.
This is of course the result Vv^e should expect to get if a branch were
formed at every node. But while the latter is far from being the case
in detail, the branches are so distributed as to give an average result
of much the same kind. The regularity of the increase in mean position
is shown in fig. 24.

(b) Neither the mean nor the modal position of the branches falls
at the node we should expect if a branch occurred at each node. Thus
first branches occur on the average at the third node, instead of the
first, fifth branches at the seventh or eighth node instead of the fifth,
and so on. In the case of the Series IV plants the mean position of the
branches is quite uniformly two nodes in advance.

(c) Each of the first five branches shows about the same degree of
variation in its position. In the Series IV plants the variation apparently
decreases somewhat in the branches beyond the second, but the differ-
ences are so small that no stress can be laid on them.

(d) The tendency for branches to occur at odd nodes with greater
frequency than at even, which was observed in the case of the first
branches, does not appear to hold for other branches, nor beyond the
third node.

b

^

/

^

}

.- ;

L
O

o

w

ft)

^^ • 1

/

^ ^JL

8^

c
c

s

2

i

3

(

1

I

i

4-

5

Branch

Fig. 24.— Graphs of the means given in table 64. Series I, II, and III com-
bined, > ■ ■nil I ; Series IV, o — — — - .

120

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

We may turn now to another question with reference to the distri-
bution of branches, which may be put in this way: What proportion
of first branches have their succeeding (i. e., second) branch on the
node immediately distad of that on which they themselves are borne,
and in what proportion of cases are there one or more nodes without
branches intervening between the two? Similarly, what are the propor-
tions of cases for second and third, third and fourth, etc., branches,
where successive branches are borne on contiguous nodes? The data
on these questions have been extracted for the first to the sixth secondary
branches in the first four series and are shown in table 64. In this table
a double column is given to each series, one half being headed + (plus)
and the other half — (minus) . Whenever the succeeding branch to the
one designated in the first (left-hand) column of the table was borne on
the next node it was entered in the + (plus) column. When branchless
nodes intervened between the two an entry was made in the — (minus)
column. Unless at least three whorls of leaves were formed beyond
a branch no record was made regarding it. That is, only cases were
included where it was possible for a succeeding branch to have been
formed.

Table 64. — Position of succeeding branches (secondary).

Branch.

Proportionate frequency of occurrence of the
succeeding branch at the next node.

Series I.

Series II.

Series III.

Series I,II

and III

combined

Series IV

+

8
7
5
4
4
6

23

10

7
5
3

1

+

23

21

13

9

11

7

25
8
5
5
2
2

+

18

16

12

7

3

5

7
2
i
3
2
0

+

49
44
30
20
18
18

55
20
16
13
7
3

+

31
30
27
18
20
19

30
17
11
11
2
0

First

Second

Third

Fourth

Fifth

Sixth

This table shows that there is a decided difference between first
branches and those farther distad on the axis, in respect to the matter
under consideration. Thus we see that there is more than an even
chance that one or more branchless nodes will follow the first branch,
while for all the others the chance of a succeeding branch occurring on
the next node is greater than the chance that it will not so occur.
Further, this chance increases the farther out on the axis we go. This
is shown most clearly if we calculate the percentage (in the total number)
of the cases in which the immediately succeeding branch occurs at the
next node. This has been done for the last two columns of the table
(Series I, II, and III combined and Series IV), with the results shown

POSITION OF BRANCHES.

121

in table 65. It will be understood that the figures in this table are the
percentages which the + (plus) entries in table 64 are of the sum of
the + (plus) and — (minus) entries for each branch.

Table %^.— Percentage of cases in which succeeding branch occurs at the next node.

Branch.

Series I, II

and III
combined.

Series IV.

Branch.

Series I, II

and III
combined.

Series IV.

First

47.1
68.8
65.2

50.8
63.8
71.0

1
Fourth

66.7
75.0
85.7

62.1

90.9

100.0

Second

1 Fifth

Third

Sixth

These results speak for themselves. They show that the branch
production becomes more regular and orderly the farther out on an axis
we go. As the plant grows it tends with ever-increasing certainty to
produce a branch at each node. After a time it does this with very
remarkable — almost perfect — precision.

Table 66. — Frequency distribution for the occurrence of secondary branches.

Branch.

First

Second..

Third

Fourth..

Fifth

Sixth

Seventh .
Eighth..
Ninth....
Tenth....

Number of branches in 5 nodes following designated branch.

Series I, II, and III combined"

0

10
2

1

11
7
5
6
5
3
2
2
1
1

Series IV.

12
4
4
2
2
2
2
1
1

The fact that the tendency towards the production of a branch at
every node increases the farther distad on the axis we go is brought
out in another way by the data given in tables QiQ and 67. In these
tables are given distributions showing the frequency with which different
numbers of branches occur in the five nodes immediately following any
designated branch. Thus, considering the portion of a plant diagram-
matically represented in fig. 25, in which ah is a portion of a main stem,
say, and xy a primary branch bearing secondaries i to x, we see that
in the five nodes following the first branch only one branch occurs.
Similarly in the 5 nodes immediately following the second branch only
one occurs. In the 5 nodes immediately following a, the third, there
are 3 branches; h, the fourth, there are 4 branches; c, the fifth, there

122

VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

are 4 branches, etc. Clearly the maximum number of branches possible
is 5. Proceeding in the way just illustrated for the first 10 secondary
branches on all plants of Series I, II, III, and IV, the results shown in
table Q6 were obtained. No branch was counted which did not have
at least 7 whorls distad of it on the primary axis.

Fia. 25. — Diagram of a portion of a plant. I, "first" branch; II "second" branch; III, "third" branch
etc. These secondaries are for convenience represented without leaves. Further explanation in
the text.

We see at once from the table that the more distad the branch with
which we are dealing lies the larger is the number of branches occurring
in the next 5 nodes. When we get beyond the sixth branch less than
four occur very rarely. These tables show very plainly the gradual
approach towards a condition where each node bears a branch. It will
help to make the point still clearer if we examine the means of the
arrays in table 67.

Table 67. — Mean number of branches in the five nodes immediately following

a designated branch.

Branch.

First...
Second
Third...
Fourth
Fifth...

Series I, II,

and III

Series IV.

combined.

2.41

3.10

3.25

3.92

3.56

4.31

3.57

4.53

3.67

4.54

Branch.

Sixth....
Seventh
Eighth.,
Ninth...
Tenth...,

Series I. II,

and III I Series IV.
combined.

3.86
4.60
4.20
4.75
4.75

4.58
4.36
4.25
4.40
5.00

The increase in the mean number of branches is evident. A little
inspection shows, however, that this increase is not uniform in all parts
of the table, but is on the whole distinctly more rapid in the first few
branches than in the later ones. In other words, the increment in the
character under discussion becomes smaller and smaller for the successive
branches. This at once suggested that the law of change here might

POSITION OF BRANCHES.

123

be a logarithmic one, similar to what we have previously demonstrated
for the growth of leaves. Accordingly the data of this table were fitted
with logarithmic curves of the type used in the previous cases, by the
same method. The resulting equations were.

Series I, II, and III combined, Y = 2.4061 + 2.2194 log a; (I)

Series IV, Y = 3.8440 + .8931 log (x-.S) . . (II)

in which Y denotes mean number of branches occurring in the five
nodes immediately following any designated branch, the ordinal position
of which is given by x. Calculating the values of Y for values of x
from 1 to 10 we have the results shown in fig. 26.

V

/
/
.,-./

4.5

r ^ ,,-■

^

^

r^

7^"'

3.5

c

■ t
,' 1

>

> ^'"^^ —

^;^

^

'

'

f

2.5

/

r^

2.0

1.5

5 6

Branch

Fig. 26.— Curves showing mean number of branches (ordinates) In five nodes Imme-
diately following a designated branch (abscissas) on a primary axis. Series I, II,
and III combined r-\ Series IV .

The numbers with which we are dealing here are so small relatively
that great regularity in the observations can not be expected. There
can be no doubt, however, that the general trend of the observations is
adequately represented by the logarithmic curves. This means that
the growth processes concerned in branch production, so far as may be
judged from our present material, follow a logarithmic law. The mean
number of branches formed in a constant number of nodes (5) increases

124 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

the farther out on an axis we proceed, but the rate of increase varies
inversely as the distance from the proximal end of the system. We
thus get further evidence, from an entirely different source, of the
generality of our first law of growth in Ceratophyllum. The data fur-
nish positive evidence that branch production and leaf production by the
growing plant follow the same law.

SUMMARY.

A study of branch production leads to the conclusion that '*as the
plant grows it tends with ever-increasing certainty to produce a branch
at each node." In other words, as an axis of the plant grows longer
the morphogenetic processes concerned in the production of lateral
branches work, so to speak, more smoothly, and attain their results with
greater regularity and constancy. But this is precisely the same conclu-
sion to which we came with reference to the production of leaf-whorls
(p. 106, mpra). It was there shown that the leaf-number per whorl
approaches more and more closely to a constant value the longer an axis
grows. Putting both sets of results together, we now see that the plant
as it grows tends to produce one or more branches and a constant num-
ber of leaves at each node. When a bud begins its growth it does not do
either of these things with anything approaching certainty or regularity,
but the longer it grows the more regular do the results become, until
finally they are almost mathematically precise. Objectively it ''profits
by its experience" just as does an animal in its behavior. The results
which have been obtained in this section of the paper show that our two
laws of growth (pp. 88 and 106 supra) operate in branch production as well
as in leaf production.

GENERAL DISCUSSION OP RESULTS. 125

GENERAL DISCUSSION OF RESULTS.

In bringing this paper to a close it appears desirable to discuss to
some extent certain general aspects of the work as a whole, and to
consider them on the theoretical side. Speaking broadly, the most
significant result of this work appears to the writer to lie in the fact
that, proceeding by quantitative analytical methods, it has been possible
to formulate two laws of growth, which serve to describe with a very
high degree of (a) precision, (6) completeness, and (c) generality the
observed results of those processes of morphogenesis which in the
growing Ceratophyllum plant lead to differentiation of parts. Further,
it has been shown by direct appeal to statistics that the characteristic
features of the variation of Ceratophyllum are obviously the result of
the fact that the organism grows in accordance with these laws. By
this, of course, is not meant that any kind of "explanation" of the
origin of variation in Ceratophyllum has been gained. What has been
gained, though, is the knowledge that, so far as our present material,
which includes a reasonably wide range of conditions as to habitat, time
of collection, etc., is concerned, the difference between two sets of
individuals in respect to their variation constants are capable of prac-
tically complete interpretation solely in terms of the two laws of growth.
The results actually observed are such as would be expected to arise in a
system differentiating in accordance with our two growth laws.

The first of these laws of growth was stated on page 88 in the fol-
lowing way: "The mean number of leaves per whorl increases with
each successive whorl, and in such a way that not only does the absolute
increment diminish, but also the rate of increase diminishes, as the
ordinal number of the whorl measured from a fixed point increases."
If we let y stand for number of leaves in the whorl, and x denote the
position in a series of successively formed whorls, then we find that y is
a simple logarithmic function of x as follows:

y = A + Clog {x — a)

where A, C, and « are constants. The leaf whorls become differentiated
with growth according to a logarithmic law. Also, as was shown in the
last section (p. 125) , if we let y denote the number of lateral^branches
found in a given number of nodes, and x as before the position of these
particular nodes in the whole series, again we find y a simple logarithmic
function of x. So that, in general terms, we see that in Ceratophyllum
growth, whether expressed in the formation of leaf whorls or of lateral
branches, takes place in such a way that the product increases at the
same proportionate rate that the logarithm of the position in the whole
series of products increases.

126 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

The second law of growth was stated in the following words (p. 106,
supra) : "As whorls are successively produced by a growing bud they
are formed with ever-increasing constancy to their type, the ultimate
limit towards which the process is tending being absolute constancy."

This means that there is a steady diminution of variability accom-
panying the repetition of the morphogenetic process of whorl production.
As has been indicated in the preceding section of the paper, the same
rule holds with reference to branch production.

The two laws evidently have one important point in common, namely,
they both express the fact that the form or character of a structure
produced at any point on the plant is in part directly related to or
determined by the previous morphogenetic history of the individual.
Successively formed structures develop in such a way that from a purely
objective point of view it appears as if the growing point on any axis
were influenced in its formative activities by the previous "experience"
through which the protoplasm of which it is composed has passed.
Especially is this true with respect to the phenomena embodied in the
second law. As has been shown above (p. 107) , in the detailed discussion
of this law morphogenetic products are progressively "better" formed,
that is, closer to type, with each successive production. The theoretical
importance of the demonstration of such a law of morphogenesis is
evident.

It is the belief of the writer that both of these laws which have been
demonstrated for Ceratophyllum have a wide generality in other organ-
isms. From considerations of space it is impossible here to present
detailed evidence for this belief, but a few points may be very briefly
mentioned. We may consider each of the laws separately. In attempt-
ing to form an idea as to how generally a logarithmic law holds in the
growth of other organisms than Ceratophyllum, we are unfortunately
met at once by the difficulty that there have been comparatively f ew
quantitatively exact studies of growth ever made. Furthermore, it is
to be noted that so far as is known to the writer, there has hitherto
been no extended study of exactly the same phase of the growth problem
as that with which we have here dealt. In the present investigation
we have dealt with what may be called intra-individual or organal
growth, that is, with the growth differentiation of a series of successively
produced, generally "like" organs of the same individual. In contrast
to this previous studies of growth have usually dealt with what may be
called individual growth, that is, with the growth change in the same
organ or character in successive stages of life history of the individual
or individuals. The two points of view stand to each other in the same
relation that intra-individual variation does to intra-racial variation.

GENERAL DISCUSSION OF RESULTS. 127

It seems somewhat remarkable that so important a matter for the
understanding of many problems of morphology as is the study of post-
embryonic growth from both of the points of view noted should have
been so much neglected. On the botanical side there is a good deal
of literature dealing with special phases of the subject, but for our
present point of view most of this material has little direct bearing.
On the zoological side the principal work is due to anthropologists who
have studied post-embryonic growth in man. In this field the available
evidence regarding individual growth, so far as it goes, appears to be in
good accord with what we have found in Ceratophyllum for organal
growth. Thus, for example, Pearson ( :04) has shown that the growth
in auricular height of the head in children follows a logarithmic
curve, and in a recent memoir by Lewenz and Pearson ( :04) it is stated
that such a curve has been found to represent the growth changes in
other characters. Probably the most thorough and in all respects the
best study of growth in any other animal than man which has been
published is the classical investigation of Minot ('91) on growth in the
guinea-pig. Speaking of his statistics Minot (p. 148) says: "They
demonstrate two fundamental facts: First, the rate of growth dimin-
ishes almost uninterruptedly from the time onwards when the animal
recovers from the post-natal loss of weight; second, the diminution is
rapid at first, but slower afterwards." It will be seen that these
statements exactly agree with those we have made above for growth
in Ceratophyllum. That is, it would appear that the "individual"
growth of the guinea-pig follows a logarithmic law. A careful study
of Minot's data indicates that this is in fact the case. There is a great
need for special investigations of growth directed towards determining
exactly the laws which the changes follow. From such investigations
we may hope to get some idea of the extent to which a logarithmic law
is general. In any event, it is clear that such a growth law is not entirely
unique in Ceratophyllum. On the contrary one has been convinced by
going over the older material available in the literature, which it would
take too much space to cite in detail here, that a logarithmic law is
probably very general for growth in both plants and animals,* and for
"individual" as well as "organal" growth.

There can be no doubt that what has been found in Ceratophyllum
with reference to the variation of repeated parts is simply an example

*It has doubtless occurred to the reader that this logarithmic law of growth super-
ficially resembles in form the well-known Weber-Fechner law regarding the quantitative
relation of stimulation and sensation, as it was formulated by Fechner. Unfortunately
later research in physiological psychology has shown that Fechner' s statement of the
law either does not hold at all, or at most only in a. very limited range of cases.

128 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

in a single case of a very general and fundamental biological law. This
law has been differently stated, according to the particular class of
phenomena in which it is seen to be operating. Thus Jennings ( :05) ,
dealing with the facts of behavior, calls it the ''law of the readier reso-
lution of physiological states" and formulates it in the way which has
already been quoted (p. 108). He gives a number of examples showing
the evidence in favor of the law from the behavior of lower organisms
and says (p. 485) : ' 'In view of the facts, it is probable that the law is
a general one and that it will be demonstrated in some form for other
lower organisms." He also suggests the probability that the essential
principle embodied in the law will be found to operate in morphogenetic
processes. Definite statistical proof that such is in fact the case has
been given by the present paper. Possibly a word of explanation is
necessary to bring out the fact that the essential principle in the law
as stated by Jennings is the same as that which underlies our law of
diminishing variability. This will be clear if we consider in a little
more detail the facts of behavior which led to the formulation of the law
from that point of view. By analyzing certain phenomena of behavior
in detail, Jennings {loc. cit, p. 481) shows that:

In the lowest organisms we find individual adjustment or regulation on the basis
of the three following facts :

(1) Definite internal processes are occurring in organisms.

(2) Interference with these processes causes a change of behavior and varied
movements, subjecting the organism to many different conditions.

(3) One of these conditions relieves the interference with the internal processes,
so that the changes in behavior cease, and the relieving condition is thus retained.
* * *

Now an additional factor enters the problem. By the process which we have just
considered, the organism reaches in time a movement that brings relief from the
interfering conditions. This relieving process becomes fixed through the operation of
a certain law which appears to hold throughout organic activities. This law may
be stated as follows: An action performed or a physiological state reached is
performed or reached more rapidly after one or more repetitions, so that in time it
becomes "habitual."

While, as Jennings points out, this statement of the law is not entirely
adequate, yet it emphasizes the point at which comparison between the
facts of behavior and morphogenesis may most easily be made. If in
the production of successive whorls on an axis of a Ceratophyllum plant
the variation about the type for each whorl diminishes, while, as has
been shown, the type of the whorls changes at an ever-decreasing rate,
it merely means that the production of a particular type of whorl tends,
speaking in purely descriptive terms, to become "habitual." In both
the psychological and morphogenetic cases there is a tendency to produce

GENERAL DISCUSSION OF RESULTS. 129

a stereotyped result with ever-increasing precision and constancy.
Again, to take another example, it is obvious, as Jennings has pointed
out, that ' 'the operations of this law are seen on a vast scale in higher
organisms, where they constitute what we commonly call memory,
association, habit, and the basis of intelligence," If we consider for a
moment the case of memory in man, it will be still further clear that
there is objectively a fundamental similarity between one characteristic
of this psychological phenomenon and such facts of morphogenesis as
we have detailed for Ceratophyllum. Suppose a man sets to work to
memorize a number of lines of poetry, and tests his acquirements by
attempting to repeat the lines after each successive reading. The
result will be something like this: When he attempts to repeat the lines
after the first reading he will make a number of mistakes, or ' 'devia-
tions from the type" which is given by the exact cext. On repeating
the extract after the second reading the number of "errors" or "devia-
tions" will tend to be fewer; after a third reading still fewer, and
so on until finally there are no "deviations, " or, in other words, the
"type" is reproduced exactly at each successive repetition. Now, what
do we find in Ceratophyllum? When the first whorl on an axis is pro-
duced we see, if we examine a large number of such whorls, that devi-
ations from the type are produced relatively very frequently; second
whorls exhibit a smaller number of such deviations; third whorls a still
smaller number, and so on until we reach a condition of minimum
variability, or in other words, a condition in which the type is produced
each time with great precision and constancy. What goes on in the
case of the memorizing and in the case of the growing plant may be
objectively described in the same terms, the principal difference being
that in the former example the type is absolutely fixed and constant,
while in the latter it changes slowly. These illustrations will suffice to
show that v/hat we have called the law of diminishing variability
operates in psychological as well as morphological phenomena.

In attempting to determine how generally this law of diminishing
variability holds in respect to processes of growth, one is met as before
with the difficulty that there have been but few investigations which
have brought to light direct evidence on this point. It should be kept
in mind that the conditions under which we should expect this law to
show its operation in the clearest and most unequivocal form are those
in which we have the production of a series of not greatly differentiated
parts or characters. Obviously these conditions are best realized on the
botanical side in plants having a type of structure similar to that of
Ceratophyllum, and on the zoological side in animals built up on a simple
metameric plan. So far as the writer knows, no systematic investiga-

130 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

tion of the comparative variability of a series of metamerically repeated
and slightly differentiated organs or characters in an animal form has
ever been made. Hence it is hopeless at present to look to this source
for confirmatory evidence for the law. On the botanical side the situation
is somewhat better. Thus, for example, in a recent study of "Stages in
the Development of Sium cicutsefolium' ' Shull (:05) has investigated the
relative variability of the successive "nepionic" leaves in this form.
In his fig. 1 (p. 9) he gives the frequency polygons for variations in the
first eight leaves. These polygons very obviously substantiate his con-
clusion that "there is a progressive lessening of the variability from the
first leaf onward. " In this case we clearly have a direct confirmation
of the law of diminishing variability. Similarly, in the papers of Cush-
man (:02, :03, and :04) there are several statements which appear to
indicate that in many of the large number of plants studied by this
worker the variation diminished in successive nepionic leaves. In a less
direct way the operation of this law is to be seen, I believe, in a wide
variety of morphological phenomena, including particularly what we have
called above "individual" growth. Thus we have one example of it
in the well-known fact that embryonic characters are much more vari-
able than adult characters. That this is true has been directly proven
by an elaborate system of measurements on duck embryos made some
years ago by Fischel ('96).

Minot found the same thing in his guinea-pig measurements. On
this point he says (loc. cit, p. 140):

This diminution of variability with age is demonstrable in the growth of other
mammals, hence it probably occurs in all. We ;are led by this to put the question
whether all variability of higher animals does not lessen with the age of the individual.
In view of the extreme variations of structures which occur in all vertebrate embryos,
and which, as all embryologists know familiarly, are far greater and more frequent
than the variations of the adult, we are justified in asserting that there is a diminution
of variability with age.

More evidence in the same direction might be cited, but it is unneces-
sary. If what has frequently been asserted should be proven by exact
measurements, namely, that phylogenetically "young" organs are
more variable than "old" organs, this again would fall in line with
the general law of diminishing variability. It has recently been
shown by Shull (:05a) that mutant forms of (Enothera are more vari-
able than the parent form. One might hazard the suggestion that
this fact is another expression of the operation of this same law. The
matter could easily be tested by a biometrical investigation of mutants
and their descendants extending through a considerable period of
time.

GENERAL DISCUSSION OF RESULTS. 131

Purely as a working hypothesis, to be tested and limited by further
investigation, the law of diminishing variability may be stated in the
most general foi-m to cover the facts in different fields, as follows: In
a continuous series of biological phenomena in which the same or homol-
ogous processes are repeated, the variation exhibited in the results of these
processes diminishes with successive repetitions.

In the opinion of the writer any attempt to develop a detailed theo-
retical explanation of why the two laws of growth which we have found
in Ceratophyllum come to operate, would at present be premature.
Just now there is a much greater need for quantitatively definite facts
than for theories in the field of morphogenesis. What is, however, of
great importance on the theoretical side is to see exactly the nature of
the fundamental problem on which the interpretation of these growth
laws depends. We have seen that there is a definite functional relation
between the morphogenetic activity of a growing bud at any given time
and its previous activity or "experience. " In what way this functional
relation is brought about is the fundamental problem which lies before
us if we are to interpret our laws of growth. It will probably have oc-
curred already to the reader that our results are a very clear cut exam-
ple of the general principles which Semon ( :04) has recently developed
at great length. The influence of earlier upon later whorl production
might very well be described as the result of "engrammatic" action on
the bud in the formation of first and succeeding whorls. In fact, the
whole of our results seem to form a most striking and complete illus-
tration of the working out of Semon 's principles in a particular case
of ontogeny. But one can not escape the feeling that to attempt to
interpret the facts in this way is simply to redescribe them in a new
terminology without any substantial gain. Semon's rather obvious
argument to meet such a criticism, which he of course foresees, that all
science is only description, does not adequately remove the difficulty.
What we require in cases like that with which we are here dealing is a
description in terms of known physiological principles, and this the
"mnemnic" terminology does not seem to provide. It seems to the
writer that a more promising hypothesis on the basis of which to inter-
pret such morphogenetic phenomena as those which have been set forth
for Ceratophyllum might be developed along the lines suggested
in recent papers by Holmes ( :04) and Schieff erdecker ( :04) . Such an
interpretation has been worked out by the writer for use as a working
hypothesis in further investigations in this field which are now in
progress, but until it has been tested it hardly seems worth while to
publish it.

132 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

Finally, in closing, I desire to emphasize the great importance in any
study of variation of the analysis of the intra-individual variability of
the characters which are to form the basis of intra-racial investigations.
We have seen in a repeated character of the same individual a whole
series of variation constants appearing in an orderly manner and repre-
senting degrees of variation from practically zero up, degrees of skewness
from positive through symmetry to negative, etc. It is obvious that on
any problems of intra-racial variation we could with Ceratophyllum get
widely divergent results by taking whorls from different parts of the
plant. Ordinarily when an investigation of a problem of geographical
variation, or of natural selection, or of almost any phase of intra-racial
variation is made on the basis of characters like the leaf- whorls of Cera-
tophyllum, only a few are taken from each individual. Either no attention
at all is paid to differentiation in the characters, or at most only the
characters in the immediate neighborhood of those chosen are examined,
and if there is no differentiation easily detectable among them, the
conclusion is reached that any factor of this kind may be neglected. It
seems to me that the results of the present paper show that the question
of whether or not there is a differentiation within the character group
chosen for investigation is not necessarily the important thing at all.
If one takes the 10 most distal whorls on Ceratophyllum plants, there is
substantially no differentiation in respect to leaf-number. Yet, to con-
clude, when this had been ascertained by superficial examination, as it
could be, that these whorls, since they were undifferentiated and there-
fore could be considered homogeneous material, might be taken to
represent the conditions of the individual as a whole in an intra-racial
investigation, would lead to absolutely fallacious results. It is true
that there is no marked differentiation among these distal whorls, but
the very reason that there is not is that a perfectly definite and orderly
process of differential development has led as an end result to the con-
tinued production of substantially the same type of whorl, regardless of
environment or other influences. These whorls are not greatly differ-
entiated, but neither are they truly representative of the individual.
If characters of this kind are to be used as the basis of investigations of
problems of intra-racial variability, it clearly is absolutely necessary
that the laws according to which the characters are differentiated during
the development of the individual must first be ascertained if valid
results are to be obtained. When this has been done we may turn to
intra- and inter-racial problems, and, using the methods which have
been developed by Pearson for dealing with differentiated characters,
hope to reach definite conclusions.

GENERAL SUMMARY. 133

GENERAL SUMMARY.

This paper deals with a biometrical analysis of intra-individual
variability and differentiation in Ceratophyllum. The characters prin-
cipally dealt with are (a) the number of leaves in the whorl; (6) the
position of the whorl on the plant; (c) the size of the various divisions
of the plant; and (d) the position of branches. Some of the chief results
as to fact may be summarized as follows:

(1) Dealing with the intra-individual variation in leaf-number per
whorl it is found that the whorls borne on the different axial divisions
of the plant (main stem., primary, secondary, etc., branches) are dis-
tinctly differentiated in respect to both type and variability.

(2) The mean number of leaves per whorl is highest in the whorls
on the most central division of the plant (the main stem) and decreases
regularly as we pass to more peripheral divisions.

(3) The whorls on the main stem are the least variable in leaf-number,
and the variation increases regularly in the more peripheral divisions,
till a maximum is reached in secondary-branch whorls. The variation
then tends to diminish in the whorls on higher-order branches.

(4) More than half of the total number of whorls are borne on pri-
mary branches. Of the remaining whorls somewhat more are borne on
secondary branches than on the main stem. Tertiary and quaternary
branches bear relatively few whorls.

(5) Primary branches are absolutely and relatively more variable in
size than are secondaries.

(6) There is a relatively high degree of correlation between the
number of leaves in the whorl and its position on any axis of the plant.

(7) The degree of this correlation is lowest in the most central
division of the plant (main stem) and increases as we pass to the more
peripheral divisions.

(8) The regression of leaf-number on position is not linear, but log-
arithmic.

(9) This leads to what we have called the "first law of growth" in
Ceratophyllum, which may be stated as follows: On any axial division
of the plant the mean number of leaves per whorl increases with each
successive whorl in such a way that both the absolute increment and
the rate of increase diminish as the distance (in units of nodes) of the
whorl from a fixed point increases.

(10) Branches which show an excess (or defect) from the average
in the number of leaves in the first whorl tend to have the succeeding
whorls greater (or less) than their respective averages. This tendency
diminishes as we go distad on the plant.

134 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM.

(11) On any axial division of the plant as whorls are successively
produced by a growing bud they are formed with ever-increasing con-
stancy to their type. This we have called the ' 'second law of growth
in Ceratophyllum. " The growing-point appears to be influenced in its
morphogenetic activity by its previous experience.

(12) These two laws operate in branch production as well as in leaf
production in Ceratophyllum.

Summarized statements of the detailed results and discussions of
the conclusions drawn from them will be found on pp. 26, 54-57, 88-92,
95, 106-108, 115, and 124 and need not be repeated here.

The results of this study emphasize the necessity for a knowledge
of the laws of diif erentiation within the individual of characters as a
preliminary to the use of these characters in the study of the problems
of evolution.

LITERATURE CITED. 135

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